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

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..

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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

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 Design and Discovery of Novel Materials (MARVEL),  Ecole Polytechnique Fe´de´rale de Lausanne, Lausanne, Switzerland

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 formal-ism for a many-electron system, the self-interaction of the Coulomb repul-sion is offset by a similar term in the exchange component, but thecancelation is incomplete when semi-local approximations to DFT, such

as the local-density approximation (LDA) and generalized gradient imation (GGA), are adopted The residual self-interaction is troublesome inmany applications of the LDA-DFT For example, it produces, for the case

approx-of a neutral atom, a one-electron potential with an exponential tail instead approx-ofthe correct (1/r) asymptotic form leading to serious problems in the calcu-lated energies Of the numerous attempts to address this deficiency, the SICscheme proposed by Perdew and Zunger in 1981 (PZ SIC) has received agreat deal of attention Early applications of PZ SIC to simple atoms andmolecules demonstrated significant improvements over the uncorrectedLDA calculations However, one serious drawback is that the PZ SICHamiltonian 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 ofthe orbitals A consequence of this is the necessity of introducing two sets oforbitals (referred to as the canonical and localized orbitals by Pederson et al.)which greatly increases the computational labor As researchers moved on tothe more complex systems and simulation of dynamic processes, the need for

an effective means to handle the problems of self-interactions becomes morepressing during the past 10 years One of us (C.C.L.) is fortunate to haveenjoyed a long and fruitful association with Dr Mark R Pederson whowas an early explorer of the applications of the PZ SIC to molecules whileworking on his doctoral dissertation and is actively involved in the currentsurge of research efforts to deal with the problems of self-interactions Withthe invaluable advice and assistance from Dr Pederson, we have compiledthe 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 1introduces the PZ SIC in general terms and compares the ious degrees of success (also the lack of it) in different kinds of calculations.Applications of the SIC to study the electronic structure of substitutional

var-xiii

impurity atoms in ionic crystals constitute the main theme of the secondchapter Addressed inChapter 3is the broad area of spin-dependent phe-nomena in nanostructures Specifically, an important question is how wellone can predict the behaviors of a few spins in a given environment by means

of first-principles theoretical treatments The computational challenge ofsuch analyses is discussed pointing toward possible future directions forimproving 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 set scheme is discussed by the authors of Chapter 4 They introduced anaverage-density SIC procedure which drastically simplifies the computa-tional work In Chapter 5, a new way to incorporate the PZ SIC to theLDA is presented The self-interaction-corrected functional employed here

two-is still dependent on the individual orbitals but two-is made to conform to theKoopmans theorem Successful applications of such Koopmans-compliantfunctionals as presented in this chapter are indicative of the potential power

of this approach As mentioned earlier, one manifestation of the interaction errors is that for a neutral system the LDA fails to reproducethe (1/r) behavior of the potential seen by an electron at a large distance.Using an effective local potential may provide an alternative avenue toreduce the self-interaction errors, and construction of optimal local poten-tials for this purpose is the main theme ofChapter 6 In contrast to worksbased on PZ SIC that result in orbital-specific potentials,Chapter 7presentsSIC with one global multiplicative potential along with a discussion of therelations of this approach to the method of optimized effective potential.Extension to the time-dependent formulation is also discussed In spite ofits success in many areas, the PZ SIC has been criticized for the undesirableorbital dependence in the functional which spoiled the invariance under aunitary transformation of the orbitals inherent in the general DFT, not tomention the complications, both conceptual and computational, resultingfrom this orbital dependence Concluding this sequence of articles onself-interactions,Chapter 8reviews the recent work on recasting the PZ SIC

self-in terms of the Fermi orbitals which restores the unitary self-invariance This stepputs a constraint on the original PZ SIC formalism, but bypasses the need forthe localization equations and the two-set procedure Results of the newSIC calculations and future outlook are also discussed therein

A quarter century after quantum coherence effects such as dark states andelectromagnetically induced transparency (EIT) have become mainstream,they are finally applied in solid-state systems in a wider context To see such

effects in typical semiconductor environments is at the same time among themost desired and the most difficult Steel shows inChapter 9recent advanceswith quantum dot exciton artificial atoms and successes, and shows amongother things typical EIT and dark-state signatures.

In Chapter 10, Murillo and Bergeson present a new kind of plasmasformed by photoionization of laser-cooled atoms These ultracold neutralplasmas are strongly coupled systems and are particularly well suited to studymany-body interactions in atomic and molecular processes like thermaliza-tion, three-body recombination, and collisional ionization The authorsbegin with an introduction to the strong coupling parameter as an index

of classification and then focus the discussion on generating strongly coupledplasmas using calcium atoms in a magneto-optical trap Molecular dynamicssimulations provide insight into electron screening Techniques such as mul-tiple ionization to higher ionization states, Rydberg atom dynamics, anddirect laser cooling of the ions for producing strongly coupled plasmas arealso discussed

In parallel with the great progresses associated to the tools developed byatomic, molecular, and optical physics, in the last few years an importanttrend was established by the solid-state community: apply those sophisti-cated tools to solid systems where the complexity is not too large and insteaddescriptions in terms of few atom-like objects can be applied This approachwas exemplified in Chapter 9 Along a similar vein, the contribution in

Chapter 11 by Singh, Chu, Lukin, and Yelin targets the control of thenuclear spins modifying the optical excitation of a single electronic spinfor the case of nitrogen-vacancy color centers in diamond Owing toimpressive technological advances, it is today possible to monitor asingle-color center that represents a single atomic-like system This centerinteracts with the nuclear spin of the surrounding crystals, between tensand hundreds of the13C isotope within the diamond The control of thosenuclear spins is essential for the application of the color center electronic spinqubit, for instance, for quantum information The authors of the presentcontribution present the control achieved by applying the coherent popu-lation trapping approach originally developed for the laser cooling of atomsand ions That method is successfully applied to the cooling and the real-time projective measurement of the nuclear spin environment surroundingthe electronic spin

In a combination of quantum optics and thermodynamics, Klimovsky, Niedenzu, and Kurizki asked the question in Chapter 12

Gelbwaser-whether quantum mechanics can allow to violate any of the laws of

xv

Preface

thermodynamics While this article does not claim to answer this wide andimpactful question fully, it turns out that the structure of the system-plus-bath of a quantum mechanical heat engine allows in certain aspects toimprove on the classical Carnot limit In their article, the authors reviewthe questions how partially and fully quantum systems behave, systems thatare driven steady state or periodically modulated, Markovian and non-Markovian systems, and systems that are stripped down to the qubit stage.The editors would like to thank all the contributing authors for theircontributions and for their cooperation in assembling this volume Theyare especially grateful to Dr Mark R Pederson for his help in organizingthe first eight chapters Sincere appreciation is also extended to Ms HeleneKabes at Elsevier for her untiring assistance throughout the preparation ofthis volume.

