Physical Chemistry Third Edition

842 20 The Electronic States of Diatomic Molecules

density for the simple LCAOMO wave function of Eq. (20.3-5) by integrating the square

of the wave function over the space coordinates of electron 2. We can omit the spin

functions and the integration over spin coordinates since this integration would give a

factor of unity:

ρMO

∫

ψσg 1 s(1)^2 ψσg 1 s(2)^2 d^3 r 2 ψσg 1 s(1)^2



1

2 + 2 S

[ψ 1 sA(1)^2 + 2 ψ 1 sA(1)ψ 1 sB(1)+ψ 1 sB(1)^2 ] (20.3-11)

We can interpret the three terms in this expressions as follows: The first term is the

contribution to the probability from the atomic orbital centered on nucleus A and the

last term is the contribution from the atomic orbital centered on nucleus B. The second

term represents the contribution of the overlap of the two atomic orbitals.

We now perform the same integration on the square of the simple valence-bond

wave function of Eq. (20.3-7):

ρVBc^2 VB

∫[

ψ 1 sA(1)^2 ψ 1 sB(2)^2 + 2 ψ 1 sA(1)ψ 1 sB(2)ψ 1 sB(1)ψ 1 sA(2)

+ψ 1 sB(1)^2 ψ 1 sA(2)^2

]

d^3 r 2

c^2 VB[ψ 1 sA(1)^2 + 2 Sψ 1 sA(1)ψ 1 sB(1)+ψ 1 sB(1)^2 ] (20.3-12)

The difference between this result and the molecular orbital result is the factorS(the

overlap integral) in the second term of the valence-bond result. The probability density

in the overlap region is smaller than with the LCAOMO function, since the overlap

integralSis smaller than unity. However, the electron moves over the entire molecule

in much the same way as with the LCAOMO wave function.

The distinction between the valence-bond method and the LCAOMO method for

the hydrogen atom disappeared when improvements were made to the simple versions

of each method. Further improvements have been made to both methods. A generalized

valence-bond method has been developed by Goddard and his collaborators.^7 In this

method, the atomic orbitals such as those in Eq. (20.3-7) are replaced by functions that

are optimized by minimizing the energy. The two electrons in a chemical bond also do

not have to occupy the same space function. Further improvements have been made.^8

Diatomic Helium

In the Born–Oppenheimer approximation the zero-order electronic Hamiltonian oper-

ator for diatomic helium consists of four hydrogen-molecule-ion-like (HMIL) Hamil-

tonian operators:

̂H(0)ĤHMIL(1)+ĤHMIL(2)+̂HHMIL(3)+ĤHMIL(4) (20.3-13)

where the HMIL Hamiltonian is

ĤHMIL−h ̄

2

2 m

∇^2 +

e^2

4 πε 0

(

−

Z

rA

−

Z

rB

)

(20.3-14)

(^7) W. J. Hunt, P. J. Hay, and W. A. Goddard,J. Chem. Phys., 57 , 738 (1972); W. A. Goddard, T. H. Dunning,
W. J. Hunt, and P. J. Hay,Acc. Chem. Res., 6 , 368 (1973).
(^8) I. N. Levine,op. cit., p. 612ff (note 4).