Physical Chemistry Third Edition

886 21 The Electronic Structure of Polyatomic Molecules

functions for different resonance structures is the mathematical expression of reso-

nance. Neither formula alone represents the structure of the molecule, and the linear

combination corresponds to a “blending” of the two structural formulas, giving six

C–C bonds that are intermediate between single and double bonds.

Because a more flexible variation trial function must give a lower energy than a less

flexible function, the optimized energy calculated from the function of Eq. (21.6-1)

will be lower than that with a wave function corresponding to one resonance struc-

ture. In this case the two coefficientscIandcIIwill be equal to each other when opti-

mized. Various other resonance structures have been constructed for benzene, including

some with “long bonds” across the ring. Terms in the wave function corresponding to

such resonance structures would have smaller coefficients after optimization, and we

omit them.

The difference between the variational energy calculated with a wave function

including resonance and that calculated with a single resonance structure is called the

resonance energy. The same name is sometimes applied to the difference between the

correct nonrelativistic ground-state energy and that calculated with a single resonance

structure. An experimental estimate of the resonance energy for benzene is obtained

from the difference between the enthalpy change of hydrogenation of benzene and

three times the enthalpy change of hydrogenation of ethylene.

EXAMPLE21.9

Estimate the resonance energy of benzene using the thermodynamics techniques of Part 1.

Solution

We calculate enthalpy changes of hydrogenation of benzene at 298 K using enthalpy changes

of formation:

C 6 H 6 (g)+3H 2 (g)−→C 6 H 12 (g)

∆H◦∆fH◦(C 6 H 12 )−∆fH◦(C 6 H 6 )−3(0)

− 123 .1 kJ mol−^1 −(82.93 kJ mol−^1 )− 206 .0 kJ mol−^1

We do the same for ethylene:

C 2 H 4 (g)+H 2 (g)−→C 2 H 6 (g)

∆H◦∆fH◦(C 2 H 6 )−∆fH◦(C 2 H 4 )−3(0) 137 .15 kJ mol−^1

Resonance energy of benzene3(137.15 kJ mol−^1 )− 206 .0 kJ mol−^1

205 kJ mol−^1

The LCAOMO Treatment of Delocalized Bonding

To treat molecules such as benzene or butadiene in the LCAOMO method we must

abandon our policy of including only two atomic orbitals in our molecular orbitals.

In a complete Hartree–Fock–Roothaan treatment or variational treatment, we would

choose a basis set of atomic orbitals and would use linear combinations that contain

all of the basis functions. Delocalized canonical orbitals would result, as with BeH 2.