Chemistry Reference
In-Depth Information
where to first order in
dc
(a column of infinitesimal variation of coeffi-
cients)
d
H
¼ dc
Hcþ c
Hdc;
d
M
¼ dc
1
c þ c
1
dc
ð
4
:
42
Þ
being stationary against arbitrary varia-
tions in the coefficients yields the equation
The necessary condition for
«
d« ¼
0
) d
H
«d
M
¼
0
ð
4
:
43
Þ
and in matrix form
dc
ðH«
1
Þc þ c
ðH«
1
Þdc ¼
0
ð
4
:
44
Þ
Because matrix
H
is Hermitian, the second term in (4.44) is simply
the complex conjugate of the first, so that, since
dc
is arbitrary, the
condition (4.43) takes the matrix form
ðH«
1
Þc ¼ 0 ) Hc ¼ «c
ð
4
:
45
Þ
which is the eigenvalue equation for matrix
H
(Equation 2.27). The
variational determination of the linear coefficients under the constraint
of orthonormality of the basis functions in the Ritz method is, therefore,
completely equivalent to the problem of diagonalizing matrix
H
. Follow-
ing what was said there, the homogeneous system (4.45) has nontrivial
solutions if and only if
jH«
1
j¼
0
ð
4
:
46
Þ
The solution of the secular Equation 4.46 yields as best values for the
variational energy (4.35) the N real roots, which are usually ordered in
ascending order:
«
1
«
2
«
N
ð
4
:
47
Þ
c
1
; c
2
; ...; c
N
ð
4
:
48
Þ
w
1
; w
2
; ...; w
N
ð
4
:
49
Þ
