Computational Chemistry

function system examined here and for the generalization (see below) tonbasis
functions. Equations4.46and4.48form a system of simultaneous linear equations:

ðH 11 "ES 11 Þc 1 þðH 12 "ES 12 Þc 2 ¼ 0

ðH 21 "ES 21 Þc 1 þðH 22 "ES 22 Þc 2 ¼ 0

(4.49)

The pattern is that the subscripts correspond to the row and column in which they
lie; this is literally true for the matrices and determinants we will consider later, but
even for the system of equations4.49we note that in the first equation (“row 1”), the
coefficient ofc 1 has the subscripts 11 (row 1, column 1) and the coefficient ofc 2 has
the subscripts 12 (row 1, column 2), while in the second equation (“row 2”) the
coefficient ofc 1 has the subscripts 21 (row 2, column 1) and the coefficient ofc 2 has
the subscripts 22 (row 2, column 2).
The system of equations4.49are called secular equations, because of a supposed
resemblance to certain equations in astronomy that treat the long-term motion of the
planets; from the Latinsaeculum, a long period of time (not to be confused with
secular meaning worldly as opposed to religious, which is from the Latinsecularis,
worldly, temporal). From the secular equations we can find the basis function
coefficientsc 1 andc 2 , and thus the MOsc, since thec’s and the basis functionsf
make up the MOs (Eq.4.41). The simplest, most elegant and most powerful way to
get the coefficients and energies of the MOs from the secular equations is to use
matrix algebra (Section 4.3.3). The following exposition may seem a little involved,
but it must be emphasized that in practice the matrix method is implemented
automatically on a computer, to which it is highly suited.
The secular equations4.49are equivalent to the single matrix equation

H 11 "ES 11 H 12 "ES 12

H 21 "ES 21 H 22 "ES 22



c 1

c 2



¼

0

0



(4.50)

Since theH"ESmatrix is anHmatrix minus anESmatrix, and since theES
matrix is the product of anSmatrix and the scalarE, Eq.4.50can be written:

H 11 H 12

H 21 H 22



"

S 11 S 12

S 21 S 22



E

c 1

c 2



¼

0

0



(4.51)

which can be more concisely rendered as

½H"SEŠc¼ 0 (4.52)

and Eq.4.52can be written

Hc¼SEc (4.53)

4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 123