Physical Chemistry Third Edition

20.2 LCAOMOs. Approximate Molecular Orbitals That Are Linear Combinations of Atomic Orbitals 837

Exercise 20.7

Draw sketches of the orbital regions for the functions in Eqs. (20.2-9) and (20.2-10). Argue that

the designationsσgandσ∗uare correct.

Normalization of the LCAOMOs

Normalization of theψ 1 σgLCAOMO means that

1 |cg|^2

∫

(ψ 1 sA+ψ 1 sB)(ψ 1 sA+ψ 1 sB)d^3 r (20.2-11)

The 1satomic orbitals are real, so we omit the complex conjugate symbols. We choose

the normalization constantcgto be real so that

1 c^2 g

∫

(ψ 1 sA 2 + 2 ψ 1 sAψ 1 sB+ψ 1 sB 2 )d^3 r (20.2-12)

The atomic orbitalsψ 1 sAandψ 1 sBare normalized, so that the first term and the last

term in the integral will each yield unity when the integration is done. The second term

gives an integral that is denoted byS 1 s 1 s:

∫

ψ 1 sAψ 1 sBd^3 rS 1 s 1 s (20.2-13)

The integralS 1 s 1 sis called anoverlap integralbecause its major contribution comes

from the overlap region, inside which both factors are significantly different from zero.

Its value depends onrAB, approaching zero if the two nuclei are very far apart and

approaching unity whenrABapproaches zero, because it then becomes the same as a

normalization integral. Similar overlap integrals can be defined for other pairs of atomic

orbitals. For any pair of normalized atomic orbitals, the values of overlap integrals must

lie between−1 and+1, and approach zero asrABis made large. If we had an overlap

integral between a 1sanda2pzorbital or any other pair of different orbitals it would

approach zero asrABapproaches zero, because it would then become the same as an

orthogonality integral.

We now have

1 c^2 g(1+ 2 S+1)c^2 g(2+ 2 S) (20.2-14)

where we omit the subscripts on the overlap integral. The normalized LCAOMO is

ψσg 1 s

1

√

2 + 2 S

(ψ 1 sA+ψ 1 sB) (20.2-15)

where we choose the positive square root.