Antisymmetry ofMany-electron Wavefunctions - Methods of Molecular Quantum Mechanics - page 90

ð dx 3

x 0 1 x 0 2 Þ¼ N ð N 1 Þ r 2 ð x 1 ; x 2 ; ; ... dx N Yð x 1 ; x 2 ; x 3 ; ...; x N Þ

ð 6 : 14 Þ

ð x 0 1 x 0 2

Y ; ; x 3 ; ...; x N Þ

where the first set of variables in r 1 and r 2 comes from Y and the second

. Functions (6.13) and (6.14) have only a mathematical meaning,
and are needed when a differential operator (like r

from Y

s. 3 All
physical meaning is carried instead by their diagonal elements ð x 0 1 ¼ x 1 Þ
and ð x 0 1 ¼ x 1

2 ) acts on the

r

x 0 2 ¼ x 2 Þ , which are the one- and two-electron distribution ;

functions

x 1 Þ¼ N ð dx 2 dx 3 ...

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

ð 6 : 15 Þ

ð dx 3

r 2 ð x 1 ; x 2 ; x 1 ; x 2 Þ¼ N ð N 1 Þ ... dx N Yð x 1 ; x 2 ; x 3 ; ...; x N Þ

ð 6 : 16 Þ

Y ð x 1 ; x 2 ; x 3 ; ...; x N Þ

having the following conservation properties:
ð dx 1

r 1 ð x 1 ; x 1 Þ¼ N ð 6 : 17 Þ

the total number of electrons and
ð dx 2 r 2 ð x 1 ;

x 2 ; x 1 ; x 2 Þ¼ð N 1 Þr 1 ð x 1 ; x 1 Þ;

ð dx 1 dx 2

ð 6 : 18 Þ

r 2 ð x 1 ; x 2 ; x 1 ; x 2 Þ¼ N ð N 1 Þ

the total number of indistinct pairs.
The physical meaning of the distribution functions (6.15) and (6.16) is
as follows:

r 1 ð x 1 ; x 1 Þ dx 1 ¼ probability of finding an electron at dx 1 ð 6 : 19 Þ

3 We recall that the operator r 2 acts only on

Y and not on Y

.