Density functional theory is the workhorse of chemistry and materials science, and novel density functional approximations are published every year. To become available in program packages, the novel density functional approximations (DFAs) need to be (re)implemented. However, according to our experience as developers of Libxc [Lehtola et al., SoftwareX 7, 1 (2018)], a constant problem in this task is verification due to the lack of reliable reference data. As we discuss in this work, this lack has led to several non-equivalent implementations of functionals such as Becke–Perdew 1986, Perdew–Wang 1991, Perdew–Burke–Ernzerhof, and Becke’s three-parameter hybrid functional with Lee–Yang–Parr correlation across various program packages, yielding different total energies. Through careful verification, we have also found many issues with incorrect functional forms in recent DFAs. The goal of this work is to ensure the reproducibility of DFAs. DFAs must be verifiable in order to prevent the reappearance of the above-mentioned errors and incompatibilities. A common framework for verification and testing is, therefore, needed. We suggest several ways in which reference energies can be produced with free and open source software, either with non-self-consistent calculations with tabulated atomic densities or via self-consistent calculations with various program packages. The employed numerical parameters—especially the quadrature grid—need to be converged to guarantee a ≲0.1 μEh precision in the total energy, which is nowadays routinely achievable in fully numerical calculations. Moreover, as such sub-μEh level agreement can only be achieved when fully equivalent implementations of the DFA are used, the source code of the reference implementation should also be made available in any publication describing a new DFA.
REFERENCES
1.
P.
Hohenberg
and W.
Kohn
, “Inhomogeneous electron gas
,” Phys. Rev.
136
, B864
–B871
(1964
).2.
W.
Kohn
and L. J.
Sham
, “Self-consistent equations including exchange and correlation effects
,” Phys. Rev.
140
, A1133
–A1138
(1965
).3.
W.
Kohn
, “Nobel Lecture: Electronic structure of matter–wave functions and density functionals
,” Rev. Mod. Phys.
71
, 1253
–1266
(1999
).4.
A. D.
Becke
, “Perspective: Fifty years of density-functional theory in chemical physics
,” J. Chem. Phys.
140
, 18A301
(2014
).5.
R. O.
Jones
, “Density functional theory: Its origins, rise to prominence, and future
,” Rev. Mod. Phys.
87
, 897
–923
(2015
).6.
N.
Mardirossian
and M.
Head-Gordon
, “Thirty years of density functional theory in computational chemistry: An overview and extensive assessment of 200 density functionals
,” Mol. Phys.
115
, 2315
–2372
(2017
).7.
A.
Jain
, Y.
Shin
, and K. A.
Persson
, “Computational predictions of energy materials using density functional theory
,” Nat. Rev. Mater.
1
, 15004
(2016
).8.
R. N.
Straker
, Q.
Peng
, A.
Mekareeya
, R. S.
Paton
, and E. A.
Anderson
, “Computational ligand design in enantio- and diastereoselective ynamide [5 + 2] cycloisomerization
,” Nat. Commun.
7
, 10109
(2016
).9.
S.
Lehtola
, C.
Steigemann
, M. J. T.
Oliveira
, and M. A. L.
Marques
, “Recent developments in LIBXC—A comprehensive library of functionals for density functional theory
,” SoftwareX
7
, 1
–5
(2018
).10.
P.
Verma
, Y.
Wang
, S.
Ghosh
, X.
He
, and D. G.
Truhlar
, “Revised M11 exchange-correlation functional for electronic excitation energies and ground-state properties
,” J. Phys. Chem. A
123
, 2966
–2990
(2019
).11.
S.
Jana
, K.
Sharma
, and P.
Samal
, “Improving the performance of Tao–Mo non-empirical density functional with broader applicability in quantum chemistry and materials science
,” J. Phys. Chem. A
123
, 6356
–6369
(2019
).12.
A.
Patra
, S.
Jana
, H.
Myneni
, and P.
Samal
, “Laplacian free and asymptotic corrected semilocal exchange potential applied to the band gap of solids
,” Phys. Chem. Chem. Phys.
21
, 19639
–19650
(2019
).13.
J. W.
Furness
, A. D.
Kaplan
, J.
Ning
, J. P.
Perdew
, and J.
Sun
, “Accurate and numerically efficient r2SCAN meta-generalized gradient approximation
,” J. Phys. Chem. Lett.
11
, 8208
–8215
(2020
).14.
S.
Lehtola
and M. A. L.
Marques
, “Meta-local density functionals: A new rung on Jacob’s ladder
,” J. Chem. Theory Comput.
17
, 943
–948
(2021
).15.
S.
Jana
, S. K.
Behera
, S.
Śmiga
, L. A.
Constantin
, and P.
