Chemistry Reference
In-Depth Information
1.2.3 Orthogonality
If
ð
dx
c
hcjwi¼
ð
x
Þwð
x
Þ¼
0
ð
1
:
6
Þ
then we say that
w
is orthogonal (
?
)to
c
.If
0
0
hc
jw
i¼
S
ð6¼
0
Þ
ð
1
:
7
Þ
0
are not orthogonal, but can be orthogonalized by choosing
the linear combination (Schmidt orthogonalization):
w
0
and
then
c
0
0
0
c ¼ c
;
w ¼
N
ðw
S
c
Þ;
hcjwi¼
0
ð
1
:
8
Þ
Þ
1
=
2
is the normalization factor. In fact, it is easily seen
whereN
¼ð
1
S
2
0
and
0
are normalized to 1:
that, if
c
w
0
0
0
hcjwi¼
N
hc
jw
S
c
i¼
N
ð
S
S
Þ¼
0
ð
1
:
9
Þ
1.2.4 Set of Orthonormal Functions
Let
fw
k
ð
x
Þg ¼ ðw
1
w
2
...w
k
...w
i
...Þ
ð
1
:
10
Þ
be a set of functions. If
hw
k
jw
i
i¼d
ki
k
;
i
¼
1
;
; ...
ð
1
:
11
Þ
2
where
d
ki
is the Kronecker delta (1 if i
¼
k,0ifi
6¼
k), then the set is said to
be orthonormal.
1.2.5 Linear Independence
A set of functions is said to be linearly independent if
X
k
w
k
ð
x
Þ
C
k
¼
0 with
;
necessarily
;
C
k
¼
0 for any k
ð
1
:
12
Þ
For a set to be linearly independent, it will be sufficient that the
determinant of themetricmatrixM(see Chapter 2) be different fromzero:
