Page 247 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

238 Approximation methods

If we substitute equation (9.8) into (9.6) and de®ne H ij by
^
H ij h÷ i jHj÷ j i (9:10)
we obtain
N N
X X
c i c j H ij
iˆ1 jˆ1
E ˆ
N N
X X
c i c j S ij
iˆ1 jˆ1
or
N N N N
X X X X
E c i c j S ij ˆ c i c j H ij (9:11)
iˆ1 jˆ1 iˆ1 jˆ1
To ®nd the values of the parameters c i in equation (9.8) which minimize E ,
we differentiate equation (9.11) with respect to each coef®cient c k (k ˆ 1, 2,
... , N)
0 1 0 1
N
N
N
N
N
N
@E X X @ X X @ X X
c i c j S ij ‡ E @ c i c j S ij A ˆ @ c i c j H ij A
@c k @c k @c k
iˆ1 jˆ1 iˆ1 jˆ1 iˆ1 jˆ1
and set (@E =@c k ) ˆ 0 for each value of k. The ®rst term on the left-hand side
vanishes. The remaining two terms may be combined to give
0 1
N
N
N
N
@ X X X X @c i @c j
@ c i c j (H ij ÿ E S ij ) A ˆ c j ‡ c i (H ij ÿ E S ij )
@c k @c k @c k
iˆ1 jˆ1 iˆ1 jˆ1
N N
X X
ˆ (ä ik c j ‡ c i ä jk )(H ij ÿ E S ij )
iˆ1 jˆ1
N N
X X
ˆ c j (H kj ÿ E S kj ) ‡ c i (H ik ÿ E S ik )
jˆ1 iˆ1
ˆ 0
where we have noted that (@c i =@c k ) ˆ ä ik because the coef®cients c i in equa-
tion (9.8) are independent of each other. If we replace the dummy index j by i
and note that H ik ˆ H ki and S ik ˆ S ki because the functions ÷ i are real, we
obtain a set of N linear homogeneous simultaneous equations
N
X
c i (H ki ÿ E S ki ) ˆ 0, k ˆ 1, 2, ... , N (9:12)
iˆ1