Recently developed locally scaled self-interaction correction (LSIC) is a one-electron SIC method that, when used with a ratio of kinetic energy densities (zσ) as iso-orbital indicator, performs remarkably well for both thermochemical properties as well as for barrier heights overcoming the paradoxical behavior of the well-known Perdew–Zunger self-interaction correction (PZSIC) method. In this work, we examine how well the LSIC method performs for the delocalization error. Our results show that both LSIC and PZSIC methods correctly describe the dissociation of H2+ and He2+ but LSIC is overall more accurate than the PZSIC method. Likewise, in the case of the vertical ionization energy of an ensemble of isolated He atoms, the LSIC and PZSIC methods do not exhibit delocalization errors. For the fractional charges, both LSIC and PZSIC significantly reduce the deviation from linearity in the energy vs number of electrons curve, with PZSIC performing superior for C, Ne, and Ar atoms while for Kr they perform similarly. The LSIC performs well at the endpoints (integer occupations) while substantially reducing the deviation. The dissociation of LiF shows both LSIC and PZSIC dissociate into neutral Li and F but only LSIC exhibits charge transfer from Li+ to F at the expected distance from the experimental data and accurate ab initio data. Overall, both the PZSIC and LSIC methods reduce the delocalization errors substantially.

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.
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
).
4.
I.
Lindgren
, “
A statistical exchange approximation for localized electrons
,”
Int. J. Quantum Chem.
5
,
411
420
(
1971
).
5.
J. P.
Perdew
, “
Orbital functional for exchange and correlation: Self-interaction correction to the local density approximation
,”
Chem. Phys. Lett.
64
,
127
130
(
1979
).
6.
J. P.
Perdew
and
A.
Zunger
, “
Self-interaction correction to density-functional approximations for many-electron systems
,”
Phys. Rev. B
23
,
5048
(
1981
).
7.
A.
Zunger
,
J. P.
Perdew
, and
G. L.
Oliver
, “
A self-interaction corrected approach to many-electron systems: Beyond the local spin density approximation
,”
Solid State Commun.
34
,
933
936
(
1980
).
8.
R. A.
Heaton
,
J. G.
Harrison
, and
C. C.
Lin
, “
Self-interaction correction for density-functional theory of electronic energy bands of solids
,”
Phys. Rev. B
28
,
5992
(
1983
).
9.
M. R.
Pederson
,
R. A.
Heaton
, and
C. C.
Lin
, “
Local-density Hartree-Fock theory of electronic states of molecules with self-interaction correction
,”
J. Chem. Phys.
80
,
1972
1975
(
1984
).
10.
J. P.
Perdew
,
A.
Ruzsinszky
,
J.
Sun
, and
M. R.
Pederson
, “
Paradox of self-interaction correction: How can anything so right be so wrong?
,”
Adv. At., Mol., Opt. Phys.
64
,
1
14
(
2015
).
11.
T.
Schmidt
and
S.
Kümmel
, “
One- and many-electron self-interaction error in local and global hybrid functionals
,”
Phys. Rev. B
93
,
165120
(
2016
).
12.
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
).
13.
A.
Ruzsinszky
,
J. P.
Perdew
,
G. I.
Csonka
,
O. A.
Vydrov
, and
G. E.
Scuseria
, “
Density functionals that are one- and two- are not always many-electron self-interaction-free, as shown for H2+, He2+, LiH+, and Ne2+
,”
J. Chem. Phys.
126
,
104102
(
2007
).
14.
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
).
15.
A. D.
Dwyer
and
D. J.
Tozer
, “
Dispersion, static correlation, and delocalisation errors in density functional theory: An electrostatic theorem perspective
,”
J. Chem. Phys.
135
,
164110
(
2011
).
16.
C.
Li
,
X.
Zheng
,
N. Q.
Su
, and
W.
Yang
, “
Localized orbital scaling correction for systematic elimination of delocalization error in density functional approximations
,”
Natl. Sci. Rev.
5
,
203
215
(
2018
).
17.
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
).
18.
A.
Ruzsinszky
,
J. P.
Perdew
,
G. I.
Csonka
,
O. A.
Vydrov
, and
G. E.
Scuseria
, “
Spurious fractional charge on dissociated atoms: Pervasive and resilient self-interaction error of common density functionals
,”
J. Chem. Phys.
125
,
194112
(
2006
).
