A property of exact density functional theory is linear fractional charge behavior as electrons are added or removed from a molecule. Typical density functional approximations (DFAs) exhibit delocalization error, which overstabilizes this fractional charge. Conversely, solvent corrections have been shown to erroneously destabilize this fractional charge. This work will show that an implicit solvent correction with a tuned dielectric can be used as an ad hoc correction to offset the delocalizing character of DFAs and achieve linear fractional charge behavior. While desirable, in principle, we find that this linear charge behavior degrades the vertical ionization energies reported by DFAs. Our results reveal that the localizing character of the solvent correction and the Hartree–Fock (HF) exchange offset each other. This helps explain the decreased ratios of HF exchange to DFA exchange in long-range hybrid tuning studies that use a solvent correction.

1.
W.
Kohn
and
L. J.
Sham
, “
Self-consistent equations including exchange and correlation effects
,”
Phys. Rev.
140
,
A1133
(
1965
).
2.
J.
Sun
,
A.
Ruzsinszky
, and
J. P.
Perdew
, “
Strongly constrained and appropriately normed semilocal density functional
,”
Phys. Rev. Lett.
115
,
036402
(
2015
).
3.
W. J.
Hehre
,
R.
Ditchfield
,
L.
Radom
, and
J. A.
Pople
, “
Molecular orbital theory of the electronic structure of organic compounds. V. Molecular theory of bond separation
,”
J. Am. Chem. Soc.
92
,
4796
4801
(
1970
).
4.
S. E.
Wheeler
,
K. N.
Houk
,
P. v. R.
Schleyer
, and
W. D.
Allen
, “
A hierarchy of homodesmotic reactions for thermochemistry
,”
J. Am. Chem. Soc.
131
,
2547
2560
(
2009
).
5.
A. D.
Becke
, “
A new mixing of Hartree–Fock and local density-functional theories
,”
J. Chem. Phys.
98
,
1372
(
1993
).
6.
A.
Savin
, in
Recent Developments and Applications of Modern Density Functional Theory
, edited by
J. M.
Seminario
(
Elsevier
,
Amsterdam
,
1996
).
7.
P. M. W.
Gill
,
R. D.
Adamson
, and
J. A.
Pople
, “
Coulomb-attenuated exchange energy density functionals
,”
Mol. Phys.
88
,
1005
1009
(
1996
).
8.
H.
Iikura
,
T.
Tsuneda
,
T.
Yanai
, and
K.
Hirao
, “
A long-range correction scheme for generalized-gradient-approximation exchange functionals
,”
J. Chem. Phys.
115
,
3540
3544
(
2001
).
9.
S.
Grimme
, “
Semiempirical hybrid density functional with perturbative second-order correlation
,”
J. Chem. Phys.
124
,
034108
(
2006
).
10.
A. I.
Liechtenstein
,
V. I.
Anisimov
, and
J.
Zaanen
, “
Density-functional theory and strong interactions: Orbital ordering in Mott-Hubbard insulators
,”
Phys. Rev. B
52
,
R5467
(
1995
).
11.
S. L.
Dudarev
,
G. A.
Botton
,
S. Y.
Savrasov
,
C. J.
Humphreys
, and
A. P.
Sutton
, “
Electron-energy-loss spectra and the structural stability of nickel oxide: An LSDA + U study
,”
Phys. Rev. B
57
,
1505
(
1998
).
12.
A. D.
Becke
and
E. R.
Johnson
, “
Exchange-hole dipole moment and the dispersion interaction revisited
,”
J. Chem. Phys.
127
,
154108
(
2007
).
13.
J. F.
Janak
, “
Proof that δEδn = ɛi in density-functional theory
,”
Phys. Rev. B
18
,
7165
(
1978
).
14.
J. P.
Perdew
,
R. G.
Parr
,
M.
Levy
, and
J. L.
Balduz
, Jr.
, “
Density-functional theory for fractional particle number: Derivative discontinuities of the energy
,”
Phys. Rev. Lett.
49
,
1691
(
1982
).
15.
W.
Yang
,
Y.
Zhang
, and
P. W.
Ayers
, “
Degenerate ground states and a fractional number of electrons in density and reduced density matrix functional theory
,”
Phys. Rev. Lett.
84
,
5172
(
2000
).
16.
Y.
Zhang
and
W.
Yang
, “
A challenge for density functionals: Self-interaction error increases for systems with a noninteger number of electrons
,”
J. Chem. Phys.
109
,
2604
(
1998
).
17.
P.
Mori-Sánchez
,
A. J.
Cohen
, and
W.
