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An efficient algorithm to calculate three-electronintegrals for Gaussian-type orbitals using numericalintegrationMooses M. Mehine a , Sergio A. Losilla a & Dage Sundholm aa Department of Chemistry , University of Helsinki , Helsinki , FinlandAccepted author version posted online: 29 Apr 2013.Published online: 08 May 2013.

To cite this article: Mooses M. Mehine , Sergio A. Losilla & Dage Sundholm (2013) An efficient algorithm to calculate three-electron integrals for Gaussian-type orbitals using numerical integration, Molecular Physics: An International Journal at theInterface Between Chemistry and Physics, 111:16-17, 2536-2543, DOI: 10.1080/00268976.2013.793847

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Molecular Physics, 2013Vol. 111, Nos. 16–17, 2536–2543, http://dx.doi.org/10.1080/00268976.2013.793847

INVITED ARTICLE

An efficient algorithm to calculate three-electron integrals for Gaussian-type orbitals usingnumerical integration

Mooses M. Mehine, Sergio A. Losilla and Dage Sundholm∗

Department of Chemistry, University of Helsinki, Helsinki, Finland

(Received 21 February 2013; final version received 28 March 2013)

A novel method to numerically calculate three-electron integrals of explicitly correlated approaches has been developed andimplemented. Coulomb operators of inter-electronic interactions are re-expressed as an integral identity, which is discretised.The discretisation of the auxiliary dimension separates the Cartesian x, y and z dependencies, transforming the integralsof Gaussian-type orbitals to a linear sum of products of three-dimensional intermediate integrals. The intermediate s-typeintegrals can be calculated analytically, whereas integrals of the higher angular-momentum functions are computed usingrecursion formulae. The three-electron integrals are obtained by two-dimensional numerical integration of the discretisedauxiliary dimensions of the integral transformation of the Coulomb operators. Common sets of quadrature points and weightsfor all integrals can be used after a coordinate transformation. Calculations indicate that it is possible to achieve an overallaccuracy of 10−15 Eh using the numerical approach. The same approach can be employed for calculating more generalthree-electron integrals in so far the operator can be accurately expanded in Gaussian-type geminals.

Keywords: three-electron integrals; numerical integration

1. Introduction

In his seminal paper from 1985, Prof Kutzelnigg showedthe importance of including inter-electronic coordinates inthe ansatz for the wave function [1]. The idea of using an ex-plicitly correlated ansatz for the wave function was not new,since Hylleraas as well as James and Coolidge showed in theearly days of quantum mechanics that accurate and compactwave functions can be obtained that way [2–6]. However, thecomputational complexity of the early implementations ofexplicitly correlated methods was huge, rendering routinecalculations difficult. Boys et al. suggested the transcorre-lated method with a similarity-transformed explicitly cor-related Hamiltonian [7–9]. The main disadvantage with thetranscorrelated method is that the non-symmetric Hamilto-nian involves calculating a huge number of three-electronintegrals. On the other hand, no four-electron integrals ap-pear in the transcorrelated Hamiltonian [10]. Kutzelniggand Klopper’s article from 1991 can nevertheless be con-sidered to be the beginning of modern explicitly correlatedapproaches [11]. The key feature of their R12 formulation isthat three-electron and four-electron integrals are expressedas sums of products of two-electron integrals obtained frominsertions of completeness relations. The use of auxiliarybasis sets in the resolution of the identity approximationrendered explicitly correlated calculations with standardbasis sets feasible [12,13]. In the implementation of themore recent F12 methods, similar ideas to avoid many-

∗Corresponding author. Email: sundholm@chem.helsinki.fi

electron integrals are used [14,15]. However, calculationsof many-electron integrals for more general Jastrow factorsalso require advanced computational approaches such asexpansions of operators in Gaussian geminals and numeri-cal integration techniques similar to the ones employed indensity functional theory calculations [15,16].

