52 The Hartree–Fock method
4.4 Many-electron systems and the Slater determinant
In the helium problem, we could make use of the fact that in the ground state the
required antisymmetry is taken care of by the spin part of the wave function, which
drops out of the Schrödinger equation. If it is not the ground state we are after, or if
more than two electrons are involved, antisymmetry requirements affect the orbital
part of the wave function, and in the next two sections we shall consider a more
general approach to an independent electron Hamiltonian, taking this antisymmetry
into account. In the present section we consider a particular class of antisymmetric
many-electron wave functions and in the next section we shall derive the equations
obeyed by them.
When considering a many-electron problem, it must be remembered that elec-
trons are identical particles. This is reflected in the form of the Hamiltonian: for
example in (4.2), interchanging electronsiandjdoes not change the Hamiltonian
and the same holds for the independent particle Hamiltonian(4.3). We say that the
Hamiltonian commutes with the particle-exchange operator,Pij. This operator acts
on a many-electron state and it has the effect of interchanging the coordinates of
particlesiandj. For anN-particle state^3
Pij(x 1 ,x 2 ,...,xi,...,xj,...,xN)=(x 1 ,x 2 ,...,xj,...,xi,...,xN). (4.22)In this equation,xiis again the combined spin and orbital coordinate:
xi=(ri,si). (4.23)AsPijis an Hermitian operator which commutes with the Hamiltonian, the eigen-
states of the Hamiltonian are simultaneous eigenstates ofPijwith real eigenvalues.
Furthermore, asP^2 ij=1 (interchanging a pair twice in a state brings the state back
to its original form), its eigenvalue is either+1or−1. It is an experimental fact
that for particles with half-integer spin (fermions) the eigenvalue of the permutation
operator is always−1, and for particles with integer spin (bosons) it is always+1. In
the first case, the wave function is antisymmetric with respect to particle exchange
and in the second case it is symmetric with respect to this operation. As electrons
have spin-1/2, the wave function of a many-electron system is antisymmetric with
respect to particle exchange.
Let us forget about antisymmetry for a moment. For the case of an independent-
particle Hamiltonian, which is a sum of one-electron Hamiltonians as in (4.3), we
can write the solution of the Schrödinger equation as a product of one-electron
(^3) In order to clarify the role of the coordinatesi, we note that for a single electron the wave function can be
written as atwo-spinor, that is, a two-dimensional vector, ands, which is a two-valued coordinate, selects one
component of this spinor. When dealing with more particles (N), the two-spinors combine into a 2N-dimensional
one and the combined coordinatess 1 ,...,sNselect a component of this large spinor (which depends on the
positionsri).