Computational Chemistry

HF theory (Sections 5.2.3.1–5.2.3.6) starts with a total wavefunction or molecu-
lar wavefunctioncwhich is a Slater determinant made of “component” wavefunc-
tions or MOsc. InSection 5.2.3.1we approached HF theory by considering the
Slater determinant for a four-electron system:

C¼

1

ffiffiffiffi

4!

p

c 1 ð 1 Það 1 Þ c 1 ð 1 Þbð 1 Þ c 2 ð 1 Það 1 Þ c 2 ð 1 Þbð 1 Þ

c 1 ð 2 Það 2 Þ c 1 ð 2 Þbð 2 Þ c 2 ð 2 Það 2 Þ c 2 ð 2 Þbð 2 Þ

c 1 ð 3 Það 3 Þ c 1 ð 3 Þbð 3 Þ c 2 ð 3 Það 3 Þ c 2 ð 3 Þbð 3 Þ

c 1 ð 4 Það 4 Þ c 1 ð 4 Þbð 4 Þ c 2 ð 4 Það 4 Þ c 2 ð 4 Þbð 4 Þ

ð 5 : 163 ¼ 5 : 10 Þ

To construct the HF determinant we used only occupied MOs: four electrons
require only two spatial “component” MOs,c 1 andc 2 , and for each of these there
are two spin orbitals, created by multiplyingcby one of the spin functionsaorb;
the resulting four spin orbitals (c 1 a,c 1 bc 2 a,c 2 b) are used four times, once with
each electron. The determinantC, the HF wavefunction, thus consists of the four
lowest-energy spin orbitals; it is the simplest representation of the total wavefunc-
tion that is antisymmetric and satisfies the Pauli exclusion principle (Section 5.2.2),
but as we shall see it is not a complete representation of the total wavefunction.
In the Roothaan–Hall implementation of ab initio theory each “component”cis
composed of a set of basis functions (Sections 5.2.3.6and5.3):

ci¼

Xm

s¼ 1

csifs i¼ 1 ; 2 ; 3 ;...;mðcomponent MOsÞð 5 : 164 ¼ 5 : 52 Þ

Now note that there is no definite limit to how many basis functionsf 1 ,f 2 ,...
can be used for our four-electron calculation; although only two spatialc’s,c 1 and
c 2 , (i.e. fourspinorbitals) arerequiredto accommodate the four electrons of thisc,
the total number ofc’s can be greater. Thus for the hypothetical H–H–H–H an
STO-3G basis gives fourc’s, a 3–21G basis gives eight, and a 6–31G** basis gives
20 (Section 5.3.3). The idea behind CI is that a better total wavefunction, and from
this a better energy, results if the electrons are confined not just to the four spin
orbitalsc 1 a,c 1 bc 2 a,c 2 b,but are allowed to roam over all, or at least some, of the
virtual spin orbitalsc 3 a,c 3 b,c 4 a,...,cmb. To permit this we could writeCas a
linear combination of determinants

c¼c 1 D 1 þc 2 D 2 þc 3 D 3 þ(((þciDi ð 5 : 165 Þ

whereD 1 is the HF determinant of Eq. (5.163¼5.10) andD 2 ,D 3 , etc. correspond to
the promotion of electrons into virtual orbitals, e.g. we might have

Di¼

1

ffiffiffiffi

4!

p

c 1 ð 1 Það 1 Þ c 1 ð 1 Þbð 1 Þ c 3 ð 1 Það 1 Þ c 2 ð 1 Þbð 1 Þ

c 1 ð 2 Það 2 Þ c 1 ð 2 Þbð 2 Þ c 3 ð 2 Það 2 Þ c 2 ð 2 Þbð 2 Þ

c 1 ð 3 Það 3 Þ c 1 ð 3 Þbð 3 Þ c 3 ð 3 Það 3 Þ c 2 ð 3 Þbð 3 Þ

c 1 ð 4 Það 4 Þ c 1 ð 4 Þbð 4 Þ c 3 ð 4 Það 4 Þ c 2 ð 4 Þbð 4 Þ

(^)
(^)
ð 5 : 166 Þ
270 5 Ab initio Calculations