The purpose of the first part of the computational laboratory is to optimise several molecules so that they may be compared in subsequent parts of this exercise. Molecules are optimised by Gaussview using the B3LYP method and various input basis sets. A molecule is optimised when its ideal energy, as interpreted by Gaussview, is achieved ie. when Gaussview is able to minimise the potential energy (find the most negative energy value with the smallest gradient close to zero) of the molecule. The molecule is in its equilibrium geometry!
Optimisation of BH3
Initially, the bond lengths of each B-H bond were intentionally offset as 1.53, 1.54 and 1.55 Å respectively while conserving the trigonal planar geometry of the molecule. An optimisation was subsequently carried out.
Optimisation file of BH3 with 3-21G basis set can be found here.
Optimised BH3 Summary Table
Summary Data
Convergence
Jmol
BH3 Optimised 3-12G
Geometry Data
Geometry Data
NH3BH3
r(B-H) Å
1.19
r(B-H) Å
1.19
r(B-H) Å
1.19
θ(H-B-H) degrees(º)
120
It is observed that despite the initial offset of the bond lengths, the optimisation of the molecule has equalised the bond lengths of each B-H bond to 1.19 Å.
Day 2
Optimisation of BH3
Optimisation file of BH3 with 6-31G(d,p) basis set can be found here.
Optimised BH3 with 6-31G (d,p) Summary Table
Summary Data
Convergence
Jmol
BH3 Optimised 6-31G (d,p)
Optimisation of GaBr3
Optimisation file of GaBr3 with LANL2DZ basis set can be found here. DOI:10042/195225
Optimised GaBr3 with LANL2DZ Summary Table
Summary Data
Convergence
Jmol
GaBr3 Optimised LANDL2DZ
Optimisation of BBr3
Optimisation file of BBr3 with GEN basis set can be found here. DOI:10042/195231
Optimised BBr3 with GEN Summary Table
Summary Data
Convergence
Jmol
BBr3 Optimised GEN
As evidenced by the above calculations, all molecules have been optimised and their stationary points have been found.
Geometry Comparison
Geometry Data
BH3
BBr3
GaBr3
r(E-X) Å
1.19
1.93
2.35
θ(X-E-X) degrees(º)
120
120
120
Geometry Analysis : It is expected, purely on the rationale of increasing the size of the constituent atoms (atomic and van der waals radii), one would expect greater bond lengths between the central and surrounding atoms as we move across from BH3 to BBr3 to GaBr3. Furthermore, the bond angles would not expect to change between the three compounds as the central atoms of B and Ga are in the same group. Therefore, the overall structure of the molecule is not majorly affected as the type of bonding interaction and the directionality of orbitals involved in the bonding is the same. These postulations are indeed shown to be the case as demonstrated from the table above.
Firstly, it is observed that all bond angles are the same at 120º demonstrating that the trigonal planar geometry is preserved across all three molecules. Subsequently, Hydrogen has a significantly smaller atomic radius of 25 pm than Bromine of 115 pm. Therefore, the B-H bond length of borane will be smaller than the B-Br bond length of borontribromide because the Br atoms cannot lie closer to the central B atom. Similarly, as the central atom is changed from B to Ga, the same rationale can be applied with the atomic radius of B (85 pm) smaller than that of Ga (130 pm). This data also suggests the decreasing strengths of the bonds moving across the molecules in the table due to shorter bond lengths and therefore higher bond order. This would be expected as the less diffuse orbitals of the smaller H and B atoms have better overlap (and therefore better bonding interactions) than those of B with the larger Br orbitals. However, this is misleading may be misleading as one must also consider the electronegativity of the constituent atoms.
Br is significantly more electronegative than H. Therefore, the greater difference in electronegativity between B and Br may result in a more stable bond. This should be able to be determined by comparing the energies of each molecule but different basis sets were used in the optimisation process; These energies therefore cannot be compared. An explanation for why this is the case is given further on.
