An integral scheme for the efficient evaluation of two-center integrals over contracted solid harmonic Gaussian functions is presented. Integral expressions are derived for local operators that depend on the position vector of one of the two Gaussian centers. These expressions are then used to derive the formula for three-index overlap integrals where two of the three Gaussians are located at the same center. The efficient evaluation of the latter is essential for local resolution-of-the-identity techniques that employ an overlap metric. We compare the performance of our integral scheme to the widely used Cartesian Gaussian-based method of Obara and Saika (OS). Non-local interaction potentials such as standard Coulomb, modified Coulomb, and Gaussian-type operators, which occur in range-separated hybrid functionals, are also included in the performance tests. The speed-up with respect to the OS scheme is up to three orders of magnitude for both integrals and their derivatives. In particular, our method is increasingly efficient for large angular momenta and highly contracted basis sets.
REFERENCES
1.
R.
Ahlrichs
, M.
Bär
, M.
Häser
, H.
Horn
, and C.
Kölmel
, “Electronic structure calculations on workstation computers: The program system turbomole
,” Chem. Phys. Lett.
162
, 165
–169
(1989
).2.
J.
Hutter
, M.
Iannuzzi
, F.
Schiffmann
, and J.
VandeVondele
, “CP2K: Atomistic simulations of condensed matter systems
,” Wiley Interdiscip. Rev.: Comput. Mol. Sci.
4
, 15
–25
(2014
).3.
Dalton, a molecular electronic structure program, Release Dalton2016.X, 2015, see http://daltonprogram.org.
4.
H.-J.
Werner
, P. J.
Knowles
, G.
Knizia
, F. R.
Manby
, and M.
Schütz
, “Molpro: A general-purpose quantum chemistry program package
,” Wiley Interdiscip. Rev.: Comput. Mol. Sci.
2
, 242
–253
(2012
).5.
M. W.
Schmidt
, K. K.
Baldridge
, J. A.
Boatz
, S. T.
Elbert
, M. S.
Gordon
, J. H.
Jensen
, S.
Koseki
, N.
Matsunaga
, K. A.
Nguyen
, S.
Su
, T. L.
Windus
, M.
Dupuis
, and J. A.
Montgomery
, “General atomic and molecular electronic structure system
,” J. Comput. Chem.
14
, 1347
–1363
(1993
).6.
M. J.
Frisch
, G. W.
Trucks
, H. B.
Schlegel
, G. E.
Scuseria
, M. A.
Robb
, J. R.
Cheeseman
, G.
Scalmani
, V.
Barone
, B.
Mennucci
, G. A.
Petersson
, H.
Nakatsuji
, M.
Caricato
, X.
Li
, H. P.
Hratchian
, A. F.
Izmaylov
, J.
Bloino
, G.
Zheng
, J. L.
Sonnenberg
, M.
Hada
, M.
Ehara
, K.
Toyota
, R.
Fukuda
, J.
Hasegawa
, M.
Ishida
, T.
Nakajima
, Y.
Honda
, O.
Kitao
, H.
Nakai
, T.
Vreven
, J. A.
Montgomery
, Jr., J. E.
Peralta
, F.
Ogliaro
, M.
Bearpark
, J. J.
Heyd
, E.
Brothers
, K. N.
Kudin
, V. N.
Staroverov
, R.
Kobayashi
, J.
Normand
, K.
Raghavachari
, A.
Rendell
, J. C.
Burant
, S. S.
Iyengar
, J.
Tomasi
, M.
Cossi
, N.
Rega
, J. M.
Millam
, M.
Klene
, J. E.
Knox
, J. B.
Cross
, V.
Bakken
, C.
Adamo
, J.
Jaramillo
, R.
Gomperts
, R. E.
Stratmann
, O.
Yazyev
, A. J.
Austin
, R.
Cammi
, C.
Pomelli
, J. W.
Ochterski
, R. L.
Martin
, K.
Morokuma
, V. G.
Zakrzewski
, G. A.
Voth
, P.
Salvador
, J. J.
Dannenberg
, S.
Dapprich
, A. D.
Daniels
, Ö.
Farkas
, J. B.
