Next: Bibliography Up: intro_estruc Previous: Simplified Notation for the

# Matrix Elements of the Hamiltonian and One- and Two-Electron Integrals

Now that we have a much more compact notation for the electronic Hamiltonian, we need to discuss how to evaluate matrix elements of this Hamiltonian in a basis of -electron Slater determinants . A matrix element between Slater determinants and will be written , where we have dropped the el'' subscript on because we will discuss the electronic Hamiltonian exclusively from this point. Because the Hamiltonian, like most operators in quantum mechanics, is a Hermitian operator, .

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,

 (35)

where and are the two orbitals which is allowed to be different.

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,

 (36)

There are other ways to write this integral. The form above, which places the complex conjugate terms on the left, is called physicist's notation,'' and is usually written in a bra-ket form.

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.

Next: Bibliography Up: intro_estruc Previous: Simplified Notation for the
David Sherrill 2003-08-07