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To understand why the exclusion principle isn't violated in this system, you really need to shift from valence bond theory to molecular orbital theory. The $$p_{z}$$ orbitals in benzene combine according to the $$D_{6h}$$ symmetry of the molecule to generate a set of bonding molecular orbitals (MOs) (which are lower in energy than the isolated $$p_{z}$$ orbitals) and a set of antibonding molecular orbitals (which are higher in energy). Bonding orbitals are generated from sets of $$p_{z}$$ orbitals which are mostly or totally in phase, whereas the highest energy antibonding orbital of the set is composed of $$p_{z}$$ orbitals that are totally out of phase with respect to their adjacent neighbors.

Here, for instance, is the MO that is generated from the totally in-phase set (I calculated this at a spin-restricted RI-BP86/6-311G* level of theory in ORCA(1) and visualised the isosurfaces in VMD(2)):

Now, this MO is shared equally across all of the carbons, i.e. it is not localised. However it only contains 2 electrons and thus satisfies the exclusion principle. (F'x mentions spin-orbitals - this is where the orbitals are split on the basis of their spin, which entails a single electron per orbital. Benzene is a closed-shell singlet so we would expect the $$\alpha$$ and $$\beta$$ spin orbitals to be spatially indistinguishable.)

Not to sound like a walking advertisment for Housecroft and Sharpe, but H&S has a great visual introduction to MO theory.

## P.S.

It was suggested in chat that I expand upon the generated orbitals - here's one of the two highest occupied molecular orbitals, from the same calculation.

As you can see, there's a big node along one of the mirror planes. This orbital is higher in energy than the earlier example because it's composed of two antiphase sets of 3 $$p_{z}$$ orbitals, so this orbital has antibonding character for two of the carbon-carbon pairs. It should be easy to extrapolate to the highest lying MO of this type - antibonding all the way around.

(1) Neese, F. ORCA – an ab initio, Density Functional and Semiempirical program package, Version 2.6. University of Bonn, 2008

(2) Humphrey, W., Dalke, A. and Schulten, K., 'VMD - Visual Molecular Dynamics', J. Molec. Graphics 1996, 14.1, 33-38.

To understand why the exclusion principle isn't violated in this system, you really need to shift from valence bond theory to molecular orbital theory. The $$p_{z}$$ orbitals in benzene combine according to the $$D_{6h}$$ symmetry of the molecule to generate a set of bonding molecular orbitals (MOs) (which are lower in energy than the isolated $$p_{z}$$ orbitals) and a set of antibonding molecular orbitals (which are higher in energy). Bonding orbitals are generated from sets of $$p_{z}$$ orbitals which are mostly or totally in phase, whereas the highest energy antibonding orbital of the set is composed of $$p_{z}$$ orbitals that are totally out of phase with respect to their adjacent neighbors.

Here, for instance, is the MO that is generated from the totally in-phase set (I calculated this at a spin-restricted RI-BP86/6-311G* level of theory in ORCA(1) and visualised the isosurfaces in VMD(2)):

Now, this MO is shared equally across all of the carbons, i.e. it is not localised. However it only contains 2 electrons and thus satisfies the exclusion principle. (F'x mentions spin-orbitals - this is where the orbitals are split on the basis of their spin, which entails a single electron per orbital. Benzene is a closed-shell singlet so we would expect the $$\alpha$$ and $$\beta$$ spin orbitals to be spatially indistinguishable.)

Not to sound like a walking advertisment for Housecroft and Sharpe, but H&S has a great visual introduction to MO theory.

(1) Neese, F. ORCA – an ab initio, Density Functional and Semiempirical program package, Version 2.6. University of Bonn, 2008

(2) Humphrey, W., Dalke, A. and Schulten, K., 'VMD - Visual Molecular Dynamics', J. Molec. Graphics 1996, 14.1, 33-38.

To understand why the exclusion principle isn't violated in this system, you really need to shift from valence bond theory to molecular orbital theory. The $$p_{z}$$ orbitals in benzene combine according to the $$D_{6h}$$ symmetry of the molecule to generate a set of bonding molecular orbitals (MOs) (which are lower in energy than the isolated $$p_{z}$$ orbitals) and a set of antibonding molecular orbitals (which are higher in energy). Bonding orbitals are generated from sets of $$p_{z}$$ orbitals which are mostly or totally in phase, whereas the highest energy antibonding orbital of the set is composed of $$p_{z}$$ orbitals that are totally out of phase with respect to their adjacent neighbors.

Here, for instance, is the MO that is generated from the totally in-phase set (I calculated this at a spin-restricted RI-BP86/6-311G* level of theory in ORCA(1) and visualised the isosurfaces in VMD(2)):

Now, this MO is shared equally across all of the carbons, i.e. it is not localised. However it only contains 2 electrons and thus satisfies the exclusion principle. (F'x mentions spin-orbitals - this is where the orbitals are split on the basis of their spin, which entails a single electron per orbital. Benzene is a closed-shell singlet so we would expect the $$\alpha$$ and $$\beta$$ spin orbitals to be spatially indistinguishable.)

Not to sound like a walking advertisment for Housecroft and Sharpe, but H&S has a great visual introduction to MO theory.

## P.S.

It was suggested in chat that I expand upon the generated orbitals - here's one of the two highest occupied molecular orbitals, from the same calculation.

As you can see, there's a big node along one of the mirror planes. This orbital is higher in energy than the earlier example because it's composed of two antiphase sets of 3 $$p_{z}$$ orbitals, so this orbital has antibonding character for two of the carbon-carbon pairs. It should be easy to extrapolate to the highest lying MO of this type - antibonding all the way around.

(1) Neese, F. ORCA – an ab initio, Density Functional and Semiempirical program package, Version 2.6. University of Bonn, 2008

(2) Humphrey, W., Dalke, A. and Schulten, K., 'VMD - Visual Molecular Dynamics', J. Molec. Graphics 1996, 14.1, 33-38.

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To understand why the exclusion principle isn't violated in this system, you really need to shift from valence bond theory to molecular orbital theory. The $$p_{z}$$ orbitals in benzene combine according to the $$D_{6h}$$ symmetry of the molecule to generate a set of bonding molecular orbitals (MOs) (which are lower in energy than the isolated $$p_{z}$$ orbitals) and a set of antibonding molecular orbitals (which are higher in energy). Bonding orbitals are generated from sets of $$p_{z}$$ orbitals which are mostly or totally in phase, whereas the highest energy antibonding orbital of the set is composed of $$p_{z}$$ orbitals that are totally out of phase with respect to their adjacent neighbors.

Here, for instance, is the MO that is generated from the totally in-phase set (I calculated this at a spin-restricted RI-BP86/6-311G* level of theory in ORCA(1) and visualised the isosurfaces in VMD(2)):

Now, this MO is shared equally across all of the carbons, i.e. it is not localised. However it only contains 2 electrons and thus satisfies the exclusion principle. (F'x mentions spin-orbitals - this is where the orbitals are split on the basis of their spin, which entails a single electron per orbital. Benzene is a closed-shell singlet so we would expect the $$\alpha$$ and $$\beta$$ spin orbitals to be spatially indistinguishable.)

Not to sound like a walking advertisment for Housecroft and Sharpe, but H&S has a great visual introduction to MO theory.

(1) Neese, F. ORCA – an ab initio, Density Functional and Semiempirical program package, Version 2.6. University of Bonn, 2008

(2) Humphrey, W., Dalke, A. and Schulten, K., 'VMD - Visual Molecular Dynamics', J. Molec. Graphics 1996, 14.1, 33-38.