PHYSICAL REVIEW B 15 NOVEMBER 2000-IIVOLUME 62, NUMBER 20
Neutral atoms in ionic lattices: Excited states of KCl:Ag0
C. Sousa, C. de Graaf, and F. IllasDepartament de Quı´mica Fısica i Centre Especial de Recerca en Quı´mica Teo`rica, Universitat de Barcelona, Martı´ i Franques 1,
08028 Barcelona, Spain
M. T. BarriusoDepartamento de Fı´sica Moderna, Facultad de Ciencias, Universidad de Cantabria, 39005 Santander, Spain
J. A. Aramburu and M. Moreno*Departamento de Ciencias de la Tierra y Fı´sica de la Materia Condensada, Facultad de Ciencias, Universidad de Cantabria,
39005 Santander, Spain~Received 22 December 1999!
The optical-absorption spectrum of a cationic Ag0 atom in a KCl crystal has been studied theoretically bymeans of a series of cluster models of increasing size. Excitation energies have been determined by means ofa multiconfigurational self-consistent field procedure followed by a second-order perturbation correlation treat-ment. Moreover results obtained within the density-functional framework are also reported. The calculationsconfirm the assignment of bands I and IV to transitions of the Ag-5s electron into delocalized states withmainly K-4s,4p character. Bands II and III have been assigned to internal transitions on the Ag atom, whichcorrespond to the atomic Ag-4d to Ag-5s transition. We also determine the lowest charge transfer~CT!excitation energy and confirm the assignment of band VI to such a transition. The study of the variation of theCT excitation energy with the Ag-Cl distanceR gives additional support to a large displacement of the Cl ionsdue to the presence of the Ag0 impurity. Moreover, from the present results, it is predicted that on passing toNaCl:Ag0 the CT onset would be out of the optical range while the 5s-5p transition would undergo a redshiftof 0.3 eV. These conclusions, which underline the different character of involved orbitals, are consistent withexperimental findings. The existence of a CT transition in the optical range for an atom inside an ionic host isexplained by a simple model, which also accounts for the differences with the more common 3d systems. Thepresent study sheds also some light on theR dependence of thes2-sp transitions due tos2 ions like Tl1.
edceXd
s0–dn
thing
isonubtveil-bealnfo
s-
t an
ro.the
riesarefirst
heon-ed to
n-the
s
o-nsi-
I. INTRODUCTION
A wide variety of defects and impurities can be forminside insulating lattices like alkali halides. For instanmetal impurities are formed by adding metal halides as AgCuX, and TlX to an alkali halide. The positively chargemetals (Ag1, Cu1, etc.! can occupy alkali cation positionwith a typical concentration of one metal cation over 2010 000 alkali ions. Thereafter, other defects can be createx irradiation, which causes the formation of free positive anegative charge carriers. The irradiated electrons cantrapped either in halide vacancies to formF andF8 centers,or creating neutral metal centers in cationic positions. Onother hand, the holes can be trapped by the anions origing the so calledVK centers, and also by the cations givinparamagnetic centers~Ag21, Cu21!.1
The formation of neutral atoms like Ag0,Hg0,Cu0,Tl0 asimpurities occupying cationic positions in ionic lattices,less common, but has been proved to exist through electrparamagnetic resonance and electron-nuclear doresonance.2 In the case of Ag0 in KCl, the hyperfine constanis only about 5% smaller than the corresponding free silatom value,2 showing that the electronic configuration of sver is 4d105s.1 Nevertheless, the system cannot be descriin terms of a free Ag atom. For instance the opticabsorption spectrum of KCl:Ag0 shows the presence of aintense band at 6.3 eV while it appears at 5.7 eV
PRB 620163-1829/2000/62~20!/13366~10!/$15.00
,
bydbe
eat-
icle
r
d-
r
KBr:Ag0.3 Based on this shift, this band was tentatively asigned to a Cl2→Ag0 charge-transfer transition.4 Moreover,an absorption peak appears in the optical spectrum aenergy well below the lowest atomic 5s→5p transition.
The whole optical absorption spectrum of KCl:Ag0 in theviolet-ultraviolet (V-UV) region,3–6 shows six peaks and fofive of them an assignment has been proposed by Moren4,7
Peaks I and VI, the most intense ones, do not depend ontemperature. The intensity of the rest of the bands valinearly with the temperature denoting that these bandsphonon assisted. Based on these considerations, theband at 2.92 eV was assigned to aa1g* →t1u* transition com-ing from the Ag atomic 5s→5p transition. The antibondinga1g* level was estimated to lie about 2.6–2.9 eV below tbottom of the conduction band, and therefore, it was ccluded that thet1u* level lies within the conduction band. Thsecond and third bands at 4.11 and 4.73 eV were assigneeg* →a1g* and t2g* →a1g* transitions, respectively. These trasitions are not electric dipole allowed, in agreement withsmall intensity observed for these bands. Band IV~at 5.35eV! was related to the silver atomic transition 4d105s1
→4d106s1, and finally, the intense band VI at 6.30 eV wainterpreted as the first charge-transfer~CT! band and as-signed by Moreno to at1u→a1g* transition.
