Computational Chemistry

density around the helium nucleus falls off more quickly with distance than does
that around the hydrogen nucleus (Fig.5.8).
We have a geometry and a basis set, and wish to do an SCF calculation on HHeþ
with both electrons in the lowest MO,c 1 , i.e. on thesingletground state. In general,
SCF calculations proceed from specification of geometry, basis set, charge and
multiplicity. The multiplicity is a way of specifying the number of unpaired
electrons:

Multiplicity¼S¼ 2 sþ 1 ð 5 : 99 Þ

wheres¼total number of unpaired electron spins (each electron has a spin of)½),
taking each unpaired spin as +½. Figure5.9shows some examples of the specifica-
tion of charge and multiplicity. By default an SCF calculation is performed on the
ground stateof specified multiplicity, i.e. the MO’s are filled fromc 1 up to give the
lowest-energy state of that multiplicity.
Step 2– Calculating the integrals
Having specified a Hartree–Fock calculation on singlet HHe+, with H#He¼
0.800 A ̊ (1.5117 bohr), using an STO-1G basis set, the most straightforward way
to proceed is to now calculate all the integrals, and the orthogonalizing matrixS#1/2

x

y

z

electron

r

basis function f 1 centered on atomic nucleus 1

basis function f 2 centered on atomic nucleus 2

r–R 1

r–R 2

R 1

R 2

f 3

f 4

Fig. 5.7 A four-atom molecule in a coordinate system. Only one of possibly many electrons is
shown. The basis functionsfare one-electron functions, usually centered on atomic nuclei.R 1 ,
R 2 , etc., are vectors representing thex, y, zcoordinates (conveniently as 3'1 column matrices;
Section 4.3.3) of the nuclei (“of the atoms”), andris a vector representing thex, y, zcoordinates of
an electron. The distances of the electron from the centers of the various basis functions are the
absolute values of the various vector differences: |r#R 1 |, |r#R 2 |, etc. For a particular molecular
geometry,R 1 ,R 2 , etc. are fixed and enter the functionsf 1 ,f 2 , etc., only parametrically, i.e. to
denote where thef’s are centered;ris the variable in these functions, which are thusf(x, y, z).
Several basis functions may be centered on each nucleus

5.2 The Basic Principles of the ab initio Method 215