# Density of classical states in quantum theory

Let's first treat electrons as classical objects. I can evaluate the classical energy of each state in a configurational space (3N real numbers and, say, spins) using just Coulomb's law.

Then I calculate the electronic wavefunction of the ground state, $\phi_0(\mathbf{r})$, in each point of the configurational space (classical state), which will give me the (square of amplitude...) probability density of each such classical state.

Combining these two results, I get the "density of classical states" of the ground state electronic wavefunction - the probability as a function of the classical energy, $c(E)$. I think this $c(E)$ might have some general features which might be used for speeding up post-Hartree-Fock (HF) quantum chemistry calculations.

I am not a theoretical physicists, therefore I do not know, how to ask Google / WoK, etc. Is there any study about this topic? Is there such an approach, and if so, what are its merits?

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 You probably mean square not square root. What is "post-HF"? The ground state wavefunction is very complicated, and you would probably want to study density functional theory. – Ron Maimon Apr 20 '12 at 9:05 DFT is an approximation often called an expensive RNG. I talk about some precise stuff, exact wavefunction. Not those coupled cluster or PIMC approximations. – Boris Apr 24 '12 at 14:23 @Boris: DFT is exact. There is no approximation, until you introduce one for the exchange energy. And DFT is quite precise --- and that's from a theorist who dislikes it on aesthetic grounds, but grudgingly admits that its real world utility is undisputable. – genneth Apr 24 '12 at 15:28 @Boris: What is "post HF" though? – Ron Maimon Apr 24 '12 at 16:35 @genneth: DFT is exact only theoretically. noone has ever found correct functional. it is like telling that using configurational integral is exact solution, but it is just rephrasing the problem. – Boris May 9 '12 at 12:32
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The above is for the ground state, that is the state of lowest energy, that is the lowest eigenvalue of the Hamiltonian. In general the system will have a complicated spectrum of excitations (think of simple atoms, like Helium, or Lithium, where the spectrum is already very rich, although you only have a few electrons). Due to the confinement, the spectrum will likely be discrete, i.e. will consist of (irregularly spaced) points on the real line. You might form a histogram of this set, and obtain an averaged distribution of energies = eigenvalues of Hamiltonian''; for one particle in a box with periodic "holes" you obtain a form of "crystalline band structure" in this way. For your fluid -- it will be rather difficult to obtain this information (I do not know how to compute such a spectrum in general, i.e. without assuming, that the system is rarefied, and that the interactions are weak).