Advances in atomic, molecular, and optical physics, volume 64

EDITORS
ENNIO ARIMONDO
University of Pisa
Pisa, Italy
CHUN C. LIN
University of Wisconsin Madison,
Madison, WI, USA
SUSANNE F. YELIN
University of Connecticut
Storrs, CT, USA

EDITORIAL BOARD
P.H. BUCKSBAUM
SLAC
Menlo Park, California
C. JOACHAIN
Universite Libre de Bruxelles,
Brussels, Belgium
J.T.M. WALRAVEN
University of Amsterdam
Amsterdam, The Netherlands


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CONTRIBUTORS
Tunna Baruah (153)
Department of Physics, University of Texas El Paso, El Paso, Texas, USA
Scott D. Bergeson (223)

Department of Physics and Astronomy, Brigham Young University, Provo, Utah, USA
Giovanni Borghi (105)
Theory and Simulations of Materials (THEOS), and National Center for Computational
 cole Polytechnique Fe´de´rale de
Design and Discovery of Novel Materials (MARVEL), E
Lausanne, Lausanne, Switzerland
Carlo Maria Canali (29)
Department of Physics and Electrical Engineering, Linnæus University, Kalmar, Sweden
Yiwen Chu (273)
Department of Applied Physics, Yale University, New Haven, Connecticut, USA
Ismaila Dabo (105)
Department of Materials Science and Engineering, Materials Research Institute, and Penn
State Institutes of Energy and the Environment, The Pennsylvania State University,
Pennsylvania, USA
Phuong Mai Dinh (87)
CNRS, and Universite´ de Toulouse, UPS, Laboratoire de Physique The´orique (IRSAMC),
Toulouse Ce´dex, France
David Gelbwaser-Klimovsky (329)
Department of Chemical Physics, Weizmann Institute of Science, Rehovot, Israel
Nikitas Gidopoulos (129)
Department of Physics, Durham University, Durham, United Kingdom
Koblar Alan Jackson (15)
Physics Department and Science of Advanced Materials Program, Central Michigan
University, Mt. Pleasant, Michigan, USA
Nathan Daniel Keilbart (105)
Department of Materials Science and Engineering, Materials Research Institute, and Penn
State Institutes of Energy and the Environment, The Pennsylvania State University,
Pennsylvania, USA
Stephan Ku¨mmel (143)
Theoretische Physik IV, Universita¨t Bayreuth, Bayreuth, Germany

Gershon Kurizki (329)
Department of Chemical Physics, Weizmann Institute of Science, Rehovot, Israel
Nektarios N.N. Lathiotakis (129)
Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation,
Athens, Greece, and Max Planck Institute of Microstructure Physics, Halle (Saale), Germany
ix


x

Contributors

Mikhail Lukin (273)
Department of Physics, Harvard University, Cambridge, Massachusetts, USA
Michael S. Murillo (223)
New Mexico Consortium Los Alamos, New Mexico, USA
Ngoc Linh Nguyen (105)
Theory and Simulations of Materials (THEOS), and National Center for Computational
 cole Polytechnique Fe´de´rale de
Design and Discovery of Novel Materials (MARVEL), E
Lausanne, Lausanne, Switzerland
Wolfgang Niedenzu (329)
Department of Chemical Physics, Weizmann Institute of Science, Rehovot, Israel
Mark R. Pederson (1, 29, 153)
Department of Chemistry, Johns Hopkins University, Baltimore, Maryland, USA
John P. Perdew (1)
Department of Physics, and Department of Chemistry, Temple University, Philadelphia,
Pennsylvania, USA
Anna Pertsova (29)
Department of Physics and Electrical Engineering, Linnæus University, Kalmar, Sweden

Nicolas Poilvert (105)
Department of Materials Science and Engineering, Materials Research Institute, and Penn
State Institutes of Energy and the Environment, The Pennsylvania State University,
Pennsylvania, USA
Paul-Gerhard Reinhard (87)
Institut fu¨r Theoretische Physik, Universita¨t Erlangen, Erlangen, Germany
Ivan Rungger (29)
School of Physics, AMBER and CRANN Institute, Trinity College, Dublin, Ireland
Adrienn Ruzsinszky (1)
Department of Physics, Temple University, Philadelphia, Pennsylvania, USA
Stefano Sanvito (29)
School of Physics, AMBER and CRANN Institute, Trinity College, Dublin, Ireland
Swati Singh (273)
ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts, and
Department of Physics, University of Connecticut, Storrs, Connecticut, USA
Duncan G. Steel (181)
Departments of EECS and Physics, University of Michigan, Ann Arbor, Michigan, USA
Jianwei Sun (1)
Department of Physics, Temple University, Philadelphia, Pennsylvania, USA
Eric Suraud (87)
CNRS; Universite´ de Toulouse, UPS, Laboratoire de Physique The´orique (IRSAMC),
Toulouse Ce´dex, France, and Physics Department, University at Buffalo, The State
University New York, Buffalo, New York, USA


Contributors

xi

Marc Vincendon (87)

CNRS, and Universite´ de Toulouse, UPS, Laboratoire de Physique The´orique (IRSAMC),
Toulouse Ce´dex, France
Kevin Wang (105)
Department of Materials Science and Engineering, Materials Research Institute, and Penn
State Institutes of Energy and the Environment, The Pennsylvania State University,
Pennsylvania, USA
Susanne Yelin (273)
Department of Physics, University of Connecticut, Storrs, Connecticut, and Department of
Physics, Harvard University, Cambridge, Massachusetts, USA


