Computational Chemistry

(h/2p,m,e, and 4pe 0 ) are the units of angular momentum, mass, charge, and
permitivity in the system of atomic units. In this system Eq.5.4becomes

1

2

r^21 #

1

2

r^22 #

Z

r 1

Z

r 2

þ

1

r 12



C¼EC ð 5 : 5 Þ

Using atomic units simplifies writing quantum-mechanical expressions, and also
means that the numerical (in these units) results of calculations are independent of
the currently accepted values of physical constants in terms of kilograms, cou-
lombs, meters, and seconds (of course, when we convert from atomic to SI units we
must use accepted SI values ofm,e, etc). The atomic units of energy and length are
particularly important to us. We can get the atomic unit of a quantity by combining
h/2p,m,e, and 4pe 0 to give the expression with the required dimensions. The
atomic units of length and energy, the bohr and the hartree, turn out to be:

Length: 1 bohr¼a 0 ¼ 4 pe 0 (h/2p)^2 /me^2 ¼e 0 h^2 /pme^2 ¼0.05292 nm¼0.5292 A ̊

Energy: 1 hartree¼Eh(or h)¼e^2 /4pe 0 a 0 ; 1 h/particle¼2625.5 kJ mol#^1

The bohr is the radius of a hydrogen atom in the Bohr model (Section 4.2.5), or the
most probable distance of the electron from the nucleus in the fuzzier Schr€odinger
picture (Section 4.2.6). The hartree is the energy needed to move a stationary electron
1 bohr distant from a proton away to infinity. The energy of a hydrogen atom, relative
to infinite proton/electron separation as zero, is#½hartree: the potential energy is
#1 h and the kinetic energy (always positive) is 0.5 h. Note that atomic units derived
by starting with the old Gaussian system (cm, grams, statC) differ by a 4pe 0 factor
from the SI-derived ones.
The Hamiltonian

H^¼#^1

2

r^21 #

1

2

r^22 #

Z

r 1

Z

r 2

þ

1

r 12

$ð 5 : 6 Þ

consists of five terms, signifying (Fig.5.1) from left to right: the kinetic energy of
electron 1, the kinetic energy of electron 2, the potential energy of the attraction
of the nucleus (charge Z¼2) for electron 1, the potential energy of the attraction of
the nucleus for electron 2, and the potential energy of the repulsion between
electron 1 and electron 2. Actually this is not theexactHamiltonian, for it neglects
effects due to relativity and to magnetic interactions such as spin–orbit coupling
[ 4 ]; these effects are rarely important in calculations involving lighter atoms,
say those in the first two or three full rows of the periodic table (up to about
chlorine or bromine). Relativistic quantum chemical calculations will be briefly
discussed later. The wavefunctioncis the “total”, overall wavefunction of the atom
and can be approximated, as we will see later for molecular HF calculations, as a
combination of wavefunctions for various energy levels. The problem with solving
Eq.5.5exactly arises from the 1/r 12 term. This makes it impossible to separate the

178 5 Ab initio Calculations