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Programming a 2D Ising Model/Locating the Curie temperature

2026-03-05T10:35:12Z

Jbettenc: Fix typo

'''<big><span style="color:blue; ">This is the eighth (and final) section of the Ising model experiment. You can return to the previous page, [[Programming a 2D Ising Model/Determining the heat capacity|Determining the heat capacity]], or go back to the [[Programming a 2D Ising Model|Introduction]].</span></big>'''

You should have seen in the previous section that the heat capacity becomes strongly peaked in the vicinity of the critical temperature and that the peak became increasingly sharply peaked as the system size was increased — in fact, Onsager proved that in an infinite system the heat capacity should diverge at ''T'' = ''T<sub>C</sub>''. In our finite systems, however, not only does the heat capacity not diverge, the Curie temperature changes with system size! This is known as a ''finite size effect''.

It can be shown, however, that the temperature at which the heat capacity has its maximum must scale according to <math>T_{C, L} = \frac{A}{L} + T_{C,\infty}</math>, where ''T<sub>C,L</sub>'' is the Curie temperature of an ''L''×''L'' lattice, ''T<sub>C,∞</sub>'' is the Curie temperature of an infinite lattice, and ''A'' is a constant in which we are not especially interested. This scaling equation comes from expanding in the limit of large ''L''.

'''<big>TASK 8a</big>: Reference data has been generated for longer simulations than would be possible on the college computers in Python. Each file contains six columns: ''T'', ''E'', ''E''<sup>2</sup>, ''M'', ''M''<sup>2</sup>, ''C'' (NOTE: these final five quantities are normalised per spin, unlike your Python data), and you can read them with the NumPy <code>loadtxt</code> function as before. For each lattice size, plot the reference data against your data. For ''one'' lattice size, save a SVG or PNG image of this comparison and add it to your report — add a legend to the graph to label which is which (check if you need a [https://matplotlib.org/stable/api/_as_gen/matplotlib.axes.Axes.legend.html reminder about use of legend]).'''

==Polynomial fitting==

To find the temperature at which the heat capacity and susceptibilities have their maxima, we are going to fit a polynomial to the data in the critical region. NumPy provides the useful [http://docs.scipy.org/doc/numpy/reference/generated/numpy.polyfit.html <code>polyfit</code>] and [http://docs.scipy.org/doc/numpy/reference/generated/numpy.polyval.html#numpy.polyval <code>polyval</code>] functions for this purpose. The following code is written for the heat capacities - try to also repeat this for the susceptibility! The usage is as follows:

<pre>
data = np.loadtxt("...") #assume data is now a 2D array containing two columns, T and C
T = data[:,0] #get the first column
C = data[:,1] # get the second column

#first we fit the polynomial to the data
fit = np.polyfit(T, C, 3) # fit a third order polynomial

#now we generate interpolated values of the fitted polynomial over the range of our function
T_min = np.min(T)
T_max = np.max(T)
T_range = np.linspace(T_min, T_max, 1000) #generate 1000 evenly spaced points between T_min and T_max
fitted_C_values = np.polyval(fit, T_range) # use the fit object to generate the corresponding values of C
</pre>

'''<big>TASK 8b</big>: Write a script to read the data from a particular file, and plot ''C'' and ''χ'' vs ''T'', as well as a fitted polynomial. Try changing the degree of the polynomial to improve the fit — in general, it might be difficult to get a good fit! Attach a SVG or PNG image of an example fit to your report.'''

===Fitting in a particular temperature range===

Rather than fit to all of the data, we might want to fit in only a particular range. NumPy provides a very powerful way to index arrays based on certain conditions. For example, if we want to extract only those data points which are between a particular ''T''<sub>min</sub> and ''T''<sub>max</sub>, we can use the following syntax:

<pre>
data = np.loadtxt("...") #assume data is now a 2D array containing two columns, T and C
T = data[:,0] #get the first column
C = data[:,1] # get the second column

Tmin = 0.5 #for example
Tmax = 2.0 #for example

selection = np.logical_and(T > Tmin, T < Tmax) #choose only those rows where both conditions are true
peak_T_values = T[selection]
peak_C_values = C[selection]
</pre>

'''<big>TASK 8c</big>: Modify your script from the previous section. You should still plot the whole temperature range, but fit the polynomial only to the peak! You should find it easier to get a good fit when restricted to this region.'''

===Finding the peak in C===

Your fitting script should now generate two variables: <code>peak_T_range</code>, containing 1000 equally spaced temperature values between ''T''<sub>min</sub> and ''T''<sub>max</sub> (whatever you chose those values to be), and fitted_''C''_values, containing the fitted heat capacity at each of those points. Use the NumPy <code>max</code> function to find the maximum in ''C''. If you store the maximum value of ''C'' in the variable <code>Cmax</code>, you can use the following notation to find the corresponding temperature:

<pre>
Cmax = np.max(...)
Tmax = peak_T_range[fitted_C_values == Cmax]
</pre>

'''<big>TASK 8d</big>: Find the temperature at which the maximum in ''C'' occurs for each datafile that you were given or generated with Python. Make a text file containing two columns: the lattice side length (2,4,8, etc.), and the temperature at which ''C'' is a maximum. This is your estimate of ''T<sub>C</sub>'' for that side length. Make a plot that uses the scaling relation given above to determine ''T<sub>C,∞</sub>''. By doing a little research, you should be able to find the theoretical exact Curie temperature for the infinite 2D Ising lattice. How does your value compare to this? Are you surprised by how good/bad the agreement is? Attach a SVG or PNG image of this final graph to your report, and discuss briefly what you think the major sources of error are in your estimate.'''

'''<big><span style="color:blue; ">This is the eighth (and final) section of the Ising model experiment. You can return to the previous page, [[Programming a 2D Ising Model/Determining the heat capacity|Determining the heat capacity]], or go back to the [[Programming a 2D Ising Model|Introduction]].</span></big>'''

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2026-03-02T18:24:15Z

Jbettenc: Replace QR code with new Main_Page URL

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2026-03-02T18:12:59Z

Jbettenc: QR code encoding the home page URL for ChemWiki

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Programming a 2D Ising Model

2026-02-25T18:57:18Z

Jbettenc: /* Getting Help */ Small fixes in the schedule

'''<big>This is the Ising Model experiment (aka programming for simple simulations). This is a core experiment for year 3 students on the Chemistry with Molecular Physics degree.</big>'''

==Introduction==

In this exercise, you are going to use the Python that you learned in the first year to write a code to perform Monte-Carlo simulations of the 2D [http://en.wikipedia.org/wiki/Ising_model Ising model], a set of spins on a lattice which is used to model ferromagnetic behaviour, and also to analyse the results of the simulation to find the heat capacity of the system and the Curie temperature — the temperature below which the system is able to maintain a spontaneous magnetisation.

A more extensive introduction to the Ising Model will be given in the next page. This page gives some practical information about how the experiment.

==Assessment==
At the end of this experiment you must submit a report, writing up your findings. Each section of the experiment has a number of tasks that you should complete, labelled '''<big>TASK</big>'''. If this is a mathematical exercise, your report should contain a short summary of the solution. If it is a graphical exercise, you should include the relevant image. Your report should explain briefly what you did in each stage of the experiment, and what your findings were.

The report should be submitted as a PDF file to Turnitin, created with your favourite word processor or LaTeX, and should contain the annotated source code of the Python scripts that your write during the experiment. For clarity, code included in the report should be written in a monospaced font (e.g. Courier), which in LaTeX can be easily achieved by placing your code inside a <code>\texttt{}</code> command. '''The code, except where relevant to discussion, can be placed in an appendix.'''

As well as including relevant source code in your report, you should also submit a bundle with all your scripts and analysis code as a separate Blackboard submission.

===On the use of AI tools===
The code you write for this experiment is assessed, and as per the [https://bb.imperial.ac.uk/bbcswebdav/pid-3344481-dt-content-rid-21115105_1/xid-21115105_1 department's guidelines on the use of AI], it is not acceptable to use generative AI tools to write computer code for this assignment.

Writing computer programs yourself enhances your logical thinking and problem solving skills. Using generative AI tools to write your code makes you dumb. Don't choose to be dumb, don't use AI to write code for you.

==Getting Help==

The Graduate Teaching Assistants demonstrating this experiment are Fionn Carman and Lei Li. Demonstrators will be available in room 232A at the following times:

{| class="wikitable"
|+ Demonstrator timetable
! !! Monday !! Tuesday !! Wednesday !! Thusrsday !! Friday
|-
| Morning || 10:00 - Experiment intro<br /> 10:00-13:00 - Li || 10:00-13:00 - Fionn|| || 10:00-12:00 - João || 10:00-13:00 - Fionn
|-
| Afternoon || 14:00-17:00 - Li|| 14:00-17:00 - Fionn|| || || 14:00-17:00 - Fionn
|}

If you have questions outside these times, please use the Discussion Board on Blackboard.

João and Giuseppe are the members of academic staff responsible for this experiment. You are welcome to address your questions and comments to them.

==Structure of this Experiment==

This experimental manual has been broken up into a number of subsections. Direct links to each of them may be found below. You should attempt them in order, and you should complete all of them to finish the experiment.

# [[Programming_a_2D_Ising_Model/Introduction_to_the_Ising_model|Introduction to the Ising model]]
# [[Programming_a_2D_Ising_Model/Calculating the energy and magnetisation|Calculating the energy and magnetisation]]
# [[Programming_a_2D_Ising_Model/Introduction to Monte Carlo simulation|Introduction to the Monte Carlo simulation]]
# [[Programming_a_2D_Ising_Model/Accelerating the code|Accelerating the code]]
# [[Programming_a_2D_Ising_Model/The effect of temperature|The effect of temperature]]
# [[Programming_a_2D_Ising_Model/The effect of system size|The effect of system size]]
# [[Programming_a_2D_Ising_Model/Determining the heat capacity|Determining the heat capacity]]
# [[Programming_a_2D_Ising_Model/Locating the Curie temperature|Locating the Curie temperature]]

Programming a 2D Ising Model

2026-02-25T18:29:57Z

Jbettenc: /* Getting Help */ Typo

'''<big>This is the Ising Model experiment (aka programming for simple simulations). This is a core experiment for year 3 students on the Chemistry with Molecular Physics degree.</big>'''

==Introduction==

In this exercise, you are going to use the Python that you learned in the first year to write a code to perform Monte-Carlo simulations of the 2D [http://en.wikipedia.org/wiki/Ising_model Ising model], a set of spins on a lattice which is used to model ferromagnetic behaviour, and also to analyse the results of the simulation to find the heat capacity of the system and the Curie temperature — the temperature below which the system is able to maintain a spontaneous magnetisation.

A more extensive introduction to the Ising Model will be given in the next page. This page gives some practical information about how the experiment.

==Assessment==
At the end of this experiment you must submit a report, writing up your findings. Each section of the experiment has a number of tasks that you should complete, labelled '''<big>TASK</big>'''. If this is a mathematical exercise, your report should contain a short summary of the solution. If it is a graphical exercise, you should include the relevant image. Your report should explain briefly what you did in each stage of the experiment, and what your findings were.

The report should be submitted as a PDF file to Turnitin, created with your favourite word processor or LaTeX, and should contain the annotated source code of the Python scripts that your write during the experiment. For clarity, code included in the report should be written in a monospaced font (e.g. Courier), which in LaTeX can be easily achieved by placing your code inside a <code>\texttt{}</code> command. '''The code, except where relevant to discussion, can be placed in an appendix.'''

As well as including relevant source code in your report, you should also submit a bundle with all your scripts and analysis code as a separate Blackboard submission.

===On the use of AI tools===
The code you write for this experiment is assessed, and as per the [https://bb.imperial.ac.uk/bbcswebdav/pid-3344481-dt-content-rid-21115105_1/xid-21115105_1 department's guidelines on the use of AI], it is not acceptable to use generative AI tools to write computer code for this assignment.

Writing computer programs yourself enhances your logical thinking and problem solving skills. Using generative AI tools to write your code makes you dumb. Don't choose to be dumb, don't use AI to write code for you.

==Getting Help==

The Graduate Teaching Assistants demonstrating this experiment are Fionn Carman and Lei Li. Demonstrators will be available in room 232A at the following times:

{| class="wikitable"
|+ Demonstrator timetable
! !! Monday !! Tuesday !! Wednesday !! Thusrsday !! Friday
|-
| Morning || 10:00 - Experiment intro<br /> 10:00-13:00 - Lei || 10:00-13:00 - Fionn|| || 10:00-13:00 - João || 10:00-13:00 - Fionn
|-
| Afternoon || 14:00-17:00 - Lei|| 14:00-17:00 - Fionn|| || || 14:00-17:00 - Fionn
|}

If you have questions outside these times, please use the Discussion Board on Blackboard.

João and Giuseppe are the members of academic staff responsible for this experiment. You are welcome to address your questions and comments to them.

==Structure of this Experiment==

This experimental manual has been broken up into a number of subsections. Direct links to each of them may be found below. You should attempt them in order, and you should complete all of them to finish the experiment.

# [[Programming_a_2D_Ising_Model/Introduction_to_the_Ising_model|Introduction to the Ising model]]
# [[Programming_a_2D_Ising_Model/Calculating the energy and magnetisation|Calculating the energy and magnetisation]]
# [[Programming_a_2D_Ising_Model/Introduction to Monte Carlo simulation|Introduction to the Monte Carlo simulation]]
# [[Programming_a_2D_Ising_Model/Accelerating the code|Accelerating the code]]
# [[Programming_a_2D_Ising_Model/The effect of temperature|The effect of temperature]]
# [[Programming_a_2D_Ising_Model/The effect of system size|The effect of system size]]
# [[Programming_a_2D_Ising_Model/Determining the heat capacity|Determining the heat capacity]]
# [[Programming_a_2D_Ising_Model/Locating the Curie temperature|Locating the Curie temperature]]

Programming a 2D Ising Model

2026-02-25T18:29:28Z

Jbettenc: /* Getting Help */ Update demonstrators rota

'''<big>This is the Ising Model experiment (aka programming for simple simulations). This is a core experiment for year 3 students on the Chemistry with Molecular Physics degree.</big>'''

==Introduction==

In this exercise, you are going to use the Python that you learned in the first year to write a code to perform Monte-Carlo simulations of the 2D [http://en.wikipedia.org/wiki/Ising_model Ising model], a set of spins on a lattice which is used to model ferromagnetic behaviour, and also to analyse the results of the simulation to find the heat capacity of the system and the Curie temperature — the temperature below which the system is able to maintain a spontaneous magnetisation.

A more extensive introduction to the Ising Model will be given in the next page. This page gives some practical information about how the experiment.

==Assessment==
At the end of this experiment you must submit a report, writing up your findings. Each section of the experiment has a number of tasks that you should complete, labelled '''<big>TASK</big>'''. If this is a mathematical exercise, your report should contain a short summary of the solution. If it is a graphical exercise, you should include the relevant image. Your report should explain briefly what you did in each stage of the experiment, and what your findings were.

The report should be submitted as a PDF file to Turnitin, created with your favourite word processor or LaTeX, and should contain the annotated source code of the Python scripts that your write during the experiment. For clarity, code included in the report should be written in a monospaced font (e.g. Courier), which in LaTeX can be easily achieved by placing your code inside a <code>\texttt{}</code> command. '''The code, except where relevant to discussion, can be placed in an appendix.'''

As well as including relevant source code in your report, you should also submit a bundle with all your scripts and analysis code as a separate Blackboard submission.

===On the use of AI tools===
The code you write for this experiment is assessed, and as per the [https://bb.imperial.ac.uk/bbcswebdav/pid-3344481-dt-content-rid-21115105_1/xid-21115105_1 department's guidelines on the use of AI], it is not acceptable to use generative AI tools to write computer code for this assignment.

Writing computer programs yourself enhances your logical thinking and problem solving skills. Using generative AI tools to write your code makes you dumb. Don't choose to be dumb, don't use AI to write code for you.

==Getting Help==

The Graduate Teaching Assistants demonstrating this experiment are Fionn Carman and Lei Li. Demonstrators will be available in room 232A at the following times:

{| class="wikitable"
|+ Demonstrator timetable
! !! Monday !! Tuesday !! Wednesday !! Thusrsday !! Friday
|-
| Morning || 10:00 - Experiment intro<br /> 10:00-13:00 - Lei || 10:00-13:00 - Fionn|| || 10:00-13:00 - João || 10:00-13:00 - Fionn
|-
| Afternoon || 14:00-17:00 - Li|| 14:00-17:00 - Fionn|| || || 14:00-17:00 - Fionn
|}

If you have questions outside these times, please use the Discussion Board on Blackboard.

João and Giuseppe are the members of academic staff responsible for this experiment. You are welcome to address your questions and comments to them.

==Structure of this Experiment==

This experimental manual has been broken up into a number of subsections. Direct links to each of them may be found below. You should attempt them in order, and you should complete all of them to finish the experiment.

# [[Programming_a_2D_Ising_Model/Introduction_to_the_Ising_model|Introduction to the Ising model]]
# [[Programming_a_2D_Ising_Model/Calculating the energy and magnetisation|Calculating the energy and magnetisation]]
# [[Programming_a_2D_Ising_Model/Introduction to Monte Carlo simulation|Introduction to the Monte Carlo simulation]]
# [[Programming_a_2D_Ising_Model/Accelerating the code|Accelerating the code]]
# [[Programming_a_2D_Ising_Model/The effect of temperature|The effect of temperature]]
# [[Programming_a_2D_Ising_Model/The effect of system size|The effect of system size]]
# [[Programming_a_2D_Ising_Model/Determining the heat capacity|Determining the heat capacity]]
# [[Programming_a_2D_Ising_Model/Locating the Curie temperature|Locating the Curie temperature]]

Template:Cite journal

2026-02-24T13:52:18Z

Jbettenc: Fix spacing

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2026-02-24T13:48:10Z

Jbettenc: Add edition field

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== Se also ==
* [[Template:Cite journal]]

</noinclude>

Template:Cite book

2026-02-24T13:45:22Z

Jbettenc: Fix template

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Jbettenc: Align with RSC.

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* [[Template:Cite book]]

</noinclude>

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== See also ==
* [[Template:Cite book]]

</noinclude>

Programming a 2D Ising Model/Calculating the energy and magnetisation

2026-02-17T19:35:38Z

Jbettenc: Add comment about expecting some tests to fail.

'''<big><span style="color:blue; ">This is the second section of the Ising model experiment. You can return to the previous page, [[Programming a 2D Ising Model/Introduction to the Ising model|Introduction to the Ising model]], or jump ahead to the next section, [[Programming a 2D Ising Model/Introduction to Monte Carlo simulation|Introduction to Monte Carlo simulation]].</span></big>'''

==Getting the files for the experiment==

'''Various scripts have been prepared to assist you with this experiment, and you can obtain them by downloading the files from the Blackboard.'''

We recommend you open the extracted folder with JupyterLab, which has an integrated terminal where you can run the simulation scripts.

==Modifying the files==

The file '''IsingLattice.py''' contains a Python class called IsingLattice. This class will be the basis for our model of the system, and is restricted to modelling the two dimensional case. It contains a number of stub methods that you will complete as you work through the experiment.

===A note about Python Classes===

The file IsingLattice.py makes use of a Python feature called ''class''. You should be familiar with the idea that all Python ''objects'' have an associated ''type'' -- you have already encountered objects such as integers, string, lists, and NumPy arrays. Classes allow us to define new types of object. In this case, we define a type called <code>IsingLattice</code>:

<code>
class IsingLattice:
</code>

Each class is allowed to have certain attributes and functions (known as methods) associated with it. We can see, for example, that each <code>IsingLattice</code> object has the methods <code>energy()</code>, <code>magnetisation()</code>, <code>montecarlostep(T)</code>, and <code>statistics()</code> (the method <code>__init__</code> will be explained shortly). Each of these methods takes a special first argument called self, which is a variable pointing to the current copy of IsingLattice. To use the new IsingLattice object that we have defined, we use code like:

<pre>
import IsingLattice

il = IsingLattice(5, 5) #create an IsingLattice object with 5 row and 5 columns

energy = il.energy() #call the energy() method of our IsingLattice object
</pre>

When writing code "within" the class definition, all attributes and methods belonging to the class must be accessed using the "self." notation. You can see an example of this in the <code>montecarlostep()</code> method:

<pre>
energy = self.energy()
</pre>

This line creates a variable called energy which contains whatever value was returned by the method <code>energy()</code> which is part of the <code>IsingLattice</code> class. If you just want to define a local variable instead, like an iterator in a loop, the <code>self.</code> notation is not required.

<pre>
def magnetisation(self):
# we have to use self.lattice, because the lattice variable is part of the object, not local to this function
for i in self.lattice:
...
# we use i, not self.i
# we don't need to access i outside this function
</pre>

Have a look at the other files that you have been given to see how this IsingLattice object is used.

===The Constructor===

The only method currently filled in is <code>__init__</code>. This is a special method called a ''constructor''. Whenever a new IsingLattice object is created, using code like <code>il = IsingLattice(5,5)</code>, this method is called automatically. It takes two arguments (excluding "self"), which are simply the number of rows and columns that our lattice will have. The attribute <code>self.lattice</code> is then created, which is a NumPy array with the requested number of rows and columns. The function <code>np.random.choice</code> is used to create a random collection of spins for our initial configuration. You can see the documentation for the function [https://numpy.org/doc/stable/reference/random/generated/numpy.random.choice.html numpy.random.choice].

'''<big>TASK 2a</big>: complete the functions <code>energy()</code> and <code>magnetisation()</code>, which should return the energy of the lattice and the total magnetisation, respectively. In the <code>energy()</code> function you may assume that ''J''=1.0 at all times (in fact, we are working in ''reduced units'' in which all energies are a multiple of ''J'', but there will be more information about this in later sections). Do not worry about the efficiency of the code at the moment — we will address the speed in a later part of the experiment. Paste your code in your report and describe how it works.'''

==Testing the files==

When you have completed the energy() and magnetisation() methods, we need to test that they work correctly.
We have provided you with a script, '''ILcheck.py''', which will create three IsingLattice objects and check their energies and magnetisations, displaying these on a graph. One configuration corresponds to the energy minimum, one to the energy maximum, and one to an intermediate state.

'''<big>TASK 2b</big>: Run the ILcheck.py script from the a terminal using the command'''<pre>python ILcheck.py</pre>'''The displayed window has a series of control buttons in the bottom left, one of which will allow you to export the figure as a SVG or PNG image. Save an image of the ILcheck.py output, and include it in your report.'''

There are also some unit tests available in the <code>test_energy.py</code> file. These tests are designed to be run by simply running the [https://docs.pytest.org pytest] command in your terminal,
<pre>pytest</pre>
You should make sure your code passes the tests for the <code>energy()</code> and <code>magnetisation()</code> methods at this time. The file contains tests for methods that will only be implemented later on, so some failures are expected for now.
These sorts of test are useful because they are easier to write and faster to run and can be made to run automatically.

You can also debug and test your program on a Jupyter notebook or IPython console by importing your IsingLattice class, creating one object, and inspecting its properties. If you do this, we recommend that you start your session running the commands:

<pre>
%load_ext autoreload
%autoreload 2
</pre>

This will ensure that when you try executing your code, you always run the most recent version (please note this isn't fool proof and if you find problems you may need to restart the kernel)!

'''<big><span style="color:blue; ">This is the second section of the Ising model experiment. You can return to the previous page, [[Programming a 2D Ising Model/Introduction to the Ising model|Introduction to the Ising model]], or jump ahead to the next section, [[Programming a 2D Ising Model/Introduction to Monte Carlo simulation|Introduction to Monte Carlo simulation]].</span></big>'''

Main Page

2026-02-17T19:12:04Z

Jbettenc: Some cleanup shifting material not used in current curriculum

[[Image:QR_complab.png|right|QR]]
This is a communal area for documenting teaching and laboratory courses. To [[admin:add|add to any content on these pages]], you will have to log in using your Imperial College account. This page has {{DOI|dh4g}}
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== First Year ==
*[[it:it_facillities|Email and IT@www.ch.imperial.ac.uk]]: A summary of available IT resources
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*[http://www.huntresearchgroup.org.uk/teaching/year2a_lab_start.html Inorganic Computational Chemistry Computational Laboratory]
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*[[mod:intro|Information needed for the course]]


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*[[mod:latebreak|Late breaking news]]

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# [[mod:3D|3D-printable models]]
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Programming a 2D Ising Model/Introduction to the Ising model

2026-02-17T18:43:21Z

Jbettenc: Replace <math> for inline expressions that can be written inline

'''<big><span style="color:blue; ">This is the first section of the Ising model experiment. You can return to the [[Programming_a_2D_Ising_Model|introduction page]], or jump ahead to the next section, [[Third year CMP compulsory experiment/Calculating the energy and magnetisation|Calculating the energy and magnetisation]].</span></big>'''

==The model==
The Ising model is a simple physical model proposed in the 1920s to understand the behaviour of ferromagnets. In most materials, the magnetic dipoles of the atoms have no preferred orientation, and the bulk material has no net magnetic moment. In certain materials, however, there is preferential alignment of the magnetic dipoles along certain directions. These materials, such as iron, exhibit an overall magnetic moment, and are called ferromagnetic.

The phenomenon is based on competition between two physical effects — energy minimisation and entropy maximisation. On the one hand, parallel alignment of magnetic moments is energetically favourable. On the other hand, aligning the system in this way reduces the entropy.

In the Ising model, we consider a collection of spins, ''s<sub>i</sub>'', each of which can either have the value "up" (+1) or "down" (-1), arranged on a lattice. In one dimension, this takes the form of a "chain" of adjacent spins, ''s<sub>1</sub>'' to ''s<sub>N</sub>''. In two dimensions we have a grid of cells, each of which contains one spin. In three dimensions, the system is a cuboid, and so on. The arrangements for 1, 2 and 3 dimensions are illustrated in '''figure 1'''.

[[File:ThirdYearCMPExpt-IsingSketch.png|300px|thumb|right|'''Figure 1''': Illustration of an Ising lattice in one (N=5), two (N=5×5), and three (N=5×5×5) dimensions. Red cells indicate the "up" spin", and blue cells indicate the "down" spin.]]

The rules of the model are simple:

* When no magnetic field is applied, the only energetic contribution comes from the interaction between spins in ''adjacent'' lattice cells
* The total interaction energy is defined as <math>- \frac{1}{2} J \sum_i^N \sum_{j\ \in\ \mathrm{neighbours}\left(i\right)} s_i s_j</math>, where ''J'' is a constant which controls the strength of interaction and the notation ''j'' &in; neighbours(''i'') indicates that spin ''j'' must lie in a lattice cell adjacent to spin ''i''. The factor of one half is included to make sure that we do not count each spin-spin interaction twice.
* Periodic boundary conditions are applied. In other words, the "final" spin in a particular direction is considered to be adjacent to the "first" spin in that direction. In '''figure 2''', for example, cell 1 has neighbour interactions with cells 2, 3, 4, and 7, while cell 9 interacts with cells 3, 6, 7, and 8.

[[File:ThirdYearCMPExpt-IsingNumberedCell.png|300px|thumb|right|'''Figure 2''': Illustration of a 2D Ising lattice, with numbered cells to indicate the neighbour interactions.]]

'''<big>TASK 1a</big>: Show that the lowest possible energy for the Ising model (with ''J''>0) is ''E'' = -''DNJ'', where ''D'' is the number of dimensions and ''N'' is the total number of spins. What is the multiplicity of this state? Calculate its entropy.'''

==Phase Transitions==

The Ising model is interesting because it is one of the simplest physical models to display a phase transition (so long as we are in two dimensions or more). The temperature of the system controls the balance between the energetic driving force, which wants the system to adopt the lowest energy configuration that you imagined in the previous section, and the entropic driving force, which seeks to maximise the number of configurations available to the system.

'''<big>TASK 1b</big>: Imagine that the system is in the lowest energy configuration. To move to a different state, one of the spins must spontaneously change direction ("flip"). What is the change in energy if this happens (''D''=3, ''N''=1000)? How much entropy does the system gain by doing so?'''

At low temperatures, little thermal energy is available to overcome such energy barriers, and we expect the system to adopt the lowest energy (the energetic driving force will dominate). Under these conditions, we expect a large number of parallel spins, and we can characterise this by considering the total ''magnetisation'' of the system: <math>M = \sum_i s_i</math>. At higher temperatures, however, enough thermal energy will be available that spins can flip relatively freely. Under these conditions, the entropic driving force should dominate. The transition temperature between the two regimes is called the ''Curie temperature'', ''T<sub>C</sub>''. Below ''T<sub>C</sub>'', we observe the highly ordered magnetised phase, while above ''T<sub>C</sub>'' a disordered, demagnetised, phase emerges.

'''<big>TASK 1c</big>: Calculate the magnetisation of the 1D and 2D lattices in figure 1. What magnetisation would you expect to observe for an Ising lattice with ''D''=3, ''N''=1000 at absolute zero?'''

We are interested, therefore, in determining the magnetisation ''M'', and energy ''E'', of the system as functions of temperature ''T''. We can write statistical mechanical equations for these quantities:

<math>\left\langle M\right\rangle_T = \frac{1}{Z}\sum_\alpha M\left(\alpha\right) \exp \left\{-\frac{E\left(\alpha\right)}{k_BT}\right\} </math>

<math>\left\langle E\right\rangle_T = \frac{1}{Z}\sum_\alpha E\left(\alpha\right) \exp \left\{-\frac{E\left(\alpha\right)}{k_BT}\right\} </math>

where ''α'' is used as a shorthand to represent all spins in the system, and ''Z'' is the partition function,
<math>Z = \sum_\alpha \exp \left\{-\frac{E\left(\alpha\right)}{k_BT}\right\} </math>.

It was discovered relatively quickly after the model was first proposed that the partition function in the one dimensional case can be solved analytically (though doing so is beyond the scope of this experiment):

<math>Z\left(T, N\right) = \left[2\cosh\left(\frac{J}{k_BT}\right)\right]^N</math>

Much later, [http://en.wikipedia.org/wiki/Lars_Onsager Lars Onsager] established that the two dimensional partition function can also be found analytically, though the mathematics is significantly more involved and the resulting expression is correspondingly much more complicated. In dimensions greater than two, no analytical solution is known.

Luckily for us, the problem can instead be tackled by numerical methods, and it is this that will be the focus of the experiment. We will discuss how the equations for ''M'' and ''E'' can be solved, but first we must set up a Python script to model the lattice.

'''<big><span style="color:blue; ">This is the first section of the Ising model experiment. You can return to the [[Programming_a_2D_Ising_Model|introduction page]], or jump ahead to the next section, [[Third year CMP compulsory experiment/Calculating the energy and magnetisation|Calculating the energy and magnetisation]].</span></big>'''

Programming a 2D Ising Model/Locating the Curie temperature

2026-02-17T16:59:00Z

Jbettenc: Replace <math> for inline expressions that can be written inline, and highlight variable names in code

'''<big><span style="color:blue; ">This is the eighth (and final) section of the Ising model experiment. You can return to the previous page, [[Programming a 2D Ising Model/Determining the heat capacity|Determining the heat capacity]], or go back to the [[Programming a 2D Ising Model|Introduction]].</span></big>'''

You should have seen in the previous section that the heat capacity becomes strongly peaked in the vicinity of the critical temperature and that the peak became increasingly sharply peaked as the system size was increased — in fact, Onsager proved that in an infinite system the heat capacity should diverge at ''T'' = ''T<sub>C</sub>''. In our finite systems, however, not only does the heat capacity not diverge, the Curie temperature changes with system size! This is known as a ''finite size effect''.

It can be shown, however, that the temperature at which the heat capacity has its maximum must scale according to <math>T_{C, L} = \frac{A}{L} + T_{C,\infty}</math>, where ''T<sub>C,L</sub>'' is the Curie temperature of an ''L''×''L'' lattice, ''T<sub>C,∞</sub>'' is the Curie temperature of an infinite lattice, and ''A''> is a constant in which we are not especially interested. This scaling equation comes from expanding in the limit of large ''L''.

'''<big>TASK 8a</big>: Reference data has been generated for longer simulations than would be possible on the college computers in Python. Each file contains six columns: ''T'', ''E'', ''E''<sup>2</sup>, ''M'', ''M''<sup>2</sup>, ''C'' (NOTE: these final five quantities are normalised per spin, unlike your Python data), and you can read them with the NumPy <code>loadtxt</code> function as before. For each lattice size, plot the reference data against your data. For ''one'' lattice size, save a SVG or PNG image of this comparison and add it to your report — add a legend to the graph to label which is which (check if you need a [https://matplotlib.org/stable/api/_as_gen/matplotlib.axes.Axes.legend.html reminder about use of legend]).'''

==Polynomial fitting==

To find the temperature at which the heat capacity and susceptibilities have their maxima, we are going to fit a polynomial to the data in the critical region. NumPy provides the useful [http://docs.scipy.org/doc/numpy/reference/generated/numpy.polyfit.html <code>polyfit</code>] and [http://docs.scipy.org/doc/numpy/reference/generated/numpy.polyval.html#numpy.polyval <code>polyval</code>] functions for this purpose. The following code is written for the heat capacities - try to also repeat this for the susceptibility! The usage is as follows:

<pre>
data = np.loadtxt("...") #assume data is now a 2D array containing two columns, T and C
T = data[:,0] #get the first column
C = data[:,1] # get the second column

#first we fit the polynomial to the data
fit = np.polyfit(T, C, 3) # fit a third order polynomial

#now we generate interpolated values of the fitted polynomial over the range of our function
T_min = np.min(T)
T_max = np.max(T)
T_range = np.linspace(T_min, T_max, 1000) #generate 1000 evenly spaced points between T_min and T_max
fitted_C_values = np.polyval(fit, T_range) # use the fit object to generate the corresponding values of C
</pre>

'''<big>TASK 8b</big>: Write a script to read the data from a particular file, and plot ''C'' and ''χ'' vs ''T'', as well as a fitted polynomial. Try changing the degree of the polynomial to improve the fit — in general, it might be difficult to get a good fit! Attach a SVG or PNG image of an example fit to your report.'''

===Fitting in a particular temperature range===

Rather than fit to all of the data, we might want to fit in only a particular range. NumPy provides a very powerful way to index arrays based on certain conditions. For example, if we want to extract only those data points which are between a particular ''T''<sub>min</sub> and ''T''<sub>max</sub>, we can use the following syntax:

<pre>
data = np.loadtxt("...") #assume data is now a 2D array containing two columns, T and C
T = data[:,0] #get the first column
C = data[:,1] # get the second column

Tmin = 0.5 #for example
Tmax = 2.0 #for example

selection = np.logical_and(T > Tmin, T < Tmax) #choose only those rows where both conditions are true
peak_T_values = T[selection]
peak_C_values = C[selection]
</pre>

'''<big>TASK 8c</big>: Modify your script from the previous section. You should still plot the whole temperature range, but fit the polynomial only to the peak! You should find it easier to get a good fit when restricted to this region.'''

===Finding the peak in C===

Your fitting script should now generate two variables: <code>peak_T_range</code>, containing 1000 equally spaced temperature values between ''T''<sub>min</sub> and ''T''<sub>max</sub> (whatever you chose those values to be), and fitted_''C''_values, containing the fitted heat capacity at each of those points. Use the NumPy <code>max</code> function to find the maximum in ''C''. If you store the maximum value of ''C'' in the variable <code>Cmax</code>, you can use the following notation to find the corresponding temperature:

<pre>
Cmax = np.max(...)
Tmax = peak_T_range[fitted_C_values == Cmax]
</pre>

'''<big>TASK 8d</big>: Find the temperature at which the maximum in ''C'' occurs for each datafile that you were given or generated with Python. Make a text file containing two columns: the lattice side length (2,4,8, etc.), and the temperature at which ''C'' is a maximum. This is your estimate of ''T<sub>C</sub>'' for that side length. Make a plot that uses the scaling relation given above to determine ''T<sub>C,∞</sub>''. By doing a little research, you should be able to find the theoretical exact Curie temperature for the infinite 2D Ising lattice. How does your value compare to this? Are you surprised by how good/bad the agreement is? Attach a SVG or PNG image of this final graph to your report, and discuss briefly what you think the major sources of error are in your estimate.'''

'''<big><span style="color:blue; ">This is the eighth (and final) section of the Ising model experiment. You can return to the previous page, [[Programming a 2D Ising Model/Determining the heat capacity|Determining the heat capacity]], or go back to the [[Programming a 2D Ising Model|Introduction]].</span></big>'''

Programming a 2D Ising Model/Determining the heat capacity

2026-02-17T16:43:23Z

Jbettenc: Replace <math> for inline expressions that can be written inline

'''<big><span style="color:blue; ">This is the seventh section of the Ising model experiment. You can return to the previous page, [[Programming a 2D Ising Model/The effect of system size|The effect of system size]], or jump ahead to the next section, [[Programming a 2D Ising Model/Locating the Curie temperature|Locating the Curie temperature]].</span></big>'''

==Calculating the heat capacity==

As we have seen, increasing the temperature above ''T<sub>C</sub>'' induces a phase transition — the magnetisation of the system rapidly drops, but it can be hard to use this information to pinpoint the Curie temperature itself. As well as demonstrating the closed form solution to the partition function that we mentioned in the introduction, Lars Onsager also demonstrated that the heat capacity of the 2D Ising model should become very strongly peaked at the phase transition temperature (in fact, when ''T'' = ''T<sub>C</sub>'' exactly, ''C'' diverges).

'''<big>TASK 7a</big>: By definition, <math>C = \frac{\partial \left\langle E\right\rangle}{\partial T}</math>. From this, show that <math>C = \frac{\mathrm{Var}[E]}{k_B T^2}</math>, where <math>\mathrm{Var}[E]</math> is the variance in <math>E</math>.'''

Using this relation, and the data that you generated in the previous sections, you can now determine the heat capacity of your lattice as a function of temperature and system size. in a similar way, we can also define another quantity called the magnetic susceptibility, ''χ'', which depends on the average properties of the order parameter of the system:

<math>\chi = \beta(\left\langle M^2\right\rangle - \left\langle M\right\rangle^2)</math>

which will behave in a similar way to the heat capacity. Here <math>\beta = \frac{1}{k_BT}</math> is the inverse of the thermodynamic temperature, often used in statistical thermodynamics.

'''<big>TASK 7b</big>: Write a Python script to make a plot showing the heat capacity and susceptibility versus temperature for each of your lattice sizes from the previous section. You may need to do some research to recall the connection between the variance of a variable, Var[''X''], the mean of its square &lang;''X''<sup>2</sup>&rang;, and its squared mean &lang;''X''&rang;<sup>2</sup>. You may find that the data around the peak is very noisy — this is normal, and is a result of being in the critical region. As before, use the plot controls to save a SVG or PNG image of your plot and attach this to the report.'''

'''<big><span style="color:blue; ">This is the seventh section of the Ising model experiment. You can return to the previous page, [[Programming a 2D Ising Model/The effect of system size|The effect of system size]], or jump ahead to the next section, [[Programming a 2D Ising Model/Locating the Curie temperature|Locating the Curie temperature]].</span></big>'''

Programming a 2D Ising Model/The effect of temperature

2026-02-17T16:36:30Z

Jbettenc: Replace <math> for inline expressions that can be written inline, and highlight variable names in code

'''<big><span style="color:blue; ">This is the fifth section of the Ising model experiment. You can return to the previous page, [[Programming a 2D Ising Model/Accelerating the code|Accelerating the code]], or jump ahead to the next section, [[Programming a 2D Ising Model/The effect of system size|The effect of system size]].</span></big>'''

We now have enough code to start to investigate the phase behaviour of our model. In the first instance, we are going to make plots of the average energy and magnetisation of the system as a function of temperature, to get an idea of where the change between energetic and entropic domination takes place, but we first have to make a small change to the code that averages the energy and magnetisation.

==Correcting the Averaging Code==
As you saw in the previous part of the experiment, when we begin from a random configuration, it takes a few Monte Carlo cycles to reach the equilibrium state of the system at a particular temperature. This manifests itself as a relatively sharp drop in the energy of the system in the first few hundred or thousand steps, before the energy starts to fluctuate around a constant value. If we want our average properties to be correct, we should only average in the region where the ''average'' energy and magnetisation are constant.

'''<big>TASK 5a</big>: The script ILfinalframe.py runs for a given number of cycles at a given temperature, then plots a depiction of the ''final'' lattice state as well as graphs of the energy and magnetisation as a function of cycle number. This is much quicker than animating every frame! Experiment with a few different temperatures and lattice sizes. Approximately how many cycles (remember, a cycle is ''N''<sub>spins</sub> steps) are needed for the system to go from its random starting position to the equilibrium state? Modify your <code>__init__()</code>, <code>statistics()</code> and <code>montecarlostep()</code> methods so that the first ''N'' cycles of the simulation are ignored when calculating the averages. Be careful of off-by-one errors. You should state in your report what period you chose to ignore, and include graphs from ILfinalframe.py to illustrate your motivation in choosing this number.'''
'''

==Running over a range of temperatures==

The script ILtemperaturerange.py performs simulations of a given length over a range of temperatures. The temperature range is determined by the parameters given to the <code>np.arange</code> function near the top of the file. Starting from a uniformly aligned lattice, a simulation of length <code>n_steps</code> will be performed at temperature <code>T_cold</code>, and the average energy and magnetisation will be recorded. A new simulation of length <code>n_steps</code> will then be performed at temperature <code>T_cold + dT</code>, and so on until all temperatures in the range have been simulated. This results in a NumPy array of five columns: the temperature, then the four statistical quantities: <code>ave_energies</code>, <code>ave_sqenergies</code> (squared energies), <code>ave_magnetisations</code>, <code>ave_sqmagnetisations</code> (squared magnetisations).

