Antisymmetry ofMany-electron Wavefunctions - Methods of Molecular Quantum Mechanics - page 92

the number of electrons with spin a and

ð dr 1

r 1 ð r 1 ; r 1 Þ¼ N b ð 6 : 26 Þ

the number of electrons with spin b . Therefore, the sum

r 1 Þþr 1 ð r 1

r 1 Þ¼r 1 ð r 1

P ð r 1 ; ; ; r 1 Þ ð 6 : 27 Þ

is the electron density, as observed, for example, from the X-ray diffrac-
tion spectra of polycyclic hydrocarbons (Bacon, 1969), with

P ð r 1 ; r 1 Þ dr 1 ¼ probability of finding at dr 1 an electron with either spin

ð 6 :

28 Þ

whereas the difference

r 1 Þr 1 ð r 1 r 1 Þ¼r 1 ð r 1 Q ð r 1 ; ; ; r 1 Þ ð 6 : 29 Þ

is the spin density, with

Q ð r 1 ; r 1 Þ dr 1 ¼ probability of finding at dr 1 an excess of spin a over spin b

ð 6 : 30 Þ

P and Q satisfy the following conservation relations:

ð dr 1 P ð r 1

; r 1 Þ¼ N a þ N b ¼ N ð 6 : 31 Þ

the total number of electrons and

ð dr 1 Q ð r 1

; r 1 Þ¼ N a N b ¼ 2M S ð 6 : 32 Þ

where

N a N b
2

M S ¼ ð 6 : 33 Þ