ENNIOARIMONDO

CHUNC LIN

SUSANNEF YELIN

Department of Chemistry, Temple University, Philadelphia, Pennsylvania, USA

{Department of Chemistry, Johns Hopkins University, Baltimore, Maryland, USA

1 Corresponding author: e-mail address: mpeder10@jhu.edu

Contents

4 SIC: How Can Anything So Right Be So Wrong? (Conclusions) 8

exchange-Advances in Atomic, Molecular, and Optical Physics, Volume 64 # 2015 Elsevier Inc.

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 aformally exact construction of the ground-state energy and electron densityfor a system of electrons with mutual Coulomb repulsion in the presence of amultiplicative scalar external potential The construction proceeds by solv-ing self-consistent one-electron equations for the occupied Kohn–Shamorbitals, fictional objects used to build up the electron density, and the non-interacting part of the kinetic energy The many-electron effects are incor-porated via the exchange-correlation energy as a functional of the density,Exc[n",n#], and its functional derivative or exchange-correlation potentialvxcσ ([n",n#];r) In practice, the exchange-correlation energy has to be approx-imated This approach is very widely used for the computation of atoms,molecules, and condensed matter, because of its useful balance betweencomputational efficiency and accuracy

The exact exchange-correlation energy is defined (Gunnarsson andLundqvist, 1976; Langreth and Perdew, 1975,1977) so that

U½n + Exc½n",n# ¼

Z 1 0

dλhΨλj ^VeejΨλi: (1)Here,

U½n ¼12

We can writeExc as the sum of exchange and correlation energies, wherethe exchange energyExis defined by

U½n + Ex½n",n# ¼ hΨ0j ^VeejΨ0i: (3)TypicallyΨ0is a single Slater determinant of Kohn–Sham orbitals, andExdiffers from Hartree–Fock exchange only via the small difference betweenKohn–Sham and Hartree–Fock orbitals.

The exchange energy and the correlation energy are nonpositive Theyarise because, as an electron moves through the density, it creates arounditself exchange and correlation holes (Gunnarsson and Lundqvist, 1976)which reduce its repulsion energy with the other electrons The exchangehole arises from self-interaction correction and wavefunction antisymmetryunder particle exchange, and its density integrates to 1, while the corre-lation hole arises from Coulomb repulsion, and its density integrates to 0.While the exchange-correlation energy can be a small fraction of the totalenergy, it is nature’s glue (Kurth and Perdew, 2000) that creates most of thebinding of one atom to another in a molecule or solid

For any spin-up one-electron density n1(r), the Coulomb repulsionoperator vanishes so

The functionalExcof Eq (1) is defined for ground-state spin-densities, but ithas a natural continuation to all fully-spin-polarized one-electron densities,given by Eqs (4) and (5), since the Coulomb repulsion operator vanishes forall such densities This continuation is not only natural but also physical:

It makes the solutions of the Kohn–Sham orbital equations exact forone-electron systems, not only in their ground states but also in theirexcited states and time-dependent states It is also the choice made in theHartree–Fock and self-interaction-corrected Hartree approximations.Approximate functionals that satisfy Eqs (4) and (5) are said to be one-electronself-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 self-interaction error

Semilocal approximations have single-integral form,

Excsl½n",n# ¼

Z

d3rnEslxcðn",n#,rn",rn#,τ",τ#Þ, (6)and are popular because of their computational efficiency The original localspin-density approximation (Gunnarsson and Lundqvist, 1976; Kohn and

3

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

Sham, 1965) uses only the spin-density arguments, the generalized gradientapproximation (GGA) (Becke, 1988;Langreth and Mehl, 1983;Lee et al.,

1988;Perdew and Wang, 1986;Perdew et al., 1996) adds the spin-densitygradients, 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

Eq (4), because of the full nonlocality of U[n], and only the meta-GGAcan satisfy Eq (5) Hybrid functionals (Becke, 1993; Ernzerhof andScuseria, 1999;Stephens et al., 1994) add an exact-exchange ingredient, e.g.,

In 1981, Perdew and Zunger (1981) (PZ) proposed to correct anyapproximate functional by subtracting its self-interaction error, orbital byorbital:

ExcSICapprox¼ Eapprox

xc ½n",n#  Σoccupied

iσ fU½niσ + Eapprox

xc ½niσ, 0g: (9)They applied this correction to the local spin-density approximation(LSDA) for atoms, achieving remarkable improvements Atoms are rela-tively easy, because their Kohn–Sham orbitals are localized and can be useddirectly to construct the single-orbital densitiesniσin Eq (9) But generally,and especially in molecules and solids where the Kohn–Sham orbitals candelocalize, Perdew and Zunger realized that the

should be constructed from a set of localized orbitals that are equivalentunder unitary transformation to the set of occupied Kohn–Sham orbitals,

in order to achieve size-consistency The best choice of unitary

transformation remained somewhat unclear Pederson and collaborators(Pederson and Lin, 1988;Pederson et al., 1984,1985) found a useful unitarytransformation that minimized the SIC-LSDA total energy, and performedearly SIC calculations for molecules But a choice that is guaranteed toachieve size-consistency for all approximations is the set of Fermi-L€owdinorbitals (Pederson, 2015;Pederson et al., 2014) constructed from the Kohn–Sham single-particle density matrix Since the latter is a functional of thedensity, 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 (becausethe exchange-correlation potential in the unified Hamiltonian is not a mul-tiplication operator), but that is for computational and not fundamental rea-sons While in the limit of a small number of atoms or a small number ofatoms per unit cell, solution in the canonical-orbital space may be techni-cally easier to program, for large system sizes greater computational effi-ciency, by solution in the localized-orbital space, may be achieved sincethe resulting equations lead to an explicitly sparse matrix for at least most

of the electronic orbitals in the problem The Fermi-L€owdin-orbital-basedmethod (Pederson, 2015; Pederson et al., 2014) has one advantage overmethods based upon the localization equations (Dabo et al., 2014;

Pederson and Lin, 1988;Pederson et al., 1984, 1985) or symmetry tions (Dinh et al., 2014; Messud et al., 2008), in that there is a clear cutdescription, naturally arising from the construction of Fermi Orbitals, onhow to simultaneous vary and relocalize the localizing orbitals even inthe limit that the self-interaction correction vanishes

condi-The PZ SIC makes the approximate functional exact for any electron density There are other ways to achieve that, but the PZ way is

one-a good one for the following reone-ason: The dominone-ant pone-art of theexchange-correlation energy is typically exchange, and the exact exchangeenergy is invariant under a unitary transformation of the occupied orbitals.Localized orbitals put much of the exchange energy into the Hartree self-interaction correction,