Samal
, “Improving the applicability of the Pauli kinetic energy density based semilocal functional for solids
,” New J. Phys.
23
, 063007
(2021
).16.
S.
Jana
, S. K.
Behera
, S.
Śmiga
, L. A.
Constantin
, and P.
Samal
, “Accurate density functional made more versatile
,” J. Chem. Phys.
155
, 024103
(2021
).17.
A.
Mitrofanov
, N.
Andreadi
, V.
Korolev
, and S.
Kalmykov
, “A search for a DFT functional for actinide compounds
,” J. Chem. Phys.
155
, 161103
(2021
).18.
H.
Ma
, A.
Narayanaswamy
, P.
Riley
, and L.
Li
, “Evolving symbolic density functionals
,” Sci. Adv.
8
, eabq0279
(2022
).19.
T.
Lebeda
, T.
Aschebrock
, and S.
Kümmel
, “First steps towards achieving both ultranonlocality and a reliable description of electronic binding in a meta-generalized gradient approximation
,” Phys. Rev. Res.
4
, 023061
(2022
).20.
A. D.
Becke
, “Density-functional exchange-energy approximation with correct asymptotic behavior
,” Phys. Rev. A
38
, 3098
–3100
(1988
).21.
J. P.
Perdew
, “Density-functional approximation for the correlation energy of the inhomogeneous electron gas
,” Phys. Rev. B
33
, 8822
–8824
(1986
).22.
J. P.
Perdew
, “Electronic structure of solids ’91
,” in Chapter. Unified theory of exchange and correlation beyond the local density approximation
(Akademie Verlag
, Berlin
, 1991
), pp. 11
–20
.23.
J. P.
Perdew
, J. A.
Chevary
, S. H.
Vosko
, K. A.
Jackson
, M. R.
Pederson
, D. J.
Singh
, and C.
Fiolhais
, “Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation
,” Phys. Rev. B
46
, 6671
–6687
(1992
).24.
J. P.
Perdew
, J. A.
Chevary
, S. H.
Vosko
, K. A.
Jackson
, M. R.
Pederson
, D. J.
Singh
, and C.
Fiolhais
, “Erratum: Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation
,” Phys. Rev. B
48
, 4978
(1993
).25.
J. P.
Perdew
, K.
Burke
, and M.
Ernzerhof
, “Generalized gradient approximation made simple
,” Phys. Rev. Lett.
77
, 3865
–3868
(1996
).26.
J. P.
Perdew
, K.
Burke
, and M.
Ernzerhof
, “Errata: Generalized gradient approximation made simple
,” Phys. Rev. Lett.
78
, 1396
(1997
).27.
P. J.
Stephens
, F. J.
Devlin
, C. F.
Chabalowski
, and M. J.
Frisch
, “Ab initio calculation of vibrational absorption and circular dichroism spectra using density functional force fields
,” J. Phys. Chem.
98
, 11623
–11627
(1994
).28.
A.
Brakestad
, P.
Wind
, S. R.
Jensen
, L.
Frediani
, and K. H.
Hopmann
, “Multiwavelets applied to metal–ligand interactions: Energies free from basis set errors
,” J. Chem. Phys.
154
, 214302
(2021
).29.
F.
Neese
, F.
Wennmohs
, U.
Becker
, and C.
Riplinger
, “The ORCA quantum chemistry program package
,” J. Chem. Phys.
152
, 224108
(2020
).30.
P.
Wind
, M.
Bjørgve
, A.
Brakestad
, G. A.
Gerez S
, S. R.
Jensen
, R. D. R.
Eikås
, and L.
Frediani
, “MRChem multiresolution analysis code for molecular electronic structure calculations: Performance and scaling properties
,” J. Chem. Theory Comput.
19
, 137
–146
(2023
).31.
S.
Lehtola
and A. J.
Karttunen
, “Free and open source software for computational chemistry education
,” WIREs Comput. Mol. Sci.
12
, e1610
(2022
).32.
S.
Lehtola
, “A review on non-relativistic, fully numerical electronic structure calculations on atoms and diatomic molecules
,” Int. J. Quantum Chem.
119
, e25968
(2019
); arXiv:1902.01431.33.
S. R.
Jensen
, T.
Flå
, D.
Jonsson
, R. S.
Monstad
, K.
Ruud
, and L.
Frediani
, “Magnetic properties with multiwavelets and DFT: The complete basis set limit achieved
,” Phys. Chem. Chem. Phys.
18
, 21145
–21161
(2016
).34.
S. R.
Jensen
, S.
Saha
, J. A.
Flores-Livas
, W.
Huhn
, V.
Blum
, S.
Goedecker
, and L.
Frediani
, “The elephant in the room of density functional theory calculations
,” J. Phys. Chem. Lett.
