Computational Chemistry

5.2.3.6.5 Using the Roothaan–Hall Equations to do ab initio Calculations – an
Example

The application of the Hartree–Fock method to an actual calculation will now be
illustrated in detail with protonated helium, H–He+, the simplest closed-shell
heteronuclear molecule. This species was also used to illustrate the details of the
extended H€uckel method (EHM) in Section 4.4.2. In this simple example all the
steps were done with a pocket calculator, except for the evaluation of the integrals
(this was done with the ab initio program Gaussian 92 [ 29 ]) and the matrix
multiplication and diagonalization steps (done with the program Mathcad [ 30 ]).
Step 1– Specifying the geometry, basis set and MO occupancy
We start by specifying a geometry and a basis set. We will use the same
geometry as with the EHM, 0.800 A ̊, i.e. 1.5117 a.u. (bohr). In ab initio calculations
on molecules, the basis functions are almost alwaysGaussianfunctions (basis
functions are discussed inSection 5.3). Gaussian functions differ from the Slater
functions we used in the EHM inChapter 4in that the exponent involves thesquare
of the distance of the electron from the point (usually an atomic nucleus) on which
the function is centered:
An s-type Slater function

f¼aexpð#brÞð 5 : 95 Þ

Ans-type Gaussian function

f¼aexpð#br^2 Þð 5 : 96 Þ

In ab initio calculations the mathematically more tractable Gaussians are used to
approximate the physically more realistic Slater functions (seeSection 5.3). We use
here the simplest possible Gaussian basis set: a 1satomic orbital on each of the two
atoms, each 1sorbital being approximated by one Gaussian function. This is called
an STO-1G basis set, meaning Slater-type orbitals-one Gaussian, because we are
approximating a Slater-type 1sorbital with a Gaussian function. The best STO-1G
approximations to the hydrogen and helium 1sorbitalsin a molecular environment
[ 31 ] are

fðHÞ¼f 1 ¼ 0 :3696 expð# 0 : 4166 jr#R 1 j^2 Þð 5 : 97 Þ

fðHeÞ¼f 2 ¼ 0 :5881 expð# 0 : 7739 jr#R 2 j^2 Þð 5 : 98 Þ

where |r#Ri| is the distance of the electron infi(fis a one-electron function)
from nucleusion whichfiis centered (Fig.5.7). The larger constant in the helium
exponent as compared to that of hydrogen (0.7739 vs 0.4166) reflects the intuitively
reasonable fact that since an electron inf 2 is bound more tightly to its doubly-
charged nucleus than is an electron inf 1 is to its singly-charged nucleus, electron

214 5 Ab initio Calculations