Σoccupied

which appears naturally in the exact exchange and which PZ SIC treatsexactly, leaving only the residual interelectronic exchange energy and thecorrelation energy to be approximated semilocally PZSIC also treats theHartree and exchange-correlation energies as similarly as possible, and gives

no correction to the exact functional (Perdew and Zunger, 1981)

5

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

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 ionsA1 Given enoughflexibility in the basis set, the energy minimizes atAq, where 0< q < 1 Butreal negative ions are predicted by PZ SIC (Cole and Perdew, 1982;Perdewand Zunger, 1981) Semilocal approximations also fail for many bindingenergy curves of diatomic molecules The binding energy curve may yield

a reasonable minimum near the true equilibrium bond length, but as thebond length is stretched several errors are encountered: (1) For anA2radical(e.g.,H2+ 1), the total energy tends as the bond length is stretched to a valuethat is much more negative than the expected energy (e.g., that of a neutral

H atom) (2) For a heteronuclear diatomic AB, the selfconsistent densitytends as the bond length is stretched toA+qBq, whereq is a spurious frac-tional charge (Ruzsinszky et al., 2006), and not to the correct configuration

of neutral atoms A and B These errors, and other charge-transfer errors, arecorrected by PZ SIC (Ruzsinszky et al., 2006)

When two species react with one another, they form an intermediate ortransition state whose energy is typically higher than the energies of the reac-tants and products, producing an energy barrier for the reaction The tran-sition state tends to be loosely bound, with long bond lengths and often withradical or spin-polarized character Semilocal approximations severelyunderestimate the barrier heights, to which reaction rates are very sensitive.But PZ SIC produces much more realistic barrier heights (Patchkovskii andZiegler, 2002)

Perdew and Zunger (1981)noted that their SIC, applied to atoms withfractional electron number, led to linear or nearly linear variation of the totalenergy between adjacent integer electron numbers, and to derivative dis-continuities of the total energy at integer electron number Soon after,

Perdew et al (1982) proved that the exact total energy varies linearlybetween the integers The derivative discontinuities or cusps at the integersexplain (Perdew et al., 1982) why separated atoms and molecules are exactlyneutral and not fractionally charged This property of the exact energy hasbeen called many-electron self-interaction freedom (Cohen et al., 2007;

Ruzsinszky et al., 2006) It is well approximated in PZ SIC, not just becausethis theory is one- electron self-interaction free but more importantlybecause PZ SIC captures the correct Hartree self-interaction correction

of Eq (11) (as other one-electron self-interaction corrections may not)

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 missingfrom semilocal functionals, and the correct SIC description of charge-transfer and excitonic excitations in time-dependent DFT (Hofmann

et al., 2012b)

Finally, note that PZ SIC provides anab initio alternative to the LSDA+Umethod for the description of strongly correlated systems including materialswith 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) indensity functional theory In part, this is a consequence of the fact that it

is computationally more demanding than the semilocal functionals and isunavailable in many popular computer codes Studies of SIC-LSDA formolecules (Cole and Perdew, 1982; Pederson et al., 1984, 1985) found amoderate improvement over LSDA in atomization energies, where how-ever GGA and meta-GGA were somewhat better than SIC-LSDA Theroad more travelled climbs the ladder of approximations from LSDA toGGA 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 acomprehensive 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 functionalssuch as PBE1PBE (Ernzerhof and Scuseria, 1999), and B3LYP (Stephens

et al., 1994) For the atomization energies of the 55 molecules in theG2-1 data set (using separated neutral atoms as a reference), they found that

PZ SIC improves agreement with experiment only for LSDA, while allother functionals perform worse with SIC They also found that the self-interaction error of the valence orbitals has the same order of magnitudefor all the tested functionals They wrote that: “The performance of SIC-DFT in comparison with the regular Kohn–Sham DFT is ambivalent

On the one hand, SIC is crucial for proper description of odd-electron

7

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

systems, improves activation barriers for chemical reactions, and improvesnuclear magnetic resonance chemical shifts On the other hand, it provideslittle or no improvement for reaction energies and results in too-short bondlengths 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 local or hybrid function, impose the additional exact constraint of self-interaction freedom, and find that some properties are significantlyimproved and others are worsened as a result? The answer is not knownfor certain, but there are several different possible interpretations

semi-One possible interpretation is that most semilocal functionals are exactfor 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 betweensemilocal exchange and semilocal correlation This error cancellation isimportant in molecules, where a combination of 100% exact exchange(a ¼ 1 in Eq (8)) with semilocal correlation fails rather badly This errorcancellation occurs because the exact exchange-correlation hole is deeperand more short ranged (more local) than are the exact exchange hole orthe exact correlation hole The self-interaction error of semilocal exchangemay 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 interaction correction in many-electron regions, without changing it inone-electron regions The scaling depended upon a single parameter whichcould be chosen to yield greatly improved atomization energies and bondlengths for molecules This looked like a possible solution at the time,but later Ruzsinszky et al (2006) found that the many-electron self-interaction error returned under this scaling, and so did the spuriousfractional-charge dissociation of heteronuclear molecules The proposedexplanation was that the exact Hartree self-interaction correction of

self-Eq (11) was also lost by the scaling

A glimpse of another explanation was also presented byVydrov et al.(2006) They observed that the one-electron densities niσ of Eq (10) areoften noded, and vary rapidly near the nodes, in the sense that the dimen-sionless density gradient

and the dimensionless Laplacian q ∝ jr2nj=n5=3 for n ¼ niσ both divergethere They suggested that, while accuracy for nodeless densities increasesfrom LSDA to GGA to meta-GGA, accuracy fornoded densities may actuallydecrease along this sequence This is the interpretation we will explorefurther here

A semilocal density functional for the exchange-correlation energy can

be associated (Constantin et al., 2009) with a localized approximateexchange-correlation hole around an electron which integrates to 1 Thus

a necessary condition for the success of any semilocal approximation is thatthe exact exchange-correlation hole is similarly localized around an electron.This condition is typically met in sp-bonded molecules and solids near equi-librium But it is not met in those stretched-bond situations where SIC isgreatly needed For example, in stretchedH2+ 1, when an electron is close

to one proton half its exact exchange-correlation hole is located aroundthe 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 dict a good atomization energy (with respect to separated neutral atoms) or agood equilibrium geometry, for a molecule or solid near equilibrium Now, if

pre-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 smallenough But that in turn requires thatExcapprox[niσ] must be accurate enough

to nearly cancelU[niσ] for compact but noded one-electron densities.Recently Perdew, Ruzsinszky, Sun and Burke (Perdew et al., 2014) haveemployed rigorous results from Lieb and Oxford (Lieb and Oxford, 1981) toshow that the exact exchange energy of any spin-polarized one-electrondensity satisfies a tight lower bound that requires