'''<big>TASK 5b</big>: Use ILtemperaturerange.py to plot the average energy and magnetisation for each temperature, ''with error bars'', for an 8 × 8 lattice. Use multiple repeats to calculate these error bars. Use your intuition and results from the script ILfinalframe.py to estimate how many cycles each simulation should be. The temperature range 0.25 to 5.0 is sufficient. Use as many temperature points as you feel necessary to illustrate the behaviour, but do not use a temperature spacing larger than 0.5. The NumPy function <code>savetxt()</code> stores your array of output data on disk — you will need it later. Save the file as ''8x8.dat'' so that you know which lattice size it came from.'''

'''<big><span style="color:blue; ">This is the fifth section of the Ising model experiment. You can return to the previous page, [[Programming a 2D Ising Model/Accelerating the code|Accelerating the code]], or jump ahead to the next section, [[Programming a 2D Ising Model/The effect of system size|The effect of system size]].</span></big>'''

Programming a 2D Ising Model/Introduction to Monte Carlo simulation

2026-02-17T16:27:21Z

Jbettenc: Replace <math> for inline expressions that can be written inline

'''<big><span style="color:blue; ">This is the third section of the Ising model experiment. You can return to the previous page, [[Programming a 2D Ising Model/Calculating the energy and magnetisation|Calculating the energy and magnetisation]], or jump ahead to the next section, [[Programming a 2D Ising Model/Accelerating the code|Accelerating the code]].</span></big>'''
==Average Energy and Magnetisation==

Consider again the expressions for the average energy and magnetisation that we gave in the introduction.

<math>\left\langle M\right\rangle_T = \frac{1}{Z}\sum_\alpha M\left(\alpha\right) \exp \left\{-\frac{E\left(\alpha\right)}{k_BT}\right\} </math>

<math>\left\langle E\right\rangle_T = \frac{1}{Z}\sum_\alpha E\left(\alpha\right) \exp \left\{-\frac{E\left(\alpha\right)}{k_BT}\right\} </math>

Imagine we want to evaluate these at a particular temperature, in a system of 100 spins.

'''<big>TASK 3a</big>: How many configurations are available to a system with 100 spins? To evaluate these expressions, we have to calculate the energy and magnetisation for each of these configurations, then perform the sum. Let's be very, very, generous, and say that we can analyse 1×10<sup>9</sup> configurations per second with our computer. How long will it take to evaluate a single value of &lang;''M''&rang;<sub>''T''</sub>?'''

Clearly, we need to try a cleverer approach.

For the overwhelming majority of the possible configurations, the Boltzmann factor, <math>\exp \left\{-\frac{E\left(\alpha\right)}{k_BT}\right\}</math>, will be very small, and that state will contribute very little to the average value. We can save an enormous amount of time, and make this problem tractable, if we only consider the states whose Boltzmann factor is not so vanishingly small. This technique is called ''importance sampling'' — instead of sampling every point in phase space, we sample only those that the system is likely to occupy.

==Importance Sampling==

This is easily stated, of course, but how can we know which states contribute a lot to the average without actually calculating the energy? So far, we have been imagining generating random states of the system; that is to say, we have been choosing ''α'' from a uniform distribution in which no arrangement of spins is any more likely to be chosen than any other. If we instead had a method which generated states randomly from the probability distribution <math>\exp \left\{-\frac{E\left(\alpha\right)}{k_BT}\right\}</math>, then our problem would be solved. The method which allows us to do this was developed in the 1950s, and is called the [https://doi.org/10.1063/1.1699114 Metropolis Monte Carlo (MMC)] method, or more often simply "Monte Carlo simulation". This was one of the major breakthroughs of the early days of computational science, and in the year 2000, the IEEE named among the [https://doi.org/10.1109/MCISE.2000.814652 "Top 10 Algorithms of the 20th Century"].

The algorithm is as follows:

# Start from a given configuration of spins, ''α<sub>0</sub>'', with energy ''E<sub>0</sub>''.
# Choose a single spin '''at random''', and "flip" it, to generate a new configuration ''α<sub>1</sub>''.
# Calculate the energy of this new configuration, ''E<sub>1</sub>''.
# Calculate the energy difference between the states, ''ΔE = E<sub>1</sub> - E<sub>0</sub>''.
## If the ''ΔE ≤ 0'' (the spin flipping decreased the energy), then we '''accept''' the new configuration.
##* We set ''α<sub>0</sub> = α<sub>1</sub>'', and ''E<sub>0</sub> = E<sub>1</sub>'', and then '''go to step 5'''.
## If ''ΔE > 0'', the spin flipping increased the energy. By considering the probability of observing the starting and final states, ''α<sub>0</sub>'' and ''α<sub>1</sub>'', it can be shown that the probability for the transition between the two to occur is <math>\exp \left\{-\frac{\Delta E}{k_BT}\right\}</math>. To ensure that we only accept this kind of spin flip with the correct probability, we use the following procedure:
### Choose a random number, ''R'', in the interval [0,1).
### If <math>R \leq \exp \left\{-\frac{\Delta E}{k_BT}\right\}</math>, we '''accept''' the new configuration.
###* We set ''α<sub>0</sub> = α<sub>1</sub>'', and ''E<sub>0</sub> = E<sub>1</sub>'', and then '''go to step 5'''.
### If <math>R > \exp \left\{-\frac{\Delta E}{k_BT}\right\}</math>, we '''reject''' the new configuration.
###* ''α<sub>0</sub>'' and ''E<sub>0</sub>'' are left unchanged. '''Go to step 5'''.
# Update the running averages of the energy and magnetisation.
# Monte Carlo "step" complete, '''return to step 2'''.

Step 4 is the key to the method. By accepting moves which increase the energy only with a certain probability, we ensure that the sequence of states that we generate is correctly distributed. A transition which carries a very large energy penalty, ''ΔE'' is extremely unlikely to be selected. The use of random numbers in this step is the reason that the method acquired the name "Monte Carlo", after the casinos located there!

Note, unlike molecular dynamics, each step only moves a single atom. In Monte Carlo simulation, there is the notion of a cycle which is ''N''<sub>spins</sub>-steps, so that each spin on average has an attempted flip per cycle.

If you are interested in the mathematical details of why this procedure generates a sequence of states distributed in the correct way, consult the Monte Carlo chapter of [https://library-search.imperial.ac.uk/permalink/f/o1297h/TN_cdi_askewsholts_vlebooks_9780080519982 "Understanding Molecular Simulation"], by Frenkel and Smit, but a discussion of this is not required for the experiment. Chapter 7 of [https://library-search.imperial.ac.uk/permalink/44IMP_INST/mek6kh/alma997227534401591 "Statistical Mechanics: Theory and Molecular Simulation"] by Tuckerman also gives a good description.

==Modifying IsingLattice.py==

Our IsingLattice object contains a stub function, '''<code>montecarlostep(T)</code>''', which takes a single argument — the temperature at which we want to do the cycle. There are also attributes used to calculate the average energy and magnetisation and respective variances: <code>self.E_tally</code>, <code>self.E2_tally</code>, <code>self.M_tally</code>, <code>self.M2_tally</code>, and <code>self.n_steps</code>. You should use these values to keep a running sum of all the energies, squared energies, magnetisations, and squared magnetisations that your system has sampled, as well as the number of steps that have been performed. Finally, there is also a function <code>statistics()</code>, that takes no arguments and should return the average energy, squared energy, magnetisation, squared magnetisation and current step, '''in that order'''.

'''<big>TASK 3b</big>: Implement a single step of the above algorithm in the <code>montecarlostep(T)</code> function. This function should return the energy of your lattice and the magnetisation at the end of the cycle. Complete the <code>statistics()</code> function. This should return the following quantities whenever it is called: &lang;''E''&rang;, &lang;''E''<sup>2</sup>&rang;, &lang;''M''&rang;, &lang;''M''<sup>2</sup>&rang;, and ''N''<sub>steps</sub>, the number of Monte Carlo steps that have elapsed. Put your code for these functions in the report.'''

''Note on units:'' In our simulation we will be using an unitless scaled energy <math>\tilde{E}=\frac{E}{J}</math>, and a unitless scaled temperature <math>\tilde{T}=\frac{k_B T}{J}</math>, where ''J'' is the interaction energy between spins discussed previously. The use of these scaled quantities in practice means that in the code we can set ''k<sub>B</sub>'' to numerically equal to 1. Going forward we will drop the &tilde; from the notation of energy and temperature.

You have been provided with a script, '''ILanim.py''', which will use your <code>IsingLattice</code> object to run a Monte Carlo simulation and display the output on the screen in real time. By default, this simulation will run an 8×8 lattice at ''T''=1.0, which is below the critical temperature. You will see a representation of the lattice, and graphs of the energy and magnetisation '''per spin'''. When you close the window, the script will use your '''<code>statistics()</code>''' function to print the average energy and magnetisation from the simulation. Note, you should run this script from the command line.

'''<big>TASK 3c</big>: If ''T'' < ''T<sub>C</sub>'', do you expect a spontaneous magnetisation (i.e. do you expect &lang;''M''&rang; ≠ 0)? When the state of the simulation appears to stop changing (when you have reached an equilibrium state), use the controls to export the output to an SVG or PNG file and attach this to your report. You should also include the output from your statistics() function.'''

'''<big><span style="color:blue; ">This is the third section of the Ising model experiment. You can return to the previous page, [[Programming a 2D Ising Model/Calculating the energy and magnetisation|Calculating the energy and magnetisation]], or jump ahead to the next section, [[Programming a 2D Ising Model/Accelerating the code|Accelerating the code]].</span></big>'''

Programming a 2D Ising Model/Calculating the energy and magnetisation

2026-02-17T15:54:50Z

Jbettenc: /* The Constructor */ Use italics instead of <math> when referring to variables in line

'''<big><span style="color:blue; ">This is the second section of the Ising model experiment. You can return to the previous page, [[Programming a 2D Ising Model/Introduction to the Ising model|Introduction to the Ising model]], or jump ahead to the next section, [[Programming a 2D Ising Model/Introduction to Monte Carlo simulation|Introduction to Monte Carlo simulation]].</span></big>'''

==Getting the files for the experiment==

'''Various scripts have been prepared to assist you with this experiment, and you can obtain them by downloading the files from the Blackboard.'''

We recommend you open the extracted folder with JupyterLab, which has an integrated terminal where you can run the simulation scripts.

==Modifying the files==

The file '''IsingLattice.py''' contains a Python class called IsingLattice. This class will be the basis for our model of the system, and is restricted to modelling the two dimensional case. It contains a number of stub methods that you will complete as you work through the experiment.

===A note about Python Classes===

The file IsingLattice.py makes use of a Python feature called ''class''. You should be familiar with the idea that all Python ''objects'' have an associated ''type'' -- you have already encountered objects such as integers, string, lists, and NumPy arrays. Classes allow us to define new types of object. In this case, we define a type called <code>IsingLattice</code>:

<code>
class IsingLattice:
</code>

Each class is allowed to have certain attributes and functions (known as methods) associated with it. We can see, for example, that each <code>IsingLattice</code> object has the methods <code>energy()</code>, <code>magnetisation()</code>, <code>montecarlostep(T)</code>, and <code>statistics()</code> (the method <code>__init__</code> will be explained shortly). Each of these methods takes a special first argument called self, which is a variable pointing to the current copy of IsingLattice. To use the new IsingLattice object that we have defined, we use code like:

<pre>
import IsingLattice

il = IsingLattice(5, 5) #create an IsingLattice object with 5 row and 5 columns

energy = il.energy() #call the energy() method of our IsingLattice object
</pre>

When writing code "within" the class definition, all attributes and methods belonging to the class must be accessed using the "self." notation. You can see an example of this in the <code>montecarlostep()</code> method:

<pre>
energy = self.energy()
</pre>

This line creates a variable called energy which contains whatever value was returned by the method <code>energy()</code> which is part of the <code>IsingLattice</code> class. If you just want to define a local variable instead, like an iterator in a loop, the <code>self.</code> notation is not required.

<pre>
def magnetisation(self):
# we have to use self.lattice, because the lattice variable is part of the object, not local to this function
for i in self.lattice:
...
# we use i, not self.i
# we don't need to access i outside this function
</pre>

Have a look at the other files that you have been given to see how this IsingLattice object is used.

===The Constructor===

The only method currently filled in is <code>__init__</code>. This is a special method called a ''constructor''. Whenever a new IsingLattice object is created, using code like <code>il = IsingLattice(5,5)</code>, this method is called automatically. It takes two arguments (excluding "self"), which are simply the number of rows and columns that our lattice will have. The attribute <code>self.lattice</code> is then created, which is a NumPy array with the requested number of rows and columns. The function <code>np.random.choice</code> is used to create a random collection of spins for our initial configuration. You can see the documentation for the function [https://numpy.org/doc/stable/reference/random/generated/numpy.random.choice.html numpy.random.choice].

'''<big>TASK 2a</big>: complete the functions <code>energy()</code> and <code>magnetisation()</code>, which should return the energy of the lattice and the total magnetisation, respectively. In the <code>energy()</code> function you may assume that ''J''=1.0 at all times (in fact, we are working in ''reduced units'' in which all energies are a multiple of ''J'', but there will be more information about this in later sections). Do not worry about the efficiency of the code at the moment — we will address the speed in a later part of the experiment. Paste your code in your report and describe how it works.'''

==Testing the files==

When you have completed the energy() and magnetisation() methods, we need to test that they work correctly.
We have provided you with a script, '''ILcheck.py''', which will create three IsingLattice objects and check their energies and magnetisations, displaying these on a graph. One configuration corresponds to the energy minimum, one to the energy maximum, and one to an intermediate state.

'''<big>TASK 2b</big>: Run the ILcheck.py script from the a terminal using the command'''<pre>python ILcheck.py</pre>'''The displayed window has a series of control buttons in the bottom left, one of which will allow you to export the figure as a SVG or PNG image. Save an image of the ILcheck.py output, and include it in your report.'''

There are also some unit tests available in the <code>test_energy.py</code> file. These tests are designed to be run by simply running the [https://docs.pytest.org pytest] command in your terminal,
<pre>pytest</pre>
These sorts of test are useful because they are easier to write and faster to run and can be run automatically.

You can also debug and test your program on a Jupyter notebook or IPython console by importing your IsingLattice class, creating one object, and inspecting its properties. If you do this, we recommend that you start your session running the commands:

<pre>
%load_ext autoreload
%autoreload 2
</pre>

This will ensure that when you try executing your code, you always run the most recent version (please note this isn't fool proof and if you find problems you may need to restart the kernel)!

'''<big><span style="color:blue; ">This is the second section of the Ising model experiment. You can return to the previous page, [[Programming a 2D Ising Model/Introduction to the Ising model|Introduction to the Ising model]], or jump ahead to the next section, [[Programming a 2D Ising Model/Introduction to Monte Carlo simulation|Introduction to Monte Carlo simulation]].</span></big>'''

Programming a 2D Ising Model

2026-02-17T15:49:02Z

Jbettenc: /* Assessment */ Remove spacing

'''<big>This is the Ising Model experiment (aka programming for simple simulations). This is a core experiment for year 3 students on the Chemistry with Molecular Physics degree.</big>'''

==Introduction==

In this exercise, you are going to use the Python that you learned in the first year to write a code to perform Monte-Carlo simulations of the 2D [http://en.wikipedia.org/wiki/Ising_model Ising model], a set of spins on a lattice which is used to model ferromagnetic behaviour, and also to analyse the results of the simulation to find the heat capacity of the system and the Curie temperature — the temperature below which the system is able to maintain a spontaneous magnetisation.

A more extensive introduction to the Ising Model will be given in the next page. This page gives some practical information about how the experiment.

==Assessment==
At the end of this experiment you must submit a report, writing up your findings. Each section of the experiment has a number of tasks that you should complete, labelled '''<big>TASK</big>'''. If this is a mathematical exercise, your report should contain a short summary of the solution. If it is a graphical exercise, you should include the relevant image. Your report should explain briefly what you did in each stage of the experiment, and what your findings were.

The report should be submitted as a PDF file to Turnitin, created with your favourite word processor or LaTeX, and should contain the annotated source code of the Python scripts that your write during the experiment. For clarity, code included in the report should be written in a monospaced font (e.g. Courier), which in LaTeX can be easily achieved by placing your code inside a <code>\texttt{}</code> command. '''The code, except where relevant to discussion, can be placed in an appendix.'''

As well as including relevant source code in your report, you should also submit a bundle with all your scripts and analysis code as a separate Blackboard submission.

===On the use of AI tools===
The code you write for this experiment is assessed, and as per the [https://bb.imperial.ac.uk/bbcswebdav/pid-3344481-dt-content-rid-21115105_1/xid-21115105_1 department's guidelines on the use of AI], it is not acceptable to use generative AI tools to write computer code for this assignment.

Writing computer programs yourself enhances your logical thinking and problem solving skills. Using generative AI tools to write your code makes you dumb. Don't choose to be dumb, don't use AI to write code for you.

==Getting Help==

The Graduate Teaching Assistants demonstrating this experiment are Fionn Carman and Lei Li. Demonstrators will be available in room 232A at the following times:

{| class="wikitable"
|+ Demonstrator timetable
! !! Monday !! Tuesday !! Wednesday !! Thusrsday !! Friday
|-
| Morning || 10:00 - Experiment intro<br /> 10:00-13:00 - Fionn and Aidan || 10:00-13:00 - Ayse and Aidan|| || 10:00-13:00 - Fionn || 10:00-13:00 - Ayse and Aidan
|-
| Afternoon || 14:00-17:00 - Fionn|| 14:00-17:00 - Ayse and Aidan|| || 14:00-16:00 - Fionn and Aidan || 14:00-17:00 - Ayse
|}

If you have questions outside these times, please use the Discussion Board on Blackboard.

João and Giuseppe are the members of academic staff responsible for this experiment. You are welcome to address your questions and comments to them.

==Structure of this Experiment==

This experimental manual has been broken up into a number of subsections. Direct links to each of them may be found below. You should attempt them in order, and you should complete all of them to finish the experiment.

# [[Programming_a_2D_Ising_Model/Introduction_to_the_Ising_model|Introduction to the Ising model]]
# [[Programming_a_2D_Ising_Model/Calculating the energy and magnetisation|Calculating the energy and magnetisation]]
# [[Programming_a_2D_Ising_Model/Introduction to Monte Carlo simulation|Introduction to the Monte Carlo simulation]]
# [[Programming_a_2D_Ising_Model/Accelerating the code|Accelerating the code]]
# [[Programming_a_2D_Ising_Model/The effect of temperature|The effect of temperature]]
# [[Programming_a_2D_Ising_Model/The effect of system size|The effect of system size]]
# [[Programming_a_2D_Ising_Model/Determining the heat capacity|Determining the heat capacity]]
# [[Programming_a_2D_Ising_Model/Locating the Curie temperature|Locating the Curie temperature]]

Programming a 2D Ising Model/Calculating the energy and magnetisation

2026-02-17T15:42:33Z

Jbettenc: Remove space

'''<big><span style="color:blue; ">This is the second section of the Ising model experiment. You can return to the previous page, [[Programming a 2D Ising Model/Introduction to the Ising model|Introduction to the Ising model]], or jump ahead to the next section, [[Programming a 2D Ising Model/Introduction to Monte Carlo simulation|Introduction to Monte Carlo simulation]].</span></big>'''

==Getting the files for the experiment==

'''Various scripts have been prepared to assist you with this experiment, and you can obtain them by downloading the files from the Blackboard.'''

We recommend you open the extracted folder with JupyterLab, which has an integrated terminal where you can run the simulation scripts.

==Modifying the files==

The file '''IsingLattice.py''' contains a Python class called IsingLattice. This class will be the basis for our model of the system, and is restricted to modelling the two dimensional case. It contains a number of stub methods that you will complete as you work through the experiment.

===A note about Python Classes===

The file IsingLattice.py makes use of a Python feature called ''class''. You should be familiar with the idea that all Python ''objects'' have an associated ''type'' -- you have already encountered objects such as integers, string, lists, and NumPy arrays. Classes allow us to define new types of object. In this case, we define a type called <code>IsingLattice</code>:

<code>
class IsingLattice:
</code>

Each class is allowed to have certain attributes and functions (known as methods) associated with it. We can see, for example, that each <code>IsingLattice</code> object has the methods <code>energy()</code>, <code>magnetisation()</code>, <code>montecarlostep(T)</code>, and <code>statistics()</code> (the method <code>__init__</code> will be explained shortly). Each of these methods takes a special first argument called self, which is a variable pointing to the current copy of IsingLattice. To use the new IsingLattice object that we have defined, we use code like:

<pre>
import IsingLattice

il = IsingLattice(5, 5) #create an IsingLattice object with 5 row and 5 columns

energy = il.energy() #call the energy() method of our IsingLattice object
</pre>

When writing code "within" the class definition, all attributes and methods belonging to the class must be accessed using the "self." notation. You can see an example of this in the <code>montecarlostep()</code> method:

<pre>
energy = self.energy()
</pre>

This line creates a variable called energy which contains whatever value was returned by the method <code>energy()</code> which is part of the <code>IsingLattice</code> class. If you just want to define a local variable instead, like an iterator in a loop, the <code>self.</code> notation is not required.

<pre>
def magnetisation(self):
# we have to use self.lattice, because the lattice variable is part of the object, not local to this function
for i in self.lattice:
...
# we use i, not self.i
# we don't need to access i outside this function
</pre>

Have a look at the other files that you have been given to see how this IsingLattice object is used.

===The Constructor===

The only method currently filled in is <code>__init__</code>. This is a special method called a ''constructor''. Whenever a new IsingLattice object is created, using code like <code>il = IsingLattice(5,5)</code>, this method is called automatically. It takes two arguments (excluding "self"), which are simply the number of rows and columns that our lattice will have. The attribute <code>self.lattice</code> is then created, which is a NumPy array with the requested number of rows and columns. The function <code>np.random.choice</code> is used to create a random collection of spins for our initial configuration. You can see the documentation for the function [https://numpy.org/doc/stable/reference/random/generated/numpy.random.choice.html numpy.random.choice].

'''<big>TASK 2a</big>: complete the functions <code>energy()</code> and <code>magnetisation()</code>, which should return the energy of the lattice and the total magnetisation, respectively. In the <code>energy()</code> function you may assume that <math>J=1.0</math> at all times (in fact, we are working in ''reduced units'' in which all energies are a multiple of <math>J</math>, but there will be more information about this in later sections). Do not worry about the efficiency of the code at the moment — we will address the speed in a later part of the experiment. Paste your code in your report and describe how it works.'''

==Testing the files==

When you have completed the energy() and magnetisation() methods, we need to test that they work correctly.
We have provided you with a script, '''ILcheck.py''', which will create three IsingLattice objects and check their energies and magnetisations, displaying these on a graph. One configuration corresponds to the energy minimum, one to the energy maximum, and one to an intermediate state.

'''<big>TASK 2b</big>: Run the ILcheck.py script from the a terminal using the command'''<pre>python ILcheck.py</pre>'''The displayed window has a series of control buttons in the bottom left, one of which will allow you to export the figure as a SVG or PNG image. Save an image of the ILcheck.py output, and include it in your report.'''

There are also some unit tests available in the <code>test_energy.py</code> file. These tests are designed to be run by simply running the [https://docs.pytest.org pytest] command in your terminal,
<pre>pytest</pre>
These sorts of test are useful because they are easier to write and faster to run and can be run automatically.

You can also debug and test your program on a Jupyter notebook or IPython console by importing your IsingLattice class, creating one object, and inspecting its properties. If you do this, we recommend that you start your session running the commands:

<pre>
%load_ext autoreload
%autoreload 2
</pre>

This will ensure that when you try executing your code, you always run the most recent version (please note this isn't fool proof and if you find problems you may need to restart the kernel)!

'''<big><span style="color:blue; ">This is the second section of the Ising model experiment. You can return to the previous page, [[Programming a 2D Ising Model/Introduction to the Ising model|Introduction to the Ising model]], or jump ahead to the next section, [[Programming a 2D Ising Model/Introduction to Monte Carlo simulation|Introduction to Monte Carlo simulation]].</span></big>'''

Programming a 2D Ising Model/Calculating the energy and magnetisation

2026-02-17T15:41:01Z

Jbettenc: Improve the flow of the page, putting more enphasys on the use of ILcheck.py

'''<big><span style="color:blue; ">This is the second section of the Ising model experiment. You can return to the previous page, [[Programming a 2D Ising Model/Introduction to the Ising model|Introduction to the Ising model]], or jump ahead to the next section, [[Programming a 2D Ising Model/Introduction to Monte Carlo simulation|Introduction to Monte Carlo simulation]].</span></big>'''

==Getting the files for the experiment==

'''Various scripts have been prepared to assist you with this experiment, and you can obtain them by downloading the files from the Blackboard.'''

We recommend you open the extracted folder with JupyterLab, which has an integrated terminal where you can run the simulation scripts.

==Modifying the files==

The file '''IsingLattice.py''' contains a Python class called IsingLattice. This class will be the basis for our model of the system, and is restricted to modelling the two dimensional case. It contains a number of stub methods that you will complete as you work through the experiment.

===A note about Python Classes===

The file IsingLattice.py makes use of a Python feature called ''class''. You should be familiar with the idea that all Python ''objects'' have an associated ''type'' -- you have already encountered objects such as integers, string, lists, and NumPy arrays. Classes allow us to define new types of object. In this case, we define a type called <code>IsingLattice</code>:

<code>
class IsingLattice:
</code>

Each class is allowed to have certain attributes and functions (known as methods) associated with it. We can see, for example, that each <code>IsingLattice</code> object has the methods <code>energy()</code>, <code>magnetisation()</code>, <code>montecarlostep(T)</code>, and <code>statistics()</code> (the method <code>__init__</code> will be explained shortly). Each of these methods takes a special first argument called self, which is a variable pointing to the current copy of IsingLattice. To use the new IsingLattice object that we have defined, we use code like:

<pre>
import IsingLattice

il = IsingLattice(5, 5) #create an IsingLattice object with 5 row and 5 columns

energy = il.energy() #call the energy() method of our IsingLattice object
</pre>

When writing code "within" the class definition, all attributes and methods belonging to the class must be accessed using the "self." notation. You can see an example of this in the <code>montecarlostep()</code> method:

<pre>
energy = self.energy()
</pre>

This line creates a variable called energy which contains whatever value was returned by the method <code>energy()</code> which is part of the <code>IsingLattice</code> class. If you just want to define a local variable instead, like an iterator in a loop, the <code>self.</code> notation is not required.

<pre>
def magnetisation(self):
# we have to use self.lattice, because the lattice variable is part of the object, not local to this function
for i in self.lattice:
...
# we use i, not self.i
# we don't need to access i outside this function
</pre>

Have a look at the other files that you have been given to see how this IsingLattice object is used.

===The Constructor===

The only method currently filled in is <code>__init__</code>. This is a special method called a ''constructor''. Whenever a new IsingLattice object is created, using code like <code>il = IsingLattice(5,5)</code>, this method is called automatically. It takes two arguments (excluding "self"), which are simply the number of rows and columns that our lattice will have. The attribute <code>self.lattice</code> is then created, which is a NumPy array with the requested number of rows and columns. The function <code>np.random.choice</code> is used to create a random collection of spins for our initial configuration. You can see the documentation for the function [https://numpy.org/doc/stable/reference/random/generated/numpy.random.choice.html numpy.random.choice].

'''<big>TASK 2a</big>: complete the functions <code>energy()</code> and <code>magnetisation()</code>, which should return the energy of the lattice and the total magnetisation, respectively. In the <code>energy()</code> function you may assume that <math>J=1.0</math> at all times (in fact, we are working in ''reduced units'' in which all energies are a multiple of <math>J</math>, but there will be more information about this in later sections). Do not worry about the efficiency of the code at the moment — we will address the speed in a later part of the experiment. Paste your code in your report and describe how it works.'''

==Testing the files==

When you have completed the energy() and magnetisation() methods, we need to test that they work correctly.
We have provided you with a script, '''ILcheck.py''', which will create three IsingLattice objects and check their energies and magnetisations, displaying these on a graph. One configuration corresponds to the energy minimum, one to the energy maximum, and one to an intermediate state.

'''<big>TASK 2b</big>: Run the ILcheck.py script from the a terminal using the command'''<pre>python ILcheck.py</pre>'''The displayed window has a series of control buttons in the bottom left, one of which will allow you to export the figure as a SVG or PNG image. Save an image of the ILcheck.py output, and include it in your report.'''

There are also some unit tests available in the <code>test_energy.py</code> file. These tests are designed to be run by simply running the [https://docs.pytest.org pytest] command in your terminal,
<pre>pytest</pre>
These sorts of test are useful because they are easier to write and faster to run and can be run automatically.

You can also debug and test your program on a Jupyter notebook or IPython console by importing your IsingLattice class, creating one object, and inspecting its properties. If you do this, we recommend that you start your session running the commands:

<pre>
%load_ext autoreload
%autoreload 2
</pre>

This will ensure that when you try executing your code, you always run the most recent version (please note this isn't fool proof and if you find problems you may need to restart the kernel)!

'''<big><span style="color:blue; ">This is the second section of the Ising model experiment. You can return to the previous page, [[Programming a 2D Ising Model/Introduction to the Ising model|Introduction to the Ising model]], or jump ahead to the next section, [[Programming a 2D Ising Model/Introduction to Monte Carlo simulation|Introduction to Monte Carlo simulation]].</span></big>'''

ThirdYearMgOExpt-1415

2026-02-15T17:29:15Z

Jbettenc: /* Related literature */ Show bibliography as proper references

<blockquote>
= Thermal Expansion of MgO =
</blockquote>This is the welcome page for the computational experiment about the thermal expansion of MgO, part of the third year computational chemistry lab.

The computational experiment will be from 10:00 to 17:00 on Monday, Tuesday, Thursday and Friday. During these 4 days demonstrators will be available to answer all your questions.

'''There is an introductory lecture to the lab at 10:00 on the Start Date of your session. You will received and email with more details.'''

The deadline is at 12:00 noon on Wed of the second week following the week of the experiment.

Work submitted late will be penalised according to the [https://www.imperial.ac.uk/media/imperial-college/administration-and-support-services/registry/academic-governance/public/academic-policy/marking-and-moderation/Late-submission-Policy.pdf Late Submission Policy].

There is a link set up on [http://bb.imperial.ac.uk Blackboard] for submitting the report in PDF and the Jupyter notebooks as a .zip folder in 3rd Year Chemistry Laboratories / Y3C Third Year Computational Laboratory/.

In each exercise there are a number questions and sub-questions. The final report will be expected to contain answers to these questions.

Questions related to this computational experiment can be directed to the demonstrators, Dr Giuseppe Mallia and Prof. Nicholas Harrison.

== Introduction ==

The properties of materials (solids, liquids, gasses) are a statistical average over the many different energy states of the molecules making up the material.

The vibrational free energy of H<sub>2</sub> can be computed analytically by summing over the harmonic vibrations of the molecule.

This cannot be done by hand for a real material containing many atoms.

The aim of this experiment is to predict how a material expands, when heated up. A simple model,

based on interatomic potentials, will be adopted to describe the interactions in a MgO crystal.

You will compare two different approximations to study the thermal expansion of MgO. In the former

approximation, you will calculate the vibrational energy levels, which will then be used to compute the free energy.

In the latter approximation, you will use a technique called molecular dynamics that

will essentially reproduce vibrations as motions of the atoms.

The simulations will be performed by using the interatomic potential program gulp though a

jupyter-notebook.

== Instructions ==

You will use Microsoft Azure Lab Windows virtual machine, where everything is installed.

Below there are some steps that are necessary only in case of local installation (check with demonstrators). Move to next session.

'''For Windows users'''

Alternatively, in order to run this computational experiment, you can install anaconda on your Windows computer.

- Download the zip file from blackboard.

- Please avoid to save the zip file in a folder tree, where a folder has a name with one or more space (" ").

- Extract the files.

- Run Anaconda

- Run Jupyter Notebook

- Run the main.ipynb

- Follow the instructions in the notebook.

'''For Mac users (please get in touch with a demonstrator) - the installation depends on your Mac version'''

1. Open a terminal and type the command below:

2. sudo /bin/bash -c "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/master/install.sh)"
(https://brew.sh/)

3. brew install gcc (https://formulae.brew.sh/formula/gcc#default)

4.1 conda create --name MgOexp python=3.7.2

4.2 conda activate MgOexp

4.3 jupyter-notebook

== Previous years related contents ==

- Maths and Physics for Chemist

- Thermodynamics

- Statistical thermodynamics

- Python

Good skills on python is not a requirement. All the scripts are written in a way where you only need to write an input value (e.g. Temperature = 300 K). However, feel free to edit and play with the scripts.

== New contents ==

- Phonons and reciprocal space

- Quasi-Harmonic approximation

- Molecular dynamics

== Submission ==

The report will be submitted as a PDF document via Turnit in Blackboard.

Additionally, all the files (including the lab notebooks) will be also submitted via Blackboard as a .zip folder.

=== Write up ===

The report structure will consist of three sections:
* Introduction/Summary (Half-page)
* Questions & answers (No page limit)
* Conclusions (Half-page)

Tips to write a report:
* The golden rule: Aim for clarity
** Structured statements that flow in a logical manner.
** Good use of diagrams and appropriate level of theory.
** Careful choice of content.

* Keep your language clear and simple.
* Label all tables and figures. Labels should be self-contained, which means that tables and figures should be interpretable by themself.
* Appropriate referencing of figures and tables.
* Cite previous works (with an accepted citation style) whenever is appropriate.

Introduction/Summary:
* The purpose of the Introduction/Summary is to put the reader in the context of the experiment and to explain how the experiment was carried in the lab. It may contain a brief review of previous research, why the research was undertaken, an explanation of the techniques and why they are used and why it is important in a broader context.

Questions & Answers:
* There are a number of questions in the lab script that has to be answered in this section of the report.
* Depending on the nature of the question, it might be appropriate to use figures or tables to give a proper answer.
* It is highly encourage to rationalise the answers.

Conclusions:
* The Conclusions gives a general description of the results and findings and it should be related back to the Introduction. If appropriate, suggest improvements or additional experiments.

=== Suggested Time Frame ===

Try to finish all the calculations by Thursday. All the calculations takes seconds, however, it takes time to analyse the results and understand all the new concepts that you will learn in this lab.

=== Mark Scheme ===

The break-down for the marks for this lab are as follows:

{|class=wikitable
|-
|Introduction/Summary
|20%
|-
|Questions & Answers
|60%
|-
|Conclusions
|20%
|}

=== Plagiarism ===

Submissions are checked for plagiarism. External images may be used if correctly cited, but it's always better to create your own.

== Demonstrators ==

The demonstrators will be available via Microsoft team channel.
Feel free to contact them.

== Related literature ==

{{Cite journal | author1=Roald Hoffmann | title=How Chemistry and Physics Meet in the Solid State | journal=Angew. Chem., Int. Ed. Engl. | year=1987 | volume=26 | issue=9 | pages=846-878 | doi=10.1002/anie.198708461}}

{{Cite journal | author1=Martin T. Dove | title=Introduction to the theory of lattice dynamics | journal=Collection SFN | year=2011 | volume=12 | pages=123-159 | doi=10.1051/sfn/201112007}}

{{Cite book | author1=Martin T. Dove | title=Introduction to Lattice Dynamics | publisher=Cambridge University Press | year=1993}} {{DOI|10.1017/CBO9780511619885}} (access available [https://library-search.imperial.ac.uk/permalink/44IMP_INST/fv0fdm/cdi_nii_cinii_1130000797936518656 through the library])

== Alternative instructions for Microsoft Azure Labs Linux virtual machine ==

'''Read this section only if recommended by the demonstrators.'''

The following steps are based on ssh connection to a Linux virtual machine.

1. Click on the link in the email with subject "Register for Lab - IC_Chemistry_yr3_UK" or ""Register for Lab - IC_Chemistry_Linux_UK"

2. Start the virtual machine.

3. Click on the vertical three dots (bottom right of the virtual machine box) and rest the password.

4. Click on the computer icon at the left of the three vertical dot.

5. Copy the content of the box that will appear

For instance:
ssh -p 12345 chemistry@ml-lab-123456789.uksouth.cloudapp.azure.com

6. Modify the previous line by adding "-X" after ssh and "-L 8888:localhost:8888"

For instance:
ssh -X -p 12345 chemistry@ml-lab-123456789.uksouth.cloudapp.azure.com -L 8888:localhost:8888

7. On Windows open the prompt command and type the ssh line;
on Mac/Linux open a terminal and type the ssh line.

8. Type "yes" to answer the following question:
Are you sure you want to continue connecting (yes/no/[fingerprint])?

9. Type your password, when prompted (the password will not be displayed for security reason)
once done, press "Enter".

10. After that a prompt similar to the following one will appear

chemistry@ML-RefVm-123456:~$

type the following and press "Enter"
conda activate
the prompt will change into:
(base) chemistry@ML-RefVm-670265:~$

11. Type the following and press "Enter"
jupyter-notebook --no-browser

A message similar to the following one will be printed:
To access the notebook, open this file in a browser:
file:///home/chemistry/.local/share/jupyter/runtime/nbserver-4021-open.html
Or copy and paste one of these URLs:
http://localhost:8888/?token=123456789123456789123456789123456789123456789123
or http://127.0.0.1:8888/?token=123456789123456789123456789123456789123456789123

12. Open a tab in your browser and type last line printed above
a jupyter-notebook page will be opened.

13. Go in the folder "yr3", then "MgO-lab-master", then click on "main.ipynb"
to start your computational experiment

14. Select "Python [conda env:py37_default]" if prompted to choose a kernel.

15. To transfer the file use https://winscp.net/eng/docs/free_ftp_client_for_windows or scp on Mac/Linux.

== Additional instructions for the manual installation of the gulp program and the setup of ASE on your personal computer ==

'''Read this section only if recommended by the demonstrators.'''

The experiment uses [http://gulp.curtin.edu.au/gulp/ Gulp] for all the calculations. Jupyter Notebook and python is used as interface between the user and Gulp.

To install the lab in Windows:

1. If you do have anaconda installed you can go to point 3.

2. Download and install [https://www.anaconda.com/download/ Anaconda]

3. Download the [https://anaconda.org/conda-forge/ase ASE] package.

4. Download GULP from the [http://gulp.curtin.edu.au/gulp/request.cfm?rel=download website]. Download version 4.3 which has the executable for Windows ready to download.

5. Copy Gulp to c:\gulp-4.3.

6. Sets a system environment variable for GULP_LIB to C:\gulp-4.3\Libraries and add to C:\gulp-4.3\Exe to the PATH system variable. [https://www.java.com/en/download/help/path.xml How do I set or change the PATH system variable?]

7. Copy the ionic.lib file that you will find in the library folder you downloaded to the $GULP_LIB folder

To install the lab in Linux/Unix:

1. If you do have conda installed you can go to point 6.

2. Open a terminal and create a new conda environment: conda create -n TE_MgO

3. Activate the environment: conda activate TE_MgO

4. Install ASE: conda install -c conda-forge ase=3.19.1

5. Install Jupyter: conda install -c anaconda notebook

6. Download GULP from the [http://gulp.curtin.edu.au/gulp/request.cfm?rel=download website]. The version 5.0 only has the source files. Download this one if you feel comfortable with compiling it by yourself. Otherwise, download version 4.3 which has the executable ready to download.

7. Set $GULP_LIB and $ASE_GULP_COMMAND in your ~/.bashrc or ~/.bash_profile (see this [https://wiki.fysik.dtu.dk/ase/ase/calculators/gulp.html page] for more information)

8. Copy the ionic.lib file that you will find in the library folder you downloaded to the $GULP_LIB folder

Template:Cite book

2026-02-15T17:27:08Z

Jbettenc: Do not generate links and show example

<includeonly>{{#if: {{{author1|}}} | {{{author1}}}{{#if: {{{author2|}}} |, {{{author2}}}{{#if: {{{author3|}}} |, {{{author3}}}{{#if: {{{author4|}}} |, {{{author4}}}{{#if: {{{author5|}}} |, {{{author5}}}{{#if: {{{author6|}}} |, {{{author6}}}{{#if: {{{author7|}}} |, {{{author7}}}{{#if: {{{author8|}}} |, {{{author8}}}{{#if: {{{author9|}}} |, {{{author9}}}{{#if: {{{author10|}}} |, {{{author10}}}{{#if: {{{author11|}}} |, {{{author11}}} }} }} }} }} }} }} }} }} }} }} | missing ''author1'' }} ({{#if: {{{year|}}} | {{{year}}} | missing ''year'' }}). {{#if: {{{title|}}} | ''{{{title}}}''. | missing ''title''. }} {{#if: {{{publisher|}}} | {{{publisher}}}. }}</includeonly><noinclude>
The '''''Cite book''''' template is use to reference books.

<nowiki>{{</nowiki>Cite book
| author1 =
| author2 =
| author3 =
| title =
| publisher =
| year =
<nowiki>}}</nowiki>

== Se also ==
* [[Template:Cite journal]]

</noinclude>

ThirdYearMgOExpt-1415

2026-02-15T17:06:03Z

Jbettenc: /* Related literature */

<blockquote>
= Thermal Expansion of MgO =
</blockquote>This is the welcome page for the computational experiment about the thermal expansion of MgO, part of the third year computational chemistry lab.

