Chemistry Reference
In-Depth Information
is the (n
n) diagonal matrix of the eigenvalues pertaining to occupied
MOs.
The density matrix R in the atomic basis will be
r
0
Þ¼fð
r
Þfð
r
0
Þ
¼
x
ð
r
Þ
CC
x
ð
r
0
Þ
¼
x
ð
r
Þ
R
xð
r
0
Þ
;
R
¼
CC
ð
7
R
ð
r
;
:
34
Þ
where R is the (m
m) matrix representative of the Fock-Dirac density
matrix over the AO basis
(hence follows the analysis of the electron
distribution in the molecule
)
Mulliken population analysis, a general-
ization of the simple example given in Section 6.3). The projection
operator properties of matrix R in the AO basis can be written as
x
trRM
¼
n
;
RMR
¼
R
ð
7
:
35
Þ
Roothaan's equations (7.29) are solved iteratively from the pseudose-
cular equation
j
F
«
M
j¼
0
ð
7
:
36
Þ
which is an algebraic equation in
«
having as m roots the MO orbital
energies
«
; «
; ...; «
n
occ
j «
n
þ
1
; «
n
þ
2
; ...; «
m
unocc
ð
7
:
37
Þ
1
2
illustrated in Figure 7.1.
According to Hund's rule, for nondegenerate levels, in the molecular
ground state the first n levels are occupied by electrons with opposite spin
(bonding levels,
0) and the remaining
ð
m
n
Þ
levels are unoccupied
(empty, antibonding levels,
«
i
<
«
i
>
0). The highest occupied MO (HOMO)
is
f
n
(orbital energy
«
n
) and the lowest unoccupied MO (LUMO) is
f
n
þ
1
(orbital energy
«
n
þ
1
).
The Roothaan SCF electronic energy is given in matrix form as
E
e
¼
2
ð
dr F
ð
r
Þ
R
ð
r
ð
dr G
ð
r
Þ
R
ð
r
r
0
Þj
r
0
¼
r
r
0
Þj
r
0
¼
r
;
;
ð
7
:
38
Þ
¼
2trFR
trGR
