A seminumerical algorithm capable of performing large-scale (time-dependent) density functional theory (TD-DFT) calculations to extract excitation energies and other ground-state and excited-state properties is outlined. The algorithm uses seminumerical integral techniques for evaluating Coulomb and exchange parts for a set of density matrices as occurring in standard TD-DFT or similar methods for the evaluation of vibrational frequencies. A suitable optimized de-aliasing procedure is introduced. The latter does not depend on further auxiliary quantities and retains the symmetry of a given density matrix. The algorithm is self-contained and applicable to any orbital basis set available without the need for further auxiliary basis sets or optimized de-aliasing grids. Relativistic two-component excited-state TD-DFT calculations are reported for the first time using the developed seminumerical algorithm for standard and local hybrid density functional approximations. Errors are compared with the widely used “resolution of the identity” (RI) approximations for Coulomb (RI-J) and exchange integrals (RI-K). The fully seminumerical algorithm does not exhibit an enlarged error for standard DFT functionals compared to the RI approximation. For the more involved local hybrid functionals and within strong external fields, accuracy is even considerably improved.
REFERENCES
1.
R. A.
Friesner
, “Solution of self-consistent field electronic structure equations by a pseudospectral method
,” Chem. Phys. Lett.
116
, 39
–43
(1985
).2.
R. A.
Friesner
, “Solution of the Hartree–Fock equations by a pseudospectral method: Application to diatomic molecules
,” J. Chem. Phys.
85
, 1462
–1468
(1986
).3.
R. A.
Friesner
, “Solution of the Hartree–Fock equations for polyatomic molecules by a pseudospectral method
,” J. Chem. Phys.
86
, 3522
–3531
(1987
).4.
M. N.
Ringnalda
, M.
Belhadj
, and R. A.
Friesner
, “Pseudospectral Hartree–Fock theory: Applications and algorithmic improvements
,” J. Chem. Phys.
93
, 3397
–3407
(1990
).5.
F.
Neese
, F.
Wennmohs
, A.
Hansen
, and U.
Becker
, “Efficient, approximate and parallel Hartree–Fock and hybrid DFT calculations. A chain-of-spheres algorithm for the Hartree–Fock exchange
,” Chem. Phys.
356
, 98
–109
(2009
).6.
P.
Plessow
and F.
Weigend
, “Seminumerical calculation of the Hartree–Fock exchange matrix: Application to two-component procedures and efficient evaluation of local hybrid density functionals
,” J. Comput. Chem.
33
, 810
–816
(2012
).7.
H.
Laqua
, J.
Kussmann
, and C.
Ochsenfeld
, “Efficient and linear-scaling seminumerical method for local hybrid density functionals
,” J. Chem. Theory Comput.
14
, 3451
–3458
(2018
).8.
H.
Laqua
, T. H.
Thompson
, J.
Kussmann
, and C.
Ochsenfeld
, “Highly efficient, linear-scaling seminumerical exact-exchange method for graphic processing units
,” J. Comput. Theory Comput.
16
, 1456
–1468
(2020
).9.
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
).10.
O.
Vahtras
, J.
Almlöf
, and M. W.
Feyereisen
, “Integral approximations for LCAO-SCF calculations
,” Chem. Phys. Lett.
213
, 514
–518
(1993
).11.
K.
Eichkorn
, O.
Treutler
, H.
Öhm
, M.
Häser
, and R.
Ahlrichs
, “Auxiliary basis sets to approximate Coulomb potentials
,” Chem. Phys. Lett.
240
, 283
–290
(1995
).12.
F.
Weigend
, M.
Kattannek
, and R.
Ahlrichs
, “Approximated electron repulsion integrals: Cholesky decomposition versus resolution of the identity methods
,” J. Chem. Phys.
130
, 164106
(2009
).13.
V.
Termath
and N. C.
Handy
, “A Kohn-Sham method involving the direct determination of the Coulomb potential on a numerical grid
,” Chem. Phys. Lett.
