Q-Chem 5.0 User’s Manual

B.6 Fundamental ERI

The fundamental ERI [912] and the basis of all ERI algorithms is usually represented

  $\displaystyle  [\ensuremath{\mathbf{0}}]^{(0)}  $ $\displaystyle  \! \! \! =\! \! \!   $ $\displaystyle  [ss\vert ss]^{(0)} \nonumber  $    
  $\displaystyle  $ $\displaystyle  \! \! \! =\! \! \!   $ $\displaystyle  D_ A D_ B D_ C D_ D \int {e^{-\alpha \left| {{\rm {\bf r}}_{\rm {\bf 1}} -{\rm {\bf A}}} \right|^2}e^{-\beta \left| {{\rm {\bf r}}_{\rm {\bf 1}} -{\rm {\bf B}}} \right|^2}\left[ {\frac{1}{r_{12} }} \right]e^{-\gamma \left| {{\rm {\bf r}}_{\rm {\bf 2}} -{\rm {\bf C}}} \right|^2}e^{-\delta \left| {{\rm {\bf r}}_2 -{\rm {\bf D}}} \right|^2}d{\rm {\bf r}}_{\rm {\bf 1}} d{\rm {\bf r}}_{\rm {\bf 2}} }  $   (B.3)

which can be reduced to a one-dimensional integral of the form

  \begin{equation}  \label{eq:b4} [{\rm {\bf 0}}]^{(0)}=U(2\, {\vartheta }^2)^{1/2} \left({\frac{2}{\pi }}\right)^{1/2} \,  \int \limits _0^1 {e^{-Tu^2}du} \end{equation}   (B.4)

and can be efficiently computed using a modified Chebyshev interpolation scheme [914]. Equation (B.4) can also be adapted for the general case $[\ensuremath{\mathbf{0}}]^{(m)}$ integrals required for most calculations. Following the fundamental ERI, building up to the full bra-ket ERI (or intermediary matrix elements, see later) are the problems of angular momentum and contraction.

Note: Square brackets denote primitive integrals and parentheses denote fully-contracted integrals.