19.
A. D.
Becke
and
M. R.
Roussel
, “
Exchange holes in inhomogeneous systems: A coordinate-space model
,”
Phys. Rev. A
39
,
3761
(
1989
).
20.
A. D.
Becke
, “
Hartree–Fock exchange energy of an inhomogeneous electron gas
,”
Int. J. Quantum Chem.
23
,
1915
1922
(
1983
).
21.
T.
Tsuneda
,
M.
Kamiya
, and
K.
Hirao
, “
Regional self-interaction correction of density functional theory
,”
J. Comput. Chem.
24
,
1592
1598
(
2003
).
22.
T.
Tsuneda
and
K.
Hirao
, “
Self-interaction corrections in density functional theory
,”
J. Chem. Phys.
140
,
18A513
(
2014
).
23.
J.
Jaramillo
,
G. E.
Scuseria
, and
M.
Ernzerhof
, “
Local hybrid functionals
,”
J. Chem. Phys.
118
,
1068
1073
(
2003
).
24.
M.
Kaupp
,
H.
Bahmann
, and
A. V.
Arbuznikov
, “
Local hybrid functionals: An assessment for thermochemical kinetics
,”
J. Chem. Phys.
127
,
194102
(
2007
).
25.
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
).
26.
R.
Latter
, “
Atomic energy levels for the Thomas-Fermi and Thomas-Fermi-Dirac potential
,”
Phys. Rev.
99
,
510
(
1955
).
27.
I.
Dabo
,
A.
Ferretti
, and
N.
Marzari
, “
Piecewise linearity and spectroscopic properties from Koopmans-compliant functionals
,” in
First Principles Approaches to Spectroscopic Properties of Complex Materials
, edited by
C.
Di Valentin
,
S.
Botti
, and
M.
Cococcioni
(
Springer
,
Berlin, Heidelberg
,
2014
), pp.
193
233
.
28.
G.
Borghi
,
A.
Ferretti
,
N. L.
Nguyen
,
I.
Dabo
, and
N.
Marzari
, “
Koopmans-compliant functionals and their performance against reference molecular data
,”
Phys. Rev. B
90
,
075135
(
2014
).
29.
C. D.
Pemmaraju
,
T.
Archer
,
D.
Sánchez-Portal
, and
S.
Sanvito
, “
Atomic-orbital-based approximate self-interaction correction scheme for molecules and solids
,”
Phys. Rev. B
75
,
045101
(
2007
).
30.
G.
Li Manni
,
R. K.
Carlson
,
S.
Luo
,
D.
Ma
,
J.
Olsen
,
D. G.
Truhlar
, and
L.
Gagliardi
, “
Multiconfiguration pair-density functional theory
,”
J. Chem. Theory Comput.
10
,
3669
3680
(
2014
).
31.
N. Q.
Su
,
A.
Mahler
, and
W.
Yang
, “
Preserving symmetry and degeneracy in the localized orbital scaling correction approach
,”
J. Phys. Chem. Lett.
11
,
1528
1535
(
2020
).
32.
B. G.
Janesko
, “
Replacing hybrid density functional theory: Motivation and recent advances
,”
Chem. Soc. Rev.
50
,
8470
(
2021
).
33.
R.
Nagai
,
R.
Akashi
, and
O.
Sugino
, “
Completing density functional theory by machine learning hidden messages from molecules
,”
npj Comput. Mater.
6
,
43
(
2020
).
34.
J.
Kirkpatrick
,
B.
McMorrow
,
D. H.
Turban
,
A. L.
Gaunt
,
J. S.
Spencer
,
A. G.
Matthews
,
A.
Obika
,
L.
Thiry
,
M.
Fortunato
,
D.
Pfau
et al, “
Pushing the frontiers of density functionals by solving the fractional electron problem
,”
Science
374
,
1385
1389
(
2021
).
35.
M.
Bogojeski
,
L.
Vogt-Maranto
,
M. E.
Tuckerman
,
K.-R.
Müller
, and
K.
Burke
, “
Quantum chemical accuracy from density functional approximations via machine learning
,”
Nat. Commun.
11
,
5223
(
2020
).
36.
S.
Dick
and
M.