Here, we propose an alternative approach to computemulti-electronic integrals over Gaussian basis functions.The approach is based on our recent idea of numerical in-tegration of the auxiliary t dimension originating from theintegral transformation of the 1/r12 operator [17]. The ad-vantage of such an approach is that the integrals separate in anatural way into a sum of products of intermediate integralsin the respective Cartesian directions. The intermediate in-tegrals are easily evaluated analytically using recursion re-lations. Prefactors appearing from Gaussian products canbe used for identifying non-vanishing contributions, whichis of importance for the prescreening of the integrals. Thenumerical integration approach introduces new indices thatcan be exploited when the computations are performedon massively parallel processors (central processing unit(CPU)) or on general-purpose graphical processing units(GPGPU) [18,19].

The paper is organised as follows. In Section 2, thebasic idea is presented. Section 3 describes the detailedalgorithm including recursion relations for obtaining in-termediate integrals for higher angular-momentum basis

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Molecular Physics 2537

functions. The quadrature and the choice of integration gridpoints are discussed in Section 4. In Section 5, we demon-strate the accuracy and assess the computational costs ofthe developed numerical approach. The applicability and afuture outlook of the presented approach are discussed inSection 6.

2. Three-electron integrals

The present algorithms and methods can be employed forcalculating many different kinds of three-electron integrals.However, we limit the discussion in here to a class of three-electron Coulomb integrals that have been previously calcu-lated for the special case of single atoms [20]. More generalthree-electron integrals of explicitly correlated electronicstructure calculations can be computed in a similar fash-ion. Analogous expressions for integrals containing linearterms such as r12 can be derived by using the trivial rela-tion r12 = r2

12r−112 . Exponential-type geminal operators can

be expanded as a linear sum of Gaussian geminal func-tions yielding similar expressions as obtained for the three-electron Coulomb integrals [15,21].

Consider three-electron integrals of the type

Gabcdef =�

χ∗a (1)χ∗

b (2)χ∗c (3)r−1

12 r−113 χd (1)χe(2)χf (3)

× d3r1d3r2d3r3, (1)

where χ (r) are Cartesian Gaussian-type orbitals (GTO),

χa(ra) = Naxlaxa y

laya z

laza exp

(−ζar2a

), (2)

where ra = (xa, ya, za) = (x − Xa, y − Ya, z − Za), Ra =(Xa, Ya, Za) are the Cartesian coordinates of the centre ofthe ath primitive basis function and ζ a is its exponent. Ther−1

12 operator is defined as r−112 = |r1 − r2|−1, la = lax + lay +

laz represents the total angular momentum quantum numberand Na is the normalisation constant

Na = N0a γ (lax )γ (lay )γ (laz ), (3)

where γ (l) = [(2l − 1)!!]−1/2 and

N0a = π−3/42la+3/4ζ la/2+3/4

a . (4)

Following the general approach used when deriving algo-rithms for efficient calculations of two-electron integrals,the r−1

12 and r−113 operators are replaced by the standard

integral transformation involving Gaussian functions[8,22–29],

1

r12= 2√

π

∫ ∞

0exp (−t2r2

12) dt and

1

r13= 2√

π

∫ ∞

0exp (−s2r2

12) ds. (5)

The auxiliary integrals of the transformation are discre-tised and calculated numerically as in electronic structurecalculations and in calculations of electrostatic potentialsusing finite-element functions or wavelets [30–35]. We re-cently used a similar approach in combination with GTOsto calculate two-electron contributions to the Fock matrix[17]. The integration range for t ∈ [0, ∞[ can be divided intotwo intervals. In the first one, the integral is calculated usingquadrature, whereas in the second interval the integral is ob-tained using a very accurate analytical approximation [17].As shown in Section 4, the proposed integration scheme isequivalent to expanding the Coulomb operator as

1

r12≈ 2√

π

∑k

ωk exp(−t2

k r212

) + π

t2f

δ(r1 − r2). (6)

The last term, which accounts for the very short-rangeinteractions, can be treated as an additional quadraturepoint. The three-electron integrals Gabcdef can then beexpressed as a weighted sum of the intermediate integralsFkl

abcdef , which can be calculated analytically,

Gabcdef =∑kl

ωkτlFklabcdef . (7)

