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 *x*_{1}. Then any arbitrary function of that variable can be
expanded exactly as

(tex2html_deferredtex2html_deferred2.tex2html_deferred1(x_1) = _i a_i _i(x_1). ) |

How can we expand a function of

(tex2html_deferredtex2html_deferred2.tex2html_deferred2(x_1, x_2) = _i a_i(x_2) _i(x_1). ) |

Now note that each expansion coefficient

(tex2html_deferredtex2html_deferred2.tex2html_deferred3a_i(x_2) = _j b_ij _j(x_2). ) |

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

(tex2html_deferredtex2html_deferred2.tex2html_deferred5(x_1, x_2, ..., x_N) =
_ij...N b_ij...N _i(x_1) _j(x_2) ...
_N(x_N)
) |

However, the wavefunction must be antisymmetric with respect to the exchange of the coordinates of any two electrons

(tex2html_deferredtex2html_deferred2.tex2html_deferred6(x_1, x_2) = -(x_2, x_1)
) |

implies that

= | |||

tex2html_{d}eferredtex2html_{d}eferred2.tex2html_{d}eferred7 |
= | (tex2html_deferredtex2html_deferred2.tex2html_deferred7 & = & _j > i 2^1/2 b_ij |_i _j ) |

More generally, an arbitrary

(tex2html_deferredtex2html_deferred2.tex2html_deferred8| _j = _i^I c_ij | _i ) |

if there are

The *N*-electron basis functions
can be written as
substitutions or ``excitations'' from the Hartree-Fock
``reference'' determinant, i.e.

(tex2html_deferredtex2html_deferred2.tex2html_deferred9| = c_0 | _0 + _ra c^r_a | ^r_a + _a<b,r<s c^rs_ab | ^rs_ab + _r<s<t,a<b<c c^rst_abc | ^rst_abc + ... ) |

where means the Slater determinant formed by replacing spin-orbital

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.