Physical Chemistry Third Edition

836 20 The Electronic States of Diatomic Molecules

LCAOMOs for Additional Excited States of H+ 2

The wave functions for additional excited states of H+ 2 are approximated by

LCAOMOs using higher-energy hydrogen-like orbitals as basis functions. Two linear

combinations of 2sorbitals that are eigenfunctions of the appropriate symmetry oper-

ators are:

ψσg 2 sCg[ψ 2 s(rA)+ψ 2 s(rB)]Cg[ψ 2 sA+ψ 2 sB] (20.2-9)

ψσ∗u 2 sCu[ψ 2 s(rA)−ψ 2 s(rB)]Cu[ψ 2 sA−ψ 2 sB] (20.2-10)

Theσg 2 sorbital is a bonding orbital, and theσ∗u 2 sorbital is an antibonding orbital.

Theσg 2 sorbital energy is higher than that of theσ∗u 1 santibonding orbital since the

H+ 2 molecule ion dissociates from theσg 2 sstate to a hydrogen nucleus and a hydrogen

atom in the 2sstate, as shown schematically in Figure 20.9. In a later section of this

chapter we will obtain still more LCAOMOs, constructed as linear combinations of

the 2pand higher-energy atomic orbitals.

(a)

(b)

H 21 molecule

ion

1

H atom in

2 s state

Bare

proton

(^) g 2 s
(^) g 2 s
u 2 s
(^) g 1 s
u 1 s
EBO
/eV
R/10^210 m
1
2 13.6
2 3.4
2 3
Figure 20.9 (a) The orbital regions before and after dissociation of a hydrogen molecule
ion in theσg 2 sstate. (b) The energy levels of the LCAO molecular orbitals as a function of
internuclear distance.