230
, 17
–24
(1994
).14.
C.
van Wüllen
, “A hybrid method for the evaluation of the matrix elements of the Coulomb potential
,” Chem. Phys. Lett.
245
, 648
–652
(1995
).15.
T.
Nakajima
and K.
Hirao
, “Pseudospectral approach to relativistic molecular theory
,” J. Chem. Phys.
121
, 3438
–3445
(2004
).16.
M.
Sierka
, A.
Hogekamp
, and R.
Ahlrichs
, “Fast evaluation of the Coulomb potential for electron densities using multipole accelerated resolution of identity approximation
,” J. Chem. Phys.
118
, 9136
–9148
(2003
).17.
K.
Eichkorn
, F.
Weigend
, O.
Treutler
, and R.
Ahlrichs
, “Auxiliary basis sets for main row atoms and transition metals and their use to approximate Coulomb potentials
,” Theor. Chem. Acc.
97
, 119
–124
(1997
).18.
F.
Weigend
, “Accurate Coulomb-fitting basis sets for H to Rn
,” Phys. Chem. Chem. Phys.
8
, 1057
–1065
(2006
).19.
P.
Pollak
and F.
Weigend
, “Segmented contracted error-consistent basis sets of double-and triple-ζ valence quality for one-and two-component relativistic all-electron calculations
,” J. Chem. Theory Comput.
13
, 3696
–3705
(2017
).20.
Y.
Cao
, T.
Hughes
, D.
Giesen
, M. D.
Halls
, A.
Goldberg
, T. R.
Vadicherla
, M.
Sastry
, B.
Patel
, W.
Sherman
, A. L.
Weisman
, and R. A.
Friesner
, “Highly efficient implementation of pseudospectral time-dependent density-functional theory for the calculation of excitation energies of large molecules
,” J. Comput. Chem.
37
, 1425
–1441
(2016
).21.
D.
Rappoport
and F.
Furche
, “Property-optimized Gaussian basis sets for molecular response calculations
,” J. Chem. Phys.
133
, 134105
(2010
).22.
A.
Hellweg
and D.
Rappoport
, “Development of new auxiliary basis functions of the Karlsruhe segmented contracted basis sets including diffuse basis functions (def2-SVPD, def2-TZVPPD, and def2-QZVPPD) for RI-MP2 and RI-CC calculations
,” Phys. Chem. Chem. Phys.
17
, 1010
–1017
(2015
).23.
G. L.
Stoychev
, A. A.
Auer
, and F.
Neese
, “Automatic generation of auxiliary basis sets
,” J. Chem. Theory Comput.
13
, 554
–562
(2017
).24.
T. J.
Martinez
and E. A.
Carter
, “Pseudospectral Møller–Plesset perturbation theory through third order
,” J. Chem. Phys.
100
, 3631
–3638
(1994
).25.
R. B.
Murphy
, M. D.
Beachy
, R. A.
Friesner
, and M. N.
Ringnalda
, “Pseudospectral localized Møller–Plesset methods: Theory and calculation of conformational energies
,” J. Chem. Phys.
103
, 1481
–1490
(1995
).26.
T. J.
Martinez
and E. A.
Carter
, “Pseudospectral methods applied to the electron correlation problem
,” Mod. Electron. Struct. Theory
2
, 1132
–1165
(1995
).27.
C.
Ko
, D. K.
Malick
, D. A.
Braden
, R. A.
Friesner
, and T. J.
Martínez
, “Pseudospectral time-dependent density functional theory
,” J. Chem. Phys.
128
, 104103
(2008
).28.
T.
Petrenko
, S.
Kossmann
, and F.
Neese
, “Efficient time-dependent density functional theory approximations for hybrid density functionals: Analytical gradients and parallelization
,” J. Chem. Phys.
134
, 054116
(2011
).29.
J.
Jaramillo
, G. E.
Scuseria
, and M.
Ernzerhof
, “Local hybrid functionals
,” J. Chem. Phys.
118
, 1068
–1073
(2003
).30.
