Summarize and explain (1) - Benzene

Thanks to the double slit experiment, we already know one of the principles of quantum mechanics, namely that we are able to calculate the correct probability amplitudes for a system by describing this system as a set of partial amplitudes weighted by coefficients. To begin with, we want to consider a system with only two states with which, nevertheless, we are able to explain many phenomena in chemistry and spectroscopy. We imagine benzene with the following pair of electronic structures.

Benzene = |  |

As a symbol for the left electronic configuration, we use φ1 (or |1>), and, for the right φ2 (or |2>) respectively. A linear combination ψBenzene = a1φ1+ a2φ2 (or |Benzol>=a1|1>+a2|2>) shall describe the "real" benzene. Here, the coefficients a1 and a2 represent the relative contributions of the partial wavefunctions. Our aim is to calculate these coefficients and the possible energies of this system.

Although we do not know neither the amplitudes |1> and |2> of benzene in these two electronic configurations nor the respective energies, we pretend having both and introduce a more or less obvious aspect: Due to symmetry, both configurations have identical energies which will be denoted as Eo. Another quick assumption could be that even the coefficients a1 and a2 are the same, but we have to be aware that the probability density is proportional to the square of ψ. Thus, the sign of the coefficients remains questionable and, at best, we are able to state |a1|=|a2|. But additional mathematical aspects pave the way to a solution.

For a start, although the Hamilton operator H is undetermined, we have Schrödinger's equation for benzene.

Within this equation, the wave function ΨBenzene is replaced by the linear combination introduced above.

We multiply the equation with <1| and with <2| and apply the distributive law subsequently. As we know the states <1| and <2| to be orthonormal, the respective values of one and zero for the terms <1|1> = 1 and <1|2> = 0 yield two profoundly simplified equations.

Multiplied with <1| : |
Multiplied with <2| : |

The terms <1|H|1> and <2|H|2> are the energies of benzene in configuration |1> and |2>, for which Eo was introduced as abbreviation. The values of <1|H|2> and <2|H|1> correlate with the chance for a shift from one electronic configuration towards the other, i.e. from |1> to |2> and vice versa. Apparently, the probabilities are in both directions is the same and therefore <1|H|2> = <2|H|1> and we substitute W for both expressions. We then obtain two equations and are able to rearrange them in order to get an expression for a1/a2

By combining both equations, we eliminate the fraction a1/a2 and find the energy states EI and EII.

We draw the conclusion that there are two stable states, one of them of magnitude W below, the other of magnitude W above E0. Usually, EI is the energy of lower, EII the energy of the higher state (a convention that would favour reversing the ± sign, but we won't care for that here). In case |1> and |2> do not at all interfere with each other, EI , EII and Eo are all the same.

Instead of W, we now write -ΔE. For the energies of the two stationary states |I> and |II> we then recieve

At last, we want to characterize the shape of the electronic configurations |I> and |II>. Therefore, we insert the solutions for E into the expression found for a1/a2 derived above. This establishes a1 = ±a2 as mathematical link between the two coefficients, a relation already expected due to general considerations. If we finally introduce the conditions <I|I> = 1, and <II|II> = 1, i.e. normalize the established linear combinations, we get

All these considerations hold true not only for benzene but also for ammonia. Energy diagrams and an interesting application have been presented in the respective unit. The background to this treatment is that molecule is able to switch between two possible states:

|NH3> | |1> | |2> |
N atom | above | below | the plane the three Hydrogen atoms

Benzene has higher interaction energy since light electron can move quite fast between two possible benzene configurations. In contrast, an atom is a massive particle. Thus, the tendency for the nitrogen atom to change position and the interaction energy ΔE is found to be much smaller. The electronic transition of Benzene is reflected by absorption of ultraviolet light whereas the molecular rearrangement within the ammonia is reflected by a peak within the infrared region.

In the next unit, we are going to use the same approach to the cationic H2+ molecule and to molecular hydrogen.