PREFACE
A large part of this volume is on the subject of self-interaction corrections
(SICs) to the density functional theory (DFT). In the Hartree–Fock formalism for a many-electron system, the self-interaction of the Coulomb repulsion is offset by a similar term in the exchange component, but the
cancelation is incomplete when semi-local approximations to DFT, such
as the local-density approximation (LDA) and generalized gradient approximation (GGA), are adopted. The residual self-interaction is troublesome in
many applications of the LDA-DFT. For example, it produces, for the case
of a neutral atom, a one-electron potential with an exponential tail instead of
the correct (1/r) asymptotic form leading to serious problems in the calculated energies. Of the numerous attempts to address this deficiency, the SIC
scheme proposed by Perdew and Zunger in 1981 (PZ SIC) has received a
great deal of attention. Early applications of PZ SIC to simple atoms and
molecules demonstrated significant improvements over the uncorrected
LDA calculations. However, one serious drawback is that the PZ SIC
Hamiltonian generally depends on the individual orbital densities in contrast
to the fundamental view that the energy of the entire system is dictated only
by the total density and should be invariant under a unitary transformation of
the orbitals. A consequence of this is the necessity of introducing two sets of
orbitals (referred to as the canonical and localized orbitals by Pederson et al.)
which greatly increases the computational labor. As researchers moved on to
the more complex systems and simulation of dynamic processes, the need for

an effective means to handle the problems of self-interactions becomes more
pressing during the past 10 years. One of us (C.C.L.) is fortunate to have
enjoyed a long and fruitful association with Dr. Mark R. Pederson who
was an early explorer of the applications of the PZ SIC to molecules while
working on his doctoral dissertation and is actively involved in the current
surge of research efforts to deal with the problems of self-interactions. With
the invaluable advice and assistance from Dr. Pederson, we have compiled
the first eight chapters of this volume which provide different viewpoints
on the SIC along with discussions of both past and current work as well
as some indications on where the future might lead.
Chapter 1 introduces the PZ SIC in general terms and compares the various degrees of success (also the lack of it) in different kinds of calculations.
Applications of the SIC to study the electronic structure of substitutional
xiii


xiv

Preface

impurity atoms in ionic crystals constitute the main theme of the second
chapter. Addressed in Chapter 3 is the broad area of spin-dependent phenomena in nanostructures. Specifically, an important question is how well
one can predict the behaviors of a few spins in a given environment by means
of first-principles theoretical treatments. The computational challenge of
such analyses is discussed pointing toward possible future directions for
improving the predictive power of the DFT-based methods. As indicated
in the preceding paragraph, adaptation of the PZ SIC to the LDA resulted
in an iteration scheme involving two sets of orbitals. The use of such a twoset scheme is discussed by the authors of Chapter 4. They introduced an
average-density SIC procedure which drastically simplifies the computational work. In Chapter 5, a new way to incorporate the PZ SIC to the
LDA is presented. The self-interaction-corrected functional employed here
is still dependent on the individual orbitals but is made to conform to the

Koopmans theorem. Successful applications of such Koopmans-compliant
functionals as presented in this chapter are indicative of the potential power
of this approach. As mentioned earlier, one manifestation of the selfinteraction errors is that for a neutral system the LDA fails to reproduce
the (1/r) behavior of the potential seen by an electron at a large distance.
Using an effective local potential may provide an alternative avenue to
reduce the self-interaction errors, and construction of optimal local potentials for this purpose is the main theme of Chapter 6. In contrast to works
based on PZ SIC that result in orbital-specific potentials, Chapter 7 presents
SIC with one global multiplicative potential along with a discussion of the
relations of this approach to the method of optimized effective potential.
Extension to the time-dependent formulation is also discussed. In spite of
its success in many areas, the PZ SIC has been criticized for the undesirable
orbital dependence in the functional which spoiled the invariance under a
unitary transformation of the orbitals inherent in the general DFT, not to
mention the complications, both conceptual and computational, resulting
from this orbital dependence. Concluding this sequence of articles on
self-interactions, Chapter 8 reviews the recent work on recasting the PZ SIC
in terms of the Fermi orbitals which restores the unitary invariance. This step
puts a constraint on the original PZ SIC formalism, but bypasses the need for
the localization equations and the two-set procedure. Results of the new
SIC calculations and future outlook are also discussed therein.
A quarter century after quantum coherence effects such as dark states and
electromagnetically induced transparency (EIT) have become mainstream,
they are finally applied in solid-state systems in a wider context. To see such