3. Cartesian-separated form of the three-electronintegrals

The product of two Cartesian GTOs can be expressed asa linear combination of Cartesian GTOs by means of theGaussian product rule [36,37],

⎡⎣ ∏

ξ∈{x,y,z}ξ

laξa

⎤⎦ exp (−ξar

2a )

⎡⎣ ∏

ξ∈{x,y,z}ξ

ldξd

⎤⎦ exp (−ξdr

2d )

= Kad

⎡⎣ ∏

ξ∈{x,y,z}

laξ +ldξ∑uP =0

TP,ξ

laξ ldξ uPξ

uP

P

⎤⎦ exp (−αP r2

P ), (8)

where P is a density index representing the basis-functionpair index ad. The coordinates of the new expansion centreare RP = (ζaRa + ζdRd )/(ζa + ζd ), and the new exponentis αP = ζ a + ζ d. The T

P,ξ

laξ ldξ uPcoefficients can be computed

using the binomial theorem. The Kad pre-exponential factoris

Kad = exp

(− ζaζd

ζa + ζd

|Rd − Ra|2)

. (9)

Similar expressions are obtained for the densities Q andS of the be and cf pairs, respectively. The three-electronintegrals can then be expressed in terms of three-centre

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2538 M.M. Mehine et al.

integrals V nPQS ,

Gabcdef =Nabcdef KadKbeKcf

lax +ldx∑uP =0

lay +ldy∑vP =0

laz +ldz∑wP =0

TP,x

lax ldx uPT

P,y

lay ldy vP

× TP,z

laz ldz wP

lbx+lex∑uQ=0

lby+ley∑vQ=0

lbz +lez∑wQ=0

TQ,x

lbx lexuQT

Q,y

lby leyvQT

Q,z

lbz lezwQ

×lcx+l

fx∑

uS=0

lcy+lfy∑

vS=0

lcz+lfz∑

wS=0

TS,x

lcx lfx uS

TS,y

lcy lfy vS

TS,z

lcz lfz wS

V nPQS,

(10)

where the normalisation coefficient (Nabcdef) is given by

Nabcdef = NaNbNcNdNeNf . (11)

The expression for the three-centre integrals V nPQS in Equa-

tion (10) is

V nPQS =

�r−1

12 r−113 x

uP

1P yvP

1P zwP

1P xuQ

2QyvQ

2QzwQ

2QxuS

3SyvS

3SzwS

3S

× exp(−αP r2

1P − αQr22Q − αSr

23S

)d3r1d3r2d3r3.

(12)

The vector index is a short-hand notation for n =(uP , vP , . . . , wS).

The integral transformation of the Coulomb operatorsin Equation (5) factorises the three-centre integrals (V n

PQS)into products of (x1, x2, x3), (y1, y2, y3) and (z1, z2, z3)dependent intermediate integrals that can be analyticallyevaluated,

V nPQS = 4

π

∫ ∞

0

∫ ∞

0IuP uQuS

(t, s; αP , αQ, αS,XP ,XQ,XS)

× IvP vQvS(t, s; αP , αQ, αS, YP , YQ, YS)

× IwP wQwS(t, s; αP , αQ, αS, ZP , ZQ,ZS)dtds.

(13)

The intermediate integrals, InP nQnS(t, s; αP , αQ, αS,�P ,

�Q,�S) for � ∈ {X, Y, Z}, which can be considered asfunctions of t and s, are obtained by integrating in the cor-responding three Cartesian directions (ξ i ∈ {xi, yi, zi} andi ∈ {1, 2, 3}):

InP nQnS(t, s; αP , αQ, αS,�P ,�Q,�S)

=�

(ξ1 − �P )nP (ξ2 − �Q)nQ (ξ3 − �S)nS

× exp (−αP (ξ1 − �P )2 −αQ(ξ2 − �Q)2 −αS(ξ3 − �S)2

× exp (−t2ξ 212−s2ξ 2

13 dξ1dξ2dξ3. (14)

