We want an eigenvalue equation because (cf. Section 4.3.4) we hope to be able
to use the matrix form of a series of such equations to invoke matrix diagonalization
to get eigenvalues and eigenvectors. Equation (5.35) is not quite an eigenvalue
equation, because it is not of the form operation on function¼k'function, but
rather operation on function¼sum of (k'functions). However, by transforming
the molecular orbitalscto a new set the equation can be put in eigenvalue form
(with a caveat, as we shall see). Equation5.35represents a system of equations
F^c 1 ð 1 Þ¼#^1
2
½l 11 c 1 ð 1 Þþl 12 c 2 ð 1 Þþl 13 c 3 ð 1 Þþ(((þl 1 ncnð 1 Þ i¼ 1F^c 2 ð 1 Þ¼#^1
2½l 21 c 1 ð 1 Þþl 22 c 2 ð 1 Þþl 23 c 3 ð 1 Þþ(((þl 2 ncnð 1 Þ i¼ 2...F^cnð 1 Þ¼#^1
2
½ln 1 c 1 ð 1 Þþln 2 c 2 ð 1 Þþln 3 c 3 ð 1 Þþ(((þlnncnð 1 Þ i¼nð 5 : 37 ÞThere arenspatial orbitalscsince we are considering a system of 2nelectrons and
each orbital holds two electrons. The 1 in parentheses on each orbital emphasizes that
each of thesenequations is aone-electron equation, dealing with the same electron
(we could have used a 2 or a 3, etc.), i.e. the Fock operator (Eq.5.36) is a one-electron
operator, unlike the general electronic Hamiltonian operator of Eq.5.15, which is a
multi-electron operator (a 2nelectron operator for our specific case). The Fock
operator acts on a total ofnspatial orbitals, thec 1 ,c 2 ,...,cnin Eq.5.35.
The series of equations Eqs.5.37can be written as the single matrix equation (cf.
Chapter 4, Eq. 4.50)
F^
c 1 ð 1 Þ
c 2 ð 1 Þ
c 3 ð 1 Þ
...
cnð 1 Þ0
B
BB
B
B
@
1
C
CC
C
C
A
¼
1
2
l 11 l 12 l 13 ... l 1 n
l 21 l 22 l 23 ... l 2 n
... ... ... ...
ln 1 ln 2 ln 3 ... lnn0
BB
B
@
1
CC
C
A
c 1 ð 1 Þ
c 2 ð 1 Þ
c 3 ð 1 Þ
...
cnð 1 Þ0
B
BB
B
B
@
1
C
CC
C
C
A
ð 5 : 38 Þi.e.
F^c¼#^1
2Lc ð 5 : 39 ÞIn Eqs.5.37, each equation will be of the formF^ci¼kci, which is what we want, if
all the lij ¼ 0 except for i ¼ j (for example, in the first equation
F^cið 1 Þ¼#ð 1 = 2 Þl 11 c 1 ð 1 Þif the only nonzerolisl 11 ). This will be the case if
in Eq.5.39Lis a diagonal matrix. It can be shown thatLis diagonalizable
(Section 4.3.3), i.e. there exist matricesP,P#^1 and a diagonal matrixL^0 such that
L¼PL^0 P#^1 ð 5 : 40 Þ192 5 Ab initio Calculations