Principles - Methods of Molecular Quantum Mechanics - page 6

1.2.11 Anti-Hermitian Operators

@=@ x and r are instead anti-Hermitian operators, for which

<

@c
@ x jw

cj @w
@ x

¼

ð 1 : 21 Þ

:

hcjrwi ¼ hrcjwi

1.2.12 Expansion Theorem

Any regular (Q-class) function F(x) can be expressed exactly in the
complete set of the eigenfunctions of any Hermitian operator 3

A.If

A ¼ A

A w k ð x Þ¼ A k w k ð x Þ;

ð 1 : 22 Þ

then

F ð x Þ¼ X
k

w k ð x Þ C k ð 1 : 23 Þ

where the expansion coefficients are given by

ð dx 0

k ð x 0 Þ F ð x 0 Þ¼hw k j F i

C k ¼ w ð 1 : 24 Þ

as can be easily shown by multiplying both sides of Equation (1.23) by
w

k ð x Þ and integrating.
Some authors insert an integral sign into (1.23) to emphasize that
integration over the continuous part of the eigenvalue spectrum must be
included in the expansion. When the set of functions fw k ð x Þg is not
complete, truncation errors occur, and a lot of the literature data from
the quantum chemistry side is plagued by such errors.

1.2.13 From Operators to Matrices

Using the expansion theorem we can pass from operators (acting on
functions) to matrices (acting on vectors; Chapter 2). Consider a finite

3 A less stringent stipulation of completeness involves the approximation in themean (Margenau,
1961).