The InP nQnS(t, s) intermediate integrals are analytically

evaluated as

InP nQnS(t, s)

= π3/2 exp[−�(t, αQ)�2

PQ − �(s, αS)�2PS

]√

[αP + �(t, αQ) + �(s, αS)](αQ + t2)(αS + s2)

× nP nQnS(�(t, αQ),�(s, αS), �PQ,�PS), (15)

where the � function is defined as

�(t, αQ) = t2αQ

t2 + αQ

(16)

and the nP nQnSfunctions are polynomials of �(t, αQ),

�(s, αS), �PQ = �P − �Q and �PS = �S − �P,

nP nQnS(�Q,�S,�PQ,�PS)

=∑ijkl

cnP ,nQ,nS

ijkl �iQ�

jS�

kPQ�l

PS, (17)

where cnP ,nQ,nS

ijkl are linear expansion coefficients. InEquation (17), we have denoted �Q = �(t, αQ) and�S = �(s, αS). For brevity, the s and t dependencies havebeen omitted in Equations (17)–(20c) and (22).

Differentiating Equation (14) with respect to �P, �Q

and �S and Equation (15) with respect to �PQ and �PS, andusing the conditions

∂

∂�P

InP nQnS= − ∂

∂�PQ

InP nQnS− ∂

∂�PS

InP nQnS, (18a)

∂

∂�Q

InP nQnS= ∂

∂�PQ

InP nQnS, (18b)

∂

∂�S

InP nQnS= ∂

∂�PS

InP nQnS, (18c)

we arrive at the recursion relations for the nP nQnSpolyno-

mials,

i+1,jk = 1

2αP

[ (−∂�PQ− ∂�PS

) ijk

+ 2(�Q�PQ + �S�PS

) ijk + i i−1,jk

],

(19a)

i,j+1,k = 1

2αQ

[∂�PQ

ijk − 2�Q�PQ ijk + j i,j−1,k

],

(19b)

ij,k+1 = 1

2αS

[∂�PS

ijk − 2�S�PS ijk + k ij,k−1].

(19c)

It is evident from the recursion relations that nP nQnS=

nP nSnQwith the changes �PQ↔�PS and αQ↔αS. As the

assumed three-electron operator r−112 r−1

13 does not containany r23 dependence, the recursion relation for P slightly

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Molecular Physics 2539

differs from those for Q and S. The explicit expressions forthe first few nP nQnS

are

000(�PQ,�PS) = 1 (20a)

100(�PQ,�PS) = �Q�PQ + �S�PS

αP

(20b)

010(�PQ,�PS) = −�Q�PQ

αQ

(20c)

200(�PQ,�PS)

= αP − (�Q + �S) + 2(�Q�PQ + �S�PS)2

2α2P

(20d)

020(�PQ,�PS) = αQ − �Q + 2�2Q�2

PQ

2α2Q

(20e)

110(�PQ,�PS) = �Q − 2�2Q�2

PQ − 2�Q�S�PQ�PS

2αP αQ

(20f)

011(�PQ,�PS) = �Q�S�PQ�PS

αQαS

(20g)

111(�PQ,�PS)

= −�Q�S

[�PQ+�PS −2�PQ�PS

(�Q�PQ+�S�PS

)]2αP αQαS

.

(20h)

The three-centre V nPQS integrals can be expressed using the

nP nQnSfunctions as

V nPQS(α, t, s, RPQ, RPS)

= 4

π

∫ ∞

0

∫ ∞

0MPQS(t, s)

× uP uQuS(t, s) vP vQvS

(t, s) wP wQwS(t, s)dtds,

(21)

where

MPQS(t, s) = π9/2 exp[−�QR2PQ − �SR

2PS][

(αP + �Q + �S)(αQ + t2)(αS + s2)]3/2

.

