Computational Physics

4.4 Many-electron systems and the Slater determinant 53

states:
(x 1 ,...,xN)=ψ 1 (x 1 )···ψN(xN). (4.24)

The one-electron statesψkare eigenstates of the one-particle Hamiltonian, so they
are orthogonal. The probability density for finding the particles with specific values
x 1 ,...,xNof their coordinates is given by

ρ(x 1 ,x 2 ,...,xN)=|ψ 1 (x 1 )|^2 |ψ 2 (x 2 )|^2 ···|ψN(xN)|^2 , (4.25)

which is just the product of the one-electron probability densities. Such a probability
distribution is calleduncorrelated, and therefore we will use the term ‘uncorrelated’
for the wave function in (4.24) too.
Of course, the same state as (4.24) but with the spin-orbitals permuted, is a
solution too, as are linear combinations of several such states. But we require
antisymmetric states, and an antisymmetric linear combination of a minimal number
of terms of the form (4.24) is given by

AS(x 1 ,...,xN)=

1

√

N!

∑

P

PPψ 1 (x 1 )···ψN(xN). (4.26)

Pis a permutation operator which permutes thecoordinatesof the spin-orbitals
only, and not their labels (ifPacted on the latter too, it would have no effect at
all!); alternatively, one could havePacting on the labels only, the choice is merely
a matter of convention. The above-mentioned exchange operator is an example of
this type of operator. In(4.26), all permutations are summed over and the states
are multiplied by the signPof the permutation (the sign is+1or−1 according to
whether the permutation can be written as product of an even or an odd number of
pair interchanges respectively).
We can write (4.26) in the form of aSlater determinant:

AS(x 1 ,...,xN)=

1

√

N!

∣∣

∣∣

∣

∣∣

∣∣

ψ 1 (x 1 )ψ 2 (x 1 ) ··· ψN(x 1 )

ψ 1 (x 2 )ψ 2 (x 2 ) ··· ψN(x 2 )

..

.

..

.

..

.

ψ 1 (xN)ψ 2 (xN) ··· ψN(xN)

∣∣

∣∣

∣

∣∣

∣∣

. (4.27)

It is important to note that after this antisymmetrisation procedure the electrons are
correlated. To see this, consider the probability density of finding one electron with
coordinatesx 1 and another withx 2 :

ρ(x 1 ,x 2 )=

∫

dx 3 ···dxN|AS(x 1 ,...,xN)|^2

=

1

N(N− 1 )

∑

k,l

[|ψk(x 1 )|^2 |ψl(x 2 )|^2 −ψk∗(x 1 )ψk(x 2 )ψl∗(x 2 )ψl(x 1 )].

(4.28)