==The Free Energy and Thermal Expansion of MgO==

===Introduction===

==Phonon modes of MgO==
Phonon dispersion curves and the density of states were computed for various grid size in order to find out an appropriate grid size for the free energy calculation using the quasi-harmonic model.

{| class="wikitable"
|+ '''Phonon dispersion curve and DOS of grd size 1x1x1'''
!Grid size
!Phono dispersion curve
!Density of states
|-
|1x1x1
! align="center"| [[File:Hyl_PDC_1x1x1.png|center|350x350px]]
! align="center"| [[File:Hyl_DOS_1x1x1.png|center|400x400px]]
|}

By comparing the frequencies in the DOS and the phonon dispersion curve, it can be seen that the k point responsible for the DOS of 1x1x1 grid is at symmetry point L which corresponds to (0.5, 0.5, 0.5) fractional coordinates in reciprocal space. The intensities at the 2 lower frequencies are doubled as there are 2 overlapping acoustic branches passing through them in the phonon dispersion curve due to symmetry.
{| class="wikitable"
!Grid size
!Density of states
!Grid size
!Density of states
|-
|2x2x2
! align="center"| [[File:Hyl_PDC_2x2x2.png|center|400x400px]]
|16x16x16
! align="center"| [[File:Hyl_DOS_16x16x16.png|center|400x400px]]
|-
|3x3x3
! align="center"| [[File:Hyl_DOS_3x3x3.png|center|400x400px]]
|32x32x32
! align="center"| [[File:Hyl_DOS_32.png|center|400x400px]]
|-
|8x8x8
! align="center"| [[File:Hyl_DOS_8x8x8.png|center|400x400px]]
|64x64x64
! align="center"| [[File:Hyl_DOS_64.png|center|400x400px]]
|}

As shown in the above table, an increase in grid size leads to a smoother and more comlicated DOS due to more k points, and therefore vibrational modes, being included in the calculation. A grid size of 32x32x32 was chosen for the quasi-harmonic calculation as it provides a smooth and continuous DOS without being too computationally expensive. It produces a less noisy DOS when compared to 16x16x16 grid size as indicated by the smooth appearance. The DOS of 32x32x32 and 64x64x64 grid sizes show the same general shape with the latter one in a higher resolution.

The same grid size of 32x32x32 can be used to model CaO as the two oxides have similar primitive cells and the same number of atom per unit cell. Crystals with more than 2 atoms in their primitive cells, such as Zeolites, requires a smaller grid size as an increase in the size of primitive cell leads to a decrease in size in k-space so less k-points has to be sampled to achieve the same level of accuracy. The same applies to regular metals such as lithium as the delocalised electrons in the lattice screens repulsions between the metal cations, resulting in more dispersed bands so less k-points and therefore a smaller grid size is needed to produce a DOS with a similar level of accuracy.

=Free Energy of MgO=
The free energy of MgO was computed using the quasi-harmonic model as a sum over all the vibrational modes. The free energy initially increases with the grid size as more k points and therefore more vibrational states are sampled. The difference between the free energy calculated using different grid size decreases as the grid size increases until it reaches a constant at 32x32x32 grid.
{| class="wikitable"
!Grid size
!Helmholtz free energy (eV)
!ΔA (eV)
|-
|1x1x1
|<nowiki>-40.930301</nowiki>
|
|-
|2x2x2
|<nowiki>-40.926609</nowiki>
|3.692E-03
|-
|4x4x4
|<nowiki>-40.926450</nowiki>
|1.590E-04
|-
|8x8x8
|<nowiki>-40.926478</nowiki>
|<nowiki>-2.800E-05</nowiki>
|-
|16x16x16
|<nowiki>-40.926482</nowiki>
|<nowiki>-4.000E-06</nowiki>
|-
|32x32x32
|<nowiki>-40.926483</nowiki>
|1.000E-06
|-
|64x64x64
|<nowiki>-40.926483</nowiki>
|0
|}

The accuracy each grid size can achieve in a calculation can be estimated using ΔA, an accuracy of 1 meV corresponds to a ΔA in the order of 10-3 and can therefore be achieved by a 4x4x4 grid. An accuracy of 0.1 meV is therefore achieved by 8x8x8 grid which has a ΔA in the order of 10-4 while that of 0.5 meV is achieved by an intermediate grid size of 6x6x6.

A grid size of 32x32x32 is used for the investigation of the temperature dependence of the Helmholtz free energy, cell volume and parameter by running the calculations at 0-1000 K with an interval of 100 K.

{| class="wikitable"
! align="center"| [[File:Hyl_QH_A.png|center|530x530px]]
! align="center"| [[File:Hyl_QH_para.png|center|500x500px]]
|}

As seen from above, the Helmholtz free energy of MgO decreases with increased temperature. This trend agrees with the equation of Helmholtz free energy (A=U-TS) as an increase in temperature leads to an increase in the second term which is always positive (entropy is always positive) resulting in a more negative Helmholtz free energy, this effect is amplified by the increase in entropy caused by an increased temperature. The opposite is true for cell volume and cell parameter as they increase with an increased temperature.

