Virial Theorem and Hartree Fock

[Derivator]

Hi,

short question:

Why is the virial threorem valid in the Hartree Fock approximation?

Some authors just mention this fact incidentally, but the don't explain it.

 

[DrDu]

Werner Kutzelnigg discusses this topic at length in his books
Einführung in die Theoretische Chemie (Wiley-VCh, Weinheim)
Bd. 1. Quantenmechanische Grundlagen 1975/1992
Bd. 2. Die Chemische Bindung 1978/1994
However, they are in German.

 

[Derivator]

i'm sorry, but i don' t have access to these books. don't you know some online resources? (if they exist, they have to be hard to find since I've already googled, of course.)

 

[cgk]

There are likely derivations for that in many old quantum chemistry books (say, from the 1970s and before). That being said, I only remember reading about it in the Kutzelnigg books DrDu mentioned (mainly due to not reading many older textbooks).

 

[SpectraCat]

The derivation is given in the original works of Vladimir Fock. He published the derivation in a 1930 paper in Z. Phys. You can find the translation http://www.calameo.com/books/0000035452dd43f81845c" on p. 139 (I can't seem to link directly to the page .. you need to scroll through to it).

(Thanks to unusualname for providing the link!)

Note that this derivation explains why the virial theorem holds for a variational solution to the Schrodinger equation. In order to see why it holds specifically in the case of the HF approximation, you need to do a little more work. Try running the Fock operator through the same treatment used by Fock for the Hamiltonian. (Since the one-electron orbitals obtained from the Fock operator are orthogonal, it is sufficient to show that the virial theorem holds for just a single Fock operator .. the expectation values of the total kinetic and potential energies will just be simple sums over the one-electron expectation values.)

 

[DrDu]

The interesting point that Kutzelnigg discusses is that while the virial theorem holds for HF, it does not necessarily hold for LCAO. I.e. one has to scale the atomic orbitals in order to fulfill the virial theorem, something that is not usually done in calculations. Also note that the newest edition of the books is from 2003, so it is not a pre-70's book.