Virial Theorem and Hartree Fock

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    Theorem Virial theorem
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Discussion Overview

The discussion centers on the validity of the virial theorem within the Hartree Fock approximation, exploring theoretical underpinnings and references to literature on the topic. Participants seek to understand why the theorem applies in this context and inquire about resources for further reading.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the validity of the virial theorem in the Hartree Fock approximation and notes that some authors mention it without explanation.
  • Another participant references Werner Kutzelnigg's books as a source that discusses the topic in detail, although they are in German.
  • A participant expresses a lack of access to Kutzelnigg's books and requests online resources for the derivation of the theorem.
  • It is suggested that derivations may be found in older quantum chemistry textbooks, particularly from the 1970s and earlier.
  • A participant mentions that Vladimir Fock's original works contain the derivation of the theorem, specifically in a 1930 paper, and provides a link to a translation.
  • There is a note that the derivation explains the theorem's validity for variational solutions to the Schrödinger equation, but additional work is needed to understand its application in the Hartree Fock approximation.
  • Another participant points out that while the virial theorem holds for Hartree Fock, it may not hold for Linear Combination of Atomic Orbitals (LCAO) without scaling the atomic orbitals, which is often overlooked in calculations.
  • It is mentioned that the latest edition of Kutzelnigg's books is from 2003, indicating that they are not outdated.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of the virial theorem in various contexts, particularly between Hartree Fock and LCAO methods. The discussion remains unresolved regarding the specifics of the theorem's validity in these frameworks.

Contextual Notes

Some participants note the potential difficulty in finding accessible resources and the reliance on older literature for derivations. There is also an acknowledgment of the need for further work to clarify the theorem's application specifically to the Hartree Fock approximation.

Derivator
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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.
 
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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.
 
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.)
 
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).
 
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 Schrödinger 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.)
 
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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.
 

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