Efficient Method for Calculating Effective Core Potential IntegralsPublished as part of The Journal of Physical Chemistry virtual special issue “Manuel Yanez and Otilia Mo Festschrift”.
Simon C. McKenzie,† Evgeny Epifanovsky,‡ Giuseppe M. J. Barca,† Andrew T. B. Gilbert,†
and Peter M. W. Gill*,†
†Research School of Chemistry, Australian National University, Canberra, Australian Capital Territory 2601, Australia‡Q-Chem Inc., 6601 Owens Drive, Pleasanton, California 94588, United States
*S Supporting Information
ABSTRACT: Effective core potential (ECP) integrals are amongthe most difficult one-electron integrals to calculate due to theprojection operators. The radial part of these operators mayinclude r0, r−1, and r−2 terms. For the r0 terms, we exploit a simpleanalytic expression for the fundamental projected integral toderive new recurrence relations and upper bounds for ECPintegrals. For the r−1 and r−2 terms, we present a reconstructionmethod that replaces these terms by a sum of r0 terms and showthat the resulting errors are chemically insignificant for a range ofmolecular properties. The new algorithm is available in Q-Chem5.0 and is significantly faster than the ECP implementations in Q-Chem 4.4, GAMESS (US) and Dalton 2016.
1. INTRODUCTION
Applying ab initio quantum chemical methods to heavyelements encounters two main challenges: large numbers ofelectrons and increasingly significant relativistic effects. Effectivecore potentials (ECPs) partially address both problems byexplicitly modeling only the valence electrons and byincorporating scalar-relativistic corrections.1 The downside tousing ECPs is that they introduce unprojected and projectedone-electron integrals, the latter being difficult to evaluate.Several authors have developed methods to compute these
problematic projected integrals. Barthelat et al.2 derived analyticexpressions by using the direct differentiation approach ofBoys.3 However, the resulting expressions rapidly becomeunwieldy, limiting their approach to low orders of angularmomentum on the projector and basis set. The method ofMcMurchie and Davidson4 is perhaps the most widely used dueto its reliability and generality. They factor the integrals intoangular and radial parts and treat the latter via asymptotic andpower series expansions, which are relatively expensive toevaluate. In separate works, Kolar5 and Bode and Gordon6
developed recurrence relations (RRs) for evaluating compo-nents of the radial part of the projected integral. However, asignificant number of radial components is required for eachprojected integral, making the method expensive. Pelissier etal.7 expressed the projected integrals as linear combinations ofoverlap integral products, but this approach introduces lineardependencies that lead to numerical instabilities. Skylaris et al.8
and Flores-Moreno et al.9 introduced adaptive quadratures tocalculate the projected integrals, but both the cost and accuracyof their methods can be difficult to predict. Hu et al. calculated
ECP integrals over solid harmonic rather than CartesianGaussian-type orbitals, computing the radial part via RRs.10
While the efficient calculation of integrals is desirable, it iseven more important to avoid calculating negligible integrals.This can be achieved by using upper bounds, but curiously,there has been little work on developing rigorous upper boundsfor either projected or unprojected ECP integrals. Formally, thetotal number of these integrals grows as MN( )2 , where M isthe number of ECP centers and N is the number of basisfunctions. However, unless the basis functions and ECP arespatially close, the resulting integral will be negligible, and as aresult, the number of signif icant ECP integrals grows as only
M( ). Song et al. recently derived an upper bound but failed todemonstrate this ideal M( ) scaling. Moreover, the computa-tion of their bound required an irksome minimization over anumerical grid.11 Shaw and Hill developed a rigorous bound forthe radial part of ECP integrals but, once again, failed todemonstrate the ideal M( ) scaling.12
In the following section, we develop efficient RRs to calculateboth projected and unprojected ECP integrals. Our approachrelies on the vast simplifications that occur when the potentialscontain only r0 terms, in particular, for the fundamentalprojected integral.2 In section 3, we present a powerful upperbound for these integrals that achieves M( ) scaling, and insection 4, we show how potentials with radial powers other
Received: December 25, 2017Revised: February 20, 2018Published: February 21, 2018
Article
pubs.acs.org/JPCACite This: J. Phys. Chem. A 2018, 122, 3066−3075
than r0 can be handled via a reconstruction process. Finally, insection 5, we outline an algorithm that couples these three ideastogether, and in section 6, we present comparative timingresults for a slab of platinum atoms.