(22)

By inserting Equation (21) into Equation (10), one obtainsthe following expression for the three-electron integrals:

Gabcdef = 4

πKadKbeKcf Nabcdef

∫ ∞

0dt

∫ ∞

0dsMPQS(t, s)

×⎡⎣lax +ldx∑

uP =0

lbx+lex∑uQ=0

lcx+lfx∑

uS=0

TP,x

lax ldx uPT

Q,x

lbx lexuQT

S,x

lcx lfx uS

uP uQuS(t, s)

⎤⎦

×

⎡⎢⎣

lay +ldy∑vP =0

lby+ley∑vQ=0

lcy+lfy∑

vS=0

TP,y

lay ldy vPT

Q,y

lby leyvQT

S,y

lcy lfy vS

vP vQvS(t, s)

⎤⎥⎦

×⎡⎣laz +ldz∑

wP =0

lbz +lez∑wQ=0

lcz+lfz∑

wS=0

TP,z

laz ldz wPT

Q,z

lbz lezwQT

S,z

lcz lfz wS

wP wQwS(t, s)

⎤⎦ .

(23)

To avoid redundant operations when computing the inte-grals in a shell, i.e., all integrals arising from a sextet ofshells of basis functions, the expression in Equation (23)can be reorganised into

Gabcdef = 4

πKadKbeKcf N0

abcdef

∫ ∞

0

∫ ∞

0MPQS(t, s)

× �x

lax lbx lcx ldx lex l

fx(t, s)�y

lay lby lcy ldy ley l

fy

(t, s)

× �z

laz lbz lcz ldz lez l

fz

(t, s)dtds (24)

= 4

π

∫ ∞

0

∫ ∞

0W (t, s)dtds,

with N0abcdef = N0

a N0b N0

c N0d N0

e N0f . The six-index function

�ξ

laξ lbξ lcξ ldξ leξ l

fξ

(t, s) is computed as

�ξ

laξ lbξ lcξ ldξ leξ l

fξ

(t, s)

=laξ +ldξ∑nP =0

lbξ +leξ∑nQ=0

lcξ +lfξ∑

nS=0

TP,ξ

laξ ldξ nPT

Q,ξ

lbξ leξ nQT

S,ξ

lcξ lfξ nS

nP nQnS(t, s),

(25)

where the angular momentum dependency of the normali-sation constants of the GTOs are assimilated into the linearexpansion coefficients of the product of the Gaussian func-tions,

TP,ξ

laξ ldξ nP= γ (laξ )γ (ldξ )T P,ξ

laξ ldξ nP, (26)

and analogously for TQ,ξ

lbξ leξ nQand T

S,ξ

lcξ lfξ nS

. Integrals over spher-

ical basis functions can be computed by linear transforma-tion of Cartesian integrals [36]. This can be done beforecarrying out the quadrature that is described in Section 4.The calculation of three-electron integrals for spherical ba-sis functions does not incur in any additional computationalsteps.

4. Numerical integration in t and s space

It is not obvious how to integrate the expression for thethree-electron integrals in Equation (23) analytically, not

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2540 M.M. Mehine et al.

even for the simplest case with n = 0. Therefore, we re-sort to numerical procedures to compute the integrals. Thequadrature developed for two-electron integrals in our pre-vious work is here generalised into a two-dimensional form[17].

The integrals that have to be integrated numerically areof the form,

∫ ∞

0

∫ ∞

0W (t, s)dtds ∝

∫ ∞

0

∫ ∞

0MPQS(t, s)

×⎡⎣ ∏

ξ∈{x,y,z}�

ξ

laξ lbξ lcξ ldξ leξ l

fξ

(�(t, αQ),�(s, αS)

)⎤⎦ dtds.(27)

The explicit expressions for MPQS(t, s) and �ξ (t, s) aregiven in Equations (22) and (25), respectively. To achievehigh accuracy, the integration domain is divided into sev-eral subdomains. The shape of the integrand depends on theexponents of the GTOs and the distances between their cen-tres. In the previous work, we introduced a transformationof variables that rendered the use of common quadraturepoints for all GTO exponents and centres feasible [17].Here, a two-dimensional generalisation of the transforma-tion is developed.

