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Appendix. Do Complex Orbitals Resolve the Paradox of SIC?

Appendix. Do Complex Orbitals Resolve the Paradox of SIC?

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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: jacks1ka@cmich.edu



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

http://dx.doi.org/10.1016/bs.aamop.2015.06.001



#



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



Self-Interaction Correction Treatment of Substitutional Defects



17



Figure 1 Schematic energy level diagram for the NaCl:Cu+ and LiCl:Cu+ impurity systems. Examples of the nd ! (n + 1)s and (n + 1)p transitions observed in experiments

are indicated.



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 field

into the twofold eg and threefold t2g levels. The detailed nature of the

impurity states, for example, their positions relative to the host VB and

CB states, cannot be determined on the basis of experimental observations

alone. This provided ample motivation for theoretical study. But modeling these systems using uncorrected DFT fails, in large part because the

band gaps of the host crystal are so badly underestimated. In some cases,

the observed impurity transition energies are larger than the DFT

band gap.

The DFT-SIC is an ideal approach for treating the impurity problem. As

mentioned above, it was clear that use of the SIC could help to open the

band gap. Further, because the positions of electron energy levels appeared

to be more physically meaningful in SIC calculations for atoms, it was reasonable to expect that the impurity levels would be more properly placed

relative to the host energy bands in a DFT-SIC calculation than in

uncorrected DFT. Jackson and Lin addressed two systems, NaCl:Cu+ and

LiCl:Ag+ ( Jackson and Lin, 1988, 1990). These calculations are described

in the following sections. Erwin and Lin 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 pure

alkali-halide calculation; and (iii) an embedded-cluster approach to the

impurity crystal. The computational machinery needed to implement

DFT-SIC in each of these settings is reviewed in the following sections.



18



Koblar Alan Jackson



2. FREE-ION CALCULATIONS

While atomic orbitals are highly localized, they do not automatically

satisfy the LE and therefore are not the variationally correct LO; however,

Pederson et al. (1985) showed that manifestly satisfying the LE leads to only

very small changes in the SIC total energy and in the orbital energies. In

other words, the orbitals in a free atom calculation obtained by solving

the SIC equations are good approximations of the LO without the extra

computational steps needed to satisfy the LE. Therefore, to obtain the wave

functions and orbital energies for a free ion, one simply solves the SIC

equations:

X

À

Á

hi ϕi ¼ h0 + ViSIC ϕi ¼

λij ϕj

(1)

j



where h0 is the one-electron Hamiltonian for uncorrected DFT and

Z

r 0 ị

SIC

Vi ẳ À dr0 i 0 À Vxc ½ρi Š

jr À r j



(2)



is the SIC potential for orbital i with ρi ¼ jϕi j2 . The local density form of

DFT with the exchange-only version of vxc was used in all calculations

described in this section:

 1=3

1

6

r ịị =3 :

vxc ẵr ị ẳ





(3)



The ϕs must be orthonormal. To insure this, a “unified Hamiltonian” is

used (Harrison et al., 1983):

hu ¼



N À

X

Á

^ i P^i + P^i hi O

^ + Oh

^ exc O;

^

P^i hi P^i + Oh



(4)



i¼1



where the projection operator P^i is defined as:

!

Z

0

0

0

dr ϕi ðr Þgðr Þ

P^i gðr Þ ¼ ϕi ðr Þ



(5)



^ projects out of the space of the

for an occupied orbital ϕi, and the operator O

N occupied orbitals:



Self-Interaction Correction Treatment of Substitutional Defects



^ ¼ ^1 À

O



N

X



P^i :



19



(6)



i¼1



With these definitions, the eigenvalue equation

hu ϕi ¼ Ei ϕi



(7)



is equivalent to Eq. (1) at self-consistency, with εi ¼ λii. Since in Eq. (7) the

ϕs are eigenfunctions of the same operator, they are automatically

orthogonal.

The transition energy for an electron moving from state a to state b can be

approximated (Harrison et al., 1983) using the orbital energies:

ΔEa, b ¼ Eb À Ea



(8)



Ea ¼ hψ a j ha j ψ a i



(9)



Eb ¼ hψ b j ha j ψ b i:



(10)



where



and



The last term is obtained by setting hexc equal to ha in hu. To understand

this choice, note that ha includes the interactions of an electron with the

nucleus and the N À 1 electrons excluding the one described by ϕa. This

is precisely what the electron described by ϕb would experience in the

excited state, neglecting any relaxation of the remaining N À 1 orbitals.

Including orbital relaxation has a relatively small effect on the calculated

transition 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 and

9.21 eV for Cu+. These agree well with observed values of 3.03 and

8.81 eV, respectively. For reference, the corresponding eigenvalue differences in uncorrected exchange-only DFT are 1.87 and 6.93 eV. Use of

the SIC clearly improves the agreement with experiment. For the

4d ! 5s and 4d ! 5p transitions in Ag+, the SIC calculations yield 5.50

and 10.5 eV, respectively, close to the corresponding experimental values

of 5.37 and 10.8 eV. Without the SIC, the exchange-only DFT gives

4.86 and 9.25 eV for these transitions. Again the use of SIC clearly improves

the calculated transition energies.



