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.

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
.

Supplementary Material

You do not currently have access to this content.