Chemistry Reference
In-Depth Information
where F is the same for all electrons. Despite their simple aspect, (7.21)
are complicated integro-differential
2
equations in which the operator
F depends on the {
c
i
} for which the equation should be solved, so that
they must be solved by iteration
)
hence the SCF method starting
from any convenient initial guess.
3
The iteration must be stopped
when the spin-orbitals obtained as solutions of the HF equations do
not differ appreciably from those used in the construction of
F
ðrÞ)
F
fc
i
g
. This is usually done by putting a convenient threshold
on the energies.
7.2 ROOTHAAN FORMULATION OF THE LCAO-
MO-SCF EQUATIONS
Eliminating spin from F
ð
x
Þ¼
F
ð
rs
Þ
, we obtain the
spinless
Fock
operator:
F
ð
r
Þ¼
h
þ
2J
ð
r
Þ
K
ð
r
Þ
ð
7
:
22
Þ
where
ð
dr
2
R
ð
r
2
;
r
2
Þ
J
ð
r
1
Þ¼
ð
7
:
23
Þ
r
12
ð
dr
2
R
ð
r
1
;
r
2
Þ
K
ð
r
1
Þ¼
P
r
1
r
2
ð
7
:
24
Þ
r
12
are spinless Coulomb and exchange potentials (space only), and
r
2
Þ¼
X
occ
i
f
i
ð
r
1
Þf
r
2
Þ¼r
a
ð
r
1
r
2
Þ¼r
b
ð
r
1
*
R
ð
r
1
;
;
;
i
ð
r
2
Þ
ð
7
:
25
Þ
is the Fock-Dirac density matrix for closed shells, the summation in the
last term of (7.25) being over all occupied orbitals (i
¼
1
;
2
; ...;
n).
K is the integral operator (7.16) while
r
2
2
is a differential operator.
3
Usually, H
€
uckel orbitals.
