Molecular Integrals

Molecular integrals connect a chosen orbital basis to the electronic Hamiltonian. Their mathematical definitions are simple, but normalization, index ordering, cancellation, and screening determine whether an implementation is reliable. This page uses real Cartesian Gaussian functions and atomic units; complex orbitals require explicit conjugations.

Helpful background. Basis sets introduces primitives, contractions, and angular conventions.

The four integral families

For real basis functions $\chi_\mu$, define

$$\begin{aligned} S_{\mu\nu}&=\int\chi_\mu\chi_\nu\,d^3r,\\ T_{\mu\nu}&=-\frac12\int\chi_\mu\nabla^2\chi_\nu\,d^3r,\\ V_{\mu\nu}&=-\sum_A Z_A\int \frac{\chi_\mu\chi_\nu}{|\mathbf r-\mathbf R_A|}\,d^3r. \end{aligned}$$

The one-electron matrix is $h=T+V$. Chemists’ electron-repulsion integrals are

$$(\mu\nu|\lambda\sigma)= \iint\frac{\chi_\mu(\mathbf r_1)\chi_\nu(\mathbf r_1) \chi_\lambda(\mathbf r_2)\chi_\sigma(\mathbf r_2)} {|\mathbf r_1-\mathbf r_2|}\,d^3r_1d^3r_2.$$

They describe products on each electron coordinate. Converting them to a second-quantized spin-orbital tensor requires the appropriate index permutation and spin factors. Naming both objects “two-electron integrals” does not make their memory layouts interchangeable.

Gaussian products and Boys functions

For unnormalized $s$ primitives with exponents $\alpha,\beta$, centers $\mathbf A,\mathbf B$, define $p=\alpha+\beta$, $\mu=\alpha\beta/p$, $\mathbf P=(\alpha\mathbf A+\beta\mathbf B)/p$, and $K_{AB}=e^{-\mu|\mathbf A-\mathbf B|^2}$. Then

$$S_{ab}=K_{AB}(\pi/p)^{3/2}, \qquad T_{ab}=\mu[3-2\mu|\mathbf A-\mathbf B|^2]S_{ab}.$$

Attraction to a nucleus of charge $Z$ at $\mathbf C$ is

$$V_{ab}^{(C)}=-\frac{2\pi Z}{p}K_{AB} F_0(p|\mathbf P-\mathbf C|^2),$$

where the Boys function is

$$F_n(t)=\int_0^1u^{2n}e^{-tu^2}\,du.$$

For a second pair $(c,d)$ with analogous $q,\mathbf Q,K_{CD}$,

$$(ab|cd)= \frac{2\pi^{5/2}K_{AB}K_{CD}}{pq\sqrt{p+q}} F_0\!\left(\frac{pq}{p+q}|\mathbf P-\mathbf Q|^2\right).$$

Multiply by primitive normalization factors and contraction coefficients only after fixing their convention. Angular recurrences extend these seed integrals to higher powers; Obara–Saika gives a systematic construction.

At $t=0$, $F_n(0)=1/(2n+1)$. Near zero, use a convergent series $F_n(t)=\sum_{k\geq0}(-t)^k/[k!(2n+2k+1)]$ rather than a formula that divides two small quantities. At large $t$, asymptotic scaling and carefully chosen recurrence directions avoid underflow and cancellation.

Exact checks before screening

A normalized $s$ Gaussian $g=(2\alpha/\pi)^{3/4}e^{-\alpha r^2}$, with a nucleus of charge $Z$ at its center, obeys

$$S_{gg}=1,\qquad T_{gg}=\frac{3\alpha}{2},\qquad V_{gg}=-2Z\sqrt{\frac{2\alpha}{\pi}}.$$

These elementary identities test normalization and kinetic and attraction prefactors. For real functions, exchange within either pair and exchange of the two pairs leave the electron-repulsion integral unchanged. Translate all centers by the same vector and verify that every integral is unchanged.

Direct four-index storage scales as the fourth power of basis size. Schwarz screening uses

$$|(\mu\nu|\lambda\sigma)| \leq\sqrt{(\mu\nu|\mu\nu)(\lambda\sigma|\lambda\sigma)}.$$

It bounds each neglected integral, not automatically the accumulated energy error. Tighten screening and any density-fitting or Cholesky threshold until target observables stabilize. Compare a small unscreened calculation against the approximate construction.

Exercises

Evaluate $F_0(0)$ and the first derivative $F_0'(0)$ directly from the integral, and use them to give the small-$t$ expansion.

Solution

$F_0(0)=\int_0^1du=1$ and $F_0'(0)=-\int_0^1u^2du=-1/3$. Therefore $F_0(t)=1-t/3+O(t^2)$. This is a useful branch-continuity test for numerical Boys-function evaluation.

References

• Obara, S., and A. Saika, “Efficient recursive computation of molecular integrals over Cartesian Gaussian functions,” Journal of Chemical Physics 84, 3963–3974 (1986), doi:10.1063/1.450106 — angular recurrence algorithms.
• Szabo, A., and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover, 1996 — Gaussian seed integrals and conventions.