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 $\hat{H} \Psi = E \Psi$. 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.

$n_{\alpha}$

The number of alpha electrons.

$n_{\beta}$

The number of beta electrons.

n

The number of orbitals in the one-particle basis set.

$\delta_{ij}$

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

$\hat{H}$

The exact nonrelativistic Hamiltonian operator.

${\bf H}$

The Hamiltonian matrix, i.e. the matrix form of $\hat{H}$, in whatever N-electron basis is currently being used.

H_ij

The i,j-th element of ${\bf H}$, equal to $\langle \Phi_i \vert \hat{H} \vert \Phi_j \rangle$, where $\Phi_i$ and $\Phi_j$ are N-electron CI basis functions.

${\bf x}_i$

The space and spin coordinates of particle i.

${\bf r}_i$

The spatial coordinates of particle i.

$\phi_i$

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.

$\chi_i$

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

$\vert \Phi_i \rangle$

The i-th N-electron basis function. Usually denotes a single Slater determinant, but may also be a configuration state function (CSF).

$\vert \Psi \rangle$

Usually denotes an eigenfunction of ${\bf H}$. The exact nonrelativistic wavefunction if a complete basis is used in the expansion of $\hat{H}$.

$\vert \Phi_a^r \rangle$

An N-electron basis function which differs from some reference function $\vert \Phi_0 \rangle$ by the replacement of spin-orbital a by spin-orbital r. Usually implies a single Slater determinant.

$\vert ab \ldots c \rangle$

A Slater determinant with spin-orbitals a, b, $\ldots c$ occupied, i.e.

$$\nonumber \vert\phi_a \phi_b \ldots \phi_c \rangle = \frac{1}... ...\bf x}_N) & \ldots & \phi_c({\bf x}_N) \end{array} \right\vert$$

$\langle i \vert \hat{h} \vert j \rangle$

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

$$\langle i \vert \hat{h} \vert j \rangle = \int \phi_i^{*}({\bf x}_1) \hat{h}({\bf x}_1) \phi_j({\bf x}_1) d{\bf x}_1$$ (tex2html_deferredtex2html_deferred1.tex2html_deferred1i | h | j = _i^*(x_1) h(x_1) _j(x_1) dx_1 )

$[i\vert\hat{h}\vert j]$ One-electron integral in chemists' notation, where i and j are spin-orbitals. Equivalent to $\langle i \vert \hat{h} \vert j \rangle$.

$(i\vert \hat{h} \vert j)$

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

$\langle ij \vert \vert kl \rangle$

Antisymmetrized two-electron integral, equal to $\langle ij \vert kl \rangle - \langle ij \vert lk \rangle$.

$\langle ij \vert kl \rangle$

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

$$\langle ij \vert kl \rangle = \int \phi_i^{*}({\bf x}_1) \phi... ...2}} \phi_k({\bf x}_1) \phi_l({\bf x}_2) d{\bf x}_1 d{\bf x}_2$$ (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

$$[ij\vert kl]= \int \phi_i^{*}({\bf x}_1) \phi_j({\bf x}_1) ... ...\phi_k^{*}({\bf x}_2) \phi_l({\bf x}_2) d{\bf x}_1 d{\bf x}_2$$ (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

$$(ij\vert kl) = \int \phi_i^{*}({\bf r}_1) \phi_j({\bf r}_1) ... ...\phi_k^{*}({\bf r}_2) \phi_l({\bf r}_2) d{\bf r}_1 d{\bf r}_2$$ (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 )

$a_{i}^{\dagger}$

Second-quantized creation operator for orbital i.

a_i

Second-quantized annihilation operator for orbital i.