8
, 1449
–1457
(2017
); arXiv:1702.00957.35.
S.
Lehtola
, “Fully numerical Hartree–Fock and density functional calculations. II. Diatomic molecules
,” Int. J. Quantum Chem.
119
, e25944
(2019
); arXiv:1810.11653.36.
S.
Lehtola
, “Fully numerical Hartree–Fock and density functional calculations. I. Atoms
,” Int. J. Quantum Chem.
119
, e25945
(2019
); arXiv:1810.11651.37.
A.
Brakestad
, S. R.
Jensen
, P.
Wind
, M.
D’Alessandro
, L.
Genovese
, K. H.
Hopmann
, and L.
Frediani
, “Static polarizabilities at the basis set limit: A benchmark of 124 species
,” J. Chem. Theory Comput.
16
, 4874
–4882
(2020
).38.
S.
Lehtola
, “Polarized Gaussian basis sets from one-electron ions
,” J. Chem. Phys.
152
, 134108
(2020
); arXiv:2001.04224.39.
S.
Lehtola
, “Meta-GGA density functional calculations on atoms with spherically symmetric densities in the finite element formalism
,” J. Chem. Theory Comput.
19
, 2502
–2517
(2023
); arXiv:2302.06284.40.
S.
Lehtola
and M. A. L.
Marques
, “Many recent density functionals are numerically ill-behaved
,” J. Chem. Phys.
157
, 174114
(2022
).41.
S.
Lehtola
, “Atomic electronic structure calculations with Hermite interpolating polynomials
,” J. Phys. Chem. A
127
, 4180
–4193
(2023
); arXiv:2302.00440.42.
J. P.
Perdew
and K.
Schmidt
, “Jacob’s ladder of density functional approximations for the exchange-correlation energy
,” AIP Conf. Proc.
577
, 1
–20
(2001
).43.
J.-D.
Chai
and M.
Head-Gordon
, “Systematic optimization of long-range corrected hybrid density functionals
,” J. Chem. Phys.
128
, 084106
(2008
).44.
Y.-S.
Lin
, G.-D.
Li
, S.-P.
Mao
, and J.-D.
Chai
, “Long-range corrected hybrid density functionals with improved dispersion corrections
,” J. Chem. Theory Comput.
9
, 263
–272
(2013
); arXiv:1211.0387.45.
Y.
Zhang
, X.
Xu
, and W. A.
Goddard
, “Doubly hybrid density functional for accurate descriptions of nonbond interactions, thermochemistry, and thermochemical kinetics
,” Proc Natl. Acad. Sci.
106
, 4963
–4968
(2009
).46.
A. D.
Becke
, “A multicenter numerical integration scheme for polyatomic molecules
,” J. Chem. Phys.
88
, 2547
–2553
(1988
).47.
R.
Neumann
and N. C.
Handy
, “Investigations using the Becke–Roussel exchange functional
,” Chem. Phys. Lett.
246
, 381
–386
(1995
).48.
E. I.
Proynov
and D. R.
Salahub
, “Simple but efficient correlation functional from a model pair-correlation function
,” Phys. Rev. B
49
, 7874
–7886
(1994
).49.
E. I.
Proynov
and D. R.
Salahub
, “Erratum: Simple but efficient correlation functional from a model pair-correlation function [Phys. Rev. B 49, 7874 (1994)]
,” Phys. Rev. B
57
, 12616
(1998
).50.
E.
Proynov
and J.
Kong
, “Analytic form of the correlation energy of the uniform electron gas
,” Phys. Rev. A
79
, 014103
(2009
).51.
E.
Proynov
and J.
Kong
, “Erratum: Analytic form of the correlation energy of the uniform electron gas [Phys. Rev. A 79, 014103 (2009)]
,” Phys. Rev. A
95
, 059904
(2017
).52.
There’s a missing factor of two for the kinetic energy of the homogeneous electron gas in Ref. 12.
54.
The description of the relPBE0 functional in Ref. 17 omits that the LDA and GGA contributions to correlation need to be scaled by factors of ≈0.390 518 and ≈0.609 48, respectively.
55.
J.
Heyd
, G. E.
Scuseria
, and M.
Ernzerhof
, “Hybrid functionals based on a screened Coulomb potential
,” J. Chem. Phys.
118
, 8207
(2003
).56.
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
).57.
To reproduce the reference implementation of Ref. 19, an additional factor of 22/3 needs to be added in the definition of α.
58.
J.
Perdew
, S.
Kurth
, A.
Zupan
, and P.
Blaha
, “Accurate density functional with correct formal properties: A step beyond the generalized gradient approximation
,” Phys. Rev. Lett.
82
, 2544
–2547
(1999
).59.
M.
Seidl
, J. P.
Perdew
, and S.
