One example that comes to mind is the idea of UHF (unrestricted Hartree-Fock) which allows one to consider breaking spin rotational symmetry. States are then no longer purely spin singlet or triplet. One improves the variational energy but has a wave function which is qualitatively wrong.
Friday, May 7, 2010
Getting the right answer for the wrong reason?
As I emphasized in a previous post, getting a "good" ground state energy with a variational wave function is not necessarily and indication that the wave function itself is very accurate. Computational quantum chemistry (for some good reasons) is fixated on calculating energies. However, I would like to see some concrete criteria and indications or anecdotes concerning the reliability (or lack thereof) of the variational energy.
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There are many UHF solutions to most problems. In one of Fukutome's early papers, he points out something that I had not (indeed, it would seem most have not) considered, although in retrospect it seems obvious. The UHF state (indeed, any SCF state) can be thought of as an ensemble, with its own generalized partition function. If there are multiple solutions with comparable energies, then they will make approximately equal contributions to the partition function. Choosing one solution amongst a family of solutions with comparable energies amounts to arbitrarily choosing a sub-ensemble.
ReplyDeleteOne envisions a new way of constructing states by identifying families of low-lying solutions and using these to construct an "inferred" Hilbert space with the right symmetry. In principle, this should be possible (Naimark's theorem?). The key would be to exhaust the solutions within a given energy range, and it is not clear how this could be done without just "trying" again and again. This would yield a particularly "Bayesian" approach to quantum chemistry, wherein finding a new low-lying solution led to an "update" of the "currently known best" state.
There has been recent development in the area of identifying multiple UHF solution from Head-Gordon and Gill's groups, but I am not sure that they are thinking what I am thinking about this.
Another really good question is: what is the entropy of the ensemble corresponding to a particular UHF state?