In the first paper on quantum mechanics, Heisenberg used matrix mechanics
to calculate the frequencies and intensities of spectral lines
[2]. Later, when Schrödinger discovered wave mechanics,
it was quickly shown that the Schrödinger and Heisenberg approaches are
mathematically equivalent [3,4].
Given the ease with which
matrices may be implemented on a computer, it is entirely natural to
attempt to solve the molecular time-independent Schrödinger equation
using matrix mechanics.
Matrix mechanics requires that we choose a vector space for the expansion
of the problem. For the case of an N-electron molecule, our wavefunction
must be expanded in a basis of N-particle functions (the nuclei need not be
considered in the electronic wavefunction, if we have invoked the
Born-Oppenheimer approximation). How do we construct the N-particle basis
functions? Here we follow the arguments of Szabo and Ostlund
[1],
p. 60. Assume we have a complete set of functions 
of a
single variable 
. Then any arbitrary function of that variable can be
expanded exactly as
How can we expand a function of two variables 
and 
which have the same domain?
If we hold 
fixed, then
Now note that each expansion coefficient 
is a function of a
single variable, which can be expanded as
Substituting this expression into the one for 
, we now have
a process which can obviously be extended for 
.
Let us now collect the spin and space coordinates of an electron into a
variable 
. We can write a spin orbital as 
. The
result analogous to equation ( 2.4) for a system of N
electrons is
However, the wavefunction must be antisymmetric
with respect to the
exchange of the coordinates of any two electrons
For the two-particle case, the requirement
implies that 
and 
, or
More generally, an arbitrary N-electron wavefunction
can be expressed
exactly as a linear combination of all possible N-electron Slater
determinants formed from a complete set of spin orbitals 
.
If we solve the matrix mechanics problem 
in a complete basis of N-electron functions as just described,
we will obtain all electronic eigenstates of the system exactly.
If our N-electron basis functions are denoted 
, the
eigenvectors of 
are given as
if there are I possible N-electron basis functions (I will be
infinite if we actually have a complete set of one electron
functions 
).
The matrix 
is constructed so that
for 
.
The matrix elements 
may be written in terms of one- and
two-electron integrals according to
``Slater's rules,''
as discussed in section 2.4.
The N-electron basis functions 
can be written as
substitutions or ``excitations'' from the Hartree-Fock
``reference'' determinant, i.e.\
where 
means the Slater determinant formed by replacing
spin-orbital a in 
with spin orbital r, etc. Every
N-electron Slater determinant can be described by the set of N spin
orbitals from which it is formed, and this set of orbital occupancies is
often referred to as a ``configuration.'' Thus the ``configuration
interaction'' method is, in its most straigtforward implementation, nothing
more or less than the matrix mechanics solution of the time-independent
non-relativistic electronic Schrödinger equation 
. One of the great strengths of the CI method is its generality; the
formalism applies to excited states, to open-shell systems, and to systems
far from their equilibrium geometries. By contrast,
traditional single-reference perturbation theory
and coupled-cluster
approaches generally assume that the reference
configuration is dominant, and they may fail when it is not.
In practice, one does not have a complete set of one-particle basis
functions 
; typically one assumes that the incomplete
one-electron basis set is large enough to give useful results, and the CI
procedure is not modified. The quality of the one-particle basis set can
be checked by comparing the results of
calculations using progressively larger basis sets.
It is also possible to reduce the size of the N-electron basis set. If
we desire only wavefunctions of a given spin and/or spatial symmetry, as is
usually the case, we need include only those N-electron basis functions
of that symmetry, since the Hamiltonian matrix is block-diagonal according
to space and spin symmetries. This point is discussed further in section
4.1.
If one performs the matrix mechanics calculation using a given set of
one-particle functions 
and all possible N-electron
basis functions 
(possibly symmetry-restricted), the
procedure is called ``full CI.''
The full CI corresponds to solving
Schrödinger's equation exactly within the space spanned by the
specified one-electron basis. If the one-electron basis is complete
(it never is in practice, but it may be in theory), then the procedure
is called a ``complete CI'' [5].
Unfortunately, even with an incomplete one-electron basis, a full CI is computationally intractable for any but the smallest systems, due to the vast number of N-electron basis functions required (the size of the CI space is discussed in section 4.4). The CI space must be reduced somehow---hopefully in such a way that the approximate CI wavefunction and energy are as close as possible to the exact values. The effective reduction of the CI space is a major concern in CI theory, and we will discuss some of the more popular approaches in these notes.
By far the most common CI approximation is the truncation of the CI space expansion according to excitation level relative to the reference state (equation 2.9). The widely-employed CI singles and doubles (CISD) wavefunction includes only those N-electron basis functions which represent single or double excitations relative to the reference state. Since the Hamiltonian operator includes only one- and two-electron terms, only singly and doubly excited configurations can interact directly with the reference, and they typically account for about 95% of the correlation energy in small molecules at their equilibrium geometries [6]. Truncation of the CI space according to excitation class is discussed more thoroughly in section 4.2.