What is a bond? : A bond is commonly denoted by a line joining two atoms and signifies some kind of interaction between the two atoms connected by the line. The question posed is a challenging one to answer due to the variety of types of chemical bonding in existence that are unique from one another in some way. Classically, a bond can be defined as a favourable, electrostatic interaction or attraction between two atoms. Chemical bonding takes many forms and is defined by the degree to which the interactions between the atoms takes place. The simplest examples of these interactions are a result of the sharing of electrons or electron density between the atoms via orbitals (covalent bonding). The other is the outright donation of electrons from one atom to another where the attraction is between the separated charges (ionic bonding). In reality, there is a spectrum between the two types of bonding described here and where the bonding lies along the scale is determined by a combination of the size and electronegativity of the atom. A bond is considered to have formed if there is a net reduction in the potential energy of the system or molecule. This was alluded to earlier when describing the process by which Gaussview optimises molecules. However, static structures often drawn for molecules are simplifications of a more dynamic picture as bonds may vibrate within molecules. Therefore, it is an important distinction to note that what is depicted when the bonds of a molecule are drawn is simply the representation of the molecule in its equilibrium state, that with its potential energy minimised.
What has been described above is merely a simplistic summary of chemical bonding. More complex forms of chemical bonding do exist and indeed recently, Manz et al. have reported an entirely new type of chemical bonding which circumvents the need for the minimisation of the potential energy of the molecule described before by stabilisation of the vibrational zero point energy instead.
How much energy is there in a strong, medium and weak bond? : The two types of bonding described above, ionic and covalent, take place within molecules (intramolecular) and are often considered examples of strong types of bonding. For example, the bonding between the two nitrogens within a molecule of di-nitrogen, N2, is incredibly strong with a triple bond strength equivalent to 945 kJ mol-1, one of the strongest known. Consequently, N2 is incredibly stable and unreactive. An example of a medium bond is a simple single C-C bond of an akyl group equivalent to 348 kJ mol-1, a third that of N2. Finally, bonding between different molecules (intermolecular) such as hydrogen bonding, dipole-dipole interactions and Van der Waal's forces, are considered to be weak. This type of bonding works on the basis of the influence of the electron density or dipole moment within a given molecule on that belonging to molecules adjacent to it. Hydrogen bonding is the strongest of the three types of intermolecular bonds just listed and a single hydrogen bond is of the order of 15 kJ mol-1.
In some structures gaussview does not draw in the bonds where we expect, why does this NOT mean there is no bond? : Gaussview is programmed to possess a pre-determined notion of the value a bond length or a bond energy. In some cases when a poor calculation has been done or the nature of bonding is complex (such as in the instance of 3c-2e bonding), values that have been determined do not fall within the range utilised by Gaussview to determine the presence of a bond. Therefore, Gaussview will not draw the expected bond.
Day 3
Frequency Analysis for BH3
In addition to molecule optimisation, a frequency analysis must also be undertaken in order to determine that the energy state achieved in the optimisation is indeed a minima, in which case the molecule given will be a stable species. Conversely, a zero gradient for the potential energy of a molecule may also represent a maximum, whereby the molecule is in a transient, fleeting state ie. transition state. The former scenario is desired in order to perform meaningful comparisons between the molecules. A minimum is achieved when the frequency values returned by the operation are all positive. Furthermore, the 6 low frequencies (attributed to the motion of the center of mass of the molecule) should be comparably lower than that of the remaining vibrational modes of the molecule as determined by the 3N-6 rule.
It should be noted that the 3N-6 rule predicts 6 vibrational modes for a molecule of borane, BH3. However, in the infrared spectra above, only 3 peaks are observed. This result can be explained by the selection rules for infrared spectroscopy that require a change in dipole moment of the molecule upon vibration in order to be visible. Symmetrical vibrations such as that at 2583 cm-1 of BH3 do not induce a change in the dipole moment. Therefore, it would not appear in the IR spectra and is consequently not predicted by Gaussview. Fewer than 6 peaks are also not observed because some vibrations are isoenergetic and therefore degenerate with others. Therefore only a single peak, representing both vibrational modes, is observed.
Frequency Analysis for GaBr3
Frequency calculation for GaBr3 can be found here.
