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# Introduction and Notation

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 understand--and 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 commonly-used 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 one-particle 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 N-electron basis is currently being used.

Hij
The i,j-th element of , equal to , where and are N-electron CI basis functions.

The space and spin coordinates of particle i.

The spatial coordinates of particle i.

The i-th one-particle basis function (orbital). Usually denotes a spin-orbital obtained from a Hartree-Fock procedure. May also be written simply as i.

The i-th one-particle basis function (orbital). Usually denotes an atomic spin-orbital.

The i-th N-electron 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 N-electron basis function which differs from some reference function by the replacement of spin-orbital a by spin-orbital r. Usually implies a single Slater determinant.

A Slater determinant with spin-orbitals a, b, occupied, i.e.

One-electron integral in physicists' notation (i and j are spin-orbitals). More explicitly, this is

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

One-electron integral in chemists' notation, where i and j are spin-orbitals. Equivalent to .

One-electron integral in chemists' notation (i and j are spatial orbitals).

Antisymmetrized two-electron integral, equal to .

A simple two-electron integral, in physicists' notation, where i, j, k, and l are spin-orbitals. 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 )

[ ij|kl ] A simple two-electron integral in chemists' notation, where i, j, k, and l are spin-orbitals. This is

 (tex2html_deferredtex2html_deferred1.tex2html_deferred3[ij|kl] = _i^*(x_1) _j(x_1) 1r_12 _k^*(x_2) _l(x_2) dx_1 dx_2 )

(ij|kl)
A simple two-electron in chemists' notation where i, j, k, and l are spatial orbitals. This is

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

Second-quantized creation operator for orbital i.

ai
Second-quantized annihilation operator for orbital i.

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C. David Sherrill
2000-04-18