Computational Chemistry

Note that the transpose arises from twisting the matrix around to interchange

rows and columns. Clearly the transpose of a symmetric matrixAis the same

matrixA. For complex-number matrices, the analogue of the transpose is the

conjugate transposeA{; to get this formA*, the complex conjugate ofA, by

converting each complex number elementaþbiinAto its complex conjugate

a–bi, then switch the rows and columns ofA* to get (A*)T¼A{. Physicists call

A{theadjointofA, but mathematicians use adjoint to mean something else.

7. An orthogonal matrix is a square matrix whose inverse is its transpose: ifA"^1 ¼AT
   thenAis orthogonal. Examples:

A 1 ¼

1 =

p

2 " 1 =

p

2

1 =

p

21 =

p

2



A 2 ¼

1 =

p

6 " 1 =

p

2 " 1 =

p

3

2 =

p

60 1=

p

3

1 =

p

61 =

p

2 " 1 =

p

3

0

@

1

A

We saw that for the inverse of a matrix,A"^1 A¼AA"^1 ¼ 1 , so for an orthogonal
matrixATA¼AAT¼ 1 , since here the transpose is the inverse. Check this out for
the matrices shown. The complex analogue of an orthogonal matrix is aunitary
matrix; its inverse is its conjugate transpose.
The columns of an orthogonal matrix are orthonormal vectors. This means that if
we let each column represent a vector, then these vectors are mutually orthogonal
and each one is normalized. Two or more vectors are orthogonal if they are
mutually perpendicular (i.e. at right angles), and a vector is normalized if it is of
unit length. Consider the matrixA 1 above. If column 1 represents the vectorv 1 and
column 2 the vectorv 2 , then we can picture these vectors like this (the long side of a
right triangle is of unit length if the squares of the other sides sum to 1):

x

y

0

v 2 v 1

(^11)
2
(^1)
2
1
2
1

• 2

(^1)
v 1 =^2
(^1)
2
1 _

• 21 _
  v 2 =
  2

1 _

The two vectors are orthogonal: from the diagram the angle between them
is clearly 90since the angle each makes with, say, thex-axis is 45. Alternatively,
the angle can be calculated from vector algebra: the dot product (scalar product) is

v 1 +v 2 ¼jv 1 kv 2 jcosy

where |v| (“mod v”) is the absolute value of the vector, i.e. its length:

jvj¼ðv^2 xþv^2 yÞ^1 =^2 ðorðv^2 xþv^2 yþv^2 zÞ^1 =^2 for a 3D vectorÞ:

Each vector is normalized, i.e.jv 1 j¼jv 2 j¼ð^12 þ^12 Þ^1 =^2 ¼1.

114 4 Introduction to Quantum Mechanics in Computational Chemistry