2. RECURRENCE RELATIONSThe unnormalized primitive Gaussian-type orbital
α
| =
= − − − − | − |x A y A z A
a a r
r A
] ( )
( ) ( ) ( ) exp( )xa
ya
za 2x y z
(1)
is defined by its exponent α, center A = (Ax,Ay,Az), angularmomentum vector a = (ax,ay,az), and total angular momentuma = ax + ay + az. A primitive shell, |a], is written in nonbold typeand comprises all Gaussian basis functions of the same totalangular momentum, exponent, and center.A contracted Gaussian-type orbital is a linear combination of
primitives
∑| ⟩ = |=
Da a]i
K
i i1
a
(2)
where Ka is the degree of contraction and Di is a contractioncoefficient. Without loss of generality, we can assume that theECP is centered at the origin and has the form
∑ ∑= + | ⟩ ⟨ |=
−
=−
U r Y U r YU r( ) ( ) ( )Ll
L
m l
l
lm l lm0
1
(3)
where UL(r) and Ul(r) are radial potentials that have the form
∑ η= −=
U r D r r( ) exp( )k
K
kn
k1
2k
(4)
ηk is the ECP exponent, and ∑m |Ylm⟩⟨Ylm| is the sphericalharmonic projector of angular momentum l. Most commonly,nk = −2, −1, or 0, and L rarely exceeds 5. For the remainder ofthe paper, we are concerned with primitive integrals and,therefore, drop the contraction index k, i.e., n ≡ nk.Primitive matrix elements of this ECP in the Gaussian basis
are constructed from unprojected integrals
∫ η| = −r ra b a r b r r[ ] ( ) exp( ) ( ) dLn 2
(5)
and projected integrals
∫ ∑π η| = − ⟨ | ⟩⟨ | ⟩∞
+r r Y Y ra b a b[ ] 4 exp( ) dln
mlm lm
0
2 2
(6)
A class of integrals is written in nonbold type, i.e., either[a|b]L or [a|b]l, and comprises all integrals over the basisfunctions in shells |a] and |b]. These are built up from themomentum-less fundamental integrals [0|0]L and [0|0]l.
13
Contracted analogues of these quantities are written usingangle brackets.2.1. Unprojected Integrals. The trivial identity
− = − + −t B t A A B( ) ( ) ( )t t t t (7)
where t ∈ {x,y,z}, yields the horizontal recurrence relation(HRR)14
⟨ | + ⟩ = ⟨ + | ⟩ + − ⟨ | ⟩A Ba b 1 a 1 b a b( )t L t L t t L (8)
where 1t is the unit 3-vector in the tth direction. The HRRshifts angular momentum from center A to center B, andtherefore, we need only a vertical recurrence relation (VRR)that builds angular momentum on center A.The unprojected fundamental integral is the three-center
overlap
∫ η α β
π ζ
α β ζ ζ
| = − − | − | − | − |
= Γ +
× − − + − −
− +⎜ ⎟
⎜ ⎟
⎛⎝
⎞⎠
⎛⎝
⎞⎠
r r
n
A B R Fn
R
0 0 r A r B r[ ] exp( ) d
23
2
exp( )2
,32
,
Ln
n
2 2 2
( 3)/2
2 2 211
2
(9)
where ζ = α + β + η and R = |R| = |(αA + βB)/ζ| are theexponent and center of the product Gaussian and 1F1 is theconfluent hypergeometric function.15
We have derived VRRs for general n, but we have notpursued these because they are much more complicated thanthe n = 0 case, where the hypergeometric reduces to unity, theintegral factorizes, and the subsequent recursion simplifiesdramatically. For n = 0, one finds
which is a simplification of the McMurchie−Davidson one-center RR for two-electron repulsion integrals.16 A near-optimalstrategy for its use is known.17
