Wednesday, 13 July 2016

Comparing Wavefunctions by their Overlap

Ever since starting in quantum chemistry I have been trying to avoid looking at orbitals. One reason is laziness. I just do not like sitting there clicking and waiting for all the orbitals to be rendered (even though this can be improved by using the proper scripts and programs). The other reason is a formal one: Orbitals, being one-body functions, can never tell us the whole truth about the many-body wavefunctions. Even worse, the same wavefunction may appear differently depending on the orbital set used to describe it (canonical orbitals, natural orbitals, natural transition orbitals, ...)

Assume that we performed two computations with different computational methods. When we look at the results we find out that, both, the molecular orbitals and wavefunction listings changed between the calculations. Does this mean that the two computations produced different wavefunctions? Not necessarily! The changes in the orbitals might be compensated by changes in the wavefunction expansion, at least in part. If we want to compare such wavefunctions we have to take into account the changes in the MOs and wavefunctions in a consistent fashion. In our newest Communication in J. Chem. Phys. "Unambiguous comparison of many-electron wavefunctions through their overlaps" we suggest to use the many-body wavefunction overlap for this task, i.e. the scalar product in the full many-body Hilbert space.

The outcome looks like this:

What we are doing is computing the 4 lowest excited states with CASSCF(12,9) and with MR-CIS(12,12). And then we compute the overlaps for all pairs of states and summarize them in pie charts. Every chart corresponds to one CASSCF wavefunction and the colors correspond to the MR-CIS wavefunctions. For example the second chart tells us that the Ψ1' wavefunction at CASSCF has a 67% overlap with the Ψ1 wavefunction at MR-CIS. But there are also smaller contributions of the Ψ2 and Ψ3 wavefunctions (as seen by the green and yellow bits).

The analysis gives us a quick overview of the relations between the wavefunctions computed at the different levels. There are two immediate conclusions: First, the overall state ordering is the same for both methods. Second, the wavefunctions are generally quite different, as seen by the large chunks of the pies missing.

The overlap code described here is actually a side product of a development we did for the nonadiabatic dynamics program SHARC (see this post). It will be released from the SHARC homepage as a standalone module, I hope soon ...

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