Concise Physical Chemistry

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

322 QUANTUM MOLECULAR MODELING

Atoms

antibonding

bonding

FIGURE 20.3 Bonding and antibonding solutions for the H+ 2. One electron in the lower

(bonding) orbital of H+ 2 is indicated by an arrow.

20.3 HIGHER MOLECULAR ORBITAL CALCULATIONS

In obtaining an approximate solution to the molecular Schr ̈odinger equation, the

many-electron wave function
(ri), which is a function of all radial distance vectors

ri, is broken up into orbitalsψi:

(ri)=(n!)−^1 /^2 det[(ψ 1 α)(ψ 1 β)(ψ 2 α)...]

The orbitalsψ 1 α, ψ 1 β, ψ 2 α,.. .accommodate single electrons. The symbolsαand

βdesignate opposite spins and “det” indicates a Slater determinant. Because of

spin pairing, each orbital can contain two electrons; hence the minimum number of

molecular orbitalsψ 1 ,ψ 2 ,ψ 3 ,...is one-half the number of electrons.

In 1951 Roothaan further divided single-electron molecular orbitalsψiintolinear

combinationsof basis functionsχμ:

ψi=

∑N

μ= 1

cμiχμ

(μ= 1 , 2 , 3 ,...N), where N>n. Having selected a basis setχμ, one wishes to find

the coefficientscμi.Alargecμimeans that the corresponding basis vector makes an

important contribution to the total molecular orbital, while a smallcμimeans that the

corresponding basis vector makes a small contribution. This gives a set ofalgebraic

equations in place of the set of coupled differential equations in the original problem.

Roothaan’s equations can be written in matrix form as

FC=SCE

whereCis the column vector of coefficients,Eis the diagonal matrix of energies,

with elementsEij=εiδij, the elements ofSareSμν=

∫

χμχνdτ, andFis the Fock

matrix.^2

Elements in theFmatrix are

Fμν=Hμν+

∑

λσ

Pλσ[(μν|λσ)−(μλ|νσ)/2]

(^2) It is unfortunate thatFis used to represent both the force field matrix in molecular mechanics and the
Fock matrix in quantum mechanics. Be careful not to confuse the two.