2.2. Projected Integrals. Unfortunately, the presence ofthe spherical harmonic projectors means that the HRR cannotbe used with projected ECP integrals. Hence, we require a VRRto build angular momentum on center A as well as asubsequent VRR for building angular momentum on centerB. The angular integrations in [0|0]l are elementary, and uponperforming these, one obtains4
∫
α β
π ζ α β
| = − − + ·
× −∞
+
⎜ ⎟⎛⎝
⎞⎠A B l P
AB
r r i Ar i Br r
0 0A B
[ ] exp( )(2 1)
4 exp( ) (2 ) (2 ) d
l l
nl l
2 2
0
2 2
(12)
where Pl is a Legendre polynomial and il is a modified sphericalBessel function of the first kind.15
For most values of n, the radial integral in eq 12 leads toawkward combinations of confluent hypergeometric functions,but in the n = 0 case, it can be shown that
| = + ·⎜ ⎟⎛⎝
⎞⎠Gi T l P
AB0 0
A B[ ] ( )(2 1)l l l (13)
where
π ζ α β= − ′ − ′G A B( / ) exp( )3/2 2 2(14)
αβζ
=TAB2
(15)
α′ = α(ζ − α)/ζ and β′ = β(ζ − β)/ζ. This result wasdiscovered by Barthelat et al.2 in 1977 but has not received theattention that it deserves, except for by Hu et al.10
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It is convenient to generalize eq 13 by defining the three-index auxiliary fundamental
| = + + +GA B I L0 0[ ]lm p q p m
m p q l l( , , ) 2 2
(16)
where
= −I T i T( )nn
n (17)
= + ·⎜ ⎟⎛⎝
⎞⎠L n T P
ABA B
(2 1)nn
n (18)
In and Ln satisfy the two-term RRs
= + +− +I n I T I(2 1)n n n12
1 (19)
= ++
· −−+ −⎜ ⎟⎛
⎝⎞⎠L
nn
TAB
Ln
nTL
A B2 31 2 1n n n1 1 (20)
and the fundamental integral in eq 16 enjoys the derivativeproperty
∑
∑
αα ζ
ζαβ
ζ α
βζ
αβζ
αβζ
∂∂
| = − |
+ | + |
+ + |
− |
+ − +
=
⌊ − ⌋
− −+ + +
=
⌊ − ⌋ +
− −+ + + +
⎡⎣⎢⎢
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎤⎦⎥⎥
AA
A pA
lB
A
0 0 0 0
0 0 0 0
0 0
0 0
2[ ] [ ]
2[ ] [ ]
(2 1) 2[ ]
2[ ]
tlm p q
t lm p q
tlm p q t
lm p q
tk
l k
l km k p k q
tk
l k
l km k p k q
( , , ) ( , , )
2
2( , 1, ) ( , 1, 1)
0
( 1)/2 2
2 1( , , 1)
0
( 2)/2 2 1
2 2( 1, , 1)
(21)
Using eqs 16 and 21, Ahlrichs’ method18 yields the 6 + 2l termVRR
∑
∑
∑
∑
α ζζ
αζ
βζ α
ζβ
ζ
α
βζ
αβζ
αβζ
βαβζ
ααβζ
+ | = − |
+ | + |
+ − | + − |
+ − |
+ + |
− |
+ | −
− − |
+ − +
+
− +
=
⌊ − ⌋
− −+ + +
=
⌊ − ⌋ +
− −+ + + +
=
⌊ − ⌋
− −+ + +
=
⌊ − ⌋ +
− −+ + + +
⎡⎣⎢⎢
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎤⎦⎥⎥
A
A pA
a a
pa
lB
A
b
a
a 1 b a b
a b a b
a 1 b a 1 b
a 1 b
a b
a b
a b 1
a 1 b
[ ] [ ]
2[ ] [ ]
2[ ] [ ]
2[ ]
(2 1) 2[ ]
2[ ]
22
[ ]
22
[ ]
t lm p q
t lm p q
tlm p q t
lm p q
tt l
m p q tt l
m p q
tt l
m p q
tk
l k
l km k p k q
tk
l k
l km k p k q
t
k
l k
t l km k p k q
t
k
l k
t l km k p k q
( , , ) ( , , )
2( 1, , ) ( , 1, 1)
( , , )2
2( 1, , )
2( , 1, 1)
0
( 1)/2 2
2 1( , , 1)
0
( 2)/2 2 1
2 2( 1, , 1)
0
( 1)/2 2
2 1( , , 1)
0
( 2)/2 2 1
2 2( 1, , 1)
(22)
which is a key result of this paper. This can be used, togetherwith the analogous RR for building momentum on center B, toconstruct the target class of projected integrals [a|b]l ≡[a|b]l
(0,0,0).
