Chemistry Reference
In-Depth Information
Ð
dx
w
ð
x
;
c
Þ
H
wð
x
;
c
Þ
Ð
dx
w
ð
x
;
c
Þwð
x
;
c
Þ
¼ «ð
c
Þ
«½¼
ð
4
:
4
Þ
d
dc
¼
0
)
c
min
ð
4
:
5
Þ
provided
!
c
min
>
d
2
«
dc
2
0
ð
4
:
6
Þ
In this way, we obtain the best approximation compatiblewith the form
assumed for the approximate trial function. Increasing the number of
flexible parameters increases the accuracy of the variational result.
2
For N variational parameters,
f
c
g¼ð
c
1
;
c
2
; ...;
c
N
Þ
, Equations 4.4
and 4.5 must be replaced by
«½w¼«ð
c
1
;
c
2
; ...;
c
N
Þ
ð
4
:
7
Þ
q
c
1
¼
q
«
q
«
q
c
2
¼¼
q
«
q
c
N
¼
0
)
c
ð
best
Þ
f
g
ð
4
:
8
Þ
«ð
best
Þ¼«ð
c
1
;
c
2
; ...;
c
0
N
Þ
ð
4
:
9
Þ
wð
best
Þ¼wð
x
;
c
1
;
c
2
; ...;
c
0
N
Þ
ð
4
:
10
Þ
where
f
c
ð
best
Þg ¼ ð
c
1
;
c
2
; ...;
c
0
N
Þ
is the set of N optimized parameters.
For working approximations, we must resort to some basis set of
regular functions (such as the STOs or GTOs of Chapter 3), introducing
either
(i) nonlinear
(orbital exponents) or
(ii)
linear variational
parameters.
2
Using appropriate numerical methods, it is feasible today to optimize variational wavefunctions
containing millions of terms, such as those encountered in the configuration interaction tech-
niques of Chapter 8.