The latter conclusion is a little surprising, as when iselectronic cations are considered, the energy of CT tra
tions decreases as far as the nominal charge increasesligand is kept.8,9 Moreover, at first sight the conclusion appears to be hardly compatible with experimental findingsdivalent 3d impurities in LiX ~X5Cl;Br! lattices.10,11 In par-ticular, the energy of the first CT transition due to a Mn1
impurity in LiCl is described~for the casen52) by thefollowing empirical law:
ECT~eV!5C2I ~M1! ~1!
derived by Simonetti and McClure.11 In this expression,I (M1) refers to the ionization potential of the free M1 ionandC is an empirical ‘‘constant’’ whose value was foundbe 22.4 eV. If we accept that the same law is valid for a Mn1
impurity with nÞ2 then one would expectECT'20 eV forKCl:Ag0 having in mind thatI (Ag2)51.2 eV.12 Despite thisbig discrepancy, previous self-consistent charge extenHuckel ~SCCEH! and multiple scattering Xa ~MS-Xa! cal-culations on KCl:Ag0, supported13 that the intense band a6.3 eV can be indeed compatible with a CT assignmeThese calculations also showed that the main contributiothe t1u* level is of K-4s,4p character, which supports thassignment of this level to lie within the conduction banMoreover, they indicate that bands II and III can both arfrom 4d10a1g* 24d9(a1g* )2 transitions though the experimental energy difference between both peaks~0.62 eV! canhardly be assigned to the 10Dq separation betweeneg* (4d)and t2g* (4d) one electron orbitals.
To gain a better insight into the rich optical absorptispectrum of KCl:Ag0, it is necessary to reproduce the maexperimental features through more powerful and reliatheoretical methods. To achieve this goal the knowledgethe ground-state equilibrium geometry is crucial as it fudetermines the energy of peaks observed in the optiabsorption spectrum. This important prerequisite has bsolved in a previous paper by means of total-energy calctions and the analysis of experimental superhyperfine tenBy both procedures the equilibrium metal-ligand distanceReis found to be equal to 3.7 Å, a result which concurs wearly estimations.13–15 Moreover it is found that this 18%relaxation of ligands is accompanied by an 8% outwarelaxation of first K1 ions along^100& directions.
Having in mind this relevant information, the present pper is devoted to explain the optical absorption spectrumKCl:Ag0 by means of reliable methods that include electrcorrelation in all electronic states. For this goal we firsttempt anab initio theoretical study where transition energiare computed at the equilibrium geometry of the ground s~as derived in Ref. 16!. To establish the character of thtransitions observed in the optical-absorption spectrum,design various cluster models of increasing size. Excitaenergies are computed by constructing highly correlawave functions based on the multiconfigurational seconsistent field procedure/second-order perturbation corrtion treatment~CASSCF/CASPT2! methodology17,18 imple-mented in theMOLCAS-4 package.19 Furthermore, we analyzethe dependence of the transition energies with the Ag-Cltance in order to show the importance of an accurate demination of the relaxation of the lattice around the Ag0 im-purity. This analysis gives additional support for the laroutward relaxation of the Cl atoms estimated in previostudies.13–16,20
the
n
ed
t.to
.e
eof
l-ena-or.
s
-f
n-
te
end-la-
s-r-
s
The optical transitions of KCl:Ag0 have also been calculated using density-functional theory~DFT! methods on clus-ters of different size. Some results obtained throughMS-Xa method are also reported. It will be shown that themethods quantitatively reproduce the main features ofoptical-absorption spectrum due to KCl:Ag0. The weaker de-pendence of the computational effort on the size ofsystem21 makes DFT certainly interesting to study large sytems that are not routinely accessible to handle with trational quantum chemical techniques. Recent developmentime-dependent DFT~Refs. 22 and 23! may cause thismethod to become an attractive alternative to study optproperties.
The paper is organized as follows: In the next sectiongive information about the basis sets, cluster models, embding schemes, and the computational methods applied instudy of the excited states. In Sec. III, we first calibrateab initio methods applying them to the free silver atomthereafter we discuss excitation energies and character ofinal states for each separate band observed in the optabsorption spectrum of KCl:Ag0. In addition, the variation ofthe excitation energies with the Ag-Cl distance is discusin this section. Section III is concluded with a brief discusion of the results reached in the DFT frameworkKCl:Ag0. In Sec. IV we focus on the comparison with othsystems. In particular from theR dependence of calculateexcitations the optical transition energies for NaCl:Ag0 areestimated. Finally, we list the conclusions.
II. COMPUTATIONAL INFORMATION
We study the ground and excited states related to thetical transitions mentioned in the introduction in orderextractab initio estimates of excitation energies and to gainsight into the character of these states. For this purposeapply a series of cluster models of increasing size. Thelowing six cluster models are studied@see Figs. 1~a!–1~f!#:
where shells of ions around the impurity are progressivadded in all three directions except for the largest cluster,which the 2 K1 ions in the fourth shell, the 8 Cl2 in the fifth,and the 8 K1 in the sixth shell are only added in one of thdirections giving rise to a noncubic cluster@see Fig. 1~f!#.The Ag0 valence electrons (4p64d105s1) are described witha (11s,8p,7d)/@4s,4p,4d# basis set, while for the Cl2 andK1 valence electrons (3s2, 3p6, and 3p6, respectively! weuse a (7s,7p)/@3s,3p# basis set. The core electrons of theatoms are represented by Cowan-Griffin relativistic effectcore potentials ~ECP! which include scalar relativisticeffects.24 For the largest cluster we reduce the basis set of8 potassium ions in the outermost shell to@1s,2p# to makethe calculations feasible.
To account for the short-range repulsion of the rest oflattice, all six clusters are surrounded byab initio embeddingmodel potentials~AIEMP!.25,26 The atoms contained in acube of length 2a ~beinga the lattice parameter of the KC
ures.
13 368 PRB 62C. SOUSAet al.