Fernandez-Serra
, “
Highly accurate and constrained density functional obtained with differentiable programming
,”
Phys. Rev. B
104
,
L161109
(
2021
).
37.
R. R.
Zope
,
Y.
Yamamoto
,
C. M.
Diaz
,
T.
Baruah
,
J. E.
Peralta
,
K. A.
Jackson
,
B.
Santra
, and
J. P.
Perdew
, “
A step in the direction of resolving the paradox of Perdew-Zunger self-interaction correction
,”
J. Chem. Phys.
151
,
214108
(
2019
).
38.
Y.
Yamamoto
,
T.
Baruah
,
P.-H.
Chang
,
S.
Romero
, and
R. R.
Zope
, “
Self-consistent implementation of locally scaled self-interaction-correction method
,”
J. Chem. Phys.
158
,
064114
(
2023
).
39.
S.
Akter
,
Y.
Yamamoto
,
C. M.
Diaz
,
K. A.
Jackson
,
R. R.
Zope
, and
T.
Baruah
, “
Study of self-interaction errors in density functional predictions of dipole polarizabilities and ionization energies of water clusters using Perdew-Zunger and locally scaled self-interaction corrected methods
,”
J. Chem. Phys.
153
,
164304
(
2020
).
40.
S.
Akter
,
Y.
Yamamoto
,
R. R.
Zope
, and
T.
Baruah
, “
Static dipole polarizabilities of polyacenes using self-interaction-corrected density functional approximations
,”
J. Chem. Phys.
154
,
114305
(
2021
).
41.
P.
Mishra
,
Y.
Yamamoto
,
P.-H.
Chang
,
D. B.
Nguyen
,
J. E.
Peralta
,
T.
Baruah
, and
R. R.
Zope
, “
Study of self-interaction errors in density functional calculations of magnetic exchange coupling constants using three self-interaction correction methods
,”
J. Phys. Chem. A
126
,
1923
1935
(
2022
).
42.
P.
Mishra
,
Y.
Yamamoto
,
J. K.
Johnson
,
K. A.
Jackson
,
R. R.
Zope
, and
T.
Baruah
, “
Study of self-interaction-errors in barrier heights using locally scaled and Perdew–Zunger self-interaction methods
,”
J. Chem. Phys.
156
,
014306
(
2022
).
43.
S.
Akter
,
J. A.
Vargas
,
K.
Sharkas
,
J. E.
Peralta
,
K. A.
Jackson
,
T.
Baruah
, and
R. R.
Zope
, “
How well do self-interaction corrections repair the over-estimation of molecular polarizabilities in density functional calculations?
,”
Phys. Chem. Chem. Phys.
23
,
18678
(
2021
).
44.
S.
Romero
,
T.
Baruah
, and
R. R.
Zope
, “
Spin-state gaps and self-interaction-corrected density functional approximations: Octahedral Fe(II) complexes as case study
,”
J. Chem. Phys.
158
,
054305
(
2023
).
45.
M. R.
Pederson
,
A.
Ruzsinszky
, and
J. P.
Perdew
, “
Communication: Self-interaction correction with unitary invariance in density functional theory
,”
J. Chem. Phys.
140
,
121103
(
2014
).
46.
W. L.
Luken
and
D. N.
Beratan
, “
Localized orbitals and the Fermi hole
,”
Theor. Chim. Acta
61
,
265
281
(
1982
).
47.
W. L.
Luken
and
J. C.
Culberson
, “
Localized orbitals based on the Fermi hole
,”
Theor. Chim. Acta
66
,
279
293
(
1984
).
48.
C. M.
Diaz
,
T.
Baruah
, and
R. R.
Zope
, “
Fermi-Löwdin-orbital self-interaction correction using the optimized-effective-potential method within the Krieger-Li-Iafrate approximation
,”
Phys. Rev. A
103
,
042811
(
2021
).
49.
S.
Romero
,
Y.
Yamamoto
,
T.
Baruah
, and
R. R.
Zope
, “
Local self-interaction correction method with a simple scaling factor
,”
Phys. Chem. Chem. Phys.
23
,
2406
2418
(
2021
).
50.
P.
Bhattarai
,
B.
Santra
,
K.
Wagle
,
Y.
Yamamoto
,
R. R.
Zope
,
A.