20



Koblar Alan Jackson



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. In

general, finding the exact WF is a difficult problem; however, Heaton and

Lin (1984) and Erwin and Lin (1988) described a method for obtaining simple approximate Wannier functions for alkali-halides that can be traced to

the atomic-like character of the density in these solids. For core energy

bands, 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 contributions from the six nearest-neighbor alkali sites. The SIC orbital energies for

the VB and CB states were only weakly sensitive to the precise form of the

WF (Erwin and Lin, 1988).

With the definition of the approximate WF as the LO, the corresponding

SIC potentials (VSIC

i ) can be computed and the SIC equations formulated for

the CO as follows (Pederson et al., 1984):

À

Á

(11)

hi ψ i ¼ h0 + ΔViSIC ψ i ¼ Ei ψ i

where

ΔViSIC ψ i ¼



X



Uijσ{ VjSIC ϕj :



(12)



j



Here U is the unitary transformation connecting the WF (ϕ) and the

Bloch functions (ψ). Because the charge densities of neighboring ions have

little overlap, Heaton and Lin (1984) showed that ΔVnSIC for the VB states

could be expressed to good approximation as a simple density-weighted sum

involving the approximate Wannier charge densities, ηn (the bar represents

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

 SIC

V SIC

n ẳ V n ẵn :

P SIC

V r R ịn r R ị

SIC

:

(13)

Vn rị ẳ ν nP

υ ηn ðr À Rν Þ

This potential was used for all VB states, i.e., for all k-points and all

sub-bands.

The unified Hamiltonian formalism is used to obtain the self-consistent

solution of Eq. (11). For the perfect crystal calculation hexc ¼ h0, where h0



21



Self-Interaction Correction Treatment of Substitutional Defects



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

of Pure Alkali-Halide Crystals

LSD (eV)

LSD-SIC (eV)

Expt. (eV)



LiCl



5.8



11.1



9.4a



NaCl



4.7



9.6



8.6b



a



Baldini and Bossachi (1970).

Nakai and Sagawa (1969).

The computed values were obtained using an exchange-only version of Vxc.



b



is the uncorrected LSD Hamiltonian. This reflects the delocalized nature

of the CB states.

The self-consistent eigenvalues of Eq. (11) represent the calculated band

structure for the perfect crystal. Using the exchange-only version of vxc, the

results for the fundamental band gaps for LiCl and NaCl obtained in both

uncorrected LSD and the corresponding LSD-SIC are given in Table 1

and compared to experiment. It is clear that the SIC reverses the underestimation of the band gap by LSD. A different choice of the exchangecorrelation potential can bring the LSD-SIC value of the gap into better

agreement with the experimental value (Erwin and Lin, 1988).

The self-consistent pure crystal charge density can be decomposed by

curve fitting the total density into a lattice summation of localized densities:

X

AH rị ẳ

ẵA ðr À Rν Þ + ρH ðr À Rν À tފ;

(14)

ν



where ρA and ρH are densities associated with the alkali and halide sites,

respectively, Rν covers all the alkali sites in the crystal, and t connects an

alkali ion to a nearest-neighbor halide. The fits are constrained such that

ρA and ρH integrate to the expected number of electrons for the respective

free ions.



4. EMBEDDED-CLUSTER APPROACH TO ISOLATED

IMPURITIES

In the impurity crystal, the transition metal ion occupies an alkali site

and 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 of

the substitutional site. The goal in treating the impurity system is to accurately represent the changes brought about by the impurity over a wide

enough region of the solid to capture all the effects of the perturbation.



22



Koblar Alan Jackson



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 ClÀ ions in

green (light gray in the print version). The electronic basis sets on the atoms in the interior of the cluster have significant variational freedom, while those on the atoms near

the surface are minimal basis sets.



To do this, we used an embedded-cluster approach. The method solves the

full Hamiltonian of the infinite solid using an orbital basis for the electronic

states that extends only over a finite spatial region in the vicinity the impurity. By carefully choosing the basis the electronic charge density within the

cluster region is faithfully reproduced (Heaton et al., 1985).

The cluster includes the impurity ion at its center and host crystal ions

extending out to the seventh symmetry shell around the impurity, or the

(220) shell (in units of the nearest-neighbor separation in the perfect

alkali-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 orbitals

taken from the respective free-ion calculations. They are expressed as linear

combinations of Gaussian-type functions. In addition, extra single Gaussiantype orbitals are placed at the impurity site and on the atoms in the first three

nearest-neighbor shells of the cluster to increase the variational freedom of

the 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 sites

external to the embedded cluster and prevents the formation of unphysical

“ghost” states (Heaton and Lin, 1984).



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