Preface

xv

effects in typical semiconductor environments is at the same time among the

most desired and the most difficult. Steel shows in Chapter 9 recent advances
with quantum dot exciton artificial atoms and successes, and shows among
other things typical EIT and dark-state signatures.
In Chapter 10, Murillo and Bergeson present a new kind of plasmas
formed by photoionization of laser-cooled atoms. These ultracold neutral
plasmas are strongly coupled systems and are particularly well suited to study
many-body interactions in atomic and molecular processes like thermalization, three-body recombination, and collisional ionization. The authors
begin with an introduction to the strong coupling parameter as an index
of classification and then focus the discussion on generating strongly coupled
plasmas using calcium atoms in a magneto-optical trap. Molecular dynamics
simulations provide insight into electron screening. Techniques such as multiple ionization to higher ionization states, Rydberg atom dynamics, and
direct laser cooling of the ions for producing strongly coupled plasmas are
also discussed.
In parallel with the great progresses associated to the tools developed by
atomic, molecular, and optical physics, in the last few years an important
trend was established by the solid-state community: apply those sophisticated tools to solid systems where the complexity is not too large and instead
descriptions in terms of few atom-like objects can be applied. This approach
was exemplified in Chapter 9. Along a similar vein, the contribution in
Chapter 11 by Singh, Chu, Lukin, and Yelin targets the control of the
nuclear spins modifying the optical excitation of a single electronic spin
for the case of nitrogen-vacancy color centers in diamond. Owing to
impressive technological advances, it is today possible to monitor a
single-color center that represents a single atomic-like system. This center
interacts with the nuclear spin of the surrounding crystals, between tens
and hundreds of the 13C isotope within the diamond. The control of those
nuclear spins is essential for the application of the color center electronic spin
qubit, for instance, for quantum information. The authors of the present
contribution present the control achieved by applying the coherent population trapping approach originally developed for the laser cooling of atoms
and ions. That method is successfully applied to the cooling and the realtime projective measurement of the nuclear spin environment surrounding
the electronic spin.

In a combination of quantum optics and thermodynamics, GelbwaserKlimovsky, Niedenzu, and Kurizki asked the question in Chapter 12
whether quantum mechanics can allow to violate any of the laws of


xvi

Preface

thermodynamics. While this article does not claim to answer this wide and
impactful question fully, it turns out that the structure of the system-plusbath of a quantum mechanical heat engine allows in certain aspects to
improve on the classical Carnot limit. In their article, the authors review
the questions how partially and fully quantum systems behave, systems that
are driven steady state or periodically modulated, Markovian and nonMarkovian systems, and systems that are stripped down to the qubit stage.
The editors would like to thank all the contributing authors for their
contributions and for their cooperation in assembling this volume. They
are especially grateful to Dr. Mark R. Pederson for his help in organizing
the first eight chapters. Sincere appreciation is also extended to Ms. Helene
Kabes at Elsevier for her untiring assistance throughout the preparation of
this volume.
ENNIO ARIMONDO
CHUN C. LIN
SUSANNE F. YELIN


CHAPTER ONE

Paradox of Self-Interaction
Correction: How Can Anything
So Right Be So Wrong?
John P. Perdew*,†, Adrienn Ruzsinszky*, Jianwei Sun*,

Mark R. Pederson{,1
*Department of Physics, Temple University, Philadelphia, Pennsylvania, USA
†
Department of Chemistry, Temple University, Philadelphia, Pennsylvania, USA
{
Department of Chemistry, Johns Hopkins University, Baltimore, Maryland, USA
1
Corresponding author: e-mail address:

Contents
1. Introduction
2. What Is Right About PZ SIC?
3. What Is Wrong About PZ SIC?
4. SIC: How Can Anything So Right Be So Wrong? (Conclusions)
Acknowledgments
Appendix. Do Complex Orbitals Resolve the Paradox of SIC?
References

2
6
7
8
10
11
12

Abstract
Popular local, semilocal, and hybrid density functional approximations to the exchangecorrelation energy of a many-electron ground state make a one-electron self-interaction
error which can be removed by its orbital-by-orbital subtraction from the total
energy, as proposed by Perdew and Zunger in 1981. This makes the functional exact

for all one-electron ground states, but it does much more as well: It greatly improves
the description of negative ions, the dissociation curves of radical molecules and of
all heteronuclear molecules, the barrier heights for chemical reactions, charge-transfer
energies, etc. PZ SIC even led to the later discovery of an exact property, the derivative
discontinuity of the energy. It is also used to understand strong correlation, which is
beyond the reach of semilocal approximations. The paradox of SIC is that equilibrium
properties of molecules and solids, including atomization energies and equilibrium
geometries, are at best only slightly improved and more typically worsened by it, especially as we pass from local to semilocal and hybrid functionals which by themselves
provide a ladder of increasing accuracy for these equilibrium properties. The reason
for this puzzling ambivalence remains unknown. In this speculative chapter, we suggest
that the problem arises because the uncorrected functionals provide an inadequate
description of compact but noded one-electron orbital densities. We suggest that a
meta-generalized gradient approximation designed to satisfy a tight lower bound on
Advances in Atomic, Molecular, and Optical Physics, Volume 64
ISSN 1049-250X
/>
#

2015 Elsevier Inc.
All rights reserved.

1


2

John P. Perdew et al.

the exchange energy of a one-electron density could resolve the paradox, providing
after self-interaction correction the first practical “density functional theory of almost

everything.”

1. INTRODUCTION
Kohn–Sham density functional theory (Kohn and Sham, 1965) is a
formally exact construction of the ground-state energy and electron density
for a system of electrons with mutual Coulomb repulsion in the presence of a
multiplicative scalar external potential. The construction proceeds by solving self-consistent one-electron equations for the occupied Kohn–Sham
orbitals, fictional objects used to build up the electron density, and the noninteracting part of the kinetic energy. The many-electron effects are incorporated via the exchange-correlation energy as a functional of the density,
Exc[n",n#], and its functional derivative or exchange-correlation potential
vσxc([n",n#];r). In practice, the exchange-correlation energy has to be approximated. This approach is very widely used for the computation of atoms,
molecules, and condensed matter, because of its useful balance between
computational efficiency and accuracy.
The exact exchange-correlation energy is defined (Gunnarsson and
Lundqvist, 1976; Langreth and Perdew, 1975, 1977) so that
Z 1
(1)
U½nŠ + Exc ½n" ,n# Š ¼
dλhΨλ jV^ ee jΨλ i:
0