The computational experiment will be from 10:00 to 17:00 on Monday, Tuesday, Thursday and Friday. During these 4 days demonstrators will be available to answer all your questions.

'''There is an introductory lecture to the lab at 10:00 on the Start Date of your session. You will received and email with more details.'''

The deadline is at 12:00 noon on Wed of the second week following the week of the experiment.

Work submitted late will be penalised according to the [https://www.imperial.ac.uk/media/imperial-college/administration-and-support-services/registry/academic-governance/public/academic-policy/marking-and-moderation/Late-submission-Policy.pdf Late Submission Policy].

There is a link set up on [http://bb.imperial.ac.uk Blackboard] for submitting the report in PDF and the Jupyter notebooks as a .zip folder in 3rd Year Chemistry Laboratories / Y3C Third Year Computational Laboratory/.

In each exercise there are a number questions and sub-questions. The final report will be expected to contain answers to these questions.

Questions related to this computational experiment can be directed to the demonstrators, Dr Giuseppe Mallia and Prof. Nicholas Harrison.

== Introduction ==

The properties of materials (solids, liquids, gasses) are a statistical average over the many different energy states of the molecules making up the material.

The vibrational free energy of H<sub>2</sub> can be computed analytically by summing over the harmonic vibrations of the molecule.

This cannot be done by hand for a real material containing many atoms.

The aim of this experiment is to predict how a material expands, when heated up. A simple model,

based on interatomic potentials, will be adopted to describe the interactions in a MgO crystal.

You will compare two different approximations to study the thermal expansion of MgO. In the former

approximation, you will calculate the vibrational energy levels, which will then be used to compute the free energy.

In the latter approximation, you will use a technique called molecular dynamics that

will essentially reproduce vibrations as motions of the atoms.

The simulations will be performed by using the interatomic potential program gulp though a

jupyter-notebook.

== Instructions ==

You will use Microsoft Azure Lab Windows virtual machine, where everything is installed.

Below there are some steps that are necessary only in case of local installation (check with demonstrators). Move to next session.

'''For Windows users'''

Alternatively, in order to run this computational experiment, you can install anaconda on your Windows computer.

- Download the zip file from blackboard.

- Please avoid to save the zip file in a folder tree, where a folder has a name with one or more space (" ").

- Extract the files.

- Run Anaconda

- Run Jupyter Notebook

- Run the main.ipynb

- Follow the instructions in the notebook.

'''For Mac users (please get in touch with a demonstrator) - the installation depends on your Mac version'''

1. Open a terminal and type the command below:

2. sudo /bin/bash -c "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/master/install.sh)"
(https://brew.sh/)

3. brew install gcc (https://formulae.brew.sh/formula/gcc#default)

4.1 conda create --name MgOexp python=3.7.2

4.2 conda activate MgOexp

4.3 jupyter-notebook

== Previous years related contents ==

- Maths and Physics for Chemist

- Thermodynamics

- Statistical thermodynamics

- Python

Good skills on python is not a requirement. All the scripts are written in a way where you only need to write an input value (e.g. Temperature = 300 K). However, feel free to edit and play with the scripts.

== New contents ==

- Phonons and reciprocal space

- Quasi-Harmonic approximation

- Molecular dynamics

== Submission ==

The report will be submitted as a PDF document via Turnit in Blackboard.

Additionally, all the files (including the lab notebooks) will be also submitted via Blackboard as a .zip folder.

=== Write up ===

The report structure will consist of three sections:
* Introduction/Summary (Half-page)
* Questions & answers (No page limit)
* Conclusions (Half-page)

Tips to write a report:
* The golden rule: Aim for clarity
** Structured statements that flow in a logical manner.
** Good use of diagrams and appropriate level of theory.
** Careful choice of content.

* Keep your language clear and simple.
* Label all tables and figures. Labels should be self-contained, which means that tables and figures should be interpretable by themself.
* Appropriate referencing of figures and tables.
* Cite previous works (with an accepted citation style) whenever is appropriate.

Introduction/Summary:
* The purpose of the Introduction/Summary is to put the reader in the context of the experiment and to explain how the experiment was carried in the lab. It may contain a brief review of previous research, why the research was undertaken, an explanation of the techniques and why they are used and why it is important in a broader context.

Questions & Answers:
* There are a number of questions in the lab script that has to be answered in this section of the report.
* Depending on the nature of the question, it might be appropriate to use figures or tables to give a proper answer.
* It is highly encourage to rationalise the answers.

Conclusions:
* The Conclusions gives a general description of the results and findings and it should be related back to the Introduction. If appropriate, suggest improvements or additional experiments.

=== Suggested Time Frame ===

Try to finish all the calculations by Thursday. All the calculations takes seconds, however, it takes time to analyse the results and understand all the new concepts that you will learn in this lab.

=== Mark Scheme ===

The break-down for the marks for this lab are as follows:

{|class=wikitable
|-
|Introduction/Summary
|20%
|-
|Questions & Answers
|60%
|-
|Conclusions
|20%
|}

=== Plagiarism ===

Submissions are checked for plagiarism. External images may be used if correctly cited, but it's always better to create your own.

== Demonstrators ==

The demonstrators will be available via Microsoft team channel.
Feel free to contact them.

== Related literature ==

{{Cite journal | author1=M.T.Dove | title=Introduction to the theory of lattice dynamics | journal=Collection SFN | year=2011 | volume=12 | pages=123-159 | doi=10.1051/sfn/201112007}}

[https://onlinelibrary.wiley.com/doi/epdf/10.1002/anie.198708461 How Chemistry and Physics meet in the Solid State by Roald Hoffman]

[https://www.neutron-sciences.org/articles/sfn/pdf/2011/01/sfn201112007.pdf Introduction to the theory of Lattice Dynamics]

Introduction to Lattice Dynamics. Dove, Martin T. (ebook available in the library)

== Alternative instructions for Microsoft Azure Labs Linux virtual machine ==

'''Read this section only if recommended by the demonstrators.'''

The following steps are based on ssh connection to a Linux virtual machine.

1. Click on the link in the email with subject "Register for Lab - IC_Chemistry_yr3_UK" or ""Register for Lab - IC_Chemistry_Linux_UK"

2. Start the virtual machine.

3. Click on the vertical three dots (bottom right of the virtual machine box) and rest the password.

4. Click on the computer icon at the left of the three vertical dot.

5. Copy the content of the box that will appear

For instance:
ssh -p 12345 chemistry@ml-lab-123456789.uksouth.cloudapp.azure.com

6. Modify the previous line by adding "-X" after ssh and "-L 8888:localhost:8888"

For instance:
ssh -X -p 12345 chemistry@ml-lab-123456789.uksouth.cloudapp.azure.com -L 8888:localhost:8888

7. On Windows open the prompt command and type the ssh line;
on Mac/Linux open a terminal and type the ssh line.

8. Type "yes" to answer the following question:
Are you sure you want to continue connecting (yes/no/[fingerprint])?

9. Type your password, when prompted (the password will not be displayed for security reason)
once done, press "Enter".

10. After that a prompt similar to the following one will appear

chemistry@ML-RefVm-123456:~$

type the following and press "Enter"
conda activate
the prompt will change into:
(base) chemistry@ML-RefVm-670265:~$

11. Type the following and press "Enter"
jupyter-notebook --no-browser

A message similar to the following one will be printed:
To access the notebook, open this file in a browser:
file:///home/chemistry/.local/share/jupyter/runtime/nbserver-4021-open.html
Or copy and paste one of these URLs:
http://localhost:8888/?token=123456789123456789123456789123456789123456789123
or http://127.0.0.1:8888/?token=123456789123456789123456789123456789123456789123

12. Open a tab in your browser and type last line printed above
a jupyter-notebook page will be opened.

13. Go in the folder "yr3", then "MgO-lab-master", then click on "main.ipynb"
to start your computational experiment

14. Select "Python [conda env:py37_default]" if prompted to choose a kernel.

15. To transfer the file use https://winscp.net/eng/docs/free_ftp_client_for_windows or scp on Mac/Linux.

== Additional instructions for the manual installation of the gulp program and the setup of ASE on your personal computer ==

'''Read this section only if recommended by the demonstrators.'''

The experiment uses [http://gulp.curtin.edu.au/gulp/ Gulp] for all the calculations. Jupyter Notebook and python is used as interface between the user and Gulp.

To install the lab in Windows:

1. If you do have anaconda installed you can go to point 3.

2. Download and install [https://www.anaconda.com/download/ Anaconda]

3. Download the [https://anaconda.org/conda-forge/ase ASE] package.

4. Download GULP from the [http://gulp.curtin.edu.au/gulp/request.cfm?rel=download website]. Download version 4.3 which has the executable for Windows ready to download.

5. Copy Gulp to c:\gulp-4.3.

6. Sets a system environment variable for GULP_LIB to C:\gulp-4.3\Libraries and add to C:\gulp-4.3\Exe to the PATH system variable. [https://www.java.com/en/download/help/path.xml How do I set or change the PATH system variable?]

7. Copy the ionic.lib file that you will find in the library folder you downloaded to the $GULP_LIB folder

To install the lab in Linux/Unix:

1. If you do have conda installed you can go to point 6.

2. Open a terminal and create a new conda environment: conda create -n TE_MgO

3. Activate the environment: conda activate TE_MgO

4. Install ASE: conda install -c conda-forge ase=3.19.1

5. Install Jupyter: conda install -c anaconda notebook

6. Download GULP from the [http://gulp.curtin.edu.au/gulp/request.cfm?rel=download website]. The version 5.0 only has the source files. Download this one if you feel comfortable with compiling it by yourself. Otherwise, download version 4.3 which has the executable ready to download.

7. Set $GULP_LIB and $ASE_GULP_COMMAND in your ~/.bashrc or ~/.bash_profile (see this [https://wiki.fysik.dtu.dk/ase/ase/calculators/gulp.html page] for more information)

8. Copy the ionic.lib file that you will find in the library folder you downloaded to the $GULP_LIB folder

Template:Cite journal

2026-02-15T17:05:06Z

Jbettenc: Do not create links for authors, title, journal and year

<includeonly>{{#if: {{{author1|}}} | {{{author1}}} | missing ''author1'' }}{{#if: {{{author2|}}} |, {{{author2}}}{{#if: {{{author3|}}} |, {{{author3}}}{{#if: {{{author4|}}} |, {{{author4}}}{{#if: {{{author5|}}} |, {{{author5}}}{{#if: {{{author6|}}} |, {{{author6}}}{{#if: {{{author7|}}} |, {{{author7}}}{{#if: {{{author8|}}} |, {{{author8}}}{{#if: {{{author9|}}} |, {{{author9}}}{{#if: {{{author10|}}} |, {{{author10}}}{{#if: {{{author11|}}} |, {{{author11}}} }} }} }} }} }} }} }} }} }} }} ({{#if: {{{year|}}} | {{{year}}} | missing ''year'' }}). {{#if: {{{title|}}} | "{{{title}}}". | missing ''title''. }} {{#if: {{{journal|}}} | ''{{{journal}}} (journal)|{{{journal}}}'' | missing ''journal'' }} {{#if: {{{volume|}}} | '''{{{volume}}}''' | missing ''volume'' }}{{#if: {{{issue|}}} | ({{{issue}}}) }}: {{#if: {{{pages|}}} | {{{pages}}}. | missing ''pages''. }} {{#if: {{{doi|}}} | doi: [http://dx.doi.org/{{{doi}}} {{{doi}}}]. }} {{#if: {{{pmid|}}} | [[PMID]]: [http://www.ncbi.nlm.nih.gov/pubmed/{{{pmid}}} {{{pmid}}}]. }} {{#if: {{{url1|}}} | [{{{url1}}}]. }}</includeonly><noinclude>
The '''''Cite journal''''' template is used to reference a journal article.

<nowiki>{{</nowiki>Cite journal
| author1 =
| author2 =
| author3 =
| title =
| journal =
| year =
| month =
| volume =
| issue =
| pages =
| doi =
| pmid =
| url1 =
<nowiki>}}</nowiki>

== See also ==
* [[Template:Cite book]]

</noinclude>

File:CH3Cl freq.log

2025-11-28T12:54:15Z

Jbettenc: Jbettenc uploaded a new version of File:CH3Cl freq.log

== Summary ==
Log of a Gaussian frequency calculation at HF/3-21g level of theory for chloro-methane.

File:HOD.freq.molden

2025-11-28T12:42:46Z

Jbettenc: Jbettenc uploaded a new version of File:HOD.freq.molden

Molden file resulting from a frequency calculation on HDO with Hartree-Fock and a minimal basis set

File:HOD.freq.molden

2025-11-28T12:38:09Z

Jbettenc: Jbettenc reverted File:HOD.freq.molden to an old version

Molden file resulting from a frequency calculation on HDO with Hartree-Fock and a minimal basis set

File:HOD.freq.molden

2025-11-28T12:34:47Z

Jbettenc: Jbettenc uploaded a new version of File:HOD.freq.molden

Molden file resulting from a frequency calculation on HDO with Hartree-Fock and a minimal basis set

File:H2O.freq.molden

2025-11-28T12:31:20Z

Jbettenc: Jbettenc uploaded a new version of File:H2O.freq.molden

Molden file with H2O vibrational frequencies and normal modes obtained from a Hartree-Fock calculation with a minimum basis set.

Mod:dont panic

2025-11-28T11:42:42Z

Jbettenc: Jbettenc moved page Rep:Mod:dont panic to Mod:dont panic without leaving a redirect: Revert

See also: [[Mod:org-startup|1C comp-lab startup]],[[Mod:timetable-1C|Timetable]],[[mod:laptop|Laptop use]], [[mod:programs|Programs]], [[mod:organic|Module 1C Script]], [[Mod:toolbox|Module 1C Toolbox]], [[Mod:writeup|Writing up]], [[Mod:dont_panic|Don't panic]].

[[Image:A_beginners_guide.jpg|center|800px]]
[[Image:dont_panic.jpg|center|200px]]

So you've done your calculations as instructed, and got your optimised structures with a whole lot of properties to boot. What else is expected of you? What does analysis exactly mean, anyway? Don't panic! Here you'll find most of what you need to know to survive Module 1 and beyond.

==Analysis==

"How much analysis is expected from us?", I hear you ask. The answer is: AS MUCH AS 60% OF YOUR MARK! Getting the lowest in energy conformers is quite a challenge which will test your chemical knowledge and patience to the limit, but it wouldn't do you much good if you can't extract chemical information from the exercise. There are several types of information which can be discussed from the calculated structures of compounds or transition states in organic chemistry. The most common ones are listed below:

*Energies
*Bond angles and dihedral angles
*Molecular orbitals
*Bond strength, bond lengths and vibrational frequencies

====Energies====

Energies show how stable your structure is and when it comes to energy, the lower the better. Watch out when you compare energies of different structures, unless the number and types of atoms involved are the same, the comparison isn't valid. Under equilibrium, species of different energies will follow the same relationship as that between Gibbs free energy and equilibrium constant.

Comparing absolute energies between different computational methods is impossible, as they are calculated as the sum of different factors. However, the difference in energy between isomeric transition states calculated by different methods has often been compared with experimentally measured selectivity to judge the accuracy of computational methods.

====Bond angles and dihedral angles====

These values are very useful in intepreting the breakdown components of energy from Molecular Mechanics calculation. They're still useful in quantum mechanics, although there won't be any tangible result you can directly relate them to. A strained structure is a strained structure regardless of how you calculate it.

====Bond strength, bond lengths and vibrational frequencies====

A bond is defined and displayed differently by Molecular Mechanics and quantum mechanics techniques. Be mindful as what you see in the visualisation doesn't necessarily be what your calculation results mean. The strength of a bond, and to some extent its multiplicity, is best judged by examining the bond length and its vibrational frequencies.

====Molecular orbitals====

MOs are only accessible via quantum mechanics. In organic chemistry, we're mostly concerned with the frontier orbitals. Examining their position, shape and symmetry often gives clues about the nature of the MOs (bonding vs antibonding, σ-π interaction, etc.), and more importantly the reactivity and selectivity of the molecule in reactions.

==Data presentation==

So you aren't as good at writing wikis as the geeks who live in a garage, who shower once a month and are on a permanent supplement course of caffeine. Neither are we!

There is no doctor-ordered style for your report and marks won't be given for visual effects or creative designs of your wiki pages. Instead, we want clear, concise communication from you so that we can fully appreciate your work. One golden rule must be remembered: IF WE CAN'T SEE IT (CLEARLY), WE CAN'T MARK IT. The same goes for your analysis. If your visual aid helps the readers understand your points, by all means include it.

Another issue, which is most relevant with the mini projects, is the context of your work. Tell us why and how. More importantly, tell us EXACTLY WHAT QUESTION ARE YOU TRYING TO ANSWER WITH YOUR CALCULATION, other than for getting marks, of course.

Occasionally, we'll contact you during the marking for your original files if we find some of your results interesting, or if we feel there wasn't enough information in your wiki page. Remember to keep your files, and name them in a way that the future you won't struggle to comprehend.

==Methods of calculation==

====Molecular Mechanics====

An excellent description of molecular mechanics has already been included at [[Mod:mechanics|this page]]. Here we'll simply summarise that it's basically balls and sophisticated springs. It's fast, cheap to compute but has to rely on carefully developed force field information (the anharmonic oscillator parameters). Thus, molecular mechanics can only handle structures it has been taught to handle and those unfortunately don't include organometallic compounds.

Of particular note, bonds are treated as springs and have to be specified in the starting structure. As a results, molecular mechanics performs poorly when it comes to electronic interactions, or bond forming-breaking processes. In these cases, molecular mechanics is often employed to clean up the structure, before a more appropriate method is applied.

====Semi-empirical methods====

To cut computational cost in quantum mechanics, approximations were made to simplify the Schrödinger equation. Semi-empricial molecular orbitals methods were born. They're still fast, albeit at the cost of accuracy, compared to ''ab initio'' methods. Semi-quantitative description of electronic distribution, molecular structure, MOs and energies can be quickly derived using these methods. They're also capable of calculations for organometallic compounds.

Being quantum mechanic techniques, they can model electronic effect, orbital interactions or hydrogen bondings, bond formation/breaking, and transition states (all the things Molecular Mechanics can't do!).

====''Ab initio'' calculations====

These fully-fledged quantum mechanic techniques are the current last words, if not the only words, in computational chemistry. They can also handle every chemical structure you can come up with, given an infinite amount of time. The price is that you'll need a supercomputer with 'brain the size of a planet', and the geeks, who write wikis for breakfast, to run it. Users are protected by a web-based or a console-based submission system. Submitted jobs will join queue and occasionally get trapped in an endless loop when you will have to contact the aforementioned IT experts to intervene.

One can optimise structure (can be quite time consumming, depending on how many electrons you have in your structure), calculate energy (enthalpy, entropy) in gas and liquid phases. Recent advances allow fairly accurate prediction of NMR chemical shifts, CD spetrum and optical rotation, as well as IR vibrational frequencies. Bewarned: optical rotation and IR vibrational frequencies calculations are time consumming and shouldn't be carried out unless there's good justification (the submission deadline on Friday is infinitesimally close compared to infinity!).

==Starting structure==

Just as in life, your starting point affects your destination in molecular modelling. A poor starting structure will give you a poorly optimised end-product. Your software doesn't come with an AI (yet!) and so your chemical intuition will have to suffice. Crystal structures, if found, are good places to start as they are at least real minima in solid phase. Most of your calculations, however, will be in gas phase or in solvents.

==Thermodynamics vs kinetics==

Reactions under equilibrium are under thermodynamic control and the more stable product will be formed predominantly, hence the term 'thermodynamic product'. When the reaction is irreversible (very slow reverse reaction), the distribution of products, i.e. the selectivity, is dictated by the relative energies of the corresponding transition states, which also means the relative rates of different pathways (Arrhenius equation), and not that of the products. The major product in this case is called 'kinetic product'.

The arbitrary end-point to a reaction also gives rise to a grey area in which the reaction mixture is approaching but has not yet reached equilibrium the “control” is effectively a mixture. One can use this knowledge to manipulate the outcome of the reaction. All reactions are initially under kinetic control, when no product means no reverse reaction.

'Thermodynamic product' and 'kinetic product' refers to the energy of different species (products and transition states), and therefore, are not mutually exclusive. It is NOT possible to define the kinetic product with knowledge of the thermodynamic product and vice versa. In many reactions, both of these are the same.

==Other things==

If you have questions about anything not covered on this page, talk to us. In fact, talk to us in any case. Mini project, especially, shouldn't even be attempted before some interaction with us. Computational chemistry can get extremely complicated very quickly and a chat with us would prevent you from realising you've bitten up more than you can chew the night before submission. This of course in theory can't happen because you've been following our advice to start your mini project as early as Monday of the second week.

See also: [[Mod:org-startup|Synthesis comp-lab startup]],[[Mod:timetable-1C|Timetable]],[[mod:laptop|Laptop use]], [[mod:programs|Programs]],[[mod:organic|Module 1-C]],[[Mod:writeup|Writing up]], [[Mod:dont_panic|Don't panic]].

Oh SH6 orbitals

2025-10-29T19:43:37Z

Jbettenc:

Right-click on the applet to control options.

<jmol>
<jmolApplet>
<name>SH6</name>
<title>SH6 valence molecular orbitals</title>
<color>white</color>
<size>800</size>
<script>
axes on
mo titleFormat "Energy=%E%U|?Symmetry=%S|?Occupancy=%O"
</script>
<uploadedFileContents>SH6.molden</uploadedFileContents>
</jmolApplet>
<jmolMenu>
<target>SH6</target>
<item>
<script>
mo delete
</script>
<text>No MO</text>
</item>
<item>
<script>
mo 1
</script>
<text>MO 1</text>
</item>
<item>
<script>
mo 2
</script>
<text>MO 2</text>
</item>
<item>
<script>
mo 3
</script>
<text>MO 3</text>
</item>
<item>
<script>
mo 4
</script>
<text>MO 4</text>
</item>
<item>
<script>
mo 5
</script>
<text>MO 5</text>
</item>
<item>
<script>
mo 6
</script>
<text>MO 6</text>
</item>
<item>
<script>
mo 7
</script>
<text>MO 7</text>
</item>
<item>
<script>
mo 8
</script>
<text>MO 8</text>
</item>
<item>
<script>
mo 9
</script>
<text>MO 9</text>
</item>
<item>
<script>
mo 10
</script>
<text>MO 10</text>
</item>
</jmolMenu>
</jmol>

[[:File:SH6.molden | Download file]] to view orbitals with [http://avogadro.cc Avogadro] or [http://jmol.sourceforge.net Jmol].

[[D4h_H4_fragment_orbitals| H<sub>4</sub> fragment orbitals]] <> [[Oh_H6_fragment_orbitals| H<sub>6</sub> fragment orbitals]]

Oh H6 fragment orbitals

2025-10-29T19:41:11Z

Jbettenc:

Right-click on the applet to control options.

<jmol>
<jmolApplet>
<name>H6_fragment</name>
<title>fragment orbitals</title>
<color>white</color>
<size>800</size>
<script>
axes on
label on
label "%[element]%[atomno]"
mo titleFormat "Energy=%E%U|?Symmetry=%S|?Occupancy=%O"
</script>
<uploadedFileContents>H6.molden</uploadedFileContents>
</jmolApplet>
<jmolMenu>
<target>H6_fragment</target>
<item>
<script>
mo delete
</script>
<text>No MO</text>
</item>
<item>
<script>
mo 1
</script>
<text>MO 1</text>
</item>
<item>
<script>
mo 2
</script>
<text>MO 2</text>
</item>
<item>
<script>
mo 3
</script>
<text>MO 3</text>
</item>
<item>
<script>
mo 4
</script>
<text>MO 4</text>
</item>
<item>
<script>
mo 5
</script>
<text>MO 5</text>
</item>
<item>
<script>
mo 6
</script>
<text>MO 6</text>
</item>
</jmolMenu>
</jmol>

[[:File:H6.molden | Download file]] to view orbitals with [http://avogadro.cc Avogadro] or [http://jmol.sourceforge.net Jmol].

[[D4h_H4_fragment_orbitals| H4 fragment orbitals]].

D4h H4 fragment orbitals

2025-10-29T19:32:23Z

Jbettenc:

Right-click on the applet to control options.

<jmol>
<jmolApplet>
<name>H4_fragment</name>
<title>fragment orbitals</title>
<color>white</color>
<size>800</size>
<script>
label "%[element]%[atomno]"
axes on
mo titleFormat "Energy=%E%U|?Symmetry=%S|?Occupancy=%O"
</script>
<uploadedFileContents>H4.molden</uploadedFileContents>
</jmolApplet>
<jmolMenu>
<target>H4_fragment</target>
<item>
<script>
mo delete
</script>
<text>No MO</text>
</item>
<item>
<script>
mo 1
</script>
<text>MO 1</text>
</item>
<item>
<script>
mo 2
</script>
<text>MO 2</text>
</item>
<item>
<script>
mo 3
</script>
<text>MO 3</text>
</item>
<item>
<script>
mo 4
</script>
<text>MO 4</text>
</item>
</jmolMenu>
</jmol>

[[:File:H4.molden | Download file]] to view orbitals with [http://avogadro.cc Avogadro] or [http://jmol.sourceforge.net Jmol].

Main Page

2025-09-17T13:51:21Z

Jbettenc: /* First year */ Declutter bu removing section header

[[Image:QR_complab.png|right|QR]]
This is a communal area for documenting teaching and laboratory courses. To [[admin:add|add to any content on these pages]], you will have to log in using your Imperial College account. This page has {{DOI|dh4g}}
== Remote Working ==
#[[Mod:Rem|Remote access to College computers]]
#[[Mod:HPC-add|Remote use of Gaussian]]
#[[Mod:support|Remote support]]

== ORCID Identifiers and Research Data Publication ==
#[[orcid|Getting an ORCID identifier]] (two stages)
#[[rdm:synthesis-lab|Publishing NMR instrument data]] (three stages)
#[[rdm:xray-data|Publishing crystal structure instrument data]]

= Laboratories and Workshops =
== First Year ==
=== [[it:it_facillities|Email and IT@www.ch.imperial.ac.uk]]: A summary of available IT resources ===

===First Year Chemical Information Lab 2015 ===
*[[it:intro-2011|Introduction]]
*[[it:lectures-2011|Lectures]]
*[[it:coursework-2011|Coursework]]
*[[it:assignment-2011|Assignment for the course]]
*[[it:software-2011|List of software for CIT]]
*[[it:searches-2011|Search facilities for CIT]]
*[[Measurement_Science_Lab:_Introduction|Measurement Science Lab Course]]
===[[organic:conventions|Conventions in organic chemistry]]===
===[[organic:arrow|Reactive Intermediates in organic chemistry]]===
===[[organic:stereo|Stereochemical models]] ===
== Second Year ==
=== Computational Chemistry Lab ===
*Electronic States and Bonding - computational laboratory (see blackboard).

== Third Year ==


=== Computational Chemistry Lab ===
* [[Mod:ts_exercise|Transition states and reactivity.]]
* [[ThirdYearMgOExpt-1415|MgO thermal expansion]]
* [[Third_year_simulation_experiment|Simulation of a simple liquid]]
* [[Programming_a_2D_Ising_Model|Programming a 2D Ising Model (CMP only)]]

= Material from previous years =
== First year ==
*[[1da-workshops-2013-14|Introduction to Chemical Programming Workshop (2013)]]

== Second year ==

*[http://www.huntresearchgroup.org.uk/teaching/year2a_lab_start.html Inorganic Computational Chemistry Computational Laboratory]
*[[CP3MD| Molecular Reaction Dynamics]]

=== Modelling Workshop ===
*[[Coursework]]
*[[Second Year Modelling Workshop|Instructions]] and [[mod:further_coursework|Further optional coursework]]
*[[it:conquest|Conquest searches]]



== Third Year ==

*[[TrendsCatalyticActivity|Understanding trends in catalytic activity for hydrogen evolution]]
*[[mod:inorganic|Bonding and molecular orbitals in main group compounds]]

===Synthesis and computational lab ===
*[[Mod:organic|Synthesis and computational lab]]


*[[mod:intro|Information needed for the course]]


*[[mod:writeup|Report writing and submission]]

*[[mod:latebreak|Late breaking news]]

*[[mod:programs|General program instructions:]]
**[[mod:gaussview|The Gaussview/Gaussian suite]]
**[[Mod:scan|Submitting jobs to the chemistry high-performance-computing resource]]

== ChemDraw/Chemdoodle Hints ==
#[[IT:chemdraw|Useful hints for using ChemDraw/ChemDoodle]]

== Tablet Project ==
# [[tablet:tablet|Tablet Pilot Project]]
== 3D ==
# [[mod:3D|3D-printable models]]
# [[mod:stereo|Lecture Theatre Stereo]]
== Online materials for mobile devices ==
# [[ebooks:howto|How to get eBooks]]
# [https://play.google.com/store/apps/details?id=com.blackboard.android&hl=en Blackboard mobile learn for Android]
# [https://itunes.apple.com/us/app/blackboard-mobile-learn/id376413870?mt=8 Blackboard mobile learn for iOS]
# [https://itunes.apple.com/gb/course/id562191620 Pericylic reactions in iTunesU ] (download [https://itunes.apple.com/gb/app/itunes-u/id490217893?mt=8 App] first)
# [https://itunes.apple.com/gb/course/id562191825 Conformational analysis in iTunesU] (download [https://itunes.apple.com/gb/app/itunes-u/id490217893?mt=8 App] first)
# [https://itunes.apple.com/gb/course/id562191342 A library of mechanistic animations in iTunesU] (download [https://itunes.apple.com/gb/app/itunes-u/id490217893?mt=8 App] first)
# [[IT:panopto|How to compress and disseminate Panopto lecture recordings]]

= PG =

# [[rdm:intro|Data management]]
= DOI =
*[[template:doi]] This page has {{DOI|dh4g}}

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Mod:inorganic

2025-09-17T13:48:56Z

Jbettenc: /* Basic instructions on how to build a wiki page */ Use internal wiki link

See also: [[Mod:timetable|Timetable]],[[mod:laptop|Laptop use]],[[Mod:lectures|Intro lecture]], [[mod:programs|Programs]], [[mod:organic|Module 1]], [[Mod:inorganic|Module 2]], [[Mod:phys3|Module 3]],[[Mod:writeup|Writing up]]
== Bonding and Molecular Orbitals in Main Group Compounds ==

[http://www.huntresearchgroup.org.uk/teaching/year3_lab_start.html Inorganic Experiments]
these experiments are run by Dr Hunt, please follow the link to her teaching web-site

== Basic instructions on how to build a wiki page ==
[[Title=Mod:inorganic_wiki_page_instructions|How to build a wiki page]]

----
See also: [[Mod:timetable|Timetable]],[[mod:laptop|Laptop use]],[[Mod:lectures|Intro lecture]], [[mod:programs|Programs]], [[mod:organic|Module 1]], [[Mod:inorganic|Module 2]], [[Mod:phys3|Module 3]],[[Mod:writeup|Writing up]]

Mod:phys3

2025-09-17T13:45:25Z

Jbettenc: Jbettenc moved page Rep:Mod:phys3 to Mod:phys3 without leaving a redirect: Revert: this was not a report

See also: [[Mod:intro|General info]],  [[mod:programs|Programs]],  [http://www.gaussian.com/g_tech/gv5ref/gv5ref_toc.htm Gaussian Online User Manual] | [http://faculty.ycp.edu/~jforesma/educ/visual/index.html Visualization Tutorials]
= Module 3 =

In this set of computational experiments, you will characterise transition structures on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition reactions.

There are two parts:

a) tutorial material: how to use the programs and methods,

b) more challenging examples, with guidelines but fewer explicit instructions.



<font color="#0000FF">''As a guideline, you should aim to complete part (a) in the first week of the experiment. Do not rush the tutorial material: once you have a good understanding of the computational techniques and how to work with the codes, you will find the remaining parts more straightforward and be better able to solve any problems you encounter. Try to write up sections on your wiki in outline as you go. Include the tutorial material in your write-up. If you are having problems, talk them through with a demonstrator or staff at the first opportunity.''</font>

In the second year physical chemistry laboratory, you may have carried out dynamics calculations using model potential energy surfaces to explore transition states. In that computational experiment, the total energy could quickly be calculated for different geometries of a triatomic system using an analytical function of the atomic coordinates (for more information, see for example [http://books.google.com/books?id=T8IZ1aa_FRkC&pg=RA1-PA36&lpg=RA1-PA36&dq=%22lake+eyring%22&source=web&ots=OXY00lSZ7D&sig=Ld_MTNwNjUDNGzB_5w1IxaMBMPU&hl=en&sa=X&oi=book_result&resnum=7&ct=result here] and [http://www.rsc.org/ejarchive/DC/1979/DC9796700007.pdf here]).

In this experiment, you will be studying transition structures in larger molecules. There are no longer fitted formulae for the energy, and the molecular mechanics / force field methods that work well for structure determination cannot be used (in general) as they do not describe bonds being made and broken, and changes in bonding type / electron distribution. Instead, we use molecular orbital-based methods, numerically solving the Schrodinger equation, and locating transition structures based on the local shape of a potential energy surface. As well as showing what transition structures look like, reaction paths and barrier heights can also be calculated.

Information that you might find useful in your wiki writeup is given in the [[Mod:Cheatsheet|cheatsheet]].

==The Cope Rearrangement Tutorial==



<font color="#0000FF">''This part of the module is described as a 'tutorial' because it's an introduction to various computational techniques for locating transition structures on potential energy surfaces. It's different to the [[mod:gv_basic |GaussView Tutorial]] you can work through: it's an exercise where you're given specific instructions, then see if you can follow them, and also determine whether there are problems or better ways of carrying the exercise out. Please include this part in your write-up. Marks will be given for correct answers, the documentation showing how you got these, discussion, and how you went about solving any problems you encountered.''</font>

In this tutorial we will use the Cope rearrangement of 1,5-hexadiene as an example of how to study a chemical reactivity problem.

Your objectives are to locate the low-energy minima and transition structures on the C<sub>6</sub>H<sub>10</sub> potential energy surface, to determine the preferred reaction mechanism.

[[Image:pic1.jpg|right|thumb|Cope rearrangement]]

This [3,3]-sigmatropic shift rearrangement has been the subject of numerous experimental and computational studies (e.g. Houk et al. {{DOI|10.1021/ja00101a078}}), and for a long time its mechanism (concerted, stepwise or dissociative) was the subject of some controversy. Nowadays it is generally accepted that the reaction occurs in a concerted fashion via either a "chair" or a "boat" transition structure, with the "boat" transition structure lying several kcal/mol higher in energy. The B3LYP/6-31G* level of theory has been shown to give activation energies and enthalpies in remarkably good agreement with experiment. In this tutorial we will show how these can be calculated using Gaussian.

<br />

{| align="center"
|- align="center"
|
[[Image:pic2a.jpg]]
|
[[Image:pic2b.jpg]]
|- align="center"
| align="center" | ''Chair Transition State''
| align="center" | ''Boat Transition State''
|}

<br />

===Optimizing the Reactants and Products===

'' In this section you will learn how to optimize a structure, symmetrize it to find its point group, calculate and visualize vibrational frequencies and correct potential energies in order to compare them with experimental values. It is assumed that you are already familiar with using the builder in GaussView. ''

<br />

(a) Using GaussView, draw a molecule of 1,5-hexadiene with an "anti" linkage (approximately a.p.p conformation) for the central four C atoms . Clean the structure using the '''Clean''' function under the '''Edit''' menu.

Now we will optimize the structure at the '''HF/3-21G''' level of theory. Select '''Gaussian''' under the '''Calculate''' menu, click on the '''Job Type''' tab and choose '''Optimization'''. The default method should already be Hartree Fock and the default basis set is 3-21G, so there should be no need to change these. You can check this by clicking on the '''Method''' tab. Submit the job by clicking on the '''Submit''' button at the bottom of the window and give the job a meaningful name (e.g. react_anti).

When the job has finished, you will be asked if you want to open a file. Select '''Yes''' and choose the checkpoint (chk) file with the name of the job you have just run (e.g. react_anti.chk). This checkpoint file is a binary file that stores data calculated by Gaussian. The name of the chk file should have been assigned by default. Once the file has been opened, click on the '''Summary''' button under the '''Results''' menu and make a note of the energy.


Does your final structure have symmetry? Select '''Symmetrize''' under the '''Edit''' menu (note that sometimes it is necessary to relax the search criteria under the '''Point Group''' menu). Make a note of the point group.

'''Warning:''' Clicking '''Symmetrize''' will change the geometry of your structure and cause a lot of the data to be discarded (including the energy data in the summary window). Instead, simply make a note of the point group that appears in the centre left of the '''Point Group Symmetry''' window. If it doesn't appear, incrementally relax the tolerance (centre of window) until it does. [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 16:27, 14 October 2015 (BST)

<br />

(b) Now draw another molecule of 1,5-hexadiene with a "gauche" linkage for the central four C atoms. Would you expect this structure to have a lower or a higher energy than the anti structure you have just optimized? Optimize the structure at the '''HF/3-21G''' level of theory and compare your final energy with that obtained in (a). Again, check if the molecule has symmetry and make a note of the point group.

<br />

(c) Normally, calculated activation energies and enthalpies use the lowest energy conformation of a reactant molecule as a reference. What would you have expected the lowest energy conformer of 1,5-hexadiene to be? Explain your reasoning, draw this structure and optimize it (if you have not already). A table containing the low energy conformers of 1,5-hexadiene and their point groups is shown in [[Mod:phys3#Appendix 1|Appendix 1]]. Draw gauche3 from the appendix and optimize it (again, if you have not already) and explain why this is the lowest conformer.

<br />

(d) Compare the structures that you have optimized in a) and b) with those in the table and see if you can identify your structure.

<br />

(e) Draw the C<sub>i</sub> ''anti2'' conformation of 1,5-hexadiene (unless you have already located it). Optimize it at the '''HF/3-21G''' level of theory and make sure it has C<sub>i</sub> symmetry. Compare your final energy to the one given in the table.


<br />

(f) When you are happy that your structure is the same as the one in the table, reoptimize it at the '''B3LYP/6-31G*''' level (6-31G* is equivalent to 6-31G(d) by selecting '''DFT''' under the '''Method''' menu and '''B3LYP''' from the box with the functionals on the right-hand side. Now select '''Link 0''' and change the name of the chk file to the name of the DFT optimization that you are about to run. Note that it is always advisable to do this when re-using or modifying existing structures to ensure that the original chk file is not overwritten. Run the job and make a note of the energy. Now compare the final structures from the '''HF/3-21G''' calculation with that at the higher level of theory. How much does the overall geometry change?

'''Note:''' It is meaningless to compare the energy of different levels of theory. [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 16:10, 14 October 2015 (BST)

<br />

(g) The final energies given in the output file represent the energy of the molecule on the bare potential energy surface. To be able to compare these energies with experimentally measured quantities, they need to include some additional terms, which requires a frequency calculation to be carried out. The frequency calculation can also be used to characterize the critical point, i.e. to confirm that it is a minimum in this case: that all vibrational frequencies are real and positive.

Starting from your optimized B3LYP/6-31G* structure, run a frequency calculation at the same level of theory. You can do this by selecting '''Frequency''' under the '''Job Type''' tab. Ensure that the method is still correctly specified under the '''Method''' tab (''caution: on Windows, sometimes 'scrf=(solvent=water,check)' is incorrectly added!'') and then change the name of the chk file under the '''Link 0''' tab to the name of the frequency job that you are about to run. Run the job. Once the job has finished, open the log file this time. Select '''Vibrations''' under the '''Results''' menu. A list of all the vibrational frequencies modes should appear. Check that there are no imaginary frequencies, only real ones. You can visualize some of these vibrations under this menu and simulate the infrared spectrum.


Now, select '''View File''' under the '''Results''' menu and open the output file in the visualizer. Scroll down to the section beginning '''Thermochemistry'''. Under the vibrational temperatures a list of energies should be printed. Make a note of (i) the sum of electronic and zero-point energies, (ii) the sum of electronic and thermal energies, (iii) the sum of electronic and thermal enthalpies, and (iv) the sum of electronic and thermal free energies. The first of these is the potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE), the second is the energy at 298.15 K and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>), the third contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions, and the last includes the entropic contribution to the free energy (G = H - TS). It is important to make sure that you select the correct energy/enthalpy term to compare to your experimental values. Note that these corrections can also be calculated at other temperatures using the '''Temperature''' option in Gaussian, If you have time, try to re-calculate these quantities at 0 K as shown in the [[mod:gv_advanced | Advanced GaussView Tutorial]].

<br />

===Optimizing the "Chair" and "Boat" Transition Structures ===

'' In this section you will learn how to set up a transition structure optimization (i) by computing the force constants at the beginning of the calculation, (ii) using the redundant coordinate editor, and (iii) using QST2. You will also visualize the reaction coordinate and run the IRC (Intrinisic Reaction Coordinate) and calculate the activation energies for the Cope rearrangement via the "chair" and "boat" transition structures. ''

<br />

The "chair" and "boat" transition structures for the Cope rearrangement are shown in [[Mod:phys3#Appendix 2|Appendix 2]]. Both consist of two C<sub>3</sub>H<sub>5</sub> allyl fragments positioned approximately 2.2 Å apart, one with C<sub>2h</sub> symmetry and the other with C<sub>2v</sub> symmetry.

<br />

(a) Draw an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) and optimize it using the '''HF/3-21G''' level of theory. Your structure should look like one half of the transition structures shown below.

