Appendix. Do Complex Orbitals Resolve the Paradox of SIC?
Tải bản đầy đủ - 0trang
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.
REFERENCES
Becke, A., 1988. Density-functional exchange approximation with correct asymptotic
behavior. Phys. Rev. A 38, 3098–3100.
Becke, A., 1993. Density-functional thermochemistry. 3. The role of exact exchange.
J. Chem. Phys. 98, 5648–5652.
Becke, A., Roussel, M., 1989. Exchange-holes in inhomogeneous systems: A coordinate
space model. Phys. Rev. A 39, 3761–3767.
Cococcioni, M., de Gironcoli, S., 2005. Linear response approach to the calculation of the
effective interaction parameters in the LDA+U method. Phys. Rev. B 71, 035105.
Cohen, A., Mori-Sanchez, P., Yang, W., 2007. Development of exchange-correlation
functionals with minimal many-electron self-interaction error. J. Chem. Phys. 126,
191109.
Cole, L., Perdew, J., 1982. Calculated electron affinities of the elements. Phys. Rev. A
25, 1265–1271.
Constantin, L., Perdew, J., Pitarke, J., 2009. Exchange-correlation hole of a generalized gradient approximation for solids and surfaces. Phys. Rev. B 79, 075126.
Dabo, I., Ferretti, A., Marzari, N., 2014. Piecewise linearity and spectroscopic properties
from Koopmans-compliant functionals. Top. Curr. Chem. 347, 193.
Dinh, P.M., Gao, C.Z., Kluăpfel, P., Reinhard, P.G., Suraud, E., Vincendon, M., Wang, J.,
Zhang, F.S., 2014. A density functional theory study of Na(H2O)n: an example of the
impact of self-interaction corrections. Eur. Phys. J. D 68 (8), 239.
Ernzerhof, M., Scuseria, G., 1999. Assessment of the Perdew-Burke-Ernzerhof exchangecorrelation functional. J. Chem. Phys. 110, 5029–5036.
Gunnarsson, O., Lundqvist, B., 1976. Exchange and correlation in atoms, molecules, and
solids by spin-density functional formalism. Phys. Rev. B 13, 43744398.
Hofmann, D., Kuămmel, S., 2012. Integer particle preference during charge transfer in
Kohn-Sham theory. Phys. Rev. B 86, 201109.
Hofmann, D., Kluăpfel, S., Kluăpfel, P., Kuămmel, 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., Kuămmel, 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, 650653.
Kluăpfel, S., Kluăpfel, P., Jonsson, H., 2011. Importance of complex orbitals in calculating the
self-interaction-corrected ground state of atoms. Phys. Rev. A 84, 050501.
Paradox of Self-Interaction Correction: How Can Anything So Right Be So Wrong?
13
Kluăpfel, S., Kluăpfel, P., Jonsson, H., 2012. The effect of the Perdew-Zunger self-interaction
correction to density functionals on the energetics of small molecules. J. Chem. Phys.
137, 124102.
Kohn, W., Sham, L., 1965. Self-consistent equations including exchange and correlation
effects. Phys. Rev. 140, A1133–A1138.
Kurth, S., Perdew, J., 2000. Role of the exchange-correlation energy: nature’s glue. Int. J.
Quantum Chem. 77, 814–818.
Langreth, D., Mehl, M., 1983. Beyond the local-density approximation in calculations of
ground-state electronic properties. Phys. Rev. B 28, 1809–1834.
Langreth, D., Perdew, J., 1975. Exchange-correlation energy of a metallic surface. Solid State
Commun. 17, 1425–1429.
Langreth, D., Perdew, J., 1977. Exchange-correlation energy of a metallic surface: wavevector analysis. Phys. Rev. B 15, 2884–2901.
Lee, C., Yang, W., Parr, R., 1988. Development of the Colle-Salvetti functional into a functional of the electron density. Phys. Rev. B 37, 785–789.
