next up previous contents index
Next: Why Configuration Interaction? Up: Fundamental Concepts Previous: Fundamental Concepts

Scope of the Method

Configuration interaction (CI) is a method for solving the nonrelativistic Schrödinger equation

 \begin{displaymath}
\hat{H} \Psi({\bf r},{\bf R}) =
\left\{
\sum_A \frac{1}{...
...j}} \right\} \Psi({\bf r},{\bf R})
= E \Psi({\bf r},{\bf R})
\end{displaymath} (tex2html_deferredtex2html_deferred2.tex2html_deferred5  H (r,R) = { _A 12 M_A ^2_A + _i 12 ^2_i + _A>B Z_A Z_BR_AB + _Ai Z_Ar_Ai + _i>j 1r_ij } (r,R) = E (r,R) )

where i,j denote electrons and A,B denote nuclei, with $r_{ij} = \vert{\bf r}_i - {\bf r}_j \vert$, $R_{Ai} = \vert{\bf R}_A - {\bf
r}_i\vert$, and $R_{AB} = \vert{\bf R}_A - {\bf R}_B \vert$. Typical applications of the CI method employ the Born-Oppenheimer approximation, whereby the the motions of the electrons are treated as uncoupled from those of the nuclei. Thus the ``electronic'' Shrödinger equation is solved at discrete sets of fixed nuclear positions

 \begin{displaymath}
\hat{H}_e \Psi_e({\bf r};{\bf R})
= \left\{ -\frac{1}{2} \...
...\Psi_e({\bf r};{\bf R}) = E_e({\bf R}) \Psi_e({\bf r};{\bf R})
\end{displaymath} (tex2html_deferredtex2html_deferred2.tex2html_deferred6 H_e _e(r;R) = { -12 _i ^2_i - _A,i Z_AR_Ai + _i>j 1r_ij } _e(r;R) = E_e(R) _e(r;R) )

The Born-Oppenheimer approximation is invoked so often in computational quantum chemistry that the subscripts in the preceeding equation are usually suppressed and the equation is written simply as $\hat{H} \Psi = E \Psi$. However, it is important to remember that the electronic energy Ee is an artifact of the Born-Oppenheimer approximation and is not as physically meaningful as the total energy of a system. Within the Born-Oppenheimer approximation, we estimate the total energy by adding the nuclear-nuclear repulsion energy and the nuclear kinetic energy to the total electronic energy Ee of equation (2.7).

While the CI method can be extended to incorporate some relativistic effects (e.g. spin-orbit terms), this is not generally done; these notes will be concerned only with the nonrelativistic Hamiltonian (2.7).


next up previous contents index
Next: Why Configuration Interaction? Up: Fundamental Concepts Previous: Fundamental Concepts
C. David Sherrill
2000-04-18