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

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

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

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This further increases the advantage of RI procedures.

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## Introduction

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The evaluation of electron repulsion integrals (ERI) over basis functions

*a*,*b*,*c*,*d*(*ab*|*cd*) = ∫*a*(*r*_{1})*b*(*r*_{1})*r*_{12}^{−1}*c*(*r*_{2})d(*r*_{2})d*τ*is 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–Saika

^{1}(OS) procedure, and finally presents new vertical recursion relations for spherical harmonics as auxiliary functions.
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An introduction of the RI technique

^{2–7}conveniently starts from the definition〈*f*|*g*〉 = ∫*f*(*r*_{1})*r*_{12}^{−1}*g*(*r*_{2})d*τ*which 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*M*_{PQ}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*(**)which has an error (***r**O*||δ*ρ*||^{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*(*N*^{4}) effort to evaluate (*ab*|*cd*),*N*= number of basis functions, is thus reduced to*O*(*N*^{3}).
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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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Multipole moment methodology can also be exploited within RI, thus combining the advantages of both procedures.

^{15}
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### 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)|*l*_{a}〉 =*a*(**) = (***r**x*−*A*_{x})^{lx}(*y*−*A*_{y})^{ly}(*z*−*A*_{z})^{lz}e^{−α|r − A|2}**= (***l**l*_{x},*l*_{y},*l*_{z}),*l*=*l*_{x}+*l*_{y}+*l*_{z}.With the shorthand notation*a*(**) = |***r**l*_{a}〉 we drop the parameters**and***A**α*.*b*(**) = |***r**l*_{b}〉 will be similarly specified by*l*_{b},*β*,**, and the auxiliary function***B**P*(**) = |***r***〉 by***L***,***L**γ*,**, and we write (***C**ab*|*P*) = (*l*_{a}*l*_{b}|**).It is sufficient to consider (***L**l*_{a}**0**|**) since the general case is recovered by the horizontal recursion relation,***L**e.g*.
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(

*l*_{a}(*l*_{b}+**1**_{i}|**) = ((***L**l*_{a}+**1**_{i})*l*_{b}|**) + (***L**A*_{i}−*B*_{i})(*l*_{a}*l*_{b}|**), with***L***1**_{i}= (δ_{ix}, δ_{iy}, δ_{iz},) for*i*=*x*,*y*,*z*.
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Although we consider Cartesian Gaussians, eqn. (7), we will assume that |

**〉 is always transformed later on to reduced (real) spherical harmonics comprising 5, 7, 9 components for d, f, g sets.***L*
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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

*c*_{L}of (*l*_{a}*l*_{b}|**) do not contain components of s type for a d set, or of p type for an f set,***L**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

**, since this can hardly be confused with the auxiliary functions):(***P***00**|**0**)^{(m)}= 2π^{5/2}(*ζ*+*γ*)^{1/2}(*ζγ*)^{−1}e^{−(αβ/ζ)|A−B|2}*F*_{m}(*x*)*x*=*ρ*|**−***P***|***Q*^{2}The true ERI has index*m*= 0, the recursion then requires*m*> 0 for intermediate quantities,*i.e.**m*= 0 to (*l*_{a}+*l*_{b}+*L*) for the start, eqn. (10).
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In general the easiest way is to first get the necessary (

*l***0**|**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 |

**〉 are (transformed to) spherical harmonics.***L*
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This implies the following asymptotic decay of integrals:(

*l***0**|**) ∝ |***L***−***P***|***C*^{−L−1}for |**−***P***| → ∞.(***C**l***0**|**) may vanish even faster if***L**l***0**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 (

*l***0**|**0**)^{(0)}∝ |**−***P***|***C*^{−1}The third term on the rhs of eqn. (12) maintains this asymptotic behavior, which in the final integral,*m*= 0, would lead to (*l***0**|**) ∝ |***L***−***P***|***C*^{−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 ((1/2*η*)(*l***0**|**0**)^{(m)}− (*ρ*/2*η*^{2})(*l***0**|**0**)^{(m+1)}) to the integrals involving Cartesian functions d_{x2}, d_{y2}, d_{z2}, and this cancels if one goes over to d_{x2−y2}and d_{3z2−r2}.
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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 (*l***0**|**0**)^{(L)}, then gets (*l***0**|**1**)^{(L−1)},*etc*., until the final integral (*l***0**|**)***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*L*≤*m*≤ (*L*+*l*_{a}+*l*_{b}).
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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: d_{x2}, d_{y2}, d_{z2}, and d_{xy}suffice to get a complete f shell, and six f components for a g set,*etc*.
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For the final batch, (

*l***0**|(**+***L***1**_{i}))^{(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 (

**−***l***1**_{i})**0**|**)***L*^{(1)}separately and to add the sum later on.
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This offers the advantage that ((

**−***l***1**_{i})**0**|**)***L*^{(1)}has fewer components than (*l***0**|(**+***L***1**_{i})).
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## 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 (

*l***0**|**) occurs only with a single***L**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 computed

^{17}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

*l*_{a}+*l*_{b}+*L*≤ 4, has been implemented in TURBOMOLE^{18}and will be included in the next release.
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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.
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