Lehtola, S., Jo´nsson, H., 2014a. Variational self-consistent implementation of the PerdewZunger self-interaction correction with complex optimal orbitals. J. Chem. Theory
Comput. 10, 5324–5337.
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.
J. Quantum Chem. 19, 427–439.
Messud, J., Dinh, P.M., Reinhard, P.G., Suraud, E., 2008. On the exact treatment of time
dependent self-interaction correction. Ann. Phys. (N.Y.) 324, 955.
Patchkovskii, S., Ziegler, T., 2002. Improving difficult reaction barriers with selfinteraction-corrected density functional theory. Chem. Phys. 116, 7806–7813.
Pederson, M., 2015. Fermi orbital derivatives in self-interaction corrected density-functional
theory: Applications to closed shell atoms. J. Chem. Phys. 142, 064112.
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 correction with unitary invariance in density functional theory. J. Chem. Phys. 140, 12110.
Perdew, J., 1990. Size-consistency, self-interaction correction, and derivative discontinuity
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 approximations 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.
14
John P. Perdew et al.
Perdew, J., Burke, K., Ernzerhof, M., 1996. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868.
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 constraints in density functional theory. J. Chem. Phys. 140, 18A533.
Polo, V., Kraka, E., Cremer, D., 2002. Electron correlation and self-interaction error of density functional theory. Mol. Phys. 100, 17711790.
Polo, V., Graăfenstein, J., Kraka, E., Cremer, D., 2003. Long-range and short-range correlation effects as simulated by Hartree-Fock, local density approximation, and generalized
gradient approximation. Theor. Chem. Acc. 109, 22–35.
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 vibrational absorption and circular dichroism spectra using density functional force fields.
J. Phys. Chem. 98, 11623–11627.
Sun, J., Pederson, M., 2015. Applications of self-interaction correction to uniform electron
gas. Unpublished.
Sun, J., Xiao, B., Ruzsinszky, A., 2012. Communication: Effect of the orbital overlap dependence on the meta-generalized gradient approximation. J. Chem. Phys. 137, 051101.
Sun, J., Xiao, B., Fang, Y., Haunschild, R., Ruzsinszky, A., Csonka, G., Perdew, J., 2013.
Density functionals that recognize covalent, metallic, and weak bonds. Phys. Rev. Lett.
111, 106401.
Sun, J., Perdew, J., Ruzsinszky, A., 2015a. Semilocal density functional obeying a stronglytightened bound for exchange. Proc. Natl. Acad. Sci. USA 112, 685–689.
Sun, J., Ruzsinszky, A., Perdew, J., 2015b. Strongly constrained, and appropriately normed
semilocal density functional. Phys. Rev. Lett. (to appear). arXiv:1504.03028.
Sun, J., Yang, Z., Peng, H., Ruzsinszky, A., Perdew, J., 2015c. Locality of exchange and
correlation for compact one- and two-electron densities. (Unpublished).
Tao, J., Perdew, J., Staroverov, V., Scuseria, G., 2003. Climbing the density functional ladder: Nonempirical meta-generalized gradient approximation designed for molecules and
solids. Phys. Rev. Lett. 91, 146401.
Toher, C., Filippetti, A., Sanvito, S., Burke, K., 2005. Self-interaction errors in densityfunctional calculations of electronic transport. Phys. Rev. Lett. 95, 146402.
Van Voorhis, T., Scuseria, G., 1998. A novel form for the exchange-correlation functional.
J. Chem. Phys. 109, 406–410.
Vydrov, O., Scuseria, G., 2004. Effect of the Perdew-Zunger self-interaction correction on
the thermochemical performance of approximate density functionals. J. Chem. Phys.
121, 8187–8193.
Vydrov, O., Scuseria, G., Perdew, J., Ruzsinszky, A., Csonka, G., 2006. Scaling down the
Perdew-Zunger self-interaction correction in many-electron systems. J. Chem. Phys.
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
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 tÞ;
(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).