Concise Physical Chemistry

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

SEMIEMPIRICAL METHODS 323

defined by

Hμν=

∫

χμHˆχνdτ, Pμν= 2

∑n

1

cμicνi

and

(μν|λσ)=

∫∫

χμ(1)χν(1)×

1

r 12

χλ(2)χσ(2)dτ 1 dτ 2

(μλ|νσ)=

∫∫

χμ(1)χλ(1)×

1

r 12

χν(2)χσ(2)dτ 1 dτ 2

The matrix elementsHμνare elements of the core Hamiltonian that would be imposed

by the nuclei on each electron in the absence of all other electrons, and elementsεi

of the diagonal matrixEare one-electron energies. Many computer routines exist for

multiplication, inversion, and diagonalization of matrices.

The integrals(μν|λσ)and(μλ|νσ)are difficult to evaluate, which caused a bifur-

cation of the field of molecular orbital studies into subdisciplines calledsemiempirical

andab initio. Research groups led by Dewar and by Stewart were devoted to obtaining

solutions by substituting empirical constants intoFμν. A second approach was fol-

lowed by groups led by Pople, Gordon, and others, who used very efficient computer

codes and relied upon the increasing power of contemporary computing machines to

solve the integrals inFμν. In general, the rule of speed vs. accuracy applies. Semiem-

pirical substitution is faster, hence applicable to larger molecules.Ab initiomethods

are more accurate but they are very expensive in computer resources.

20.4 SEMIEMPIRICAL METHODS

Solution of the Schrodinger equation requires evaluation of many integrals. A large ̈

proportion of these integrals make a very small contribution to molecular energy and

enthalpy. When they are dropped, the calculation is simplified in the hope that the

sacrifice in accuracy will be small. Dropping the integrals in the equation set leaves

onlyHμνin place of the Fock matrix elementsFμν. Droppingsomeintegrals and

replacing others with empirical parameters gives legitimate Hamiltonian elements

but elements that are approximate because of the use of empirical parameters. They

are elementsHμνof asemiempiricalHamiltonian operator. The general rule is that if

you are modifying theFmatrix to obtain an approximate Hamiltonian, the method is

semiempirical. If you are working with the fullFmatrix and attempting to approach

a complete basis set, the method isab initio. Neither method is exact.

The most important steps in development of a computer-based semiempirical

method were in deciding which of the many integrals (μν|λσ) and(μλ|νσ)in a

polyatomic molecule can be dropped, which of the integrals must be retained and

parameterized, and how they should be parameterized.Neglect of differential overlap