Here,
U½nŠ ¼

1
2

Z

Z
d3 r

d3 r 0

nðrÞnðr0 Þ
,
jr À r0 j

(2)

is the Hartree electrostatic self-repulsion energy of the total electron density
n(r) ¼ n"(r) + n#(r), the sum of up-spin and down-spin contributions. V^ ee is
the electron–electron Coulomb repulsion operator. And Ψλ is the groundstate wavefunction for electrons with interaction λV^ ee and with density
nðrÞ ¼ hΨλ j^
n ðrÞjΨλ i independent of coupling constant λ. The spindependent external scalar potential vσλ (r) varies between the Kohn–Sham
effective potential at λ ¼ 0 and the physical external potential at λ ¼ 1.
We can write Exc as the sum of exchange and correlation energies, where
the exchange energy Ex is defined by


Paradox of Self-Interaction Correction: How Can Anything So Right Be So Wrong?

U½nŠ + Ex ½n" , n# Š ¼ hΨ0 jV^ ee jΨ0 i:

3

(3)

Typically Ψ0 is a single Slater determinant of Kohn–Sham orbitals, and Ex
differs from Hartree–Fock exchange only via the small difference between
Kohn–Sham and Hartree–Fock orbitals.

The exchange energy and the correlation energy are nonpositive. They
arise because, as an electron moves through the density, it creates around
itself exchange and correlation holes (Gunnarsson and Lundqvist, 1976)
which reduce its repulsion energy with the other electrons. The exchange
hole arises from self-interaction correction and wavefunction antisymmetry
under particle exchange, and its density integrates to À 1, while the correlation hole arises from Coulomb repulsion, and its density integrates to 0.
While the exchange-correlation energy can be a small fraction of the total
energy, it is nature’s glue (Kurth and Perdew, 2000) that creates most of the
binding of one atom to another in a molecule or solid.
For any spin-up one-electron density n1(r), the Coulomb repulsion
operator vanishes so
U½n1 Š + Ex ½n1 ,0Š ¼ 0,
Ec ½n1 , 0Š ¼ 0:

(4)
(5)

The functional Exc of Eq. (1) is defined for ground-state spin-densities, but it
has a natural continuation to all fully-spin-polarized one-electron densities,
given by Eqs. (4) and (5), since the Coulomb repulsion operator vanishes for
all such densities. This continuation is not only natural but also physical:
It makes the solutions of the Kohn–Sham orbital equations exact for
one-electron systems, not only in their ground states but also in their
excited states and time-dependent states. It is also the choice made in the
Hartree–Fock and self-interaction-corrected Hartree approximations.
Approximate functionals that satisfy Eqs. (4) and (5) are said to be one-electron
self-interaction-free (Perdew and Zunger, 1981). For other functionals,
the numerical values of the right-hand sides are self-interaction errors (SIE)
for exchange and correlation, respectively, and their sum is the total selfinteraction error.
Semilocal approximations have single-integral form,

Z
sl
Exc ½n" ,n# Š ¼ d3 rnEslxc ðn" ,n# , rn" , rn# ,τ" , τ# Þ,
(6)
and are popular because of their computational efficiency. The original local
spin-density approximation (Gunnarsson and Lundqvist, 1976; Kohn and


4

John P. Perdew et al.

Sham, 1965) uses only the spin-density arguments, the generalized gradient
approximation (GGA) (Becke, 1988; Langreth and Mehl, 1983; Lee et al.,
1988; Perdew and Wang, 1986; Perdew et al., 1996) adds the spin-density
gradients, and the meta-GGA (Becke and Roussel, 1989; Perdew et al.,
1999; Sun et al., 2012, 2013; Tao et al., 2003; Van Voorhis and Scuseria,
1998) adds the positive kinetic energy densities
1
τσ ðrÞ ¼ Σoccupied
jrψ ασ j2
α
2

(7)

of the Kohn–Sham orbitals ψ ασ. The exchange-correlation energy per particle
Eslxc may be constructed to satisfy exact constraints on Exc, and the addition of
more arguments permits the satisfaction of more constraints with resulting
greater accuracy. For some GGA s (e.g., PBE) and meta-GGA s (e.g., TPSS),

this construction is nonempirical. But no semilocal functional can satisfy
Eq. (4), because of the full nonlocality of U[n], and only the meta-GGA
can satisfy Eq. (5). Hybrid functionals (Becke, 1993; Ernzerhof and
Scuseria, 1999; Stephens et al., 1994) add an exact-exchange ingredient, e.g.,
hybrid
Exc
¼ ð1 À aÞExsl + aExexact + Ecsl :

(8)

Typically they achieve higher accuracy through empirical selection of the
mixing parameter (e.g., 0.25) and not through satisfaction of additional
constraints.
In 1981, Perdew and Zunger (1981) (PZ) proposed to correct any
approximate functional by subtracting its self-interaction error, orbital by
orbital:
occupied

SICÀapprox
approx
Exc
¼ Exc
½n" , n# Š À Σiσ

approx
fU½niσ Š + Exc
½niσ , 0Šg :

(9)

They applied this correction to the local spin-density approximation
(LSDA) for atoms, achieving remarkable improvements. Atoms are relatively easy, because their Kohn–Sham orbitals are localized and can be used
directly to construct the single-orbital densities niσ in Eq. (9). But generally,
and especially in molecules and solids where the Kohn–Sham orbitals can
delocalize, Perdew and Zunger realized that the
niσ ðrÞ ¼ jϕiσ ðrÞj2

(10)

should be constructed from a set of localized orbitals that are equivalent
under unitary transformation to the set of occupied Kohn–Sham orbitals,
in order to achieve size-consistency. The best choice of unitary


Paradox of Self-Interaction Correction: How Can Anything So Right Be So Wrong?