Yang
, “
Many-electron self-interaction error in approximate density functionals
,”
J. Chem. Phys.
125
,
201102
(
2006
).
18.
A. J.
Cohen
,
P.
Mori-Sánchez
, and
W.
Yang
, “
Fractional spins and static correlation error in density functional theory
,”
J. Chem. Phys.
129
,
121104
(
2008
).
19.
A. J.
Cohen
,
P.
Mori-Sánchez
, and
W.
Yang
, “
Insights into current limitations of density functional theory
,”
Science
321
,
792
(
2008
).
20.
X.
Zheng
,
M.
Liu
,
E. R.
Johnson
,
J.
Contreras-García
, and
W.
Yang
, “
Delocalization error of density-functional approximations: A distinct manifestation in hydrogen molecular chains
,”
J. Chem. Phys.
137
,
214106
(
2012
).
21.
E. R.
Johnson
,
A.
Otero-de-la-Roza
, and
S. G.
Dale
, “
Extreme density-driven delocalization error for a model solvated-electron system
,”
J. Chem. Phys.
139
,
184116
(
2013
).
22.
E. R.
Johnson
,
M.
Salamone
,
M.
Bietti
, and
G. A.
DiLabio
, “
Modeling noncovalent radical–molecule interactions using conventional density-functional theory: Beware erroneous charge transfer
,”
J. Phys. Chem. A
117
,
947
952
(
2013
).
23.
A.
Otero-De-La-Roza
,
E. R.
Johnson
, and
G. A.
DiLabio
, “
Halogen bonding from dispersion-corrected density-functional theory: The role of delocalization error
,”
J. Chem. Theory Comput.
10
,
5436
5447
(
2014
).
24.
S. R.
Whittleton
,
X. A.
Sosa Vazquez
,
C. M.
Isborn
, and
E. R.
Johnson
, “
Density-functional errors in ionization potential with increasing system size
,”
J. Chem. Phys.
142
,
184106
(
2015
).
25.
D. R.
Lonsdale
and
L.
Goerigk
, “
The one-electron self-interaction error in 74 density functional approximations: A case study on hydrogenic mono- and dinuclear systems
,”
Phys. Chem. Chem. Phys.
22
,
15805
15830
(
2020
).
26.
J. P.
Perdew
, “
Density functional theory and the band gap problem
,”
Int. J. Quantum Chem.
28
,
497
(
1985
).
27.
L.-J.
Yu
,
S. G.
Dale
,
B.
Chan
, and
A.
Karton
, “
Benchmark study of DFT and composite methods for bond dissociation energies in argon compounds
,”
Chem. Phys.
531
,
110676
(
2020
).
28.
L. M.
LeBlanc
,
S. G.
Dale
,
C. R.
Taylor
,
A. D.
Becke
,
G. M.
Day
, and
E. R.
Johnson
, “
Pervasive delocalisation error causes spurious proton transfer in organic acid–base Co-crystals
,”
Angew. Chem.
130
,
15122
(
2018
).
29.
M.-C.
Kim
,
E.
Sim
, and
K.
Burke
, “
Understanding and reducing errors in density functional calculations
,”
Phys. Rev. Lett.
111
,
073003
(
2013
).
30.
J. P.
Perdew
and
A.
Zunger
, “
Self-interaction correction to density-functional approximations for many-electron systems
,”
Phys. Rev. B
23
,
5048
(
1981
).
31.
A. D.
Becke
, “
Density-functional thermochemistry. III. The role of exact exchange
,”
J. Chem. Phys.
98
,
5648
(
1993
).
32.
J.
Heyd
,
G. E.
Scuseria
, and
M.
Ernzerhof
, “
Hybrid functionals based on a screened Coulomb potential
,”
J. Chem. Phys.
118
,
8207
(
2003
).
33.
O. A.
Vydrov
,
J.
Heyd
,
A. V.
Krukau
, and
G. E.
Scuseria
, “
Importance of short-range versus long-range Hartree-Fock exchange for the performance of hybrid density functionals
,”
J. Chem. Phys.
125
,
074106
(
2006
).
34.
O. A.
Vydrov
and
G. E.
Scuseria
, “
Assessment of a long-range corrected hybrid functional
,”
J. Chem. Phys.
125
,
234109
(
2006
).
35.
U.
Salzner
and
R.
Baer
, “
Koopmans’ springs to life
,”
J. Chem. Phys.
131
,
231101
(
2009
).
36.
R.
Baer
,
E.
Livshits
, and
U.
