Chemistry Reference
In-Depth Information
For linearly independent functions, we have
∀
k
=
l
:
f
k
|
f
l
=
0
(2.1)
The basis is orthonormal if all vectors are in addition normalized to
+
1:
∀
k
:
f
k
|
f
k
=
1
(2.2)
This result can be summarized with the help of the Kronecker delta,
δ
ij
, which is
zero unless the subscript indices are identical, in which case it is unity. Hence, for
an orthonormal basis set,
f
k
|
f
l
=
δ
kl
(2.3)
In quantum mechanics, the bra-function of
f
k
is simply the complex-conjugate func-
tion,
f
k
, and the bracket or scalar product is defined as the integral of the product of
the functions over space:
f
k
f
l
dV
f
k
|
f
l
=
(2.4)
One thus also has
f
k
|
f
l
=
f
l
|
f
k
(2.5)
2.2 Linear Operators and Transformation Matrices
A linear operator is an operator that commutes with multiplicative scalars and is
distributive with respect to summation: this means that when it acts on a sum of
functions, it will operate on each term of the sum:
Rc
c R
|
f
k
=
|
f
k
(2.6)
R
|
f
l
=
R
f
k
+
R
f
k
+|
|
|
f
l
If the transformations of functions under an operator can be expressed as a map-
ping of these functions onto a linear combination of the basis vectors in the function
space, then the operator is said to leave the function space
invariant
. The corre-
sponding coefficients can then be collected in a transformation matrix. For this pur-
pose, we arrange the components in a row vector,
(
)
, as agreed
upon in Chap.
1
. This row precedes the transformation matrix. The usual symbols
are
R
for the operator and
|
f
1
,
|
f
2
,...,
|
f
n
D
(R)
for the corresponding matrix:
⎛
⎞
R
|
f
n
=
|
f
n
⎝
D
⎠
f
1
|
f
2
···|
f
1
|
f
2
···|
(R)
