transpose ofP, PT(or more generally, what computational chemists call the
conjugate transposeA{– see transpose, above). Thus
ifA¼01
10
thenP¼
0 :707 0: 707
0 : 707 " 0 : 707
;D¼
10
0 " 1
;P"^1 ¼
0 :707 0: 707
0 : 707 " 0 : 707
(In this simple example the transpose ofPhappens to be identical withP). In the
spirit of numerical methods 0.707 is used instead of 1/
pffi- A matrix likeAabove,
for which the premultiplying matrixPis orthogonal (and so for whichP"^1 ¼PT) is
said to beorthogonally diagonalizable. The matrices we will use to get molecular
orbital eigenvalues and eigenvectors are orthogonally diagonalizable. A matrix is
orthogonally diagonalizable if and only if it is symmetric; this has been described as
“one of the most amazing theorems in linear algebra” (see Roman S (1988) An
introduction to linear algebra with applications. Harcourt Brace, Orlando, p 408)
because the concept of orthogonal diagonalizability is not simple, but that of a
symmetric matrix is very simple.
4.3.3.6 Determinants
A determinant is asquarearray of elements that is a shorthand way of writing a sum
of products; if the elements are numbers, then the determinant is a number. Examples:
a 11 a 12
a 21 a 22(^)
¼a 11 a 22 "a 12 a 21 ;
52
43
(^)
¼^5 ð^3 Þ"^2 ð^4 Þ¼^7
As shown here, a 2&2 determinant can be expanded to show the sum it
represents by “cross multiplication”. A higher-order determinant can be expanded
by reducing it to smaller determinants until we reach 2&2 determinants; this is
done like this:
213 0
173 5
346 1
182 " 2
(^)
(^)
¼ 2
73 5
46 1
82 " 2
(^)
(^)
" 1
13 5
36 1
12 " 2
(^)
(^)
þ 3
17 5
34 1
18 " 2
(^)
(^)
" 0
173
346
182
Here we started with element (1,1) and moved across the first row. The first of
the above four terms is 2 times the determinant formed by striking out the row and
116 4 Introduction to Quantum Mechanics in Computational Chemistry