Salzner
, “
Tuned range-separated hybrids in density functional theory
,”
Annu. Rev. Phys. Chem.
61
,
85
109
(
2010
).
37.
T.
Stein
,
H.
Eisenberg
,
L.
Kronik
, and
R.
Baer
, “
Fundamental gaps in finite systems from eigenvalues of a generalized Kohn-Sham method
,”
Phys. Rev. Lett.
105
,
266802
(
2010
).
38.
N.
Sai
,
P. F.
Barbara
, and
K.
Leung
, “
Hole localization in molecular crystals from hybrid density functional theory
,”
Phys. Rev. Lett.
106
,
226403
(
2011
).
39.
S.
Refaely-Abramson
,
R.
Baer
, and
L.
Kronik
, “
Fundamental and excitation gaps in molecules of relevance for organic photovoltaics from an optimally tuned range-separated hybrid functional
,”
Phys. Rev. B
84
,
075144
(
2011
).
40.
L.
Kronik
,
T.
Stein
,
S.
Refaely-Abramson
, and
R.
Baer
, “
Excitation gaps of finite-sized systems from optimally tuned range-separated hybrid functionals
,”
J. Chem. Theory Comput.
8
,
1515
1531
(
2012
).
41.
S.
Refaely-Abramson
,
S.
Sharifzadeh
,
M.
Jain
,
R.
Baer
,
J. B.
Neaton
, and
L.
Kronik
, “
Gap renormalization of molecular crystals from density-functional theory
,”
Phys. Rev. B
88
,
081204
(
2013
).
42.
S.
Zheng
,
E.
Geva
, and
B. D.
Dunietz
, “
Solvated charge transfer states of functionalized anthracene and tetracyanoethylene dimers: A computational study based on a range separated hybrid functional and charge constrained self-consistent field with switching Gaussian polarized continuum models
,”
J. Chem. Theory Comput.
9
,
1125
1131
(
2013
).
43.
T. B.
de Queiroz
and
S.
Kümmel
, “
Charge-transfer excitations in low-gap systems under the influence of solvation and conformational disorder: Exploring range-separation tuning
,”
J. Chem. Phys.
141
,
084303
(
2014
).
44.
H.
Phillips
,
Z.
Zheng
,
E.
Geva
, and
B. D.
Dunietz
, “
Orbital gap predictions for rational design of organic photovoltaic materials
,”
Org. Electron.
15
,
1509
1520
(
2014
).
45.
S.
Refaely-Abramson
,
M.
Jain
,
S.
Sharifzadeh
,
J. B.
Neaton
, and
L.
Kronik
, “
Solid-state optical absorption from optimally tuned time-dependent range-separated hybrid density functional theory
,”
Phys. Rev. B
92
,
081204
(
2015
).
46.
T. B.
de Queiroz
and
S.
Kümmel
, “
Tuned range separated hybrid functionals for solvated low bandgap oligomers
,”
J. Chem. Phys.
143
,
034101
(
2015
).
47.
Z.
Zheng
,
J.-L.
Brédas
, and
V.
Coropceanu
, “
Description of the charge transfer states at the pentacene/C60 interface: Combining range-separated hybrid functionals with the polarizable continuum model
,”
J. Phys. Chem. Lett.
7
,
2616
2621
(
2016
).
48.
D.
Neuhauser
,
E.
Rabani
,
Y.
Cytter
, and
R.
Baer
, “
Stochastic optimally tuned range-separated hybrid density functional theory
,”
J. Phys. Chem. A
120
,
3071
3078
(
2016
).
49.
H.
Sun
,
S.
Zhang
,
C.
Zhong
, and
Z.
Sun
, “
Theoretical study of excited states of DNA base dimers and tetramers using optimally tuned range-separated density functional theory
,”
J. Comput. Chem.
37
,
684
693
(
2016
).
50.
H.
Sun
,
S.
Ryno
,
C.
Zhong
,
M. K.
Ravva
,
Z.
Sun
,
T.
Körzdörfer
, and
J.-L.
Brédas
, “
Ionization energies, electron affinities, and polarization energies of organic molecular crystals: Quantitative estimations from a polarizable continuum model (PCM)-tuned range-separated density functional approach
,”
J. Chem. Theory Comput.
12
,
2906
2916
(
2016
).
51.
M.
Rubesova
,
E.
Muchova
, and
P.
Slavicek
, “
Optimal tuning of range-separated hybrids for solvated molecules with time-dependent density functional theory
,”
J. Chem. Theory Comput.
13
,
4972
4983
(
2017
).
52.
A.