We note that these RRs relate integrals over differentprojectors, e.g., a d-projected class is constructed from d-, p-,and s-projected classes. It is also worth noting that, althoughour RR has three auxiliary indices, the index decrements in thethird and sixth terms cause many of the intermediates todisappear, unlike other multi-index RRs.19
3. UPPER BOUNDSIf we can show that an ECP integral is smaller than an upperbound that, in turn, is smaller than a cutoff threshold τ, theECP integral can be safely ignored. It follows that the idealupper bound is both strong (i.e., similar in magnitude to theintegral) and simple (i.e., much cheaper than the integral).20
For a given ECP and shell ⟨a|, it is possible to form a two-center upper bound B2 for all possible ECP classes ⟨a|b⟩, andthis can be used to eliminate many ECP−shell pairs. Similarly,for a given ECP, shell ⟨a|, and shell |b⟩, one can form a (tighter)three-center upper bound B3 for the particular class ⟨a|b⟩, andthis can be used to eliminate many ECP−shell−shell triples.This two-layer strategy is analogous to the shell pair and shellquartet screening used for electron repulsion integrals.13
We derive our two- and three-center upper bounds usingshell-bounding gaussians (SBGs),21 and this allows us to exploitthe simple analytic expressions for n = 0 fundamental integrals.
3.1. Shell-Bounding Gaussians. An SBG a is an s-typeGaussian that bounds all of the functions in a primitive shell |a],i.e.
≥ || | ∀ | ∈ |a ar a a( ) ] ] ] (23)
If the shell is unnormalized, one can show that
α = − | − |a Nr r A( ) exp( )a2
(24)
ασ=
⎛⎝⎜
⎞⎠⎟N
ae2a
a
a/2
(25)
where α = (1 − σa)α is an effective exponent and σa is anarbitrary parameter whose value can be chosen to minimize theresulting upper bound.21 Because SBGs bound an entire shell ofbasis functions, they lead naturally to more efficient classbounds rather than integral bounds. In the following sections,we present key formulas for a variety of upper bounds.Additional details can be found in the Appendix.
3.2. Unprojected Integrals. 3.2.1. Two-Center Bounds.The largest integral
| | | = | | | | ∈ | | ∈ |a b a ba b a b[ ] max [ ] ] ], ] ]L L (26)
in an unprojected integral class possesses the two-center classbound
| | | ≤ = a b B a Q[ ] [ ] [ ]L L L2 (27)
where
αηα η
= − +
⎛⎝⎜
⎞⎠⎟a N
A[ ] expL a
2
(28)
πα
= +
+
+
⎛⎝⎜
⎞⎠⎟Q
ae
( 3) ( /3)(2 )
a
a a
3 3
3
1/2
(29)
a is the largest angular momentum and α is the smallestexponent in the basis set.The optimal σ minimizes [a ]L and exactly satisfies
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However, for computational efficiency, we use the inexpensivebut accurate approximation
σ α ηα η α η
= ++ +
aA a
( )2 ( ( ))a
2
2 2 (31)
Only shells for which [a ]L ≥ τ/Q need to be retained forsubsequent processing, and in this way, most of the shells thatare far from the ECP are eliminated.3.2.2. Three-Center Bounds. An unprojected integral class
possesses the tighter three-center class bound
| | | ≤ = a b B N N ab[ ] [ ] [ ]L L a b L3 (32)
where
π ζ α β ζ = − − + ab A B R[ ] ( / ) exp( )L3/2 2 2 2
(33)
ζ = α + β + η and R = |R| = |(αA + βB)/ζ |. Ideally, σa and σbwould be obtained by minimizing [B3]L. However, the resultingexpression is too expensive to be used as a bound; therefore, weuse the values obtained from the two-center bound (i.e., eq 31).The primitive [B3]L bound can be naively extended to obtain
a contracted bound for the unprojected integrals
∑ ∑ ∑|⟨ | ⟩ | ≤ | | | | | |a b D D D B([ ] )Li
K
ij
K
jk
K
k L ijk3
a b L
(34)
but this requires K( )3 computational work. An K( )contracted upper bound can be obtained by following theapproach of Barca et al.21
|⟨ | ⟩ | ≤ ⟨ ⟩ = ⟨ ⟩⟨ ⟩⟨ ⟩a b B a b abL L L3 (35)
where
∑⟨ ⟩ =a D Ni
K
i a
a
i(36)
∑α β π η ζ⟨ ⟩ = − − ab A B D Rexp( ) ( / ) exp( )Lk
