Computational Chemistry

from the relative positions (throughSij) of the orbitals or basis functions and
the ionization energies of these orbitals (Section 4.4.1); in neither case isHij
calculated from first principles. Section 4.3.4, Eqs. 4.44 imply thatHijis:

Hij¼

Z

fiH^fjdv $ð 5 : 3 Þ

In ab initio calculationsHijis calculated from Eq.5.3by actually performing the
integration using explicit mathematical expressions for the basis functionsfiandfj
and the Hamiltonian operatorH^; of course the integration is done by a computer
following a detailed algorithm. How this algorithm works will now be outlined.

5.2.2 The Hartree SCF Method........................................

The simplest kind of ab initio calculation is a Hartree–Fock (HF) calculation.
Modern molecular HF calculations grew out of calculations first performed on
atoms by Hartree^1 in 1928 [ 3 ]. The problem that Hartree addressed arises from
the fact that for any atom (or molecule) with more thanoneelectron an exact
analytic solution of the Schr€odinger equation (Section 4.3.2) is not possible,
because of the electron–electron repulsion term(s). Thus for the helium atom the
Schr€odinger equation (cf. Section 4.3.4, Eqs. 4.36 and 4.37) is, in SI units

h^2

8 p^2 m

ðr^21 þr^22 Þ#

Ze^2

4 pe 0 r 1

Ze^2

4 pe 0 r 2

þ

e^2

4 pe 0 r 12



C¼EC ð 5 : 4 Þ

Heremis the mass (kg) of the electron,eis the charge (coulombs, positive) of
the proton (¼minus the charge on the electron), the variablesr 1 ,r 2 , andr 12 are the
distances (m) of electrons 1 and 2 from the nucleus, and from each other,Z¼2 is
the number of protons in the nucleus, ande 0 is something called the permitivity of
empty space; the factor 4pe 0 is needed to make SI units consistent. The force (N)
between chargesq 1 andq 2 separated byrisq 1 q 2 /4pe 0 r^2 , so the potential energy (J)
of the system isq 1 q 2 /4pe 0 r(energy is force'distance).
Hamiltonians can be written much more simply by usingatomic units. Let’s take
Planck’s constant, the electron mass, the proton charge, and the permitivity of space
as the building blocks of a system of units in whichh/2p,m,e, and 4pe 0 are
numerically equal to 1 (i.e.h¼ 2 p,m¼1,e¼1, ande 0 ¼1/4p; the numerical
values of physical constants are always dependent on our system of units). These

(^1) Douglas Hartree, born Cambridge, England, 1897. Ph.D. Cambridge, 1926. Professor applied
mathematics, theoretical physics, Manchester, Cambridge. Died Cambridge, 1958.
5.2 The Basic Principles of the ab initio Method 177