Foresman
, J. V.
Ortiz
, J.
Cioslowski
, and D. J.
Fox
, gaussian 09, Revision X, Gaussian, Inc., Wallingford, CT, 2009
.7.
J.-P.
Piquemal
, G. A.
Cisneros
, P.
Reinhardt
, N.
Gresh
, and T. A.
Darden
, “Towards a force field based on density fitting
,” J. Chem. Phys.
124
, 104101
(2006
).8.
G. A.
Cisneros
, J.-P.
Piquemal
, and T. A.
Darden
, “Generalization of the Gaussian electrostatic model: Extension to arbitrary angular momentum, distributed multipoles, and speedup with reciprocal space methods
,” J. Chem. Phys.
125
, 184101
(2006
).9.
D.
Elking
, T.
Darden
, and R. J.
Woods
, “Gaussian induced dipole polarization model
,” J. Comput. Chem.
28
, 1261
–1274
(2007
).10.
N.
Gresh
, G. A.
Cisneros
, T. A.
Darden
, and J.-P.
Piquemal
, “Anisotropic, polarizable molecular mechanics studies of inter- and intramolecular interactions and ligand-macromolecule complexes. A bottom-up strategy
,” J. Chem. Theory Comput.
3
, 1960
–1986
(2007
).11.
D. M.
Elking
, G. A.
Cisneros
, J.-P.
Piquemal
, T. A.
Darden
, and L. G.
Pedersen
, “Gaussian multipole model (GMM)
,” J. Chem. Theory Comput.
6
, 190
–202
(2010
).12.
G. A.
Cisneros
, “Application of Gaussian electrostatic model (GEM) distributed multipoles in the AMOEBA force field
,” J. Chem. Theory Comput.
8
, 5072
–5080
(2012
).13.
A. C.
Simmonett
, F. C.
Pickard
, H. F.
Schaefer
, and B. R.
Brooks
, “An efficient algorithm for multipole energies and derivatives based on spherical harmonics and extensions to particle mesh Ewald
,” J. Chem. Phys.
140
, 184101
(2014
).14.
R.
Chaudret
, N.
Gresh
, C.
Narth
, L.
Lagardère
, T. A.
Darden
, G. A.
Cisneros
, and J.-P.
Piquemal
, “S/G-1: An ab initio force-field blending frozen Hermite Gaussian densities and distributed multipoles. Proof of concept and first applications to metal cations
,” J. Phys. Chem. A
118
, 7598
–7612
(2014
).15.
T. J.
Giese
, M. T.
Panteva
, H.
Chen
, and D. M.
York
, “Multipolar Ewald methods, 1: Theory, accuracy, and performance
,” J. Chem. Theory Comput.
11
, 436
–450
(2015
).16.
P.
Koskinen
and V.
Mäkinen
, “Density-functional tight-binding for beginners
,” Comput. Mater. Sci.
47
, 237
–253
(2009
).17.
N.
Bernstein
, M. J.
Mehl
, and D. A.
Papaconstantopoulos
, “Nonorthogonal tight-binding model for germanium
,” Phys. Rev. B
66
, 075212
(2002
).18.
T. J.
Giese
and D. M.
York
, “Improvement of semiempirical response properties with charge-dependent response density
,” J. Chem. Phys.
123
, 164108
(2005
).19.
T. J.
Giese
and D. M.
York
, “Charge-dependent model for many-body polarization, exchange, and dispersion interactions in hybrid quantum mechanical/molecular mechanical calculations
,” J. Chem. Phys.
127
, 194101
(2007
).20.
D.
Golze
, M.
Iannuzzi
, M.-T.
Nguyen
, D.
Passerone
, and J.
Hutter
, “Simulation of adsorption processes at metallic interfaces: An image charge augmented QM/MM approach
,” J. Chem. Theory Comput.
9
, 5086
–5097
(2013
).21.
T. J.
Giese
and D. M.
York
, “Ambient-potential composite Ewald method for ab initio quantum mechanical/molecular mechanical molecular dynamics simulation
,” J. Chem. Theory Comput.
12
, 2611
–2632
(2016
).22.
A.
Sodt
and M.