K
k k k k2 2 3/2 2L
(37)
α is the smallest effective exponent in the contracted shell,
ζ α β η = + +k k, and α β ζ = | | = | + |R R A B( )/k k k . Theoptimal σa in Nai of eq 36 is calculated using the smallestexponent in the contracted ECP.3.3. Projected Integrals. To construct upper bounds for
projected integral classes, we begin by bounding the projectorby its absolute value and, subsequently, by a scaled s-projector
| ⟩ ≤ || |⟩ ≤ + | ⟩Y Y l Y(2 1)lm lm 00 (38)
3.3.1. Two-Center Bounds. A projected integral classpossesses the two-center class bound
| | | ≤ = + a b B l a Q[ ] [ ] (2 1) [ ]l l l22
(39)
where [a ]l ≡ [a ]L is given by eq 28 and Q is defined in eq 29.3.3.2. Three-Center Bounds. A projected integral class
possesses the tighter three-center class bound
| | | ≤ = + a b B l N N ab[ ] [ ] (2 1) [ ]l l a b l32
(40)
where
π η α β α ζ
β ζ
= − − +
+ ab A B A
B f T
[ ] ( / ) exp( /
/ ) ( )l
3/2 2 2 2 2
2 2 (41)
=>
≤
⎪
⎪
⎧⎨⎩
f TT T T
T( )
exp( )/(2 ) 1
sinh(1) 1 (42)
αβζ
= TAB2
(43)
and we have made use of the inequality i0(x) ≤ exp(x)/(2x).The σa and σb that appear in Na and Nb are determinedindependently by considering the interaction of each SBG withthe ECP projector exp(−ηr2)|Y00⟩, and because Y00 isspherically symmetric, they are given by eq 31.As in the unprojected case, a contracted three-center bound
for the projected class is
|⟨ | ⟩ | ≤ ⟨ ⟩ = + ⟨ ⟩⟨ ⟩⟨ ⟩a b B l a b ab(2 1)l l l32
(44)
where
∑α β π η α ζ
β ζ
⟨ ⟩ = − − | |
+
ab A B D A
B f T
exp( ) ( / ) exp( /
/ ) ( )
lk
K
k k k
k k
2 2 3/2 2 2
2 2
l
(45)
αβ
ζ =
T
AB2k
k (46)
3.4. Performance. To investigate the performance of thenew upper bounds, we considered slabs of platinum atoms, asingle unit cell thick and of varying sizes, (Figure 1) with theSRSC22−28 ECP and basis set.
As can be seen in Figure 2, the number of ECP centers, M,increases linearly with the size of the Pt slab, and the totalnumber of ECP classes grows as MN( )2 , where N is thenumber of basis functions. Asymptotically, the number ofsignif icant ECP classes increases only linearly, and this ideal
M( ) scaling can already be observed for Pt256, which has amaximum dimension of 27.5 Å and 104 basis functions. Thethree-center screening overestimates the number of significantclasses by a factor of approximately four, but it still has thecorrect M( ) scaling. The two-center screening is weaker andoverestimates the number of significant classes by a factor of
Figure 1. Pt36 slab constructed from face-centered unit cells.
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about 25. It also predicts M( ) significant classes, although itapproaches this scaling more slowly.
4. ECP RECONSTRUCTIONThe efficient method to evaluate ECP projected integralsoutlined in sections 2 and 3 applies only to radial potentialsU(r) where n = 0. While this is the case for some ECPs, such asthose in the Stuttgart−Cologne family,22−28 many ECPs haveradial potentials with n = −2 and −1 terms, such asLANL2DZ,29−31 HWMB,29 SBKJC,32−34 CRENBL,35−40 andCRENBS.36,37,40,41 Examples of these ECP radial potentials forthe d-projector on a Pt atom are shown in Figure 3. The
significant variation between these radial potentials at small r isstriking and highlights the nonuniqueness of ECPs.42 Despitethis variation, all of these ECPs produce similar chemistry,indicating that the resulting valence orbitals are insensitive tothe behavior of the ECP potential at small r.43 Furthermore, theperformance of the SRLC and SRSC ECPs, which only includen = 0 terms, indicates that the inclusion of n = −2 and −1 termsis by no means essential.
In this section, we propose a method for reconstructing the n= −2 and −1 radial potentials using only n = 0 terms to allowour RRs and bounds to be applied more generally.