5

transformation remained somewhat unclear. Pederson and collaborators
(Pederson and Lin, 1988; Pederson et al., 1984, 1985) found a useful unitary
transformation that minimized the SIC-LSDA total energy, and performed
early SIC calculations for molecules. But a choice that is guaranteed to
achieve size-consistency for all approximations is the set of Fermi-L€
owdin
orbitals (Pederson, 2015; Pederson et al., 2014) constructed from the Kohn–
Sham single-particle density matrix. Since the latter is a functional of the
density, the PZ SIC energy is too. In practice, it might be easier to find
PZ SIC canonical orbitals that are not exactly Kohn–Sham orbitals (because
the exchange-correlation potential in the unified Hamiltonian is not a multiplication operator), but that is for computational and not fundamental reasons. While in the limit of a small number of atoms or a small number of
atoms per unit cell, solution in the canonical-orbital space may be technically easier to program, for large system sizes greater computational efficiency, by solution in the localized-orbital space, may be achieved since

the resulting equations lead to an explicitly sparse matrix for at least most
of the electronic orbitals in the problem. The Fermi-L€
owdin-orbital-based
method (Pederson, 2015; Pederson et al., 2014) has one advantage over
methods based upon the localization equations (Dabo et al., 2014;
Pederson and Lin, 1988; Pederson et al., 1984, 1985) or symmetry conditions (Dinh et al., 2014; Messud et al., 2008), in that there is a clear cut
description, naturally arising from the construction of Fermi Orbitals, on
how to simultaneous vary and relocalize the localizing orbitals even in
the limit that the self-interaction correction vanishes.
The PZ SIC makes the approximate functional exact for any oneelectron density. There are other ways to achieve that, but the PZ way is
a good one for the following reason: The dominant part of the
exchange-correlation energy is typically exchange, and the exact exchange
energy is invariant under a unitary transformation of the occupied orbitals.
Localized orbitals put much of the exchange energy into the Hartree selfinteraction correction,
occupied

ÀΣiσ

U½niσ Š,

(11)

which appears naturally in the exact exchange and which PZ SIC treats
exactly, leaving only the residual interelectronic exchange energy and the
correlation energy to be approximated semilocally. PZSIC also treats the
Hartree and exchange-correlation energies as similarly as possible, and gives
no correction to the exact functional (Perdew and Zunger, 1981).


6

John P. Perdew et al.

2. WHAT IS RIGHT ABOUT PZ SIC?
Semilocal approximations can work reasonably well for neutral atoms
A, but they typically do not bind negative atomic ions AÀ1. Given enough
flexibility in the basis set, the energy minimizes at AÀq, where 0 < q < 1. But
real negative ions are predicted by PZ SIC (Cole and Perdew, 1982; Perdew
and Zunger, 1981). Semilocal approximations also fail for many binding
energy curves of diatomic molecules. The binding energy curve may yield
a reasonable minimum near the true equilibrium bond length, but as the
bond length is stretched several errors are encountered: (1) For an A2 radical
(e.g., H2+ 1 ), the total energy tends as the bond length is stretched to a value
that is much more negative than the expected energy (e.g., that of a neutral
H atom). (2) For a heteronuclear diatomic AB, the selfconsistent density
tends as the bond length is stretched to A+q BÀq, where q is a spurious fractional charge (Ruzsinszky et al., 2006), and not to the correct configuration
of neutral atoms A and B. These errors, and other charge-transfer errors, are
corrected by PZ SIC (Ruzsinszky et al., 2006).
When two species react with one another, they form an intermediate or
transition state whose energy is typically higher than the energies of the reactants and products, producing an energy barrier for the reaction. The transition state tends to be loosely bound, with long bond lengths and often with
radical or spin-polarized character. Semilocal approximations severely
underestimate the barrier heights, to which reaction rates are very sensitive.
But PZ SIC produces much more realistic barrier heights (Patchkovskii and
Ziegler, 2002).
Perdew and Zunger (1981) noted that their SIC, applied to atoms with
fractional electron number, led to linear or nearly linear variation of the total
energy between adjacent integer electron numbers, and to derivative discontinuities of the total energy at integer electron number. Soon after,
Perdew et al. (1982) proved that the exact total energy varies linearly
between the integers. The derivative discontinuities or cusps at the integers
explain (Perdew et al., 1982) why separated atoms and molecules are exactly

neutral and not fractionally charged. This property of the exact energy has
been called many-electron self-interaction freedom (Cohen et al., 2007;
Ruzsinszky et al., 2006). It is well approximated in PZ SIC, not just because
this theory is one- electron self-interaction free but more importantly
because PZ SIC captures the correct Hartree self-interaction correction
of Eq. (11) (as other one-electron self-interaction corrections may not).


Paradox of Self-Interaction Correction: How Can Anything So Right Be So Wrong?

