Antisymmetry ofMany-electron Wavefunctions - Methods of Molecular Quantum Mechanics - page 89

(iv) Ground state of the LiH molecule (N ¼ 4)

1 þ

2 2 2

S Þ¼jj 1 s 1 s 2 s 2 sjj S ¼ M S ¼ 0

Yð 1 s s ;

where 1 are the first two MOs obtained, in a first

approximation, by the linear combination of the basic AOs (1s,
2s) on Li and (1s) on H.

s and 2 s

All these electronic states are described by a single Slater determinant. But:

(v) First excited 1s2s state of the He atom (N ¼ 2)

8
<

jj 1s 2s jj S ¼ 1 ; M S ¼ 1

1
p jj 1s2

3 S Þ¼

s jjþ jj 1 s2s jj S ¼ 1 ; M S ¼ 0

Yð 1s 2s ;

:

jj 1

s jj S ¼ 1

;

M S ¼ 1

s2

Only the components of the triplet with j M S j¼ S are described by
a single Slater determinant, whereas the component with M S ¼ 0
requires two Slater determinants.

6.3 DISTRIBUTION FUNCTIONS

The passage from the abstract mathematical space of the N-electron
wavefunction

Y

to the physical four-dimensional space including spin is
done by the distribution functions. Since the Hamiltonian operator for a
many-electron atom or molecule contains only symmetrical sums of one-
electron and two-electron operators, the most important distribution
functions for us will be those involving one and two electrons respectively.

6.3.1 One- and Two-electron Distribution Functions

The distribution functions are the diagonal elements of the one- and two-
electron density matrices (McWeeny, 1960) defined as

x 0 1 Þ¼ N ð dx 2 dx 3 ð x 0 1

r 1 ð x 1 ; ... dx N Yð x 1 ; x 2 ; ...; x N ÞY ; x 2 ; ...; x N Þ

ð 6 : 13 Þ