# Virial Theorem and Hartree Fock

• Derivator
In summary, the virial theorem applies to the Schrodinger equation in the Hartree Fock approximation, but it does not always hold in more accurate calculations.
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.

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 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.)

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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.

## 1. What is the Virial Theorem and how does it relate to quantum mechanics?

The Virial Theorem is a mathematical equation that relates the average kinetic energy of a system to its potential energy. In quantum mechanics, it is used to determine the average energy of a bound system, such as an atom or molecule.

## 2. How is the Virial Theorem used in molecular dynamics simulations?

In molecular dynamics simulations, the Virial Theorem is used to calculate the forces between particles in a system. This allows for the prediction of the system's behavior over time.

## 3. What is the Hartree Fock method and how does it differ from other quantum mechanical methods?

The Hartree Fock method is a quantum mechanical approach to solving the Schrödinger equation for a multi-electron system. It differs from other methods in that it uses a self-consistent field approximation, where the electron density is determined by the average of all electron positions, rather than the position of each individual electron.

## 4. What are the limitations of the Hartree Fock method?

The Hartree Fock method does not take into account electron correlation, which is the repulsion between electrons in a system. This can lead to inaccurate predictions for systems with strong electron correlation, such as transition metal complexes.

## 5. How does the Hartree Fock method contribute to our understanding of chemical bonding?

The Hartree Fock method is used to calculate the electronic energies and wavefunctions of molecules, which are crucial for understanding chemical bonding. It allows us to predict the geometry, stability, and reactivity of molecules, providing valuable insights into the nature of chemical bonds.

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