7

Related to the derivative discontinuity is the correct SIC description
of electron transport through molecular wires (Hofmann and Ku¨mmel,
2012; Toher et al., 2005), which displays Coulomb blockade effects missing
from semilocal functionals, and the correct SIC description of chargetransfer and excitonic excitations in time-dependent DFT (Hofmann
et al., 2012b).
Finally, note that PZ SIC provides an ab initio alternative to the LSDA+U
method for the description of strongly correlated systems including materials
with open shells of localized d and f electrons (Cococcioni and de Gironcoli,
2005; Hughes et al., 2007).

3. WHAT IS WRONG ABOUT PZ SIC?
PZ SIC is “the road less travelled” (Pederson and Perdew, 2012) in
density functional theory. In part, this is a consequence of the fact that it
is computationally more demanding than the semilocal functionals and is
unavailable in many popular computer codes. Studies of SIC-LSDA for
molecules (Cole and Perdew, 1982; Pederson et al., 1984, 1985) found a
moderate improvement over LSDA in atomization energies, where however GGA and meta-GGA were somewhat better than SIC-LSDA. The
road more travelled climbs the ladder of approximations from LSDA to

GGA to meta-GGA and/or hybrid functionals, without SIC.
Vydrov and Scuseria implemented a version of self-consistent PZ SIC in
a developmental version of the GAUSSIAN code. In 2004, they made a
comprehensive study (Vydrov and Scuseria, 2004) of the performance of
PZ SIC for molecules. They applied SIC to LSDA, to several GGAs such
as PBE (Perdew et al., 1996), and BLYP (Becke, 1988; Lee et al., 1988)
and meta-GGAs such as PKZB (Perdew et al., 1999), TPSS (Tao et al.,
2003), VSXC (Van Voorhis and Scuseria, 1998), and to hybrid functionals
such as PBE1PBE (Ernzerhof and Scuseria, 1999), and B3LYP (Stephens
et al., 1994). For the atomization energies of the 55 molecules in the
G2-1 data set (using separated neutral atoms as a reference), they found that
PZ SIC improves agreement with experiment only for LSDA, while all
other functionals perform worse with SIC. They also found that the selfinteraction error of the valence orbitals has the same order of magnitude
for all the tested functionals. They wrote that: “The performance of SICDFT in comparison with the regular Kohn–Sham DFT is ambivalent.
On the one hand, SIC is crucial for proper description of odd-electron


8

John P. Perdew et al.

systems, improves activation barriers for chemical reactions, and improves
nuclear magnetic resonance chemical shifts. On the other hand, it provides
little or no improvement for reaction energies and results in too-short bond
lengths in molecules.”

4. SIC: HOW CAN ANYTHING SO RIGHT BE SO WRONG?
(CONCLUSIONS)
How can it be that we can start from a sophisticated and accurate semilocal or hybrid function, impose the additional exact constraint of selfinteraction freedom, and find that some properties are significantly
improved and others are worsened as a result? The answer is not known

for certain, but there are several different possible interpretations.
One possible interpretation is that most semilocal functionals are exact
for a uniform or slowly varying electron density, and this constraint may
be lost when we make the PZ SIC. Existing studies (Pederson et al.,
1989; Sun and Pederson, 2015) do not suggest that the error so introduced
is large. Moreover, we do not expect it to be very important for molecules,
which are not close to the uniform or slowly varying limit.
Another possibility is that SIC upsets the error cancellation between
semilocal exchange and semilocal correlation. This error cancellation is
important in molecules, where a combination of 100% exact exchange
(a ¼ 1 in Eq. (8)) with semilocal correlation fails rather badly. This error
cancellation occurs because the exact exchange-correlation hole is deeper
and more short ranged (more local) than are the exact exchange hole or
the exact correlation hole. The self-interaction error of semilocal exchange
may mimic the long-range or nondynamic correlation in a molecule (Polo
et al., 2002, 2003).
In 2006, Vydrov et al. (2006) proposed a way to scale down the PZ selfinteraction correction in many-electron regions, without changing it in
one-electron regions. The scaling depended upon a single parameter which
could be chosen to yield greatly improved atomization energies and bond
lengths for molecules. This looked like a possible solution at the time,
but later Ruzsinszky et al. (2006) found that the many-electron selfinteraction error returned under this scaling, and so did the spurious
fractional-charge dissociation of heteronuclear molecules. The proposed
explanation was that the exact Hartree self-interaction correction of
Eq. (11) was also lost by the scaling.


Paradox of Self-Interaction Correction: How Can Anything So Right Be So Wrong?

9

A glimpse of another explanation was also presented by Vydrov et al.
(2006). They observed that the one-electron densities niσ of Eq. (10) are
often noded, and vary rapidly near the nodes, in the sense that the dimensionless density gradient
s ∝ jrnj=n4=3

(12)

and the dimensionless Laplacian q ∝ jr2 nj=n5=3 for n ¼ niσ both diverge
there. They suggested that, while accuracy for nodeless densities increases
from LSDA to GGA to meta-GGA, accuracy for noded densities may actually
decrease along this sequence. This is the interpretation we will explore
further here.
A semilocal density functional for the exchange-correlation energy can
be associated (Constantin et al., 2009) with a localized approximate
exchange-correlation hole around an electron which integrates to À 1. Thus
a necessary condition for the success of any semilocal approximation is that
the exact exchange-correlation hole is similarly localized around an electron.
This condition is typically met in sp-bonded molecules and solids near equilibrium. But it is not met in those stretched-bond situations where SIC is
greatly needed. For example, in stretched H2+ 1 , when an electron is close
to one proton half its exact exchange-correlation hole is located around
the distant other proton. Only a fully nonlocal approximation, like SIC,
can describe such a situation correctly.
The previous paragraph explains how a good semilocal functional can predict a good atomization energy (with respect to separated neutral atoms) or a
good equilibrium geometry, for a molecule or solid near equilibrium. Now, if
we apply the PZ self-interaction correction of Eq. (9), these good results will
be preserved if the self-interaction errors of the valence electrons are small
approx
enough. But that in turn requires that Exc
[niσ ] must be accurate enough
to nearly cancel U[niσ ] for compact but noded one-electron densities.