4.1. Method. To model a generic n = −2 term by a sum ofthree n = 0 terms, we minimized the integral
∫ ∑ λ= − − −∞
−
=
⎡⎣⎢⎢
⎤⎦⎥⎥Z r r c r w r r rexp( ) exp( ) ( ) d
kk k2
0
2 2
1
32
22
(47)
and chose the weight function w(r) = r4 that gave the best fit tothe chemically important regions of those we tested. Theoptimal scale factors that resulted were λ1 = 1.5, λ2 = 5.4, and λ3= 30.5.To model a generic n = −1 term by a sum of three n = 0
terms, we minimized the integral
∫ ∑ μ= − − −∞
−
=
⎡⎣⎢⎢
⎤⎦⎥⎥Z r r d r w r r rexp( ) exp( ) ( ) d
kk k1
0
1 2
1
32
22
(48)
with the same weight function. The optimal scale factors thatresulted were μ1 = 1.2, μ2 = 2.8, and μ3 = 11.6.Armed with these results, we can reconstruct an ECP using a
three-step protocol:
(1) For each n = −2 term with exponent η, form n = 0 termswith exponents λ1η, λ2η, λ3η.
(2) For each n = −1 term with exponent η, form n = 0 termswith exponents μ1η, μ2η, μ3η.
(3) Combine the n = 0 terms from steps 1 and 2 with any n =0 terms in the original potential and use these as a basisto least-squares fit the original radial potential with thesame weight function.
The fitting process is straightforward and performed at runtimeto allow any ECP with n = −2 and −1 terms to bereconstructed. Linear dependence is uncommon, but to guardagainst numerical problems, the least-squares equations aresolved using the Moore−Penrose pseudoinverse.
Figure 2. Scaling of significant ECP classes with increasing size of the Pt slab. The SRSC ECP was used and the threshold set to 10−8.
Figure 3. SRSC, LANL2DZ, CRENBS, and SBKJC radial potentialsfor the d-projector, U2(r), on Pt.
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4.2. Reconstructed ECP Performance. Figure 4 shows anexample of a reconstructed SBKJC ECP using the aboveprocedure. The original SBKJC d-projector for Pt has one n = 0term and one n = −2 term. The latter yields three new n = 0terms, resulting in a 4-fold contracted potential. We choose toshow an SBKJC ECP because this family of ECPs is challengingto reconstruct. This is due to their small degree of contraction,K, which results in less flexibility for the fit. For r ≤ 0.1, thereconstructed ECP is qualitatively incorrect, but this region ischemically unimportant. For r > 0.1, the reconstruction is veryaccurate and the potentials become essentially indistinguish-able. The differences are smaller than those between potentialsfrom different ECPs, as shown in Figure 3.To test further the effects of the reconstruction procedure,
we performed energy, geometry, and frequency calculations onseveral diatomic molecules using the original and reconstructedECPs. The reconstruction changes the total energy of an atomby an average of 2 mEh. However, because this arises from theinner core region, it cancels almost perfectly when relativeenergies are calculated.All calculations were run at the HF level of theory and used
the 3-21G basis set on nonmetal atoms. The results are shownin Table 1. The mean deviations between the original andreconstructed ECPs for dissociation energies, equilibrium bonddistances, and fundamental vibrational frequencies are 0.3 kJmol−1, 0.08 pm, and 0.6 cm−1, respectively. These deviationsare orders of magnitude smaller than the differences betweendifferent ECPs and are of the same order of magnitude as thoseintroduced by the quadrature methods used by Flores-Morenoet al.9 In this sense, we claim that the ECP reconstructions areaccurate.
5. COMPUTATIONAL DETAILSThe present method has been implemented within Q-Chem’snew integrals library, LIBQINTS, replacing the previous ECPimplementation.44 LIBQINTS provides a general framework toboth compute and validate molecular integrals, and becausevalidation is a key part of our approach, we have providedsuitable reference data in the Supporting Information.We first compute and store the basis constant Q and the ⟨a ⟩
and ⟨a ⟩L upper bound factors for each contracted shell. Theseare computed over batches of similar basis function and ECPprojector angular momentum pairs (a,l) to allow forvectorization and efficient memory usage.13
As with the calculation of the upper bound factors, the ECPintegrals are also processed in batches of similar shell andprojector angular momenta, (a,l,b). For each batch, we loopover the ECP centers, shells |a⟩, and shells |b⟩ in the batch.Inside of these loops, the ECP integral screener forms a list
of significant ECP−shell pairs using either the ⟨B2⟩L or the⟨B2⟩l bounds for unprojected and projected integrals,respectively. These significant ECP−shell pairs are combinedto form a list of ECP−shell−shell triples, which are furtherscreened using either the ⟨B3⟩L or ⟨B3⟩l bounds. The advantageof this approach is that the bulk of insignificant integrals arescreened by the less expensive ECP−shell screener beforeproceeding to the stronger, but more expensive, ECP−shell−shell screener. The surviving ECP−shell−shell triples areordered so that a ≥ b in preparation for the VRRs.The computation of the ECP integrals is divided into
unprojected and projected cases. For the unprojected integrals,we first construct the required fundamental integrals (eq 10),then build angular momentum on center A using the VRR (eq
Figure 4. Comparison of the SBKJC radial potential for the d-projector U2(r) on Pt with its reconstructed form.
