Rep : Mod:inorganicjk1911

This structure does not have the required D6h point group, so would give the wrong MOs. D6h symmetry is imposed and the structure is subjected to further optimisation.

Summary

It can be seen from the energies of the optimised structures that the one with D6h geometry is lower in energy, so it should be a more accurate representation of the real structure.

The D6h structure is subjected to frequency analysis.

Summary

Low frequencies ---   -2.5530   -2.5530   -0.0089   -0.0042   -0.0041   10.3930
 Low frequencies ---  413.9723  413.9723  621.1358

All low frequencies are low, and all "real" frequencies are positive, so an energy minimum has been found.

Optimised Structure of Benzene (D6h)

Pentahelicene

Results

C-C bond length = 1.40 Å
C-H bond length = 1.09 Å
Energy(631G-(d,p)) = -232.25821389 a.u.

MO Calculation Benzene

The MOs were calculated from the optimised D6h structure of Benzene.

Summary

Checkpoint File

Charge Distribution

C is more electronegative than H, so the partial charge on C is negative.

MO Diagram for Benzene

Central Section of the MO diagram of Benzene
File:Huge MO diagram.pdf

The diagram is arranged in two columns, 3e_2g is above 1a_2u. The computed MOs are shown side by side with their LCAO approximations.

Most of the LCAO approximations describe the "real" MOs quite accurately, albeit in a qualitative way. There are however some exceptions towards the top of the MO diagram. The most notable of these is the 3e_2g set, in which the p orbitals overlap in a way that isn't obvious from the LCAO diagram, resulting in an unusual shape in the MO.

The definition of aromaticity is complex. On a basic level, 2D aromaticity involves the cyclic delocalisation of electrons in a planar system. Hückel's rule states that the number of delocalised π electrons must be equal to (4n + 2) where n is an integer value. Aromaticity also grants the molecule certain physical properties, such as equal bond lengths and unusual stability compared to acyclic analogues. Furthermore, the cyclic delocalisation of electrons causes a diamagnetic ring current which isn't present in acyclic isomers. An important extension of Hückel's rule is that all bonding MOs in the molecule are fully occupied, and all antibonding orbitals are empty.

The MO diagram of benzene makes it possible to visualise some of the important aspects of aromaticity. For example, the 1a_2u orbital illustrates perfectly the cyclic π delocalisation present in the molecule. The electrons in that orbital are completely delocalised above and below the plane of the molecule. There are three occupied π MOs in benzene, and since their wavefunctions all overlap there is no reason why electrons can't travel between the three orbitals provided they have the correct spin. The MO diagram also illustrates how benzene conforms to Hückel's rule: each MO formally contains 2 electrons, and there are three π MOs, making 6π electrons overall. Therefore n=1, and the molecule is aromatic.

Another feature of aromaticity that the MO diagram illustrates is the orbital occupation criteria: all of the bonding orbitals are fully occupied (up to and including the 3e_1g set), and all antibonding orbitals are empty (1e_2u onwards). All bonding orbitals have negative orbital energy values and all antibonding orbitals have positive energy values. Because all bonding orbitals are filled and all antibionding orbitals are empty the molecule is electronically as stable as is possible.

In addition to the π orbitals, some σ orbitals also display delocalisation. 2a_1g for example is delocalised over the entire molecule, and even several higher energy σ orbitals display considerable delocalisation, such as 3a_1g and 3e_1u. It is possible that these σ MOs can overlap with the π MOs and increase the delocalisation of the molecule. This would explain why all the C-C bond lengths are identical despite the fact that not all of the MOs are contribute exactly the same amount of electron density to every C-C bond.

Boratabenzene

Boratabenzene was drawn in a rough geometry and then subjected to geometry optimisation using a 631G-(d,p) basis set.

Summary

This structure does not have the desired C_2v point group, so it is symmetrised and subjected to further optimisation using the same basis set. This time the molecule has the correct point group.

Optimised Structure of Boratabenzene (C_2v)

Pentahelicene

It can be seen from the energies in the item tables that the structure of boratabenzene with C_2v symmetry is slightly higher in energy than the unsymmetrised structure. This indicates that the symmetrised structure is not fully optimised, and a better basis set or a tighter convergence criteria would be needed to make the C_2v molecule lower in energy.

To check that an energy minimum has been found the optimised C_2v structure of boratabenzene was subjected to frequency analysis using the 631G-(d,p) basis set.

Summary

Full mass-weighted force constant matrix:
 Low frequencies ---  -13.1280   -0.0011   -0.0010   -0.0009   15.5233   18.3505
 Low frequencies ---  371.3341  404.2351  565.2516

All the low frequencies are reasonably low, and all the "real" frequencies are positive. This indicates that an energy minimum has been found.

Results

C-C bond length(adjacent to B) = 1.40 Å
C-C bond length(opposite to B) = 1.41 Å
C-H bond length(opposite to B) = 1.09 Å
C-H bond length(all others) = 1.10 Å
B-H bond length = 1.22 Å
Energy(631G-(d,p)) = -219.02052965 a.u.

The MOs and charge distribution were calculated from the optimised boratobenzene structure.

Summary

Charge Distribution

NBO Charge Distribution of Boratobenzene

Pyridinium

As for boratabenzene, the pyridinium ion was generated in a rough geometry, optimised using a 631G-(d,p) basis set, symmetrised to C_2v, and re-optimised.

Initial Optimisation Summary

C_2v symmetry is now imposed and the structure is optimised again.

Summary

Once again the molecule with C_2v symmetry is slightly higher in energy than the unsymmetrised molecule. A better basis set with a tighter convergence criteria would optimise the molecule to a more accurate minimum, with the expected C_2v symmetry.

Frequency Analysis was carried out on the molecule with C_2v symmetry to check whether it is an energy minimum.

Summary

Low frequencies ---   -7.2801   -0.0008    0.0004    0.0006   17.3641   18.5605
 Low frequencies ---  392.5055  404.0587  620.4783

All low frequencies are reasonably low; a more accurate basis set with a tighter convergence criteria would give lower low frequencies, but for the qualitative comparison of MOs, this geometry is sufficiently accurate. All "real" frequencies are positive, indicating that an energy minimum has been reached.

Optimised Structure of Pyridinium

Pentahelicene

Results

C-N bond length = 1.35 Å
N-H bond length = 1.02 Å
C-C bond length (adjacent to N) = 1.38 Å
C-C bond length (opposite N) = 1.40 Å
C-H bond length (opposite N) = 1.09 Å
C-H bond length (all others) = 1.08 Å

Charge Distribution

NBO Charge Distribution of Pyridinium

Borazine

Borazine was created in a rough geometry and the first optimisation was carried out.

Summary

The point group of this structure is not D3h, so the D3h symmetry is assigned, and a second optimisation is carried out.

Summary

As for the other heteroaromatic molecules, the molecule with the "real" symmetry is higher in energy than the one with lower symmetry, but the "real" higher symmetry structure will be used for calculating MOs.

The D3h structure is subjected to frequency analysis to confirm that it is an energy minimum.

Summary

 Low frequencies ---  -12.1146  -11.8808   -8.1025   -0.0097    0.0464    0.0700
 Low frequencies ---  289.1792  289.1896  403.7656