Let us examine the �(u, α) function in more detail,where u is a common notation for t and s. For α > 0 andu ≥ 0, �(u, α) is a monotonically growing function of u,with �(u, α) ≈ u2 when u � √

α and �(u, α) ≈ √α when

t � √α. Because �ξ (t, s) is a polynomial of �(t, αQ) and

�(s, αS),∏

ξ�ξ is a smooth function, which behaves like a

polynomial of t2 and s2 when both t � √αP or s � √

αQ.∏ξ�

ξ is practically constant when both t � √αP and s �√

αQ.The integrand factor MPQS(t, s) can be separated into

three functions,

exp[ − �(t, αQ)R2

PQ

](αQ + t2)3/2︸ ︷︷ ︸

exp[ − �(s, αS)R2

PS

](αS + s2)3/2︸ ︷︷ ︸

(αP + �(t, αQ) + �(s, αS))−3/2︸ ︷︷ ︸ .

A(t ; αQ,RPQ

)A(s; αS, RPS) B(t, s; αP , αQ, αS)

A(u; α, R) is a monotonically decaying bell-shaped function.Its width decreases with increasing R and α. For sufficientlylarge u, it can be expressed as

A(u; α,R) = exp(−αR2

) [u−3 + α

(αR2 − 3

2

)u−5

+O (u−7

) ]. (28)

B(t, s; αP, αQ, αS) is a two-dimensional bell-shaped functionthat decreases monotonically, i.e., B(t + δ, s) < B(t, s)

and B(t, s + δ) < B(t, s) for any δ > 0. The width inthe t direction decreases when increasing αP and αQ, andthe width in the s direction decreases analogously withincreasing αP and αS. Close to the t and s axes, i.e., when

t � (α−1

P + α−1Q

)−1/2or s � (

α−1P + α−1

S

)−1/2, B(t, s) can

be approximated with a two-dimensional polynomial of tand s. As s and t grow, B(t, s) approaches the constant (αP

+ αQ + αS)−3/2.For a fixed value of s, integration in t can be split into

three parts as in our previous work [17].

(1) In t ∈ [0, tl], W(t, s) can be approximated witha polynomial that can be integrated using Gauss–Legendre quadrature,

∫ tl

0W (t, s)dt ≈

Nlin∑k=1

ωkW (tk, s).

(2) In [tl, tf], A(t) decays slowly. B(t, s) and∏

ξ�ξ (�(t,

αQ), �(s, αS)) are practically constant in this inter-val. W(t, s) can be accurately integrated over a longrange with Gauss–Legendre quadrature in logarith-mic coordinates,∫ tf

tl

W (t, s)dt ≈Nlog∑k=1

ωketkW (etk , s),

where tk and ωk are regular Gauss–Legendre pointsand weights in the interval [log (tl), log (tf)].

(3) In the [tf, ∞[ interval, A(t ; α,R) ≈ exp( − αQ

R2PQ

)t−3 + O (

t−5)

and∏

ξ�ξ (�(t, αQ), �(s, αS))

is practically constant. The integral can thereforebe computed analytically with very good accuracyas∫ ∞

tf

W (t, s)dt ≈ 1

2t2f

limt→∞ t3W (t, s) + O (

t−4f

).

(29)

limt → ∞t3W(t, s) is easily obtained as

limt→∞ t3W (t, s) = exp

(−αQR2PQ

)A(s; αS, RPS)

(αP + αQ + �(s, αS))3/2

×∏ξ

�ξ (αQ,�(s, αS)) (30)

because limt → ∞�(t, α) = α.

The integration in the last interval can be regarded asevaluating W(t, s) at some special quadrature point t∞ such

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Molecular Physics 2541

that W(t∞, s) ≡ limt → ∞t3W(t, s), multiplied by the weightω∞ = (2t2

f )−1. This yields a quadrature rule with Ntot =Nlin + Nlog + 1 points {tk: k ∈ {1, . . ., Nlin + Nlog, ∞}},

∫ ∞

0W (t, s)dt =

∑k

ωkW (tk, s), (31)

where the linear and logarithmic quadrature rules have beencombined into a quadrature of Ntot points.