The spectra for molecules are fairly similar showing 3 peaks. The reason only 3 peaks are observed in the spectra was elaborated on earlier for BH3. This is similarly the case for GaBr3. Furthermore, the number of vibrational modes are also the same as would be expected by the 3N-6 rule. Unsurprisingly, the types of each vibrations are also the same as both molecules are of the same symmetry, geometry and point group; only the constituent atoms have changed. However, there ends the similarities between frequency analyses of the molecules.
It is observed that the frequency values are considerably larger for BH3 in general as well as within each type of vibrational mode as compared to GaBr3. This is because the heavier atoms of Ga and Br vibrate slower (as a result of the change in reduced mass and force constant of the molecule) and therefore at a lower frequency than that of the lighter constituent atoms of BH3. Therefore, the frequencies associated with the vibrations of GaBr3 will be smaller. Additionally, there has also been a reordering of modes, particularly the A2" umbrella motion shown below. This vibration is lower in energy than the two degenerate bending modes for BH3 but higher for GaBr3. It would be expected that a more significant proportion of the vibration is largely attributed to the displacement of the lighter atoms in the molecule. In the case of BH3, it is the surrounding hydrogens that are responsible; for GaBr3, it is the central Ga atom. This is easily observed by the displacement vectors in the depictions below and demonstrates the reordering of modes.
BH3
GaBr3
Why must you use the same method and basis set for both the optimisation and frequency analysis calculations?: The accuracy of the calculations performed by Gaussview is defined by the input basis set used. Different basis sets possess different degrees of errors associated with the calculation. If using different basis sets, the accuracy will be different and comparison of calculations between molecules is not possible.
Are there any significant differences between the real and LCAO MOs? What does this say about the accuracy and usefulness of qualitative MO theory?: At a glance, it is noticeable from the MO diagram above that the MOs computed by Gaussview and those as predicted by the Linear Combination of Atomic Orbitals (LCAO) theory are in relatively good agreement. However, both approaches have their own respective drawbacks. Quantum mechanic calculations have shown that bonding orbitals often possess more character from the fragment orbitals which they are closer in energy to ie the orbital coefficient will be larger. For the orbitals computed by Gaussian, it is not easily determinable which orbital fragment has a greater orbital contribution. This is evidenced by the 2a1' bonding orbital in the diagram above where the computed orbitals blend the electron density into one another. However, for this same reason, computed orbitals are indeed more representative of the actual orbitals. Gaussian is better able to produce a space-filling model of the orbitals involved that the simple LCAO depictions cannot. This is particularly the case for the higher unoccupied bonding orbitals where the computed depictions are askew from those depicted using LCAO in order to minimise unfavourable interactions. In summary, LCAO provides a good qualitative representation of MOs from which contributions by different orbital fragments can be gauged as well as the rough ordering of the MOs. However, they do not give the most accurate representation of how MOs actually look like when the interact with each other.
Optimisation of NH3
Optimisation file of NH3 with 6-31G (d,p) basis set can be found here.
How do these MOs relate to the common conception of aromaticity? Hint 2: aromaticity relates to delocalisation. Hint 3: aromaticity relates to the total pi electron density, and MOs contain formally only 2 electrons each.
Boratabenzene
Optimisation of Boratabenzene
Optimisation file of Boratabenzene with 6-31G (d,p) basis set can be found here.
Optimised Boratabenzene Summary Table
Summary Data
Convergence
Jmol
Boratabenzene Optimised 6-31G (d,p)
Geometry Data
Geometry Data
NH3BH3
r(B-H) Å
1.22
r(C1,5-H) Å
1.10
r(C2,4-H) Å
1.10
r(C3) Å
1.09
r(B-C1,5) Å
1.51
r(C1,5-C2,4) Å
1.40
r(C2,4-C3) Å
1.41
θ(C2-C3-C4) degrees(º)
120
θ(C3-C2,4-C1,5) degrees(º)
122
θ(C2,4-C1,5-B) degrees(º)
120
θ(C1,5-B-C5,1) degrees(º)
115
θ(H-C/H-B) degrees(º)
180
Frequency Analysis for Boratabenzene
Frequency calculation for Boratabenzene can be found here.
