Self-consistent-field Calculations and Model Hamiltonians - Methods of Molecular Quantum Mechanics - page 101

and x 0 , defined as

x 0 Þ¼ X

N
i ¼ 1 c i ð x Þc

*

i ð x 0 Þ

rð x ; ð 7 : 3 Þ

which is called the Fock-Dirac density matrix, and whose formal
mathematical properties are

ð dx ð dx

x 0 Þj x 0 ¼ x ¼ rð x ; rð x ; x Þ¼ N conservation ð 7 : 4 Þ

ð dx 00

x 00 Þrð x 00 x 0 Þ¼rð x x 0 Þ

rð x ; ; ; idempotency ð 7 : 5 Þ

r 0 ð x x 0 Þ¼rð x x 0 Þ

; ; invariance ð 7 : 6 Þ

r

0 is the result of any unitary transformation among its spin-

where

orbitals.
Mathematically speaking,

x 0 Þ is the kernel of the integral operator rð x ;

ð dx 0

x 0 Þ P xx 0

rð x Þ¼ rð x ; ð 7 : 7 Þ

which transforms a function F(x) into

ð dx 0 ð dx 0

x 0 Þ P xx 0 F ð x Þ¼

x 0 Þ F ð x 0 Þ rð x Þ F ð x Þ¼ rð x ; rð x ; ð 7 : 8 Þ

a new function of x, namely

ð dx 0 ð dx 0 X

i ð x 0 Þ F ð x 0 Þ¼ X

N
i ¼ 1 c i ð x Þc

N
i ¼ 1 jc i ihc i j F i
ð 7

x 0 Þ F ð x 0 Þ¼

*

r F ¼ rð x ;

: 9 Þ

r
is a projection operator in Fock space which projects out of the arbitrary
regular function F(x) its N-components along the basic vectors

wherewe introduced theDirac notation for the last term. In otherwords,

jc i i ,

i ¼ 1 ; 2 ; ...; N.