FIG. 1. Cluster models of Ag0 impurity in KCl at the optimized geometry. Only the atoms explicitly treated are included in the figThe black sphere represents the Ag0 impurity, the dark gray spheres refer to K ions, and light gray spheres to Cl ions. Froma to f thefollowing clusters are shown:~a! AgCl6, ~b! AgCl6K12, ~c! AgCl6K12K6, ~d! AgCl6K12Cl8, ~e! AgCl6K12Cl8K6, and ~f!AgCl6K12Cl8K6Cl8K8.
s
p
lly
theube
crystal, 6.293 Å! centered in the impurity plus the atomlocated at the~3/2,0,0!, ~3/2,1/2,0!, and~3/2,1/2,1/2! uniquelattice positions are described as AIEMP’s. These modeltentials have been optimized to represent the K1 and Cl2
ions in the pure KCl crystal as described in Ref. 27. Fina
o-
,
the long-range electrostatic interaction is included inmodel by an array of Evjen point charges to complete a cof 8a around the Ag0 impurity. Although all clusters but thelargest one exhibitsOh symmetry, we only exploit theD2hpoint-group symmetry.
TABLE I. Experimental and theoretical transition energies, TE~in eV!, of the Ag atom. The CASSCFwave functions are constructed with an active space that includes 12 orbitals—the 5 Ag~4d! orbitals, 5 Ag(d8), Ag~5s!, and Ag~5d!—and 11 electrons. CASPT2 correlates the 4p, 4d, and 5s, electrons. TE averagemeans the weighted average of the spin-orbit splitted components of the2D and 2P states.
Configuration
Experiment Theory
State TE TEaverage
State CASSCF CASPT2
4d10 5p1 2P1/2 3.66 3.74 2P 3.68 3.792P3/2 3.78
4d9 5s2 2D5/2 3.75 3.97 2D 4.31 4.002D3/2 4.30
ontrre
arre
inis
s-getha
reivay4llyn-coal
chat-taI
,eaeardthsi,th
er-a
cesu
r,
thesp-set
n.ted
12
e
ri-f the
lit-tserbit
or-el
hela-
achhe-
Excitation energies are estimated as vertical transitifrom the optimized ground-state geometry. This geomehas been determined from a restricted open-shell HartFock calculation for a AgCl6K6 cluster embedded inAIEMP’s and point charges. Because of the large outwrelaxation, both the Ag-Cl and Cl-K distances along the thaxes have been relaxed. The resulting Ag-Cl distance3.707 Å, i.e., an outward relaxation of about 20% comparto the ideal KCl structure. The optimized Cl-K distance2.995 Å, which is only 5% smaller than the ideal Cl-K ditance in the KCl crystal. Further details concerning theometry relaxation can be found in Ref. 16. The rest ofatoms of the clusters, the AIEMP’s, and the point chargesfixed at the lattice positions of the KCl crystal.
Multireference wave functions for the states consideare constructed in a CASSCF calculation with an actspace containing 11 electrons distributed in all possible wover 11 orbitals. For all states, we include the five Ag-dorbitals and a correlating set of five virtual orbitals, usuareferred to as Ag-4d8. As has been shown before, this esures a correct treatment of the large dynamical electronrelation effects in the TMd shell, whereas a perturbationtreatment tends to overestimate these correlation effects.28–31
Note that this is especially important for transitions in whithe number ofd-electrons changes. For the ground st(a2A1g), the 2Eg , the 2T2g , and theb2A1g states, associated to bands II, III, and IV, respectively, we add an orbiof a1g symmetry. For the two2T1u states related to bandsand VI, the eleventh orbital is oft1u symmetry. Remainingmostly dynamical, electron correlation effects are includby second-order perturbation theory, CASPT2, in whichvalence electrons are correlated. In this perturbational trment the CASSCF wave function is taken as a zeroth-owave function. The character of the open shell orbital ofexcited states is studied by Mulliken population analy~MPA!. Although it is well known that it has its limitationsthis analysis gives a reasonable qualitative insight intocharacter of the orbitals.
DFT calculations have been performed with the Amstdam density functional code32,33 using the local density approximation exchange-correlation functional. Triple zeta bsis sets~quality IV! are employed to describe the valenelectrons, whereas the core electrons are kept frozen. Reare reported for the 39-atom cluster, AgCl6K12Cl8K6Cl6
22
described in the previous paper.16 The details of the
sye-
deisg
-ere
des
r-
e
l
dllt-
eres
e
-
-
lts
MS-Xa calculations for the 81-atom clusteAgCl6K12Cl8K6Cl2K24
41, are also given there.
III. RESULTS AND DISCUSSION
A. Free silver atom
Before attempting the study of the optical spectra ofsilver impurity in KCl, we calculate the transition energiefor a free silver atom to calibrate the computational aproach. The Ag atom is described with the ECP and basismentioned before in the computational informatioCASSCF wave functions for the ground- and lowest-excistates are constructed by distributing 11 electrons overorbitals (Ag-4d, 4d8, 5s, and 5p). CASPT2 accounts forremaining electron correlation. Table I lists thCASSCF/CASPT2 excitation energies of the2P ~comingfrom a 4d105p1 configuration! and 2D ~arising from a4d95s2 configuration! excited states with respect to the2S(4d105s1) ground state. A direct comparison with expemental excitation energies cannot be made because olarge spin-orbit splitting present in the silver atom~0.12 eVfor the 2P state and 0.55 eV for the2D). Therefore, resultsare compared with a weighted average of the spin-orbit spted components of the2D and 2P states. Table I shows thaour CASPT2 excitation energies are within 0.05 eV of theweighted averages. This indicates that, except for spin-oeffects, the computational approach~ECP, basis set andCASSCF/CASPT2 method! is well calibrated and can beused in the system of interest, KCl:Ag0. Note that CASPT2lowers the excitation energy for the2D state, while a smallincrease is observed for the2P state. The CASSCF wavefunctions already include the major part of the electron crelation effects caused by the Ag-4d electrons, and hence thstabilization of the2D state is caused by the differentiaelectron correlation effects of the 5s electrons treated byCASPT2. The slight destabilization of the transition to t2P state must be ascribed to slightly larger electron corretion effects in the 5s than in the 5p orbital.
