A.2 Calculating Electron Repulsion Integrals (ERIs)A.2.2 Data Structures: Shell Pairs and Quartets A.2.4 Efficiency of ERI Evaluation

A.2.3 Survey of ERI Evaluation

(September 1, 2024)

Various aspects of the ERI evaluation problem are described in the following subsections.

A.2.3.1 Fundamental ERI

The fundamental ERI $[ss|ss]^{(0)}\equiv[\bm{0}]^{(0)}$ , which is the basis of all ERI algorithms, is usually represented as ⁴²⁰ Gill P. M. W.
Adv. Quantum Chem.
(1994), 25, pp. 141. Link

[\bm{0}]^{(0)}=D_{A}D_{B}D_{C}D_{D}\int e^{-\alpha|\mathbf{r}_{1}-\mathbf{A}|^% {2}}e^{-\beta|\mathbf{r}_{1}-\mathbf{B}|^{2}}\left(\frac{1}{r_{12}}\right)e^{-% \gamma|\mathbf{r}_{2}-\mathbf{C}|^{2}}e^{-\delta|\mathbf{r}_{2}-\mathbf{D}|^{2% }}\;d\mathbf{r}_{1}d\mathbf{r}_{2}

(A.3)

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

[\bm{0}]^{(0)}=U(2\,{\vartheta}^{2})^{1/2}\left({\frac{2}{\pi}}\right)^{1/2}\,% \int_{0}^{1}e^{-Tu^{2}}du

(A.4)

and can be efficiently computed using a modified Chebyshev interpolation scheme. ⁴¹⁵ Gill P. M. W., Johnson B. G., Pople J. A.
Int. J. Quantum Chem.
(1991), 40, pp. 745. Link Equation (A.4) can also be adapted for the general case $[\mathbf{0}]^{(m)}$ 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.

A.2.3.2 Angular Momentum Problem

The fundamental integral is essentially an integral without angular momentum (i.e., it is an integral of the type $[ss|ss]$ ). Angular momentum, usually depicted by $L$ , has been problematic for efficient ERI formation, evident in the above time line. Initially, angular momentum was calculated by taking derivatives of the fundamental ERI with respect to one of the Cartesian coordinates of the nuclear center. This is an extremely inefficient route, but it works and was appropriate in the early development of ERI methods. Recursion relations ⁹³⁷ Obara S., Saika A.
J. Chem. Phys.
(1986), 84, pp. 3963. Link ^, ⁹³⁸ Obara S., Saika A.
J. Chem. Phys.
(1988), 89, pp. 1540. Link and the tensor equations ²² Adams T. R., Adamson R. D., Gill P. M. W.
J. Chem. Phys.
(1997), 107, pp. 124. Link are the basis for the modern approaches.

A.2.3.3 Contraction Problem

The contraction problem may be described by considering a general contracted ERI of $s$ -type functions derived from the STO-3G basis set. Each basis function has degree of contraction $K=3$ . Thus, the ERI may be written

\displaystyle\begin{aligned} \displaystyle(ss|ss)&\displaystyle=\sum_{i=1}^{3}% \sum_{j=1}^{3}\sum_{k=1}^{3}\sum_{\ell=1}^{3}D_{Ai}D_{Bj}D_{Ck}D_{D\ell}\\ &\displaystyle\qquad\times\int e^{-\alpha_{i}|\mathbf{r}_{1}-\mathbf{A}|^{2}}e% ^{-\beta_{j}|\mathbf{r}_{1}-\mathbf{B}|^{2}}\left(\frac{1}{r_{12}}\right)e^{-% \gamma_{k}|\mathbf{r}_{2}-\mathbf{C}|^{2}}e^{-\delta_{\ell}|\mathbf{r}_{2}-% \mathbf{D}|^{2}}d\mathbf{r}_{1}d\mathbf{r}_{2}\\ &\displaystyle=\sum_{i=1}^{3}\sum_{j=1}^{3}\sum_{k=1}^{3}\sum_{\ell=1}^{3}[s_{% i}s_{j}|s_{k}s_{\ell}]\end{aligned}

(A.5)

and requires 81 primitive integrals for the single ERI. The problem escalates dramatically for more highly contracted sets (STO-6G, 6-311G) and has been the motivation for the development of techniques for shell-pair modeling, in which a second shell-pair is constructed with fewer primitives that the first, but introduces no extra error relative to the integral threshold sought.

The Pople-Hehre axis-switch method ¹⁰³⁹ Pople J. A., Hehre W. J.
J. Comput. Phys.
(1978), 27, pp. 161. Link is excellent for high contraction low angular momentum integral classes.

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A.2.3 Survey of ERI Evaluation

A.2.3.1 Fundamental ERI

A.2.3.2 Angular Momentum Problem

A.2.3.3 Contraction Problem