[[File:Hyl_QH_vol.png|500x500px]]

The thermal expansion coefficient was calculated using the equation <math>\alpha = \frac{1}{V_0}\,\left(\frac{\partial V}{\partial T}\right)_p</math> where <math>\left(\frac{\partial V}{\partial T}\right)_p</math> is the gradient of the line of best fit in the cell volume against temperature graph. This gradient is divided by the initial cell volume in order to obtain an intrinsic property which is independent of the amount of matter.Using the method mentioned above, the thermal expansion coefficient was found to be 2.124x10-5 K-1 which is a close approximate to the literature value (3.706x10-5 K-1).

==Molecular Dynamics==
The thermal expansion coefficient can also be found using molecular dynamics in which all atoms in a system are given a velocity and allowed to move accordingly following Newton's second law (F=ma). The loss of periodicity of movements means that a larger number of cell is required for this method and a supercell of 32 MgO units is used.

[[File:Hyl_MD_vol.png|500x500px]]

As with the quasi-harmonic model, the thermal expansion coefficient can be found by dividing the gradient of the change in volume with respect to temperature by the initial cell volume. The coefficient obtained is 2.654x10-5 K-1 and is a closer match to the literature value.

[[File:Hyl_QH+MD_vol.png|500x500px]]

File:Hyl MD vol.png

2017-01-27T10:35:53Z

Hyl314:

File:Hyl QH+MD vol.png

2017-01-27T10:35:30Z

Hyl314:

Rep:Mod:hyl314MgO

2016-12-16T10:41:34Z

Hyl314:

==Introduction==
The lowest energy species in a reaction profile is known as a minimum while the highest energy species is the transition state. Both species correspond to a turning point on the profile where gradient = 0 (dx/dy=0) with positive and negative gradients respectively.
Gaussview was used to model the structure and energy of the reactants, transition states (cannot be obtained experimentally) and products of different reactions to predict the thermodynamic and kinetic products. Molecular orbital diagrams have also been drawn using information obtained from the calculations.

==Exercise 1: Reaction of Butadiene with Ethylene==

===Bond Length===
Diels- Alder reaction is a [4+2] cycloaddition that proceeds via a concerted mechanism through a cyclic transition state. As seen in the below reaction scheme, all the double bonds in the reactants lengthened from ~1.3 Å to ~1.5 Å in the product as the bond order decreased to one. The opposite is true for the single bond in butadiene which reduced in length from ~1.5 Å to ~1.3 Å as the double bond was formed. The intermediate bond lengths between single and double bonds in the transition state indicates partial single/ double bond character caused by bond breaking and forming in progress. All bond lengths in reactants and product show good agreement with literature (sp3-sp3 = 1.45 Å, sp3-sp2 = 1.50 Å, sp2-sp2 = 1.33 Å). Bonding interaction between the terminal Cs of the reatants is confirmed as the bond length of the partially formed C-C bond is shorted than 2 x the Van de Waals radius (1.7 Å) of C.

[[File:Hyl314 Ex1 RS+bond length.png|thumb|800px|Figure1. Reaction Scheme of Diels-Alder reaction between butadiene and ethylene with bond lengths|center]]

===MO Diagram===
The MO diagram of the reaction between Butadiene and Ethylene is shown in Figure1, note that only the HOMO and LUMO of the reactants are shown in the diagram.

[[File:hyl314_Ex1_MO1.png|thumb|500px|Figure2. MO diagram of Diels-Alder reaction between butadiene and ethylene|center]]

Symmetry of the orbitals are labelled as g (gerade, symmetric with respect to inversion) or u (ungerade, asymmetric with respect to inversion). Reactions will only happen between 2 orbitals if the overlap intergral is non-zero which occurs when the orbitals have the same symmetry label. Therefore, only orbitals with the same symmetry label can react (e.g. g-g, u-u are allowed; u-g = forbidden) as illustrated in the MO diagram above.

[[File:Hyl314_Ex1_SM_MO1.png|thumb|1000px|Figure3. HOMO and LUMO of butadiene, ethylene and transition state|center]]

===Vibrations===
The imaginary frequency at -948.63 cm-1 corresponds to the reaction path at the transition state which shows a synchronous bond formation, which agrees with the concerted mechanism
[[File:hyl314_Ex1_ts_vib1.gif|thumb|500px|Figure4. Transition state vibration (Please double click to see animation)|center]]

The lowest positive frequency at 144.94 cm-1 however is asynchronous as shown below.
[[File:hyl314_Ex1_ts_vib2.gif|thumb|500px|Figure5. First positive vibration (Please double click to see animation)|center]]

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

===MOs===
MOs of the 1,3-dioxole and cyclohexadiene are shown below, symmetry was labeled using the MO diagram in exercise 1 with 1,3-dioxole acting as the dienophile and cyclohexadiene as the diene.

