Direct calculation of the Coulomb matrix: Slater-type orbitals

Abstract

It is proved that the evaluation of the Coulomb potential and the calculation of its matrix elements can be carried out in separate steps whose costs formally increase as the square of the number of basis functions. The resulting method for computing the Coulomb matrix is reported, and its main features are tested with a trial program for Slater functions. A comparison of the Coulomb matrices obtained with this method and those computed from the repulsion integrals shows that the current procedure is potentially exact, highly accurate in practice, and much less expensive. The effects of basis product cutoff and long-range separation have been analyzed finding that the method tends to linear scaling with the size of the system. Moreover, the storage requirements are very low since two-electron integrals are completely absent, and it is well suited to be used in the density functional context.

Appendix

Let ρ^A(r _A ) and ρ^C(r _C ) be two charge distributions fully enclosed by two spheres centered at R _A and R _C and with radii R ^A and R ^B, and let R _AC ≡| R _C  − R _A | ≥R ^A + R ^C. Its electrostatic interaction energy, E ^AC, can be obtained by combining the bipolar expansion of |r − r′|⁻¹:

$$ \begin{aligned} {\frac{1} {| {\bf r} - {\bf r}'|}} & =\sum_{l=0}^\infty \sum_{m=-l}^l (2 - \delta_{m0}) \, {\frac{(l-|m|)!} {(l+|m|)!}} \, z_{l}^{m}({\bf r}_A) \, \sum_{L=0}^\infty \sum_{M=-L}^L (2 - \delta_{M0}) \, {\frac{(L-|M|)!} {(L+|M|)!}}\\ & \times z_{L}^{M}({\bf r}'_C) \, {\frac{\sqrt{\pi} \, (l+L-1/2)!} {(l-1/2)! \, (L-1/2)!}} \, \sum_{m'} \alpha_{l+L, m'}^{lm LM} \, {\frac{z_{l+L}^{m'}({\bf R}_{AC})} {R_{AC}^{2l+2L+1}}} \end{aligned} $$

(42)

with the expansions in spherical harmonics of ρ^A(r _A ) and ρ^C(r _C ). This gives:

$$ \begin{aligned} E^{AC} & = \sum_{l=0}^\infty \sum_{m=-l}^l Q_{lm}^A \sum_{L=0}^\infty \sum_{M=-L}^L Q_{LM}^C\\ & \times {\frac{\sqrt{\pi} \, (l+L-1/2)!} {(l-1/2)! \, (L-1/2)!}} \, \sum_{m'} \alpha_{l+L, m'}^{lm LM} \, {\frac{z_{l+L}^{m'}({\bf R}_{AC})} {R_{AC}^{2l+2L+1}}} \end{aligned} $$

(43)

where α ^lmLM_{l+L, m'} are the coefficients for the decomposition of products of spherical harmonics into spherical harmonics. The sum on m′, determined by $\alpha_{l+L, m'}^{lm LM}$, contains two terms at most, and $Q_{lm}^A$ and $Q_{LM}^C$ are the multipolar moments related to the radial factors by:

$$ Q_{lm}^A = {\frac{4 \pi} {2l + 1}} \, \int\limits_0^{R^A} {\rm d}r \, r^{2l+2} \, f_{lm}^A(r) $$

(44)

with a likewise expression for $Q_{LM}^C$.

In practice, the ρ^A(r _A ) fragments extend to infinite but, since the radial factors have a quick decay (at least, exponential) one can estimate—for a fixed threshold—the size of the sphere that encloses the nonnegligible part of ρ^A(r _A ). In a previous work [3], we showed that the ρ^A(r _A ) given by DAM looks very much like the densities of the corresponding isolated atom, so that a conservative criterion would be to take for atoms in molecules the size of the spheres of the isolated atoms. This point has been tested and verified finding that a radius between 5 and 6 bohr should give sufficient accuracy but, for prudential reasons, we take 6.5 bohr in this trial program.

Once this radius is fixed, C is considered as a neighbor of A if their spheres intersect each other (R _AC <  13 bohr), and nonneighbor (or far away from A) otherwise. Notice that the set of atoms placed far away from A generates a constant potential region where ρ^A(r _A ) is nonnegligible. This is the $B_{lm}^A$ constant appearing in Eqs. 23–26 which, according to Eq. 43 reads:

$$ B_{lm}^A = \sqrt{\pi} \, \sum_{L=0}^{L_{\rm max}} {\frac{(-1)^L \, (L+l-1/2)!} { (L-1/2)! \, (l-1/2)!}} \, \sum_{M=-L}^L \sum_{m'} c_{L+l m'}^{LM \, l m} \, \sum_C Q_{LM}^C\,{\frac{z_{L+l}^{m'}({\bf R}_{AC})} {R_{AC}^{2L+2l+1}}} $$

(45)

where L _max is the order of the highest atomic multipolar moment considered.