We consider the evaluation of (ab|P) for the usual choice of atom-centered GTOs (Gaussian type orbitals)|la〉 = a(r) = (x − Ax)lx(y − Ay)ly(z − Az)lze−α|r − A|2l = (lx,ly,lz), l = lx + ly + lz.With the shorthand notation a(r) = |la〉 we drop the parameters A and α. b(r) = |lb〉 will be similarly specified by lb, β, B, and the auxiliary function P(r) = |L〉 by L, γ, C, and we write (ab|P) = (lalb|L).It is sufficient to consider (la0|L) since the general case is recovered by the horizontal recursion relation, e.g. (la(lb + 1i|L) = ((la + 1i)lb|L) + (Ai − Bi)(lalb|L), with 1i = (δix, δiy, δiz,) for i = x,y,z.
Although we consider Cartesian Gaussians, eqn. (7), we will assume that |L〉 is always transformed later on to reduced (real) spherical harmonics comprising 5, 7, 9 components for d, f, g sets. This can be done explicitly by a transformation step, which is advantageous if integrals have to be transformed into an MO representation as in correlated treatments. For HF or DFT it is easier to ensure simply that contraction coefficients cL of (lalb|L) do not contain components of s type for a d set, or of p type for an f set, etc. The choice of reduced auxiliary functions is not only more aesthetic; it also improves numerical stability.
We demonstrate simplifications of integral evaluations for the OS scheme since this has been considered and implemented. The relevant equations and definitions are, if we stay close to the nomenclature of OS (keeping their P, since this can hardly be confused with the auxiliary functions):(00|0)(m) = 2π5/2(ζ + γ)1/2(ζγ)−1e−(αβ/ζ)|A−B|2Fm(x)x = ρ|P − Q|2The true ERI has index m = 0, the recursion then requires m > 0 for intermediate quantities, i.e.m = 0 to (la + lb + L) for the start, eqn. (10). In general the easiest way is to first get the necessary (l0|0)(m), eqn. (11), and then to increase L by means of eqn. (12). The last step typically dominates by far the evaluation of a complete integral batch.
The recursive increase of L can be simplified. The first term on the rhs of eqn. (12) clearly vanishes since Q = C. This is trivial and the corresponding term has only been included to show differences to the general case (ab|cd). Next we exploit that components of |L〉 are (transformed to) spherical harmonics. This implies the following asymptotic decay of integrals:(l0|L) ∝ |P − C|−L−1 for |P − C| → ∞.(l0|L) may vanish even faster if l0 does not include a partial wave of s character but this is of no concern for the present considerations. Withone can identify all contributions to the final integral that vanish too slowly, and these terms can be neglected since they cancel in the transformation to spherical harmonics. From eqns. (10) and (11) one has (l0|0)(0) ∝ |P − C|−1The third term on the rhs of eqn. (12) maintains this asymptotic behavior, which in the final integral, m = 0, would lead to (l0|L) ∝ |P − C|−1 in contradiction to eqn. (17). This implies that the third term on the rhs of eqn. (12) can be neglected since it cancels after transformation. The fourth term on the rhs of eqn. (12) has the same structure as the third and cancellation applies here as well. That the third and fourth term cannot matter can also be seen directly. For L = 2, i.e. a set of d functions, these terms give identical contributions ((1/2η)(l0|0)(m) − (ρ/2η2)(l0|0)(m+1)) to the integrals involving Cartesian functions dx2, dy2, dz2, and this cancels if one goes over to dx2−y2 and d3z2−r2. The same reasoning applies for sets of f, g, etc., functions.
We thus can replace the five-term recursion in eqn. (12) by a two-term recursion Since the index m on the lhs is connected only with (m + 1) on the rhs, one starts the recursion with (l0|0)(L), then gets (l0|1)(L−1), etc., until the final integral (l0|L)(0) is reached. For each L one has only a single m value and this index can simply be implied with corresponding savings in memory and overhead necessary to implement eqn. (20). The reduction in the index range of m following from eqn. (20) also leads to a reduced index range in the recursion eqn. (11), and eqn. (10) is required only for L ≤ m ≤ (L + la + lb). It should be mentioned that there is an alternative to eqn. (12),[16] which is more efficient for four-center integrals, especially for large angular momentum functions. This advantage is lost for the present case of three-center integrals.
Two features are relevant for an implementation of eqn. (20). For the intermediate integrals, m > 0, one does not need all components of a shell: dx2, dy2, dz2, and dxy suffice to get a complete f shell, and six f components for a g set, etc. For the final batch, (l0|(L + 1i))(0) , one will not use eqn. (20) directly if integrals over contracted GTOs are computed. It is more efficient to accumulate (l − 1i)0|L)(1) separately and to add the sum later on. This offers the advantage that ((l − 1i)0|L)(1) has fewer components than (l0|(L + 1i)).