The key property at the heart of all discussions on charge distribution within a molecule is the electronegativity of the constituent atoms. Electronegativity is often simply defined as an atom's affinity for a pair of electrons in a covalent bond. This is important in this context as all molecules in consideration are organic molecules containing intramolecular covalent bonding between the atoms via, as what has been emphasised thus far, orbital interactions. According to the Pauling Scale, the electronegativity values for the 4 types of atoms involved are 2.20, 2.55, 2.04 and 3.04 for H, C, B and N respectively.
Firstly, we will look at benzene as a reference and as a standard moiety to study the cyclic, aromatic properties of the systems. Benzene is perfectly symmetrical with 6 carbons bound together in a ring via 6 sigma C-C bonds. Each carbon is also bound to one hydrogen. Additionally, each carbon is sp2 hybridised and possesses a single unpaired electron in the remaining un-hybridised p-borbital orthogonal to the ring. These 6 single unpaired electrons are then delocalised in the ring resulting in an aromatic compound. Due to its symmetry, benzene has a uniform distribution of electronic charge; This is to say, each carbon possesses the same charge as every other. his is similarly the case for the hydrogens. It is observed that the carbons possess a lower electronic charge density than the hydrogens. This is in agreement with carbon being more electronegative as well as the high electron density as a result of the delocalised electrons in the ring.
When moving from benzene to boratabenzene, one of the carbons of the ring is substituted with a boron atom. Hence,the element of symmetry is loss and the uniform distribution of electronic charge across the carbons is lost. Boron is less electronegative than both carbon and hydrogen. Immediately, we may observe that the boron atom of the molecule possesses the lowest charge density of 0.202 (ie. the most electropositive value) compared to all the constituent atoms in the molecule. Subsequently, we can see that there is significant build up of negative charge on the carbons of the ring, in particular at the ortho- (-0.588) and para- (-0.348) positions. This is not surprising as carbon is the most electronegative atom in this system. The build of charge at the two positions mentioned is a result of the formal negative charge of the molecule which can be pushed around the ring. A series of resonance forms describing this process will clearly demonstrate the build up of charge at these positions. Consequently, the hydrogens bound at these positions are comparably more electropositive than those at the other positions. It is noted that the hydrogens bound to the carbons are once again electropositive as was the case in benzene. However, for the reason just described the result is that some variation in the values between the hydrogens is also present. Most notably, the hydrogen bound to the boron atom actually possesses a slightly negative value of -0.096. This is because in this instance, hydrogen is actually more electronegative than the boron and therefore holds a higher share of the electron density.
When moving from benzene to pyridinium, one of the carbons of the ring is substituted with a nitrogen atom instead. Similarly with boratabenzene, the symmetry of the molecule is loss and as expected, the distribution of charge across the molecule will once again be affected. Nitrogen is considerably more electronegative than the 3 atoms previously discussed. Therefore, one observes that it possesses a great deal of the electron density with a value of -0.476. As with boratabenzene before, a series of resonance structures pushing the formal positive charge around the ring will demonstrate a build up of delta positive charge at the ortho- and para- positions. It is therefore unsurprising that this is where the carbons in possession of the least amount of negative charge are found with values of -0.122 and 0.071. In contrast to the hetereoatom-bound hydrogen in boratazene, that of pyridinium possesses an exceptionally high electropositive charge of 0.483. This is testament to the high electronegativity of nitrogen which pulls a lot of the charge density from the hydrogen. Again, the remaining hydrogens possess charge density values in reflection of the carbons they are bound to as well as the break in symmetry.
Borazine varies from benzene in that the 6 membered ring alternating between boron and nitrogen atoms and each similarly bound to a hydrogen. From the above discussion, it is unsurprising to observe the polarisation of electron density at the nitrogens (-1.102) leaving considerable electropositive build up on the borons (0.747). As with the heteroatom-bound hydrogens before, those bonded to nitrogens possess an overall electropositive charge (0.432) and those bonded to borons with an overall electronegative charge distribution (-0.077).