T. M.
Maier
, H.
Bahmann
, and M.
Kaupp
, “Efficient semi-numerical implementation of global and local hybrid functionals for time-dependent density functional theory
,” J. Chem. Theory Comput.
11
, 4226
–4237
(2015
).31.
M.
Kehry
, Y. J.
Franzke
, C.
Holzer
, and W.
Klopper
, “Quasirelativistic two-component core excitations and polarisabilities from a damped-response formulation of the Bethe–Salpeter equation
,” Mol. Phys.
1
–16
(published online, 2020).32.
J.
Gao
, W.
Zou
, W.
Liu
, Y.
Xiao
, D.
Peng
, B.
Song
, and C.
Liu
, “Time-dependent four-component relativistic density-functional theory for excitation energies. II. The exchange-correlation kernel
,” J. Chem. Phys.
123
, 054102
(2005
).33.
F.
Wang
, T.
Ziegler
, E.
van Lenthe
, S.
van Gisbergen
, and E. J.
Baerends
, “The calculation of excitation energies based on the relativistic two-component zeroth-order regular approximation and time-dependent density-functional with full use of symmetry
,” J. Chem. Phys.
122
, 204103
(2005
).34.
D.
Peng
, W.
Zou
, and W.
Liu
, “Time-dependent quasirelativistic density-functional theory based on the zeroth-order regular approximation
,” J. Chem. Phys.
123
, 144101
(2005
).35.
R.
Bast
, H. J. A.
Jensen
, and T.
Saue
, “Relativistic adiabatic time-dependent density functional theory using hybrid functionals and noncollinear spin magnetization
,” Int. J. Quantum Chem.
109
, 2091
–2112
(2009
).36.
M.
Kühn
and F.
Weigend
, “Implementation of two-component time-dependent density functional theory in turbomole
,” J. Chem. Theory Comput.
9
, 5341
–5348
(2013
).37.
F.
Egidi
, S.
Sun
, J. J.
Goings
, G.
Scalmani
, M. J.
Frisch
, and X.
Li
, “Two-component noncollinear time-dependent spin density functional theory for excited state calculations
,” J. Chem. Theory Comput.
13
, 2591
–2603
(2017
).38.
C.
Holzer
and W.
Klopper
, “Ionized, electron-attached, and excited states of molecular systems with spin–orbit coupling: Two-component GW and Bethe–Salpeter implementations
,” J. Chem. Phys.
150
, 204116
(2019
).39.
R.
Bauernschmitt
and R.
Ahlrichs
, “Treatment of electronic excitations within the adiabatic approximation of time dependent density functional theory
,” Chem. Phys. Lett.
256
, 454
–464
(1996
).40.
R. E.
Stratmann
, G. E.
Scuseria
, and M. J.
Frisch
, “An efficient implementation of time-dependent density-functional theory for the calculation of excitation energies of large molecules
,” J. Chem. Phys.
109
, 8218
–8224
(1998
).41.
O.
Treutler
and R.
Ahlrichs
, “Efficient molecular numerical integration schemes
,” J. Chem. Phys.
102
, 346
–354
(1995
).42.
J. E.
Bates
and F.
Furche
, “Harnessing the meta-generalized gradient approximation for time-dependent density functional theory
,” J. Chem. Phys.
137
, 164105
(2012
).43.
P.
Salek
, T.
Helgaker
, and T.
Saue
, “Linear response at the 4-component relativistic density-functional level: Application to the frequency-dependent dipole polarizability of Hg, AuH and PtH2
,” Chem. Phys.
311
, 187
–201
(2005
).44.
F.
Neese
, “Prediction of molecular properties and molecular spectroscopy with density functional theory: From fundamental theory to exchange-coupling
,” Coord. Chem. Rev.
253
, 526
–563
(2009
).45.
M.
Häser
and R.
Ahlrichs
, “Improvements on the direct SCF method
,” J. Comput. Chem.
10
, 104
–111
(1989
).46.
H.