B. CASSCFÕCASPT2 approach to the optical spectrumof KCl:Ag 0
KCl presents a face-centered-cubic structure where eion is surrounded by six nearest neighbors in a local octadral symmetry. The Ag0 impurity substitutes a K1 ion andthe crystal field splits the Ag-4d orbitals into two sets of
Cl
13 370 PRB 62C. SOUSAet al.
TABLE II. CASSCF and CASPT2 excitation energies~in eV! relative to the2A1g ground state for different cluster models using a Ag-distance of 3.707 Å and a Cl-K distance of 2.995 Å.
State AgCl6 AgCl6K12 AgCl6K12K6 AgCl6K12Cl8 AgCl6K12Cl8K6 AgCl6K12Cl8K2Cl8K8 Exp.
orbitals ofeg(dx22y2,d3z22r 2) andt2g(dxy ,dyz ,dxz) symme-try. In this section we study the five assigned bands obsein the spectrum using the cluster models and computatioapproach commented before. Table II summarizes the etation energies of the different bands obtained with thedifferent clusters. In order to analyze the optical spectrumthe Ag0 impurity in KCl we study and interpret each banseparately.
1. Band I at 2.92 eV
The band appearing at 2.92 eV has been assigneda1g* →t1u* transition, which in the free Ag atom corresponto a 5s→5p transition. The entry in Table II for the2T1ustate shows the variation of the excitation energies on gofrom the smallest cluster, AgCl6, to the larger ones. Theexcitation energy of the2A1g→2T1u transition decreasewith increasing the cluster size. This reduction is accomnied by a delocalization of the open shell orbital in the ecited state for the larger clusters. Indeed, this orbital hastrong Ag-5p character for the smallest AgCl6 cluster andextends over the K ions as the size of the cluster increaThe inclusion of the 12 K surrounding the basic AgCl6 clus-ter already leads to a significant delocalization; the Agpcharacter of the open shell orbital of the excited state drto 77% and the excitation energy decreases about 0.2Adding six K in the^1,0,0& directions to the AgCl6K12 clus-ter, does not significantly affect the excitation energy andcharacter of the open shell orbital. Instead, inclusion ofnext shell of 8 Cl ions causes a stabilization of 0.1 eV athe open shell orbital now has an important contribut~;31%! of the K-4s,4p orbitals. The fact that this delocaization does not occur in the AgCl6K12 cluster indicates thathe Cl ions surrounding the K need to be included explicin the cluster and that the description of these ions with bAIEMP’s introduces a too strong repulsion that preventspopulation of the K-4s and 4p orbitals in the AgCl6K12 clus-ter.
The next entry in Table II (AgCl6K12Cl8K6) shows againthat incorporation of six K in the1,0,0& directions to theAgCl6K12Cl8 cluster does not change the general imagethe transition unless the surrounding Cl2 ions are explicitlyincluded in the cluster. However, this leads to a clusterunmanageable size if the cluster is extended in all threerections. From the previous clusters, we observe that the
edalci-xf
a
g
--a
s.
sV.
eed
ree
f
fi-r-
bitals of the K1 ions located at the~1/2,1/2,0! and~1/2,0,1/2!positions are more populated than those at the~0,1/2,1/2! inagreement with thet1u(x) symmetry of the open shell orbital. Therefore, we extend the AgCl6K12Cl8 cluster with the2 K1 ions at the~61,0,0! positions and all Cl2 and K1 near-est neighbors in the sameyz plane. This large cluster is referred to as AgCl6K12Cl8K2Cl8K8 @see Fig. 1~f!# and per-forming ab initio calculations with such a cluster is at thlimit of present day computational resources. For this reasonly CASSCF calculations has been performed. HowevCASPT2 hardly affects the excitation energies; as canseen in Table II, the effect is never larger than 0.05 eV. Wthe largest cluster we find a further delocalization of the opshell orbital over the K1 ions in the second, fourth, and sixtshell surrounding the impurity and a further decrease ofexcitation energy to 3.44 eV, slowly approaching the expemental value of 2.92 eV. The contribution of the Ag-p or-bital to the open shell orbital further decreases to appromately 55%.
In summary, the final state connected to the2A1g→2T1utransition is as delocalized as the size of the cluster allohence, to have a correct description and an accurate extion energy of this state, much larger cluster modelsneeded. Nevertheless, our calculations confirm the deloized character of this state as a transition into the conducband of predominantlyK-4s and 4p character as proposeby Cabriaet al.13,20 The delocalized character of the finstate explains the lower-excitation energy compared tocorresponding transition in the free atom. The effect of sporbit coupling on the transition energy is estimated tovery small. Spin-orbit coupling splits the correspondiatomic transition by 0.1 eV~see Table I!, but for theKCl:Ag0 system, the splitting is expected to be smaller bcause of the delocalized character of the2T1u state.
2. Bands II and III at 4.10 and 4.73 eV
Bands II and III in the absorption spectrum have beassigned toeg* →a1g* and t2g* →a1g* transitions, respectivelySCCEH and MS-Xa calculations performed by Cabriet al.13,20 showed that theeg* and t2g* orbitals involved insuch transitions, have almost pure Ag-4d character while thea1g* orbital can be considered as a Ag-5s orbital.