MO Comparison
Comparison 1
Benzene
Boratabenzene
Pyridinium
Borazine
MO
Energy / au
-0.35994
-0.13210
-0.64064
-0.36129
Order
17th
17th
17th
17th
Degenerate?
No
No
No
No
This set of MOs depict the delocalised π system described before. In all cases, only the pz-orbitals of the 6 members of the ring are involved, found above and below the plane of the ring. As a result of this, a nodal plane is found in the place of the ring. The interaction between the 6 members is the through space bonding of the in phase lobes of the orbitals. No anti-bonding components are involved.
The shapes of the orbitals are all fairly similar with the differences between them being the skewing of the orbital lobes in certain directions, towards certain atoms. The skewing is easily explained by the charge distribution describe before. In the case of benzene, the surface describing the orbitals is uniform due to the high symmetry. In the case of boratabenzene, electron density is skewed away from the electropositive boron atom and this is show by the boron in the depiction being slightly exposed. This is also explained by the relative energies of boron and carbon in an orbital diagram. As this is a bonding MO in consideration, the boron orbital fragments, which is less electronegative, will be higher in energy and therefore contribute less to final bonding MO. In contrast, the orbital surface for pyridinium is skewed toward and thickest at the highly electronegative nitrogen atom leaving the carbons on the opposite end of the ring exposed. Such is the extent of this polarisation and build up of electron density at nitrogen that the hydrogen bound to the nitrogen appears to have an orbital contribution of its own to the overall MO. This is, in reality, not the case. Once again, the contribution to the final bonding MO will be greater from the nitrogen orbital fragment as it is more electronegative and lower in energy. Finally, for borazine, the orbitals are in effect a superposition of the MOs at the B-H and N-H units of boratabenzene and pyridinium respectively and is almost an exact representation of the charge distribution described before in the NBO analysis.
The energies of each MO can also be explained by the same rationale employed when describing the orbital surfaces. Boron is more electropositive than carbon, the MO is therefore higher in energy at -0.13210 AU than that of benzene at -0.35994 AU. Conversely for nitrogen, the high electronegativity stabilises the molecule and brings the energy down to -0.64064 AU. In the case of borazine, the effects of the heteroatoms once again combine to cancel each other out and is therefore found somewhere between boratabenzene and pyridinium at -0.36129, similar to benzene.
Comparison 2
Benzene
Boratabenzene
Pyridinium
Borazine
MO
Energy / au
-0.84671
-0.06437
1.21402
-0.88851
Order
7th
7th
7th
7th
Degenerate?
No
No
No
No
These MOs are in fact very similar to those described in Comparison 1. The major difference between them is that the orbital involved are the s-orbitals and not the p-orbitals. The same rationales used before can be used again here to explain the depiction of the orbitals as well as the energies of the orbitals which are considerably lower in energy because the MOs are comprised of lower energy s-orbitals. The most interesting aspect of these MOs is that the orbitals are considerably more diffuse than that suggested by LCAO. The s-orbitals also appear to fill across the centre of molecule rather than localised simply along the sigma framework. This perhaps suggests there may be some contribution of this MO to the delocalisation of electrons in the aromaticity.
Comparison 3
Benzene
Boratabenzene
Pyridinium
Borazine
MO
Energy / au
-0.24960
0.01094
-0.50847
-0.27590
Order
21st
21st
20th
21st
Degenerate?
Yes
No
No
Yes
This MO under consideration is the HOMO of each system with exception to pyridinium. Firstly we can describe this MO of benzene with a LCAO approach. The MO consists of perpendicular pz orbitals on each member of the ring where three orbitals on each side are in phase with one another but not with respect to the three orbitals on the other side. As a result of this, an additional nodal plane is present perpendicular to the plane of the ring where the two triplets of orbitals meet.