Horn
, H.
Weiß
, M.
Háser
, M.
Ehrig
, and R.
Ahlrichs
, “Prescreening of two-electron integral derivatives in SCF gradient and Hessian calculations
,” J. Comput. Chem.
12
, 1058
–1064
(1991
).47.
F.
Furche
, B. T.
Krull
, B. D.
Nguyen
, and J.
Kwon
, “Accelerating molecular property calculations with nonorthonormal Krylov space methods
,” J. Chem. Phys.
144
, 174105
(2016
).48.
R. M.
Parrish
, E. G.
Hohenstein
, and T. J.
Martínez
, “Balancing the block Davidson–Liu algorithm
,” J. Chem. Theory Comput.
12
, 3003
–3007
(2016
).49.
T. H.
Thompson
and C.
Ochsenfeld
, “Integral partition bounds for fast and effective screening of general one-, two-, and many-electron integrals
,” J. Chem. Phys.
150
, 044101
(2019
).50.
F.
Jensen
, “The basis set convergence of spin-spin coupling constants calculated by density functional methods
,” J. Chem. Theory Comput.
2
, 1360
–1369
(2006
).51.
F.
Jensen
, “Basis set convergence of nuclear magnetic shielding constants calculated by density functional methods
,” J. Chem. Theory Comput.
4
, 719
–727
(2008
).52.
F.
Jensen
, “Segmented contracted basis sets optimized for nuclear magnetic shielding
,” J. Chem. Theory Comput.
11
, 132
–138
(2015
).53.
Y. J.
Franzke
, R.
Treß
, T. M.
Pazdera
, and F.
Weigend
, “Error-consistent segmented contracted all-electron relativistic basis sets of double- and triple-zeta quality for NMR shielding constants
,” Phys. Chem. Chem. Phys.
21
, 16658
–16664
(2019
).54.
Y. J.
Franzke
, L.
Spiske
, P.
Pollak
, and F.
Weigend
, “Segmented contracted error-consistent basis sets of quadruple-valence quality for one- and two-component relativistic all-electron calculations
,” J. Chem. Theory Comput.
16
, 5658
–5674
(2020
).55.
R.
Izsák
and F.
Neese
, “An overlap fitted chain of spheres exchange method
,” J. Chem. Phys.
135
, 144105
(2011
).56.
R.
Izsák
, F.
Neese
, and W.
Klopper
, “Robust fitting techniques in the chain of spheres approximation to the Fock exchange: The role of the complementary space
,” J. Chem. Phys.
139
, 094111
(2013
).57.
A.
Yachmenev
and W.
Klopper
, “A composite density-fitting+numerical integration-approximation for electron-repulsion integrals
,” Mol. Phys.
111
, 1129
–1142
(2013
).58.
J. P.
Perdew
, K.
Burke
, and M.
Ernzerhof
, “Generalized gradient approximation made simple
,” Phys. Rev. Lett.
77
, 3865
–3868
(1996
).59.
J. P.
Perdew
, K.
Burke
, and M.
Ernzerhof
, “Erratum: Generalized gradient approximation made simple [Phys. Rev. Lett. 77, 3865 (1996)]
,” Phys. Rev. Lett.
78
, 1396
(1997
).60.
C.
Adamo
and V.
Barone
, “Toward reliable density functional methods without adjustable parameters: The PBE0 model
,” J. Chem. Phys.
110
, 6158
–6170
(1999
).61.
J.-D.
Chai
and M.
Head-Gordon
, “Long-range corrected hybrid density functionals with damped atom–atom dispersion corrections
,” Phys. Chem. Chem. Phys.
10
, 6615
–6620
(2008
).62.
A. V.
Arbuznikov
and M.
Kaupp
, “Importance of the correlation contribution for local hybrid functionals: Range separation and self-interaction corrections
,” J. Chem. Phys.
136
, 014111
(2012
).63.
F.
Weigend
and R.
Ahlrichs
, “Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy
,” Phys. Chem. Chem. Phys.
