Chemistry Reference
In-Depth Information
C r
1
; ...; r
N
; R
1
; ...R
M
;
ð
t
Þ ¼ C
ð
rfg; Rfg;
t
Þ;
(7)
depends on all nuclear coordinates, the coordinates of the electrons, and time (if an
absolute time frame is assumed). Since we are interested in the calculation of the
stationary electron density, we look for a time-independent wave function that
depends only parametrically on the nuclear coordinates. After the separation of
time (by a product ansatz) and of the nuclear coordinates (Born-Oppenheimer
approximation [
24
-
26
]), one arrives at the time-independent electronic wave func-
tion
C
m
({
r
i
}) for electronic state
m
, which has to be approximated. A simple
product ansatz (Hartree product) of one-electron functions (orbitals) violates the
Pauli exclusion principle, because the wave function is no longer antisymmetric
with respect to the exchange of any two electronic coordinates. A corrected ansatz
explicitly implements the Pauli exclusion principle and can also be expressed as
a normalized determinant of a set of all
N
occupied orbitals,
c
k
1
rð Þ
c
k
1
rðÞ
Þ¼A
Y
N
1
N
.
.
.
Y
k
ð
rfg
c
k
i
rðÞ¼
p
.
.
;
(8)
!
i¼
1
c
k
N
rð Þ
c
k
N
rðÞ
with
A
denoting the antisymmetrization operator given by
!
X
X
1
N
A
¼
p
1
P
ij
þ
P
ijk
;
(9)
!
ij
ijk
where
P
ij
stands for all permutations of the two electrons
i
and
j
,
P
ijk
for all possible
permutations of the electrons
i
,
j
,and
k
, and so forth. The so-called Slater determinant
Y
k
contains either the
N
orbitals with the lowest energy (corresponding to the ground
state in a one-determinant picture) or orbitals with higher orbital energy. In practical
applications, a basis set is introduced to represent the one-electron functions.
In general, linear combinations of Slater determinants can be set up to yield
eigenfunctions of the squared spin operator. These linear combinations are called
configuration state functions (CSFs)
X
F
l
¼
b
lk
Y
k
;
(10)
k
with {
b
lk
} being known expansion coefficients. The single-determinant or single-
CSF ansatz can be improved by subsequently adding more and more CSFs. If all
(infinitely many) possible determinants are considered in the linear combination,
one obtains the so-called full configuration interaction (FCI) wave function
!
X
X
X
A
Y
N
C
FCI
m
ð
rfg
Þ¼
c
lm
F
l
ð
rfg
Þ¼
c
lm
b
lk
c
k
i
rðÞ
;
(11)
l
l
k
i¼
1