Boruah
,
M. P.
Borpuzari
,
Y.
Kawashima
,
K.
Hirao
, and
R.
Kar
, “
Assessment of range-separated functionals in the presence of implicit solvent: Computation of oxidation energy, reduction energy, and orbital energy
,”
J. Chem. Phys.
146
,
164102
(
2017
).
53.
M.
Alipour
and
Z.
Safari
, “
Photophysics of OLED materials with emitters exhibiting thermally activated delayed fluorescence and used in hole/electron transporting layer from optimally tuned range-separated density functional theory
,”
J. Phys. Chem. C
123
,
746
761
(
2018
).
54.
M.
Alipour
, “
Dipole moments of molecules with multi-reference character from optimally tuned range-separated density functional theory
,”
J. Comput. Chem.
39
,
1508
1516
(
2018
).
55.
A. J.
Lee
,
M.
Chen
,
W.
Li
,
D.
Neuhauser
,
R.
Baer
, and
E.
Rabani
, “
Dopant levels in large nanocrystals using stochastic optimally tuned range-separated hybrid density functional theory
,”
Phys. Rev. B
102
,
035112
(
2020
).
56.
S. G.
Dale
and
E. R.
Johnson
, “
Counterintuitive electron localisation from density-functional theory with polarisable solvent models
,”
J. Chem. Phys.
143
,
184112
(
2015
).
57.
O. S.
Bokareva
,
G.
Grell
,
S. I.
Bokarev
, and
O.
Kühn
, “
Tuning range-separated density functional theory for photocatalytic water splitting systems
,”
J. Chem. Theory Comput.
11
,
1700
1709
(
2015
).
58.
S.
McKechnie
,
G. H.
Booth
,
A. J.
Cohen
, and
J. M.
Cole
, “
On the accuracy of density functional theory and wave function methods for calculating vertical ionization energies
,”
J. Chem. Phys.
142
,
194114
(
2015
).
59.
R.
Garrick
,
A.
Natan
,
T.
Gould
, and
L.
Kronik
, “
Exact generalized Kohn-Sham theory for hybrid functionals
,”
Phys. Rev. X
10
,
021040
(
2020
).
60.
V.
Barone
and
M.
Cossi
, “
Quantum calculation of molecular energies and energy gradients in solution by a conductor solvent model
,”
J. Phys. Chem. A
102
,
1995
(
1998
).
61.
M.
Cossi
,
N.
Rega
,
G.
Scalmani
, and
V.
Barone
, “
Energies, structures, and electronic properties of molecules in solution with the C-PCM solvation model
,”
J. Comput. Chem.
24
,
669
(
2003
).
62.
R.
Di Remigio
,
A. H.
Steindal
,
K.
Mozgawa
,
V.
Weijo
,
H.
Cao
, and
L.
Frediani
, “
PCMSolver: An open-source library for solvation modeling
,”
Int. J. Quantum Chem.
119
,
e25685
(
2019
).
63.
R. M.
Parrish
,
L. A.
Burns
,
D. G. A.
Smith
,
A. C.
Simmonett
,
A. E.
DePrince
 III
,
E. G.
Hohenstein
,
U.
Bozkaya
,
A. Y.
Sokolov
,
R.
Di Remigio
,
R. M.
Richard
 et al, “
Psi4 1.1: An open-source electronic structure program emphasizing automation, advanced libraries, and interoperability
,”
J. Chem. Theory Comput.
13
,
3185
(
2017
).
64.
G.
Scalmani
and
M. J.
Frisch
, “
Continuous surface charge polarizable continuum models of solvation. I. General formalism
,”
J. Chem. Phys.
132
,
114110
(
2010
).
65.
A. V.
Marenich
,
C. J.
Cramer
, and
D. G.
Truhlar
, “
Universal solvation model based on solute electron density and on a continuum model of the solvent defined by the bulk dielectric constant and atomic surface tensions
,”
J. Phys. Chem. B
113
,
6378
(
2009
).
66.
A. D.
Becke
, “
Correlation energy of an inhomogeneous electron gas: A coordinate-space model
,”
J. Chem. Phys.
88
,
1053
(
1988
).
67.
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
(
1988
).
68.
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
).
69.
T. H.
Dunning
, Jr.
, “
Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen
,”
J. Chem. Phys.
90
,
1007
(
1989
).
70.
T.
Körzdörfer
,
J. S.
Sears
,
C.
Sutton
, and
J.-L.
Brédas
, “
Long-range corrected hybrid functionals for π-conjugated systems: Dependence of the range-separation parameter on conjugation length
,”
J. Chem. Phys.
135
,
204107
(
2011
).