Now open a new GaussView window by going to the '''File''' menu and selecting '''New''' and then '''Create MolGroup'''. Copy the optimized allyl structure from the first calculation by selecting '''Copy''' under the '''Edit''' menu, and then paste it twice into the new window by selecting '''Paste''' and then '''Append Molecule'''. Now orient the two fragments so that they look roughly like the chair transition state below by using the '''Shift Alt keys + Left Mouse button''' to translate one fragment with respect to the other and the '''Alt key + Left Mouse button''' to rotate it. The distance between the terminal ends of the allyl fragments should be approximately 2.2 Å apart. Save this structure to a Gaussian input file with a meaningful name (e.g. chair_ts_guess).

'''Note:''' If you're having trouble getting the bond lengths close enough to each other, an exact match can be achieved with '''Symmetrize'''. Get as close as you can to the structure, then open the '''Point Group''' window in the '''Edit''' menu. Relax the tolerance until the appropriate symmetry is displayed on the left and click '''Symmetrize'''. You should now be able to match the bond lengths. [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 16:37, 14 October 2015 (BST)

We are now going to optimize this transition state manually in two different ways. Transition state optimizations are more difficult than minimizations because the calculation needs to know where the negative direction of curvature (i.e. the reaction coordinate) is. If you have a reasonable guess for your transition structure geometry, then normally the easiest way to produce this information is to compute the force constant matrix (also known as the Hessian) in the first step of the optimization which will then be updated as the optimization proceeds. This is what we will try to do in the next section. However, if the guess structure for the transition structure is far from the exact structure, then this approach may not work as the curvature of the surface may be significantly different at points far removed from the transition structure. In some cases, a better transition structure can be generated by freezing the reaction coordinate (using '''Opt=ModRedundant''' and minimizing the rest of the molecule. Once the molecule is fully relaxed, the reaction coordinate can then be unfrozen and the transition state optimization is started again. One advantage of doing this, is that it may not be necessary to compute the whole Hessian once this has been done, and just differentiating along the reaction coordinate might give a good enough guess for the initial force constant matrix. This can save a considerable amount of time in cases where the force constant calculation is expensive.

<br />

(b) Use Hartree Fock and the default basis set 3-21G for parts (b) to (f).

Create a new MolGroup ('''File → New → Create MolGroup''') and copy and paste your guess structure into the window. Now set up a Gaussian optimization for a transition state. Go to the '''Gaussian''' menu under '''Calculate''' and click on the '''Job Type''' tab. Select '''Opt+Freq''' and then change '''Optimization to a Minimum''' to '''Optimization to a TS (Berny)'''. Choose to calculate the force constants '''Once''' and in the Additional keyword box at the bottom, type '''Opt=NoEigen'''. The latter stops the calculation crashing if more than one imaginary frequency is detected during the optimization which can often happen if the guess transition structure is not good enough. Submit the job. If the job completes successfully, you should have optimized to the structure shown in [[Mod:phys3#Appendix 2|Appendix 2]] and the frequency calculation should give an imaginary frequency of magnitude 818 cm<sup>-1</sup>. Animate the vibration and ensure that it is the one corresponding to the Cope rearrangement.

When performing an optimisation and subsequent frequency calculation, the geometry is at a minimum energy when all the vibrational frequencies are positive. When checking for a TS, there must be only one imaginary frequency. Explain briefly in simple terms, what does this imaginary frequency represent, and where does it come from?
Hint: think of the role of the force constant in a one dimensional quantum harmonic oscillator, covered in the QM3 course

<br />

(c) Now we will try optimizing the transition structure again using the frozen coordinate method. Create a new MolGroup ('''File → New → Create MolGroup''') and copy and paste your guess structure into the window again. Now select '''Redundant Coord Editor''' from the '''Edit''' menu. Click on the highlighted file icon at the top left-hand corner (Create a New Coordinate) and a line should appear below saying '''Add Unidentified (?, ?, ?, ?)'''. Now go back to the GaussView window and select two of the terminal carbons from the allyl fragments which form/break a bond during the rearrangement. Return to the coordinate editor and select '''Bond''' instead of '''Unidentified''' and select '''Freeze Coordinate''' instead of '''Add'''. Now click on the icon again to generate another coordinate. This time select the opposite two terminal atoms and again select '''Bond''' and '''Freeze Coordinate'''. Click OK. Now set up the optimization as if it were a minimum and you should see the option '''Opt=ModRedundant''' already included in the input line. Submit the job.

'''Note:''' GaussView allows you to produce an input file with the frozen coordinate specified as e.g. <tt>B 5 1 2.200000 F</tt>. Unfortunately, a recent update to the Gaussian program means it does not recognise this syntax, and just ignores this line. This means that the coordinate ends up being optimised rather than frozen. Therefore do not use this method, but ensure the guess structure has suitable guess transition bond distances(~2.2 Å) using the ''Modify Bond'' tool in GaussView --[[User:Rzepa|Rzepa]] 14:39, 29 October 2012 (UTC)

'''Note 2:''' If you set the coordinate distance (i.e. B 5 1 2.2 B) and freeze it (i.e. B 5 1 F) as separate inputs to the modredundant editor, it appears to work as expected. (Lee)

'''Note 3:''' If you are having trouble with this section (bonds not freezing) open your input file in Notepad or Notepad++ and go to the bottom of the file. You will see the lines that you would expect to freeze the bonds there. Make sure the line matches the notation in Note 2 (B 5 1 F, for example will freeze the bond between atoms 5 and 1). [[User:Tam10|Tam10]] ([[User talk:Tam10|talk]]) 16:09, 14 October 2015 (BST)

<br />

(d) When the job has finished, open the chk file. You should find that the optimized structure looks a lot like the transition you optimized in section (b), except the bond forming/breaking distances are fixed to 2.2 Å. Now we are going to optimize them too. Open the '''Redundant Coord Editor''' from the '''Edit''' menu again and create a new coordinate as before by clicking on the icon, Select one of the bonds that was previously frozen and this time choose '''Bond''' instead of '''Unidentified''' and '''Derivative''' instead of '''Add'''. Repeat the procedure for the other bond. This time you need to set up a transition state optimization but we are not going to calculate the force constants as we did in section (b) (so we leave this option as '''Never'''), instead we will use a normal guess Hessian modified to include the information about the two coordinates we are differentiating along. Change the name of the chk file in '''Link 0''' if you do not want to write over the previous calculation and submit the job. When the calculation has finished, open the chk file, check the bond forming/bond breaking bond lengths and compare the structure to the one you optimized in section (b).

'''Note''' For calculations using the redundant coord editor. Once the calculations have been run; due a bug which was caused by a recent update in gaussian. You may get an error saying 'bad data line x' in the .chk point file. by convention gausview uses the .chk to open for the next step, but this will not open due to the error, the output is a binary file, so it becomes very difficult to find and fix. This file contains more data, particularly orbital data. However for this section we just need the geometries to continue which are sorted in the human readable log file which is still produced with no error in the output. Therefor if you get this error continue to the next step with log file. A new .chk file will be produced in the next step. [[User:NF710|NF710]] ([[User talk:NF710|talk]]) 14:01, 29 October 2015 (BST)

<br />

(e) Now we will optimize the boat transition structure. We will do this using the '''QST2''' method. In this method, you can specify the reactants and products for a reaction and the calculation will interpolate between the two structures to try to find the transition state between them. You must make sure that your reactants and products are numbered in the same way. Therefore, although our reactants and products are both 1,5-hexadiene, we will need to manually change the numbering for the product molecule so that it corresponds to the numbering obtained if our reactant had rearranged.

e.g.

<center>[[Image:pic3.jpg|200px]]</center>

Open the chk file corresponding to the optimized C<sub>i</sub> reactant molecule (''anti2'' in [[Mod:phys3#Appendix 1|Appendix 1]]). Now open a second window and create a new MolGroup. Copy the optimized reactant molecule into the new window. In the same window, now select '''File → New → Add to MolGroup'''. The original molecule should disappear and a green circle should appear at the top left-hand corner with a '''2''' next to it. Clicking on the down arrow by the '''2''' will take you back to the original window and you will see your molecule again. This is how we read multiple geometries into GaussView. Go back to window '''2''', and copy and paste the reactant molecule a second time. This is going to be the product molecule and will be the molecule on which we need to change the numbering. If you now click on the icon showing two molecules side by side, then you can view both molecules simultaneously.

Now go to the '''View''' menu and select '''Labels''' so that you can see the numbering on both structures. Orient the two structures separately so they look something like the following:

{| align="center"
|- align="center"
|
[[Image:pic4a.jpg|200px]]
|
[[Image:pic4b.jpg|200px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

<br />

Now click on the product structure. Go to the '''Edit''' menu and select '''Atom List'''. Starting from Atom 1 on the reactant, go through and renumber all the atoms on the Product so that they match the reactant molecule, e.g. for the numbering above you would start by changing atom '''6''' on the product molecule to atom '''3'''. The other atom numbers will update as you do this so make sure you do it in the correct order. (Note: Start numbering at 1 and go up, i.e. do not order 5 before 3 to avoid problems with the numbers updating --[[User:Jrc10|Jrc10]] 15:13, 31 January 2013 (UTC)) At the end, the numbering on your two molecules should correspond to each other in the following way:

<br />

{| align="center"
|- align="center"
|
[[Image:pic5a.jpg|200px]]
|
[[Image:pic5b.jpg|200px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

<br />

Now we will set up the first '''QST2''' calculation. Go to the '''Gaussian''' menu and select '''Job Type''' as '''Opt+Freq''', and optimize to a transition state. This time you will have two options - '''TS (Berny)''' which we used in the previous calculations and '''TS (QST2)'''. Select '''TS (QST2)'''. Submit the job.

You will find that the job fails. To see why, open the chk file you created and view the structure. You will see that it looks a bit like the chair transition structure but more dissociated. In fact when the calculation linearly interpolated between the two structures, it simply translated the top '''allyl''' fragment and did not even consider the possibility of a rotation around the central bonds. It is clear that the QST2 method is never going to locate the boat transition structure if we start from these reactant and product structures.

Now go back to the original input file where you set up your QST2 calculation. We will now modify the reactant and product geometries so that they are closer to the boat transition structure. Click on the reactant molecule first and select the central '''C-C-C-C''' dihedral angle (i.e. '''C2-C3-C4-C5''' for the molecule above) and change the angle to 0<sup>o</sup>. Then select the inside '''C-C-C''' (i.e. '''C2-C3-C4''' and '''C3-C4-C5''' for the molecule above) and reduce them to 100<sup>o</sup>. Do the same for the product molecule. Your reactant and product molecules should now look like the following:

<br />

{| align="center"
|- align="center"
|
[[Image:pic6a.jpg|200px]]
|
[[Image:pic6b.jpg|200px]]
|- align="center"
| align="center" | ''Reactant''
| align="center" | ''Product''
|}

<br />

Set up the QST2 calculation again, renaming both the chk file under '''Link 0''' and the input file. Run the job again. This time it should converge to the boat transition structure. Check that there is only one imaginary frequency and visualize its motion.

The object of this exercise is to illustrate that although the QST2 method is has some advantages because it is fully automated, it can often fail if your reactants and products are not close to the transition structure. There is another method, the '''QST3''' method, that allows you to input the geometry of a guess transition structure also and this can often be more reliable. If you have time, you can try generating a guess boat transition structure and see if you can get the calculation to converge using the original reactant and product molecules. Remember to check the atom numbers in the transition structure are in the right order.

<br />

(f) Take a look at your optimized chair and boat transition structures. Which conformers of 1,5-hexadiene do you think they connect? You will find that it is almost impossible to predict which conformer the reaction paths from the transitions structures will lead to. However, there is a method implemented in Gaussian which allows you to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This is called the '''Intrinsic Reaction Coordinate''' or '''IRC''' method. This creates a series of points by taking small geometry steps in the direction where the gradient or slope of the energy surface is steepest.

Open the chk file for one of your optimized chair transition structures. Under the '''Gaussian''' menu, select '''IRC''' under the '''Job Type''' tab. You will be presented with a number of options. The first is to decide whether to compute the reaction coordinate in one or both directions. As our reaction coordinate is symmetrical, we will only choose to compute it in the forward direction. Normally you would do both forward and reverse, either in one job or in two separate jobs. You are also given the option to calculate the force constants once, at every step along the IRC or to read them from the chk file, in this case choose calculate always. You would use the latter option if you have previously run a frequency calculation.

The final option to consider is the number of points along the IRC. The default is '''6''' but this is normally never enough. Let's change this to 50 and see how the calculation progresses. Change the name of the chk file under '''Link 0''' and submit the job. The job will take a while so now is a good time to take a coffee break...

When the IRC calculation has finished, open the chk file with all the intermediate geometries and see how the calculation has progressed. You will find that it hasn't reached a minimum geometry yet. This leaves you three options: (i) you can take the last point on the IRC and run a normal minimization; (ii) you can restart the IRC and specify a larger number of points until it reaches a minimum; (iii) you can redo the IRC specifying that you want to compute the force constants at every step. There are advantages and disadvantages to each of these approaches. Approach (i) is the fastest, but if you are not close enough to a local minimum, you may end up in the wrong minimum. Approach (ii) is more reliable but if too many points are needed, then you can also veer off in the wrong direction after a while and end up at the wrong structure. Approach (iii) is the most reliable but also the most expensive and is not always feasible for large systems. You can try any or all of these approaches and see which conformation you end up in.

<br />

(g) Finally we need to calculate the activation energies for our reaction via both transition structures. To do this we will need to reoptimize the chair and boat transition structures using the '''B3LYP/6-31G*''' level of theory and to carry out frequency calculations. You can start from the HF/3-21G optimized structures. Once the calculations have converged, compare both the geometries and the difference in energies between the reactants and transition states at the two levels of theory. What you should find is that the geometries are reasonably similar, but the energy differences are markedly different.

As a consequence of this, it is often more computational efficient to map the potential energy surface using the low level of theory first and then to reoptimize at the higher level as we have done in this exercise.

The experimental activation energies are 33.5 ± 0.5 kcal/mol via the chair transition structure and 44.7 ± 2.0 kcal/mol via the boat transition structure at 0 K. If you take the values computed at 0 K, how close are they to the experimental values? You can also find the energies with thermal correction at 298.15 K under the Thermochemistry data in the output file. If you have time, you can recompute them at higher temperature. Alternatively, you can use the utility program '''FreqChk''' to obtain energies at a different temperature. This only requires the chk file from a frequency calculation and allows you to retrieve frequency and thermochemistry data as well as calculating them with an alternate temperature, pressure, scale factor, and/or isotope substitutions. The '''FreqChk''' utility program can be accessed from '''Gaussian09W'''. Launch '''Gaussian09W'''. Select '''utilities''' from the menu and click on '''FreqChk''' to launch the utility program. You will be prompted for a chk file. Follow the instructions from this [http://www.gaussian.com/g_tech/g_ur/u_freqchk.htm web link] to proceed.

<br />

=== Appendix 1 ===

<br />

'''Note:''' 3D models of all the conformers in the table below are given '''[[Mod:phys3_appendix1|here]]'''.

<br />

{| border="1" cellpadding="5"
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Relative Energy/kcal/mol'''
|- align="center"
| ''gauche''
|
[[Image:gauche1.jpg|150px]]
| C<sub>2</sub>
| -231.68772
| 3.10
|- align="center"
| ''gauche2''
|
[[Image:gauche2.jpg|150px]]
| C<sub>2</sub>
| -231.69167
| 0.62
|- align="center"
| ''gauche3''
|
[[Image:gauche3.jpg|150px]]
| C<sub>1</sub>
| -231.69266
| 0.00
|- align="center"
| ''gauche4''
|
[[Image:gauche4.jpg|150px]]
| C<sub>2</sub>
| -231.69153
| 0.71
|- align="center"
| ''gauche5''
|
[[Image:gauche5.jpg|150px]]
| C<sub>1</sub>
| -231.68962
| 1.91
|- align="center"
| ''gauche6''
|
[[Image:gauche6.jpg|150px]]
| C<sub>1</sub>
| -231.68916
| 2.20
|- align="center"
| ''anti1''
|
[[Image:anti1.jpg|150px]]
| C<sub>2</sub>
| -231.69260
| 0.04
|- align="center"
| ''anti2''
|
[[Image:anti2.jpg|150px]]
| C<sub>i</sub>
| -231.69254
| 0.08
|- align="center"
| ''anti3''
|
[[Image:anti3.jpg|150px]]
| C<sub>2h</sub>
| -231.68907
| 2.25
|- align="center"
| ''anti4''
|
[[Image:anti4.jpg|150px]]
| C<sub>1</sub>
| -231.69097
| 1.06
|}

<br />

=== Appendix 2 ===

{| cellpadding="5"
|- align="center"
|
[[Image:appendix2a.jpg|300px]]
|- align="center"
| ''C<sub>2h</sub> Chair Transition State''
|- align="center"
|- align="center"
|- align="center"
|
[[Image:appendix2b.jpg|300px]]
|- align="center"
| ''C<sub>2v</sub> Boat Transition State''
|}

<br />

=== Results Table ===

<br />

''' Summary of energies (in hartree) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466705
| align="center" | -231.461346
| align="center" | -234.556983
| align="center" | -234.414919
| align="center" | -234.408998
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450929
| align="center" | -231.445300
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539539
| align="center" | -231.532566
| align="center" | -234.611710
| align="center" | -234.469203
| align="center" | -234.461856
|}

<br /> *1 hartree = 627.509 kcal/mol <br /><br />

''' Summary of activation energies (in kcal/mol) '''

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.70
| align="center" | 44.69
| align="center" | 34.06
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.96
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

<br />

==The Diels Alder Cycloaddition==

''In this exercise, you will characterise transition structures using any of the methods described above in the tutorial: the choice is up to you. In addition, you will look at the shape of some of the molecular orbitals. To help you structure your report, there is a data/discussion sheet at the end of this section.''

<br>

[[Image:mb_da1.jpg |right|thumb|Diels Alder cycloaddition]]
The Diels Alder reaction belongs to a class of reactions known as pericyclic reactions. The π orbitals of the dieneophile are used to form new σ bonds with the π orbitals of the diene. Whether or not the reactions occur in a concerted stereospecific fashion ('''allowed''') or not ('''forbidden''') depends on the number of π electrons involved. In general the HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other reactant to form two new bonding and anti-bonding MOs. The nodal properties allow one to make predictions according to the following rule:

''If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is '''allowed'''.''

''The HOMO-LUMO can only interact when there is a significant overlap density. If the orbitals have different symmetry properties then no overlap density is possible and the reaction is '''forbidden'''.''

If the dieneophile is substituted, with substituents that have π orbitals that can interact with the new double bond that is being formed in the product, then this interaction can stabilise the regiochemistry (i.e. head to tail versus tail to head) of the reaction. In this exercise you will study the nature of the transition structure for the Diels Alder reaction, both for the prototypical reaction and for the case where both diene and dieneophile carry substituents, and where secondary orbital effects are possible. Clearly, the factors that control the nature of the transition state are quantum mechanical in origin and thus we shall use methods based upon quantum chemistry.

<br>

Shown on the right is a diagram of the transition state for the Diels-Alder reaction between ethylene and butadiene. The ethylene approaches the cis form of butadiene from above.
[[Image:mb_da2.jpg |right|thumb|Ethylene+Butadiene cycloaddition]]

Before beginning our quantitative study, it is helpful to discuss the interaction of the π orbitals in a simple qualitative way.

''You will confirm some of these considerations in your computations.''

The principal orbital interactions involve the π/ π* orbitals of ethylene and the HOMO/LUMO of butadiene. It is referred to as [4s + 2s] since one has 4 π orbitals in the π system of butadiene. The orbitals of ethylene and butadiene and ethylene can be classified as symmetric '''s''' or anti-symmetric '''a''' with respect to the plane of symmetry shown.

The HOMO of ethylene and the LUMO of butadiene are both '''s''' (symmetric with respect to the reflection plane) and the LUMO of ethylene and the HOMO of butadiene are both '''a'''. Thus it is the HOMO-LUMO pairs of orbital that interact, and energetically, the HOMO of the resulting adduct with two new σ bonds is '''a'''.

<br>

=== Exercise ===

Use the the AM1 semi-empirical molecular orbital method for these calculations.

i) Use GaussView to build cis butadiene, and optimize the geometry using Gaussian. Plot the HOMO and LUMO of cis butadiene and determine its symmetry (symmetric or anti-symmetric) with respect to plane.

''There are two ways to do this in GaussView. One is: Select '''Edit→MOs'''. Select the HOMO and the LUMO from the MO list (highlights it yellow). Click the button '''Visualise''' (not Calculation), then '''Update'''. Alternately, having calculated the surface for this orbital, you can display it in the main GaussView window for the molecule, from the '''Results→Surfaces''' menu. Select '''Surface Actions→Show Surface'''. Having displayed the surface this way, you can also select '''View→Display Format→Surface''', and change '''Solid''' to '''Mesh'''.''

<br>

ii) Computation of the Transition State geometry for the prototype reaction and an examination of the nature of the reaction path.

[[Image:mb_da3.jpg |right|thumb|]]

The transition structure has an envelope type structure, which maximizes the overlap between the ethylene π orbitals and the π system of butadiene. One way to obtain the starting geometry is to build the bicyclo system (b) and then remove the -CH<sub>2</sub>-CH<sub>2</sub>- fragment. One must then guess the interfragment distance (dashed lines) and optimize the structure, but use any method you wish, based on the tutorial above, to characterise the transition structure. Confirm you have obtained a transition structure for the Diels Alder reaction!

[[Image:mb_da4.jpg |right|thumb|guessing the transition structure]]

Once you have obtained the correct structure, plot the HOMO as in (i). Rotate the molecule so that the symmetry and nodal properties of the system can be interpreted, and save a copy of the image.

<br>

(iii) To Study the regioselectivity of the Diels Alder Reaction

Cyclohexa-1,3-diene '''1''' undergoes facile reaction with maleic anhydride '''2''' to give primarily the endo adduct. The reaction is supposed to be kinetically controlled so that the exo transition state should be higher in energy.

[[Image:Bearpark_pic_edit_by_jm906.JPG |right|thumb|regioslectivity]]

Locate the transition structures for both 3 and 4. Compare the energies of the endo and exo forms.

Measure the bond lengths of the partly formed σ C-C bonds and the other C-C distances. Make a sketch with the important bond lengths. Measure the orientation, (C-C through space distances between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH<sub>2</sub>-CH<sub>2</sub>- for the exo and the “opposite” -CH=CH- for the endo). The structure must be a compromise between steric repulsions of the -CH<sub>2</sub>-CH<sub>2</sub>- fragment and the maleic anhydride for the exo versus secondary orbital interactions between the π systems of -CH=CH- and -(C=O)-O-(C=O)- fragment for the endo.

Plot the HOMO as in the previous exercise. Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system. What can you conclude about the so called “secondary orbital overlap effect”?

<br>

=== Discussion ===

Screen images can be saved from the GaussView '''File''' menu.

''For cis butadiene and ethylene'': <br>
Plot the HOMO and LUMO and determine the symmetry (symmetric or anti-symmetric) with respect to the plane.

''For the ethylene+cis butadiene transition structure'':<br>
Sketch HOMO and LUMO, labeling each as symmetric or anti symmetric.

Show the geometry of the transition structure, including the bond-lengths of the partly formed σ C-C bonds.

What are typical sp<SUP>3</SUP> and sp<SUP>2</SUP> C-C bondlengths? What is the van der Waals radius of the C atom? What can you conclude about the C-C bond length of the partly formed σ C-C bonds in the TS.

Illustrate the vibration that corresponds to the reaction path at the transition state.
Is the formation of the two bonds synchronous or asynchronous?
How does this compare with the lowest positive frequency?

Is the HOMO at the transition structure '''s''' or '''a'''?

Which MOs of butadiene and ethylene have been used to form this MO?
Explain why the reaction is allowed.

''For the cyclohexa-1,3-diene reaction with maleic anhydride'':<br>
Give the relative energies of the exo and endo transition structures.
Comment on the structural difference between the endo and exo form. Why do you think that the exo form could be more strained?
Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system. What can you conclude about the so-called “secondary orbital overlap effect”?
(There is some discussion of this in Ian Fleming's book 'Frontier Orbitals and Organic Chemical Reactions').

''Further discussion'':<br>
What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time.
(e.g. {{DOI|10.1021/jo0348827}})

----


© 2008-2013, Imperial College London

Mod:stereo

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= '''Stereoscopic (3D) Projection''' {{DOI|bp9p}} =

An overview of stereo formats can be found at https://www.sview.ru/en/help/input/ LT B10 (MSRH, Chemistry) is equipped with a stereoscopic projector configuration used with polarized glasses stored in the projection booth, which allow viewing of 3D images and video. You will need swipe-card access to enter. Check that the battery in the glasses has charge before attempting to use!

To start, select both projectors from the lectern’s control panel by first selecting the PC as the output and then selecting “'''Enable 3D'''”. In order to view the output, the applications that will be used to display 3D content must be set up to use the '''Quad-Buffer OpenGL''' viewing method.

If you are aware of any interesting software applications not listed below, do please consider adding them to this list.

== Testing Stereo projection ==
#Download Stereoscopic Player: http://www.3dtv.at/Downloads/Index_en.aspx
# To play sample stereo videos, install http://www.microsoft.com/windows/windowsmedia/player/11/default.aspx)
#Get a stereo jpg: http://www.chasm.com/images.htm
#Install the Stereoscopic Player, open truck.jpg, and choose side-by-side, left eye first for the image format.
#In the View menu, change the stereo view method to OpenGl quad-buffer stereo.

== [http://accelrys.com/products/collaborative-science/biovia-discovery-studio/visualization-download.php Discovery Studio (DS)] ==

Discovery Studio is a comprehensive software suite for analyzing and modelling molecular structures, sequences, and other data of relevance to life science researchers. The product includes functionality for viewing and editing data along with tools for performing basic data analysis.

This application is already setup to output via the default
setting of the graphics adapter. Please ensure that:

• Hardware Stereo Type is selected in the Molecule Window section of the program preferences as shown below (Click Edit and then Preferences):

[[Image:Ds_image_1.jpg|800px]]

• There is a tick in the Stereo option in the View menu as below.

[[Image:Ds_image_2.jpg|800px]]

== [http://www.cambridgesoft.com/support/ProductHomePage.aspx?KBCatID=112 Chem3D] ==

Chem3D is a powerful desktop modelling program that enables synthetic chemists and biologists to generate 3D models of small molecules
and biochemical compounds. College has a site-wide license.

To access the Chem3D program settings, click on
'''File''', then '''Preferences''' to display the options on the right. Ensure that the Use Back Buffer for Refresh checkbox is ticked, then click Apply and then OK (see below):

[[File:Chem_bio3d_image1.jpg|800px]]

Use the OpenGL tab

[[File:Chem_bio3d_image2.jpg|800px]]

== [http://www.ks.uiuc.edu/Research/vmd VMD] ==

VMD is a molecular visualization program for displaying, animating, and analyzing large bio molecular systems using 3-D graphics. Pleaseuse the display settings shown in the image below to render output via the 3D projector:

[[Image:Vmd_image1.jpg|800px]]

== [http://www.ccdc.cam.ac.uk/solutions/csd-system/components/mercury CCDC Mercury] ==

This is a component of the Cambridge Structural Database System (CSD database) which provides a range of tools for 3D structure visualization, the exploration of crystal packing and the statistical analysis of CSD search data. The Mercury viewer is free. The CSD database has a College-wide license.

To access the Mercury program display settings, click the top menu and select '''Display/Display options/Stereo'''. Ensure '''hardware stereo''' is enabled (see below):

[[File:Mercury_image1.jpg|800px]]

To set the 3D effect to emerge out of the screen, set the focal point to a +ve value. Adjustments can also be made with the Separation to create the optimal effect.

== [http://www.knotplot.com/ Knotplot] ==

This programs allows 3D visualization of an enormous variety of knots and other topological objects. To initialise stereo, the program has to be started from the command line rather than the more usual start menu or application double click. A shortcut which invokes this flag is available on the desktop. A college-wide license could be purchased upon demand.

<code>cmd: knotplot -stereo</code>

== [http://www.3dtv.at/products/player/index_en.aspx Stereoscopic Player] ==
Stereoscopic Player is a 3D movie and stereo JPEG player. It allows you to
play stereoscopic videos and DVDs (external decoder required) and also allows you to watch live video from a capture device. It can handle most media formats, for example AVI, MPEG, WMV, ASF and MOV. Videos can be coded in several different stereoscopic formats.

Please use the display settings shown in the image below to render output from Stereoscopic player via the 3D projector.

YouTube has a 3D video selection, see https://www.youtube.com/results?search_query=yt3d

[[File:Stereoscopic_image1.jpg|400px]]
=== XStereo-player ===

Another program that supports a variety of 3D formats, including the Fuji 3D camera, is http://urixblog.com/en/xstereo-player-menu/xstereo-player-osx/ (Free). A paid for upgrade allows export of various formats, including 3D-TV.

=== Bino Stereo player ===

See http://bino3d.org/ and http://bino3d.org/3d-videos.html

== [http://mipav.cit.nih.gov MIPAV] Medical images ==

The MIPAV (Medical Image Processing, Analysis, and Visualization) application enables quantitative analysis and visualization of medical images of numerous modalities such as PET, MRI, CT, or microscopy. Using MIPAV's standard user-interface and analysis tools, researchers at remote sites (via the internet) can easily share research data and analyses, thereby enhancing their ability to research, diagnose, monitor, and treat medical disorders.

After starting the application (Start > All Programs > MIPAV), you will see the following splash screen:

[[File:mipav1.jpg|300px]]

Followed by two windows as shown below:

[[File:mipav2.jpg|400px]]

Select Help > MIPAV Options. The MIPAV Options dialog box opens.
Click on the Other tab and ensure that the GPU Computing enabled checkbox is ticked. If it is greyed out, there may be a problem with the graphics card, graphics driver or both and GPU volume rendering will not be possible.

To display an image series in S3D, open the image series first. Click File > Open image (A) from disk
as shown below (Sample images were downloaded from http://www.cancerimagingarchive.net/ )

[[File:mipav3.jpg|400px]]

Select the first image in the series as below (ensure that the Open as multifile option is ticked) and click on Open

[[File:mipav4.jpg|400px]]

You will then see the following: Click on the GPU-based Volume Renderer v1.0 button (circled in red below)

[[File:mipav5.jpg|400px]]

You will see a dialog showing the rendering progress…

[[File:mipav6.jpg|200px]]

Followed by a window with _clone appended in the title bar as shown below:

[[File:mipav7.jpg|400px]]

Click the Renderer mode control button (circled in red above). This will display the following dialogue:
Select Quad Buffer as the Stereo Mode in the Display Components section as shown here.

[[File:mipav8.jpg|400px]]

Click on the Surface Volume renderer button (circled in red below)

[[File:mipav9.jpg|400px]]

You will now be able to rotate the 3D image with the mouse and adjust opacity to show/hide different parts of the image (note the additional Opacity and Slices tabs).

Useful links
# http://www.cancerimagingarchive.net/
# http://mipav.cit.nih.gov/documentation/presentations/visualization.pdf
# http://mipav.cit.nih.gov/documentation/userguide/Volume1/MIPAV_VisualizationTools.pdf
# http://mipav.cit.nih.gov/documentation/userguide/MIPAVUsersGuideVolume1.pdf
# http://mipav.cit.nih.gov/pubwiki/index.php/Volume_Renderer
# http://mipav.cit.nih.gov/pubwiki/index.php/FAQ:_How_do_I_setup_and_use_Active_stereo_display_in_MIPAV%3F Instructions for enabling Stereo.

== Autodesk VRED Professional 2016 ==
VRED (Virtual Reality EDitor) 3D visualization software helps automotive designers and digital marketers create product renderings, design reviews, and virtual
prototypes. Please see http://www.autodesk.com/products/vred/features/vred/all/gallery-view for more detailed comparison of different VRED products.

VRED Professional is used in the automotive industry to create high-end visualisations and virtual prototypes. It is currently installed in RCS1 LTC. To launch the application, double-click on the icon on the desktop.
[[File:VRED icon.png]]

You will then see the following splash screen:

[[File:VRED splash.png|200px]]

To view examples, click on File > Open Examples

[[File:VRED open1.png|400px]]

Select and open the armchair file for example:

[[File:VRED open2.png|400px]]

To enable stereo visualisation, click on Visualisation > Stereo > Double Buffered

[[File:VRED stereo1.png|400px]]

Eye separation and Zero parallax distance adjustments can be made in the Stereo Settings menu.
Click Visualization > Stereo > Stereo Settings as shown below.

[[File:VRED stereo3.png|400px]]

[[File:VRED stereo2.png|400px]]

Eye Separation: Sets viewer’s eye left and right eye distance in millimeters.
Zero Parallax Distance: Sets viewer’s focus distance. This distance is the distance to a plane where the images for both eyes match.
Disable Stereo: Disables stereo rendering.
Auto: Calculates the Eye distance depending on the view distance.

The average human eye separation distance (also known as interocular or interpupillary distance (IPD)) is about 63.5mm (2.5inches).
In stereo viewing, if an object appears to be on the projection plane there is zero parallax. Negative parallax occurs when the object appears in front of the projection screen and positive parallax when the object appears behind the screen.

Full-screen view is available as a main menu bar item

[[File:VRED fullscreen.png]]

Mod:3D

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= 3D-Printable chemistry models =

[http://www.shapeways.com Shapeways] is one commercial company that can produce '''full-colour''' 3D-printed models. As we start to generate such content, it is desirable to try to keep track of what is available from local activity (the Shapeways catalogue already contains much chemistry). The list below illustrates some of this activity:

# Chiral [https://www.shapeways.com/model/1347220/goctoh.html?li=aeTabs molecular-scale metal] wire, [http://www.ch.imperial.ac.uk/rzepa/blog/?p=8337 More details].
# Left handed [http://shpws.me/p2vr DNA] and right handed [http://shpws.me/p2vy DNA] duplex (CGCG)
# Six π-MOs of Benzene: http://shpws.me/oYY1 http://shpws.me/oYY6 http://shpws.me/oYYa (occupied) http://shpws.me/oYYh http://shpws.me/oYYk http://shpws.me/oYYp (unoccupied)
# Four π-MOs of cis-butadiene: http://shpws.me/oYZa http://shpws.me/oYZd (HOMO),http://shpws.me/oYZn (LUMO), http://shpws.me/oYZp
# Four π-MOs of cis-dimethyl hexatriene: http://shpws.me/pacC http://shpws.me/pacD (HOMO-1), http://shpws.me/pacE (HOMO), http://shpws.me/pacJ (LUMO)
# The MOs of Copper phthalocyanine [http://shpws.me/p0AC HOMO], mauveine ([http://shpws.me/p0Az HOMO] and [http://shpws.me/p0AB LUMO]) and an [http://shpws.me/oYsx octaphyrin] (corresponding to the [http://www.ch.ic.ac.uk/rzepa/artwork/ artwork] on the student resource-centre windows)
# The [http://shpws.me/p3e6 NBO orbitals] showing the overlap leading to the gauche effect in 1,2-difluoroethane.
# Molecular orbital for [http://shpws.me/oTLF {144}-annulene] indicating a topological linking number of 18.[http://www.ch.imperial.ac.uk/rzepa/blog/?p=9322 More details].
# Transition state for [http://shpws.me/p0DQ cyclohexene oxide+CO<sub>2</sub>] co-polymerisation using the Williams Zn-catalyst
----
{| class="wikitable" border="1"
|+ Transition states for asymmetric epoxidation
|-
| [http://shpws.me/p0AF Shi Fructose-catalyst] || [http://shpws.me/oZ0I Jacobsen Mn-catalyst]
|-
| [http://shpws.me/p0Bz Sharpless Ti-catalyst] || [http://shpws.me/p0D0 Sodeoka-Hii Pd-catalyst]
|}
----
== Instructions on how to generate 3D-printable models (orbitals) using Jmol. ==
# Run a QM calculation, select the desired orbital and create a Gaussian cube file containing the wavefunction (medium or fine grid).
# Acquire [http://sourceforge.net/projects/jmol/files/ Jmol 13.3] or greater.
# Run '''Jmol.jar''' and open '''File/console'''
# >load bd_mol15.cub
# >cpk 200
# >set bondradiusmilliangstroms 500
# >isosurface sign color yellow green cutoff 0.02 bd_mol15.cub
# >set exportscale 12.0
# >write bd_mol15.wrl
# Upload the .wrl file to the manufacturer

You should replace the values above with ones relevant to your model. The isosurface cutoff can be adjusted to produce a good-looking model. Similarly the bond radii and the atom size (CPK). The exportscale controls the cost of the resulting model. If printed in full-colour sandstone, a model with largest dimension ~15 cm is going to be a reasonable cost.

An alternative output format which is accepted by [http://www.sculpteo.com/en/ some manufacturers] is '''.obj'''. This produces not a single file but two or even three. These have to be zipped into a compressed archive before uploading.

== Instructions on how to generate 3D-printable models (ball&stick) using Jmol. ==
# Acquire suitable coordinates. If from a crystal structure, it is probably best to save as a .pdb file rather than .cif. Ensure you edit the connectivity to avoid any components which are disconnected from the main model. If necessary, connect them using an explicit bond.
# Acquire [http://sourceforge.net/projects/jmol/files/ Jmol 13.3] or greater.
# Run '''Jmol.jar''' and open '''File/console'''
# >load Z-CGCG.pdb
# >cpk 150
# >set bondradiusmilliangstroms 300
# >set exportscale 8.0
# >write Z-CGCG.wrl
# Upload the .wrl file to the manufacturer

An alternative output format which is accepted by [http://www.sculpteo.com/en/ some manufacturers] is '''.obj'''. This produces not a single file but two or even three. These have to be zipped into a compressed archive before uploading.
== Instructions on how to generate 3D-printable models (proteins) using PyMol. ==
# [[File:Pymol1.jpg|130px|right]] Drag-n-drop a .pdb file onto the main Pymol Window (MacPymol tested). It will open in standard wireframe mode.
# Edit off any water molecules and other disconnected units (skill needed here!)
# From the all/S(how) button (right), select '''surface'''
# From the all/C(olour) button, select '''spectrum''' (or other)
# From '''File/Save image as/VRML 2''', save a .wrl file.
# Scale this file as per below using a text-editor to open the .wrl file.

== Instructions on how to generate 3D-printable models (proteins) using Chimera. ==

[[File:1a30.jpeg|right|thumb|HIV protease rendered using Chimera]][http://www.cgl.ucsf.edu/chimera/download.html Chimera] has similar functionality to Pymol.
# '''File/open''' to load a .pdb file
# '''Tools/Structure editing/Minimise structure''' offers editing tools to trim off solvent, non-complexed ions, incomplete side chains etc.
# '''Actions/Surface/Show''' to produce a smooth Connolly surface.
# '''tools/Depiction/Rainbow''' and select eg '''Residue'''.
# '''File/Export-Scene''' and select eg VRML.
# One can also load VRML (.wrl) files produced by other programs as a check.

== Scaling a VRML file ==
*Open the .wrl file in any simple text editor, and BEFORE the objects begin add lines:
<pre>Transform {
children [</pre>
*At the very end of the file add
<pre>]#end of children
scale 2.0 2.0 2.0
}#end of Transform</pre>
----

An example might look like (the addition is just before the Shape command) for a scaling factor of 2.0 in all directions

<pre>#VRML V2.0 utf8
Viewpoint {
position 0 0 160.36393738
orientation 1 0 0 0
description "Z view"
fieldOfView 0.465421
}
DirectionalLight {
direction -0.348155 -0.348155 -0.870
}
NavigationInfo {
headlight TRUE
type "EXAMINE"
}
Transform {
children [
Shape {
.....
.....
]
}
}
] #end of children
scale 2.0 2.0 2.0
} #end of transform</pre>

== Software ==

A variety of software for viewing, validating and editing the 3D models is available.
# [http://www.cgl.ucsf.edu/chimera/download.html Chimera] is very good for viewing a .wrl model on your computer after generation.
# [http://www.sculpteo.com/en/ This company] has a useful web site where you can colour-code the robustness of your model to manufacture. Red=likely to break; Green=robust, and hence adjust the bond thickness etc to the appropriate values.

== Useful links ==

[http://www.macinchem.org/reviews/3D/3dprinting.php Macinchem] have a very good collection of pointers.

Mod:writeup

2025-09-17T13:41:42Z

Jbettenc: Jbettenc moved page Rep:Mod:writeup to Mod:writeup without leaving a redirect: Revert: this was not a report

See also: [[Mod:Cheatsheet|Cheatsheet]].

== The expected length of the report ==
A Wiki does not have pages as such. But as a very rough guide, expect to produce something the equivalent of about six printed pages (although you can invoke ''pop-ups'' and the like which make a page count only very approximate). Use graphics reasonably sparingly, and to the point.

= Why Wiki? =
Since everyone is used to using [http://www.wordonwiki.com Microsoft Word], why do we [[talks:rzepa2011|use a Wiki]] for this course? Well, the Wiki format has several advantages.
#A full revision and fully dated history across sessions is kept (Word only keeps this during a session). This is more suited for laboratory work, where you indeed might need to go back to a particular day and experiment to check your notes.
#The (chemistry) Wiki allows you to include molecule coordinates, vibrations and MO surfaces which can be rotated and inspected, along with other chemical extensions. Word does not offer this.
#The Wiki allows you to include "zoomable" graphics in the form of SVG (which Gaussview generates), and access to the 17-million large [http://commons.wikimedia.org/wiki/Main_Page WikiCommons] image library, as well as access to the Wikipedia InterWiki.
#The [[w::Help:Template|template]] concept allows pre-formated entry. There are lots of powerful [[w:Category:Chemical_element_symbol_templates|chemical templates]] available.
#Autonumbered referencing, and particularly cross-referencing, is actually easier than using Word.
#You (and the graders) can access your report anywhere online, it is not held on a local hard drive which you may not have immediate access to.
#It has automatic date and identity stamps for ALL components, which means we can assess that part of the report handed in by any deadline, and deal separately with anything which has a date-stamp past a given deadline. A Word document has only a single date-and-time stamp and so deadlines must apply to the whole document.
#And we have been using Wikis for course work since '''2006''', so there is lots of expertise around!
#And finally, Wiki is an example of a [http://en.wikipedia.org/wiki/Markdown MarkDown] language, one designed to facilitate writing using an easy-to-read, easy-to-write plain text format (with the option of converting it to structurally valid XHTML).