Ruzsinszky
,
K. A.
Jackson
, and
J. P.
Perdew
, “
Exploring and enhancing the accuracy of interior-scaled Perdew-Zunger self-interaction correction
,”
J. Chem. Phys.
154
,
094105
(
2021
).
51.
J. M.
Leonard
and
W. L.
Luken
, “
Quadratically convergent calculation of localized molecular orbitals
,”
Theor. Chim. Acta
62
,
107
132
(
1982
).
52.
U.
Lundin
and
O.
Eriksson
, “
Novel method of self-interaction corrections in density functional calculations
,”
Int. J. Quantum Chem.
81
,
247
252
(
2001
).
53.
O. A.
Vydrov
,
G. E.
Scuseria
,
J. P.
Perdew
,
A.
Ruzsinszky
, and
G. I.
Csonka
, “
Scaling down the Perdew-Zunger self-interaction correction in many-electron regions
,”
J. Chem. Phys.
124
,
094108
(
2006
).
54.
Y.
Yamamoto
,
S.
Romero
,
T.
Baruah
, and
R. R.
Zope
, “
Improvements in the orbitalwise scaling down of Perdew–Zunger self-interaction correction in many-electron regions
,”
J. Chem. Phys.
152
,
174112
(
2020
).
55.
S.
Romero
,
Y.
Yamamoto
,
T.
Baruah
, and
R. R.
Zope
, “
Complexity reduction in self-interaction-free density functional calculations using the Fermi-Löwdin self-interaction correction method
,” arXiv:2308.04664 (
2023
).
56.
P.-O.
Löwdin
, “
On the non-orthogonality problem connected with the use of atomic wave functions in the theory of molecules and crystals
,”
J. Chem. Phys.
18
,
365
375
(
1950
).
57.
Z.-h.
Yang
,
M. R.
Pederson
, and
J. P.
Perdew
, “
Full self-consistency in the Fermi-orbital self-interaction correction
,”
Phys. Rev. A
95
,
052505
(
2017
).
58.
M. R.
Pederson
, “
Fermi orbital derivatives in self-interaction corrected density functional theory: Applications to closed shell atoms
,”
J. Chem. Phys.
142
,
064112
(
2015
).
59.
M. R.
Pederson
and
T.
Baruah
, “
Self-interaction corrections within the Fermi-orbital-based formalism
,”
Adv. At., Mol., Opt. Phys.
64
,
153
180
(
2015
).
60.
R. R.
Zope
,
T.
Baruah
, and
K. A.
Jackson
, FLOSIC 0.2, based on the NRLMOL code of M. R. Pederson.
61.
Y.
Yamamoto
,
L.
Basurto
,
C. M.
Diaz
,
R. R.
Zope
, and
T.
Baruah
, Flosic software public release, based on the NRLMOL code of M. R. Pederson.
62.
P.
Bhattarai
,
K.
Wagle
,
C.
Shahi
,
Y.
Yamamoto
,
S.
Romero
,
B.
Santra
,
R. R.
Zope
,
J. E.
Peralta
,
K. A.
Jackson
, and
J. P.
Perdew
, “
A step in the direction of resolving the paradox of Perdew–Zunger self-interaction correction. II. Gauge consistency of the energy density at three levels of approximation
,”
J. Chem. Phys.
152
,
214109
(
2020
).
63.
J.
Tao
,
V. N.
Staroverov
,
G. E.
Scuseria
, and
J. P.
Perdew
, “
Exact-exchange energy density in the gauge of a semilocal density-functional approximation
,”
Phys. Rev. A
77
,
012509
(
2008
).
64.
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
).
65.
D.
Porezag
and
M. R.
Pederson
, “
Optimization of Gaussian basis sets for density-functional calculations
,”
Phys. Rev. A
60
,
2840
(
1999
).
66.
M. R.
Pederson
and
K. A.
Jackson
, “
Variational mesh for quantum-mechanical simulations
,”
Phys. Rev. B
41
,
7453
7461
(
1990
).
67.
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
2608
(
1998
).
68.
A.
Ruzsinszky
,
J. P.
Perdew
, and
G. I.
Csonka
, “
Binding energy curves from nonempirical density functionals. I. Covalent bonds in closed-shell and radical molecules
,”
J. Phys. Chem. A
109
,
11006
11014
(
2005
).
69.