As for the previous band, Table II shows the excitatienergies of the2A1g→2Eg and 2A1g→2T2g transitions at the
CASSCF~11/11! and CASPT2 level. In the first place, wobserve that the excitations energies at a given level ofculation hardly change with increasing cluster size. Thislustrates that the excitations are rather local and MPA shthat the excited state open shell orbital indeed has a stAg-4d character. It is well known that transitions involvind electrons in transition-metal compounds are very localiand can be accurately described with relative small clumodels.34–37 As in the free Ag atom, CASPT2 lowers thCASSCF excitation energies, but does not affect2Eg-2T2g splitting. Moreover, the effect of the dynamicelectron correlation on the excitation energies doeschange from cluster to cluster; a constant lowering of;0.5eV is observed.
Second, the splitting between the2Eg and 2T2g states isvery small, only 0.07 eV, whereas the experimental seption between bands II and III is 0.62 eV. The differenbetween experiment and the calculated value must becribed to the lack of spin-orbit coupling effects in our calclations. However, these effects can readily be estimated fthe free atom because of the very localized character oftransitions. Table I shows a2D3/2-
2D5/2 separation of 0.55eV, which is almost identical to the splitting of bands II anIII for KCl:Ag 0. Therefore, we conclude that the separatbetween bands II and III is mainly due to spin-orbit effecwhile crystal-field effects are much less important. To proerly describe the final states of bands II and III, calculatiothat explicitly include spin-orbit coupling are indispensabNevertheless, our calculations corroborate the assignmethese bands to transitions in which an electron is transfefrom the Ag-4d shell to an orbital with mainly Ag-5s char-acter.
3. Band IV at 5.35 eV
The calculated excitation energy connected to thea2A1g→b2A1g transition, which in an isolated atom correspondsa 5s→6s transition, rapidly reduces as the size of the clusincreases~see Table II!. The excitation energy decreasesmost 6 eV on going from the smallest AgCl6 cluster to thenext one with the 12 K1 next neighbors included. Transitioenergies for the AgCl6K12K6 cluster have not been calculatesince the calculations for the previous three bands cleindicate that inclusion of the six K1 in the ^1,0,0& directionsdoes not significantly influence the excitation energies. Tsame holds for the AgCl6K12Cl8K6 cluster, which gives simi-lar results as the AgCl6K12Cl8 cluster model. For the lattecluster, excitation energies are about 2 eV smaller thanthe AgCl6K12 cluster and 0.4 eV larger than the one obtainfor the largest cluster studied, AgCl6K12Cl8K2Cl8K8. Thisindicates that the excitation energy seems to rapidly cverge with the cluster size and tends to the experimevalue of 5.35 eV.
Again, for the largest cluster, no CASPT2 estimate cangiven because of the very large computational demandsuch a calculation. However, the results of the other clusindicate that electron correlation tends to lower the CASSvalue by a small amount, and therefore, we expectCASPT2 will lower the CASSCF excitation energy for thlargest cluster, 5.59 eV, approaching the experimental vaeven more. Concerning the nature of the open shell orbitathe excited state, MPA shows that this orbital extends
l--s
ng
der
e
ot
a-
s-
me
-s.ofd
or
-
ly
e
ord
-al
eofrsFat
einll
over the cluster. The highly delocalized character meansthis orbital belongs to the conduction band of the crysHence, it is not appropriate to relate the open shell orbitathe final state to a 6s-like atomic orbital. Spin-orbit effectsfor this transition are not important since only unpaired eltrons ofs character are involved.
4. Band VI at 6.3 eV
Band VI is the last band observed in the opticaabsorption spectrum and carries large intensity. It has bassigned to a charge transfer~CT! transition from at1uCl-3porbital to aa1g* orbital with strong Ag-5s character.4,13 Atfirst sight, the appearance of local CT transitions at suchenergies might be surprising because the Ag impuchanges from a neutral state to a negatively charged swhereas a positively charged oxidation state is more comon for silver.
Excitation energies of this transition obtained for differeclusters are reported in Table II@see entry2T1u~CT!#. As forthe previous band, only the cluster models, which introdunew effects, are taken into account. In all clusters the oshell of the excited state,2T1u , has a strong Cl-3p characterand the2A1g→2T1u transition is ascribed to a local chargtransfer transition. The excitation energy for the smallcluster is very high, larger than 10 eV. Obviously, suchsmall cluster cannot account for an excitation in which a his formed on the edge of the cluster. The ions in the dircluster environment are described with model potentialsdo not allow these ions to respond to the changes incharge distribution between Ag and Cl. This artifact seembe largely removed in the next cluster that includes the 1ions surrounding the Cl ions; the excitation energy expeences an important lowering of about 4 eV. The fact thatexcitation energy does not change anymore for the larclusters considered shows that the polarization of the latin response to the CT transition is not as long ranged asionization processes. Two reasons can be given to rationathis observation. In the first place, the transfer of an electfrom the ligand-p shell to the Ag-5s orbital is accompaniedby a redistribution of the density in the closed-shell orbitaThis causes the hole created on the ligands to be effectiscreened and therefore the net transfer of charge is smthan one. For the transition studied here, the net tranestimated by MPA is;0.5 electrons. The second reason fthe rather short-ranged character of the lattice relaxationbe found in the fact that electron reorganization occwithin a local region. This means that the atoms located fther away from the impurity do not experience large chanin the Coulomb potential due to a local CT transition.
Note that the effect of CASPT2 is much larger than fany of the other states considered. The large stabilizatiothe CT state by CASPT2 has been observed before byleijns et al.38 in the study of the local charge transfer prcesses in NiO. Two effects contribute to the differential eletron correlation. The first one is caused by the different Agsand Cl-p occupations in the ground and excited state, whtends to stabilize the state with highers occupation, as hasbeen shown by Sousaet al. in the study of the copper halidmolecules.39 The second effect is the recuperation of thebital relaxation effects.35,40–42In the CASSCF wave functionof the 2T1u~CT! state, an electron is removed from a molec
e
13 372 PRB 62C. SOUSAet al.
