Sigma-SCF

In quantum chemistry, people find ground state solutions to the electronic Schrodinger equation by practicing the “golden rule” of lowest energy. The variational principle tells us that the energy of an approximate wave function is always higher than that of the exact ground state. Based on this theorem, people have come up with a large number of approaches to parameterizing a wave function and turning the solution of Schrodinger equation to a much easier energy minimization problem.

When it comes to excited states, however, the golden rule does not apply any more. Unlike ground states, excited states are saddle points instead of minima in energy landscape and thus hardly located by conventional minimization schemes. One of the key observations we have made is that energy variance, unlike energy, is at minimum for every state. As a consequence, the lowest energy principle should be replaced by the lowest variance principle.

This new scheme, however, faces an immediate challenge: if every state is a (local) minimum, how can we differentiate them? One can imagine that, without any guidance, this new method will generate solutions by luck. This issue has brought us to our second observation that using a direct energy-targeting functional gives us a good guess for an excited state near a chosen energy, which can then be used in variance minimization. We hence name this two-step method sigma-SCF since people usually denote energy variance using sigma-2.