Concise Physical Chemistry

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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.

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