The two-site Hubbard model and photochemistry

At the cake meeting I gave an informal talk on how a two-site Hubbard-Holstein model can illuminate some basic and important concepts in the photo-isomerisation of simple molecules such as ethylene. A previous post discusses how the two-site Hubbard models illustrates many basic concepts in quantum chemistry and many-body theory.

Here are a few of the key references and ideas I drew upon in my talk.

A PRA from 2000 (and largely uncited) by Aalberts et al.
Quantum coherent dynamics of molecules: A simple scenario for ultrafast photoisomerization

1. It points out that photoisomerisation only occurs if there are "steric" interactions. i.e. the sigma bonds (not included in the Hubbard model) do not favour a planar arrangement for the molecule. Thus, the ground state is only planar due to the delocalisation energy associated with the pi electrons included in the Hubbard model.

2. This then leads to a twist angle (phi) dependence of the energies of three singlet states similar to that shown below from ab initio calculations in
Photoinduced dynamics of the valence states of ethene: A six-dimensional potential-energy surface of three electronic states with several conical intersection
by Robert P. Krawczyk, Alexandra Viel, Uwe Manthe, and Wolfgang Domcke

Note that the potential energies surfaces of the S1 (V) and S2 (Z) states touch when the molecule is twisted (phi=90 degrees).

This paper also gives a complete parameterisation of an effective 3x3 matrix Hamiltonian for these three low-lying singlet states. The paper never mentions it but this can be compared to the corresponding Hamiltonian for the two-site Hubbard-Holstein model to extract parameters.

3. How does one get a conical intersection between the S0 and S1 (N and V) states?
One must introduce an asymmetry between the energies of the pi orbitals localised on the two carbon atoms. This can be done by pyramidalization, where the hydrogen atoms are moved out of plane (HOOP=hydrogen out of plane) so that the sp2 hybridisation of the sigma orbital is distorted towards the sp3 hybridisation (pyramid) characteristic of methane. The graph below shows the ab initio calculation of the eigenenergies for a twisted geometry (phi=90 degrees) as a  function of the pyramidalisation angle.

In the Hubbard Holstein model the co-ordinate on the horizontal axis is q=q1-q2, the difference in the on-site vibrational mode co-ordinate between the two sites.

Some of the above connections are aided by the classic paper
Neutral and Charged Biradicals, Zwitterions, Funnels in S1, and Proton Translocation: Their Role in Photochemistry, Photophysics, and Vision
Vlasta Bonačić-Koutecký, Jaroslav Koutecký, and Josef Michl

and a 1985 Journal of Chemical Education paper
Electronic structure in pi systems. Part I. Huckel theory with electron repulsion 
Marye Anne Fox and F. A. Matsen

The former does not mention the Hubbard model, but the latter does.