=Report Preparation =

== Before you start writing ==
Before you start writing, you might wish to read this article<ref name="adma.200400767">G.M. Whitesides, "Whitesides' Group: Writing a Paper", ''Advanced Materials'', '''2004''', ''16'', 1375–137 {{DOI|10.1002/adma.200400767}}</ref> (or perchance this advice<ref name="ac2000169">R. Murray, "Skillful writing of an awful research paper", ''Anal. Chem.'', '''2011''', ''83'', 633. {{DOI|10.1021/ac2000169}}</ref>). In your report you should discuss your evaluation of each of the techniques you use here. You should include at least '''three literature references in addition to the ones given here'''. You might also want to check the [[Mod:latebreak|late breaking news]] to see if there are any helpful hints about the project you might want to refer to. You will be writing your report in Wiki format, and it is best to do this continually as you do the experiment. In effect, your Wiki report is also your laboratory manual.

#[[File:Wiked.jpg|right|400px|WikED editor]]Open Firefox as a Web browser.
#There should be a tab for the course Wiki, but if not, use the URL '''www.ch.ic.ac.uk'''
#*[[Image:exception1.jpg|right|thumb|Security exception]]If the browser asks you to add a security exception, do so and proceed to '''view/confirm''' the certificate.
#You can view the Wiki without logging in, but to create a report, you will have to login as yourself. Check '''Remember my login on this computer'''
#Before you start, you might want to visit the [[Special:Preferences|preferences]] page to customise the Wiki for yourself.
#Follow the procedures below. Check that the WikED icon is present at the top, just to the right of the log out text (ringed in red). If you want a minimalist editing interface, click this icon to switch it off. A [http://en.wikipedia.org/wiki/Wikipedia:Cheatsheet cheatsheet] summarises the commands with a [[It:projects|playpen]] for playing. You can write your report by simply typing the appropriate text as shown in the cheatsheet, or by using the WikEd buttons in Word-style composition.

=== Assigning your report an identifier ===
#*[[Image:New_report.jpg|right|400px|Creating a report page]] In the address box, type something like
#**'''wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:{{fontcolor1|yellow|black|XYZ1234}}'''
#*The characters '''Mod''' indicate a report associated with the modelling course, and '''{{fontcolor1|yellow|black|XYZ1234}}''' is your secret password for the report. It can be any length, but do not make it too long! It should then tell you there is no text in this page. If not, try another more unique password. You should now click on the '''edit this page''' link to start. Use a different address for each module of the course you are submitting.
#*It is a {{fontcolor1|yellow|black|good idea}} to add a bookmark to this page, so that you can go back to it quickly.
==== Assigning your report a persistent (DOI-style) identifier.====
Use [https://data.hpc.imperial.ac.uk/publish/url/ this tool] to assign a shorter identifier for your report (one that can be invoked using eg <nowiki>{{DOI|shortidentifier]]</nowiki>.

=== Converters to the Wiki format ===
#Convert a Word document. Open it in '''OpenOffice''' (rather than the Microsoft version) and '''export''' as Mediawiki. Open the resulting .txt file in eg WordPad, select all the text, copy, and then paste this into the Wiki editing page. You will still have to upload the graphical images from the original Word document separately.
#There is also a [http://labs.seapine.com/htmltowiki.cgi HTML to Wiki] converter which you can use to import HTML code from an existing Web page into a (Media)Wiki.

== Basic editing ==
An [[Title%3DMod:inorganic_wiki_page_instructions|introductory tutorial]] is available which complements the information here.
*A [http://en.wikipedia.org/wiki/Wikipedia:Cheatsheet cheatsheet] summarises the commands with a [[It:projects|playpen]] for playing. You can write your report by simply typing the appropriate text as shown in the cheatsheet, or by using the WikEd buttons in Word-style composition.
*[[Image:report12345.jpg|right|thumb|The editing environment]]You will need to create a separate report page on this Wiki for each module of the course. Keep its location private (i.e. do not share the URL with others).
*The WikED toolbar along the top of the page has a number of tools for:
**adding citation references,
**superscript and subscripting (the H<sub>2</sub>O WikEd symbol will automatically do this for a formula),
**creating tables
**adding links (Wiki links are internal, External links do what they say on the tin)
**# local to the wiki, as <nowiki>[[mod:writeup|text of link]]</nowiki>
**# remote, as <nowiki>[http://www.webelements.com/ text of link]</nowiki>
**# Interwiki, as <nowiki>[[w:Mauveine|Mauveine]]</nowiki>
**# DOI links are invoked using the DOI template <nowiki>{{DOI|..the doi string ..}}</nowiki> or the more modern form <nowiki>[[doi:..the dpi string..]]</nowiki>
**# Links to an Acrobat file you have previously uploaded to the Wiki can be invoked using this template: <nowiki>{{Pdf|tables_for_group_theory.pdf|...description of link ...}}</nowiki>
**# There are lots of other [[Special:UncategorizedTemplates|templates]] to make your life easier such as the [[w:Template:Chembox|ChemBox]]
**If you need some help, invoke it from the left hand side of this page.
*Upload all graphics files also with unique names (so that they do not conflict with other people's names). If you are asked to replace an image, '''REFUSE''' since you are likely to be over-writing someone else's image!
** Invoke such an uploaded file as <nowiki>[[image:nameoffile.jpg|right|200px|Caption]] </nowiki>
**We support WikiComons, whereby images from the [http://commons.wikimedia.org/wiki/Main_Page content (of ~10 million files)] from [http://meta.wikimedia.org/wiki/Wikimedia_Commons Wikimedia Commons Library] can be referenced for your own document. If there is a name conflict, then the local version will be used before the Wiki Commons one.
***To find a file, go to [http://commons.wikimedia.org/wiki/Main_Page Commons]
***Find the file you want using the search facility
***Invoke the top menu, '''use this file in a Wiki''', and copy the string it gives you into your Wiki page
*** <nowiki>[[File:Armstrong Edward centric benzene.jpg|thumb|Armstrong Edward centric benzene]]</nowiki>
*Colour can be added (sparingly) using this {{fontcolor1|yellow|black|text fontcolor}} template. (invoked as <nowiki>{{fontcolor1|yellow|black|text fontcolor}}</nowiki> )
*Save and preview constantly (this makes a new version, which you can always revert to). It goes without saying that you should not reference this page from any other page, or indeed tell anyone else its name.
*'''Important:''' Every 1-2 hours, you might also want to make a [[Mod:writeup#Backing_up_your_report|backup of your report]]. This is particularly important when adding Jmol material, since any error in the pasted code can result in XML errors. The current Wiki version does not flag these errors properly, but instead just hangs the page. Whilst you can try to [[Mod:writeup#Fixing_broken__Pages|repair the page]] as described below, it is much safer to also have a backup!
*You should get into the habit of recording results, and appropriate discussion, soon after they are available, in the manner of a laboratory note book.

== More Editing features ==
=== Handling References/citations with a DOI ===
This section shows how literature citations<ref name="jp027596s">W. T. Klooster , T. F. Koetzle , P. E. M. Siegbahn , T. B. Richardson , and R. H. Crabtree, "Study of the N-H···H-B Dihydrogen Bond Including the Crystal Structure of BH<sub>3</sub>NH<sub>3</sub> by Neutron Diffraction", ''J. Am. Chem. Soc.'', '''1999''', ''121'', 6337–6343. {{DOI|10.1021/ja9825332}}</ref> can be added to text<ref name="dataset1"> Henry S. Rzepa, "Gaussian Job Archive for C4H6NO3S(1-)", 2013. {{DOI|10.6084/m9.figshare.679974}}</ref> using the '''<nowiki>{{DOI|value}}</nowiki>''' (digital object identifier) template to produce a nice effect. Citations can be easily added from the WikED toolbar.
*The following text is added to the wiki, exactly as shown here: '''<nowiki><ref name="ja9825332">W. T. Klooster , T. F. Koetzle , P. E. M. Siegbahn , T. B. Richardson , and R. H. Crabtree, "Study of the N-H···H-B Dihydrogen Bond Including the Crystal Structure of BH<sub>3</sub>NH<sub>3</sub> by Neutron Diffraction", ''J. Am. Chem. Soc.'', '''1999''', ''121'', 6337–6343.{{DOI|10.1021/ja9825332}}</ref></nowiki>'''
*Giving a reference a unique identifier, such as '''<nowiki><ref name="ja9825332"></nowiki>''' allows you to refer to the same footnote again by using a ref tag with the same name. The text inside the second tag doesn't matter, because the text already exists in the first reference. You can either copy the whole footnote, or you can use a terminated empty ref tag that looks like this: '''<nowiki><ref name="ja9825332" /></nowiki>'''.
*Collected citations will appear below wherever you place the '''<nowiki><references /></nowiki>''' tag, as here. If you forget to include this tag, the references will not appear!
==== Including the DOI for your experiment data ====
The datasets associated with your experiment can be given a DOI by '''publishing''' any entry in the [https://scanweb.cc.imperial.ac.uk/uportal2/ SCAN Portal]. You can include this DOI as a normal citation.<ref name="dataset1"> Henry S. Rzepa, "Gaussian Job Archive for C4H6NO3S(1-)", 2013. {{DOI|10.6084/m9.figshare.679974}}</ref>

==== Additional citation handling ====
* A macro-based reference formatting program has been developed in Microsoft Excel to not only produce the wiki code for direct pasting into your report, but that '''''also''''' formats text for placing in documents, such as synthesis lab reports. This program is available [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:Reference_Formatting_Program here].
* A '''[[Template:Cite_journal|Cite journal]] template''' is installed for anyone who wants to experiment
----

References and citations
<references />

----

=== Using an iPad ===

[https://itunes.apple.com/gb/app/wiki-edit/id391012741?mt=8 Wiki Edit] for IOS allows a Wiki to be edited using an iPad. You can dictate your text using '''Siri''' if your speed at '''thumb-typing''' is not what it should be.

= Bringing your report to life =
== Basic JSmol ==
You can use coordinate files created as part of your work (in CML or Molfile format) to insert rotating molecules for your page.
#Using your graphical program (ChemBio3D or Gaussview), '''save''' your molecule as an '''MDL File''', which has the extension '''.mol''', or as '''chemical markup language''', which has the extension '''.cml'''.
#Or, if your calculation ran on the SCAN batch system, '''publish''' the calculation, and in the resulting deposited space, download the .cml or the '''logfile.out''' file to be found there (the latter should be used for vibrations only).
#On the Wiki, '''Upload File''' (from the left hand panel) and select the molecule file you have just placed on your hard drive as above.
#On your Wiki page, insert <nowiki><jmolFile text="Explanatory text for link">BCl3-09.log</jmolFile></nowiki> where in this example, BCl3-09.log is the just uploaded file.
##The should produce <jmolFile text="this link">BCl3-09.log</jmolFile>. When clicked, it will open up a separate floating window for your molecule.
##Further actions upon the loaded molecule (such as selecting a vibrational mode and animating the vibration) are done by right-mouse clicking in the Jmol window.

#When using animations, please let them pop up in a separate window using the <jmolAppletButton> function. Your browser won't slow down and you will make your life so much simpler. =)
##Read more on how to do that [http://wiki.jmol.org/index.php/MediaWiki#Jmol_applet_in_a_popup_window here ]. --[[User:Rea12|Rea12]] 20:58, 8 September 2014 (BST)

== Advanced JSmol ==

A much more powerful invocation is as follows. The following allows a molecule to be directly embedded into the report, and it also shows how to put a script in to control the final display.
{| class="wikitable" border="2"
|-
! copy/paste either of the two sections below into your own Wiki
|-
|<pre><jmol>
<jmolApplet>
<title>Pentahelicene</title><color>white</color>
<size>150</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>yourmolecule.cml</uploadedFileContents>
</jmolApplet>
</jmol>


<jmol>
<jmolApplet>
<title>Vibration</title><color>white</color><size>200</size>
<script>frame 8;vectors 4;vectors scale 5.0;color vectors red;vibration 10;
</script><uploadedFileContents>BCl3-09.log</uploadedFileContents>
</jmolApplet>
</jmol></pre>
|-
! First molecule (if you see yellow below, then check the [[Mod:latebreak#Firefox.2C_Java_and_the_Wiki|late breaking news]])
|-
|<jmol>
<jmolApplet>
<title>Pentahelicene</title><color>white</color><size>300</size>
<script>select atomno=3,atomno=4,atomno=5; color purple;measure 3 5;measure 5 4;</script>
<uploadedFileContents>pentahelicene.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
! Second molecule (if you see yellow below, then check the [[Mod:latebreak#Firefox.2C_Java_and_the_Wiki|late breaking news]])

|-
|<jmol>
<jmolApplet>
<title>Vibration</title><color>white</color><size>300</size>
<script>frame 8;vectors 4;vectors scale 5.0;color vectors red;vibration 10;
</script>
<uploadedFileContents>BCl3-09.log</uploadedFileContents>
</jmolApplet>
</jmol>
|}
Every time you embed a molecule in a Wiki page in the above manner, the Web browser must set aside memory. Too many molecules, and the memory starts to run out, and the browser may slow down significantly. So use the feature sparingly, only including key examples where some structural feature would benefit from the rotational capabilities.
It is [http://chemapps.stolaf.edu/jmol/docs/ possible] to add many other commands to the JSmol container above. For example, '''<nowiki><script>select atomno=3,atomno=4,atomno=5; color purple;measure 3 5;measure 5 4;</script></nowiki>''' will colour atoms 3 4 and 5 (obtained by mouse-overs) purple, and then measure the length of the 3-5 bond. Further examples of how to invoke Jmol are [http://www.mediawiki.org/wiki/Extension:Jmol#Installing_Jmol_Extension found here], and a comprehensive list [http://chemapps.stolaf.edu/jmol/docs/ given here].

== Incorporating orbital/electrostatic potential isosurfaces ==
The procedure is as follows
# Run a Gaussian calculation on the SCAN
# When complete, select ''Formatted checkpoint file'' from the output files and download
# Double click on the file to load into Gaussview
##To generate a molecule orbital, '''Edit/MOs''' and select (= yellow) your required orbitals.
##* '''Visualise''' and '''Update''' to generate them
## To generate an electrostatic potential, '''Results/Surfaces and Contours''', then '''Cube Actions/New Cube/Type=ESP'''. This will take 2-3 minutes to generate
##* In '''Surfaces available''' pre-set the Density to '''0.02''' and then '''Surface Actions/New Surface'''. Try experimenting with the value of Density for the best result. Save the cube as per below.
# In '''Results/Surfaces and contours''' from the '''cubes available''' list, select one and '''Cube actions/save cube'''
# Invoke [http://www.ch.ic.ac.uk/rzepa/cub2jvxl/ this page] and you will be asked to select your cube file,
# followed by three file save dialogs, one for the coordinates (.xyz), one for the MO surface (.jvxl) and a shrink-wrapped bundle (.pngj).
# Insert the following code into your Wiki, replacing the file name with your own choice from the preceding file save dialogs.
<pre><nowiki><jmol>
<jmolApplet>
<title>Orbital</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/1/1b/AHB_mo22-2.cub.jvxl" translucent;</script>
<uploadedFileContents>AHB_mo22.xyz</uploadedFileContents>
</jmolApplet>
</jmol></nowiki>
</pre>
# [[image:absolute_path.jpg|right|350px]]Next, upload these two files into the Wiki (one file at a time, the multiple file uploader does not seem to work for this task)
## Now for the tough bit. You will need to find the absolute path for the .jvxl file. Above, this appears as images/1/1b/AHB_mo22-2.cub.jvxl
## Just after uploading a .jvxl file, you will see a response as shown on the right.
## Right click on '''Edit this file using an external application'''. You can used any text editor (Wordpad etc).
## This text file will contain something like: <br /><tt>; or go to the URL <nowiki>https://wiki.ch.ic.ac.uk/wiki/images/1/1b/AHB_mo22-2.cub.jvxl</nowiki></tt>
#Select just the string '''images/1/1b/AHB_mo22-2.cub.jvxl''' and paste it in as shown above:
#You should get something akin to:
<jmol>
<jmolApplet>
<title>Orbital</title><color>white</color><size>300</size>
<script>isosurface color orange purple "images/1/1b/AHB_mo22-2.cub.jvxl" translucent;
</script>
<uploadedFileContents>AHB_mo22.xyz</uploadedFileContents>
</jmolApplet>
</jmol>
* You can superimpose two surfaces. Change the script contents above to append a second surface to the first: <pre><script>isosurface color orange purple "images/4/42/AHB_mo22.jvxl" translucent;isosurface append color red blue "images/1/1b/AHB_mo22-2.cub.jvxl" translucent;</script></pre>
* The four colours used in this line can be changed to whatever you consider appropriate.
=== An alternative way of loading surfaces ===
This method avoids the need to specify paths to files as seen above. Instead uses the '''.pngj''' bundle which contains all necessary information and can be invoked by
<pre><nowiki><jmolFile text="just a link">CF3_mo30.cub.pngj</jmolFile></nowiki></pre> which produces <jmolFile text="just a link">CF3_mo30.cub.pngj</jmolFile>.
# It only supports one surface (you cannot superimpose two orbitals)
# You can also load other surfaces, such as <jmolFile text="molecular electrostatic potentials">Checkpoint_60018_esp.cub.pngj‎</jmolFile> generated from a cube of electrostatic potential (ESP) values created using Gaussview as follows:
## Download .fchk file from SCAN
## Open using Gaussview
## '''Results/Surfaces & contours/Cube actions/New Cube/ESP''' and then '''cube actions/save cube''' which is how the above was generated. You may have to play around with the value of the density (~0.02).

=== MOPAC orbitals ===
# Run MOPAC from ChemBio3D, selecting '''Compute Properties/Molecular Surface''' from the '''properties''' pane, and in the '''General pane''' specify a location for the output and deselect '''Kill temporary files''' if not already so.
# Upload the '''.mgf''' file so produced to the Wiki
# Invoke as follows ('''MO 35;''' means the 35th most stable orbital for that molecule).
<pre><nowiki><jmol>
<jmolApplet>
<title>MOPAC Orbital</title><color>white</color><size>300</size>
<script>MO 35;mo fill nomesh translucent;</script>
<uploadedFileContents>test.mgf</uploadedFileContents>
</jmolApplet>
</jmol></nowiki></pre>
<jmol>
<jmolApplet>
<title>MOPAC Orbital</title><color>white</color><size>200</size>
<script>MO 35;mo fill nomesh translucent;</script>
<uploadedFileContents>test.mgf</uploadedFileContents>
</jmolApplet>
</jmol>

== Enhancing your report with Equations ==
*You may have need to express some [http://meta.wikimedia.org/wiki/Help:Formula equations] on the Wiki. This is currently supported only using a notation derived from '''LaTeX''', and as with the Jmol insertion above, is enabled within a '''<nowiki><math>\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}</math></nowiki>''' field inserted using the default editor (the SQRT(n) button), and producing the following effect:
<math>\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}</math>

The requisite syntax can be produced by using
*MathType as an equation editor (used standalone or in Word). It places the required LaTeX onto the clipboard for pasting into the Wiki).
*[http://www.lyx.org/ Lyx] which is a free stand-alone editor.
*A more general solution to this problem is simply to create a graphical image of your equation, and insert that instead as a picture.

== Inserting Tables ==
Instead of inserting screenshots of Excel, tables can be produced using MediaWiki markup (see [http://www.mediawiki.org/wiki/Help:Tables this page]), where you can also find lots of examples of different styles of table. However, this can be quite time consuming when you have a lot of tabulated data and need to copy it from somewhere like Excel into ChemWiki. Instead of typing it by hand, you can save your Excel worksheet as a comma separated file (.csv) and then use this [[http://area23.brightbyte.de/csv2wp.php CSV to MediaWiki markup]] convertor. Note that cells starting with "-", e.g. for negative numbers, need a space inserted between the - and | in the output otherwise MediaWiki interprets it as a new row.

Another web-based utility is available called [http://excel2wiki.net/ Excel2Wiki] which can be used to generate MediaWiki code from an Excel table.

== SVG (for display of IR/NMR/Chiroptical Spectra) ==
[[File:IR.svg|300px|right|SVG]]SVG stands for '''scaleable-vector-graphics'''. Its advantage is well, that it scales properly (but it has many others, including the ability to make simple edit to captions etc using Wordpad or similar). From your point of view, it is readily generated using Gaussview. If you view an '''IR/NMR/UV-vis/IRC/Scan/''' spectrum in this program, it allows you to export the spectrum as SVG (right-mouse-click on the spectrum to pull down the required menu). Upload this file, and invoke it as <nowiki>[[File:IR.svg|200px|right|SVG]]</nowiki> If you open eg IR.svg in Wordpad (or other text editor), you can edit the captions, font sizes etc (its fairly obvious). Oh, you will need to use a web browser that actually displays SVG. Internet Explorer 8 does not (9 is supposed to). Use Firefox/Chrome/Safari etc.

== Chemical Templates ==
An example is the [[w:Template:Chembox|ChemBox]]. Volunteers needed to test/extend these!

= Backing up your report =
[[Image:export1.jpg|left|250px]][[Image:Export2.jpg|right|200px]]Invoke [[Special:Export|this utility]] to back your project up. In the box provided, enter e.g. '''Mod:wzyz1234''' being the password for your report. This will generate a page (right) which can be saved using the Firefox '''File/Save_Page_as''' menu. Specify '''Web Page, XML only''' as the format, and add .xml to the file suffix. You might want to do this eg on a daily basis to secure against corruption. This is in addition to the notes for how to repair broken pages.

The same file can now be reloaded using [[Special:Import|Import]].

= Fixing broken Pages =
There are several ways in which a page can break.

*We have had instances of people inserting a corrupted version of the Jmol lines into their project, resulting in a '''XML error''' or '''Database error'''. Recovering from such an error is not simple. So we do ask that you carefully check what you are pasting into the Wiki, and that its form is exactly as shown above. For example, below is a real example of inducing such an error. Can you see where the fault lies? (Answer: the <nowiki><jmol></nowiki> tag is not matched by <nowiki></jmol></nowiki>. If tags are not balanced, XML errors will occur).
<pre><jmol><jmolApplet><title>equatorial</title><color>white</color>
<size>200</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Pl506_14_equatorial.mol</uploadedFileContents></pre>
*Another example might be wish to indicate a citation using <nowiki><ref>...details </ref></nowiki> but in fact end up entering <nowiki><ref> ...</nowiki> (i.e. missing out the <nowiki></ref></nowiki>)
*If you do encounter such an error, try invoking your project as '''<nowiki>https://www.ch.ic.ac.uk/wiki/index.php?title=Mod:wzyz1234&action=history</nowiki>''' and edit and then save an uncorrupted version. You will need to be already logged in before you attempt to view the history in this way, since logging in '''after''' you invoke the above will return you not to the history, but to the corrupted page (Hint: it sometimes helps to check '''Remember my login on this computer''' as you log in). For example, the history for this page can be seen [https://www.ch.imperial.ac.uk/wiki/index.php?title=Mod:writeup&action=history here]. You can eg load this preceeding page, and then use it to replace '''writeup''' with your own project address.
*If the preceeding does not work try the instructions shown [[mod:fix|here]].

= Feedback =
Each Wiki page has a discussion section, including your submitted report page. 

See also: [[Mod:Cheatsheet|cheatsheet]]

Mod:support

2025-09-17T13:41:05Z

Jbettenc: Jbettenc moved page Rep:Mod:support to Mod:support without leaving a redirect: Revert: this was not a report

= Remote Support =

There are various ways of providing support to someone else remotely. This is one method: https://appleinsider.com/articles/21/01/10/how-to-provide-remote-mac-tech-support-using-google-chrome-remote-desktop

Mod:HPC-add

2025-09-17T13:40:16Z

Jbettenc: Jbettenc moved page Rep:Mod:HPC-add to Mod:HPC-add without leaving a redirect: Revert: this was not a report

= Breaking news. The configuration mechanism has changed =
Whilst we wait for new instructions, please do not attempt the below. Currently, it will not work.

== Configuring the Imperial College HPC Resource to run Gaussian Calculations using an ELN ==

Main Page

2025-09-17T13:37:10Z

Jbettenc: Move non-active experiments to "Material from previous years" section

[[Image:QR_complab.png|right|QR]]
This is a communal area for documenting teaching and laboratory courses. To [[admin:add|add to any content on these pages]], you will have to log in using your Imperial College account. This page has {{DOI|dh4g}}
== Remote Working ==
#[[Mod:Rem|Remote access to College computers]]
#[[Mod:HPC-add|Remote use of Gaussian]]
#[[Mod:support|Remote support]]

== ORCID Identifiers and Research Data Publication ==
#[[orcid|Getting an ORCID identifier]] (two stages)
#[[rdm:synthesis-lab|Publishing NMR instrument data]] (three stages)
#[[rdm:xray-data|Publishing crystal structure instrument data]]

= Laboratories and Workshops =
== First Year ==
=== [[it:it_facillities|Email and IT@www.ch.imperial.ac.uk]]: A summary of available IT resources ===

===First Year Chemical Information Lab 2015 ===
*[[it:intro-2011|Introduction]]
*[[it:lectures-2011|Lectures]]
*[[it:coursework-2011|Coursework]]
*[[it:assignment-2011|Assignment for the course]]
*[[it:software-2011|List of software for CIT]]
*[[it:searches-2011|Search facilities for CIT]]
*[[Measurement_Science_Lab:_Introduction|Measurement Science Lab Course]]
===[[organic:conventions|Conventions in organic chemistry]]===
===[[organic:arrow|Reactive Intermediates in organic chemistry]]===
===[[organic:stereo|Stereochemical models]] ===
== Second Year ==
=== Computational Chemistry Lab ===
*Electronic States and Bonding - computational laboratory (see blackboard).

== Third Year ==


=== Computational Chemistry Lab ===
* [[Mod:ts_exercise|Transition states and reactivity.]]
* [[ThirdYearMgOExpt-1415|MgO thermal expansion]]
* [[Third_year_simulation_experiment|Simulation of a simple liquid]]
* [[Programming_a_2D_Ising_Model|Programming a 2D Ising Model (CMP only)]]

= Material from previous years =
== First year ==
===Introduction to Chemical Programming Workshop 2013===
*[[1da-workshops-2013-14|Workshop script]]
== Second year ==

*[http://www.huntresearchgroup.org.uk/teaching/year2a_lab_start.html Inorganic Computational Chemistry Computational Laboratory]
*[[CP3MD| Molecular Reaction Dynamics]]

=== Modelling Workshop ===
*[[Coursework]]
*[[Second Year Modelling Workshop|Instructions]] and [[mod:further_coursework|Further optional coursework]]
*[[it:conquest|Conquest searches]]



== Third Year ==

*[[TrendsCatalyticActivity|Understanding trends in catalytic activity for hydrogen evolution]]
*[[mod:inorganic|Bonding and molecular orbitals in main group compounds]]

===Synthesis and computational lab ===
*[[Mod:organic|Synthesis and computational lab]]


*[[mod:intro|Information needed for the course]]


*[[mod:writeup|Report writing and submission]]

*[[mod:latebreak|Late breaking news]]

*[[mod:programs|General program instructions:]]
**[[mod:gaussview|The Gaussview/Gaussian suite]]
**[[Mod:scan|Submitting jobs to the chemistry high-performance-computing resource]]

== ChemDraw/Chemdoodle Hints ==
#[[IT:chemdraw|Useful hints for using ChemDraw/ChemDoodle]]

== Tablet Project ==
# [[tablet:tablet|Tablet Pilot Project]]
== 3D ==
# [[mod:3D|3D-printable models]]
# [[mod:stereo|Lecture Theatre Stereo]]
== Online materials for mobile devices ==
# [[ebooks:howto|How to get eBooks]]
# [https://play.google.com/store/apps/details?id=com.blackboard.android&hl=en Blackboard mobile learn for Android]
# [https://itunes.apple.com/us/app/blackboard-mobile-learn/id376413870?mt=8 Blackboard mobile learn for iOS]
# [https://itunes.apple.com/gb/course/id562191620 Pericylic reactions in iTunesU ] (download [https://itunes.apple.com/gb/app/itunes-u/id490217893?mt=8 App] first)
# [https://itunes.apple.com/gb/course/id562191825 Conformational analysis in iTunesU] (download [https://itunes.apple.com/gb/app/itunes-u/id490217893?mt=8 App] first)
# [https://itunes.apple.com/gb/course/id562191342 A library of mechanistic animations in iTunesU] (download [https://itunes.apple.com/gb/app/itunes-u/id490217893?mt=8 App] first)
# [[IT:panopto|How to compress and disseminate Panopto lecture recordings]]

= PG =

# [[rdm:intro|Data management]]
= DOI =
*[[template:doi]] This page has {{DOI|dh4g}}

= Accessibility on this site =

* The Department of Chemistry wants as many people as possible to be able to use this website. The site hopes to maintain WCAG 2.1 AA standards, but it is not always possible for all our content to be accessible.

=== Technical information about this website’s accessibility ===

* Imperial College London is committed to making its website accessible in accordance with the Public Sector Bodies (Websites and Mobile Applications) (No. 2) Accessibility Regulations 2018.

* This website is compliant with the Web Content Accessibility Guidelines version 2.1 AA standard.

=== Reporting accessibility issues ===

* If you need information on this website in a different format or if you have any issues accessing the content then please contact [gmallia at imperial.ac.uk]. I will reply as soon as possible.

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If you’re not happy with how we respond to your complaint, contact the Equality Advisory and Support Service (EASS).

=== Last updated ===

This statement was prepared in September 2020 (rechecked in May 2021).

Rep talk:FabianThomasThirdYearLiquidFlowSimulation

2025-09-17T13:27:55Z

Jbettenc: Jbettenc moved page Talk:MRD:FabianThomasThirdYearLiquidFlowSimulation to Rep talk:FabianThomasThirdYearLiquidFlowSimulation without leaving a redirect: Move to report namespace

Overall: Good attempt at the lab although the VACF was not attempted. Be careful not to become too colloquial in your writing and read through to check for grammar. Your style is pretty good apart from the grammar and sometimes vague statements - keep it up and read more before your MSci proj, it helps :^) Mark: 68/100

== Abstract ==

<span style="color:red">Not a bad attempt at writing an abstract - hardware, unless it's revolutionary, doesn't need a mention aka HPC.</span>

Lennard-Jones simulations of solids, liquids, and gases were run through LAMMPS on the imperial HPC service. How number density varied against temperature and at different pressures, radial distribution functions of L-J solids, liquids, and gases, Mean Squared Displacements, Diffusion coefficients, and the variation of heat capacity with temperature were investigated in order to understand more about phase properties, and specifically those that can be modelled using a L-J simulation. <ref> Jean-Pierre Hansen and Loup Verlet, Phys. Rev. 184, 151</ref> For the equation of state, number density increased with pressure, but was inversely proportional to an increasing temperature. Heat capacity was determined to be inversely proportional to temperature due to a change in density of states for a fixed NVT ensemble. The radial distribution function allowed for analysis of solid, liquid, and gas simulations of 3375 atoms and allowed for the probing of the structure of each phase, along with details about the co-ordination spheres and numbers of nearest neighbors. Diffusion coefficients were found using the Mean Squared Displacements <span style="color:red">gram not capitalised</span> for solids, liquids and gases, along with simulations with one million atoms for each phase as well, with values for <math>D</math> of<math>-1 \times 10^{-17}</math>,<math>1.1 \times 10^{-10}</math> and <math> 3.61 \times 10^{-9} \ m^2 s^{-1}</math> for the solid liquid and gas phases. <span style="color:red">tick</span>

== Introduction ==

<span style="color:red">Very good introduction Fabian - hit the nail on the head :^)</span>
Phase changes occur every day in daily life, and are common place in the world around us. From boiling water to be used in cooking or in steam generators to produce electricity, to instant heat packs, where sodium acetate is super cooled, and upon perturbation instantly raises the pack’s temperature significantly to its freezing point at 54 degrees. These are all every day examples of how phase changes are used and how research into phase changes can provide new avenues for products and areas of research such as new instant-heat pack materials, all based on previous phase change research.

Phase change materials; not unlike sodium acetate, is a large area of research today. Materials capable of storing large amounts of energy through phase changes such as the solid to liquid transition or solid to solid transitions, <ref>Belén Zalba, José M Marı́n, Luisa F. Cabeza, Harald Mehling, Review on thermal energy storage with phase change: materials, heat transfer analysis and applications, In Applied Thermal Engineering, Volume 23, Issue 3, 2003, Pages 251-283</ref> with applications such as improved solar cell energy storage<ref>Murat Kenisarin, Khamid Mahkamov, Solar energy storage using phase change materials, In Renewable and Sustainable Energy Reviews, Volume 11, Issue 9, 2007, Pages 1913-1965</ref>, and materials able to change state for useful purposes in textiles<ref>S. Mondal, Phase change materials for smart textiles – An overview, In Applied Thermal Engineering, Volume 28, Issues 11–12, 2008, Pages 1536-1550</ref> these simulations carried out can further progress phase-change transition research, and can be used to go on to model more complex systems for use with this research.

Modelling theses systems using L-J simulations provides a fast and easy method to compute the phase change temperatures, order, and structure of the phases transitioned through. Thus simplifying the need to synthesise the molecule beforehand to view the properties of the system. Some of the systems with the largest heat of fusion are systems such as Lithium or water, which are very easily simulated through L-J models and provide an excellent way to probe the phase diagrams of these systems. The ease of determining the crystal structure of solids, is very largely beneficial for the phase change materials, as it allows for easy categorisation of the solid phases, such as ice which has over ten different solid crystalline phases depending on the pressure and temperature of the system, which is modelled accurately and quickly using an L-J simulation through use of radial distribution functions. Heat capacity measurements of phases are of a large interest in phase-change materials as materials with large heat capacities coupled with a large latent heat of fusion allows for the storage of very large amount of energy in a small amount of solid. But all these simulations lead to further questions, how far can L-J simulations go in helping to discover phase change materials, and how accurately can the simulations predict real heat capacities, structures and transitions of unmade molecules?

== Methods ==

<span style="color:red">A good computational methods section should be structured model, params, forcefield, params. Then describe what you did with this model. Not bad though. </span>

All simulations were run as Lennard-Jones systems through the Imperial College HPC facility <ref>https://www.imperial.ac.uk/admin-services/ict/self-service/research-support/hpc/</ref> using the LAMMPS runtime option, with all graphs and respective data produced and analysed using Origin Pro <ref>http://www.originlab.com/Origin</ref>.
Equation of State simulations were run using an NpT ensemble. Using dimensionless L-J reduced units <ref>http://cbio.bmt.tue.nl/pumma/index.php/Manual/ReducedUnits</ref>, pressures <math>p^*</math> of 1.8 and 2.6 were used, along with temperatures above the critical temperature <math>T^*</math> = 1.5 of 2, 2.5, 3, 3.5 and 4. A timestep of 0.0025 was used, as it was found in the earlier stages of the experiment to be as accurate as a timestep of 0.001, but would run the simulation 250% faster in comparison. As it was a liquid that was to be simulated, a simple cubic lattice was used for the simulation.

Heat Capacity simulations were run at <math>\rho^*</math> = 0.2 and 0.8, and <math>T^*</math> = 2.0, 2.2, 2.4, 2.6, 2.8 in L-J reduced units. A simple cubic crystal was melted under microcanonical NVE conditions. As heat capacity, shown below, is directly proportional to the variance of energy of the system if temperature is held constant, after melting, the system was put under an NVT ensemble to hold the temperature constant but allow the energy to vary.

<center><math>C_V = \frac{\partial E}{\partial T} = \frac{\mathrm{Var}\left[E\right]}{k_B T^2} = N^2\frac{\left\langle E^2\right\rangle - \left\langle E\right\rangle^2}{k_B T^2}</math></center>

Where <math>C_V</math> is Heat Capacity, <math>\mathrm{Var}\left[E\right]</math> is the variance in Energy, <math>k_b</math> is Boltzmann's constant, <math>\left\langle E^2\right\rangle</math> is mean squared energy, <math>\left\langle E\right\rangle^2</math> is mean energy squared, and <math>N^2</math> is the number of atoms in the simulation squared.

A factor of <math>N^2</math> is added to the equation, as LAMMPS divides all energies output by the number of particles in the system, therefor giving <math>\frac{E}{N}</math> for any value of <math>E</math> measured. Using this equation, the heat capacity was found and <math>\frac {C_V}{V}</math> was plotted against <math>T^*</math> to view how the heat capacity varied with respect to temperature.

Using VMD software <ref>http://www.ks.uiuc.edu/Research/vmd/</ref>, the radial distributions functions for the solid, liquid, and vapour phases of a L-J fluid were calculated, setting selection 1, and 2 to ‘all’, and delta r to 0.05. Using these plots the co-ordination sphere radii and numbers of the solid, liquid, and vapour phases were determined.

Using the equation <math>D = \frac{1}{6}\frac{\partial\left\langle r^2\left(t\right)\right\rangle}{\partial t}</math> and the relationship between D and Mean-Squared Displacement <math>\langle (x(t)-x_0)^2 \rangle = 2Dt</math>, <math>D</math> was calculated for solid, liquid, and vapour phases by extracting the gradient from the plot of Mean-Squared Displacement against time, giving a gradient of <math>2D</math>.

== Results and Discussion ==

=== Equation of State ===

[[File:Ft1015_Standard_Error_plots_graph.PNG|480px|thumb|left|'''Figure 1''': A plot of average density against average critical temperature of the system with standard error error bars]]

From figure 1, we can see that number density <math> \frac{N}{V}</math> is inversely proportional to <math>T</math>. This can be rationalised by looking at the results of the L-J reduced gas law <math> \frac{N}{V} = \frac{p}{T}</math>

The discrepancies between the idealised and simulated number density results can be attributed to the simulation being non-ideal. The basis of a L-J system is that the particles interact with each other, with a short range repulsive term, and a longer range attractive term. The ideal gas law however, assumes that there are no interactions between particles and thus, due to the short range highly repulsive term, the particles are further apart than in a system with no interactions and therefore the number density of the simulated systems are lower than their idealised counterparts.<span style="color:red">tick</span>

As the temperature is increased, the kinetic energy of the particles increases leading to an increase in the average distance between particles in order to reduce repulsions between particles to a minimum. This can be seen in Charles’ Lawː
<center><math> \frac{N_1}{T_1} = \frac{N_2}{V_2} </math></center>
As the temperature is increased to <math>T_2</math>, <math>V_2</math> must increase, thus leading to a decrease in number density. As the particles are further away from each other on average, the particles interact with one another less, and so become more ‘ideal’, reducing the difference between the simulated result and the gas law result.<span style="color:red">tick</span>

This result is the opposite for lower temperatures. As the temperature is reduced, so is the kinetic energy, and therefore the particles are less able to overcome the attractive and repulsive forces between them. This leads to larger interactions between the particles due to their increased proximity to each other, and so simulated results deviate from the gas law more at lower temperatures. From Charles’ Law again, we can see that the volume of the cell will decrease as temperature decreases, and so number density will increase at lower temperatures.<span style="color:red">tick</span>

From the graphs, we can see that, at higher pressures the deviations from ideality are larger than that of a lower pressure simulation. This arises from the fact that at higher pressures, the number density is higher due to decreased cell volume, leading to decreased distances between particles and therefore increased interactions between them as they collide with each other more frequently. This increase in interactions between particles leads to a large deviation from ideality and so a larger difference between simulated and ideal law data for higher pressures. <span style="color:red">tick, what about the variation of density with temperature?</span>

=== Heat Capacity ===

[[File:ft1015HeatCapacity.png|480px|thumb|left|'''Figure 2''': A plot of <math> {C_{V}}/{V} </math> against <math> {T*} </math> for two different densities]]
[[File:Ft1015HeatCapacityScript.PNG|250px|thumb|right|'''Figure 3''': An image of a sample of the script used for one of the heat capacity calculations]]

Heat capacity defined in the equation below, varies directly with the variance in energy of a system held at constant temperature, but is essentially a measure of how easily a substance can be heated up. Thus, a substance with a larger heat capacity will require more energy to heat up than one with a lower heat capacity.<span style="color:red">tick</span>

<center><math>C_V = \frac{\partial E}{\partial T} = \frac{\mathrm{Var}\left[E\right]}{k_B T^2} = N^2\frac{\left\langle E^2\right\rangle - \left\langle E\right\rangle^2}{k_B T^2}</math></center>

The odd relationship between <math>\frac{C_V}{V}</math> and temperature <math> T^*</math> can be described through use of the equipartition theorem and the Boltzmann Distribution <span style="color:red">in part!</span>

The large jumps in heat capacity seen in both data sets at <math>T^* = 2.6</math> can be explained through phase changes. As the particles simulated are totally symmetric there are no extra degrees of freedom to be reached through an increase in temperature as only translational modes are active through equipartition theorem, therefore the jump in heat capacity must be due to a phase change. <span style="color:red">tick</span> Assuming a second order phase transition; as an NVT ensemble is used, and in a first order transition density is discontinuous which is impossible for a system with fixed <math>N</math> and <math>V</math>, Heat capacity rises to infinity until after the transition has been completed, after which it lowers again, which can be seen at <math>T^* = 2.8</math>, due to thermal energy needing to be expended in order to complete the phase transition. <span style="color:red">tick, ok good</span>

The decrease of heat capacity with temperature increase excluding phase changes, can be explained through the Density of States. Similar to the anharmonic oscillator model for electron excitation for an atom or particles where as the energy levels are increased, the spacing between the levels decrease, as the temperature of our system is increased the spacing between energy levels contracts, and thus the higher energy levels come down in energy and become more accessible. This leads to more modes available for promotion in the system, thus ease of promotion of an atom increases, leading to a reduced energy needed to promote and atom and so heat capacity decreases with temperature. <span style="color:red">tick, good</span>

This can be viewed form a different angle with the Boltzmann Distributionː

<center><math>\frac{N_i} N = \frac{\exp(-E_i/kT) } { \sum_j \exp(-E_j/kT) }</math></center>

For a specified energy gap, the higher in energy the energy state is, the less likely it is to be populated. If the energy gap is decreased, higher energy states are more likely to be populated, and are therefore more easily reached, and thus the ease of promotion is increased for a decrease in energy gap between states and so heat capacity falls.<span style="color:red">tick</span>

=== Structural Properties of the Radial Distribution Function ===
[[File:ft1015G(r).png|400px|thumb|right|'''Figure 4''': ft1015G(r).png]]

[[File:ft1015G(r)N.png|400px|thumb|left|'''Figure 4''': ft1015G(r)N.png]]
Using VMD <ref>http://www.ks.uiuc.edu/Research/vmd/</ref> the RDFs for a solid, liquid, and gas were found and investigated.