The orbital composition of this MO is similar to that described in Comparison 2 with only p-orbitals present. However, this MO has an additional anti-bonding interaction and therefore the energy of this orbital is slightly higher and computed to be -0.24690. The energy of the boratabenzene HOMO is 0.01095, higher than the HOMO of benzene and slightly positive making it anti-bonding. This high energy HOMO therefore makes boratabenzene more prone to electrophilic attack. The fact that it is anti-bonding also explains the larger orbital surface around the boron atom despite it being more electropositive. Because the energy of this MO is so high, it is closer in energy to the boron orbital fragment and therefore possesses greater character from this fragment. Next, it should be noted that this MO for pyridinium has in fact been reordered. In benzene, the HOMO consists of two degenerate MOs. In the instance of pyridinium, the MO in consideration is not in fact the HOMO as a result of the degeneracy that has been broken. It is the 20th MO as a result of participation of the nitrogen heteroatom orbital fragment to the final MO and therefore the energy is lowered. The other associated degenerate orbital in benzene excludes two carbon fragment contributions to the MO and involves only the p-orbitals from 4 carbons. In the case of pyridinium, one of the two atoms excluded from contributing to the final MO is the nitrogen. This MO therefore does not benefit from the low energy nitrogen. Considering this, the broken degeneracy of boratabenzene can also be explained. However, the contribution of the boron atom results in the MO being pushed up in energy to the HOMO instead of down with the nitrogen. We are in reality describing the same MO despite the difference in their order.
Finally for borazine, the imbalance between the orbital surfaces on both sides of the molecule is again explained by different contributions from the orbital fragments that define the MO in the LCAO. Because the orbital fragments are asymmetrical, one fragment possesses two stabilising nitrogens compared to only one on the other. The fragment with two nitrogens is therefore lower in energy and has a greater contribution to the MO. This is why the orbital lobe on one side of the nodal plane is larger than on the other.
Effects of Substituition
As has widely been discussed thus far, minute alterations of the structure of the benzene molecule leads to a change in the MOs. These alterations arise out of the presence of different atoms in the molecule but also the resulting optimised geometries these atoms impose. This is clearly demonstrated by the geometry data collected from the optimisations of the 4 molecules performed earlier:
Geometry Data
Benzene
Boratazene
Pyridinium
Borazine
r(C-H) Å
1.09
n/a
n/a
n/a
r(B-H) Å
n/a
1.22
n/a
1.19
r(N-H) Å
n/a
n/a
1.02
1.01
r(B-N) Å
n/a
n/a
n/a
1.43
r(C-C) Å
1.40
n/a
n/a
n/a
r(C1,5-H) Å
n/a
1.10
1.08
n/a
r(C2,4-H) Å
n/a
1.10
1.08
n/a
r(C3-H) Å
n/a
1.09
1.09
n/a
r(N-C1,5) Å
n/a
n/a
1.35
n/a
r(B-C1,5) Å
n/a
1.51
n/a
n/a
r(C1,5-C2,4) Å
n/a
1.40
1.38
n/a
r(C2,4-C3) Å
n/a
1.41
1.40
n/a
θ(C-C-C) degrees(º)
120
n/a
n/a
n/a
θ(B-N-B) degrees(º)
n/a
n/a
n/a
123
θ(N-B-N) degrees(º)
n/a
n/a
n/a
117
θ(C2-C3-C4) degrees(º)
n/a
120
120
n/a
θ(C3-C2,4-C1,5) degrees(º)
n/a
122
119
n/a
θ(C2,4-C1,5-B) degrees(º)
n/a
120
n/a
n/a
θ(C1,5-N-C5,1) degrees(º)
n/a
n/a
123
n/a
θ(H-C/H-N/H-B) degrees(º)
180
180
180
180
This change in geometry ultimately affects the point group allocation of the molecule:
Molecule
Point Group
Benzene
D6h
Boratabenzene
C2v
Pyridinium
C2v
Borazine
D3h
As a result of the variation of point group, the MOs formed for these molecules will differ in symmetry labels and so will also have different structures as has been evident in the comparisons carried out above.
As was described in Comparison 3 for boratabenzene and pryidinium, disruption of the symmetry can result in the disruption of degenerate orbitals. The orbitals compared are originally degenerate in benzene but due to the change in symmetry, the molecules are no longer degenerate and have different energies. This disruption may also affect the reactivity of a molecule as the ordering of the orbitals may influence which orbital eventually becomes the HOMO.
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