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
one-electron integral,
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Likewise, the operator allows up to two orbitals to be different
in Slater determinants
and
before matrix
element
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,
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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.