Head-Gordon
, “Hartree-Fock exchange computed using the atomic resolution of the identity approximation
,” J. Chem. Phys.
128
, 104106
(2008
).23.
S. F.
Manzer
, E.
Epifanovsky
, and M.
Head-Gordon
, “Efficient implementation of the pair atomic resolution of the identity approximation for exact exchange for hybrid and range-separated density functionals
,” J. Chem. Theory Comput.
11
, 518
–527
(2015
).24.
A. C.
Ihrig
, J.
Wieferink
, I. Y.
Zhang
, M.
Ropo
, X.
Ren
, P.
Rinke
, M.
Scheffler
, and V.
Blum
, “Accurate localized resolution of identity approach for linear-scaling hybrid density functionals and for many-body perturbation theory
,” New J. Phys.
17
, 093020
(2015
).25.
S. V.
Levchenko
, X.
Ren
, J.
Wieferink
, R.
Johanni
, P.
Rinke
, V.
Blum
, and M.
Scheffler
, “Hybrid functionals for large periodic systems in an all-electron, numeric atom-centered basis framework
,” Comput. Phys. Commun.
192
, 60
–69
(2015
).26.
M.
Guidon
, F.
Schiffmann
, J.
Hutter
, and J.
VandeVondele
, “Ab initio molecular dynamics using hybrid density functionals
,” J. Chem. Phys.
128
, 214104
(2008
).27.
J. P.
Perdew
, K.
Burke
, and M.
Ernzerhof
, “Generalized gradient approximation made simple
,” Phys. Rev. Lett.
77
, 3865
–3868
(1996
).28.
M.
Ernzerhof
, J. P.
Perdew
, and K.
Burke
, “Coupling-constant dependence of atomization energies
,” Int. J. Quantum Chem.
64
, 285
–295
(1997
).29.
M.
Ernzerhof
and G. E.
Scuseria
, “Assessment of the Perdew-Burke-Ernzerhof exchange-correlation functional
,” J. Chem. Phys.
110
, 5029
–5036
(1999
).30.
A. D.
Becke
, “Density-functional thermochemistry. III. The role of exact exchange
,” J. Chem. Phys.
98
, 5648
–5652
(1993
).31.
C.
Lee
, W.
Yang
, and R. G.
Parr
, “Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density
,” Phys. Rev. B
37
, 785
–789
(1988
).32.
S. H.
Vosko
, L.
Wilk
, and M.
Nusair
, “Accurate spin-dependent electron liquid correlation energies for local spin density calculations: A critical analysis
,” Can. J. Phys.
58
, 1200
–1211
(1980
).33.
J.
Heyd
, G. E.
Scuseria
, and M.
Ernzerhof
, “Hybrid functionals based on a screened Coulomb potential
,” J. Chem. Phys.
118
, 8207
–8215
(2003
).34.
J.
Heyd
, G. E.
Scuseria
, and M.
Ernzerhof
, “Erratum: Hybrid functionals based on a screened Coulomb potential [J. Chem. Phys. 118, 8207 (2003)]
,” J. Chem. Phys.
124
, 219906
(2006
).35.
A. V.
Krukau
, O. A.
Vydrov
, A. F.
Izmaylov
, and G. E.
Scuseria
, “Influence of the exchange screening parameter on the performance of screened hybrid functionals
,” J. Chem. Phys.
125
, 224106
(2006
).36.
A. J.
Cohen
, P.
Mori-Sánchez
, and W.
Yang
, “Development of exchange-correlation functionals with minimal many-electron self-interaction error
,” J. Chem. Phys.
126
, 191109
(2007
).37.
E. J.
Baerends
, D. E.
Ellis
, and P.
Ros
, “Self-consistent molecular Hartree-Fock-Slater calculations I. The computational procedure
,” Chem. Phys.
2
, 41
–51
(1973
).38.
C. F.
Guerra
, J. G.
Snijders
, G.
te Velde
, and E. J.
Baerends
, “Towards an order-N DFT method
,” Theor. Chem. Acc.
99
, 391
–403
(1998
).39.
G.
te Velde
, F. M.
Bickelhaupt
, E. J.
