Computational Chemistry

and

f 2 ðHe1sÞ¼

z^3 He

p

! 1 = 2

e"zHjr"RHej (4.109)

Reasonable values [ 60 ] arezH¼1.24 Bohr"^1 andzHe¼2.0925 Bohr"^1 , ifris in

atomic units, a.u. (see Section 5.2.2); 1 a.u.¼0.5292 A ̊. The overlap integrals are

S 11 ¼S 22 ¼1 (as must be the case iff 1 andf 2 are normalized)

and S 12 ¼S 21 ¼0.435 (for all well-behaved functions

R

f 1 f 2 dq¼

R

f 2 f 1 dq).

The overlap matrix is thus

S¼

10 : 435

0 :435 1



(4.110)

3. Fock matrix
   We need the matrix elementsH 11 ¼H 22 andH 12 ¼H 21 , where the integrals
   Hij¼<fijH^jfj>are not actually calculated from first principles but rather are
   estimated with the aid of overlap integrals and orbital ionization energies:

fijH^jfi

¼"Ii

fijH^jfj

DE

¼"

1

2

KSijðIiþIjÞ

Using simply the ionization energies [cf. 58 ]:

IðHÞ¼I 1 ¼ 13 :6 eV, IðHeÞ¼I 2 ¼ 24 :6 eV

Hoffmann used in his initial calculations [ 56 a]K¼1.75.

So

H 11 ¼ 13 :6 eV

H 12 ¼H 21 ¼"^1 = 2 ð 1 : 75 Þð 0 : 435 Þð 13 : 6 þ 24 : 6 Þ¼" 14 : 5

H 22 ¼" 24 : 6

And the Fock matrix is

H¼

" 13 : 6 " 14 : 5

" 14 : 5 " 24 : 6



(4.111)

4.4 The Extended H€uckel Method 161