Computational Chemistry

We now have the energy levels ("25.5 and"5.95 eV), but the eigenvectors

ofC^0 must be transformed to give us the coefficients of the original, nonortho-

gonal basis functions.

7. Transformation ofC^0 toC
   Using Eq.4.100,(C¼S"1/2C^0 ):

C¼

1 : 083 " 0 : 248

" 0 :248 1: 083



0 :436 0: 899

0 : 900 " 0 : 437



¼

0 :249 1: 082

0 : 867 " 0 : 696



S"^1 =^2 C^0

(4.117)

Note that unlike the case in the SHM, the sum of the squares of thec’s for an MO

does not equal 1, since overlap integralsSijfor basis functions on different atoms

are not set equal to 0; in other words, the basis functions are not assumed to be

orthogonal, and the overlap matrix is not a unit matrix. Thus forc 1 :

c 1 ¼c 1 f 1 þc 2 f 2 ; so

Z

c^21 dv¼

Z

c^21 f^21 þ 2 c 1 c 2 f 1 f 2 þc^22 f^22

'(

dv¼ 1

since the probability of finding an electron inc 1 somewhere in space is 1. The basis
functionsfare normalized, so

c^21 þ 2 c 1 c 2 S 12 þc^22 ¼ 1 ; i.e:

c^21 þc^22 ¼ 1 " 2 c 1 c 2 S 12

not¼1 as on the simple H€uckel method.

4.4.3 The Extended Huckel Method – Applications€

The EHM was initially applied to the geometries (including conformations) and
relative energies of hydrocarbons [ 56 a], but the calculation of these two basic
chemical parameters is now much better handled by semiempirical methods like
AM1 and PM3 (Chapter 6) and by ab initio (Chapter 5) and DFT (Chapter 7)
methods. The main use of the EHM nowadays is to study large, extended systems
[ 62 ] like polymers, solids and surfaces. Indeed, of four papers by Hoffmann and
coworkers in theJournal of the American Chemical Societyin 1995, using the
EHM, three applied it to such polymeric systems [ 63 ]. The ability of the method to
illuminate problems in solid-state science makes it useful to physicists. Even when
not applied to polymeric systems, the EHM is frequently used to study large,

4.4 The Extended H€uckel Method 163