TABLE III. CASPT2 excitation energies~in eV! relative to the2A1q ground state as function of thAg0-Cl distance,R ~in Å!, for the AgCl6K12Cl8 cluster. The Ag-K4 distance is 6.70 Å.
lar orbital that is a linear combination of two Cl atomic obitals. This implies that only part of the orbital relaxation dto the removal of an electron is included in this wave funtion. CASPT2 has the complete orbital relaxation includup to first order and this causes a further stabilization ofcharge-transfer state.
The calculated excitation energy for the CT state is cverged with the cluster size. We observe that this transitiogenuinely local and can be accurately described with a rasmall cluster. The excitation energy is computed to beeV, which compares very well with the experimental trantion energy of band VI, 6.3 eV. Spin-orbit effects are retively unimportant as the open shell orbital in the excitstate is highly localized on the Cl atom. The atomic sporbit splitting for Cl is only 0.1 eV, and hence, it is noexpected that the2T1u~CT! state is significantly affected bspin-orbit coupling effects.
5. Influence of the Ag0-Cl distance on the excitation energies
It is well known that various spectroscopic propertistrongly depend on the metal-ligand distance in transitimetal compounds. By studying the dependence of the etation energies on the local geometry around the Ag0 impu-rity, further insight into the electronic structure of thKCl:Ag0 system can be obtained. This dependence is ecially relevant for the present case, since the size of theward relaxation of the Cl ions due to the presence of the0
atom is not definitely settled. The excitation energies oftransitions discussed before are calculated at four diffeAg-Cl distances using a cluster that accurately describeselectronic states involved in the transitions, namely,AgCl6K12Cl8 model. Beside the optimized distance, we cosider distances of 3.146 Å corresponding to the K-Cl dtance in the pure crystal, a distance of 3.427 Å, which isminimum of the potential-energy surface varying only tAg-Cl distance, and finally, we also apply a Ag-Cl distanof 3.8 Å, slightly larger than the optimized value. In all thecases the 6 K1 ions at the axis neighboring the Cl ions aplaced at their optimized position as described in the coputational information.
Table III, which lists the CASPT2 results, shows that texcitation energies of the delocalized states (2T1u andb2A1g) increase with the Ag-Cl distance. This is not surpring as the delocalization, which causes a stabilization ofexcited state, is facilitated at smaller distances. The tendefor the localized states@ 2Eg , 2T2g , and 2T1u~CT!# is theopposite; with increasing distance the excitation energiesminishes. The effect of the ligand field on the2Eg and 2T2g
-de
-iser4--
-
-i-
e-t-
enthee--e
-
-ecy
i-
states decreases with the distance, which explains the tof the transition energies for these two states towardsatomic value. On the other hand, the explanation ofvariation for the2T1u~CT! state is less obvious. This behaior has been previously observed for many octahedralcompounds and has been rationalized by a more pronoudestabilization of the one-electron levels associated withmetal than those associated with the ligand when the meligand distance decreases.43,44
The dependence of the CT transition energy with the dtance gives us an additional way to approximately estimthe Ag-Cl distance. The other bands are less suitable for san analysis since they are either delocalized or lack the efof spin-orbit coupling. Table III shows that the calculated Ctransition energy only coincides with the experimental vawhen the outward relaxation of the Cl ions surroundingsilver impurity is of the order of 20%, in excellent agreemewith previous work.13–16
C. DFT approach to the optical spectrum of KCl:Ag0
Representative DFT and MS-Xa results on optical transi-tions due to KCl:Ag0 are summarized in Table IV. It can bseen that the main experimental features are also quatively reproduced by DFT and MS-Xa calculations, althoughthe agreement with experiment is not as good as forCASSCF/CASPT2 method. It is worth noting that in thpresent case, DFT results lead to an underestimation ofeV of the CT energy. The results collected in Table IV habeen derived maintaining all ions~except the Cl2 ligands! atthe equilibrium positions for the perfect KCl lattice. It iworth noting that the energies of optical transitions are litdependent on the positions of these ions. For instance,energy of the charge-transfer transition experiences a vation less than 1% when the first K1 ions along^100& direc-tions are placed at the equilibrium geometry found for tAg0 center.
In principle two allowed CT transitions can be observin the spectrum of as1 ion in an octahedral site: First,transition in which an electron is transferred from at1u(p)orbital with dominant ligand-pp character, and second, froma t1u(s) orbital, mainly built fromps ligand orbitals. It hasbeen shown45,46 that for transitions like the twot1u→a1g*transitions of KCl:Ag0 the oscillator strength directly depends on the amount ofs admixture in the ligand orbital. FoKCl:Ag0 the present DFT calculations indicate that thesadmixture for the so-calledt1u(s) orbital is about nine timesbigger than fort1u(p). This situation is quite different to
TABLE IV. LDA and MS-Xa transition energies~in eV! at two different Ag-Cl distances. LDA calculations are performed with a 39-atoms cluster whereas MS-Xa results are reported for a 81-ions cluster. Tresults shown have been calculated placing all ions~except Cl2 ligands! at the equilibrium positions corresponding to the perfect KCl lattice.
in insulating lattices.45–47 Such a difference can be ascribeto the relative closeness of the 5p orbital of Ag0 in thepresent case, which tends to destroy the significants-p hy-bridization in thet1u orbitals found for CuCl4
22, TlCl642, or
CrF632 units.46
The results collected in Table IV indicate that the sepation between the two CT transitions is around 0.25 ewhich is comparable to the experimental bandwidth.48 De-spite this fact it is expected that the CT peak observedperimentally arises essentially from thet1u(s)→a1g* transi-tion, because of the small oscillator strength of thet1u(p)→a1g* CT transition.