Esl x

Eunif x

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

meta-Esl x

Eunif x

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

τWjrnj2

/8n ¼ τ and jn" n#j/n ¼ 1, a meta-GGA can be constructed

to satisfy the tight bound of Eq (13) for all one-electron densities Anotheroften-neglected exact constraint (Perdew et al., 2014), which makes theratio on the left-hand side of Eq (13) vanish likes1/2as s ! 1, can also

be imposed and might also reduce the self-interaction error for compactnoded orbital densities Applying PZ SIC to a meta-GGA that satisfies theseadded constraints could give a theory of almost everything that works forboth stretched and equilibrium bonds (The long-range van der Waals inter-action could not be captured by such a theory, butintermediate-range van derWaals might still be usefully described (Sun et al., 2012,2013))

We cannot yet say if this dream will ever be realized But a tantalizingclue 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+0.12 hartree for the LSDA (ELSDAx

E unif

x ¼ 1ð21 =3Þ), and  0.14 hartree for thePBE GGA and TPSS meta-GGA (EunifEsl

x  1:804ð21 =3Þ) So it is tempting toimagine that the valence-shell self-interaction error could be very smallfor a meta-GGA satisfying the tight bound of Eq (13) for all spin-polarizedone-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 ofexchange and correlation (Sun et al., 2015c)

ACKNOWLEDGMENTS

This work was supported by the National Science Foundation under Grant No

DMR-1305135 (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.

APPENDIX DO COMPLEX ORBITALS RESOLVE THEPARADOX OF SIC?

The orbitals ϕiσ(r) used to make the self-interaction correction inEqs (9) and (10) have traditionally been chosen to be real Real orthonormalorbitals are necessarily noded, and noded orbitals present a special challenge

to existing semilocal functionals It has been further noted (Pederson andPerdew, 2012) that any set of real orbitals that satisfy all the localization equa-tions (Pederson and Lin, 1988; Pederson et al., 1984,1985) are stationarywith respect to an infinitesimal complex unitary transformation on any pair

of localized orbitals

Recently the Jo´nsson group at the University of Iceland has proposedthat 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 coincidewith the node of the imaginary part) This eliminates the nodes of the orbitaldensities, which are then less challenging to existing semilocal functionals Infact, it has been found that the total energies of atoms within the TPSS meta-GGA are worsened by PZ SIC with real orbitals, but slightly improved by

PZ SIC with complex orbitals (Lehtola and Jo´nsson, 2014a,b) and furthernoted (Hofmann et al., 2012a) that complex orbitals may not eliminate allthe nodes of the orbital densities, but reduce the number of nodal planesand particularly eliminate the nodes in the most energetically importantregions of space

Complex orbitals bring another benefit in variational approaches thatchoose the orbitalsϕiσ(r) to minimize the SIC energy: The extra variationalfreedom can further lower the SIC energy This increases the chance thatlocalized orbitals can be found variationally Recall that SIC is size-consistentonly when all the occupiedϕiσ(r) are localized Going to complex orbitals isthus also helpful for size-consistency within a variational approach like thosenext 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 SICenergy will be minimized by localized orbitals for all possible systems We donot know of any cases where the energy-minimizing orbitals are not local-ized in SIC-LSD, but this is not guaranteed for higher level functionalswhere the self-interaction correction from a localized orbital can be positive

As an example, consider the Ar atom, where (Vydrov and Scuseria, 2004)

11

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

the self-interaction correction from the localized valence orbitals is 0.12hartree for LSDA but +0.14 hartree for the TPSS meta-GGA Then in ahighly expanded lattice of Ar atoms the energy- minimizing SIC valenceorbitals will be localized atomic orbitals in SIC-LSDA but delocalized Blochorbitals (with zero self-interaction correction) in SIC-TPSS; the valenceself-interaction correction to the energy will be present in the SIC-TPSSsingle 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 antee size-consistency (Perdew, 1990) for all possible systems

Cole, L., Perdew, J., 1982 Calculated electron affinities of the elements Phys Rev A

Ernzerhof, M., Scuseria, G., 1999 Assessment of the Perdew-Burke-Ernzerhof correlation functional J Chem Phys 110, 5029–5036.

exchange-Gunnarsson, O., Lundqvist, B., 1976 Exchange and correlation in atoms, molecules, and solids by spin-density functional formalism Phys Rev B 13, 4374–4398.

Hofmann, D., Ku¨mmel, S., 2012 Integer particle preference during charge transfer in Kohn-Sham theory Phys Rev B 86, 201109.

Hofmann, D., Klu¨pfel, S., Klu¨pfel, P., Ku¨mmel, S., 2012a Using complex degrees of freedom in the Kohn-Sham self-interaction correction Phys Rev A 85, 062514 Hofmann, D., K€orzd€orfer, T., Ku¨mmel, S., 2012b Kohn-Sham self-interaction correction

in real time Phys Rev Lett 108, 14601–14605.

Hughes, I., Daene, M., Ernst, A., Hergert, W., Luders, M., Poulter, J., Staunton, J., Svane, A., Szotek, Z., Temmerman, W., 2007 Lanthanide contraction and magnetism

in the heavy rare earth elements Nature 446, 650–653.

Klu¨pfel, S., Klu¨pfel, P., Jo´nsson, H., 2011 Importance of complex orbitals in calculating the self-interaction-corrected ground state of atoms Phys Rev A 84, 050501.

Klu¨pfel, S., Klu¨pfel, P., Jo´nsson, H., 2012 The effect of the Perdew-Zunger self-interaction correction to density functionals on the energetics of small molecules J Chem Phys.

Perdew-Lehtola, S., Jo´nsson, H., 2014b Erratum: Variational self-consistent implementation of the Perdew-Zunger self-interaction correction with complex optimal orbitals J Chem Theory Comput 11, 839.

Lieb, E., Oxford, S., 1981 Improved lower bound on the indirect coulomb energy Int.

self-Pederson, M., Lin, C., 1988 Localized and canonical atomic orbitals in self-interaction corrected local density functional formalism J Chem Phys 88, 1807–1817.

Pederson, M., Perdew, J., 2012 Self-interaction correction in density functional theory: The road less traveled Psi-k Newslett 109, 77–100.

Pederson, M., Heaton, R., Lin, C., 1984 Local-density Hartree-Fock theory of electronic states of molecules with self-interaction correction J Chem Phys 80, 1972–1975 Pederson, M., Heaton, R., Lin, C., 1985 Density functional theory with self-interaction correction: Application to the lithium molecule J Chem Phys 82, 2688–2699 Pederson, M., Heaton, R., Harrison, J., 1989 Metallic state of the free-electron gas within the self-interaction-corrected local-spin-density approximation Phys Rev B 39, 1581–1586.

