Convert a one-electron into a two-electron term

Photo Credit: Theodore Chasseriau - The Two Sisters

Convert a one-electron into a two-electron term

In quantum chemistry the Hamiltonian is usually split into a one-electron term and a two-electron term,

It has always seemed to me a bit silly to do this, because the one-electron term can be converted into a two-electron term by the following trick:

The sum is over all electrons \( N \). Now the one electron term has been written as a two electron term, and it can be incorporated into the “normal” two-electron term. The matrix elements of this “converted” one-electron term in the atomic orbital basis are

Here \( \B{h} \) and \( \B{S} \) are the usual core and overlap matrices, respectively. It is easy to include them into the normal two electron integrals, since these matrices are usually be stored in the memory of the computer.

When is the trick useful?

This trick becomes very useful when dealing with correlated methods. In that case, we need to worry about one-electron matrix elements at all, which considerably simplifies the equations. Also, the terms pertaining to “single excitations” in the correlated methods may effectively be combined with, or derived from, the “double excitations”. Effectively, the correlated methods only need to worry about double excitations.

Pretty cool, huh?

Caveat

Note: this does not help when you want to consider specific subterms involving single excitations, as in the CCSD(T) theory. In that case, using this trick, we would obtain these triples corrections as part of the CCSD(Q) method. In view of the way that the one electron terms actually “belong” to the two electron terms, perhaps this is actually a theoretically more satisfactory theory—if not actually more practical.