The Radial Distribution Function, RDF, describes for a central particle how density varies as a function of distance <math>r</math>. The RDF can be used to find the co-ordination spheres of solids, liquids and gases, along with the distance each sphere lies from the central particle.

Integrating the RDF gives the number density RDF, which allows users to find out how many particles lie within each co-ordination sphere, and when used in conjunction to the normal RDF, the structure of that phase.<span style="color:red">tick</span>

Looking first at the RDF for the solid, the large peaks relate to particles being found at that distance from the central atom. These peaks relate to the co-ordination spheres around the atom in the solid phase. The peaks are sharper than the other phases due to the crystal structure of the solid phase and that the particles are fixed in place inside the lattice. Because of the crystal structure and it’s short- and long-range order, the peaks are sharp due to distinct spheres of co-ordination found at each distance. Looking at the first three peaks, these relate to the first three co-ordination spheres or nearest neighbours of particles around the central atom. The troughs in the RDF relate to the empty space between the co-ordination spheres and lack of particles between the spheres. Looking at the first three peaks in the RDF and number integrated RDF, we can say that the first three co-ordination spheres contain 12, 6, and 24 atoms respectively. <span style="color:red">tick</span>

<center>
{| class="wikitable"
|-
! r
! Integrated g(r)
! Co-ordination Number
|-
| 1.175
| 11.85813
| 12
|-
| 1.675
| 18.44288
| 6
|-
| 1.975
| 42.26256
| 24
|}
</center>

The lattice spacing is the distance between repeat lattice cells and is present in systems with repeating units, and therefore repeating lattice cells. For the FCC lattice in the solid case, the lattice spacing was 1.175 in L-J reduced units. <span style="color:red">tick</span>

The liquid phase in comparison to the solid phase has a highly reduced order due to the lack of long range order and rigidity in the liquid phase, but more order than the gas phase as one can see in the increased amount of peaks that the liquid phase has in comparison to the gas phase which shows more co-ordination spheres in the liquid phase. As liquid particles are not held tightly in place, but can vibrate and move, the co-ordination peaks are broad due to the fact that the particles are slightly spread out across the co-ordination shell as they are bound, but not tightly. This peak occurs around <math>r = 1</math>, with further co-ordination shells smaller than the closer shells as the short range order of a liquid breaks down at longer range. The troughs, similarly to the solid are due to no particles being in-between the shells and their atomic neighbours.<span style="color:red">tick</span>

The gas phase has the least order out of the three phases examined. This is easily seen through there being <span style="color:red">phraseology like this is a bit colloquial</span> only one peak on the RDF in comparison to the other phases. This peak relates to the very weak attractive forces between gas particles, and their nearest neighbours, effectively leading to one singular diffuse co-ordination sphere, which can be seen by the broadness of the peak. After the first coordination, there is no order relating to the central particle, and so the RDF falls to one after the peak, showing an equal likelihood of finding a gas particle at any distance after the co-ordination sphere, which further emphasises the lack of order in the gas phase.<span style="color:red">tick</span>

For all phases the RDF is zero until the first co-ordination sphere or right before. This is due to the width of the atoms in the simulation and how the atoms cannot overlap due to the strong repulsive forces. This leads the first particles to be detected by the function at the first co-ordination sphere.<span style="color:red">tick</span>

===Dynamical Properties and the Diffusion Coefficient===

[[File:ft1015GasMSD.png|400px|thumb|left|'''Figure 4''': A plot of MSD against time for the gas phase]]
[[File:ft1015GasMSD_1m.png|400px|thumb|right|'''Figure 5''': A plot of MSD against time for one million atoms in the gas phase]]
[[File:ft1015LiquidMSD.png|400px|thumb|left|'''Figure 6''': A plot of MSD against time for the liquid phase]]
[[File:ft1015LiquidMSD_1m.png|400px|thumb|right|'''Figure 7''': A plot of MSD against time for one million atoms in the liquid phase]]
[[File:ft1015SolidMSD.png|400px|thumb|left|'''Figure 8''': A plot of MSD against time for the solid phase]]
[[File:ft1015SolidMSD_1m.png|400px|thumb|right|'''Figure 9''': A plot of MSD against time for one million atoms in the solid phase]]

Using Mean Squared Displacement, MSD, in an NVT ensemble, one can determine the diffusion coefficient <math>D</math>. MSD is with <span style="color:red">? bit unclear</span>, respective to a reference position, the deviation of the position of a particle from said point over time, and can be measured through single particle tracking via photon scattering <ref>Phys. Chem. Chem. Phys.,2013, 15, 845</ref>.

The diffusion coefficient <math>D = \frac{1}{6}\frac{\partial\left\langle r^2\left(t\right)\right\rangle}{\partial t}</math><math>D</math> is directly proportional to MSD as shown below in the equation:

<center><math> MSD = \langle (x(t)-x_0)^2 \rangle = 2Dt </math></center>

Plotting MSD against time, gives a graph with a gradient of <math>2D</math> and therefore a diffusion coefficient equal to <math>\frac{gradient}{2}</math>. These plots can be seen in figures 4-9 for solid, liquid and gas phases, along with plots of data given to us of those phases with one million atoms. If data points were not used in the linear fit of a graph as they did not follow the linearity expected for the gradient were masked and are shown in red on the plot.

<center>
{| class="wikitable"
|+ style="text-align: centre;" | Diffusion Coefficient
|-
! Phase
! Simulated plots / m^2 s^-1
! one million atoms plots / m^2 s^-1
|-
| Gas
| 1.17E-10
| 3.61E-09
|-
| Liquid
| 1.03E-10
|1.1E-10
|-
| Solid
| -1.4E-17
| -1E-17
|}
</center>

Looking at each phase individually and the two plots associated with them. <span style="color:red">what? Needs to ensure you maintain the flow and importantly the subject and object of sentences!</span> Starting with the gas phase we can see that for our simulated data diffusive rates are reached immediately and stay constant throughout the simulation as there is little to no order in gas phases which was established by use of the RDF function earlier. However, for the one million atom simulation, the rate of diffusion is not initially constant but rises as the simulation progresses until system reaches equilibrium and <math>D</math> stabilizes. Thus for the one million atoms plot data points outside the linear region were excluded, as shown in the graph.<span style="color:red">tick</span>

Rates of diffusion for liquids were lower than that of gases, which is to be expected due to the short range order shown in the RDF functions, thus diffuse more slowly to keep the short-range order intact.<span style="color:red">tick</span>

Values of <math>D</math> for the solid were extremely small to the point of there being no diffusion occurring in solids. This can be explained by the highly ordered structure which only allows particles to vibrate about a fixed point, this is confirmed through RDF calculations above, with the large change in diffusion coefficients are the start of each simulation due to the system reaching equilibrium. Diffusion will only be able to occur in from a solid phase if a phase change occurred and thus diffusion would occur through a different phase.<span style="color:red">tick</span>

A change in value of <math>D</math> for the larger one million atoms in comparison to our own simulations was expected as these larger simulations are likely to be more accurate and precise due to the larger amount of averaging that is able to occur, and the larger amount of atoms lowers microscopic effects as the system move into the macro scale.<span style="color:red">tick</span>

== Conclusion ==

<span style="color:red">Be careful write correct conclusions. Also go through your writing careful - there are a few grammar mistakes throughout and never use the active tense!</span>

Through the successful modelling of a L-J system above the critical temperature, the equation of state was found at two different pressures. Number density was found to increase with pressure due to the increased repulsions between particles because of the increased number of collisions, while also found to be inversely proportional to temperature, as at higher temperatures the particles moved further apart, thus lowering the interactions between them and so lowering the number density, this was reinforced using Charles’ law. Deviations from an ideal gas were seen as the L-J system allows for attractive and repulsive forces between the particles, but at higher pressures the difference from ideality was larger due to the increased density of the particles leading to more interactions between them, thus straying from the ideal gas postulates.

It was found that Heat capacity was inversely proportional to temperature <span style="color:red">not true!</span> due to the change in density of states associated with an increase in temperature. The increasing of the density of states, and thus the lowering of energy of the higher energy levels allowed for more available energy states for promotion to, and so easier promotions of particles, thus leading to a lowered heat capacity due to the increased ease of promoting a particle to a higher energy state. A peak in the heat capacity seen at <math>T^* = 2.6 </math> was seen due to a change in phase with the heat capacity reaching infinity until the necessary amount of thermal energy had been input to the system.

The radial and integrated radial distribution functions were successfully used to model the co-ordination spheres of a solid, liquid, and gas, along with numbers for the nearest neighbours, and information on the order and structure of each phase. The gaseous phase was found to have no long range order, and only a single co-ordination shell thus leading to extremely short range order, with a broad peak suggesting a diffuse shell. The liquid phase was found to be more ordered than the gaseous phase due to the increased number of three co-ordination shells, leading to short range, but similarly to the gas no long range order in the phase. The peaks were also broadened, suggesting slightly diffusive co-ordination shells. Solids were found to be highly ordered in structure, with long range and short range order due to its crystal structure. Peaks were sharp due to the set spacings between particles, and the fact that the particles can only vibrate around their fixed positions, not move like in a liquid or gas. The co-ordination of the solid was found to be 12, 4, and 24 in the first three co-ordination shells respectively, along with a lattice spacing of 1.175.

Diffusion coefficients for solids, liquids, and gases were also found using the relationship between MSD and <math>D</math>. Solids were found to have an almost zero diffusion gradient, while liquids and gases were much higher. Data for 1000000 atoms simulations for each phase were received and plotted, leading to more accurate results for <math>D</math>, which agreed with the values of <math>D</math> received from the simulations of 3375 atoms completed by us.<span style="color:red">maintain passive voice..</span>

== References ==
<references/>

== Introduction to molecular dynamics simulation ==

=== Numerical Integration ===
'''<big>TASK</big>: Open the file HO.xls. In it, the velocity-Verlet algorithm is used to model the behaviour of a classical harmonic oscillator. Complete the three columns "ANALYTICAL", "ERROR", and "ENERGY": "ANALYTICAL" should contain the value of the classical solution for the position at time <math>t</math>, "ERROR" should contain the ''absolute'' difference between "ANALYTICAL" and the velocity-Verlet solution (i.e. ERROR should always be positive -- make sure you leave the half step rows blank!), and "ENERGY" should contain the total energy of the oscillator for the velocity-Verlet solution. Remember that the position of a classical harmonic oscillator is given by <math> x\left(t\right) = A\cos\left(\omega t + \phi\right)</math> (the values of <math>A</math>, <math>\omega</math>, and <math>\phi</math> are worked out for you in the sheet).'''

'''<big>TASK</big>: For the default timestep value, 0.1, estimate the positions of the maxima in the ERROR column as a function of time. Make a plot showing these values as a function of time, and fit an appropriate function to the data.'''

'''<big>TASK</big>: Experiment with different values of the timestep. What sort of a timestep do you need to use to ensure that the total energy does not change by more than 1% over the course of your "simulation"? Why do you think it is important to monitor the total energy of a physical system when modelling its behaviour numerically? '''

[[File:ft1015Analytical.png|480px|thumb|center|]]
[[File:ft1015Error.png|480px|thumb|center|]]
[[File:ft1015Energy5.png|480px|thumb|center|]]

The total energy of a classical harmonic system is given by <math>\frac{1}{2}mv^{2}</math> and <math>\frac{1}{2}kx^{2}</math> for the kinetic and potential energies respectively. Combing these give the total energy of the system which should be a constant for an isolated system as no energy enters or leaves the system. A time-step value of 0.2 s gives a maximum difference in energy of 1% of the total starting energy, and so ensures a good approximation to realistic behavior. The Velocity-Verlet algorithm will have fluctuations in the total energy of the system as it is only an approximation through a Taylor-series expansion, however, minimizing these fluctuations provides a more accurate simulation and therefore for a physical system would provide more accurate thermodynamic properties of the system. Decreasing the time-step, and so decreasing the energy fluctuations will provide more accurate results at the cost of increased simulation time needed for the same time frame of a system, and so is a trade-off between time and accuracy for the simulation.<span style="color:red">tick</span>

===Atomic Forces===
'''For a single Lennard-Jones interaction, <math>\phi\left(r\right) = 4\epsilon \left( \frac{\sigma^{12}}{r^{12}} - \frac{\sigma^6}{r^6} \right)</math>, find the separation, <math>r_0</math>, at which the potential energy is zero. What is the force at this separation? Find the equilibrium separation, <math>r_{eq}</math>, and work out the well depth <math>(\phi\left(r_{eq}\right))</math>. Evaluate the integrals <math>\int_{2\sigma}^\infty \phi\left(r\right)\mathrm{d}r</math>, <math>\int_{2.5\sigma}^\infty \phi\left(r\right)\mathrm{d}r</math>, and <math>\int_{3\sigma}^\infty \phi\left(r\right)\mathrm{d}r</math> when <math>\sigma = \epsilon = 1.0</math>.'''

'''For the separation <math>r_0</math>:'''
<center><math>\phi\left(r\right) = 4\epsilon \left( \frac{\sigma^{12}}{r_0^{12}} - \frac{\sigma^6}{r_0^6} \right)</math></center>

<center><math> \frac{\sigma^{12}}{r_0^{12}} - \frac{\sigma^6}{r_0^6} =0</math></center>

<center><math> \frac{\sigma^{12}}{r_0^{12}}=\frac{\sigma^6}{r_0^6} </math></center>

<center><math>\frac{\sigma^6}{r_0^6} =1</math></center>

<center><math>\sigma^6=r_0^6 </math></center>

<center><math>r_0=\sigma</math></center><span style="color:red">tick</span>

'''The force experienced at <math>r_0</math>:'''
<center><math> F = -\frac{\phi (r)}{dr} </math></center>

<center><math>= 4\epsilon \left( \frac{12\sigma^{12}}{r_0^{13}} -\frac{6\sigma^6}{r_0^7} \right)</math></center>

<center><math>= 24\epsilon \left( \frac{2\sigma^{12}}{r_0^{13}} -\frac{\sigma^6}{r_0^7} \right)</math></center>

as''' <math>r_0=\sigma</math>:'''

<center><math>= 24\epsilon \left( \frac{2\sigma^{12}}{\sigma^{13}} -\frac{\sigma^6}{\sigma^7} \right)</math></center>

<center><math>F = 24\epsilon \left( \frac{2}{\sigma} -\frac{1}{\sigma} \right)</math></center>

<center><math>= \frac{24\epsilon}{\sigma}</math></center><span style="color:red">tick</span>

E'''quilibrium separation, <math>r_{eq}</math>, is given by finding the minimum of the potential well:'''

<center><math>\frac{\phi (r)}{dr}= 24\epsilon \bigg( \frac{2\sigma^{12}}{r^{13}} -\frac{1\sigma^6}{r^7}\bigg)=0</math></center>

<center><math>\frac{2\sigma^{12}}{r^{13}} = \frac{\sigma^6}{r^7}</math></center>

<center><math>2\sigma^6=r^6</math></center>

<center><math> r_{eq} = 2^{1/6}\sigma</math></center><span style="color:red">tick</span>

'''Potential well depth:'''
<center><math>\frac{\phi (r)}{dr} = 4\epsilon \bigg( \frac{\sigma^{12}}{(2^{1/6}\sigma)^{12}} - \frac{\sigma^6}{(2^{1/6}\sigma)^{6}}\bigg)</math></center>

<center><math> = 4\epsilon \bigg( \frac{\sigma^{12}}{4\sigma^{12}} - \frac{\sigma^6}{2\sigma^{6}}\bigg)</math></center>

<center><math> = 4\epsilon \bigg( \frac{1}{4} - \frac{1}{2}\bigg)</math></center>

<center><math> = -\epsilon </math></center><span style="color:red">tick</span>

'''Integrals when <math>\sigma = \epsilon = 1.0</math>:'''
<center><math>\int_{2\sigma}^\infty \phi\left(r\right)\mathrm{d}r=4\int_{2\sigma}^\infty \bigg( \frac{1}{r^{12}} - \frac{1}{r^6}\bigg)\mathrm{d}r</math></center>

<center><math>=4\bigg[\frac{1}{-11r^{11}} + \frac{1}{5r^{5}}\bigg]_{2\sigma}^{\infty}</math></center>

<center><math>=-0.0248</math></center><span style="color:red">tick</span>

<center><math>\int_{2.5\sigma}^\infty \phi\left(r\right)\mathrm{d}r=\int_{2.5\sigma }^{\infty} 4\left ( \frac{1}{r^{12}}- \frac{1}{r^{6}}\right ) dr</math></center>

<center><math>=4\bigg[\frac{1}{-11r^{11}} + \frac{1}{5r^{5}}\bigg]_{2.5\sigma}^{\infty}</math></center>

<center><math>=-8.18\times10^{-3}</math></center><span style="color:red">tick</span>

<center><math>\int_{3\sigma}^\infty \phi\left(r\right)\mathrm{d}r=\int_{3\sigma }^{\infty} 4\left ( \frac{1}{r^{12}}- \frac{1}{r^{6}}\right ) dr</math></center>

<center><math>=4\bigg[\frac{1}{-11r^{11}} + \frac{1}{5r^{5}}\bigg]_{3\sigma}^{\infty}</math></center>

<center><math>=-3.29\times10^{-3}</math></center><span style="color:red">tick</span>

'''<big>TASK</big>: Estimate the number of water molecules in 1ml of water under standard conditions. '''

Assuming 1 mL of water has a mass of 1 g under standard conditions:

<center><math>N=\frac{M}{M(H_2 O)}N_a</math></center>

<center><math>=\frac{1\times 6.022 \times 10^{23}}{18.0}=3.35 \times 10^{22}\ </math></center><span style="color:red">tick</span>

'''<big>TASK</big>:Estimate the volume of <math>10000</math> water molecules under standard conditions.'''

Assuming standard conditionsː

<center><math>M =\frac{10000 \times M(H_2 O)}{N_a}</math></center>

<center><math>\frac{10000 \times 18.0}{6.022 \times 10^{23}}=2.989 \times 10^{-19}\ g </math></center>

<center><math>V=\frac{M}{\rho }</math></center>

<center><math>V=\frac{2.989 \times 10^{-19}}{1 \times 10^{-6}}= 2.989 \times 10^{-25}\ m^{3}</math></center><span style="color:red">tick</span>

'''<big>TASK</big>: Consider an atom at position <math>\left(0.5, 0.5, 0.5\right)</math> in a cubic simulation box which runs from <math>\left(0, 0, 0\right)</math> to <math>\left(1, 1, 1\right)</math>. In a single timestep, it moves along the vector <math>\left(0.7, 0.6, 0.2\right)</math>. At what point does it end up, ''after the periodic boundary conditions have been applied''?'''.

After the periodic boundary conditions are applied, the atom ends up atː

<center><math>\left(0.2, 0.1, 0.7\right)</math></center><span style="color:red">tick</span>

'''<big>TASK</big>: The Lennard-Jones parameters for argon are <math>\sigma = 0.34\mathrm{nm}, \epsilon\ /\ k_B= 120 \mathrm{K}</math>. If the LJ cutoff is <math>r^* = 3.2</math>, what is it in real units? What is the well depth in <math>\mathrm{kJ\ mol}^{-1}</math>? What is the reduced temperature <math>T^* = 1.5</math> in real units?

For the LJ cutoffː

<center><math>r^*=\frac{r}{\sigma}=3.2</math></center>

<center><math>3.2 \times 0.34 \times 10^{-9}=r</math></center>

<center><math>=1.09 \times 10^{-9} \ m </math></center>

For the well depth, from aboveː

<center><math>\frac{\phi (r)}{dr} = 4\epsilon \bigg( \frac{\sigma^{12}}{(2^{1/6}\sigma)^{12}} - \frac{\sigma^6}{(2^{1/6}\sigma)^{6}}\bigg)</math></center><span style="color:red">tick</span>

<center><math> = -\epsilon </math></center>

Thereforeː

<center><math> -\epsilon = -120k_B</math></center>

<center><math>=-1.657^{-21} \ J</math></center>
<center><math>=-0.9977 \ kJ \ mol^{-1}</math></center>

For the reduced temperatureː

<center><math>T^* = \frac{T}{\epsilon}</math></center>
<center><math>1.5 = \frac{T}{120}</math></center>

<center><math>T=1.5 \times 120</math></center><span style="color:red">tick</span>

<center><math>T=180 \ K</math></center><span style="color:red">tick</span>

== Equilibration ==

=== Creating the simulation box ===
'''<big>TASK</big>: Why do you think giving atoms random starting coordinates causes problems in simulations? Hint: what happens if two atoms happen to be generated close together?'''

Atoms could be generated very close to one another which would cause them to shoot out at a high speed due to the extremely large repulsive forces between the atoms. This would affect the simulation by disturbing the other atoms around it due to the high speed atoms travelling away from each other and effecting the interactions between all the other particles. By using a lattice, favorable distances can be chosen for a simulation to run smoothly and take a short amount of time to reach equilibrium.

'''<big>TASK</big>: Satisfy yourself that this lattice spacing corresponds to a number density of lattice points of <math>0.8</math>. Consider instead a face-centred cubic lattice with a lattice point number density of 1.2. What is the side length of the cubic unit cell?'''

<center><math> \frac{1}{0.8} = 1.25</math></center>

<center><math>= 1.25^\frac{1}{3} = 1.07722</math></center>

For a face-centred cubic latticeː
<center><math>x = \sqrt[3](\frac{4}{1.2})=1.4938</math></center>

<center><math>x = (\frac{4}{1.2})^\frac{1}{3} </math></center>
<center><math>x = 1.49380 </math></center><span style="color:red">tick</span>

'''<big>TASK</big>: Consider again the face-centred cubic lattice from the previous task. How many atoms would be created by the create_atoms command if you had defined that lattice instead?'''

An FCC lattice contains 4 lattice points, therefore for a box of side length 10, there are 4000 lattice points and therefore 4000 atoms will be generated.

=== Setting the properties of the atoms ===
'''<big>TASK</big>: Using the [http://lammps.sandia.gov/doc/Section_commands.html#cmd_5 LAMMPS manual], find the purpose of the following commands in the input script:'''

<pre>
mass 1 1.0
pair_style lj/cut 3.0
pair_coeff * * 1.0 1.0
</pre>

'mass 1 1.0' Relates to the mass of '''atom type''' '''1''' being '''1.0'''

'pair_style lj/cut 3.0' computes the standard LJ potential, but gives a cutoff point for the potential at point <<math>r_c</math>, with <math>r</math> being the distance between the atoms, along with having no colombic attractions between the atoms.

'pair_coeff * * 1.0 1.0' Sets the coefficents for all l,J pairs

'''<big>TASK</big>: Given that we are specifying <math>\mathbf{x}_i\left(0\right)</math> and <math>\mathbf{v}_i\left(0\right)</math>, which integration algorithm are we going to use?'''

We will be using the velocity Verlet algorithm.

[[File:ThirdYearSimulationExpt-Intro-VelocityVerlet-flowchart.svg|300px|thumb|centre|'''Figure 1''': Steps to implement the velocity Verlet algorithm.]]

=== Running the simulation ===
'''<big>TASK</big>: Look at the lines below.'''
<pre>
### SPECIFY TIMESTEP ###
variable timestep equal 0.001
variable n_steps equal floor(100/${timestep})
timestep ${timestep}

### RUN SIMULATION ###
run ${n_steps}
run 100000
</pre>
'''The second line (starting "variable timestep...") tells LAMMPS that if it encounters the text ${timestep} on a subsequent line, it should replace it by the value given. In this case, the value ${timestep} is always replaced by 0.001. In light of this, what do you think the purpose of these lines is? Why not just write:'''
<pre>
timestep 0.001
run 100000
</pre>This is so timestep is defined as a variable and so can be changed later after the code has been run so it does not need to be defined again and again throughout the code, but can be overwritten easily for use.

=== Checking equilibration ===
'''<big>TASK</big>: make plots of the energy, temperature, and pressure, against time for the 0.001 timestep experiment (attach a picture to your report). Does the simulation reach equilibrium? How long does this take? When you have done this, make a single plot which shows the energy versus time for all of the timesteps (again, attach a picture to your report). Choosing a timestep is a balancing act: the shorter the timestep, the more accurately the results of your simulation will reflect the physical reality; short timesteps, however, mean that the same number of simulation steps cover a shorter amount of actual time, and this is very unhelpful if the process you want to study requires observation over a long time. Of the five timesteps that you used, which is the largest to give acceptable results? Which one of the five is a ''particularly'' bad choice? Why?'''
[[File:Ft1015Equilibrium7.PNG|600px|thumb|centre|A zoomed in plot of the total energy vs time plot to show that the system equilibrates after 0.002 ps]]
[[File:Ft1015Energy7.PNG|600px|thumb|centre|Total energy vs time for a 0.001 time step]]
[[File:Ft1015Pressure7.PNG|600px|thumb|centre|Pressure vs time for a 0.001 time step]]
[[File:Ft1015Temperature7.PNG|600px|thumb|centre|Temperature vs time for a 0.001 time step]]
[[File:ft1015_Timestep_Graph_Plots.PNG|600px|thumb|centre|A plot five different timesteps and the energies they converge on]]
The largest timestep of 0.015 is a particularly bad choice as the energy does not converge, but increases dramatically, something which should not happen. The largest to give acceptable results is 0.0025, as though 0.001 and 0.0025 both converge to the same value, 0.0025 allows a simulation which simulates 250% more time than the 0.001 timestep. This is more practical as it allows more simulations and longer simulations to be done in the same time-frame as the smaller timestep, without losing accuracy or precision.<span style="color:red">tick</span>

== Running simulations under specific conditions ==

=== Temperature and Pressure Control ===
'''<big>TASK</big>: Choose 5 temperatures (above the critical temperature ), and two pressures (you can get a good idea of what a reasonable pressure is in Lennard-Jones units by looking at the average pressure of your simulations from the last section). This gives ten phase points — five temperatures at each pressure. Create 10 copies of npt.in, and modify each to run a simulation at one of your chosen  points. You should be able to use the results of the previous section to choose a timestep. Submit these ten jobs to the HPC portal. While you wait for them to finish, you should read the next section.'''

===Thermostats and Barostats===
'''<big>TASK</big>: We need to choose <math>\gamma</math> so that the temperature is correct <math>T = \mathfrak{T}</math> if we multiply every velocity <math>\gamma</math>. We can write two equations:'''

<math>\frac{1}{2}\sum_i m_i v_i^2 = \frac{3}{2} N k_B T</math>

<math>\frac{1}{2}\sum_i m_i \left(\gamma v_i\right)^2 = \frac{3}{2} N k_B \mathfrak{T}</math>

'''Solve these to determine <math>\gamma</math>.'''

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math> \frac{\frac{1}{2}\sum_i m_i \left(\gamma v_i\right)^2}{\frac{1}{2}\sum_i v_i^2} = \frac{\frac{3}{2} N k_B \mathfrak{T}}{\frac{3}{2} N k_B T} </math></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math> \frac{\sum_i \left(\gamma v_i\right)^2}{\sum_i v_i^2} = \frac{\mathfrak{T}}{T} </math></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math> \frac{\sum_i \gamma^2 v_i^2}{\sum_i v_i^2} = \frac{\mathfrak{T}}{T} </math></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math> \frac{\gamma^2 \sum_i v_i^2}{\sum_i v_i^2} = \frac{\mathfrak{T}}{T} </math></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math> \gamma^2 = \frac{\mathfrak{T}}{T} </math></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math> \gamma = \sqrt{\frac{\mathfrak{T}}{T}} </math></div><span style="color:red">tick</span>

=== Examining the Input Script ===
'''<big>TASK</big>: Use the manual page to find out the importance of the three numbers ''100 1000 100000''. How often will values of the temperature, etc., be sampled for the average? How many measurements contribute to the average? Looking to the following line, how much time will you simulate?'''

The first number defines the length of the timestep between data points taken for the calculation of the average, the second number relates to how many timesteps will be used to calculate the average at the specificed time, the last number specifies at what timestep will an average be taken. Thus 100 1000 100000 means that data points for the average will be taken every 100 timesteps, and 1000 will be used to calculate the average at 100000 timesteps. The time simulated will be 100000 * the timestep in seconds, which for 0.0025 s per timestep means that 250 s will be simulated.

'''<big>TASK</big>: When your simulations have finished, download the log files as before. At the end of the log file, LAMMPS will output the values and errors for the pressure, temperature, and density <math>\left(\frac{N}{V}\right)</math>. Use software of your choice to plot the density as a function of temperature for both of the pressures that you simulated. Your graph(s) should include error bars in both the x and y directions. You should also include a line corresponding to the density predicted by the ideal gas law at that pressure. Is your simulated density lower or higher? Justify this. Does the discrepancy increase or decrease with pressure?'''

[[File:ft1015_Standard_Error_plots_graph.PNG|600px|thumb|centre| A plot of average density against average critical temperature of the system with standard error error bars]]

The discrepancies between the idealised and simulated number density results can be attributed to the simulation being non-ideal. The basis of a L-J system is that the particles interact with each other, with a short range repulsive term, and a longer range attractive term. The ideal gas law however, assumes that there are no interactions between particles and thus, due to the short range highly repulsive term, the particles are further apart than in a system with no interactions and therefore the number density of the simulated systems are lower than their idealised counterparts.<span style="color:red">tick</span>

At higher pressures the deviations from ideality are larger than that of a lower pressure simulation. This arises from the fact that at higher pressures, the number density is higher due to decreased cell volume, leading to decreased distances between particles and therefore increased interactions between them as they collide with each other more frequently. This increase in interactions between particles leads to a large deviation from ideality and so a larger difference between simulated and ideal law data for higher pressures.<span style="color:red">tick</span>

== Calculating heat capacities using statistical physics ==

===Heat Capacity Calculation===
'''<big>TASK</big>: As in the last section, you need to run simulations at ten phase points. In this section, we will be in density-temperature <math>\left(\rho^*, T^*\right)</math> phase space, rather than pressure-temperature phase space. The two densities required at <math>0.2</math> and <math>0.8</math>, and the temperature range is <math>2.0, 2.2, 2.4, 2.6, 2.8</math>. Plot <math>C_V/V</math> as a function of temperature, where <math>V</math> is the volume of the simulation cell, for both of your densities (on the same graph). Is the trend the one you would expect? Attach an example of one of your input scripts to your report.'''

[[File:Ft1015HeatCapacityScript.PNG|600px|thumb|centre|An image of one of the scripts used for the heat capacity calculations]]

[[File:ft1015HeatCapacity.png|600px|thumb|centre|A plot of <math> {C_{V}}/{V} </math> against <math> {T}/{T*} </math> for two different densities]]

The decrease of heat capacity with temperature increase excluding phase changes, can be explained through the Density of States. Similar to the anharmonic oscillator model for electron excitation for an atom or particles where as the energy levels are increased, the spacing between the levels decrease, as the temperature of our system is increased the spacing between energy levels contracts, and thus the higher energy levels come down in energy and become more accessible. This leads to more modes available for promotion in the system, thus ease of promotion of an atom increases, leading to a reduced energy needed to promote and atom and so heat capacity decreases with temperature.<span style="color:red">tick</span>

The large jumps in heat capacity seen in both data sets at <math>T^* = 2.6</math> can be explained through phase changes. As the particles simulated are totally symmetric there are no extra degrees of freedom to be reached through an increase in temperature as only translational modes are active through equipartition theorem, therefore the jump in heat capacity must be due to a phase change. Assuming a second order phase transition; as an NVT ensemble is used, and in a first order transition density is discontinuous which is impossible for a system with fixed <math>N</math> and <math>V</math>, Heat capacity rises to infinity until after the transition has been completed, after which it lowers again, which can be seen at <math>T^* = 2.8</math>, due to thermal energy needing to be expended in order to complete the phase transition.<span style="color:red">tick</span>

== Structural properties and the radial distribution function ==

===Simulations in this section===

'''<big>TASK</big>: perform simulations of the Lennard-Jones system in the three phases. When each is complete, download the trajectory and calculate <math>g(r)</math> and <math>\int g(r)\mathrm{d}r</math>. Plot the RDFs for the three systems on the same axes, and attach a copy to your report. Discuss qualitatively the differences between the three RDFs, and what this tells you about the structure of the system in each phase. In the solid case, illustrate which lattice sites the first three peaks correspond to. What is the lattice spacing? What is the coordination number for each of the first three peaks?'''

[[File:ft1015G(r).png|720px|thumb|centre|'''Figure 4''': ft1015G(r).png]]

[[File:ft1015G(r)N.png|720px|thumb|centre|'''Figure 4''': ft1015G(r)N.png]]

The RDF for the gas one broad peak that drops to 1 quickly after the peak finishes, leading to there being no long range order as the probability of finding an particle outside of the first shell is equal everywhere. This shows that there is no long range order in a gas, only extremely short range order. The peak could be broad due to weak attractive effects holding the nearest neighbors close, but not tightly holding them in place.<span style="color:red">tick</span>

The liquid RDF has three broad peaks, leading to 3 co-ordination spheres of particles held tightly, but not so tightly that the particles cannot move, therefore the spheres are very lightly diffuse, but like the gas the RDF drops to 1 after the last peaking, leading to no long range order in a liquid.

The RDF of the solid has many sharp peaks corresponding to the solid atoms fixed positions and long range order in the crystal lattice.

The lattice spacing is the minimum distance between lattice unit cells which is equal to 1.175.<span style="color:red">tick</span>

<center>
{| class="wikitable"
|-
! r
! Integrated g(r)
! Co-ordination Number
|-
| 1.175
| 11.85813
| 12
|-
| 1.675
| 18.44288
| 6
|-
| 1.975
| 42.26256
| 24
|}
</center>

= Dynamical properties and the diffusion coefficient =

=== Simulations in this Section ===
'''<big>TASK</big>: In the D subfolder, there is a file ''liq.in'' that will run a simulation at specified density and temperature to calculate the mean squared displacement and velocity autocorrelation function of your system. Run one of these simulations for a vapour, liquid, and solid. You have also been given some simulated data from much larger systems (approximately one million atoms). You will need these files later.'''

=== Mean Squared Displacement ===
'''<big>TASK</big>: make a plot for each of your simulations (solid, liquid, and gas), showing the mean squared displacement (the "total" MSD) as a function of timestep. Are these as you would expect? Estimate <math>D</math> in each case. Be careful with the units! Repeat this procedure for the MSD data that you were given from the one million atom simulations.'''

From https://en.wikipedia.org/wiki/Mean_squared_displacementː

<center><math> MSD = \langle (x(t)-x_0)^2 \rangle = 2Dt </math></center>

[[File:ft1015GasMSD.png|720px|thumb|centre|'''Figure 4''': ft1015GasMSD.png]]
[[File:ft1015GasMSD_1m.png|720px|thumb|centre|'''Figure 5''': ft1015GasMSD_1m.png]]
[[File:ft1015LiquidMSD.png|720px|thumb|centre|'''Figure 6''': ft1015LiquidMSD.png]]
[[File:ft1015LiquidMSD_1m.png|720px|thumb|centre|'''Figure 7''': ft1015LiquidMSD_1m.png]]
[[File:ft1015SolidMSD.png|720px|thumb|centre|'''Figure 8''': ft1015SolidMSD.png]]
[[File:ft1015SolidMSD_1m.png|720px|thumb|centre|'''Figure 9''': ft1015SolidMSD_1m.png]]

=== EXTENSION: Velocity Autocorrelation Function ===
'''<big>TASK</big>: In the theoretical section at the beginning, the equation for the evolution of the position of a 1D harmonic oscillator as a function of time was given. Using this, evaluate the normalised velocity autocorrelation function for a 1D harmonic oscillator (it is analytic!):

<center><math>C\left(\tau\right) = \frac{\int_{-\infty}^{\infty} v\left(t\right)v\left(t + \tau\right)\mathrm{d}t}{\int_{-\infty}^{\infty} v^2\left(t\right)\mathrm{d}t}</math></center>

'''Be sure to show your working in your writeup. On the same graph, with x range 0 to 500, plot <math>C\left(\tau\right)</math> with <math>\omega = 1/2\pi</math> and the VACFs from your liquid and solid simulations. What do the minima in the VACFs for the liquid and solid system represent? Discuss the origin of the differences between the liquid and solid VACFs. The harmonic oscillator VACF is very different to the Lennard Jones solid and liquid. Why is this? Attach a copy of your plot to your writeup.'''

'''<big>TASK</big>: Use the trapezium rule to approximate the integral under the velocity autocorrelation function for the solid, liquid, and gas, and use these values to estimate <math>D</math> in each case. You should make a plot of the running integral in each case. Are they as you expect? Repeat this procedure for the VACF data that you were given from the one million atom simulations. What do you think is the largest source of error in your estimates of D from the VACF?'''

Rep:FabianThomasThirdYearLiquidFlowSimulation

2025-09-17T13:27:55Z

Jbettenc: Jbettenc moved page MRD:FabianThomasThirdYearLiquidFlowSimulation to Rep:FabianThomasThirdYearLiquidFlowSimulation without leaving a redirect: Move to report namespace

== Abstract ==
Lennard-Jones simulations of solids, liquids, and gases were run through LAMMPS on the imperial HPC service. How number density varied against temperature and at different pressures, radial distribution functions of L-J solids, liquids, and gases, Mean Squared Displacements, Diffusion coefficients, and the variation of heat capacity with temperature were investigated in order to understand more about phase properties, and specifically those that can be modelled using a L-J simulation. <ref> Jean-Pierre Hansen and Loup Verlet, Phys. Rev. 184, 151</ref> For the equation of state, number density increased with pressure, but was inversely proportional to an increasing temperature. Heat capacity was determined to be inversely proportional to temperature due to a change in density of states for a fixed NVT ensemble. The radial distribution function allowed for analysis of solid, liquid, and gas simulations of 3375 atoms and allowed for the probing of the structure of each phase, along with details about the co-ordination spheres and numbers of nearest neighbors. Diffusion coefficients were found using the Mean Squared Displacements for solids, liquids and gases, along with simulations with one million atoms for each phase as well, with values for <math>D</math> of<math>-1 \times 10^{-17}</math>,<math>1.1 \times 10^{-10}</math> and <math> 3.61 \times 10^{-9} \ m^2 s^{-1}</math> for the solid liquid and gas phases.

== Introduction ==
Phase changes occur every day in daily life, and are common place in the world around us. From boiling water to be used in cooking or in steam generators to produce electricity, to instant heat packs, where sodium acetate is super cooled, and upon perturbation instantly raises the pack’s temperature significantly to its freezing point at 54 degrees. These are all every day examples of how phase changes are used and how research into phase changes can provide new avenues for products and areas of research such as new instant-heat pack materials, all based on previous phase change research.

Phase change materials; not unlike sodium acetate, is a large area of research today. Materials capable of storing large amounts of energy through phase changes such as the solid to liquid transition or solid to solid transitions, <ref>Belén Zalba, José M Marı́n, Luisa F. Cabeza, Harald Mehling, Review on thermal energy storage with phase change: materials, heat transfer analysis and applications, In Applied Thermal Engineering, Volume 23, Issue 3, 2003, Pages 251-283</ref> with applications such as improved solar cell energy storage<ref>Murat Kenisarin, Khamid Mahkamov, Solar energy storage using phase change materials, In Renewable and Sustainable Energy Reviews, Volume 11, Issue 9, 2007, Pages 1913-1965</ref>, and materials able to change state for useful purposes in textiles<ref>S. Mondal, Phase change materials for smart textiles – An overview, In Applied Thermal Engineering, Volume 28, Issues 11–12, 2008, Pages 1536-1550</ref> these simulations carried out can further progress phase-change transition research, and can be used to go on to model more complex systems for use with this research.