Within the DFT framework the electronic relaxationthe 2T1u(s) excited state has also been analyzed. Whenground-state Kohn-Sham orbitals are used, the CT energfound to be 0.9 eV higher than when relaxed orbitalsemployed. At the same time the total charge on centraldecreases by 0.4e once the electron density is allowedrelax. The resulting net charge transfer is equal to 0.3e.This result is similar to that reached throughab initio calcu-lations ~Sec. III B! and indicates that electronic relaxatioeffects for the 2T1u(s) excited CT state of KCl:Ag0 aresmaller than those for open shell 3d ions.45,49–51
IV. COMPARISON WITH 3 d AND OTHER S1 SYSTEMS
Though the present calculations strongly support theistence of a CT transition in the optical range for KCl:Ag0 itis also necessary to understand why the widely used emcal law @Eq. ~1!# reported by Simonetti and McClure11 is nolonger valid for the present case. The main origin for sucdiscrepancy can be well accounted for through a simmodel. Let us consider a highly ionic MX6 species(X5halide) embedded in an insulating lattice. The workquired for extracting an electron from a ligand level aplacing it at the zero energy level can be approximated b
W~X!5E~X!1Uc~X!2UR~X!, ~2!
where Uc(X) 5(e2/R)@zM23.3zX# reflects the electrostatiinteraction of an electron on a ligand with all other ionsthe complex,UR(X) accounts for the interaction with threst of the ions in the lattice andE(X) means the electronaffinity of free X atom.zM andzX are the formal charges othe metal and the ligand, andR the M-X distance.
-,
x-
eisen
x-
ri-
ale
-
f
Assuming that the metal cation Mn1 becomes M(n21)1
after the CT process, the work required to carry an electfrom the cation to the zero energy level can be approximaby
W~M!5I ~M~n21!1!2Uc~M!2UR~M!. ~3!
In this expressionUc(M) 56zXe2/R and I (M(n21)1) is theionization potential of free M(n21)1 ion. Equation~3! doeshowever not consider all important facts in a CT procebecause~i! antibonding electrons spend some time onligands and~ii ! once a hole is created on the ligands, therea flow of electronic charge from the metal to the liganthrough the closed-shell orbitals. These effects lead toeffective workW(M) that is higher than that described bEq. ~3!.
If all this kind of contributions toECT are consideredthrough an additional term denoted asUR ,ECT can be de-duced by subtracting Eq.~2! and Eq.~3!:
ECT5E~X!1Q2Ur2I ~M~n21!1!, ~4!
with Q5Uc(X) 1Uc(M). If the complex is mainly ionic inits ground state one would expect thatUr.0. To derive Eq.~4! it has been assumed thatUR(M) 5UR(X). This conditionis well fulfilled for substitutional impurities in cubic lattice~see Refs. 52–54 for further discussion!. When expression~4! is compared to Eq.~1! it turns out that the empirical termC is equal toE(X) 1Q2Ur . For divalent impurities in LiClR'2.5 Å; E(Cl)53.6 eV, andQ'27 eV. The compari-son with C522.4 eV derived from experiments indicatethatUr is indeed positive and equal to'8 eV. However, it isimportant to note that this analysis also shows that the tC in Eq. ~1! cannot be considered as a constant whennature of the impurity is changed. On passing from dival3d ions to Ag0 not only changesR significantly, but also thezM involved in Uc(X). Putting nowR53.7 Å andzM50 inEqs.~2!–~4! it turns out thatQ becomes equal to 10 eV foKCl:Ag0 and henceC is much smaller than 22.4 eV. Withthis reducedC, the existence of a CT transition in the opticrange can now reasonably be understood.
As previously pointed out43,45 the termdQ/dR appears tobe a main factor controlling theR dependence of a CT transition. In the present case,dQ/dR'22.8 eV/Å, which iscomparable to thedECT/dR values derived from Table III.From this value, it is now possible to estimateECT for
ec
eAle
di
be
aV
he
a
t
lts
t.t e
train
li-
nelusasl-f
hno
e
y
rp-
itedicalrderch,een
tran-
andinlytethelues aita-V.
veAg
ethe
the
-a
nd
gn-
theandita-
up-ur-s
Tel
riety
nby
them-nto
13 374 PRB 62C. SOUSAet al.
NaCl:Ag0. AssumingRe53.37 Å, as previously reported,4
the energy of the first allowed CT transition in NaCl:Ag0 ispredicted to be 7.2 eV, i.e., a blueshift of 0.9 eV with respto what is measured in KCl:Ag0. Therefore the first allowedCT transition in NaCl:Ag0 would not be located in the moraccessible optical range but in the vacuum UV region.similar conclusion is obtained from the DFT results of TabIV though in this casedECT/dR'21.5 eV/Å. Further ex-perimental work is necessary to check this relevant pretion.
Experimentally the Ag-5s→5p transition in NaCl:Ag0 isfound within the optical range the corresponding energying equal to 2.65 eV.2 When compared to KCl:Ag0 the red-shift undergone by that transition can reasonably becounted for by the results contained in Tables III and IThese results indicate that the Ag-5s→5p transition experi-ences a redshift whenR decreases, a behavior that is just topposite to that found for a CT transition. TakingdE/dR'0.9 eV/Å for this transition from Table III, one expectsvalue of approximately 2.60 eV for NaCl:Ag0.
