Scope of the Method

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

$$\hat{H} \Psi({\bf r},{\bf R}) = \left\{ \sum_A \frac{1}{... ...j}} \right\} \Psi({\bf r},{\bf R}) = E \Psi({\bf r},{\bf R})$$ (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

$$\hat{H}_e \Psi_e({\bf r};{\bf R}) = \left\{ -\frac{1}{2} \... ...\Psi_e({\bf r};{\bf R}) = E_e({\bf R}) \Psi_e({\bf r};{\bf R})$$ (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 E_e 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 E_e 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).