[[File:hyl314_Ex2_SM_MO.png|thumb|800px|Figure1. MOs of 1,3-dioxole and cyclohexadiene with symmetry label|center]]

Diels Alder reactions can either be normal electron demand (electron rich diene and electron poor dienophile) or inverse electron demand (electron poor diene and electron rich dienophile) depending on the nature of the reactants. The HOMO and LUMO of both endo and exo TS all show a gerade symmetry, therefore must be formed by the interaction of the 1,3-dioxole HOMO and cyclohexadiene LUMO as they are the only orbitals with the correct symmetry (g) to interact. This is due to the presence of electron rich O on 1,3-dioxole which raises the energy of its HOMO and LUMO, the overlap between the cyclohexadiene LUMO and the high energy 1,3-dioxole HOMO is now better than that of the cyclohexadiene HOMO and 1,3-dioxole LUMO, making this an inverse electron demand reaction.

[[File:hyl314_Ex2_TS_MO.png|thumb|800px|Figure1. MOs of endo and exo transition states|center]]

===Reaction Energies and Stabilisation===
Kinetic product of a reaction is the one that requires the lowest activation energy while the thermodynamic product is the most stabilised (lowest energy) conformer. In this reaction, the endo product has the lowest activation energy and is also more stabilised so it is both the kinetic and thermodynamic product and will be the only product formed in the reaction. The lower activation barrier is due to the secondary orbital interaction between the O lone pair in p orbital and the empty π* orbital at the back of the diene which lowers the energy of the endo transition state. This stabilisation is absent in the exo transition state as the π* orbital and the O lone pair are not in the correct orientation to interact, resulting in a higher activation energy.

{| class="wikitable"
!
!Reactants
!Transition State
!Product
!Activation Energy
!ΔG
|-
|Endo
|<nowiki>-1313782</nowiki>
|<nowiki>-1313622</nowiki>
|<nowiki>-1313849</nowiki>
|160
|<nowiki>-67</nowiki>
|-
|Exo
|<nowiki>-1313782</nowiki>
|<nowiki>-1313614</nowiki>
|<nowiki>-1313846</nowiki>
|168
|<nowiki>-64</nowiki>
|+Table showing the energies of reactants, transition state and product of both endo and exo pathways (kJ mol-1)
|}

[[File:hyl314_Ex2_orbital_int.png|thumb|500px|Figure4. Secondary orbital interactions in TS|center]]

==Exercise 3: Diels- Alder vs Cheletropic==

Xylylene and SO2 can react through a normal Diels-Alder reaction forming either the endo- or exo- product or through a competing cheletropic pathway as shown in the reaction scheme below. The activation energy and change in Gibbs free energy of each pathway was calculated to determine the most favourable reaction.

[[File:Hyl314_Ex_03_Reactionscheme.png|thumb|500px|Figure4. Reaction scheme of possible reactions between xylylene and SO2|center]]
{| class="wikitable"

The starting material xylylene is not aromatic as it does not follow the Huckel rule (4n+2π electrons) and the abundance of π bonds make it very reactive. During the reactions, the 6-membered ring is aromatised as shown in the below animations and is therefore a lot more stable, making the reactions thermodynamically favourable. (Note that the endo reaction pathway appears to be reverse here as Gaussian cannot identify prduct and reacta

===Intrinsic Reaction Coordinate===
{| class="wikitable" border="1"
|+ Table showing the animations and intrinsic reaction pathways for different reactions
| Endo Diels Alder || Exo Diels Alder || Cheletropic
|-
| [[File:Hyl314 Ex3 DA endo IRC.gif|thumb|500px|center]] || [[File:hyl314_Ex3_DielsAlder_EXO_IRC.gif|thumb|500px|center]] || [[File:hyl314_Ex3_CH_IRC.gif|thumb|500px|center]]
|-
| [[File:hyl314_Ex3_DA_endo_reactionpath.png|thumb|500px|center]] || [[File:hyl314_Ex3_DA_exo_reactionpathway.png|thumb|500px|center]] || [[File:hyl314_Ex3_CH_reactionpath.png|thumb|500px|center]]
|}

===Reaction Profile===
As seen in the reaction profile below, the endo Diels Alder pathway has the lowest activation energy, making the endo product kinetically favourable at low temperature (kinetic product). The exo Diels Alder product is more stabilised than its endo equivalent but has a higher activation barrier and therefore would not form if the reaction is under kinetic control (non-reversible). The cheletropic product is the most stabilised but also requires the highest activation energy to form, it is therefore the thermodynamic product.

[[File:Hyl314_Ex_03_Reactionsprofile.png|thumb|500px|Figure4. Reaction profile of possible reactions between xylylene and SO2|center]]

File:Hyl314 Ex2 orbital int.png

2016-12-16T10:21:40Z

Hyl314: