# HF expectation value

1. Jan 5, 2016

### Morten

I am not sure why a factor of (½) appears in front of the summation over orbitals, i, j to N, of the Coulomb and exchange integrals in the HF energy expectation value.

2. Jan 5, 2016

### blue_leaf77

A constant factor appearing in an equation in a specialized field of physics, uncommon to the majority of physicists, is susceptible to the definition of the one who derives the said equation. I have seen several authors write their own version of Hartree-Fock energy which differ in the prefactors. It's a lot more helpful if you write the particular form which you are confused with.

3. Jan 5, 2016

### Morten

Thank you, but the thing is if the summation runs from i<j instead of i,j, something that is claimed to be equivalent in general I think, then the factor is omitted, and I am not sure how this is so.

4. Jan 5, 2016

### A. Neumaier

You may perhaps claim this, but it is not equivalent in general.

Instead of getting each pair of distinct indices once you get it twice and hence need a factor 1/2. You also need to ensure that equal indices don't make a contribution.

5. Jan 5, 2016

### Morten

Thank you. I refer to "Basic Principles and Techniques of Molecular Quantum Mechanics" by Ralph E. Christoffersen, p. 445 + 483 footnotes, when I write: "something that is claimed to be equivalent in general I think".

6. Jan 5, 2016

### blue_leaf77

If the factor of 1/2 appears, that means the index of summation must be written like $\sum_i \sum_j$ with the condition $i\neq j$ being imposed as Neumaier said. You need to check how the indices in the summation are written when 1/2 is appearing.

7. Jan 5, 2016

Thank you