Computational Chemistry

x 10

1 x 1

01 x

¼ 0 (4.80)

(compare this with the Fock matrix for the propenyl system). Solving this equation
(seeSection 4.3.3):

x 10

1 x 1

01 x

¼x

x 1

1 x

" 1

11

0 x

þ 1

1 x

01

¼xðx^2 " 1 Þ"ðx" 0 Þþ 0 ¼x^3 "x"x¼x^3 " 2 x¼0 (4.81)

This cubic can be factored (but in general polynomial equations require numeri-
cal approximation methods):

xðx^2 " 2 Þ¼0 so x¼0 and x^2 " 2 ¼0 or x¼,

p

2

From (a"E)/b¼x,E¼a"xband

x¼0 leads toE¼a

x¼þ

p

2 leads toE¼a"

p

2 b

x¼"

p

2 leads toE¼aþ

p

2 b

So we get the same energy levels as from matrix diagonalization (

pffi
2 ¼1.414).
To find the coefficients we substitute the energy levels into the secular equations;
for the propenyl system these are, projecting from the secular equations for a
two-orbital system, Eqs.4.49:

ðH 11 "ES 11 Þc 1 þðH 12 "ES 12 Þc 2 þðH 13 "ES 13 Þc 3 ¼ 0

ðH 21 "ES 21 Þc 1 þðH 22 "ES 22 Þc 2 þðH 23 "ES 23 Þc 3 ¼ 0

ðH 31 "ES 31 Þc 1 þðHS 32 "ES 32 Þc 2 þðH 33 "ES 33 Þc 3 ¼ 0

(4.82)

These can be simplified (Eqs.4.57, 4.61) to

ða"EÞc 1 þbc 2 þ 0 c 3 ¼ 0

bc 1 þða"EÞc 2 þbc 3 ¼ 0

0 c 1 þbc 2 þða"EÞc 3 ¼ 0

(4.83)

For the energy levelE¼aþ

p
2 b(MO level 1,c 1 ), substituting into the first
secular equation we get

150 4 Introduction to Quantum Mechanics in Computational Chemistry