Antisymmetry ofMany-electron Wavefunctions - Methods of Molecular Quantum Mechanics - page 87

6.2 SLATER DETERMINANTS

Generalizing to the N-electron system, we shall write the N-electron
wavefunction Y satisfying Pauli's principle as

c 1 ð x 1 Þ c 2 ð x 1 Þ c N ð x 1 Þ

1
N !

c 1 ð x 2 Þ c 2 ð x 2 Þ c N ð x 2 Þ

Yð x 1 ; x 2 ; ...; x N Þ¼ p

c 1 ð x N Þ c 2 ð x N Þ c N ð x N Þ

¼jjc 1 c 2 c N jj ð 6 : 7 Þ

a Slater determinant of order N, where rows denote space-spin coordi-
nates of the electrons (x ¼ rs) and columns the spin-orbital functions.
If the latter are orthonormal, then

Y of Equation 6.7 is normalized to 1:

ð dx 1 dx 2

hYjYi¼ ... dx N Y ð x 1 ; x 2 ; ...; x N ÞYð x 1 ; x 2 ; ...; x N Þ¼ 1 ð 6 : 8 Þ

The elementary properties of determinants introduced in Chapter 2
show that (6.7) can be equally well written as

c 1 ð x 1 Þ c 1 ð x 2 Þ c 1 ð x N Þ
c 2 ð x 1 Þ c 2 ð x 2 Þ c 2 ð x N Þ
c N ð x 1 Þ c N ð x 2 Þ c N ð x N Þ

1
N !

p

Yð x 1 ; x 2 ; ...; x N Þ¼ ð 6 : 9 Þ

where we now choose spin-orbitals as rows and electrons as columns,
having interchanged rows with columns in the original definition (6.7).
Furthermore, it is easily seen that

Y does satisfy Pauli's antisymmetry

principle:

c 1 ð x 2 Þ c 2 ð x 2 Þ c N ð x 2 Þ
c 1 ð x 1 Þ c 2 ð x 1 Þ c N ð x 1 Þ
c 1 ð x N Þ c 2 ð x N Þ c N ð x N Þ

1
N !

Yð x 2 ; x 1 ; ...; x N Þ¼ p

¼Yð x 1 ; ; ...;

x N Þ

ð 6 :

10 Þ

x 2

since this is equivalent to interchange of two rows in the determinant (6.7)
and the determinant changes sign.