MODULE 12
The Computation of Molecular Structure and Electronic Configurations
Before reading this Module you might find it useful to refer back to Module 9 where we
introduced the Hartree-Fock procedure for calculating the energies of many-electron atoms.
The ultimate aim of quantum chemistry is the solution of the non-relativistic, time-independent
Schrödinger equation for a molecular system, and to calculate the electronic structure of
polyatomic entities. All computations necessarily involve heavy usage of powerful computers.
With molecular systems everything is done within the confines of the Born-Oppenheimer
approximation wherein the electronic Schrödinger equation is solved at a fixed value of R (internuclear distance), and a succession of eigenvalues are obtained as a function of variations in the
parameter R. The electronic Schrödinger equations for the ith electron can be written as
Hˆ  j (ri ; R)  E j j (ri ; R)
(12.1)
where the j’s refer to the set of orbitals that are eigenfunctions of the hamiltonian. Solutions of
this equation are found for a set of values of R (nuclear positions). The hamiltonian is:
n N
1 n
1 1 n 1
Hˆ    i2    
2 i
2 ij rij
i
I rIi
(12.2)
where n and N are the numbers of electrons (index i) and nuclei (index I), respectively; rIi is the
set of electron-nucleon distances, and rij is the set of electron-electron distances.
We are interested here in ab initio calculations where a trial function is chosen for the
wavefunction (r;R), and the Schrödinger equation is solved using only fundamental constants
and atomic numbers. It is a serious challenge to compute E(R) to within 1 kJ mol-1 of the
experimental value.
The Hartree-Fock Self Consistent Field Method
The Formulation of the Approach
The complicating feature of all calculations is the electron-electron repulsion term (1/rij) in the
hamiltonian. We saw in Module 9 that the effect of electron correlation is difficult to estimate.
We start by supposing that the true electronic wavefunction that we seek () is similar in form to
an idealized wavefunction without electron-electron repulsion (

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The Schrödinger equation for the idealized wavefunction is given by
Ĥ 0 0  E 0 0
(12.3)
where
n
Hˆ 0   hˆi
(12.4)
i 1
The hˆi terms are core hamiltonians for the ith electrons (total n).
Equation (12.4) is an n-
electron equation that is separable into n-one electron equations, and 0 is the product of one
electron wavefunctions (orbitals), each of which is a solution to the Schrödinger equation
hˆi a0 (i)  Ea0 a0 (i)
(12.5)
where a denotes orbital index and i denotes electron index. The eigenvalue Ea0 is the energy of
an electron in orbital a. This is an independent electron model. This leads to the relationship
 0   a0 (1). b0 (2)... z0 ( n)
(12.6)
The wavefunctions depend on all electron spatial coordinates, but the nuclear locations are taken
as a parameter within the Born-Oppenheimer approximation.
So far we have confined our attention to spatial wavefunctions, without considering spin
explicitly. This can be taken into consideration by redefining equation (12.6) in terms of
spinorbitals. Remember that a spinorbital is the product of spatial and spin functions and we will
write it here as a  i ; R  where i represents the joint space and spin coordinates of electron i.
Finally we write out our orbital as a Slater determinant
 0 (  ; R)  (n!)1/ 2 det a (1)b (2)...z (n)
(12.7)
where the principal diagonal only has been written. The u functions, with u = a,b,c,...,z are
spinorbitals and are taken to be orthonormal. In equation (12.7) we have arrived at our idealized
wavefunction, which is to be our trial function and the starting point of a variational calculation.
We need to remember, however, that it is idealized and electron-electron interactions have been
ignored (independent electron approximation).
The Hartree-Fock approach will go part way to accounting for the 1/rij potentials. H-F treats
electron repulsions in an average way. Each electron is taken to be moving in the electrostatic
field of the nuclei and in the time-averaged field of the other electrons.
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The Hartree-Fock equation
The Hartree-Fock equation is defined as
Fˆ1a 1   aa 1
(12.8)
for electron 1 in orbital a of energy a, and F̂ is the so-called Fock operator. This operator is
defined as follows:
Fˆ1  hˆ1    J u (1)  K u (1)
(12.9)
u
where h1 is the core hamiltonian for electron 1 and the sum is taken over all the u (a through z)
spinorbitals. The J and K terms are the Coulomb and Exchange operators respectively. These
are defined (in au) by the equations