Modelling theses systems using L-J simulations provides a fast and easy method to compute the phase change temperatures, order, and structure of the phases transitioned through. Thus simplifying the need to synthesise the molecule beforehand to view the properties of the system. Some of the systems with the largest heat of fusion are systems such as Lithium or water, which are very easily simulated through L-J models and provide an excellent way to probe the phase diagrams of these systems. The ease of determining the crystal structure of solids, is very largely beneficial for the phase change materials, as it allows for easy categorisation of the solid phases, such as ice which has over ten different solid crystalline phases depending on the pressure and temperature of the system, which is modelled accurately and quickly using an L-J simulation through use of radial distribution functions. Heat capacity measurements of phases are of a large interest in phase-change materials as materials with large heat capacities coupled with a large latent heat of fusion allows for the storage of very large amount of energy in a small amount of solid. But all these simulations lead to further questions, how far can L-J simulations go in helping to discover phase change materials, and how accurately can the simulations predict real heat capacities, structures and transitions of unmade molecules?

== Methods ==

All simulations were run as Lennard-Jones systems through the Imperial College HPC facility <ref>https://www.imperial.ac.uk/admin-services/ict/self-service/research-support/hpc/</ref> using the LAMMPS runtime option, with all graphs and respective data produced and analysed using Origin Pro <ref>http://www.originlab.com/Origin</ref>.
Equation of State simulations were run using an NpT ensemble. Using dimensionless L-J reduced units <ref>http://cbio.bmt.tue.nl/pumma/index.php/Manual/ReducedUnits</ref>, pressures <math>p^*</math> of 1.8 and 2.6 were used, along with temperatures above the critical temperature <math>T^*</math> = 1.5 of 2, 2.5, 3, 3.5 and 4. A timestep of 0.0025 was used, as it was found in the earlier stages of the experiment to be as accurate as a timestep of 0.001, but would run the simulation 250% faster in comparison. As it was a liquid that was to be simulated, a simple cubic lattice was used for the simulation.

Heat Capacity simulations were run at <math>\rho^*</math> = 0.2 and 0.8, and <math>T^*</math> = 2.0, 2.2, 2.4, 2.6, 2.8 in L-J reduced units. A simple cubic crystal was melted under microcanonical NVE conditions. As heat capacity, shown below, is directly proportional to the variance of energy of the system if temperature is held constant, after melting, the system was put under an NVT ensemble to hold the temperature constant but allow the energy to vary.

<center><math>C_V = \frac{\partial E}{\partial T} = \frac{\mathrm{Var}\left[E\right]}{k_B T^2} = N^2\frac{\left\langle E^2\right\rangle - \left\langle E\right\rangle^2}{k_B T^2}</math></center>

Where <math>C_V</math> is Heat Capacity, <math>\mathrm{Var}\left[E\right]</math> is the variance in Energy, <math>k_b</math> is Boltzmann's constant, <math>\left\langle E^2\right\rangle</math> is mean squared energy, <math>\left\langle E\right\rangle^2</math> is mean energy squared, and <math>N^2</math> is the number of atoms in the simulation squared.

A factor of <math>N^2</math> is added to the equation, as LAMMPS divides all energies output by the number of particles in the system, therefor giving <math>\frac{E}{N}</math> for any value of <math>E</math> measured. Using this equation, the heat capacity was found and <math>\frac {C_V}{V}</math> was plotted against <math>T^*</math> to view how the heat capacity varied with respect to temperature.

Using VMD software <ref>http://www.ks.uiuc.edu/Research/vmd/</ref>, the radial distributions functions for the solid, liquid, and vapour phases of a L-J fluid were calculated, setting selection 1, and 2 to ‘all’, and delta r to 0.05. Using these plots the co-ordination sphere radii and numbers of the solid, liquid, and vapour phases were determined.

Using the equation <math>D = \frac{1}{6}\frac{\partial\left\langle r^2\left(t\right)\right\rangle}{\partial t}</math> and the relationship between D and Mean-Squared Displacement <math>\langle (x(t)-x_0)^2 \rangle = 2Dt</math>, <math>D</math> was calculated for solid, liquid, and vapour phases by extracting the gradient from the plot of Mean-Squared Displacement against time, giving a gradient of <math>2D</math>.

== Results and Discussion ==

=== Equation of State ===

[[File:Ft1015_Standard_Error_plots_graph.PNG|480px|thumb|left|'''Figure 1''': A plot of average density against average critical temperature of the system with standard error error bars]]

From figure 1, we can see that number density <math> \frac{N}{V}</math> is inversely proportional to <math>T</math>. This can be rationalised by looking at the results of the L-J reduced gas law <math> \frac{N}{V} = \frac{p}{T}</math>

The discrepancies between the idealised and simulated number density results can be attributed to the simulation being non-ideal. The basis of a L-J system is that the particles interact with each other, with a short range repulsive term, and a longer range attractive term. The ideal gas law however, assumes that there are no interactions between particles and thus, due to the short range highly repulsive term, the particles are further apart than in a system with no interactions and therefore the number density of the simulated systems are lower than their idealised counterparts.

As the temperature is increased, the kinetic energy of the particles increases leading to an increase in the average distance between particles in order to reduce repulsions between particles to a minimum. This can be seen in Charles’ Lawː
<center><math> \frac{N_1}{T_1} = \frac{N_2}{V_2} </math></center>
As the temperature is increased to <math>T_2</math>, <math>V_2</math> must increase, thus leading to a decrease in number density. As the particles are further away from each other on average, the particles interact with one another less, and so become more ‘ideal’, reducing the difference between the simulated result and the gas law result.

This result is the opposite for lower temperatures. As the temperature is reduced, so is the kinetic energy, and therefore the particles are less able to overcome the attractive and repulsive forces between them. This leads to larger interactions between the particles due to their increased proximity to each other, and so simulated results deviate from the gas law more at lower temperatures. From Charles’ Law again, we can see that the volume of the cell will decrease as temperature decreases, and so number density will increase at lower temperatures.

From the graphs, we can see that, at higher pressures the deviations from ideality are larger than that of a lower pressure simulation. This arises from the fact that at higher pressures, the number density is higher due to decreased cell volume, leading to decreased distances between particles and therefore increased interactions between them as they collide with each other more frequently. This increase in interactions between particles leads to a large deviation from ideality and so a larger difference between simulated and ideal law data for higher pressures.

=== Heat Capacity ===

[[File:ft1015HeatCapacity.png|480px|thumb|left|'''Figure 2''': A plot of <math> {C_{V}}/{V} </math> against <math> {T*} </math> for two different densities]]
[[File:Ft1015HeatCapacityScript.PNG|250px|thumb|right|'''Figure 3''': An image of a sample of the script used for one of the heat capacity calculations]]

Heat capacity defined in the equation below, varies directly with the variance in energy of a system held at constant temperature, but is essentially a measure of how easily a substance can be heated up. Thus, a substance with a larger heat capacity will require more energy to heat up than one with a lower heat capacity.

<center><math>C_V = \frac{\partial E}{\partial T} = \frac{\mathrm{Var}\left[E\right]}{k_B T^2} = N^2\frac{\left\langle E^2\right\rangle - \left\langle E\right\rangle^2}{k_B T^2}</math></center>

The odd relationship between <math>\frac{C_V}{V}</math> and temperature <math> T^*</math> can be described through use of the equipartition theorem and the Boltzmann Distribution

The large jumps in heat capacity seen in both data sets at <math>T^* = 2.6</math> can be explained through phase changes. As the particles simulated are totally symmetric there are no extra degrees of freedom to be reached through an increase in temperature as only translational modes are active through equipartition theorem, therefore the jump in heat capacity must be due to a phase change. Assuming a second order phase transition; as an NVT ensemble is used, and in a first order transition density is discontinuous which is impossible for a system with fixed <math>N</math> and <math>V</math>, Heat capacity rises to infinity until after the transition has been completed, after which it lowers again, which can be seen at <math>T^* = 2.8</math>, due to thermal energy needing to be expended in order to complete the phase transition.

The decrease of heat capacity with temperature increase excluding phase changes, can be explained through the Density of States. Similar to the anharmonic oscillator model for electron excitation for an atom or particles where as the energy levels are increased, the spacing between the levels decrease, as the temperature of our system is increased the spacing between energy levels contracts, and thus the higher energy levels come down in energy and become more accessible. This leads to more modes available for promotion in the system, thus ease of promotion of an atom increases, leading to a reduced energy needed to promote and atom and so heat capacity decreases with temperature.

This can be viewed form a different angle with the Boltzmann Distributionː

<center><math>\frac{N_i} N = \frac{\exp(-E_i/kT) } { \sum_j \exp(-E_j/kT) }</math></center>

For a specified energy gap, the higher in energy the energy state is, the less likely it is to be populated. If the energy gap is decreased, higher energy states are more likely to be populated, and are therefore more easily reached, and thus the ease of promotion is increased for a decrease in energy gap between states and so heat capacity falls.

=== Structural Properties of the Radial Distribution Function ===
[[File:ft1015G(r).png|400px|thumb|right|'''Figure 4''': ft1015G(r).png]]

[[File:ft1015G(r)N.png|400px|thumb|left|'''Figure 4''': ft1015G(r)N.png]]
Using VMD <ref>http://www.ks.uiuc.edu/Research/vmd/</ref> the RDFs for a solid, liquid, and gas were found and investigated.

The Radial Distribution Function, RDF, describes for a central particle how density varies as a function of distance <math>r</math>. The RDF can be used to find the co-ordination spheres of solids, liquids and gases, along with the distance each sphere lies from the central particle.

Integrating the RDF gives the number density RDF, which allows users to find out how many particles lie within each co-ordination sphere, and when used in conjunction to the normal RDF, the structure of that phase.

Looking first at the RDF for the solid, the large peaks relate to particles being found at that distance from the central atom. These peaks relate to the co-ordination spheres around the atom in the solid phase. The peaks are sharper than the other phases due to the crystal structure of the solid phase and that the particles are fixed in place inside the lattice. Because of the crystal structure and it’s short- and long-range order, the peaks are sharp due to distinct spheres of co-ordination found at each distance. Looking at the first three peaks, these relate to the first three co-ordination spheres or nearest neighbours of particles around the central atom. The troughs in the RDF relate to the empty space between the co-ordination spheres and lack of particles between the spheres. Looking at the first three peaks in the RDF and number integrated RDF, we can say that the first three co-ordination spheres contain 12, 6, and 24 atoms respectively.

<center>
{| class="wikitable"
|-
! r
! Integrated g(r)
! Co-ordination Number
|-
| 1.175
| 11.85813
| 12
|-
| 1.675
| 18.44288
| 6
|-
| 1.975
| 42.26256
| 24
|}
</center>

The lattice spacing is the distance between repeat lattice cells and is present in systems with repeating units, and therefore repeating lattice cells. For the FCC lattice in the solid case, the lattice spacing was 1.175 in L-J reduced units.

The liquid phase in comparison to the solid phase has a highly reduced order due to the lack of long range order and rigidity in the liquid phase, but more order than the gas phase as one can see in the increased amount of peaks that the liquid phase has in comparison to the gas phase which shows more co-ordination spheres in the liquid phase. As liquid particles are not held tightly in place, but can vibrate and move, the co-ordination peaks are broad due to the fact that the particles are slightly spread out across the co-ordination shell as they are bound, but not tightly. This peak occurs around <math>r = 1</math>, with further co-ordination shells smaller than the closer shells as the short range order of a liquid breaks down at longer range. The troughs, similarly to the solid are due to no particles being in-between the shells and their atomic neighbours.

The gas phase has the least order out of the three phases examined. This is easily seen through there being only one peak on the RDF in comparison to the other phases. This peak relates to the very weak attractive forces between gas particles, and their nearest neighbours, effectively leading to one singular diffuse co-ordination sphere, which can be seen by the broadness of the peak. After the first coordination, there is no order relating to the central particle, and so the RDF falls to one after the peak, showing an equal likelihood of finding a gas particle at any distance after the co-ordination sphere, which further emphasises the lack of order in the gas phase.

For all phases the RDF is zero until the first co-ordination sphere or right before. This is due to the width of the atoms in the simulation and how the atoms cannot overlap due to the strong repulsive forces. This leads the first particles to be detected by the function at the first co-ordination sphere.

===Dynamical Properties and the Diffusion Coefficient===

[[File:ft1015GasMSD.png|400px|thumb|left|'''Figure 4''': A plot of MSD against time for the gas phase]]
[[File:ft1015GasMSD_1m.png|400px|thumb|right|'''Figure 5''': A plot of MSD against time for one million atoms in the gas phase]]
[[File:ft1015LiquidMSD.png|400px|thumb|left|'''Figure 6''': A plot of MSD against time for the liquid phase]]
[[File:ft1015LiquidMSD_1m.png|400px|thumb|right|'''Figure 7''': A plot of MSD against time for one million atoms in the liquid phase]]
[[File:ft1015SolidMSD.png|400px|thumb|left|'''Figure 8''': A plot of MSD against time for the solid phase]]
[[File:ft1015SolidMSD_1m.png|400px|thumb|right|'''Figure 9''': A plot of MSD against time for one million atoms in the solid phase]]

Using Mean Squared Displacement, MSD, in an NVT ensemble, one can determine the diffusion coefficient <math>D</math>. MSD is with, respective to a reference position, the deviation of the position of a particle from said point over time, and can be measured through single particle tracking via photon scattering <ref>Phys. Chem. Chem. Phys.,2013, 15, 845</ref>.

The diffusion coefficient <math>D = \frac{1}{6}\frac{\partial\left\langle r^2\left(t\right)\right\rangle}{\partial t}</math><math>D</math> is directly proportional to MSD as shown below in the equation:

<center><math> MSD = \langle (x(t)-x_0)^2 \rangle = 2Dt </math></center>

Plotting MSD against time, gives a graph with a gradient of <math>2D</math> and therefore a diffusion coefficient equal to <math>\frac{gradient}{2}</math>. These plots can be seen in figures 4-9 for solid, liquid and gas phases, along with plots of data given to us of those phases with one million atoms. If data points were not used in the linear fit of a graph as they did not follow the linearity expected for the gradient were masked and are shown in red on the plot.

<center>
{| class="wikitable"
|+ style="text-align: centre;" | Diffusion Coefficient
|-
! Phase
! Simulated plots / m^2 s^-1
! one million atoms plots / m^2 s^-1
|-
| Gas
| 1.17E-10
| 3.61E-09
|-
| Liquid
| 1.03E-10
|1.1E-10
|-
| Solid
| -1.4E-17
| -1E-17
|}
</center>

Looking at each phase individually and the two plots associated with them. Starting with the gas phase we can see that for our simulated data diffusive rates are reached immediately and stay constant throughout the simulation as there is little to no order in gas phases which was established by use of the RDF function earlier. However, for the one million atom simulation, the rate of diffusion is not initially constant but rises as the simulation progresses until system reaches equilibrium and <math>D</math> stabilizes. Thus for the one million atoms plot data points outside the linear region were excluded, as shown in the graph.

Rates of diffusion for liquids were lower than that of gases, which is to be expected due to the short range order shown in the RDF functions, thus diffuse more slowly to keep the short-range order intact.

Values of <math>D</math> for the solid were extremely small to the point of there being no diffusion occurring in solids. This can be explained by the highly ordered structure which only allows particles to vibrate about a fixed point, this is confirmed through RDF calculations above, with the large change in diffusion coefficients are the start of each simulation due to the system reaching equilibrium. Diffusion will only be able to occur in from a solid phase if a phase change occurred and thus diffusion would occur through a different phase.

A change in value of <math>D</math> for the larger one million atoms in comparison to our own simulations was expected as these larger simulations are likely to be more accurate and precise due to the larger amount of averaging that is able to occur, and the larger amount of atoms lowers microscopic effects as the system move into the macro scale.

== Conclusion ==
Through the successful modelling of a L-J system above the critical temperature, the equation of state was found at two different pressures. Number density was found to increase with pressure due to the increased repulsions between particles because of the increased number of collisions, while also found to be inversely proportional to temperature, as at higher temperatures the particles moved further apart, thus lowering the interactions between them and so lowering the number density, this was reinforced using Charles’ law. Deviations from an ideal gas were seen as the L-J system allows for attractive and repulsive forces between the particles, but at higher pressures the difference from ideality was larger due to the increased density of the particles leading to more interactions between them, thus straying from the ideal gas postulates.

It was found that Heat capacity was inversely proportional to temperature due to the change in density of states associated with an increase in temperature. The increasing of the density of states, and thus the lowering of energy of the higher energy levels allowed for more available energy states for promotion to, and so easier promotions of particles, thus leading to a lowered heat capacity due to the increased ease of promoting a particle to a higher energy state. A peak in the heat capacity seen at <math>T^* = 2.6 </math> was seen due to a change in phase with the heat capacity reaching infinity until the necessary amount of thermal energy had been input to the system.

The radial and integrated radial distribution functions were successfully used to model the co-ordination spheres of a solid, liquid, and gas, along with numbers for the nearest neighbours, and information on the order and structure of each phase. The gaseous phase was found to have no long range order, and only a single co-ordination shell thus leading to extremely short range order, with a broad peak suggesting a diffuse shell. The liquid phase was found to be more ordered than the gaseous phase due to the increased number of three co-ordination shells, leading to short range, but similarly to the gas no long range order in the phase. The peaks were also broadened, suggesting slightly diffusive co-ordination shells. Solids were found to be highly ordered in structure, with long range and short range order due to its crystal structure. Peaks were sharp due to the set spacings between particles, and the fact that the particles can only vibrate around their fixed positions, not move like in a liquid or gas. The co-ordination of the solid was found to be 12, 4, and 24 in the first three co-ordination shells respectively, along with a lattice spacing of 1.175.

Diffusion coefficients for solids, liquids, and gases were also found using the relationship between MSD and <math>D</math>. Solids were found to have an almost zero diffusion gradient, while liquids and gases were much higher. Data for 1000000 atoms simulations for each phase were received and plotted, leading to more accurate results for <math>D</math>, which agreed with the values of <math>D</math> received from the simulations of 3375 atoms completed by us.

== References ==
<references/>

== Introduction to molecular dynamics simulation ==

=== Numerical Integration ===
'''<big>TASK</big>: Open the file HO.xls. In it, the velocity-Verlet algorithm is used to model the behaviour of a classical harmonic oscillator. Complete the three columns "ANALYTICAL", "ERROR", and "ENERGY": "ANALYTICAL" should contain the value of the classical solution for the position at time <math>t</math>, "ERROR" should contain the ''absolute'' difference between "ANALYTICAL" and the velocity-Verlet solution (i.e. ERROR should always be positive -- make sure you leave the half step rows blank!), and "ENERGY" should contain the total energy of the oscillator for the velocity-Verlet solution. Remember that the position of a classical harmonic oscillator is given by <math> x\left(t\right) = A\cos\left(\omega t + \phi\right)</math> (the values of <math>A</math>, <math>\omega</math>, and <math>\phi</math> are worked out for you in the sheet).'''

'''<big>TASK</big>: For the default timestep value, 0.1, estimate the positions of the maxima in the ERROR column as a function of time. Make a plot showing these values as a function of time, and fit an appropriate function to the data.'''

'''<big>TASK</big>: Experiment with different values of the timestep. What sort of a timestep do you need to use to ensure that the total energy does not change by more than 1% over the course of your "simulation"? Why do you think it is important to monitor the total energy of a physical system when modelling its behaviour numerically? '''

[[File:ft1015Analytical.png|480px|thumb|center|]]
[[File:ft1015Error.png|480px|thumb|center|]]
[[File:ft1015Energy5.png|480px|thumb|center|]]

The total energy of a classical harmonic system is given by <math>\frac{1}{2}mv^{2}</math> and <math>\frac{1}{2}kx^{2}</math> for the kinetic and potential energies respectively. Combing these give the total energy of the system which should be a constant for an isolated system as no energy enters or leaves the system. A time-step value of 0.2 s gives a maximum difference in energy of 1% of the total starting energy, and so ensures a good approximation to realistic behavior. The Velocity-Verlet algorithm will have fluctuations in the total energy of the system as it is only an approximation through a Taylor-series expansion, however, minimizing these fluctuations provides a more accurate simulation and therefore for a physical system would provide more accurate thermodynamic properties of the system. Decreasing the time-step, and so decreasing the energy fluctuations will provide more accurate results at the cost of increased simulation time needed for the same time frame of a system, and so is a trade-off between time and accuracy for the simulation.

===Atomic Forces===
'''For a single Lennard-Jones interaction, <math>\phi\left(r\right) = 4\epsilon \left( \frac{\sigma^{12}}{r^{12}} - \frac{\sigma^6}{r^6} \right)</math>, find the separation, <math>r_0</math>, at which the potential energy is zero. What is the force at this separation? Find the equilibrium separation, <math>r_{eq}</math>, and work out the well depth <math>(\phi\left(r_{eq}\right))</math>. Evaluate the integrals <math>\int_{2\sigma}^\infty \phi\left(r\right)\mathrm{d}r</math>, <math>\int_{2.5\sigma}^\infty \phi\left(r\right)\mathrm{d}r</math>, and <math>\int_{3\sigma}^\infty \phi\left(r\right)\mathrm{d}r</math> when <math>\sigma = \epsilon = 1.0</math>.'''

'''For the separation <math>r_0</math>:'''
<center><math>\phi\left(r\right) = 4\epsilon \left( \frac{\sigma^{12}}{r_0^{12}} - \frac{\sigma^6}{r_0^6} \right)</math></center>

<center><math> \frac{\sigma^{12}}{r_0^{12}} - \frac{\sigma^6}{r_0^6} =0</math></center>

<center><math> \frac{\sigma^{12}}{r_0^{12}}=\frac{\sigma^6}{r_0^6} </math></center>

<center><math>\frac{\sigma^6}{r_0^6} =1</math></center>

<center><math>\sigma^6=r_0^6 </math></center>

<center><math>r_0=\sigma</math></center>

'''The force experienced at <math>r_0</math>:'''
<center><math> F = -\frac{\phi (r)}{dr} </math></center>

<center><math>= 4\epsilon \left( \frac{12\sigma^{12}}{r_0^{13}} -\frac{6\sigma^6}{r_0^7} \right)</math></center>

<center><math>= 24\epsilon \left( \frac{2\sigma^{12}}{r_0^{13}} -\frac{\sigma^6}{r_0^7} \right)</math></center>

as''' <math>r_0=\sigma</math>:'''

<center><math>= 24\epsilon \left( \frac{2\sigma^{12}}{\sigma^{13}} -\frac{\sigma^6}{\sigma^7} \right)</math></center>

<center><math>F = 24\epsilon \left( \frac{2}{\sigma} -\frac{1}{\sigma} \right)</math></center>

<center><math>= \frac{24\epsilon}{\sigma}</math></center>

E'''quilibrium separation, <math>r_{eq}</math>, is given by finding the minimum of the potential well:'''

<center><math>\frac{\phi (r)}{dr}= 24\epsilon \bigg( \frac{2\sigma^{12}}{r^{13}} -\frac{1\sigma^6}{r^7}\bigg)=0</math></center>

<center><math>\frac{2\sigma^{12}}{r^{13}} = \frac{\sigma^6}{r^7}</math></center>

<center><math>2\sigma^6=r^6</math></center>

<center><math> r_{eq} = 2^{1/6}\sigma</math></center>

'''Potential well depth:'''
<center><math>\frac{\phi (r)}{dr} = 4\epsilon \bigg( \frac{\sigma^{12}}{(2^{1/6}\sigma)^{12}} - \frac{\sigma^6}{(2^{1/6}\sigma)^{6}}\bigg)</math></center>

<center><math> = 4\epsilon \bigg( \frac{\sigma^{12}}{4\sigma^{12}} - \frac{\sigma^6}{2\sigma^{6}}\bigg)</math></center>

<center><math> = 4\epsilon \bigg( \frac{1}{4} - \frac{1}{2}\bigg)</math></center>

<center><math> = -\epsilon </math></center>

'''Integrals when <math>\sigma = \epsilon = 1.0</math>:'''
<center><math>\int_{2\sigma}^\infty \phi\left(r\right)\mathrm{d}r=4\int_{2\sigma}^\infty \bigg( \frac{1}{r^{12}} - \frac{1}{r^6}\bigg)\mathrm{d}r</math></center>

<center><math>=4\bigg[\frac{1}{-11r^{11}} + \frac{1}{5r^{5}}\bigg]_{2\sigma}^{\infty}</math></center>

<center><math>=-0.0248</math></center>

<center><math>\int_{2.5\sigma}^\infty \phi\left(r\right)\mathrm{d}r=\int_{2.5\sigma }^{\infty} 4\left ( \frac{1}{r^{12}}- \frac{1}{r^{6}}\right ) dr</math></center>

<center><math>=4\bigg[\frac{1}{-11r^{11}} + \frac{1}{5r^{5}}\bigg]_{2.5\sigma}^{\infty}</math></center>

<center><math>=-8.18\times10^{-3}</math></center>

<center><math>\int_{3\sigma}^\infty \phi\left(r\right)\mathrm{d}r=\int_{3\sigma }^{\infty} 4\left ( \frac{1}{r^{12}}- \frac{1}{r^{6}}\right ) dr</math></center>

<center><math>=4\bigg[\frac{1}{-11r^{11}} + \frac{1}{5r^{5}}\bigg]_{3\sigma}^{\infty}</math></center>

<center><math>=-3.29\times10^{-3}</math></center>

'''<big>TASK</big>: Estimate the number of water molecules in 1ml of water under standard conditions. '''

Assuming 1 mL of water has a mass of 1 g under standard conditions:

<center><math>N=\frac{M}{M(H_2 O)}N_a</math></center>

<center><math>=\frac{1\times 6.022 \times 10^{23}}{18.0}=3.35 \times 10^{22}\ </math></center>

'''<big>TASK</big>:Estimate the volume of <math>10000</math> water molecules under standard conditions.'''

Assuming standard conditionsː

<center><math>M =\frac{10000 \times M(H_2 O)}{N_a}</math></center>

<center><math>\frac{10000 \times 18.0}{6.022 \times 10^{23}}=2.989 \times 10^{-19}\ g </math></center>

<center><math>V=\frac{M}{\rho }</math></center>

<center><math>V=\frac{2.989 \times 10^{-19}}{1 \times 10^{-6}}= 2.989 \times 10^{-25}\ m^{3}</math></center>

'''<big>TASK</big>: Consider an atom at position <math>\left(0.5, 0.5, 0.5\right)</math> in a cubic simulation box which runs from <math>\left(0, 0, 0\right)</math> to <math>\left(1, 1, 1\right)</math>. In a single timestep, it moves along the vector <math>\left(0.7, 0.6, 0.2\right)</math>. At what point does it end up, ''after the periodic boundary conditions have been applied''?'''.

After the periodic boundary conditions are applied, the atom ends up atː

<center><math>\left(0.2, 0.1, 0.7\right)</math></center>

'''<big>TASK</big>: The Lennard-Jones parameters for argon are <math>\sigma = 0.34\mathrm{nm}, \epsilon\ /\ k_B= 120 \mathrm{K}</math>. If the LJ cutoff is <math>r^* = 3.2</math>, what is it in real units? What is the well depth in <math>\mathrm{kJ\ mol}^{-1}</math>? What is the reduced temperature <math>T^* = 1.5</math> in real units?

For the LJ cutoffː

<center><math>r^*=\frac{r}{\sigma}=3.2</math></center>

<center><math>3.2 \times 0.34 \times 10^{-9}=r</math></center>

<center><math>=1.09 \times 10^{-9} \ m </math></center>

For the well depth, from aboveː

<center><math>\frac{\phi (r)}{dr} = 4\epsilon \bigg( \frac{\sigma^{12}}{(2^{1/6}\sigma)^{12}} - \frac{\sigma^6}{(2^{1/6}\sigma)^{6}}\bigg)</math></center>

<center><math> = -\epsilon </math></center>

Thereforeː

<center><math> -\epsilon = -120k_B</math></center>

<center><math>=-1.657^{-21} \ J</math></center>
<center><math>=-0.9977 \ kJ \ mol^{-1}</math></center>

For the reduced temperatureː

<center><math>T^* = \frac{T}{\epsilon}</math></center>
<center><math>1.5 = \frac{T}{120}</math></center>

<center><math>T=1.5 \times 120</math></center>

<center><math>T=180 \ K</math></center>

== Equilibration ==

=== Creating the simulation box ===
'''<big>TASK</big>: Why do you think giving atoms random starting coordinates causes problems in simulations? Hint: what happens if two atoms happen to be generated close together?'''

Atoms could be generated very close to one another which would cause them to shoot out at a high speed due to the extremely large repulsive forces between the atoms. This would affect the simulation by disturbing the other atoms around it due to the high speed atoms travelling away from each other and effecting the interactions between all the other particles. By using a lattice, favorable distances can be chosen for a simulation to run smoothly and take a short amount of time to reach equilibrium.

'''<big>TASK</big>: Satisfy yourself that this lattice spacing corresponds to a number density of lattice points of <math>0.8</math>. Consider instead a face-centred cubic lattice with a lattice point number density of 1.2. What is the side length of the cubic unit cell?'''

<center><math> \frac{1}{0.8} = 1.25</math></center>

<center><math>= 1.25^\frac{1}{3} = 1.07722</math></center>

For a face-centred cubic latticeː
<center><math>x = \sqrt[3](\frac{4}{1.2})=1.4938</math></center>

<center><math>x = (\frac{4}{1.2})^\frac{1}{3} </math></center>
<center><math>x = 1.49380 </math></center>

'''<big>TASK</big>: Consider again the face-centred cubic lattice from the previous task. How many atoms would be created by the create_atoms command if you had defined that lattice instead?'''

An FCC lattice contains 4 lattice points, therefore for a box of side length 10, there are 4000 lattice points and therefore 4000 atoms will be generated.

=== Setting the properties of the atoms ===
'''<big>TASK</big>: Using the [http://lammps.sandia.gov/doc/Section_commands.html#cmd_5 LAMMPS manual], find the purpose of the following commands in the input script:'''

<pre>
mass 1 1.0
pair_style lj/cut 3.0
pair_coeff * * 1.0 1.0
</pre>

'mass 1 1.0' Relates to the mass of '''atom type''' '''1''' being '''1.0'''

'pair_style lj/cut 3.0' computes the standard LJ potential, but gives a cutoff point for the potential at point <<math>r_c</math>, with <math>r</math> being the distance between the atoms, along with having no colombic attractions between the atoms.

'pair_coeff * * 1.0 1.0' Sets the coefficents for all l,J pairs

'''<big>TASK</big>: Given that we are specifying <math>\mathbf{x}_i\left(0\right)</math> and <math>\mathbf{v}_i\left(0\right)</math>, which integration algorithm are we going to use?'''

We will be using the velocity Verlet algorithm.

[[File:ThirdYearSimulationExpt-Intro-VelocityVerlet-flowchart.svg|300px|thumb|centre|'''Figure 1''': Steps to implement the velocity Verlet algorithm.]]

=== Running the simulation ===
'''<big>TASK</big>: Look at the lines below.'''
<pre>
### SPECIFY TIMESTEP ###
variable timestep equal 0.001
variable n_steps equal floor(100/${timestep})
timestep ${timestep}

### RUN SIMULATION ###
run ${n_steps}
run 100000
</pre>
'''The second line (starting "variable timestep...") tells LAMMPS that if it encounters the text ${timestep} on a subsequent line, it should replace it by the value given. In this case, the value ${timestep} is always replaced by 0.001. In light of this, what do you think the purpose of these lines is? Why not just write:'''
<pre>
timestep 0.001
run 100000
</pre>This is so timestep is defined as a variable and so can be changed later after the code has been run so it does not need to be defined again and again throughout the code, but can be overwritten easily for use.

=== Checking equilibration ===
'''<big>TASK</big>: make plots of the energy, temperature, and pressure, against time for the 0.001 timestep experiment (attach a picture to your report). Does the simulation reach equilibrium? How long does this take? When you have done this, make a single plot which shows the energy versus time for all of the timesteps (again, attach a picture to your report). Choosing a timestep is a balancing act: the shorter the timestep, the more accurately the results of your simulation will reflect the physical reality; short timesteps, however, mean that the same number of simulation steps cover a shorter amount of actual time, and this is very unhelpful if the process you want to study requires observation over a long time. Of the five timesteps that you used, which is the largest to give acceptable results? Which one of the five is a ''particularly'' bad choice? Why?'''
[[File:Ft1015Equilibrium7.PNG|600px|thumb|centre|A zoomed in plot of the total energy vs time plot to show that the system equilibrates after 0.002 ps]]
[[File:Ft1015Energy7.PNG|600px|thumb|centre|Total energy vs time for a 0.001 time step]]
[[File:Ft1015Pressure7.PNG|600px|thumb|centre|Pressure vs time for a 0.001 time step]]
[[File:Ft1015Temperature7.PNG|600px|thumb|centre|Temperature vs time for a 0.001 time step]]
[[File:ft1015_Timestep_Graph_Plots.PNG|600px|thumb|centre|A plot five different timesteps and the energies they converge on]]
The largest timestep of 0.015 is a particularly bad choice as the energy does not converge, but increases dramatically, something which should not happen. The largest to give acceptable results is 0.0025, as though 0.001 and 0.0025 both converge to the same value, 0.0025 allows a simulation which simulates 250% more time than the 0.001 timestep. This is more practical as it allows more simulations and longer simulations to be done in the same time-frame as the smaller timestep, without losing accuracy or precision.

== Running simulations under specific conditions ==

=== Temperature and Pressure Control ===
'''<big>TASK</big>: Choose 5 temperatures (above the critical temperature ), and two pressures (you can get a good idea of what a reasonable pressure is in Lennard-Jones units by looking at the average pressure of your simulations from the last section). This gives ten phase points — five temperatures at each pressure. Create 10 copies of npt.in, and modify each to run a simulation at one of your chosen points. You should be able to use the results of the previous section to choose a timestep. Submit these ten jobs to the HPC portal. While you wait for them to finish, you should read the next section.'''

===Thermostats and Barostats===
'''<big>TASK</big>: We need to choose <math>\gamma</math> so that the temperature is correct <math>T = \mathfrak{T}</math> if we multiply every velocity <math>\gamma</math>. We can write two equations:'''

<math>\frac{1}{2}\sum_i m_i v_i^2 = \frac{3}{2} N k_B T</math>

<math>\frac{1}{2}\sum_i m_i \left(\gamma v_i\right)^2 = \frac{3}{2} N k_B \mathfrak{T}</math>

'''Solve these to determine <math>\gamma</math>.'''

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math> \frac{\frac{1}{2}\sum_i m_i \left(\gamma v_i\right)^2}{\frac{1}{2}\sum_i v_i^2} = \frac{\frac{3}{2} N k_B \mathfrak{T}}{\frac{3}{2} N k_B T} </math></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math> \frac{\sum_i \left(\gamma v_i\right)^2}{\sum_i v_i^2} = \frac{\mathfrak{T}}{T} </math></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math> \frac{\sum_i \gamma^2 v_i^2}{\sum_i v_i^2} = \frac{\mathfrak{T}}{T} </math></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math> \frac{\gamma^2 \sum_i v_i^2}{\sum_i v_i^2} = \frac{\mathfrak{T}}{T} </math></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math> \gamma^2 = \frac{\mathfrak{T}}{T} </math></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math> \gamma = \sqrt{\frac{\mathfrak{T}}{T}} </math></div>

=== Examining the Input Script ===
'''<big>TASK</big>: Use the manual page to find out the importance of the three numbers ''100 1000 100000''. How often will values of the temperature, etc., be sampled for the average? How many measurements contribute to the average? Looking to the following line, how much time will you simulate?'''

The first number defines the length of the timestep between data points taken for the calculation of the average, the second number relates to how many timesteps will be used to calculate the average at the specificed time, the last number specifies at what timestep will an average be taken. Thus 100 1000 100000 means that data points for the average will be taken every 100 timesteps, and 1000 will be used to calculate the average at 100000 timesteps. The time simulated will be 100000 * the timestep in seconds, which for 0.0025 s per timestep means that 250 s will be simulated.

'''<big>TASK</big>: When your simulations have finished, download the log files as before. At the end of the log file, LAMMPS will output the values and errors for the pressure, temperature, and density <math>\left(\frac{N}{V}\right)</math>. Use software of your choice to plot the density as a function of temperature for both of the pressures that you simulated. Your graph(s) should include error bars in both the x and y directions. You should also include a line corresponding to the density predicted by the ideal gas law at that pressure. Is your simulated density lower or higher? Justify this. Does the discrepancy increase or decrease with pressure?'''

[[File:ft1015_Standard_Error_plots_graph.PNG|600px|thumb|centre| A plot of average density against average critical temperature of the system with standard error error bars]]

The discrepancies between the idealised and simulated number density results can be attributed to the simulation being non-ideal. The basis of a L-J system is that the particles interact with each other, with a short range repulsive term, and a longer range attractive term. The ideal gas law however, assumes that there are no interactions between particles and thus, due to the short range highly repulsive term, the particles are further apart than in a system with no interactions and therefore the number density of the simulated systems are lower than their idealised counterparts.

At higher pressures the deviations from ideality are larger than that of a lower pressure simulation. This arises from the fact that at higher pressures, the number density is higher due to decreased cell volume, leading to decreased distances between particles and therefore increased interactions between them as they collide with each other more frequently. This increase in interactions between particles leads to a large deviation from ideality and so a larger difference between simulated and ideal law data for higher pressures.

== Calculating heat capacities using statistical physics ==

===Heat Capacity Calculation===
'''<big>TASK</big>: As in the last section, you need to run simulations at ten phase points. In this section, we will be in density-temperature <math>\left(\rho^*, T^*\right)</math> phase space, rather than pressure-temperature phase space. The two densities required at <math>0.2</math> and <math>0.8</math>, and the temperature range is <math>2.0, 2.2, 2.4, 2.6, 2.8</math>. Plot <math>C_V/V</math> as a function of temperature, where <math>V</math> is the volume of the simulation cell, for both of your densities (on the same graph). Is the trend the one you would expect? Attach an example of one of your input scripts to your report.'''

[[File:Ft1015HeatCapacityScript.PNG|600px|thumb|centre|An image of one of the scripts used for the heat capacity calculations]]

[[File:ft1015HeatCapacity.png|600px|thumb|centre|A plot of <math> {C_{V}}/{V} </math> against <math> {T}/{T*} </math> for two different densities]]

The decrease of heat capacity with temperature increase excluding phase changes, can be explained through the Density of States. Similar to the anharmonic oscillator model for electron excitation for an atom or particles where as the energy levels are increased, the spacing between the levels decrease, as the temperature of our system is increased the spacing between energy levels contracts, and thus the higher energy levels come down in energy and become more accessible. This leads to more modes available for promotion in the system, thus ease of promotion of an atom increases, leading to a reduced energy needed to promote and atom and so heat capacity decreases with temperature.

The large jumps in heat capacity seen in both data sets at <math>T^* = 2.6</math> can be explained through phase changes. As the particles simulated are totally symmetric there are no extra degrees of freedom to be reached through an increase in temperature as only translational modes are active through equipartition theorem, therefore the jump in heat capacity must be due to a phase change. Assuming a second order phase transition; as an NVT ensemble is used, and in a first order transition density is discontinuous which is impossible for a system with fixed <math>N</math> and <math>V</math>, Heat capacity rises to infinity until after the transition has been completed, after which it lowers again, which can be seen at <math>T^* = 2.8</math>, due to thermal energy needing to be expended in order to complete the phase transition.

== Structural properties and the radial distribution function ==

===Simulations in this section===

'''<big>TASK</big>: perform simulations of the Lennard-Jones system in the three phases. When each is complete, download the trajectory and calculate <math>g(r)</math> and <math>\int g(r)\mathrm{d}r</math>. Plot the RDFs for the three systems on the same axes, and attach a copy to your report. Discuss qualitatively the differences between the three RDFs, and what this tells you about the structure of the system in each phase. In the solid case, illustrate which lattice sites the first three peaks correspond to. What is the lattice spacing? What is the coordination number for each of the first three peaks?'''

[[File:ft1015G(r).png|720px|thumb|centre|'''Figure 4''': ft1015G(r).png]]

[[File:ft1015G(r)N.png|720px|thumb|centre|'''Figure 4''': ft1015G(r)N.png]]

The RDF for the gas one broad peak that drops to 1 quickly after the peak finishes, leading to there being no long range order as the probability of finding an particle outside of the first shell is equal everywhere. This shows that there is no long range order in a gas, only extremely short range order. The peak could be broad due to weak attractive effects holding the nearest neighbors close, but not tightly holding them in place.

The liquid RDF has three broad peaks, leading to 3 co-ordination spheres of particles held tightly, but not so tightly that the particles cannot move, therefore the spheres are very lightly diffuse, but like the gas the RDF drops to 1 after the last peaking, leading to no long range order in a liquid.

The RDF of the solid has many sharp peaks corresponding to the solid atoms fixed positions and long range order in the crystal lattice.

The lattice spacing is the minimum distance between lattice unit cells which is equal to 1.175.

<center>
{| class="wikitable"
|-
! r
! Integrated g(r)
! Co-ordination Number
|-
| 1.175
| 11.85813
| 12
|-
| 1.675
| 18.44288
| 6
|-
| 1.975
| 42.26256
| 24
|}
</center>

= Dynamical properties and the diffusion coefficient =

=== Simulations in this Section ===
'''<big>TASK</big>: In the D subfolder, there is a file ''liq.in'' that will run a simulation at specified density and temperature to calculate the mean squared displacement and velocity autocorrelation function of your system. Run one of these simulations for a vapour, liquid, and solid. You have also been given some simulated data from much larger systems (approximately one million atoms). You will need these files later.'''

