Why Does ½ Factor in HF Expectation Value?

[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.

 

[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.

 

[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.

 

[A. Neumaier]

if the summation runs from i<j instead of i,j, something that is claimed to be equivalent in general

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.

 

[Morten]

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

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".

 

[blue_leaf77]

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.

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.

 

[Morten]

Thank you