Baerends
, C. F.
Guerra
, S. J. A.
van Gisbergen
, J. G.
Snijders
, and T.
Ziegler
, “Chemistry with ADF
,” J. Comput. Chem.
22
, 931
–967
(2001
).40.
O.
Schütt
and J.
VandeVondele
,“Machine learning adaptive basis sets for efficient large scale DFT simulation
,” J. Chem. Theory Comput.
(submitted).41.
M.
Dupuis
, J.
Rys
, and H. F.
King
, “Evaluation of molecular integrals over Gaussian basis functions
,” J. Chem. Phys.
65
, 111
–116
(1976
).42.
S.
Obara
and A.
Saika
, “Efficient recursive computation of molecular integrals over Cartesian Gaussian functions
,” J. Chem. Phys.
84
, 3963
–3974
(1986
).43.
M.
Head-Gordon
and J. A.
Pople
, “A method for two-electron Gaussian integral and integral derivative evaluation using recurrence relations
,” J. Chem. Phys.
89
, 5777
–5786
(1988
).44.
R.
Lindh
, U.
Ryu
, and B.
Liu
, “The reduced multiplication scheme of the Rys quadrature and new recurrence relations for auxiliary function based two-electron integral evaluation
,” J. Chem. Phys.
95
, 5889
–5897
(1991
).45.
P.
Bracken
and R. J.
Bartlett
, “Calculation of Gaussian integrals using symbolic manipulation
,” Int. J. Quantum Chem.
62
, 557
–570
(1997
).46.
P. M. W.
Gill
, A. T. B.
Gilbert
, and T. R.
Adams
, “Rapid evaluation of two-center two-electron integrals
,” J. Comput. Chem.
21
, 1505
–1510
(2000
).47.
R.
Ahlrichs
, “A simple algebraic derivation of the Obara-Saika scheme for general two-electron interaction potentials
,” Phys. Chem. Chem. Phys.
8
, 3072
–3077
(2006
).48.
L. E.
McMurchie
and E. R.
Davidson
, “One- and two-electron integrals over Cartesian Gaussian functions
,” J. Comput. Phys.
26
, 218
–231
(1978
).49.
T.
Helgaker
and P. R.
Taylor
, “On the evaluation of derivatives of Gaussian integrals
,” Theor. Chim. Acta
83
, 177
–183
(1992
).50.
W.
Klopper
and R.
Röhse
, “Computation of some new two-electron Gaussian integrals
,” Theor. Chim. Acta
83
, 441
–453
(1992
).51.
S.
Reine
, E.
Tellgren
, and T.
Helgaker
, “A unified scheme for the calculation of differentiated and undifferentiated molecular integrals over solid-harmonic Gaussians
,” Phys. Chem. Chem. Phys.
9
, 4771
–4779
(2007
).52.
B. I.
Dunlap
, “Three-center Gaussian-type-orbital integral evaluation using solid spherical harmonics
,” Phys. Rev. A
42
, 1127
–1137
(1990
).53.
B. I.
Dunlap
, “Direct quantum chemical integral evaluation
,” Int. J. Quantum Chem.
81
, 373
–383
(2001
).54.
B. I.
Dunlap
, “Angular momentum in solid-harmonic-Gaussian integral evaluation
,” J. Chem. Phys.
118
, 1036
–1043
(2003
).55.
A.
Hu
and B. I.
Dunlap
, “Three-center molecular integrals and derivatives using solid harmonic Gaussian orbital and Kohn-Sham potential basis sets
,” Can. J. Chem.
91
, 907
–915
(2013
).56.
T. J.
Giese
and D. M.
York
, “Contracted auxiliary Gaussian basis integral and derivative evaluation
,” J. Chem. Phys.
128
, 064104
(2008
).57.
J.
Kuang
and C. D.
Lin
, “Molecular integrals over spherical Gaussian-type orbitals: I
,” J. Phys. B: At., Mol. Opt. Phys.
30
, 2529
–2548
(1997
).58.
J.
Kuang
and C. D.
Lin
, “Molecular integrals over spherical Gaussian-type orbitals: II. Modified with plane-wave phase factors
,” J. Phys. B: At., Mol. Opt. Phys.
