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These notes attempt to present the essential ideas of configuration
interaction (CI) theory in a fairly detailed mathematical framework.
Of all the ab initio
methods, CI is probably the easiest to
understandand perhaps one of the hardest to implement efficiently
on a computer! The next two sections explain what the CI method is: the
matrix formulation of the Schrödinger equation
.
The remaining sections describe various simplifications, approximations,
and computational techniquies.
I have attempted to use a uniform notation throughout these notes.
Much of the notation is consistent with that of Szabo and Ostlund,
Modern Quantum Chemistry [1].
Below are listed several of the commonlyused symbols and their meanings.
 N
 The number of electrons in the system.

 The number of alpha electrons.

 The number of beta electrons.
 n
 The number of orbitals in the oneparticle basis set.

 Kronecker delta function, equal to one if i=j and
zero otherwise.

 The exact nonrelativistic Hamiltonian operator.

 The Hamiltonian matrix, i.e. the matrix form of
,
in whatever Nelectron basis is currently being used.
 H_{ij}
 The i,jth element of ,
equal to
,
where
and
are Nelectron CI basis functions.

 The space and spin coordinates of particle i.

 The spatial coordinates of particle i.

 The ith oneparticle basis function (orbital).
Usually denotes a spinorbital obtained from a HartreeFock
procedure. May also be written simply as i.

 The ith oneparticle basis function (orbital).
Usually denotes an atomic spinorbital.

 The ith Nelectron basis function. Usually
denotes a single Slater determinant, but may also be a configuration
state function (CSF).

 Usually denotes an eigenfunction of .
The
exact nonrelativistic wavefunction if a complete basis is used in the
expansion of .

 An Nelectron basis function which differs from
some reference function
by the replacement of spinorbital
a by spinorbital r. Usually implies a single Slater determinant.

 A Slater determinant with spinorbitals a,
b,
occupied, i.e.

 Oneelectron integral in physicists'
notation (i and j are spinorbitals). More explicitly, this is

(tex2html_deferredtex2html_deferred1.tex2html_deferred1i  h  j = _i^*(x_1) h(x_1)
_j(x_1) dx_1
) 

Oneelectron integral in chemists' notation,
where i and j are spinorbitals. Equivalent to
.

 Oneelectron integral in chemists' notation
(i and j are spatial orbitals).

 Antisymmetrized twoelectron integral,
equal to
.

 A simple twoelectron integral, in
physicists' notation, where i, j, k, and l are spinorbitals.
This is

(tex2html_deferredtex2html_deferred1.tex2html_deferred2ij  kl =
_i^*(x_1) _j^*(x_2)
1r_12 _k(x_1) _l(x_2) dx_1
dx_2
) 
 [ ijkl ]
A simple twoelectron integral in chemists' notation,
where i, j, k, and l are spinorbitals. This is

(tex2html_deferredtex2html_deferred1.tex2html_deferred3[ijkl] =
_i^*(x_1) _j(x_1)
1r_12 _k^*(x_2) _l(x_2) dx_1
dx_2
) 
 (ijkl)
 A simple twoelectron in chemists' notation where i,
j, k, and l are spatial orbitals. This is

(tex2html_deferredtex2html_deferred1.tex2html_deferred4(ijkl) =
_i^*(r_1) _j(r_1)
1r_12 _k^*(r_2) _l(r_2) dr_1
dr_2
) 

 Secondquantized creation operator for orbital i.
 a_{i}
 Secondquantized annihilation operator for orbital i.
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C. David Sherrill
20000418