Chemistry Reference
In-Depth Information
The HF equations for the occupied spatial MOs hence become
N
2
F
ð
r
Þf
i
ð
r
Þ¼«
i
f
i
ð
r
Þ
i
¼
1
;
2
; ...;
n
¼
ð
7
:
26
Þ
R has the projection properties of a density matrix in the space of
occupied MOs:
R
2
R
0
¼
R
ð
7
:
27
Þ
trR
¼
n
;
¼
R
;
Introducing a basis of m
ð
n
Þ
AOs, if C is the rectangular (m
n)
matrix of the LCAO coefficients
f
i
ð
r
Þ¼
X
m
m¼
1
x
m
ð
r
Þ
c
m
i
; fð
r
Þ¼
x
ð
r
Þ
C
ð
7
:
28
Þ
then the Roothaan pseudoeigenvalue equations for the ith MO are best
formulated in matrix form as
Fc
i
¼ «
i
Mc
i
i
¼
1
;
2
; ...;
n
ð
7
:
29
Þ
and for the whole set of the n occupied MOs
FC
¼
MC
e
ð
7
:
30
Þ
where
F
¼
x
F
x
ð
7
:
31
Þ
is the (m
m) matrix representative of the spinless Fock operator over the
m basis functions
x ¼ðx
1
x
2
...x
m
Þ
M
¼
x
x
ð
7
:
32
Þ
is the (m
m) metric matrix of the atomic basis, and
0
@
1
A
«
0
0
1
0
«
0
2
e
¼
ð
7
:
33
Þ
00
«
n