7
, 3297
–3305
(2005
).64.
F.
Weigend
and A.
Baldes
, “Segmented contracted basis sets for one-and two-component Dirac–Fock effective core potentials
,” J. Chem. Phys.
133
, 174102
(2010
).65.
M.
Schreiber
, M. R.
Silva-Junior
, S. P. A.
Sauer
, and W.
Thiel
, “Benchmarks for electronically excited states: CASPT2, CC2, CCSD, and CC3
,” J. Chem. Phys.
128
, 134110
(2008
).66.
P.
Deglmann
, F.
Furche
, and R.
Ahlrichs
, “An efficient implementation of second analytical derivatives for density functional methods
,” Chem. Phys. Lett.
362
, 511
–518
(2002
).67.
Developer’s version of TURBOMOLE V7.5 2020, a development of University of Karlsruhe and Forschungszentrum Karlsruhe GmbH 1989–2007, TURBOMOLE GmbH, since 2007; available from https://www.turbomole.org, retrieved July 16, 2020.
68.
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
).69.
S. G.
Balasubramani
, G. P.
Chen
, S.
Coriani
, M.
Diedenhofen
, M. S.
Frank
, Y. J.
Franzke
, F.
Furche
, R.
Grotjahn
, M. E.
Harding
, C.
Hättig
etal., “Turbomole: Modular program suite for ab initio quantum-chemical and condensed-matter simulations
,” J. Chem. Phys.
152
, 184107
(2020
).70.
R.
Bauernschmitt
, M.
Häser
, O.
Treutler
, and R.
Ahlrichs
, “Calculation of excitation energies within time-dependent density functional theory using auxiliary basis set expansions
,” Chem. Phys. Lett.
264
, 573
–578
(1997
).71.
P.
Deglmann
, K.
May
, F.
Furche
, and R.
Ahlrichs
, “Nuclear second analytical derivative calculations using auxiliary basis set expansions
,” Chem. Phys. Lett.
384
, 103
–107
(2004
).72.
F.
Weigend
, M.
Häser
, H.
Patzelt
, and R.
Ahlrichs
, “RI-MP2: Optimized auxiliary basis sets and demonstration of efficiency
,” Chem. Phys. Lett.
294
, 143
–152
(1998
).73.
C.
Hättig
, “Optimization of auxiliary basis sets for RI-MP2 and RI-CC2 calculations: Core–valence and quintuple-ζ basis sets for H to Ar and QZVPP basis sets for Li to Kr
,” Phys. Chem. Chem. Phys.
7
, 59
–66
(2005
).74.
A.
Hellweg
, C.
Hättig
, S.
Höfener
, and W.
Klopper
, “Optimized accurate auxiliary basis sets for RI-MP2 and RI-CC2 calculations for the atoms Rb to Rn
,” Theor. Chem. Acc.
117
, 587
–597
(2007
).75.
C.
Yu
, W.
Harbich
, L.
Sementa
, L.
Ghiringhelli
, E.
Aprá
, M.
Stener
, A.
Fortunelli
, and H.
Brune
, “Intense fluorescence of Au20
,” J. Chem. Phys.
147
, 074301
(2017
).76.
D.-e.
Jiang
, M.
Kühn
, Q.
Tang
, and F.
Weigend
, “Superatomic orbitals under spin–orbit coupling
,” J. Chem. Phys. Lett.
5
, 3286
–3289
(2014
).77.
M. W.
Heaven
, A.
Dass
, P. S.
White
, K. M.
Holt
, and R. W.
Murray
, “Crystal structure of the gold nanoparticle .
,” J. Am. Chem. Soc.
130
, 3754
–3755
(2008
).78.
N.
Asadi-Aghbolaghi
, R.
Rüger
, Z.
Jamshidi
, and L.
Visscher
, “TD-DFT+TB: An efficient and fast approach for quantum plasmonic excitations
,” J. Phys. Chem. C
124
, 7946
–7955
(2020
).79.
H.