Kurth
, “Simulation of all-order density-functional perturbation theory, using the second order and the strong-correlation limit
,” Phys. Rev. Lett.
84
, 5070
–5073
(2000
).60.
P.
Haas
, F.
Tran
, P.
Blaha
, and K.
Schwarz
, “Construction of an optimal GGA functional for molecules and solids
,” Phys. Rev. B
83
, 205117
(2011
).61.
S.
Jana
, B.
Patra
, H.
Myneni
, and P.
Samal
, “On the many-electron self-interaction error of the semilocal exchange hole based meta-GGA level range-separated hybrid with the B88 hybrids
,” Chem. Phys. Lett.
713
, 1
–9
(2018
).62.
S.
Jana
and P.
Samal
, “A meta-GGA level screened range-separated hybrid functional by employing short range Hartree–Fock with a long range semilocal functional
,” Phys. Chem. Chem. Phys.
20
, 8999
–9005
(2018
).63.
The implementation of Ref. 25 employs a more precise value for μ than the one given in the paper.
64.
C.
Adamo
and V.
Barone
, “Exchange functionals with improved long-range behavior and adiabatic connection methods without adjustable parameters: The mPW and mPW1PW models
,” J. Chem. Phys.
108
, 664
–675
(1998
).65.
In commit 4b9609d1c57 of the source code of the NWChem implementation, dated 19 Feb 2003, Edoardo Apra comments that Adamo has confirmed that there is a typo in the JCP paper; b = 0.004 26 instead of 0.0046 given in the text of ref. 64, also the exponent is 3.72 and not 3.73 as given in the manuscript. See https://github.com/nwchemgit/nwchem/blob/master/src/nwdft/xc/xc_xmpw91.F (accessed 1 June 2022).
66.
A. D.
Boese
, N. L.
Doltsinis
, N. C.
Handy
, and M.
Sprik
, “New generalized gradient approximation functionals
,” J. Chem. Phys.
112
, 1670
(2000
).67.
The c5 parameter of HCTH/147 was given with the wrong sign in Boese et al.66 Another set of parameter values are given in Boese, Martin, and Handy 155 that agree with the ones in Ref. 66 for c1–c3 and c9–c15 but have small differences for the c4–c8 coefficients Ref. 155 gives (in addition to the correct sign for c5) one more decimal for c4–c7 than Ref. 66 but also a differently rounded value for c8. The original implementation in CADPAC appears to use still more decimals.
68.
W.-m.
Hoe
, A. J.
Cohen
, and N. C.
Handy
, “Assessment of a new local exchange functional OPTX
,” Chem. Phys. Lett.
341
, 319
–328
(2001
).69.
The parameters of Ref. 68 do not reproduce the data of the paper; see http://www.ccl.net/chemistry/resources/messages/2008/10/09.007-dir/index.html (accessed 1 June 2022).
70.
R.
Peverati
and D. G.
Truhlar
, “Exchange–correlation functional with good accuracy for both structural and energetic properties while depending only on the density and its gradient
,” J. Chem. Theory Comput.
8
, 2310
–2319
(2012
).71.
The same-spin and opposite-spin correlation coefficients are interchanged in Ref. 70.
72.
R.
Peverati
and D. G.
Truhlar
, “Screened-exchange density functionals with broad accuracy for chemistry and solid-state physics
,” Phys. Chem. Chem. Phys.
14
, 16187
–16191
(2012
).73.
The same-spin and opposite-spin correlation coefficients are interchanged in Ref. 72. Moreover, the exchange functional coefficients need to be transposed.
74.
The Jupyter notebook code of Ref. 18 employs many more decimals for the parameters than given in the paper.
75.
This error cancellation is also the reason why slightly different implementations of a DFA still tend to yield similar geometries and excitation energies, for example.
76.
J. M. L.
Martin
, C. W.
Bauschlicher
, and A.
Ricca
, “On the integration accuracy in molecular density functional theory calculations using Gaussian basis sets
,” Comput. Phys. Commun.
133
, 189
–201
(2001
); arXiv:0006069 [physics].77.
B. N.
Papas
and H. F.
Schaefer
, “Concerning the precision of standard density functional programs: Gaussian, Molpro, NWChem, Q-Chem, and Gamess
,” J. Mol. Struct.: THEOCHEM
768
, 175
–181
(2006
).78.
S. E.
Wheeler
and K. N.
Houk
, “Integration grid errors for meta-GGA-predicted reaction energies: Origin of grid errors for the M06 suite of functionals
,” J. Chem. Theory Comput.
6
, 395
–404
(2010
).79.
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
).80.
P.
Giannozzi
, O.
Baseggio
, P.
Bonfà
, D.
Brunato
, R.
Car
, I.
Carnimeo
, C.