 1
Jˆu (1)a (1)    u* (2)   u (2) d  2 a (1)


 r12 


 1
Kˆ u (1)a (1)    u* (2)   a (2) d  2 a (1)
 r12 


(12.10)
the d elements refer to combined space and spin coordinates. The Coulomb operator accounts
for electrostatic repulsion between electron 1 and one of the other electrons (#2) in equation
(12.10)).
The exchange operator represents the way the Coulomb potential is modified
according to electron correlation. Thus the sum in equation (12.9) represents the average energy
of electron 1 due to the presence of n-1 other electrons. Since Jˆa (1)a (1)  Kˆ a (1)a (1) the sum in
equation (12.9) includes contributions from all spinorbitals except the one being computed.
[Note that in Lowe 11-5 the J operator is taken twice, but only once here in equation (12.9). The
Lowe treatment is for spatial orbitals each being doubly occupied whereas here we are dealing
with spinorbitals that, by definition, carry only a single electron.]
To arrive at the expression for the spinorbitals we solve equations like (12.8), after first
evaluating the Fock operator, which depends on the spinorbitals of the n-1 other electrons, and,
as before, it seems we must know the solutions before we can find the solutions. The selfconsistent field method detailed earlier allows this to be performed. Thus the n occupied
spinorbitals in the determinental wavefunction (12.7) all contribute to the Fock operator.
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Once arrived at, the Fock Operator is like any other Hermitian operator inasmuch as it has an
infinite number of eigenfunctions, i.e. there is an infinite number of spinorbitals u having energy
u. An infinite number is an impracticable number so it is usual to find m solutions where m>n,
the number of electrons.
The m optimized HF-SCF MOs are arranged in order of increasing
energy; the n lowest are termed “occupied” and the m-n remaining are termed “virtual” orbitals.
Some Definitions
Restricted Hartree-Fock
The wavefunction is a single determinental function where  and  spin electrons have identical
spatial wavefunctions.
Unrestricted Hartree-Fock
Different spatial orbitals are allowed for the  and  spins. This is useful in free radicals where
n > nfor example.
The Roothaan-Hall equations
The HF procedure outlined above is straightforward to use in situations where the Coulomb field
is spherically symmetrical, viz., atoms. However it is not, in this form computationally feasible
for molecules that are not spherically symmetrical. Hall and Roothaan (1951) independently
devised a way out of this. They suggested using a known set of basis functions to expand the
spatial parts of the spinorbitals. This transformed the coupled HF equations into a resolvable
matrix problem.
We express each spatial wavefunction in the Slater determinant as a linear combination of a set
of M basis functions, j.
M
 i   c ji j
(12.11)
j 1
where cji are unknown coefficients (so-called MO coefficients). Substituting equation (12.11)
into the Fock equation and performing some manipulations leads to
M
M
j 1
j 1
 Fij c ja   a  Sij c ja
where
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(12.12)
Fij    i* (1) Fˆ1 i (1)dr1
Sij    i* (1) i (1)dr
Fij and S ij are elements of the Fock matrix and the overlap matrix, respectively. Equation
(12.12) is one of a set of M simultaneous equations (one for each value of i) and known as the
Roothaan-Hall equations. The entire set can be written as a single matrix equation
Fc =Sc
(12.13)
c is a M x M matrix composed of elements cja and  is an M x M diagonal matrix of the orbital
energies, a. The Roothaan-Hall equations have a non-trivial solution if, and only if, the secular
equation is satisfied, i.e.
det F-ε aS  0
The matrix elements Fij involve Coulomb and Exchange integrals which themselves depend on
the spatial wavefunctions. Therefore a direct solution evades us and we need to employ the SCF
method. The end result is a set of energies a, and coefficients, cja.
Basis Sets
At this point it is necessary to consider the basis set to use for evaluation of i. In other words,
what shall be the nature of j and how many shall be taken? The two that dominate today are
composed of either STO’s or Gaussian type orbitals. Here you are referred to Lowe, 11-8, for a
discussion of the pros and cons and for becoming familiar with the current jargon of quantum
calculations.
Table 12.1 presents some results for small molecules using HF calculations with different basis
sets.
Koopman's Theorem
Although the sum of all the one-electron energies calculated by the HF-SCF method is not yet
equal to the total electronic energy of our molecule, it is possible for relationships to physical
properties to be made. For example if we wish to know the energy of a molecule after one
electron has been removed (the radical cation), we use Koopman’s theorem. This states that the
ionization energy of an electron in an orbital (r) is the negative of the calculated one-electron
energy of that orbital (r)
I r   r
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This identity assumes that the distribution of the remaining electrons does not relax after the
removal (see the discussion in Lowe, 11-11).
Electron Correlation
At this point we have arrived at a method for evaluating our idealized function through the HF
method, which attempts to account for inter-electron repulsion through the J and K operators.
However HF relies on averages and does not consider instantaneous electrostatic interactions.
Electrons in motion tend to avoid each other. We summarize the situation by saying that HF
ignores electron correlation.
TABLE 12.1
Equilibrium bond lengths (as multiples of a0) for some polyatomic molecules, comparing
experimental values with those calculated employing two types of basis sets
compound
STO-3G
6-31G**
observed
H2
1.346
1.385
1.401
CH4
2.047
2.048
2.050
NH3
1.952
1.897
1.912
H2O
1.871
1.782
1.809
Configuration State Functions and Configuration Interaction
The HF method generates a finite set of spinorbitals when a finite basis set expansion is used (in
equation (12.11), for example). A basis set with M spatial members provides 2M different
spinorbitals. With n electrons occupying the n lowest of the set, we have generated our HF
ground state, . These n orbitals define a particular Slater determinant. However 2M-n virtual
spinorbitals remain, and many Slater determinants can be formed from the 2M total; is just
one of them. We can readily generate other determinental wavefunctions by promoting one, or
more electrons from the occupied set to the virtual orbitals. A Configuration State Function
(CSF) is a linear combination of determinental wavefunctions having the correct electronic
symmetry. A CSF containing 0 and some appropriate singly (or multiply) excited determinants
can be employed to improve the representation of the ground state function. An ab initio
calculation in which the wavefunction is expressed as a linear combination of determinants is a
Configuration Interaction (CI) calculation. CI has emerged as one of the ways of including the
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effects of electron correlation. Sections 11-14 and 11-15 in Lowe reveal many details of ab
initio calculations and some useful approximations to the SCF methods. 
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