I have been working through a really nice paper Using Valence Bond Theory to Understand Electronic Excited States: Application to the Hidden Excited State (21Ag) of C2nH2n+2 (n = 2−14) Polyenes by Wu, Danovich, Shurki, and Shaik.
[An earlier post discusses some of the interesting photophysics associated with these molecules].
Here are just a few of the key ideas. First, the ground and low lying singlet (covalent) states are written in a Rumer basis set of valence bond states [these are not orthogonal]. See R1 and R2 below for C4H6 (butadiene)
There is only one parameter in the Hamiltonian, lambda, and this is extracted from DFT based calculations. The eigenstates and energies are shown on the right.
For larger molecules one needs to include a larger number of basis states (e.g., see below for the case of hexatriene).
Simple energy correlation (Walsh) diagrams can then be used to understand how these states interact to produce the low lying excited states.
- the relative ordering of the excited states
- the hardening of the -C=C- stretch frequency in the 2Ag excited state (see an earlier post Finding the lost twin on similar physics in other molecules)
- the opposite bond alternation in the ground and 2Ag states
- the presence of conical intersections between the ground state and 2Ag potential energy surfaces that are important for non-radiative decay