Cavazzoni
, S.
de Gironcoli
, P.
Delugas
, F.
Ferrari Ruffino
, A.
Ferretti
, N.
Marzari
, I.
Timrov
, A.
Urru
, and S.
Baroni
, “Quantum ESPRESSO toward the exascale
,” J. Chem. Phys.
152
, 154105
(2020
).81.
J.
Hafner
, “Ab-initio simulations of materials using VASP: Density-functional theory and beyond
,” J. Comput. Chem.
29
, 2044
–2078
(2008
).82.
J.
Heyd
and G. E.
Scuseria
, “Assessment and validation of a screened Coulomb hybrid density functional
,” J. Chem. Phys.
120
, 7274
–7280
(2004
).83.
T. M.
Henderson
, B. G.
Janesko
, and G. E.
Scuseria
, “Generalized gradient approximation model exchange holes for range-separated hybrids
,” J. Chem. Phys.
128
, 194105
(2008
).84.
T. M.
Henderson
, A. F.
Izmaylov
, G.
Scalmani
, and G. E.
Scuseria
, “Can short-range hybrids describe long-range-dependent properties?
,” J. Chem. Phys.
131
, 044108
(2009
).85.
E. H.
Lieb
and S.
Oxford
, “Improved lower bound on the indirect Coulomb energy
,” Int. J. Quantum Chem.
19
, 427
–439
(1981
).86.
U.
Ekström
, L.
Visscher
, R.
Bast
, A. J.
Thorvaldsen
, and K.
Ruud
, “Arbitrary-order density functional response theory from automatic differentiation
,” J. Chem. Theory Comput.
6
, 1971
–1980
(2010
).87.
S.
Lehtola
, F.
Blockhuys
, and C.
Van Alsenoy
, “An overview of self-consistent field calculations within finite basis sets
,” Molecules
25
, 1218
(2020
); arXiv:1912.12029.88.
E.
Clementi
and C.
Roetti
, “Roothaan–Hartree–Fock atomic wavefunctions
,” At. Data Nucl. Data Tables
14
, 177
–478
(1974
).89.
T.
Koga
, K.
Kanayama
, S.
Watanabe
, and A. J.
Thakkar
, “Analytical Hartree–Fock wave functions subject to cusp and asymptotic constraints: He to Xe, Li+ to Cs+, H− to I−
,” Int. J. Quantum Chem.
71
, 491
–497
(1999
).90.
T.
Koga
, K.
Kanayama
, T.
Watanabe
, T.
Imai
, and A. J.
Thakkar
, “Analytical Hartree–Fock wave functions for the atoms Cs to Lr
,” Theor. Chem. Acc.
104
, 411
–413
(2000
).91.
J.
Furness
and S.
Lehtola
, “AtomicOrbitals—A Python module implementing the evaluation of accurate Hartree–Fock orbitals and the resulting electron densities for atoms under spherical symmetry
,” https://github.com/JFurness1/AtomicOrbitals/ (accessed 11 April 2022).92.
S.
Lehtola
, “Curing basis set overcompleteness with pivoted Cholesky decompositions
,” J. Chem. Phys.
151
, 241102
(2019
); arXiv:1911.10372.93.
F.
Jensen
, “Atomic orbital basis sets
,” Wiley Interdiscip. Rev.: Comput. Mol. Sci.
3
, 273
–295
(2013
).94.
J. G.
Hill
, “Gaussian basis sets for molecular applications
,” Int. J. Quantum Chem.
113
, 21
–34
(2013
).95.
W. J.
Hehre
, R. F.
Stewart
, and J. A.
Pople
, “Self-consistent molecular-orbital methods. I. Use of Gaussian expansions of Slater-type atomic orbitals
,” J. Chem. Phys.
51
, 2657
(1969
).96.
Q.
Sun
, X.
Zhang
, S.
Banerjee
, P.
Bao
, M.
Barbry
, N. S.
Blunt
, N. A.
Bogdanov
, G. H.
Booth
, J.
Chen
, Z.-H.
Cui
, J. J.
Eriksen
, Y.
Gao
, S.
Guo
, J.
Hermann
, M. R.
Hermes
, K.
Koh
, P.
Koval
, S.
Lehtola
, Z.
Li
, J.
Liu
, N.
Mardirossian
, J. D.
McClain
, M.
Motta
, B.
Mussard
, H. Q.
Pham
, A.
Pulkin
, W.
Purwanto
, P. J.
Robinson
, E.
Ronca
, E. R.
Sayfutyarova
, M.
Scheurer
, H. F.
Schurkus
, J. E. T.
Smith
, C.
Sun
, S.-N.
Sun
, S.
Upadhyay
, L. K.
Wagner
, X.
Wang
, A.