The expression in Equation (30) can be alternativelyobtained by replacing exp(−t2r2

12) in the [tf, ∞[ intervalby π3/2t−3δ(r1 − r2). Therefore, the present approach isequivalent to the expansion of the r−1

12 operator given inEquation (6). The [tf, ∞[ interval accounts for the veryshort-ranged interactions (i.e. the singularity in the r−1

12 op-erator), which would not be considered if the integrationrange were truncated at tf.

The particular value of s has practically no influenceon the domain limits tf and tl . The same procedure can beapplied when integrating W(t, s) in the s dimension yielding

∫ ∞

0

∫ ∞

0W (t, s)dtds =

∑kl

ωkτlW (tk, sl). (32)

The resulting quadrature points are the Cartesian prod-uct of the one-dimensional quadrature points {tk} × {sl}= {(tk, sl)}, and the weights are the tensor product ofthe one-dimensional weights {ωk}⊗{τ l} = {ωkτ l}. Thus,nine different integration domains arise, as illustrated in(Figure 1).

Depending on the values of the GTO exponents andcentres, different quadrature limits tl , tf , sl , and sf need tobe chosen. However, it is desirable to have an input indepen-dent quadrature scheme, which is a prerequisite in parallelcomputations. A universal quadrature can be achieved byscaling the coordinates as

W (t, s) = W

(t ′

aPQ

,s ′

aPS

)(33)

with

aPQ =(√

1αP

+ 1αQ

+ RPQ

)aPS =

(√1

αP+ 1

αS+ RPS

).

(34)

The result of the coordinate scaling is illustrated in (Figure 2). In the new coordinates, adequate universal valuesfor the integration interval limits are t ′l = s ′

l = 2.0 and t ′f =s ′

f = 104. Accordingly, the quadrature points and weightshave to be transformed using the following procedure:

(0) Generate linear and logarithmic Gauss–Legendre pointsand weights as well as the asymptotic point,{u′

1, . . . , u′Ntot−1, u

′∞} and {β ′

1, . . . , β′Ntot−1, β

′∞}.

Figure 1. Points for a quadrature with 5 + 5 + 1 points ineach dimension. Different colours represent different integrationdomains. Isolines at levels 10−2, 10−4, 10−6, etc. are shown forW(t, s)/W(0, 0) with αP = αQ = αS = 1/2 and RPQ = RPS = 0.

(1) Compute aPQ and aPS for the Gabcdef integral.(2) Obtain the quadrature points in t as {t1 =

u′1/aPQ, . . . , tNtot−1 = u′

Ntot−1/aPQ, t∞ = u′∞} and

the weights as {ω1 = β ′1/aPQ, . . . , ωNtot−1 =

β ′Ntot−1/aPQ, ω∞ = β∞a2

PQ}. The points andweights for the integration in the s dimension areobtained analogously by replacing aPQ with aPS.

(3) Compute the summation in Equation (32).(4) Go to step 1.

5. Accuracy and computational costs

To explore the accuracy and computational costs, we haveimplemented a subroutine to compute Gabcdef integrals inour sivari integral library [17]. The present version canonly compute integrals over s-type GTO’s (l = 0). How-ever, the extension to arbitrary angular momentum quantumnumbers is straightforward.

The only case known to the authors where analyticalsolutions exist is for one-centre Gabcdef integrals [20]. One-centre three-electron integrals over s-type basis functionswith identical exponents are given by

Gaaaaaa = 4ζa

3, (35)

where ζ a is the exponent of the common Gaussian s-typefunction. For exponents in the range ζ a ∈ [10−2, 104], theanalytical values can be reproduced by increasing the num-ber of quadrature points. A maximum accuracy of 14 digitsusing 64-bit arithmetics is obtained with a quadrature rule

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2542 M.M. Mehine et al.

