Talk:Molecular orbital theory

In the section labeled Derivation, I found the following statement: "It is an (unproven) assumption of the variational method that the trial function that minimizes the expectation value gives the best approximation of the exact wave function."

This NOT an unproven assumption, since the variational theorem can be easily proved. Sekhar Talluri 14:29, 12 May 2009 (UTC)

I respectfully disagree. The linear variation method converges (for infinite basis) to the exact eigenfunction of a (bounded) Hamiltonian. This result is probably what you have in mind. For non-linear variation parameters (like AO exponents) such a theorem does not hold. Then it depends on what you consider the best approximation. A famous example is the description of the hydrogen atom with a single Gaussian variation function with one non-linear variation parameter: exp(−αr²). If we minimize the energy we find α=−0.4244, if we do a least square fit of the exact wave function (which we know in this case), we find α=−0.4242. The two optimized functions are close, but not the same, and it depends on what like best (lowest energy, or optimum least squares) what you find the best approximation.

This discussion is not completely academic for working quantum chemists, because a brute force minimization of orbital exponents usually gives a poor basis. What happens is that the inner shell orbitals (which carry lots of energy but are irrelevant for chemistry) are optimized and the valence orbitals become the orphans of the optimization. --Paul Wormer 14:54, 12 May 2009 (UTC)

PS1 See also Rayleigh-Ritz method.--Paul Wormer 14:57, 12 May 2009 (UTC)

PS2 Even in the Rayleigh-Ritz method it is not clear what the best wave function is before convergence: variation gives the lowest energy, but does it give also the largest overlap with the exact wave function? --Paul Wormer 15:00, 12 May 2009 (UTC)

Thanks for the explanation - I find your perspective quite enlightening. When I first read your statement I assumed that you had stated that the variation theorem itself is unproven. On closer inspection I found that your statement only applies to the use of the variation method for finding the 'best approximation to the wavefunction.' Sekhar Talluri 15:38, 12 May 2009 (UTC)