White
, J. D.
Whitfield
, M. J.
Williamson
, S.
Wouters
, J.
Yang
, J. M.
Yu
, T.
Zhu
, T. C.
Berkelbach
, S.
Sharma
, A. Y.
Sokolov
, and G. K.-L.
Chan
, “Recent developments in the PySCF program package
,” J. Chem. Phys.
153
, 024109
(2020
); arXiv:2002.12531.97.
D. G. A.
Smith
, L. A.
Burns
, A. C.
Simmonett
, R. M.
Parrish
, M. C.
Schieber
, R.
Galvelis
, P.
Kraus
, H.
Kruse
, R.
Di Remigio
, A.
Alenaizan
, A. M.
James
, S.
Lehtola
, J. P.
Misiewicz
, M.
Scheurer
, R. A.
Shaw
, J. B.
Schriber
, Y.
Xie
, Z. L.
Glick
, D. A.
Sirianni
, J. S.
O’Brien
, J. M.
Waldrop
, A.
Kumar
, E. G.
Hohenstein
, B. P.
Pritchard
, B. R.
Brooks
, H. F.
Schaefer
, A. Y.
Sokolov
, K.
Patkowski
, A. E.
DePrince
, U.
Bozkaya
, R. A.
King
, F. A.
Evangelista
, J. M.
Turney
, T. D.
Crawford
, and C. D.
Sherrill
, “PSI4 1.4: Open-source software for high-throughput quantum chemistry
,” J. Chem. Phys.
152
, 184108
(2020
).98.
T. H.
Dunning
, “Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen
,” J. Chem. Phys.
90
, 1007
(1989
).99.
R. C.
Raffenetti
, “General contraction of Gaussian atomic orbitals: Core, valence, polarization, and diffuse basis sets; Molecular integral evaluation
,” J. Chem. Phys.
58
, 4452
–4458
(1973
).100.
T.
Hashimoto
, K.
Hirao
, and H.
Tatewaki
, “Comment on Dunning’s correlation-consistent basis sets
,” Chem. Phys. Lett.
243
, 190
–192
(1995
).101.
E. R.
Davidson
, “Comment on “Comment on Dunning’s correlation-consistent basis sets”
,” Chem. Phys. Lett.
260
, 514
–518
(1996
).102.
M.
Mansoori Kermani
and D. G.
Truhlar
, “Reproducibility of calculations on Li species with correlation-consistent basis sets
,” Chem. Phys. Lett.
825
, 140575
(2023
).103.
B. P.
Pritchard
, D.
Altarawy
, B.
Didier
, T. D.
Gibson
, and T. L.
Windus
, “New basis set exchange: An open, up-to-date resource for the molecular sciences community
,” J. Chem. Inf. Model.
59
, 4814
–4820
(2019
).104.
S.
Lehtola
, “Fully numerical calculations on atoms with fractional occupations and range-separated exchange functionals
,” Phys. Rev. A
101
, 012516
(2020
); arXiv:1908.02528.105.
G.
Scalmani
, Private communication (2022
).106.
J. P.
Perdew
, “Erratum: Density-functional approximation for the correlation energy of the inhomogeneous electron gas
,” Phys. Rev. B
34
, 7406
(1986
).107.
D. C.
Langreth
and M. J.
Mehl
, “Easily implementable nonlocal exchange-correlation energy functional
,” Phys. Rev. Lett.
47
, 446
–450
(1981
).108.
C. D.
Hu
and D. C.
Langreth
, “A spin dependent version of the Langreth–Mehl exchange-correlation functional
,” Phys. Scr.
32
, 391
–396
(1985
).109.
P. M. W.
Gill
, “A new gradient-corrected exchange functional
,” Mol. Phys.
89
, 433
–445
(1996
).110.
We have finally received a copy of Ref. 22 from colleagues abroad, because of an analogous disclaimer in a preprint of this work.
111.
J. P.
Perdew
and Y.
Wang
, “Accurate and simple analytic representation of the electron-gas correlation energy
,” Phys. Rev. B
45
, 13244
–13249
(1992
).112.
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
).113.
A. D.
Becke
, “Density-functional thermochemistry. V. Systematic optimization of exchange-correlation functionals
,” J. Chem. Phys.
107
, 8554
(1997
).114.
R.
Armiento
and A. E.
Mattsson
, “Functional designed to include surface effects in self-consistent density functional theory
,” Phys. Rev. B
72
, 085108
(2005
).115.
A. E.
Mattsson
, R.
Armiento
, J.
Paier
, G.
Kresse
, J. M.
Wills
, and T. R.
Mattsson
, “The AM05 density functional applied to solids
,” J. Chem. Phys.
128
, 084714
(2008
).116.
A. D.
Boese
and J. M. L.