Table 1. Absolute Differences in Dissociation Energies (ΔDe/kJ mol−1), Equilibrium Bond Lengths (Δre/pm), and VibrationalFrequencies (Δωe/cm
−1) between Original and Reconstructed LANL2DZ, SBKJC, and CRENBL ECPs
11), then contract the integrals, and finally shift angularmomentum from center A to B using the HRR (eq 8).
The construction of the projected fundamental integrals andthe application of the VRR (eq 22) is relatively complicated. At
Figure 5. Diagram showing how auxiliary classes are combined to form a [d|0]d class of integrals. Colors indicate projector momentum.
Figure 6. ECP integral timings for Pt slabs using the SRSC ECP and an integral threshold of 10−8.
Figure 7. ECP integral timings for Pt slabs using the SBKJC ECP. The equations show the results of linear regressions performed on the last fewdata points. Calculations were performed using the original SBKJC ECP except for the present method, which used the reconstructed SBKJC ECP.The integral threshold was set to 10−8.
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runtime, we create drivers that determine the small set ofintermediates required to build the desired integrals from theauxiliary fundamental integrals (eq 16). A diagram of the [d|0]ddriver is shown in Figure 5 and shows how these intermediatesare combined to form the target integral class. These drivers arecreated for each class, (a,l,b), of projected integrals, and ourimplementation can handle arbitrary ECP projector and basisfunction angular momenta. We compute the factors (G,Ln,Im)for the fundamental integrals separately, with the Ln and Imfactors being computed via the RRs (eqs 19 and 20). Thesefactors are combined to build the range of requiredfundamental projected integrals (eq 16). The projected VRR(eq 22) is executed for each step of the driver, first buildingangular momentum on center A and then on center B using thecorresponding VRR. Finally, the unprojected and projectedintegrals from each ECP center are accumulated into the targetECP matrix element.
6. RESULTS AND DISCUSSION
To evaluate the efficiency of the present ECP integral method,timings were carried out on the same Pt slab systems fromsection 3 using the new method as implemented in Q-Chem5.0, the old ECP method in Q-Chem 4.4,44 GAMESS (US),6,45
and Dalton 2016.46 All calculations were performed on a 2.6GHz Intel Xeon E5-2690 v4 platform using a single CPU core.Figure 6 shows timing data for Pt slabs with SRSC ECPs. Q-
Chem 4.4 and GAMESS (US) cannot handle the g-projectorsin this ECP; therefore, only Dalton 2016 and the presentmethod are shown. It is immediately obvious that the presentmethod is around 2 orders of magnitude faster than Daltonacross all system sizes considered. Dalton does not employ anyscreening and therefore calculates the full MN( )2 set of ECPintegrals, as demonstrated by the asymptotic scaling of itstimings M( )2.70 . While the pair/triple screener predicts theideal M( ) number of integrals, the present method does not
achieve this ideal asymptotic scaling in timings N( )1.68 due tothe small MN( ) cost of the screener and the MN( )2
overhead of looping over the batches.Figure 7 shows timing data for the same Pt slabs except this
time using the SBKJC ECP, which includes a maximum of f-projectors on Pt. Q-Chem 4.4, GAMESS (US) and Dalton areable to calculate ECP integrals where n ≠ 0, and therefore, theoriginal Pt SBKJC ECP with K = 1−3 was used. The presentmethod used the reconstructed Pt SBKJC ECP with K′ = 3−5.Despite the greater number of ECP primitives, the presentmethod is significantly faster than Q-Chem 4.4, GAMESS(US), and Dalton across all system sizes considered. GAMESS(US) and Dalton employ no screening, whereas Q-Chem 4.4uses shell pair screening and the present method uses pair/triple screening. The asymptotic scaling of the timings for eachmethod reflects these screening strategies.