Recently Perdew, Ruzsinszky, Sun and Burke (Perdew et al., 2014) have
employed rigorous results from Lieb and Oxford (Lieb and Oxford, 1981) to
show that the exact exchange energy of any spin-polarized one-electron
density satisfies a tight lower bound that requires
Eslx

Eunif
x

1:174ð21=3 Þ,

(13)

2 1/3
where Eunif
. This bound does not require that the onex ¼ À[3/(4π)](3π n)
electron density be a ground-state density. But the PBE GGA (Perdew et al.,


10

John P. Perdew et al.

1996) and the PKZB (Perdew et al., 1999) and TPSS (Tao et al., 2003) metaGGA s only satisfy the much looser bound
Eslx

1:804ð21=3 Þ,

Eunif
x

(14)

where the bound is approached when the reduced density gradient s of
Eq. (13) tends to infinity (as it does at the nodes of an orbital density). Since
a meta-GGA can recognize a one-electron density by the conditions
τW jrnj2/8n ¼ τ and jn"À n#j/n ¼ 1, a meta-GGA can be constructed
to satisfy the tight bound of Eq. (13) for all one-electron densities. Another
often-neglected exact constraint (Perdew et al., 2014), which makes the
ratio on the left-hand side of Eq. (13) vanish like sÀ1/2 as s ! 1, can also
be imposed and might also reduce the self-interaction error for compact
noded orbital densities. Applying PZ SIC to a meta-GGA that satisfies these
added constraints could give a theory of almost everything that works for
both stretched and equilibrium bonds. (The long-range van der Waals interaction could not be captured by such a theory, but intermediate-range van der
Waals might still be usefully described (Sun et al., 2012, 2013)).
We cannot yet say if this dream will ever be realized. But a tantalizing
clue comes from the work of Vydrov and Scuseria (2004), who presented
a figure showing the valence-shell self-interaction error for the atoms from
Na to Ar. For the argon atom, as an example, this error is approximately
ELSDA

+0.12 hartree for the LSDA ( Exunif ¼ 1ð21=3 Þ), and À 0.14 hartree for the
x

Esl

PBE GGA and TPSS meta-GGA (Eunifx
x

1:804ð21=3 Þ). So it is tempting to

imagine that the valence-shell self-interaction error could be very small
for a meta-GGA satisfying the tight bound of Eq. (13) for all spin-polarized
one-electron densities.
Work completed since submission of this chapter includes recent work
on a semilocal density functional with improved exchange descriptions (Sun
et al., 2015a), a new strongly constrained and appropriately normed (SCAN)
functional (Sun et al., 2015b) and some new insights regarding the locality of
exchange and correlation (Sun et al., 2015c).

ACKNOWLEDGMENTS
This work was supported by the National Science Foundation under Grant No. DMR1305135 (J.P.). We thank Hannes Jo´nsson and Susi Lehtola for a private communication
concerning Lehtola and Jo´nsson (2014a) and Lehtola and Jo´nsson (2014b), and Stephan
Ku¨mmel for comments and suggestions.


Paradox of Self-Interaction Correction: How Can Anything So Right Be So Wrong?

11

APPENDIX. DO COMPLEX ORBITALS RESOLVE THE
PARADOX OF SIC?
The orbitals ϕiσ (r) used to make the self-interaction correction in
Eqs. (9) and (10) have traditionally been chosen to be real. Real orthonormal
orbitals are necessarily noded, and noded orbitals present a special challenge
to existing semilocal functionals. It has been further noted (Pederson and
Perdew, 2012) that any set of real orbitals that satisfy all the localization equations (Pederson and Lin, 1988; Pederson et al., 1984, 1985) are stationary
with respect to an infinitesimal complex unitary transformation on any pair
of localized orbitals.
Recently the Jo´nsson group at the University of Iceland has proposed

that the unitary transformation from canonical to localized orbitals should
be generalized to include complex localized orbitals ϕiσ (r) (Klu¨pfel et al.,
2011, 2012; Lehtola and Jo´nsson, 2014a,b), which can eliminate the nodes
(in the sense that the node of the real part of the orbital does not coincide
with the node of the imaginary part). This eliminates the nodes of the orbital
densities, which are then less challenging to existing semilocal functionals. In
fact, it has been found that the total energies of atoms within the TPSS metaGGA are worsened by PZ SIC with real orbitals, but slightly improved by
PZ SIC with complex orbitals (Lehtola and Jo´nsson, 2014a,b) and further
noted (Hofmann et al., 2012a) that complex orbitals may not eliminate all
the nodes of the orbital densities, but reduce the number of nodal planes
and particularly eliminate the nodes in the most energetically important
regions of space.
Complex orbitals bring another benefit in variational approaches that
choose the orbitals ϕiσ (r) to minimize the SIC energy: The extra variational
freedom can further lower the SIC energy. This increases the chance that
localized orbitals can be found variationally. Recall that SIC is size-consistent
only when all the occupied ϕiσ (r) are localized. Going to complex orbitals is
thus also helpful for size-consistency within a variational approach like those
next referenced (Klu¨pfel et al., 2011, 2012; Lehtola and Jo´nsson, 2014a,b).
A remaining problem seems to be that there is no guarantee that the SIC
energy will be minimized by localized orbitals for all possible systems. We do
not know of any cases where the energy-minimizing orbitals are not localized in SIC-LSD, but this is not guaranteed for higher level functionals
where the self-interaction correction from a localized orbital can be positive.
As an example, consider the Ar atom, where (Vydrov and Scuseria, 2004)


