Physical Chemistry Third Edition

21.10 More Advanced Treatments of Molecular Electronic Structure. Computational Chemistry 905

and that do not necessarily have conjugated systems of bonds. All of the valence-shell

electrons are included in the treatment, but inner-shell electrons are assumed to be

separately treated. A minimal set of basis functions is used that includes all atomic

orbitals in the valence shell of every atom.

The procedure is based on the variation method. The Hamiltonian for the valence

electrons is assumed to be a sum of effective one-particle Hamiltonians, which are not

explicitly represented. Approximations are invoked that differ somewhat from those

of the simple Hückel method. The electron–electron repulsive energy is neglected, but

the overlap integrals are explicitly calculated. The matrix elements of the effective

Hamiltonian, Eq. (H-5) of Appendix H, are approximated using various formulas.

The elements withab(the diagonal elements) are set equal to the valence-state

ionization potential (VSIP) of the given orbital, which is the energy required to remove

one electron from the valence shell of the atom. Wolfsberg and Helmholz approximated

the off-diagonal elements by the mean of the VSIP of each orbital times a fixed constant

times the overlap integral:

Hab

1

2

K

(

Haaeff+Hbbeff

)

Sab (21.10-1)

The constantKhas been assigned various values from 1 to 3. A value of 1.75 is

common. A slightly different version is employed by Ballhausen and Gray, who use

the geometric mean instead of the arithmetic mean in approximatingHab.^12

The extended Hückel method can be used to find the conformation of lowest energy

by repeated calculations with different conformations. Reasonable results are obtained,

and the resulting conformations are sometimes used as starting points for more sophis-

ticated calculations.

Exercise 21.21

Use a software package such as CAChe or Spartan to find the equilibrium (lowest-energy)

conformation for the following molecules, using the extended Hückel method:

a.H 2 CO (formaldehyde or methanal)

b.C 2 H 6 (ethane)

c.CH 3 COCH 3 (acetone or propanone)

The Pariser–Pople–Parr method

The Pariser–Pople–Parr method is the simplest implementation of the self-consistent

field method. Like the Hückel method, it treats only the electrons in delocalized pi

orbitals in planar molecules and represents these orbitals as linear combinations of

basis functions that include only the unhybridizedporbitals. All overlap integrals are

assumed to vanish, as in the Hückel method, but not all of the integrals representing

electron–electron repulsions are assumed to vanish. Two notations are used for these

integrals:

(rt|su)〈rt|

1

r 12

|su〉

∫∫

fr∗(1)f∗t(2)

1

r 12

fs(1)fu(2)dq 1 dq 2 (21.10-2)

(^12) C. J. Ballhausen and H. B. Gray,Molecular Orbital Theory, W. A. Benjamin, New York, 1964, p. 118.