Variational principle convergence

[Manchot]

A text I am reading has used the variational principle not only to find the ground state of a system, but also to find some higher order states. (Specifically, it was used to derive the Roothaan equations, which are ultimately related to the LCAO method of orbital calculations.) I don't see how this could be valid.

For finding the ground state energy of a system, it is obvious why minimizing the expected value of the Hamiltonian gets the best approximation to the ground state. But in what sense does the variational principle converge to the right result? I thought at first that it might minimize the variance of Hamiltonian in the trial wavefunction, but I do not believe that this is the case. So, does anyone know exactly in what precise sense the variational principle finds the "best" solutions to the eigenvalue problem?

 

[soarce]

For finding the ground state energy of a system, it is obvious why minimizing the expected value of the Hamiltonian gets the best approximation to the ground state. But in what sense does the variational principle converge to the right result? I thought at first that it might minimize the variance of Hamiltonian in the trial wavefunction, but I do not believe that this is the case. So, does anyone know exactly in what precise sense the variational principle finds the "best" solutions to the eigenvalue problem?

The higher order (excited) states are local minima of the Hamiltonian, while the ground state are the global minima. If your initial guess is sufficiently close to the excited state then it will converge to the excited one.
Or you can modify the functional (the Hamiltonian) such that the excited states are also grobal minima.

There is a lot of literature on this subject, you can start with http://portal.acm.org/citation.cfm?id=587202

 

[yaychemistry]

The way I see how to Variational principle works in the Roothan equation, or in Raylieigh-Schrödinger perturbation theory, etc in order to find both gound and excited states (at least in 1-particle Hamiltonians) is this:

We agree that the variational principle makes sense that one could find the ground state wavefunction by minimizing the energy, right? I submit the strategy, without proof, that I could then start again minimizing the energy of a wavefunction, while also requiring that this new wavefunction is orthogonal to the ground state wavefunction. This will give me the first excited state. I can then find a third wavefunction through minimization by requiring that it be orthogonal to the first two, etc, until I build up an approximation to the entire spectrum.

Using a matrix technique, like the Roothan equations or Rayleigh-Schrödinger, is sort of like doing that procedure to find all of the states all at once. The eigenvalues from the Matrix equation all correspond with orthogonal wavefunctions.