12

John P. Perdew et al.

the self-interaction correction from the localized valence orbitals is À0.12
hartree for LSDA but +0.14 hartree for the TPSS meta-GGA. Then in a
highly expanded lattice of Ar atoms the energy- minimizing SIC valence
orbitals will be localized atomic orbitals in SIC-LSDA but delocalized Bloch
orbitals (with zero self-interaction correction) in SIC-TPSS; the valence
self-interaction correction to the energy will be present in the SIC-TPSS
single atom, but missing from the SIC-TPSS atom on the expanded lattice.
In contrast, while the Fermi-L€
owdin orbitals (Pederson, 2015; Pederson
et al., 2014) are real and thus noded, they are always localized and thus guarantee size-consistency (Perdew, 1990) for all possible systems.

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124, 094108.


CHAPTER TWO

Local Spin Density Treatment of
Substitutional Defects in Ionic
Crystals with Self-Interaction
Corrections
Koblar Alan Jackson1
Physics Department and Science of Advanced Materials Program, Central Michigan University, Mt. Pleasant,

Michigan, USA
1
Corresponding author: e-mail address:

Contents
1. Introduction
2. Free-Ion Calculations
3. Pure Crystal Calculation
4. Embedded-Cluster Approach to Isolated Impurities
5. Discussion
Acknowledgment
References

15
18
20
21
26
27
27

Abstract
The application of the self-interaction correction to the local density functional theory to
the problem of transition metal defects in alkali-halide crystals is reviewed. The computational machinery involves a number of approximations that are based on the localized,
atomic-like nature of the charge distributions in these systems. These allow the detailed
calculation of the variationally correct local orbitals to be circumvented and a much
more computationally convenient approach to determining the defect and host crystal
orbitals to be used. Results are presented for the NaCl:Cu+ and LiCl:Ag+ impurity
systems.

1. INTRODUCTION
The self-interaction-correction (SIC) paper of Perdew and Zunger
(1981) represented an exciting step forward for the field of density functional
theory (DFT). The SIC addressed a clear defect present in DFT and the
results presented in that work showed that the SIC is very successful when
Advances in Atomic, Molecular, and Optical Physics, Volume 64
ISSN 1049-250X
/>
#

2015 Elsevier Inc.
All rights reserved.

15


16

Koblar Alan Jackson

applied to atomic systems. However, as shown by Pederson et al. (1984,
1985) and discussed in detail elsewhere in this review, the orbital-dependent
nature of the theory makes applying DFT-SIC to multiatom systems
difficult. They showed that two sets of orbitals are required to implement
DFT-SIC. The canonical orbitals (CO) reflect the symmetry of the multiatom system and the one-electron energies corresponding to the CO represent approximate electron removal energies. The CO are connected by
a unitary transformation to the local orbitals (LO) that are the basis for
the correction terms in DFT-SIC. The variationally correct LO that
minimize the DFT-SIC total energy must also satisfy an additional set of
equations, the localization equations (LE). Simultaneously satisfying the
DFT-SIC Kohn–Sham equations with the CO and the LE with the LO

is challenging. The lack of an easily implemented solution for finding the
correct LO has prevented a more widespread use of DFT-SIC.
One detour around the LO problem is to study multiatom systems that
possess atomic-like charge densities. In an alkali-halide crystal such as NaCl,
for example, the charge density can be thought of in the first approximation
as a packing of free Na+ and ClÀ ions. The free-ion orbitals are thus good
starting points for the LO. In the mid to late 1980s, the Wisconsin group of
Lin applied DFT-SIC to a series of alkali-halide-based systems, taking
advantage of the atomic-like features (Erwin and Lin, 1988, 1989;
Harrison et al., 1983; Heaton and Lin, 1984; Heaton et al., 1985; Jackson
and Lin, 1988, 1990).
One problem that made the alkali halides interesting to study involved
the fundamental band gap energy. It is well known that the local spin density
(LSD) form of DFT underestimates the valence–conduction band gaps of
insulating solids by 30–50%. For NaCl, for example, use of exchange-only
LSD gives a band gap of 4.7 eV, compared to the measured gap of 8.6 eV.
The SIC should give a larger correction for the more localized valence band
(VB) states and thus move their energies down relative to the less-localized
conduction band (CB) states. The SIC could therefore be expected to open
the gap.
A second problem involved substitutional impurities. Transition metal
impurities in alkali-halide crystals were being studied actively in the early
1980s as prototype solid-state impurity systems (Payne et al., 1984; Pedrini
et al., 1983; Simonetti and McClure, 1977). The impurity ions introduce
unoccupied defect states into the wide band gap of the host material.
Transitions to these gap states give rise to absorption in the visible and
near u–v, whereas the onset of band gap absorption occurs at much higher