Computational Chemistry

The diagonal terms are alwaysxbut the off-diagonal terms, 1 for adjacent and
0 for nonadjacent orbital pairs, depend on the numbering (which does not affect the
results: Fig.4.25). Any specific determinant of the type in Eq.4.77can be expanded
into a polynomial of ordern(where the determinant is of ordern&n), making
Eq.4.77yield polynomial equation:

xnþa 1 xn"^1 þa 2 xn"^2 þ+++an¼ 0 (4.78)

The polynomial can be solved forxand then the energy levels can be found from
(a"E)/b¼x, i.e. from

E¼a"bx (4.79)

The coefficients can then be calculated from the energy levels by substituting
theE’s into one of the secular equations, finding the ratio of thec’s, and normalizing
to get the actualc’s. An example will indicate how the determinant method can be
implemented.
Consider the propenyl system. In the secular determinant thei,i-type interactions
will be represented byx, adjacenti, j-type interactions by 1 and nonadjacenti, j-type
interactions by 0. For the determinantal equation we can write (Fig.4.25)

1

3 2

1

2 x^10

1 x 1

01 x

3

x 11

1 x 0

10 x

= x^3 – 2x

= x^3 – 2x

x 11

1 x 1

11 x

1

23

Fig. 4.25 The determinants corresponding to different numbering patterns can seem to differ, but
on expansion they give the same polynomial

4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 149