Pederson, M., Ruzsinszky, A., Perdew, J., 2014 Communication: Self-interaction tion with unitary invariance in density functional theory J Chem Phys 140, 12110 Perdew, J., 1990 Size-consistency, self-interaction correction, and derivative discontinuity

correc-in density functional theory In: Trickey, S (Ed.), In: Density Functional Theory of Many-Fermion Systems, Advances in Quantum Chemistry, 21, pp 113–134 Perdew, J., Wang, Y., 1986 Accurate and simple density functional for the electronic exchange energy: Generalized gradient approximation Phys Rev B 33, 8800–8802 Perdew, J., Zunger, A., 1981 Self-interaction correction to density functional approxima- tions for many-electron systems Phys Rev B 23, 5048–5079.

Perdew, J., Parr, R., Levy, M., Balduz, J., 1982 Density-functional theory for fractional particle number: Derivative discontinuities of the energy Phys Rev Lett 49, 1691–1694.

13

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

Perdew, J., Burke, K., Ernzerhof, M., 1996 Generalized gradient approximation made ple Phys Rev Lett 77, 3865–3868.

sim-Perdew, J., Kurth, S., Zupan, A., Blaha, P., 1999 Accurate density functional with correct formal properties: A step beyond the generalized gradient approximation Phys Rev Lett 82, 2544–2547.

Perdew, J., Ruzsinszky, A., Sun, J., Burke, K., 2014 Gedanken densities and exact straints in density functional theory J Chem Phys 140, 18A533.

con-Polo, V., Kraka, E., Cremer, D., 2002 Electron correlation and self-interaction error of sity functional theory Mol Phys 100, 1771–1790.

den-Polo, V., Gra¨fenstein, J., Kraka, E., Cremer, D., 2003 Long-range and short-range lation effects as simulated by Hartree-Fock, local density approximation, and generalized gradient approximation Theor Chem Acc 109, 22–35.

corre-Ruzsinszky, A., Perdew, J., Csonka, G., Vydrov, O., Scuseria, G., 2006 Spurious fractional charge on dissociated atoms: Pervasive and resilient self-interaction error of common density functionals J Chem Phys 125, 194112.

Stephens, P., Devlin, F., Chabalowski, C., Frisch, M., 1994 Ab initio calculation of tional absorption and circular dichroism spectra using density functional force fields.

lad-Toher, C., Filippetti, A., Sanvito, S., Burke, K., 2005 Self-interaction errors in functional calculations of electronic transport Phys Rev Lett 95, 146402.

density-Van Voorhis, T., Scuseria, G., 1998 A novel form for the exchange-correlation functional.

Koblar Alan Jackson1

Physics Department and Science of Advanced Materials Program, Central Michigan University, Mt Pleasant, Michigan, USA

1 Corresponding author: e-mail address: jacks1ka@cmich.edu

Contents

1 INTRODUCTION

The self-interaction-correction (SIC) paper of Perdew and Zunger(1981)represented an exciting step forward for the field of density functionaltheory (DFT) The SIC addressed a clear defect present in DFT and theresults presented in that work showed that the SIC is very successful when

Advances in Atomic, Molecular, and Optical Physics, Volume 64 # 2015 Elsevier Inc.

applied to atomic systems However, as shown by Pederson et al (1984,1985)and discussed in detail elsewhere in this review, the orbital-dependentnature of the theory makes applying DFT-SIC to multiatom systemsdifficult They showed that two sets of orbitals are required to implementDFT-SIC The canonical orbitals (CO) reflect the symmetry of the multi-atom system and the one-electron energies corresponding to the CO rep-resent approximate electron removal energies The CO are connected by

a unitary transformation to the local orbitals (LO) that are the basis forthe correction terms in DFT-SIC The variationally correct LO thatminimize the DFT-SIC total energy must also satisfy an additional set ofequations, the localization equations (LE) Simultaneously satisfying theDFT-SIC Kohn–Sham equations with the CO and the LE with the LO

is challenging The lack of an easily implemented solution for finding thecorrect LO has prevented a more widespread use of DFT-SIC

One detour around the LO problem is to study multiatom systems thatpossess 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 Cl
ions The free-ion orbitals are thus goodstarting points for the LO In the mid to late 1980s, the Wisconsin group ofLin applied DFT-SIC to a series of alkali-halide-based systems, takingadvantage of the atomic-like features (Erwin and Lin, 1988, 1989;Harrison et al., 1983; Heaton and Lin, 1984; Heaton et al., 1985; Jacksonand Lin, 1988, 1990)

One problem that made the alkali halides interesting to study involvedthe fundamental band gap energy It is well known that the local spin density(LSD) form of DFT underestimates the valence–conduction band gaps ofinsulating solids by 30–50% For NaCl, for example, use of exchange-onlyLSD 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-localizedconduction band (CB) states The SIC could therefore be expected to openthe gap

A second problem involved substitutional impurities Transition metalimpurities in alkali-halide crystals were being studied actively in the early1980s as prototype solid-state impurity systems (Payne et al., 1984; Pedrini

et al., 1983; Simonetti and McClure, 1977) The impurity ions introduceunoccupied defect states into the wide band gap of the host material.Transitions to these gap states give rise to absorption in the visible andnear u–v, whereas the onset of band gap absorption occurs at much higher

energies A schematic of the relevant one-electron energy levels is given in

Fig 1 Note that the impurity ion d-states are split by the host crystal fieldinto the twofold eg and threefold t2g levels The detailed nature of theimpurity states, for example, their positions relative to the host VB and

CB states, cannot be determined on the basis of experimental observationsalone This provided ample motivation for theoretical study But model-ing these systems using uncorrected DFT fails, in large part because theband gaps of the host crystal are so badly underestimated In some cases,the observed impurity transition energies are larger than the DFTband gap

The DFT-SIC is an ideal approach for treating the impurity problem Asmentioned above, it was clear that use of the SIC could help to open theband gap Further, because the positions of electron energy levels appeared

to be more physically meaningful in SIC calculations for atoms, it was sonable to expect that the impurity levels would be more properly placedrelative to the host energy bands in a DFT-SIC calculation than inuncorrected DFT Jackson and Lin addressed two systems, NaCl:Cu+andLiCl:Ag+ (Jackson and Lin, 1988, 1990) These calculations are described

rea-in the followrea-ing sections Erwrea-in and Lrea-in also treated a similar system,NaF:Cu+(Erwin and Lin, 1989)

The ingredients needed for the impurity system calculations included(i) an accurate treatment of the free transition metal ions; (ii) the purealkali-halide calculation; and (iii) an embedded-cluster approach to theimpurity crystal The computational machinery needed to implementDFT-SIC in each of these settings is reviewed in the following sections

Figure 1 Schematic energy level diagram for the NaCl:Cu +

and LiCl:Cu+impurity tems Examples of the nd!(n+1)s and (n+1)p transitions observed in experiments are indicated.

