Computational Chemistry

SubstitutingLfrom Eq.5.40into Eq.5.39:

F^c¼#^1

2

PL^0 P#^1 c ð 5 : 41 Þ

Multiplying on the left byP#^1 and on the right byPwe get

F^P#^1 cP¼#^1

2

ðP#^1 PÞL^0 ðP#^1 cPÞ

which, sinceP#^1 P¼ 1 can be written

F^c^0 ¼#^1

2

L^0 c 0 ð 5 : 42 Þ

where

c^0 ¼P#^1 cP ð 5 : 43 Þ

We may as well remove the#1/2 factor by incorporating it intoL^0 , and we can
omit the prime fromC(had we been prescient we could havestartedthe derivation
using primes then writtenC¼P#^1 C^0 Pfor Eq.5.43). Equation5.42then becomes
(notationally anticipating the soon-to-be-apparent fact that the diagonal matrix is an
energy-level matrix)

F^c¼«c ð 5 : 44 Þ

where

e¼

ð# 1 = 2 Þl 11 00 ... 0

0 ð# 1 = 2 Þl 22 0 ... 0

... ... ... ...

000 ... ð# 1 = 2 Þlnn

0

B

B

B

@

1

C

C

C

A

ð 5 : 45 Þ

Equation5.44is the compact form of Eq.5.38. Thus

F^

c 1 ð 1 Þ

c 2 ð 1 Þ

c 3 ð 1 Þ

...

cnð 1 Þ

0

B

BB

B

B

@

1

C

CC

C

C

A

¼

e 1 00 ... 0

0 e 2 0 ... 0

... ... ... ...

00 0... en

0

BB

B

@

1

CC

C

A

c 1 ð 1 Þ

c 2 ð 1 Þ

c 3 ð 1 Þ

...

cnð 1 Þ

0

B

BB

B

B

@

1

C

CC

C

C

A

ð 5 : 46 Þ

where the superfluous double subscripts on thee’s have been replaced by single
ones. Equations5.44/5.46are the matrix form of the system of equations

5.2 The Basic Principles of the ab initio Method 193