Braunschweig
, M. A.
Celik
, K.
Dück
, F.
Hupp
, and I.
Krummenacher
, “f-Block ansa complexes in the solid state: [3]Thoro- and [3]uranocenophanes
,” Chem. Eur. J.
21
, 9339
–9342
(2015
).80.
M. W.
Hussong
, F.
Rominger
, P.
Krämer
, and B. F.
Straub
, “Isolation of a non-heteroatom-stabilized gold–carbene complex
,” Angew. Chem., Int. Ed.
53
, 9372
–9375
(2014
).81.
F.
Weigend
, “A fully direct RI-HF algorithm: Implementation, optimised auxiliary basis sets, demonstration of accuracy and efficiency
,” Phys. Chem. Chem. Phys.
4
, 4285
–4291
(2002
).82.
K.
Krause
and W.
Klopper
, “Implementation of the Bethe–Salpeter equation in the turbomole program
,” J. Comput. Chem.
38
, 383
–388
(2017
).83.
F.
Weigend
, A.
Köhn
, and C.
Hättig
, “Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations
,” J. Chem. Phys.
116
, 3175
–3183
(2002
).84.
C.
Hättig
, G.
Schmitz
, and J.
Koßmann
, “Auxiliary basis sets for density-fitted correlated wavefunction calculations: Weighted core-valence and ECP basis sets for post-d elements
,” Phys. Chem. Chem. Phys.
14
, 6549
–6555
(2012
).85.
J.
Chmela
and M. E.
Harding
, “Optimized auxiliary basis sets for density fitted post-Hartree–Fock calculations of lanthanide containing molecules
,” Mol. Phys.
116
, 1523
–1538
(2018
).86.
F.
Weigend
, “Hartree–Fock exchange fitting basis sets for H to Rn
,” J. Comput. Chem.
29
, 167
–175
(2008
).87.
T. M.
Maier
, H.
Bahmann
, A. V.
Arbuznikov
, and M.
Kaupp
, “Validation of local hybrid functionals for TDDFT calculations of electronic excitation energies
,” J. Chem. Phys.
144
, 074106
(2016
).88.
A.
Pausch
and W.
Klopper
, “Efficient evaluation of three-centre two-electron integrals over london orbitals
,” Mol. Phys.
e1736675
(published online, 2020).89.
K. G.
Dyall
, “Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the actinides Ac–Lr
,” Theor. Chem. Acc.
117
, 491
–500
(2007
).90.
D.
Peng
and M.
Reiher
, “Local relativistic exact decoupling
,” J. Chem. Phys.
136
, 244108
(2012
).91.
D.
Peng
, N.
Middendorf
, F.
Weigend
, and M.
Reiher
, “An efficient implementation of two-component relativistic exact-decoupling methods for large molecules
,” J. Chem. Phys.
138
, 184105
(2013
).92.
Y. J.
Franzke
, N.
Middendorf
, and F.
Weigend
, “Efficient implementation of one-and two-component analytical energy gradients in exact two-component theory
,” J. Chem. Phys.
148
, 104110
(2018
).93.
Y. J.
Franzke
and F.
Weigend
, “NMR shielding tensors and chemical shifts in scalar-relativistic local exact two-component theory
,” J. Chem. Theory Comput.
15
, 1028
–1043
(2019
).94.
R.
Ahlrichs
, S. D.
Elliott
, and U.
Huniar
, “Quantum chemistry: Large molecules—Small computers
,” Ber. Bunsenges. Phys. Chem.
102
, 795
–804
(1998
).95.
F.
Furche
and R.
Ahlrichs
, “Adiabatic time-dependent density functional methods for excited state properties
,” J. Chem. Phys.
117
, 7433
–7447
(2002
).96.
T.
Helgaker
, P.
Jørgensen
, and N. C.
Handy
, “A numerically stable procedure for calculating Møller-Plesset energy derivatives, derived using the theory of Lagrangians
,” Theor. Chim. Acta
76
, 227
–245
(1989
).© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.
