It would seem that computing these matrix elements would be extremely tedious, because both and are Slater determinants which expand into different products of orbitals, giving a total of different terms! However, all but of these terms go to zero because the orbitals are orthonormal. If we were computing a simple overlap like , all of the remaining terms are identical and cancel the in the denominator from the Slater determinant normalization factor.
If and differ,
then any term which places electron in some orbital
in and a different orbital
in must also
go to zero unless the integration over electron is not
just a simple overlap integral like
, but involves some operator
or . Since a particular operator can only affect
one coordinate , all the other spin orbitals for other electrons must
be identical, or the integration will go to zero (orbital orthonormality).
Hence, allows, at most, only one spin orbital to be different
in and for
to be nonzero.
Integration over the other electron coordinates will give factors
of one, resulting in a single integral over a single set of electron
coordinates for electron , which is called a
Likewise, the operator allows up to two orbitals to be different
in Slater determinants and before matrix
goes to zero. Integration over other
electron coordinates gives factors of one, leading to an integral over
two electronic coordinates only, a two-electron integral,
There is a very simple set of rules, called Slater's rules, which explain how to write matrix elements in terms of these one- and two-electron integrals. Most of quantum chemistry is derived in terms of these quantities.