Concise Physical Chemistry

c20 JWBS043-Rogers September 13, 2010 11:29 Printer Name: Yet to Come

THE HYDROGEN MOLECULE ION 321

0 5

0

0.2

EB(R)

EA(R)

R

FIGURE 20.2 Bonding and antibonding orbitals for H+ 2. The units ofEare hartrees and the

units ofRare bohr=52.9 pm.

which are combined to give

E=

J+K

1 +S

The functionsJ,K, andSare all dependent upon the internuclear separationRbecause

orbital interaction is greater when the nuclei are close together and smaller when they

are far apart. The functionE=f(R) for the positive combination of atomic orbitals

is shown by the solid curve in Fig. 20.2.

Positive nuclei attracted to an electron between them are bound to each other. The

nuclei do not crash into each other because at someR, internuclear repulsion becomes

dominant over bonding and the energy begins to rise sharply. The compromise be-

tween attractive and repulsive forces results in a minimum inE=f(R). The bond

has a definite and fixed length of 2.5 bohr=132 pm in this first approximation. To

find the energy of the H+ 2 bond at this distance, we must solve equations forS,J, and

Kto give

S= 0. 461 , J= 0. 00963 , K=− 0. 1044 , E=− 0. 065 Eh

The hartree of energy is 1 Eh=2625 kJ mol−^1. The minimum is at E=

− 0. 065 Eh=−171 kJ mol−^1. This is about 64% of the experimental value. The

quantitative result is not very good, but the qualitative result shows that the energy is

negative at 132 pm, which indicates bonding. It is noteworthy that the chemical bond

arises as a natural consequence of quantum mechanics without further assumptions.

There is a second solution forEin Fig. 20.2 which arises from the negative

combinationψH+ 2 = 1 sA− 1 sB. The energy does not go through a minimum with

thisantibondingwave function but rises monotonically with decreasing R. In H+ 2 ,

bonding and antibonding orbitals are above and below the energy of the separated

system H+and H. The higher and lower bonding and antibonding molecular orbitals

at the hond distance are often shown simply as Fig. 20.3.