K. R.
Bryenton
,
A. A.
Adeleke
,
S. G.
Dale
, and
E. R.
Johnson
, “
Delocalization error: The greatest outstanding challenge in density-functional theory
,”
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
13
,
e1631
(
2023
).
70.
J. L.
Bao
,
L.
Gagliardi
, and
D. G.
Truhlar
, “
Self-interaction error in density functional theory: An appraisal
,”
J. Phys. Chem. Lett.
9
,
2353
2358
(
2018
).
71.
T. M.
Maier
,
A. V.
Arbuznikov
, and
M.
Kaupp
, “
Local hybrid functionals: Theory, implementation, and performance of an emerging new tool in quantum chemistry and beyond
,”
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
9
,
e1378
(
2019
).
72.
O. A.
Vydrov
,
G. E.
Scuseria
, and
J. P.
Perdew
, “
Tests of functionals for systems with fractional electron number
,”
J. Chem. Phys.
126
,
154109
(
2007
).
73.
J. L.
Bao
,
Y.
Wang
,
X.
He
,
L.
Gagliardi
, and
D. G.
Truhlar
, “
Multiconfiguration pair-density functional theory is free from delocalization error
,”
J. Phys. Chem. Lett.
8
,
5616
5620
(
2017
).
74.
A. J.
Cohen
,
P.
Mori-Sánchez
, and
W.
Yang
, “
Challenges for density functional theory
,”
Chem. Rev.
112
,
289
320
(
2012
).
75.
J. F.
Janak
, “
Proof that Eni = ɛ in density-functional theory
,”
Phys. Rev. B
18
,
7165
(
1978
).
76.
A. J.
Cohen
,
P.
Mori-Sánchez
, and
W.
Yang
, “
Fractional charge perspective on the band gap in density-functional theory
,”
Phys. Rev. B
77
,
115123
(
2008
).
77.
J.
Vargas
,
P.
Ufondu
,
T.
Baruah
,
Y.
Yamamoto
,
K. A.
Jackson
, and
R. R.
Zope
, “
Importance of self-interaction-error removal in density functional calculations on water cluster anions
,”
Phys. Chem. Chem. Phys.
22
,
3789
3799
(
2020
).
78.
P.
Ufondu
,
P.-H.
Chang
,
T.
Baruah
, and
R. R.
Zope
, “
Vertical detachment energies of ammonia cluster anions using self-interaction-corrected methods
,”
J. Chem. Phys.
158
,
164308
(
2023
).
79.
R. D.
Johnson
III
, NIST computational chemistry comparison and benchmark database, NIST standard reference database number 101, Release 16a http://cccbdb.nist.gov/,
2013
; accessed March 13, 2015.
80.
C.
Li
,
R.
Requist
, and
E. K. U.
Gross
, “
Density functional theory of electron transfer beyond the Born-Oppenheimer approximation: Case study of LiF
,”
J. Chem. Phys.
148
,
084110
(
2018
).
81.
Y.
Yamamoto
,
C. M.
Diaz
,
L.
Basurto
,
K. A.
Jackson
,
T.
Baruah
, and
R. R.
Zope
, “
Fermi-Löwdin orbital self-interaction correction using the strongly constrained and appropriately normed meta-GGA functional
,”
J. Chem. Phys.
151
,
154105
(
2019
).
82.
L.
Goerigk
,
A.
Hansen
,
C.
Bauer
,
S.
Ehrlich
,
A.
Najibi
, and
S.
Grimme
, “
A look at the density functional theory zoo with the advanced GMTKN55 database for general main group thermochemistry, kinetics and noncovalent interactions
,”
Phys. Chem. Chem. Phys.
19
,
32184
32215
(
2017
).
83.
O. A.
Vydrov
and
G. E.
Scuseria
, “
Effect of the Perdew–Zunger self-interaction correction on the thermochemical performance of approximate density functionals
,”
J. Chem. Phys.
121
,
8187
8193
(
2004
).
84.
F. W.
Aquino
,
R.
Shinde
, and
B. M.
Wong
, “
Fractional occupation numbers and self-interaction correction-scaling methods with the Fermi-Löwdin orbital self-interaction correction approach
,”
J. Comput. Chem.
41
,
1200
1208
(
2020
).

Supplementary Material

You do not currently have access to this content.