sys-17

Self-Interaction Correction Treatment of Substitutional Defects

is the SIC potential for orbital i withρi¼ ϕj ji 2

The local density form ofDFT with the exchange-only version of vxc was used in all calculationsdescribed in this section:

is equivalent to Eq.(1)at self-consistency, withεi¼λii Since in Eq.(7)the

ϕs are eigenfunctions of the same operator, they are automaticallyorthogonal

The transition energy for an electron moving from state a to state b can beapproximated (Harrison et al., 1983) using the orbital energies:

is precisely what the electron described by ϕb would experience in theexcited state, neglecting any relaxation of the remaining N
1 orbitals.Including orbital relaxation has a relatively small effect on the calculatedtransition energies (Heaton et al., 1987)

Applying the formalism outlined above using the exchange-only form of

vxc, we obtained 3d!4s and 3d!4p transition energies of 3.21 and9.21 eV for Cu+ These agree well with observed values of 3.03 and8.81 eV, respectively For reference, the corresponding eigenvalue differ-ences in uncorrected exchange-only DFT are 1.87 and 6.93 eV Use ofthe SIC clearly improves the agreement with experiment For the4d!5s and 4d!5p transitions in Ag+, the SIC calculations yield 5.50and 10.5 eV, respectively, close to the corresponding experimental values

of 5.37 and 10.8 eV Without the SIC, the exchange-only DFT gives4.86 and 9.25 eV for these transitions Again the use of SIC clearly improvesthe calculated transition energies

19

Self-Interaction Correction Treatment of Substitutional Defects

3 PURE CRYSTAL CALCULATION

For a translationally periodic solid, the LO are the Wannier functions(WF), while the CO are the corresponding delocalized Bloch functions Ingeneral, finding the exact WF is a difficult problem; however,Heaton andLin (1984)andErwin and Lin (1988)described a method for obtaining sim-ple approximate Wannier functions for alkali-halides that can be traced tothe atomic-like character of the density in these solids For core energybands, the WF are simply taken to equal the corresponding free-ion orbitals.For the VB, which derives from the halide p orbitals, the approximate WF at

a given halide site remains largely free-ion-like, but includes small tions from the six nearest-neighbor alkali sites The SIC orbital energies forthe VB and CB states were only weakly sensitive to the precise form of the

contribu-WF (Erwin and Lin, 1988)

With the definition of the approximate WF as the LO, the correspondingSIC potentials (ViSIC) can be computed and the SIC equations formulated forthe CO as follows (Pederson et al., 1984):

an average over all sub-bands), and the corresponding SIC potentials,

is the uncorrected LSD Hamiltonian This reflects the delocalized nature

of the CB states

The self-consistent eigenvalues of Eq.(11)represent the calculated bandstructure for the perfect crystal Using the exchange-only version of vxc, theresults for the fundamental band gaps for LiCl and NaCl obtained in bothuncorrected LSD and the corresponding LSD-SIC are given in Table 1

and compared to experiment It is clear that the SIC reverses the timation of the band gap by LSD A different choice of the exchange-correlation potential can bring the LSD-SIC value of the gap into betteragreement with the experimental value (Erwin and Lin, 1988)

underes-The self-consistent pure crystal charge density can be decomposed bycurve fitting the total density into a lattice summation of localized densities:

ρAandρHintegrate to the expected number of electrons for the respectivefree ions

4 EMBEDDED-CLUSTER APPROACH TO ISOLATEDIMPURITIES

In the impurity crystal, the transition metal ion occupies an alkali siteand has the same +1 net charge as the alkali ion it replaces Because of this,the perturbation due to the impurity is limited to the immediate vicinity ofthe substitutional site The goal in treating the impurity system is to accu-rately represent the changes brought about by the impurity over a wideenough region of the solid to capture all the effects of the perturbation

Table 1 Computed and Observed Values (in eV) of the Fundamental Band Gap

of Pure Alkali-Halide Crystals

a Baldini and Bossachi (1970)

b

Nakai and Sagawa (1969)

The computed values were obtained using an exchange-only version of V xc

21

Self-Interaction Correction Treatment of Substitutional Defects

To do this, we used an embedded-cluster approach The method solves thefull Hamiltonian of the infinite solid using an orbital basis for the electronicstates that extends only over a finite spatial region in the vicinity the impu-rity By carefully choosing the basis the electronic charge density within thecluster region is faithfully reproduced (Heaton et al., 1985).

The cluster includes the impurity ion at its center and host crystal ionsextending out to the seventh symmetry shell around the impurity, or the(220) shell (in units of the nearest-neighbor separation in the perfectalkali-halide crystal) This includes a total of 93 atoms, as shown in

Fig 2 The rocksalt structure of the host crystal is evident in the figure

No lattice relaxation of the host crystal ions was included in the calculations.The electronic basis set for the cluster includes optimized atomic orbitalstaken from the respective free-ion calculations They are expressed as linearcombinations of Gaussian-type functions In addition, extra single Gaussian-type orbitals are placed at the impurity site and on the atoms in the first threenearest-neighbor shells of the cluster to increase the variational freedom ofthe calculations Minimal basis sets are placed on the atoms in the outer shells

of the cluster This “cushion” limits the overlap of basis functions on sitesexternal to the embedded cluster and prevents the formation of unphysical

“ghost” states (Heaton and Lin, 1984)

Figure 2 The cluster used for the impurity crystal studies The impurity ion (Cu +

or Ag+)

is shown in brown (dark gray in the print version) at the center of the cluster, the alkali ions (Na+or Li+) are depicted in blue (black in the print version), and the Cl
ions in green (light gray in the print version) The electronic basis sets on the atoms in the inte- rior of the cluster have significant variational freedom, while those on the atoms near the surface are minimal basis sets.