30
, 2549
–2567
(1997
).59.
S.
Reine
, T.
Helgaker
, and R.
Lindh
, “Multi-electron integrals
,” Wiley Interdiscip. Rev.: Comput. Mol. Sci.
2
, 290
–303
(2012
).60.
E. J.
Weniger
and E. O.
Steinborn
, “A simple derivation of the addition theorems of the irregular solid harmonics, the Helmholtz harmonics, and the modified Helmholtz harmonics
,” J. Math. Phys.
26
, 664
–670
(1985
).61.
E. J.
Weniger
, “The spherical tensor gradient operator
,” Collect. Czech. Chem. Commun.
70
, 1225
(2005
); e-print arXiv:math-ph/0505018v1.62.
E. W.
Hobson
, “On a theorem in differentiation, and its application to spherical harmonics
,” Proc. London Math. Soc.
s1-24
, 55
–67
(1892
).63.
T. J.
Giese
and D. M.
York
, “A modified divide-and-conquer linear-scaling quantum force field with multipolar charge densities
,” Many-Body Effects and Electrostatics in Biomolecules
(Pan Stanford Publishing
, Singapore
, 2016
), pp. 1
–32
.64.
M. A.
Watson
, P.
Sałek
, P.
Macak
, and T.
Helgaker
, “Linear-scaling formation of Kohn-Sham Hamiltonian: Application to the calculation of excitation energies and polarizabilities of large molecular systems
,” J. Chem. Phys.
121
, 2915
–2931
(2004
).65.
T.
Helgaker
, P.
Jørgensen
, and J.
Olsen
, Molecular Electron-Structure Theory
(Wiley
, 2012
), pp. 412
–414
.66.
H. B.
Schlegel
and M. J.
Frisch
, “Transformation between Cartesian and pure spherical harmonic Gaussians
,” Int. J. Quantum Chem.
54
, 83
–87
(1995
).67.
A.
Hu
, M.
Staufer
, U.
Birkenheuer
, V.
Igoshine
, and N.
Rösch
, “Analytical evaluation of pseudopotential matrix elements with Gaussian-type solid harmonics of arbitrary angular momentum
,” Int. J. Quantum Chem.
79
, 209
–221
(2000
).68.
J. A.
Gaunt
, “The triplets of helium
,” Philos. Trans. R. Soc., A
228
, 151
–196
(1929
).69.
Y.-L.
Xu
, “Fast evaluation of the Gaunt coefficients
,” Math. Comput.
65
, 1601
–1612
(1996
).70.
J. M.
Pérez-Jordá
and W.
Yang
, “A concise redefinition of the solid spherical harmonics and its use in fast multipole methods
,” J. Chem. Phys.
104
, 8003
–8006
(1996
).71.
H. H. H.
Homeier
and E. O.
Steinborn
, “Some properties of the coupling coefficients of real spherical harmonics and their relation to Gaunt coefficients
,” J. Mol. Struct.: THEOCHEM
368
, 31
–37
(1996
).72.
The CP2K developers group, CP2K is freely available from http://www.cp2k.org/ (accessed
August 2016
).73.
Intel® Xeon® E5–2697v3/DDR 2133.
74.
J.
VandeVondele
and J.
Hutter
, “Gaussian basis sets for accurate calculations on molecular systems in gas and condensed phases
,” J. Chem. Phys.
127
, 114105
(2007
).75.
I. N.
Bronshtein
, K. A.
Semendyayev
, G.
Musiol
, and H.
Mühlig
, Handbook of Mathematics
, 6th ed. (Springer
, 2015
), p. 1100
.76.
M.
Abramowitz
and I. A.
Stegun
, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
, 9th ed. (Dover Publications
, 1972
), p. 256
.77.
G.
Gasper
and M.
Rahman
, in Basic Hypergeometric Series
, 2nd ed., Encyclopedia of Mathematics and its Applications, edited by R. S.
Doran
, P.
Flajolet
, M.
Ismail
, T.-Y.
Lam
, and E.
Lutwak
(Cambridge University Press
, 2004
), p. XIV
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