=== Mean Squared Displacement ===
'''<big>TASK</big>: make a plot for each of your simulations (solid, liquid, and gas), showing the mean squared displacement (the "total" MSD) as a function of timestep. Are these as you would expect? Estimate <math>D</math> in each case. Be careful with the units! Repeat this procedure for the MSD data that you were given from the one million atom simulations.'''

From https://en.wikipedia.org/wiki/Mean_squared_displacementː

<center><math> MSD = \langle (x(t)-x_0)^2 \rangle = 2Dt </math></center>

[[File:ft1015GasMSD.png|720px|thumb|centre|'''Figure 4''': ft1015GasMSD.png]]
[[File:ft1015GasMSD_1m.png|720px|thumb|centre|'''Figure 5''': ft1015GasMSD_1m.png]]
[[File:ft1015LiquidMSD.png|720px|thumb|centre|'''Figure 6''': ft1015LiquidMSD.png]]
[[File:ft1015LiquidMSD_1m.png|720px|thumb|centre|'''Figure 7''': ft1015LiquidMSD_1m.png]]
[[File:ft1015SolidMSD.png|720px|thumb|centre|'''Figure 8''': ft1015SolidMSD.png]]
[[File:ft1015SolidMSD_1m.png|720px|thumb|centre|'''Figure 9''': ft1015SolidMSD_1m.png]]

=== EXTENSION: Velocity Autocorrelation Function ===
'''<big>TASK</big>: In the theoretical section at the beginning, the equation for the evolution of the position of a 1D harmonic oscillator as a function of time was given. Using this, evaluate the normalised velocity autocorrelation function for a 1D harmonic oscillator (it is analytic!):

<center><math>C\left(\tau\right) = \frac{\int_{-\infty}^{\infty} v\left(t\right)v\left(t + \tau\right)\mathrm{d}t}{\int_{-\infty}^{\infty} v^2\left(t\right)\mathrm{d}t}</math></center>

'''Be sure to show your working in your writeup. On the same graph, with x range 0 to 500, plot <math>C\left(\tau\right)</math> with <math>\omega = 1/2\pi</math> and the VACFs from your liquid and solid simulations. What do the minima in the VACFs for the liquid and solid system represent? Discuss the origin of the differences between the liquid and solid VACFs. The harmonic oscillator VACF is very different to the Lennard Jones solid and liquid. Why is this? Attach a copy of your plot to your writeup.'''

'''<big>TASK</big>: Use the trapezium rule to approximate the integral under the velocity autocorrelation function for the solid, liquid, and gas, and use these values to estimate <math>D</math> in each case. You should make a plot of the running integral in each case. Are they as you expect? Repeat this procedure for the VACF data that you were given from the one million atom simulations. What do you think is the largest source of error in your estimates of D from the VACF?'''

Rep talk:Mod:NPG11041993

2025-09-17T11:24:46Z

Jbettenc: Jbettenc moved page Talk:Wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:NPG11041993 to Rep talk:Mod:NPG11041993 without leaving a redirect: Move to report namespace

Very good exercise, complemented with much extra information and insight. Pity that some parts of the exercise were not completed. Nice go at introduction to the theory. Computational chemistry is not only limited at finding the electronic energy of the molecule (this branch is informally called quantum chemistry) as done in this exercise, it does not even need to solve the Schrödinger equation in any form: there is a lot of computational chemistry out there! When optimizing conformers, were frequencies all positive? You correctly compare the energies at different levels of theory. Are the geometries comparable? In the IRC section you mention the system goes through a syn conformation. Is there a stable syn conformer? Which stable conformer is the end of the IRC in each case? You don't seem to have reoptimised the transition state at higher level or compared the activation energies with experimental values. In your diagram showing the secondary orbital effect on the activation energy, should the products have the same energy? Missing analysis of the orbitals of the butadiene-ethene transition state and connection with reactants orbitals. Regarding your comment on bond length in the Maleic Anhydride + Cyclopentadiene reaction, why would you not expect a large bond distance at the transition state when bonds are forming? [[User:Jbettenc|Jbettenc]] ([[User talk:Jbettenc|talk]]) 11:51, 4 December 2014 (UTC)

Rep:Mod:NPG11041993

2025-09-17T11:24:46Z

Jbettenc: Jbettenc moved page Wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:NPG11041993 to Rep:Mod:NPG11041993 without leaving a redirect: Move to report namespace

==Abstract==

Computational chemistry has been developed over the last half-century and is now one of the most powerful methods of analyzing the dynamics of chemical reactions. The subject area is primarily occupied with obtaining numerical solutions to the Schrodinger equation, which allows for the calculation of energy for a given system of atoms and/or molecules. Calculations over an collection of different configurations for any such system will yield a potential energy hypersurface, which can be interrogated to give information about the dynamics of reactions and other processes.

In this experiment, a number of different computations are carried out to determine the energies and dynamics of three key areas in organic chemistry: the conformations of molecules; the properties of the cope rearrangement; and the stereoselectivity of a Diels-Alder reaction. Key concept in the theory of organic chemistry are applied and rationalized through the use of these computations. The results are then used to explain various phenomena associated with these three processes.

== Background And Motivation ==

===The Shrödinger Equation===

The use of the Schrödinger Equation has been a key aspect of the study of atoms and molecules since its inception in 1926.

Any given system of atoms (such as a molecule, or a number of molecules) has associated with it a mathematical function, the wavefunction (Ψ), which can be interrogated to give observable information about the system. This interrogation takes place through the use of operators, and will return the product of the original wavefunction and a value for the desired observable. This is known as the ''eigenvalue equation'':

<math>\hat{\Omega}\,\psi =o\cdot\psi</math> '''(1)'''

Where <math>\hat{\Omega}</math> is the operator, <math>o</math> is the value of the observable, and <math>\psi</math> is the wavefunction .

The Time independent Schrödinger Equation (TISE) uses one such operator - the Hamiltonian - to return the total energy of the system <ref>Hinchliffe, Alan. Computational Quantum Chemistry. p.3, Chichester: Wiley, 1988. Print.</ref> :

<math>\hat{H}\,\psi =E\cdot\psi </math> '''(2)'''

Exact solutions of the Schrodinger equation for one electron systems - such as the hydrogen atom and the dihydrogen cation - are known. However, analytical solutions for multiple electron systems cannot be calculated exactly and thus must be approximated. Using successively more complicated Hamiltonian operators allows for better approximation of the energy of a molecular system. These calculations cannot be easily performed by hand and are thus must be solved iteratively using computers. Performing these calculations forms the basis of computational chemistry.<ref>Hinchliffe, Alan. Computational Quantum Chemistry. p.12, Chichester: Wiley, 1988. Print.</ref>

===The Hartree Matrix Description===

Let the Hamiltonian for a multiple electron system be described as the total energy arising from three separate contributions:

* Kinetic energy
* Coulomb attraction energy
* Coulomb repulsion energies with all other electrons in the system

Thus, the Hamiltonian becomes:

<br>
<math> \hat{H}= \hat{T}+\hat{V}_{en}+\hat{V}_{ee}</math> '''(3)'''

<br>

where <math>\hat{H}</math> is the Hamiltonian and the terms on the right represent the operators for kinetic, attractive and repulsive energies respectively<ref>Hinchliffe, Alan. Computational Quantum Chemistry. p.19, Chichester: Wiley, 1988. Print.</ref>. Thus the energy of the system may be extracted by operating on some Hartree orbital <math>\left \langle \phi_i\right |</math>. This Hartree orbital can, in turn be represented using the LCAO method, becoming a linear combination of basis orbitals, thus:

<br>
<math> \left \langle \phi_i\right | = \sum_{n}C_{i,n}\cdot \chi_n </math> '''(4)'''
<br>

Substituting equation 4 into the Schrödinger equation yields the ''Hartree Matrix Equation'':

<br>
<math> \sum_{n}C_{i,n} \cdot \left \langle \chi_n \left | \hat{H} \right | \chi_m \right \rangle = E_n \cdot \sum_{n}C_{i,n} \cdot \left \langle \chi_n | \chi_m \right \rangle </math> '''(5)'''

<br>

This equation is analogous to the eigenvalue equation outlined above. Here the energy <math>E_n</math> and weighting coefficients <math>C_{i,n}</math> are the eigenvalues and eigenvectors of the equation respectively. The terms <math> \left \langle \chi_n \left | \hat{H} \right | \chi_m \right \rangle </math> and <math> \left \langle \chi_n | \chi_m \right \rangle </math> are referred to as the Hamiltonian and overlap matrices respectively.

Both the Hamiltonian matrix and <math>C_{i,n}</math> must be solved. However, in order to find <math>C_{i,n}</math>, one must solve the Hamiltonian matrix. Paradoxically, in order to solve the Hamiltonian matrix, one must know <math>C_{i,n}</math>. This can be resolved using the ''self-consistent field'' (SCF) technique, in which an initial guess for <math>C_{i,n}</math> is made. This allows for computation of the Hamiltonian matrix, which in turn allows for construction of equation 5. The equation can then be solved iteratively using the values for <math>C_{i,n}</math> derived in each previous iteration. This is repeated until the difference in values between <math>C_{i,n}</math> in the n<sup>th</sup> and (n+1)<sup>th</sup> iteration have become negligible. When this occurs, it is said that a ''self-consistent field'' has been derived. The final value of <math>C_{i,n}</math> can be plugged back into the Hartree Matrix Equation and a value for <math>E_n</math> can be extracted<ref>Hinchliffe, Alan. Computational Quantum Chemistry. p.21, Chichester: Wiley, 1988. Print.</ref>.

Additional corrections for orbital symmetry under electron interchange must be made. The combination of the Hartree Matrix Equation with symmetry corrections is referred to as the ''Hartree-Fock Approximation''. More complex mothods (such as Density Functional Theory) exist and can be used in place of the Hartree-Fock approximation.

===The Basis Orbitals (χ<sub>n</sub>)===

The basis set of orbitals (χ<sub>n</sub>) can typically be one of three types:

*Screened Hydrogenic Orbitals
*Slater-Type Orbitals
*Gaussian-Type Orbitals

Hydrogenic orbitals are typically not used in computational chemistry and thus may be ignored entirely. Slater orbitals are typically used for linear systems of molecules and diatoms, but the integrals that result from higher order or non-linear systems cannot be computed. Gaussian type orbitals (so named, because they include a gaussian distribution function e<sup>-αr<sup>2</sup></sup>) however are perfectly solvable for these systems and can be modified to ensure the presence of a derivative 'cusp' st their zero vector. These modified orbitals are referred to as ''contracted Gaussian-type orbitals'' and comprise the basis of the orbital sets used in the Gaussian software package<ref>Hinchliffe, Alan. Computational Quantum Chemistry. p.22, Chichester: Wiley, 1988. Print.</ref>.

===Generating a Potential Energy Surface===

Both the approximation method and the basis orbital set are freely selected and are collectively referred to as the ''level of theory'' at which a calculation is carried out. Larger basis sets and more intricate operators will generate more precise numerical values, but require a greater amount of processing time. Thus a balance must be struck between system resources and the desired accuracy of the answer.

Once a particular value for an equation is derived, further calculations can be conducted to find how the energy of the system would change under differing arrangements of the atoms within the system. A system with N atoms will require 3 individual co-ordinates per atom, a total of 3N co-ordinates per system. Translation and rotation of the system as a whole will not effect the energy whatsoever, and can therefore be discounted, meaning only 3N-6 co-ordinates are required. The energy can be therefore be represented as a function of these co-ordinates:

<math>E_{sys}=f(\textbf{r}_{1},\textbf{r}_{2},\cdots )</math> '''(6)'''

Where <math>f(\textbf{r}_{1},\textbf{r}_{2},\cdots )</math> represents the functional form of a 'plot' resulting from the solution of the Schrodinger equation for each individual configuration of atoms <math>\textbf{r}_{1},\textbf{r}_{2},\cdots</math>. The result is a hypersurface of dimension 3N-6 which yields the energy of the system as a function of the position vectors for each atom. We define this as the ''potential energy hypersurface'' and may utilise it to analyse the dynamics of chemical reactions. <ref>Leach, Andrew R. Molecular Modelling: Principles and Applications. p.4, Harlow, England: Prentice Hall, 2001. Print.</ref>

== Conformational study of 1,5-hexadiene ==

1,5-Hexadiene can exist in a number of different interchangeable conformations, with the dihedral angles between the central four carbon atoms providing the primary route of conformational interchange. The terminal carbon atoms are conformationally locked by virtue of their C=C bond and can thus be largely neglected. The resultant manifold conformers of 1,5-Hexadiene can therefore be broadly arranged into 3 types:<ref>Isaacs, Neil S. Physical Organic Chemistry. p.341 Harlow: Longman, 1995. Print.</ref>

<br> '''i) Syn Conformers:''' The central chain of four carbon atoms are eclipsed and have a dihedral angle of approximately 0 degrees.
<br> '''ii) Anti Conformers:''' The dihedral angle between the central carbon atoms angle is approximately 180 degrees.
<br> '''iii) Gauche Conformers:''' Central carbon atoms have a dihedral angle of 60 degrees.

The relative energies of these three classes of conformer are governed by 3 key effects: <ref>Isaacs, Neil S. Physical Organic Chemistry. p.342 Harlow: Longman, 1995. Print.</ref>

<br> '''i) σ - σ* Orbital Overlap'''

Overlap between the bonding σ orbital and an adjacent σ* anti bonding orbital results in an overall stabilization of the filled bonding orbital by an energy <math>E_{Stab}</math>, and a destabilization of the empty anti-bonding orbital by an energy <math>E_{stab}+\epsilon</math>. The result is an overall stabilization of the filled natural bonding orbital (NBO, resulting in a stabilisation of the molecule itself.

<br> '''ii) Pauli Repulsion'''

In a similar manner to the σ - σ* overlap mentioned above. If two σ bonds overlap then the resulting NBO has both bonding and anti-bonding character. The anti-bonding orbital is more destabilising than the bonding orbital is stabilising, however, and thus the result is an overall destabilization of the molecule.

Other effects - such as dispersion forces - manifest themselves in larger molecules but will largely be discounted here.

In 1,5-hexadiene, the terminal C=C bonds are fixed and therefore do not play a major role in the generation of conformers. However, the central alkyl chain is free to rotate and thus determine the overall configuration of the molecule. These four atoms can therefore be: gauche, antiperiplanar, eclipsed, or some combination of the three. We would expect a fully antiperiplanar configuration to generate the most stable species, as this would maximize σ - σ* overlap while minimizing Pauli repulsion. Conversely, the eclipsed - or synperiplanar - conformation would maximize Pauli repulsion and minimize σ - σ* overlap, resulting in maximum destabilization. The gauche conformation would lie somewhere in between the two.

=== Experimental ===

The relative stabilities of these conformers was assessed using the Gaussian software package. A 1,5-hexadiene molecule was generated with an antiperiplanar linkage between each of the 4 central carbon atoms, symmetrised and was optimised at the HF/3-21G level of theory. The resulting energy of the optimised molecule was noted. The molecule was symmetrised and re-optimised at the same level of theory, yielding a Ci point group. The process was then repeated at the same level of theory for a gauche conformer of the molecule. Vibrational calculations were then generated at the same level of theory for both molecules.

The Ci Conformer molecule was optimized again at the B3LYP/6-31G* level of theory and compared to the results of the HF/3-21G calculations.

=== Results and Discussion ===

<br>

{|
|<jmol>
<jmolApplet>
<title>Anti Hexa-1,5-diene (Ci Symmetry)</title><color>white</color>
<size>300</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>ANTI_Ci_CLN_OPT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>Gauche Hexa-1,5-diene (C2 Symmetry)</title><color>white</color>
<size>300</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>GaucheC6OPTIMISED.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|}

''Figure 1: The optimized conformers of 1,5,-hexadiene''

{|class="wikitable"
!!! colspan="3" | Energy (Hartrees)
|-
| '''Species''' || '''Absolute''' || '''+ ZPE''' || '''Thermal Correction'''
|-
|'''Gauche''' || -231.68771615 || -231.539540 || -231.527699
|-
|'''Anti (Ci)''' || -231.69253528 || -231.534385 || -231.532567
|}

''Figure 2: Relative energies of the HF/3-21G optimized species displayed in Figure 1''

<br>

Figure 2 shows the energy of the bare potential energy surface at its minimum point, along with corrections for zero-point energy (ZPE) and thermal energy input. It can be seen that the thermally corrected energy for the anti (Ci) conformation is lower than that of the Gauche conformer. This is largely in line with the predictions outlined in the introduction, where the anti configuration maximizes σ - σ* interactions while minimizing Pauli repulsions. Furthermore the trend observed in these computations is in line with those reported by Gung et al (2010).<ref>doi:10.1021/ja00111a016</ref> and in the lab script.

It has also reported that the global minimum for the system as a gauche conformer in which an interaction between a vinyl proton and an opposing π - orbital provides an additional stabilization energy.

{|class="wikitable"
!!! Colspan="7" | Energy (Hartrees)
|-
|||colspan="3"|'''HF/3-21G''' || colspan="3"|'''B3LYP/6-31G*'''
|-
|'''Species'''|| '''Absolute''' || '''+ZPE''' || '''Thermal Correction''' || '''Absolute''' || '''+ZPE''' || '''Thermal Correction'''
|-
| '''Anti Ci''' || -231.699508 || -231.539540 || -231.532567 || -234.611711 || -234.469217 || -234.461867
|-
| '''Gauche''' || -231.687716 || -231.534385 || -231.527699 || -234.607883 || -234.465193 || -234.458096
|}

''Figure 3: Comparison of the energies (absolute) obtained by the HF/3-21G and the B3LYP/6-31G* levels of theory''

Figure 1.3 shows a comparison between the results obtained for both conformers through application of the HF/3-21G and B3LYP/6-31G* levels of theory. In every case, the results obtained by through application B3LYP/6-31G* are significantly lower (by approximately 1 Hartree) than those obtained through calculation with the HF/3-21G level of theory. This suggests that the the Hartree-Fock method significantly overestimates the energy of the molecule.

== The Cope Rearrangement ==

The Cope Rearrangement is a widely studied pericyclic reaction involving the [3,3]-sigmatropic rearrangement of 1,5-dienes. The reaction can occur in both directions, with the stability of both species determining the overall position of the resulting equilibrium. The mechanism for the rearrangement has been hotly debated, although the literature generally agrees that the rearrangement takes place through a concerted diradical transition state.<ref>Clayden, Jonathan. Organic Chemistry. Oxford: Oxford UP, 2001. Print.</ref> This allows for two seperate transition structures, referred to as the 'boat-like' and 'chair-like' transition structures to take place in equal proportion. Both Transition states are 6-electron [π2s + σ2s + π2s] thermally allowed processes and thus are expected to have approximately similar energies. If all else is equal, then the chair transition state is slightly favoured over that of the boat transition state.<ref>Fleming, Ian. Pericyclic Reactions. p. 78 Oxford: Oxford UP, 1999. Print.</ref>

[[File:Cope_110493NG.jpg|center]]
<div style="text-align: center;">''Figure 2.1: The cope rearrangement, showing the boat and chair transition states''</div>

====Computing a Transition State====

From figure 2.1 it can be seen that the transition state consists of 2 allyl fragments arranged in the boat and chair conformation respectively. Thus it is a relatively trivial exercise to generate these fragments separately and arrange them into the appropriate conformation. An optimization to a transition state can therefore take place from this initial configuration.

The transition state may be defined as a saddle point along the minimum reaction pathway. Thus, it is a minimum with respect to the potential energy surface and a maximum with respect to the energy of the intrinsic reaction co-ordinate. The nature of the curvature of these points can thus be probed by computing the force constants at any particular reaction co-ordinate while observing that the force constants of the molecule are equal to the second derivative of the potential energy surface. Thus the nature of the force constant can be interrogated to yield information about any given reaction co-ordinate. Negative curvature of the potential energy surface (i.e a local maximum) for some given direction will generate a negative second derivative and thus will result in a negative force constant.<ref>Leach, Andrew R. Molecular Modelling: Principles and Applications. p.5, Harlow, England: Prentice Hall, 2001. Print.</ref>

Thus a number of different methods for computing the transition state may be utilized:

'''Method I:'''

A guess structure may be inputted and optimized to a transition state through stepwise computation of force constants. For small perturbations from the true transition structure, this method will almost always converge. However, if the guess structure is significantly different from that of the true transition structure, than no convergence will occur and the computation will fail.

'''Method II:'''

A guess structure is input, with bond making atoms set to their approximate bond making distances. These atoms are then 'frozen' (i.e locked to their present co-ordinates) and the remainder of the system is optimized. Movement of the bond making atoms is then allowed and the new structure is optimized to the transition state through calculation of force constants for all bond making atoms.

Conceptually, this optimization takes place in two phases. The first phase relaxes the molecule so the it is close to the potential energy minimum, or ''reaction channel''. The second phase 'tracks' this potential energy minimum to the local maximum (i.e.: the transition state). This therefore has the advantage of being able to arrive at a transition state for more dissociated geometries, but may still be divergent for extremely abnormal systems.

'''Method III:'''

Both the starting molecule(s) and the product are defined. Both molecules are then optimized forwards and backwards respectively to their transition states. If the starting system and the product both converge on the same transition structure, then the transition state for this particular reaction has been arrived at. This method can be implemented through the QTS2 functionality present on Gaussian.

In this exercise, all three methods will be utilized to probe the nature of the boat and chair transition states of the cope rearrangement.

=== Experimental ===

Both transition structures were treated as 2 allyl fragments. The allyl fragment was generated and optimized to a minimum using the HF/3-21G level of theory. The optimized fragment was then copied and the two resulting fragments arranged into a boat and chair conformation. These conformations were then optimized using the three different methods outlined above:

'''Method I:'''

The molecule was optimized directly to a transition state using the Berny Algorithm at the HF/3-21G level of theory. The force constant (Hessian) matrix was computed once and the keyword Opt=NoEigen was used to prevent early termination of the calculation upon detection of imaginary negative vibrational modes.

'''Method II:'''

The terminal ends of each allyl group were frozen at approximately 2.2 Å from each other. The resulting system was then optimized at the HF/3-21G level of theory, keeping the terminal ends frozen. The bonds were unfrozen and the molecule was then optimized to a transition state at the HF/3-21G level of theory using a numerical Hessian and the Berny Algorithm.

'''Method III:'''

The precise nature of the transition state was elucidated through the use of the QTS2 method. The initial and final forms of the 1,5-hexadiene were inputted and allowed allowed to converge to a transition state.

Arrival at a transition structure was confirmed by the presence of imaginary negative frequencies which imply negative curvature of the potential energy surface.

==== IRC Calculations From the Transition State ====

Further computations were performed using the IRC functionality on Gaussian with the boat and chair transition structures utilized as a starting point. Computations were performed at the HF/3-21G level of theory. Computations were carried out in the forward direction only due to symmetric nature of the rearrangement.

=== Results ===
[[File:Chair_ts_PartB_OPT_OPT_OPT.gif|thumb|Animation of the imaginary bond making vibration for the chair conformer at -817.96 wavenumbers]]
[[File:Cope_Rearrangement_Vibration.gif|thumb|Animation of the vibration for the boat conformer at -840.27]]

The vibrational modes of the Boat-like and chair-like transition states were confirmed through identification of imaginary negative wavelengths. Imaginary bond making vibrations at -817.96 cm<sup>-1</sup> and -840.27 cm<sup>-1</sup> for the chair and boat conformers respectively were identified. These are shown below.

{|class="wikitable"
! !! colspan="4" | Energy (Hartrees)
|-
|'''Method'''|| colspan="2" | '''Boat TS''' || colspan="2" | '''Chair TS'''
|-
| || +'''ZPE''' || '''Thermal Correction''' || '''+ZPE''' || '''Thermal Correction'''
|-
|'''Method I'''|| - || - || -231.466699 || -231.461340
|-
|'''Method II'''||-231.450933||-231.445305|| -231.466700 ||-231.461338
|-
|'''Method III'''|| -234.414612 || -234.408690|| - || -
|}

''Figure 2.2: Table showing the energies of the boat-like and chair-like transition states.''

Figure 2.1 shows the energies of the the boat like and chair like transition states using the three methods outlined in the experimental. Methods I and II give very similar answers for the energy of the chair and boat transition states, with a discrepancy of approximately 0.0000001 hartrees. This strongly suggests that both methods have converged to the same transition state. Furthermore, the very low level of discrepancy suggests that both methods could be used interchangeably with little detriment to the accuracy of the final answer. A significant discrepancy exists in the energies computed using method II and method III, suggesting that one method may lack accuracy. It can also be seen that the derived transition state for the chair conformer is slightly lower in energy than that of the boat conformer.

==== IRC Calculations ====

It may be observed that the boat and chair transition states both link separate conformers of hexa-1,5-diene. Thus, one particular configuration of the starting compound will generate one specific transition structure.<br> Because the reaction pathway for the cope rearrangement is symmetrical, it may be observed that the one particular configuration of hexa-1,5-diene will generate one particular transition structure and vice-versa.<br> This is illustrated below:

[[File:CHAIR_TS_IRC_NPG110493.gif|200px|thumb|right|Figure2.4: Visualisation of the IRC calculation for the chair transition state]] [[File:BOAT_IRC_Output_110493NPG.gif|200px|thumb|right|Figure2.5: Visualisation of the IRC calculation for the boat transition state]]

<br>

[[File:Configurational_preservation.jpg | center]]

<br>

<div style="text-align: center;">''Figure 2.3: Diagram illustrating how the dihedral angle of the propylene groups in hexa-1,5-diene are preserved during the cope rearrangement. In this case, the two propylene side chains are oriented in the gauche configuration. By the same rationale, two synperiplanar propylene groups will rearrange through the boat transition state. Antiperiplanar systems can be ignored as the resulting transition structure would be too high in energy to occur''</div>

This was confirmed through the 'IRC method' outlined in the experimental. The boat and chair transition states were used as starting points and the intrinsic traction pathway was calculated in the forward direction. The symmetric nature of the cope rearrangement meant that IRC calculations could be carried out in one direction only without loss of pertinent information.

Figures 2.4 and 2.5 show the outcomes of these calculations. Viewing the animation for figure 2.4, it is easily seen that the initial result of the rearrangement is a gauche conformer, which subsequently relaxes to a lower energy state. A similar outcome is seen in Figure 2.5, in which the result of the rearrangement is the synperiplanar conformer. However, the strong Pauli repulsions and favourability of σ-σ* interactions result in a relaxation away from the synperiplanar to the gauche configuration.

The results of the IRC calculations also shed light on the favourability of the chair transition state with respect to the boat transition state. In order for the reaction to proceed via the boat transition state, the starting molecule must adopt an energetically disfavored synperiplanar conformation. This provides an additional energy requirement, thus elevating the overall energy of the transition state. A higher energy transition state results in kinetically disfavored reaction. This conforms for the calculated energies of the boat and chair transition state outlined in Figure 2.2, and with the expectations outlined in the literature. The calculated activation energies (Figure 2.5) also corroborate these observations, with the chair transition state possessing a marginally lower activation energy barrier.

[[File:IRCFINAL2110493.jpg|500px|center]]

<div style="text-align: center;">''Figure 2.4: Plot of the energy of the reaction system as it moves away from the transition state.''</div>

{| class="wikitable"
!Species !! Activation Energy (Hartrees)
|-
|Boat TS || 0.08164
|-
|Chair TS || 0.07225
|}

''Figure 2.5: Activation energies for the system derived from the IRC calculations''

== The Diels-Alder Reaction ==

=== Introduction and Background ===

The Diels-Alder reaction is one particular subset of a group of reactions known as cycloadditions. Diels-Alder reactions can be characterised by the concerted mature of the bond forming and breaking, which proceeds through a cyclic, aromatic transition structure. Thus reactions of this type are typically Hückel 4n+2 processes. Photochemical 4n process are also present but are beyond the remit of this exercise.<ref>Fleming, Ian. Pericyclic Reactions. p.32 Oxford: Oxford UP, 1999. Print.</ref>

<br>

[[File:DA_Mechanism.jpg|center]]<br>
<div style="text-align: center;">''Figure 3.1: Mechanism of a Diels-Alder Cycloaddition''</div>

<br>

This affinity for 4n+2 electrons can be rationalized through consideration of the frontier orbitals involved in the reaction. In order for the concerted formation and breakage of bonds to occur, each interacting orbital must be of the correct parity to constructively interfere and thus form a bond. This can be further developed through consideration of the change of orbital parity over a mirror plane passing through the center of the molecule. A ''symmetric'' orbital system will not result in any parity change across the mirror plane, while an antisymmetric system will exhibit a parity change. Symmetric-Symmetric and Antisymmetric-Antisymmetric orbital interactions will always give rise to constructive interactions and thus allow the concerted formation of bonds. Symmetric-Antisymmetric interactions, however, will always possess one anti-bonding interaction and thus cannot result in concerted bond formation (stepwise bond formation, however, is still possible in certain cases).<ref>Fleming, Ian. Pericyclic Reactions. p.33 Oxford: Oxford UP, 1999. Print.</ref>

This leads to the so called ''Woodward-Hoffan Selection Rule'', which states that:

<div style="text-align: center;">''A pericyclic reaction is symmetry allowed if the sum total of suprafacial components involving 4j+2 π electrons '''And''' antarafacial components containing 4K π electrons is odd.''</div>

<br>

[[File:Untitled11041993NPG116.jpg|center]]<br>
<div style="text-align: center;">''Figure 3.2: diagram showing the change in orbital parity over a mirror plane (σ). On the left, two antisymmetric (A) interactions result in concerted constructive interactions. On the right, a symmetric-antisymmetric interaction precludes concerted bond formation.''</div>

<br>

It is also vital the stereochemical outcome of the reaction. Two major modes of attack may be expected, and are referred to as the ''endo'' and ''exo'' modes of attack. These designations refer to the position of the diene relative to the attacking dienophile, as shown below.<ref>Fleming, Ian. Pericyclic Reactions. p.8 Oxford: Oxford UP, 1999. Print.</ref>

<br>

[[File:Endoexoatknpg110493.jpg|center]]<br>
<div style="text-align: center;">''Figure 3.3: Diagram illustrating the difference between endo and exo attack. In the endo attack, the residual portion of the dieneophile faces inwards towards the main portion of the diene, while in an exo attack the residual group of the dieneophile faces away from the bulk of the diene system''</div>

<br>

Endo and Exo attacks will result in differing stereochemistry for their respective products. The endo product will result in a trans configuration between the residual groups on the dieneophile and the central bridging carbon chain. Conversely, a product in which the central bridging carbon chain is cis to the Diels-Alder adduct.<ref>Fleming, Ian. Pericyclic Reactions. p.8 Oxford: Oxford UP, 1999. Print.</ref>

The exo product results from an exo attack, in which the orientation of reactants is less sterically hindered and is therefore expected to dominant isomer in any reaction. However, experimentally this is often not the case. The reason for this anomalous selectivity results from the constructive overlap of orbitals on the residual group on the dieneophile with the diene system. This stabilizes the transition state, thus lowering it in energy. The result in increased favourability of the endo product. This increased favourability is referred to as Alder's ''Endo Rule''. The phenomenon of favorable orbital overlap between the diene system and dienophile resudual is dubbed the ''Secondary Orbital Effect'' <ref>doi:jo00384a016</ref>

[[File:SOOE_Diag.jpg|300px|center]]

<div style="text-align: center;">''Figure 3.4: Diagram illustrating the Secondary Orbital Effect.''</div>

Figure 3.4 illustrates this effect. In cartoon A, the residual chain is oriented inward (i.e.: the endo position), which allows constructive secondary orbital interactions (blue) to stabilize the transition state, thus lowering the overall energy of the profile (green) compared to the violet energy profile (no secondary orbital interactions). In contrast, the exo form has no secondary orbital interactions, and thus the energy of the transitions state will be correspondingly higher. Thus, the endo product will be formed more rapidly than the exo product. Further heating however will allow interconversion between the two isomers, where the more stable exo product would be expected to dominate.

=== The Butadiene + Ethene Addition ===

Figure 3.1 shows the addition of ethene to butadiene. The product of the reaction is symmetric and thus there are no stereochemical considerations to account for in the resulting Diels-Alder Adduct. As outlined above, The HOMO and LUMO of the reactant and product must be of appropriate symmetry to permit the concerted formation of bonds. Computing of the HOMO and LUMO for butadiene and ethene will allow for prediction of the orbital symmetry and thus will allow for determination of the pericyclec nature of the reaction.

====Experimental====

Butadiene and Ethene were generated separately and optimized to a minimum at the B3LYP/6-31G* level of theory. The resulting HOMO and LUMO of both molecules was generated and saved as an image file. The symmetry of these orbitals was determined and plotted on an orbital correlation diagram.

The reactant system and expected product were used as inputs in a QTS2 calculation at the HF/3-21G level of theory. The computation was found to converge to a transition state and the resulting transition structure was elucidated.

==== Results ====

[[File:Butadiene+ethen_RXN_scheme.jpg|center|800px]]

<div style="text-align: center;"> ''Figure 3.3: MO diagram showing the frontier orbitals of butadiene and ethene.''</div>

[[File:DA_RXN_bond_making_vibration_110493NPG.gif|200px|thumb|right|Figure 3.4: Animation of the imaginary bond making frequency for the Butadiene/Ethene reaction]]

Figure 3.3 illustrates the computed HOMO and LUMO of both butadiene and ethene. It can be seen that the HOMO of each molecule is of the same symmetry as the LUMO of its counterpart. Thus the reaction is expected to be symmetry allowed. We therefore expect a cyclic, aromatic transition state for this reaction, which exhibits concerted bond formation at an imaginary frequency.

The results of this experiment are displayed below. The presence of a product forming transition state was confirmed through the presence of an imaginary bond forming frequency at -XXXX cm<sup>-1</sup>. From Figure 3.4 it is easily seen that the movement of the ethene molecule and the flanking C=C substituents on the butadiene molecule were concerted. It can therefore be inferred that a pericyclic transition state has been reached for this reaction system.

{|
! Colspan="2"|Energy (Hartrees)
|-
|Absolute+ZPE || +Thermal Correction
|-
|-234.403320 || -234.396903
|}

=== Maleic Anhydride + Cyclopentadiene ===

[[file:endoexo.jpg|center]]
<div style="text-align: center;">''Figure 3.3: Reaction of Maleic Anhydride and Cyclopentadiene, displaying the exo and endo forms of the resulting Diels-Alder adduct. </div>

Figure 3.3 Shows the reaction between maleic anhydride and cyclohexadiene. The resulting Diels-Alder adduct has a bridging carbon chain and thus competition between the exo and endo forms will manifest. Due to secondary orbital interactions, the endo form is expected as the kinetic product, as predicted using Alder's Endo Rule. Thus, the endo transition state is likely to be lower in energy compared to the exo transition state. The energy of these transition states can then be probed using computational methods.

====Method====

The maleic anhydride and cyclohexadiene were generated separately and optimized at the B3LYP/6-31G** level of theory. The resulting HOMO and LUMO of both molecules was computed and saved as an image file. The symmetry of these orbitals was determined and plotted on an orbital correlation diagram.

Transition state calculations were performed using the semi-empirical AM1 Model and the ''Frozen Co-ordinate Method'' (Method II) described in Section 2. An initial guess structure for the exo and endo adducts was made. All bond making atoms were frozen through use of the 'redundant co-ordinate' functionality on Gaussian. The molecule was optimized to a minimum at the AM1 level of theory. The atoms were unfrozen and a second optimization to a transition state (Berny) was made at the same level of theory. The resulting transition structures were then optimized a second time to a TS(Berny) at the B3LYP/6-31G** level of theory, allowing free computation of force constants. The structure was found to converge and a negative frequency was found corresponding to concerted bond formation at -447.45 cm<sup>-1</sup> and -448.45cm<sup>-1</sup> for the endo and exo structures respectively. The orbitals of the resulting transition states were computed and saved. IRC calculations were then performed in the forward and reverse directions using the AM1 derived transition state at the AM1 level of theory.

====Results====
[[File: DA2OrbitalCorrelDiag.jpg|800px|center]]

<div style="text-align: center;">''Figure 3.5: Orbital Correlation diagram for the Maleic Anhydride/Cyclohexadiene reaction displaying the symmetry of the HOMO AND LUMO of each species''</div>

[[File: EXO_bond_making_NPG11041993.gif|200px|thumb|right|Figure 3.6: Animation of the bond forming frequency for the exo transition state at -448.45cm<sup>-1</sup>]]
[[File: ENDO_bond_making_vibr_110493.gif|200px|thumb|right|Figure 3.7: Animation of the bond forming frequency for the endo transition state at -447.45cm<sup>-1</sup>]]
[[File: ENDO_FULL_RXN_MOVIE110493.gif|200px|thumb|right|Figure 3.9: Animation of the Diels-Alder cycloaddition yielding the endo isomer]]
[[File: EXO_IRC_3110493.gif|200px|thumb|right|Figure 3.10: Animation of the Diels-Alder cycloaddition yielding the exo isomer]]

Figure 3.5 shows the Orbital Correlation diagram for the reaction outline in Figure 3.3. As was the case for the Butadiene/Ethene reaction, the HOMO of ane species and the LUMO of its counterpart are of the correct symmetry to react in a concerted fashion. Thus a pericyclic reaction is expected to take place for both the exo and endo forms. A visualization of the imaginary vibrations for the exo and endo transition states is shown in Figures 3.6 and 3.7. In each case, the observed vibration corresponds to concerted bond formation and breakage, thus confirming that the correct transition structure has been reached.

{|class="wikitable"
!!! Colspan="2" | Energy (Hartrees, Temperature Corrected)
|-
|'''Species'''|| '''AM1 (Semi-Empirical)''' || '''B3LYP/6-31G**'''
|-
|'''Endo-Adduct''' || 0.143677 || -612.491788
|-
|'''Exo-Adduct''' || 0.144881 || -612.487661
|}

''Figure 3.8: Tabulated thermally corrected energies of the endo and exo transition states using the AM1 and B3LYP/6-31G** levels of theory.''

Figure 3.8 displays the energies of the Endo and Exo transition state. A lower energy is observed for the endo transition state, in line with Alder's Endo Rule and the predictions outlined by the secondary orbital effect. A plot of the HOMO for the endo transition state allows for visual confirmation of this effect.

<br>

[[File:TSORBS.jpg|500px]]

''Figure 3.11: Surface plots of the HOMO and LUMO of the Exo (Right) and Endo (Left) transition states. All transition state orbitals are antisymmetic with respect to a central mirror plane.''

<br>

Figures 3.9 and 3.10 show visualizations of the Diels-Alder cycloaddition for the endo and exo forms of the adduct. The resulting secondary orbital interactions can be seen in figure 3.11. The inward facing anhydride group is able to interact with additional MO's on the cyclobenzadiene, thus generating a stabilizing effect. This lowers the energy of the corresponding transition state, thus accounting for the discrepancy in energy of the two forms.

Figure 3.12 displays the lengths of the bonds affected as the reaction system approaches the transition structure. The C-C bonds between maleic anhydride and the cyclohexadiene molecule are much longer than typical C-C bonds, which have a length of approximately 154pm. This large discrepancy is most likely an artifact of the 'Frozen Co-ordinate method' and should not be taken as indicative of the true mechanism to the reaction. An elongation of double bonds is observed, with double bond distances averaging around 140nm, part way in between those of a C-C single bond and a C-C double bond, suggesting the conversion of a double bond to a single bond. The bond lengths are similar in size to those of aromatic C-C bonds, suggesting the possibility of an aromatic transition state.

{|
|<jmol>
<jmolApplet>
<title>Endo Isomer</title><color>white</color>
<size>400</size><script>measure 2 3;measure 3 10;measure 2 13;measure 10 11;measure 11 12;measure 12 13</script>
<uploadedFileContents>ENDO-B3LYP_OPT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|<jmol>
<jmolApplet>
<title>Exo Isomer</title><color>white</color>
<size>400</size><script>measure 2 10;measure 3 13;measure 13 12;measure 12 11;measure 11 10;measure 2 3</script>
<uploadedFileContents>EXO_B3LYP_OPT.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|}

''Figure 3.11: JMOl files of the endo and exo isomer displaying the distances of the bonds that are being made and broken.''

==Conclusion==

In the first exercise, the relative stabilities of a gauche and an antiperiplanar conformation of hexa-1,5-diene were studied. The gauche and antiperiplanar conformers were found to have C2 and Ci symmetry respectively, and the gauche C2 conformer was found to be significantly higher in energy compared to the Anti Ci conformer, suggesting that the gauche conformation is significantly less stable than its counterpart. Furthermore, it was shown that the results obtained by the HF/3-21G level of theory significantly overestimate the energy of the optimized structure. This in turn suggests that more powerful methods and larger basis sets will be needed to generate more numerically precise answers.

In the second excercise, three idfferent methods were utilised to probe the nature of a transition structure. These methods were found to yield vary similar results and could therefore be used interchangeably. The activation energy for the cope rearrangement was computed and the results were found to conform to theory.

In the third exercise, two Diels-Alder reactions were studied. The transitions states for both reactions were found and confirmed. Additional calculations demonstrated the favourability of the endo transition state over the exo transition state, in keeping with Alder's rule. The Molecular Orbitals were also calculated and found to agree with the Woodward-Hoffman selection rules, in addition to permitting rationalization of the secondary orbital effect.

These exercises have demonstrated the power of computational chemistry in providing information about molecules and chemical reactions, enabling one to retrieve data about a molecule that could not be easily determined experimentally without leaving one's desk.

==Citations==
<references/>