7. CONCLUSIONS
We have presented an efficient method for computing ECPintegrals over Gaussian basis functions. The key to this newmethod is restricting the ECP radial potentials, U(r), to containonly r0 terms. This allows for a simple analytic form for theotherwise difficult projected fundamental integral. Using thisanalytic form, we have derived the first RRs in the literature todirectly build angular momentum on the Gaussian basis
functions. These RRs can handle arbitrary ECP projector andbasis function angular momenta.We have also developed powerful upper bounds to screen
ECP−shell pairs and ECP−shell−shell triples using SBGs.21
These new bounds are able to reduce the number of ECPintegrals from formally cubic scaling to the ideal linear scaling.In order to make our method more widely applicable, we
presented a procedure that converts ECPs containing r−2 andr−1 terms into reconstructed ECPs that contain only r0 terms.The discrepancies introduced by this reconstruction wereshown to be chemically insignificant and similar in magnitudeto those introduced by quadratures used in other ECPmethods.9
Our method has been implemented in the Q-Chem 5.0package, and this was used to show that our approachsignificantly reduces the cost of calculating ECP integralscompared to Q-Chem 4.4, GAMESS (US) and Dalton 2016. Inlight of the advantages of our method, we recommend only r0
terms be included in future ECPs.
■ APPENDIX
To derive our upper bounds, we exploit the absolute valueinequality (AVI)
∫ ∫≤f fr r r r( ) d ( ) d(49)
and Holder’s inequality15
∫ ∫ ∫≤ ⎡⎣⎢
⎤⎦⎥
⎡⎣⎢
⎤⎦⎥f g f gr r r r r r r( ) ( ) d ( ) d ( ) dp
pq
q1/ 1/
(50)
where p−1 + q−1 = 1.
A.1. Unprojected IntegralsUsing the AVI and SBGs, we can bound a primitiveunprojected integral by
η| | | ≤ | − | a r ba b[ ] [ exp( ) ]L2
(51)
A.1.1. Two-Center Bounds. Applying Holder’s inequality toeq 51 and taking the limit as p → ∞ and q → 0 gives theunprojected upper bound
| | | ≤ a ba b[ ] [ ] [ ]L L (52)
where [a ]L is given by eq 28 and
π β = b N[ ] ( / )b3/2
(53)
The σb that minimizes [b ] is
σ =+b
b 3b (54)
Maximizing [b ] with respect to b and β yields the Q factor (eq29), which is independent of |b].
A.1.2. Three-Center Bounds. Using the analogous funda-mental unprojected integral (eq 10) solution, eq 51 gives thethree-center bound (eq 32) for an unprojected class.
A.2. Projected Integrals
The AVI, SBGs, and the inequality | | ≤ +Y l Y(2 1)lm 00 yieldthe projected integral bound
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A.2.1. Two-Center Bounds. Applying Holder’s inequality toeq 55 gives the following projected upper bound
| | | ≤ + l a ba b[ ] (2 1) [ ] [ ]l L2
(56)
where [a ]L and [b ] are defined as in the unprojected two-centerbound derivation, eqs 28 and 53, respectively. Following thesame procedure leads to the two-center bound for a primitiveclass of projected integrals (eq 39).A.2.2. Three-Center Bounds. Applying the simple projected
fundamental integral solution eq 13 to eq 55 gives the three-center bound (eq 40) for a projected class
π η α β α ζ
β ζ
= − − +
+ ab A B A
B i T
[ ] ( / ) exp( /
/ ) ( )l
3/2 2 2 2 2
2 20 (57)
where T is defined in eq 43. Exploiting the inequality
≤>
≤
⎪
⎪
⎧⎨⎩
i xx x x
x( )
exp( )/2 1
sinh(1) 10
(58)
allows us to bound eq 57 and avoid evaluating the expensivemodified spherical Bessel function, as seen in eq 41.
■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.jpca.7b12679.
Reference ECP integral values for a variety of classes(PDF)
ORCIDGiuseppe M. J. Barca: 0000-0001-5109-4279Peter M. W. Gill: 0000-0003-1042-6331NotesThe authors declare the following competing financialinterest(s): Many of the present authors are also employedby Q-Chem Inc., in which this research is implemented.
■ ACKNOWLEDGMENTSS.C.M. thanks the Westpac Bicentennial Foundation for aFuture Leaders scholarship and the Australian Government foran Australian Government Research Training Program scholar-ship. P.M.W.G. thanks the Australian Research Council forfunding (Grants DP140104071 and DP160100246) and theNCI National Facility for generous grants of supercomputertime.
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