1
Efficient evaluation of three-center two-electron integrals over Gaussian functions

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The RI (resolution of the identity) technique achieves significant increases in efficiency for various molecular electronic structure methods.

3
This results from the approximation of four-center two-electron integrals by corresponding three-center integrals.

4
It is shown that the three-center integrals required can be evaluated with a much simpler algorithm than for the general case.

5
This further increases the advantage of RI procedures.

Introduction

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The evaluation of electron repulsion integrals (ERI) over basis functions a,b,c,d(ab|cd) = ∫ a(r1)b(r1)r12−1c(r2)d(r2)dτis a basic task in molecular electronic structure methods.

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The computation of ERIs is typically the dominant step in HF (Hartree–Fock) and DFT (density functional theory) treatments of molecules if the electronic energy is computed exactly within a basis set expansion of molecular orbitals.

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It is thus desirable to develop approximations for eqn. (1) which combine efficiency and accuracy.

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The RI (resolution of the identity) technique is a proven procedure for this purpose.

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The present paper gives a brief summary of the RI method at first, then treats the evaluation of three-center integrals within the Obara–Saika1 (OS) procedure, and finally presents new vertical recursion relations for spherical harmonics as auxiliary functions.

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An introduction of the RI technique2–7 conveniently starts from the definition〈f|g〉 = ∫ f(r1)r12−1g(r2)dτwhich fulfils all requirements of a scalar product: it is linear and positive definite, since 〈f|f〉 = 0 if and only if f = 0.

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ERIs are then simply written as (ab|cd) = 〈ab|cd〉.

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Let us now introduce a set of functions labelled P, Q and the projection operator onto this space where MPQ denotes matrix elements of the inverse of 〈P|Q〉. is optimal in the sense is usually called ‘resolution of the identity’; its insertion into eqn. (1) yields the RI approximation for ERIs (using parentheses for two-electron integrals since this is common usage for the charge density notation) We similarly get a concise expression for the Coulomb energy J of a charge distribution ρ(r)which has an error (O||δρ||2) in the sense of eqn. (4).

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The functions P, Q are usually termed auxiliary or fitting functions, since eqn. (6) implies that ρ is approximated as ρ = ρ.

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The approximation in eqn. (5) decomposes four-index ERIs into two- and three-index-terms which is its most important feature.

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The formal O(N4) effort to evaluate (ab|cd), N = number of basis functions, is thus reduced to O(N3).

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If the auxiliaries P, Q are chosen as atom-centered functions one further has to deal with three-centre integrals only, which leads to additional simplifications as will be demonstrated below.

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The errors introduced by the approximation in eqn. (5) are of no concern if optimized auxiliary functions are employed.

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This has been demonstrated in a series of investigations considering J, the HF exchange K, and correlated treatments based on second order perturbation theory.7–11

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Gains in efficiency resulting from the RI technique are most pronounced for the treatment of J.

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Other procedures have been developed for the very same purpose, such as CFMM (continuous fast multipole moment)12 in combination with the J engine method13 and Fourier techniques.14

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Multipole moment methodology can also be exploited within RI, thus combining the advantages of both procedures.15

Three-center integrals

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We consider the evaluation of (ab|P) for the usual choice of atom-centered GTOs (Gaussian type orbitals)|la〉 = a(r) = (xAx)lx(yAy)ly(zAz)lzeα|rA|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.

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(la(lb + 1i|L) = ((la + 1i)lb|L) + (AiBi)(lalb|L), with 1i = (δix, δiy, δiz,) for i = x,y,z.

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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.

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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.

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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.

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The choice of reduced auxiliary functions is not only more aesthetic; it also improves numerical stability.

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We demonstrate simplifications of integral evaluations for the OS scheme since this has been considered and implemented.

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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)|AB|2Fm(x)x = ρ|PQ|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).

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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).

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The last step typically dominates by far the evaluation of a complete integral batch.

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The recursive increase of L can be simplified.

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The first term on the rhs of eqn. (12) clearly vanishes since Q = C.

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This is trivial and the corresponding term has only been included to show differences to the general case (ab|cd).

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Next we exploit that components of |L〉 are (transformed to) spherical harmonics.

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This implies the following asymptotic decay of integrals:(l0|L) ∝ |PC|L−1 for |PC| → ∞.(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.

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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.

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From eqns. (10) and (11) one has (l0|0)(0) ∝ |PC|−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) ∝ |PC|−1 in contradiction to eqn. (17).

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This implies that the third term on the rhs of eqn. (12) can be neglected since it cancels after transformation.

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The fourth term on the rhs of eqn. (12) has the same structure as the third and cancellation applies here as well.

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That the third and fourth term cannot matter can also be seen directly.

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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 dx2y2 and d3z2r2.

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The same reasoning applies for sets of f, g, etc., functions.

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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.

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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).

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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 Lm ≤ (L + la + lb).

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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.

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This advantage is lost for the present case of three-center integrals.

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Two features are relevant for an implementation of eqn. (20).

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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.

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For the final batch, (l0|(L + 1i))(0) , one will not use eqn. (20) directly if integrals over contracted GTOs are computed.

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It is more efficient to accumulate (l1i)0|L)(1) separately and to add the sum later on.

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This offers the advantage that ((l1i)0|L)(1) has fewer components than (l0|(L + 1i)).

Summary

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We summarize the simplification resulting from the RI technique in comparison to conventional treatments based on (ab|cd).

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(i) The original five term recursion eqn. (12) is replaced by a two-term recursion eqn. (20).

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(ii) Different from the general case in eqn. (12) each intermediate batch (l0|L) occurs only with a single m value; this reduces storage and computational requirements.

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(iii) The simple form of eqn. (20) greatly facilitates the development of optimized hand-coded routines.

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We note in passing that even larger simplifications can be achieved if only two-center integrals have to be computed17 and the present case lies in between the two-center and four-center cases.

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The above described algorithm, including optimized hand-coded routines for la + lb + L ≤ 4, has been implemented in TURBOMOLE18 and will be included in the next release.

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The efficiency of the procedure can be seen from timings of an example at hand, a treatment of Cu40Sb21(PH3)28, symmetry T, with 173 atoms, 1905 contracted GTOs in an SV(P) basis,19 and 5184 (contracted) auxiliary functions.7

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The evaluation of the interelectronic Coulomb interaction J required (per iteration): 64 s in direct mode (all integrals computed on the fly); 46 s if 660 MB was used to store three-center integrals in memory; and 20 s if the far-field integrals were obtained by multipole moment techniques,15 all timings for a 2.4 GHz Xeon.