Concise Physical Chemistry

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

PERTURBATION 337

SCF averaging procedure. The difference between a Hartree–Fock energy and the

experimental energy is called thecorrelation energy. To remedy this fault,correlated

models are made up which consist of a linear combination of the Hartree–Fock

solution plus new basis functions representing singly, doubly, and so on, substituted

wave functions:

ψ=aψ 0 +

∑

ia

aaiψia+

∑

ijab

aabijψijab+···

In this equation,ij...designate occupiedspin orbitals(orbitals treated separately

according to electron spinαorβ), andab...designate excited orbitals of higher

energy thanij...calledvirtual orbitals.Virtual orbitals have small but nonzero

occupation numbers. The method is called configuration interaction (CI). The first

sum on the right above includes singly substituted orbitals (CIS). Inclusion of the

second sum on the right leads to doubly substituted orbitals (CID), while inclusion

of both sums is (CISD), and so on. A QCISD(T) method, with a 6-31G(d) basis set

QCISD(T)/6-31G(d), includes exponential terms in the expansion and is generally

considered to give a better estimate of the energy than do simple CI terms alone

(Pople, 1999).

20.16 PERTURBATION

Another method of progressing beyond the Hartree–Fock limit is by inclusion of

many-body perturbation terms(Atkins and Friedman, 1997):

E=E 0 +λiEi (i= 1 , 2 ,...)

The termsλiEiare small energies due to perturbations of the larger Hartree–Fock

calculation of the base energyE 0. The method was described by Moeller and Plesset

(1934), long before computers existed that were powerful enough to fully exploit it.

Higher values ofλiEilead to correlated energies. Perturbations ati=2or4are

calledMoeller–Plesset MP2 and MP4energies. They are used in corrected Gaussian

calculated energies, designated, for example, MP2/6-31G.

Atomic spin–orbit coupling energies E(SO) can be added (C: 0.14 mEh,

H: 0.0 mEh). A “higher level correction” (HLC) and a zero-point energyE(ZPE)

are added in the powerful combined methods called G3 or G4 calculations as well.

The zero point energy arises because the ground state of a quantum harmonic oscil-

lator is one-half quantum above the bottom of its parabolic potential well (Section

18.2). The summed zero-point energies of all atoms in a molecule, oscillating about

their equilibrium positions, isE(ZPE). The HLC, 0.009279Ehper pair of valence

electrons for a neutral molecule in the ground state, is a purely empirical factor,

parameterized so as to give the minimum discrepancy between a large test set of

accurately known experimental energies and calculated energies.