Figure 2. (a) Scaled integrands W(t,s)/W(0,0), for the parameters αP = αQ = αS = 1, RPQ = RPS = 0 (blue), αP = 10−1, αQ = αS =1, RPQ = RPS = 0 (green) and αP = αQ = αS = 1, RPQ = RPS = 5a0 (red). (b) Scaled integrands W(t′,s′)/W(0,0) for the same sets ofparameters.

with 20 linear + 27 logarithmic + 1 tail point = 48 points.The accuracy is not very sensitive with respect to the chosenintegration range limits (tl = 2 and tf = 104 as mentionedin Section 4), i.e. a sufficiently large number of points willprovide the correct result unless tl and tf are very poorlychosen.

For integrals involving s-type basis functions with dif-ferent exponents and centres, it is still possible to assessthe convergence with respect to the number of quadra-ture points. We have studied different geometric configura-tions of basis functions. In all considered cases, at most 20quadrature points are needed in the linear interval to obtaina maximum accuracy of 14 digits. Different convergencerates with respect to Nlog are observed. In the worst case wefound,1 the maximum accuracy was reached with approx-imately 70 quadrature points in total. Cheaper quadraturescan be employed when a lower accuracy is tolerable.

In (Table 1), the floating-point operation (FLOP) costsfor the different parts of the algorithm are illustrated forcalculating the three-electron integrals arising from a sextetof shells with the same L. Due to the large prefactors, the

most expensive step for L ≤ 2 is computing the nP nQnS

polynomials in Equation (17) and evaluating them at point(�PQ, �PS) (for � {X, Y, Z}). For larger L the accumulationof the results becomes the main bottleneck due to the largenumber of integrals (L + 1)6(L + 2)6/64. Although thecost for computing one shell of integrals seems bearable,the main obstacle is that the number of integral shells growsas N6

ps , where Nps is the number of primitive shells in thebasis set.

6. Conclusions

The calculations of three-electron integrals using the pre-sented method show that accurate integrals can be obtainedwith a numerical integration scheme. The discretisation ofthe auxiliary t and s dimensions leads to a separation intointermediate integrals that can be calculated analytically.The discretisation also yields indices that are available formassively parallel computation using CPUs or GPGPUs.Prefactors obtained when applying the Gaussian productrule can be used for identifying non-vanishing integrals

Table 1. CPU and memory costs for the different parts of the algorithm when integrating over all sixshells with the same L number.

Step FLOP count Memory (floats)

Computea T P,ξ , T Q,ξ , T S,ξ for ξ = x, y, z 9(L + 1)2(4L + 1) 9(L + 1)3

Computeb (�PQ, �PS) for � = X, Y, Z [416L5 + 840L4 + O(L3)]N 2tot 3(2L + 1)3

Contractc �ξ for ξ = x, y, z [2L7 + O(L6)]N 2tot (L + 1)6

Accumulated Gabcdef 5(L + 1)6(L + 2)6N 2tot/64 (L + 1)6(L + 2)6/64

a See Equation (26). b See Equation (17) c See Equation (25). d See Equation (10)

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Molecular Physics 2543

or non-vanishing contributions when contracting the three-electron integrals with density-matrix elements or otherquantities representing the wave function. The extremelyfast Gaussian-like decay of the integral values with thedistance between basis functions can be exploited to ne-glect many of the integrals. The same algorithm can also beemployed for calculating three-electron integrals for Gaus-sian geminal operators. An extension of the computationalmethod to more general three-electron Coulomb integralsand to four-electron integrals cannot be ruled out eventhough the calculation would involve a three-dimensional orhigher-dimensional numerical integration scheme becausethe number of integration points needed in each dimensionis small.

AcknowledgementsWe acknowledge Olli Lehtonen and Susi Lehtola for their helpfulcomments. This research has been supported by the Academy ofFinland through its Computational Science Research Programme(LASTU) and within project 137460. CSC – the Finnish IT Centerfor Science is thanked for computer time. We also acknowledgethe Magnus Ehrnrooth Foundation for financial support.

Note

1. Ra = Rd = (1.5a0, 0, 0), Rb = Rc = Re = Rf = (0, 0, 0),ζ a = ζ b = ζ c = ζ d = ζ e = ζ f = 10.

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