Computational Chemistry

So for example, ifSa¼5 2 1 andSb¼ 362
then SaSb¼5(3)þ2(6)þ1(2)¼ 15 þ 12 þ 2 ¼ 29
Now it’s easy to understand matrix multiplication: ifAB¼C, whereA,B, andC
are matrices, then elementi,jof the product matrixCis the series product of rowi
ofAand columnjofB. For example

AB¼

13

72



24

56



¼

1 ð 2 Þþ 3 ð 5 Þ 1 ð 4 Þþ 3 ð 6 Þ

7 ð 2 Þþ 2 ð 5 Þ 7 ð 4 Þþ 2 ð 6 Þ



¼

17 22

24 40



(With practice, you can multiply simple matrices in your head.) Note that matrix
multiplication is notcommutative:ABis notnecessarilyBA, e.g.

BA¼

24

56



13

72



¼

2 ð 1 Þþ 4 ð 7 Þ 2 ð 3 Þþ 4 ð 2 Þ

5 ð 1 Þþ 6 ð 7 Þ 5 ð 3 Þþ 6 ð 2 Þ



¼

30 14

47 27



(two matrices are identical if and only if their corresponding elements are the
same). Note that two matrices may be multiplied together only if the number of
columns of the first equals the number of rows of the second. Thus we can multiply
A(2&2)B(2&2),A(2&2)B(2&3),A(3&1)B(1&3), and so on. A useful
mnemonic is (a&b)(b&c)¼(a&c), meaning, for example thatA(2&1) times
B(1&2) givesC(2&2):

5

2



ðÞ¼ 03

5 ð 0 Þ 5 ð 3 Þ

2 ð 0 Þ 2 ð 3 Þ



¼

0 15

06



It is helpful to know beforehand the size i.e. (2&2), (3&3), whatever, of the
matrix you will get on multiplication.
To get an idea of why matrices are useful in dealing with systems of linear
equations, let’s go back to our system of equations

1. a 11 xþa 12 y¼c 1

2.

a 21 xþa 22 y¼c 2

Provided certain conditions are met this can be solved forxandy,e.g.bysolving(1)
forxin terms ofythen substituting forxin (2) etc. Now consider the equations from
the matrix viewpoint. Since

AB¼

a 11 a 12

a 21 a 22



x

y



¼

a 11 xa 12 y

a 21 xa 22 y



clearlyABcorresponds to the left hand side of the system, and the system can be
written

4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 111