Martin
, “Development of density functionals for thermochemical kinetics
,” J. Chem. Phys.
121
, 3405
(2004
); arXiv:0405158v1 [arXiv:physics].117.
E.
Fabiano
, P. E.
Trevisanutto
, A.
Terentjevs
, and L. A.
Constantin
, “Generalized gradient approximation correlation energy functionals based on the uniform electron gas with gap model
,” J. Chem. Theory Comput.
10
, 2016
–2026
(2014
); arXiv:1404.3484.118.
R.
Peverati
, Y.
Zhao
, and D. G.
Truhlar
, “Generalized gradient approximation that recovers the second-order density-gradient expansion with optimized across-the-board performance
,” J. Phys. Chem. Lett.
2
, 1991
–1997
(2011
).119.
A. D.
Becke
, “Current-density dependent exchange-correlation functionals
,” Can. J. Chem.
74
, 995
–997
(1996
).120.
T.
Schmidt
, E.
Kraisler
, A.
Makmal
, L.
Kronik
, and S.
Kümmel
, “A self-interaction-free local hybrid functional: Accurate binding energies vis-à-vis accurate ionization potentials from Kohn–Sham eigenvalues
,” J. Chem. Phys.
140
, 18A510
(2014
).121.
Y.
Zhao
, N. E.
Schultz
, and D. G.
Truhlar
, “Exchange-correlation functional with broad accuracy for metallic and nonmetallic compounds, kinetics, and noncovalent interactions
,” J. Chem. Phys.
123
, 161103
(2005
).122.
K.
Pernal
, R.
Podeszwa
, K.
Patkowski
, and K.
Szalewicz
, “Dispersionless density functional theory
,” Phys. Rev. Lett.
103
, 263201
(2009
).123.
Y.
Zhao
and D. G.
Truhlar
, “Exploring the limit of accuracy of the global hybrid meta density functional for main-group thermochemistry, kinetics, and noncovalent interactions
,” J. Chem. Theory Comput.
4
, 1849
–1868
(2008
).124.
R.
Peverati
and D. G.
Truhlar
, “Improving the accuracy of hybrid meta-GGA density functionals by range separation
,” J. Phys. Chem. Lett.
2
, 2810
–2817
(2011
).125.
R.
Peverati
and D. G.
Truhlar
, “M11-L: A local density functional that provides improved accuracy for electronic structure calculations in chemistry and physics
,” J. Phys. Chem. Lett.
3
, 117
–124
(2012
).126.
R.
Peverati
and D. G.
Truhlar
, “An improved and broadly accurate local approximation to the exchange-correlation density functional: The MN12-L functional for electronic structure calculations in chemistry and physics
,” Phys. Chem. Chem. Phys.
14
, 13171
(2012
).127.
H. S.
Yu
, X.
He
, S. L.
Li
, and D. G.
Truhlar
, “MN15: A Kohn–Sham global-hybrid exchange–correlation density functional with broad accuracy for multi-reference and single-reference systems and noncovalent interactions
,” Chem. Sci.
7
, 5032
–5051
(2016
).128.
H. S.
Yu
, X.
He
, and D. G.
Truhlar
, “MN15-L: A new local exchange-correlation functional for Kohn–Sham density functional theory with broad accuracy for atoms, molecules, and solids
,” J. Chem. Theory Comput.
12
, 1280
–1293
(2016
).129.
T.
Van Voorhis
and G. E.
Scuseria
, “A novel form for the exchange-correlation energy functional
,” J. Chem. Phys.
109
, 400
(1998
).130.
A. D.
Becke
, “A new inhomogeneity parameter in density-functional theory
,” J. Chem. Phys.
109
, 2092
–2098
(1998
).131.
A. C.
Cancio
and M. Y.
Chou
, “Beyond the local approximation to exchange and correlation: The role of the Laplacian of the density in the energy density of Si
,” Phys. Rev. B
74
, 081202
(2006
).132.
J.
Tao
, J. P.
Perdew
, V. N.
Staroverov
, and G. E.
Scuseria
, “Climbing the density functional ladder: Nonempirical meta-generalized gradient approximation designed for molecules and solids
,” Phys. Rev. Lett.
91
, 146401
(2003
).133.
J. P.
Perdew
, J.
Tao
, V. N.
Staroverov
, and G. E.
Scuseria
, “Meta-generalized gradient approximation: Explanation of a realistic nonempirical density functional
,” J. Chem. Phys.
120
, 6898
–6911
(2004
).134.
J.
Sun
, A.
Ruzsinszky
, and J. P.
Perdew
, “Strongly constrained and appropriately normed semilocal density functional
,” Phys. Rev. Lett.
115
, 036402
(2015
).135.
N.
Mardirossian
and M.
