Chemistry Reference
In-Depth Information
Functions having definite symmetry properties can be obtained
by letting suitable projection operators (projectors) act upon a function
having no specific symmetry, as the following two examples show.
First, it is known from elementary mathematics that a function f(x) can
be classified as even or odd with respect to the interchange x
)
x (for
instance, cos x or sin x). An arbitrary function without any symmetry can
always be expressed as a linear combination of an even and an odd
function as
1
2
f
ð
x
Þþ
f
ð
x
Þ
1
2
f
ð
x
Þ
f
ð
x
Þ
1
2
g
ð
x
Þþ
1
2
u
ð
x
Þ
12
f
ð
x
Þ¼
½
þ
½
¼
:
2
Þ
where we use g(x) (from German gerade) for the even function and u(x)
(German, ungerade) for the odd function. The operation (12.2) can be
aptly called the resolution of the function f(x) into its components having
definite symmetry properties, and it is immediately evident that any
integral like
¥
dxg
ð
x
Þ
u
ð
x
Þ¼
0
ð
12
:
3
Þ
¥
is identically zero.
1
Second, there is the split-shell description of the atomic or molecular
two-electron problem. We have seen in Chapters 4 and 9 that an accep-
table two-electron wavefunction for the ground states of He (
1
S) or H
2
(
1
g
) is given by
S
a
ðr
1
Þ
b
ðr
2
Þþ
b
ðr
1
Þ
a
ðr
2
Þ
2
þ
2S
2
1
2
Y ¼
p
p
½
að
s
1
Þbð
s
2
Þbð
s
1
Það
s
2
Þ
ð
12
:
4
Þ
where a
¼
1s, b
¼
1s
0
for He (
1
S) (the Eckart wavefunction), and a
¼
1s
A
,
b
¼
1s
B
for H
2
(
1
g
) (the HL wavefunction). Both wavefunctions are the
product of a symmetrical space part by an antisymmetrical spin part. The
simple product a(
r
1
)b(
r
2
) has no symmetry properties and must be acted
S
1
Changing x into
x the integral is changed into itself with a minus sign.