The electronic charge density in the impurity crystal,ρIC, can be written

as the sum of the perfect crystal density, ρPC, and a localized differencecharge,δρ:

The Kohn–Sham Hamiltonian for the impurity crystal can be similarlyexpressed in terms of the perfect crystal Hamiltonian plus a localized differ-ence potential:

We note that hPCin Eq.(16)includes contributions from an infinite array

of positive and negative host ions located outside the embedded cluster.These can be included accurately and efficiently using multipole-based tech-niques Equation(17)only includes the DFT part of the Hamiltonian Weconsider the additional SIC terms below

The initial difference charge density,δρ(0), is taken to be the differencebetween the free transition metal ion charge density and ρA(r), the alkalidensity taken from the pure crystal calculation In subsequent iterations,the wave functions obtained from hICare used to formρICand from that

δρ(1)

is found from Eq.(15) After each iteration,δρ is fit to exponential typefunctions centered at the origin and surrounding atoms.δVLSD

can then becomputed analytically, tabulated on a grid, and fitted to Gaussian-type func-tions, again centered at the origin and the positions of the surroundingatoms Hamiltonian matrix elements can then be evaluated analyticallyfor the Gaussian-type basis functions

Appropriate LO are needed to implement SIC in the impurity crystalproblem As with the other problems discussed here, free-ion orbitals can

be used as LO for the core levels of both the impurity and the host crystalions For the valence states, the proper choices are the generalized Wannierfunctions (GWF) described by Kohn and Onffroy (1973) The GWF aresite-dependent, but rapidly approach the pure crystal WF away from theimpurity

23

Self-Interaction Correction Treatment of Substitutional Defects

In practice, the eigenfunctions of the impurity-related states are, to goodapproximation, admixtures of orbitals from the impurity site and the nearest-neighbor Cl
sites In the case of NaCl:Cu+, the impurity 3d states arestrongly localized at the Cu+ site and the admixture of orbitals from theneighboring Cl
sites is very small A Mulliken analysis shows 95% and99% of the charge associated with the impurity site for the 3deg and 3dt2gstates, respectively In the case of LiCl:Ag+, the impurity states involve amuch stronger admixture of first-neighbor Cl
basis orbitals For the 4degstates, for example, a Mulliken analysis gives 62% of the charge on the impu-rity site and 36% on the first-neighbor Cl
sites For the 4dt2g states, thecomparable values are 73% and 20% This strong mixing implies bondingpartners of these states in the VB that involve a majority Cl
orbital com-plement with an admixture of impurity orbitals For the eglevel, the VB part-ner has 33% and 63% of its charge on the impurity site and the first-neighborshell For the t2glevel, the comparable values are 13% and 73%.

Approximate GWF can be found from the eigenvector coefficients ofthe impurity and VB states using the technique described by Heaton andLin (1984) We find that the resulting GWF charge densities centered on theimpurity site are very close to that of the free-ion nd densities Similarly,the GWF densities for the first-neighbor shell of Cl
ions are very close

to the free-ion 3p densities centered on those sites We therefore usedthe relevant free-ion densities to construct SIC potentials

Since the impurity nd states are localized on the impurity site and thefirst-neighbor shell, we adopted the following density-weighted form ofthe SIC potential for these states:

Given these forms for ΔVnSIC

, the SIC equations for the CO for theimpurity system can be constructed as in Eq (11) These can be solvedself-consistently using the unified Hamiltonian formalism In this case,

hexc¼ha, where “a” represents ndegor ndt2gto compute the correspondingtransition energies

The results of exchange-only LSD-SIC calculations for the related transition energies are shown in Table 2, along with thecorresponding experimental values For convenience, the correspondingfree-ion transitions are shown as well It can be seen that LSD-SIC valuesfor the transition energies are in good agreement with the experimentalvalues and capture key physical features including (i) the nd!(n+1)s tran-sition energy is increased over that of the corresponding free-ion; (ii) the

impurity-nd!(n+1)p transition energy is decreased relative to the free ion; and(iii) the crystal field splitting of the egand t2gstates is in good agreement withTable 2 Theoretical and Experimental Values of the nd!(n+1)s and nd!(n+1)p Transition Energies (in eV) for the Free Cu+and Ag+Ions and the NaCl:Cu+and LiCl:Ag+Impurity Systems

Transitions

Theory (eV) Expt (eV) Theory (eV) Expt (eV) NaCl:Cu+

observed values (0.51 vs 0.41 eV for NaCl:Cu+ and 1.33 vs 1.35 eVfor LiCl:Ag+).

The transition energies obtained using Eq.(8)involve an excited electronwith the same spin as the electron in the ground state Each energy thereforerepresents an average of the transitions to the singlet and triplet excited states

A detailed treatment of the singlet–triplet splittings in the impurity transitionenergies, along with spin–orbit effects was given byErwin and Lin (1989)forthe NaF:Cu+system A related treatment of multiplet-dependent orbital wavefunctions can be found inJackson and Lin (1989)

It is interesting to consider the results of an uncorrected DFT calculationfor the impurity system For NaCl:Cu+, the 3degand 3dt2gstates lie in the gapabove the VB edge, and the impurity 4s and 4p levels lie above the CB edge.The energy difference between the 3degand 4s states is 2.56 eV, badly under-estimating the observed value of 4.36 eV

5 DISCUSSION

Despite the success of the calculations reviewed above, the use ofLSD-SIC for treating substitutional impurities did not become widespread.There are at least two reasons for this One is that, at the time these calcu-lations were done in the late 1980s, methods for obtaining accurate totalenergies for multiatom systems were not available This limited how reliablyimpurity systems could be modeled For example, without total energies,host ion relaxation around the impurity site cannot be predicted Such relax-ation can clearly affect the degree of admixture of the impurity and host crys-tal VB states and hence the position of the impurity energy levels and thetransition energies Relaxation effects are expected to be especially impor-tant in cases where there is a mismatch between the size of the impurity andhost crystal cations

A second problem involves the approximations used for the LO and thecorresponding SIC potentials As mentioned at the outset of this chapter, theproblems discussed here were chosen because the LO could be plausiblyapproximated by orbitals from free-ion calculations These approximationswere physically motivated and were validated after the fact by the success ofthe results Yet these systems are special cases In the general case where thebonding may be a more complicated mixture of ionic, covalent, or evenmetallic character, simple approximations for the LO are lacking For suchsystems, a computationally efficient means of finding the variationally cor-rect LO remains elusive

self-Heaton, R.A., Pederson, M.R., Lin, C.C., 1987 A new density functional for fractionally occupied orbital systems with application to ionization and transition energies J Chem Phys 86, 258–267.

Jackson, K.A., Lin, C.C., 1988 Ground and excited states of the NaCl:Cu+impurity system Phys Rev B 38, 12171–12183.

Jackson, K.A., Lin, C.C., 1989 Multiplet-dependent wave functions from the density approximation with self-interaction correction Phys Rev B 39, 1557–1563 Jackson, K.A., Lin, C.C., 1990 Theory of the electronic states and absorption spectrum of the LiCl:Ag+impurity system Phys Rev B 41, 947–957.

local-spin-Kohn, W., Onffroy, J.R., 1973 Wannier functions in a simple nonperiodic system Phys Rev B 8, 2485–2495.

Nakai, S., Sagawa, T., 1969 Na+L 2,3 absorption spectra of sodium halides J Phys Soc Jpn.