71.
X. A.
Sosa Vazquez
and
C. M.
Isborn
, “
Size-dependent error of the density functional theory ionization potential in vacuum and solution
,”
J. Chem. Phys.
143
,
244105
(
2015
).
72.
L.
Kronik
and
S.
Kümmel
, “
Dielectric screening meets optimally tuned density functionals
,”
Adv. Mater.
30
,
1706560
(
2018
).
73.
T.
Koopmans
, “
Ordering of wave functions and eigenenergies to the individual electrons of an atom
,”
Physica
1
,
104
(
1933
).
74.
Z.
Zheng
,
D. A.
Egger
,
J.-L.
Brédas
,
L.
Kronik
, and
V.
Coropceanu
, “
Effect of solid-state polarization on charge-transfer excitations and transport levels at organic interfaces from a screened range-separated hybrid functional
,”
J. Phys. Chem. Lett.
8
,
3277
3283
(
2017
).
75.
K.
Garrett
,
X.
Sosa Vazquez
,
S. B.
Egri
,
J.
Wilmer
,
L. E.
Johnson
,
B. H.
Robinson
, and
C. M.
Isborn
, “
Optimum exchange for calculation of excitation energies and hyperpolarizabilities of organic electro-optic chromophores
,”
J. Chem. Theory Comput.
10
,
3821
3831
(
2014
).
76.
Z.
Zheng
,
A. K.
Manna
,
H. P.
Hendrickson
,
M.
Hammer
,
C.
Song
,
E.
Geva
, and
B. D.
Dunietz
, “
Molecular structure, spectroscopy, and photoinduced kinetics in trinuclear cyanide bridged complex in solution: A first-principles perspective
,”
J. Am. Chem. Soc.
136
,
16954
(
2014
).
77.
S.
Bhandari
,
Z.
Zheng
,
B.
Maiti
,
C.-H.
Chuang
,
M.
Porel
,
Z.-Q.
You
,
V.
Ramamurthy
,
C.
Burda
,
J. M.
Herbert
, and
B. D.
Dunietz
, “
What is the optoelectronic effect of the capsule on the guest molecule in aqueous host/guest complexes? A combined computational and spectroscopic perspective
,”
J. Phys. Chem. C
121
,
15481
15488
(
2017
).
78.
I. C.
Gerber
,
J. G.
Ángyán
,
M.
Marsman
, and
G.
Kresse
, “
Range separated hybrid density functional with long-range Hartree-Fock exchange applied to solids
,”
J. Chem. Phys.
127
,
054101
(
2007
).
79.
T.
Shimazaki
and
Y.
Asai
, “
Band structure calculations based on screened Fock exchange method
,”
Chem. Phys. Lett.
466
,
91
(
2008
).
80.
F.
Tran
and
P.
Blaha
, “
Accurate band gaps of semiconductors and insulators with a semilocal exchange-correlation potential
,”
Phys. Rev. Lett.
102
,
226401
(
2009
).
81.
M. A.
Marques
,
J.
Vidal
,
M. J.
Oliveira
,
L.
Reining
, and
S.
Botti
, “
Density-based mixing parameter for hybrid functionals
,”
Phys. Rev. B
83
,
035119
(
2011
).
82.
D.
Koller
,
P.
Blaha
, and
F.
Tran
, “
Hybrid functionals for solids with an optimized Hartree–Fock mixing parameter
,”
J. Phys.: Condens. Matter
25
,
435503
(
2013
).
83.
J. H.
Skone
,
M.
Govoni
, and
G.
Galli
, “
Self-consistent hybrid functional for condensed systems
,”
Phys. Rev. B
89
,
195112
(
2014
).
84.
S.-J.
Kim
,
S.
Lebègue
,
H.
Kim
, and
W. J.
Kim
, “
Assessment and prediction of band edge locations of nitrides using a self-consistent hybrid functional
,”
J. Chem. Phys.
155
,
024120
(
2021
).
85.
P.
Mori-Sánchez
,
A. J.
Cohen
, and
W.
Yang
, “
Localization and delocalization errors in density functional theory and implications for band-gap prediction
,”
Phys. Rev. Lett.
100
,
146401
(
2008
).
86.
V.
Vlček
,
H. R.
Eisenberg
,
G.
Steinle-Neumann
,
L.
Kronik
, and
R.
Baer
, “
Deviations from piecewise linearity in the solid-state limit with approximate density functionals
,”
J. Chem. Phys.
142
,
034107
(
2015
).
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