Head-Gordon
, “ωB97M-V: A combinatorially optimized, range-separated hybrid, meta-GGA density functional with VV10 nonlocal correlation
,” J. Chem. Phys.
144
, 214110
(2016
).136.
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
).137.
J.
Lehtola
, M.
Hakala
, A.
Sakko
, and K.
Hämäläinen
, “ERKALE-A flexible program package for X-ray properties of atoms and molecules
,” J. Comput. Chem.
33
, 1572
–1585
(2012
).138.
See https://gitlab.com/libxc/libxc/-/issues/419 (accessed 11 July 2023).
139.
A. D.
Becke
, “Density-functional thermochemistry. III. The role of exact exchange
,” J. Chem. Phys.
98
, 5648
–5652
(1993
).140.
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
).141.
Michael Frisch’s email reply to Mikael Johansson’s question on the Computational Chemistry List, see http://www.ccl.net/chemistry/resources/messages/2002/05/22.008-dir/. Accessed 26 April 2022.
142.
R. H.
Hertwig
and W.
Koch
, “On the parameterization of the local correlation functional. what is Becke-3-LYP?
,” Chem. Phys. Lett.
268
, 345
–351
(1997
).143.
F.
Bloch
, “Bemerkung zur Elektronentheorie des Ferromagnetismus und der elektrischen Leitfähigkeit
,” Z. Phys.
57
, 545
–555
(1929
).144.
P. A. M.
Dirac
, “Note on exchange phenomena in the Thomas atom
,” Math. Proc. Cambridge Philos. Soc.
26
, 376
–385
(1930
).145.
R. A.
King
, J. M.
Galbraith
, and H. F.
Schaefer
, “Negative ion thermochemistry: The sulfur fluorides SFn/ (n= 1–7)
,” J. Phys. Chem.
100
, 6061
–6068
(1996
).146.
R. A.
King
, V. S.
Mastryukov
, and H. F.
Schaefer
, “The electron affinities of the silicon fluorides SiFn (n = 1–5)
,” J. Chem. Phys.
105
, 6880
–6886
(1996
).147.
R. A.
King
, N. D.
Pettigrew
, and H. F.
Schaefer
, “The electron affinities of the perfluorocarbons C2Fn, n =1–6
,” J. Chem. Phys.
107
, 8536
–8544
(1997
).148.
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
, A.
Hellweg
, B.
Helmich-Paris
, C.
Holzer
, U.
Huniar
, M.
Kaupp
, A.
Marefat Khah
, S.
Karbalaei Khani
, T.
Müller
, F.
Mack
, B. D.
Nguyen
, S. M.
Parker
, E.
Perlt
, D.
Rappoport
, K.
Reiter
, S.
Roy
, M.
Rückert
, G.
Schmitz
, M.
Sierka
, E.
Tapavicza
, D. P.
Tew
, C.
van Wüllen
, V. K.
Voora
, F.
Weigend
, A.
Wodyński
, and J. M.
Yu
, “TURBOMOLE: Modular program suite for ab initio quantum-chemical and condensed-matter simulations
,” J. Chem. Phys.
152
, 184107
(2020
).149.
O.
Treutler
and R.
Ahlrichs
, “Efficient molecular numerical integration schemes
,” J. Chem. Phys.
102
, 346
(1995
).150.
J.
Sun
, B.
Xiao
, and A.
Ruzsinszky
, “Communication: Effect of the orbital-overlap dependence in the meta generalized gradient approximation
,” J. Chem. Phys.
137
, 051101
(2012
); arXiv:1203.2308v1.151.
J.
Sun
, J. P.
Perdew
, and A.
Ruzsinszky
, “Semilocal density functional obeying a strongly tightened bound for exchange
,” Proc. Natl. Acad. Sci. U. S. A.
112
, 685
–689
(2015
).152.
J. W.
Furness
, A. D.
Kaplan
, J.
Ning
, J. P.
Perdew
, and J.
Sun
, “Correction to ”Accurate and numerically efficient r2SCAN meta-generalized gradient approximation
,” J. Phys. Chem. Lett.
11
, 9248
(2020
).153.
V. I.
Lebedev
, “Quadratures on a sphere
,” USSR Comput. Math. Math. Phys.
16
, 10
–24
(1976
).154.
A. P.
Bartók
and J. R.
Yates
, “Regularized SCAN functional
,” J. Chem. Phys.
150
, 161101
(2019
); arXiv:1903.01007.155.
A. D.
Boese
, J. M. L.
Martin
, and N. C.
Handy
, “The role of the basis set: Assessing density functional theory
,” J. Chem. Phys.
119
, 3005
(2003
); arXiv:0305060 [physics].© 2023 Author(s). Published under an exclusive license by AIP Publishing.
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