Perdew, J.P., Zunger, A., 1981 Self-interaction correction to density-functional tions for many-electron systems Phys Rev B 23, 5048–5079.

approxima-Simonetti, J., McClure, D.S., 1977 The 3d -> 4s transitions of Cu +

in LiCl and of metal ions in crystals Phys Rev B 16, 3887–3892.

transition-27

Self-Interaction Correction Treatment of Substitutional Defects

CHAPTER THREE

Electronic Transport as a Driver for Self-Interaction-Corrected

Methods

Anna Pertsova*, Carlo Maria Canali*, Mark R Pederson†,1,

Ivan Rungger{, Stefano Sanvito{

*Department of Physics and Electrical Engineering, Linnæus University, Kalmar, Sweden

†

Department of Chemistry, Johns Hopkins University, Baltimore, Maryland, USA

{School of Physics, AMBER and CRANN Institute, Trinity College, Dublin, Ireland

1 Corresponding author: e-mail address: mpeder10@jhu.edu

Contents

3.2 Electron Transport in Layered Solid State Structures 45

4 Derivative Discontinuity of Exchange –Correlation Functional 47

4.2 Derivative Discontinuity and Atomic Self-Interaction Correction 50

5 Recent Developments: DFT-NEGF with Improved Exchange-Correlation

6.1 Master Equation Approach for Tunneling Transport 66

6.1.2 Coulomb Blockade Regime: Sequential Tunneling 68

6.2.1 Many-Body Description of Ground-State and Low-Lying Excitations 72

a molecule, or localized on an impurity or dopant The issue considered in this chapter

Advances in Atomic, Molecular, and Optical Physics, Volume 64 # 2015 Elsevier Inc.

involves taking this extreme to the nanoscale and the quest to use first-principles methods to predict and control the behavior of a few “spins” (down to 1 spin) when they are placed in an interesting environment Particular interest is on environments for which addressing these systems with external fields and/or electric or spin currents

is possible The realization of such systems, including those that consist of a core of a few transition-metal (TM) atoms carrying a spin, connected and exchanged-coupled through bridging oxo-ligands has been due to work by many experimental researchers

at the interface of atomic, molecular and condensed matter physics This chapter addresses computational problems associated with understanding the behaviors of nano- and molecular-scale spin systems and reports on how the computational com- plexity increases when such systems are used for elements of electron transport devices Especially for cases where these elements are attached to substrates with electroneg- ativities that are very different than the molecule, or for coulomb blockade systems, or for cases where the spin-ordering within the molecules is weakly antiferromagnetic, the delocalization error in DFT is particularly problematic and one which requires solutions, such as self-interaction corrections, to move forward We highlight the intersecting fields of spin-ordered nanoscale molecular magnets, electron transport, and coulomb blockade and highlight cases where self-interaction corrected methodologies can improve our predictive power in this emerging field.

In density functional theory (DFT)-based quantum electron transport ulations, one of the fundamental approximations is that the DFT eigenvaluesare assumed to correspond to quasi-particle energies, so that the Kohn–Shamstates are used to evaluate the current We discuss the range of validity ofsuch approximation and outline its limitations, both for the low bias trans-port as well as for larger applied bias voltages In particular, the effects of theself-interaction error are described, and possible practical corrections pres-ented for a set of systems

sim-One of the factors determining the conductance in molecular ojunctions is the energy level alignment between electrodes and mole-cules Within the local density approximation (LDA) or the generalizedgradient approximation (GGA) the molecular orbitals are usually too close

nan-to the metal Fermi energy, which results in overestimated conductances.Adding a self-interaction correction improves the description of theenergy level alignment, and thereby reduces the current for a given volt-age In particular, we consider application of the atomic self-interactionscheme (ASIC) In the case of molecules adsorbed at small distances tothe metal surface, one also needs to take into account that a transfer ofcharge to the molecule leaves a screening image charge on the metal sur-faces This additional screening can be captured either using the GWapproximation, or else within DFT by evaluating charge transfer energies

using a constrained DFT approach We show that for such small molecules

in close proximity to the metal surface, in order to obtain quantitativelycorrect conductances one therefore needs to include both self-interactioncorrections, as well as the energy level renormalization due to such imagecharge formation We conclude the linear response transport section bypresenting results for structures with thin films made of oxide materials,where the LDA/GGA underestimation of the band gap leads again to

an overestimation of the conductance, which we correct by applyingthe ASIC

We then discuss a fundamental property of the exact exchange tion (XC) functional, namely the derivative discontinuity, and its implica-tions for quantum transport Using a simple parametrization for the XCfunctional which restores the derivative discontinuity, we present the errorsarising in typical transport calculations for molecular junctions described byDFT with semi-local functionals Furthermore, using an effective tight-binding model, we show that a self-interaction correction scheme is able

correla-to reintroduce, in an approximate way, the derivative discontinuity As aresult, such scheme correctly reproduces the conducting regime of a weaklycoupled molecular junction As a more elaborate example of an approxima-tion to the XC functional which includes a derivative discontinuity, we con-sider the Bethe ansatz local density approximation, which is based on exactsolution of the one-dimensional Hubbard model We illustrate the perfor-mance of this functional in describing Coulomb blockade (CB) in a singlequantum dot junction, both in time-dependent and steady-state transportregimes We conclude this section with an overview of recent developments

on the use of DFT in the theory of steady-state transport though interactingnanojunctions

In the final part of the chapter, we focus on ab initio theory of tunnelingquantum transport through magnetic molecules based on a quantum masterequation approach We revisit the methodological challenges in describingthese systems using standard DFT techniques containing self-interactionerrors, particularly when the molecule acquires extra electronic charge as

a result of quantum transport in the CB regime The basics of CB physics

in single-electron-transistor devices and the quantum master equationapproach to CB transport are discussed We then review recent attempts

of constructing approximate many-body wave functions describing chargedelectronic states (both ground and excited states) of the molecule plus leadssystem, which are the basic ingredients in the theory of CB quantum trans-port based on the quantum master equation approach

31

Electronic Transport as a Driver for Self-Interaction-Corrected Methods

1 ELECTRON TRANSPORT FORMALISM

To evaluate the electronic transport properties of nanoscale devices, awidely used approach is based on the combination of DFT with the non-equilibrium Green’s functions (NEGF) formalism (Datta, 1995; Haug andJauho, 2008), which we denote as DFT-NEGF formalism (Rocha et al.,

2005,2006) This allows to calculate the charge density at an applied biasvoltage, as well as the current flowing between the two metal electrodes.Within the NEGF formalism the system is subdivided into a central part,usually called the scattering region or extended molecule (EM), and thetwo or more semi-infinite electrodes (also denoted as leads), one on the leftand one on the right of the scattering region (Fig 1)

DFT-NEGF is usually applied using a linear combination of atomicorbitals (LCAO) basis set, where the charge density in real space is expandedover the localized basis orbitals

Extended molecule

Figure 1 Schematic setup for the electron transport calculations: the system is divided into semi-infinite left and right electrodes (also denoted as leads), bridged

sub-by an extended molecule (also denoted as scattering region).