Finally, we discuss the Ag-5s→6s transition appearing a5.16 eV in NaCl:Ag0 ~Refs. 2 and 3! and thus suffering aredshift of 0.19 eV with respect to KCl:Ag0. Although fortransitions froma1g* to higher one-electron levels, the resugathered in Table III predict a redshift whenRdecreases, it isalso deriveddE(5s-6s)/dR'3.5 eV/Å, implying a redshiftclose to 1 eV. This discrepancy indicates that in orderdescribe the2A1g excited state larger clusters are needed
The present results can also be compared with recenperimental data and MS-Xa calculations on the 6s1 ion Tl21
in different alkali halides.47 It turns out that whenR de-creases, the CT transitions undergo a blueshift whereasTl-6s→6p transition experiences a redshift. This genetrend has to be related to the different character of thevolved one electron orbitals innMs1 impurities (nM55 forAg0, nM56 for Tl21) placed in insulating lattices. In a CTtransition, the electron jumps from a localized orbital ongands to an orbital with mainlynMs1 character, which islocalized to a reasonable extent on the central ion. WheRdecreases the increase of electrostatic repulsion on antron placed ina1g* increases the energy of this level and thfavors the increase of the CT transition energy. By contrthe t1u* orbital with mainlynMp character is much less locaized on the central ion thana1g* . Therefore the increase oelectrostatic repulsion for decreasingR is stronger ona1g*than ont1u* , which explains the observed redshift. Althougthis qualitative trend is followed by both impurities, the sesitivity to R changes is of course different in both cases. Finstance, for thet1u(s)→a1g* transition of TlCl6
42, MS-Xacalculations47 lead todECT/dR57.9 eV/Å. The significantincrease ofdECT/dR on passing from KCl:Ag0 to TlCl6
42
can be understood by the fact that for TlCl642 Re has been
estimated to be 2.8 Å and the nominal charge of Tl21 is notzero butzM52.
The present results are helpful to understand relevantperimental facts coming froms2 ions in insulating lattices.For instance the energy ofnMs2→nMsp optical transitionsdue to ions like Tl1 or Sn21 is also found to increase slightlalong the series NaCl-KCl-RbCl.55
t
c-
-
c-.
o
x-
hel-
ec-
t,
-r
x-
V. CONCLUSIONS
In this paper, we have presentedab initio cluster modelcalculations in order to study and analyze the optical absotion spectrum of a Ag0 impurity in a KCl lattice. We haveconstructed CASSCF wave functions for ground and excstates involved in the transitions and remaining dynamelectron correlation has been accounted for by second-operturbation theory by means of CASPT2. This approatogether with the basis sets and ECP’s applied, have bproven to be adequate to give accurate estimates of thesition energies of the free silver atom.
Five bands in the optical spectrum of the Ag0:KCl systemhave been studied in detail and characterized. The first bcorresponds to a transition to a delocalized state with maK-4s,4p character. In our cluster calculations the final stais as delocalized as the size of the cluster allows andexcitation energy slowly approaches the experimental vaof 2.92 eV as the cluster size increases. Band IV showsimilar delocalized character and convergence of the exction energy towards the experimental value of 5.35 eTherefore, states of KCl:Ag0 coming from 5p and 6s statesof free Ag0 are in fact resonant states. Bands II and III habeen confirmed to arise from transitions localized on theatom in which an electron is excited from the Ag-4d to theAg-5s orbital. Spin-orbit coupling have been found to bessential in order to explain the right splitting betweentwo bands; crystal-field effects are of minor importance.
The origin of the small band calledV and peaked at 5.6eV is not clarified through the present paper. Despite5s→6p transition of free silver atom placed at 6.0 eV,4 thiscan hardly be the origin of bandV as its intensity is temperature dependent.3 This fact strongly suggests that band V isparity forbidden transition as it happens for bands II, III, aIV.3
Finally, we have confirmed the previous tentative assiment of Moreno4 of band VI to a local CT transition. Thecalculated transition energy of 6.4 eV is converged withcluster size and in good agreement with the absorption bobserved at 6.3 eV. From the dependence of the CT exction energy with the distance, we have given additional sport to the outward displacement of 20% of the Cl ions srounding the Ag0 impurity, as proposed in previoustudies13–15 and in the preceding paper.16 A model has beenproposed to explain the surprisingly low energy of the Ctransition. In spite of the very simple nature, the modserves to understand trends in excitation energies in a vaof impurities in ionic insulators.
ACKNOWLEDGMENTS
We thank Dr. Ria Broer of the University of Groningefor stimulating discussions. The work has been financedSpanish ‘‘Ministerio de Educacio´n y Ciencia’’ under CICyTprojects PB98-1216-CO2-01 and PB98-0190. Part ofcomputer time was provided by the ‘‘Center de Supercoputaciode Catalunya,’’ C4-CESCA, through a research grafrom the University of Barcelona. C. de G. would like tthank the European Community for financial support.
*Corresponding author. Email: [email protected]. M. Stoneham,Theory of Defects in Solids; Electronic Stru
ture of Defects in Insulators and Semiconductors~Oxford Uni-versity Press, Oxford, 1996!.
2S. V. Nistor, D. Schoemaker, and I. Ursu, Phys. Status Solid185, 9 ~1994!.
3M. Saidoh and N. Itoh, J. Phys. Soc. Jpn.35, 1122~1973!.4M. Moreno, J. Phys. C13, 6641~1980!.5C. J. Delbecq, W. Hayes, M. C. M. O’Brien, and P. H. Yust
Proc. R. Soc. London, Ser. A271, 243 ~1963!.6M. Saidoh, N. Itoh, and M. Ikeya, J. Phys. Soc. Jpn.25, 1197
~1968!.7M. Moreno, J. Phys. C12, I921 ~1979!.8C. K. Jorgensen,Solid State Physics~Academic, New York,
1962!, Vol. 13, pp. 375.9C. K. Jorgensen, Prog. Inorg. Chem.12, 101 ~1970!.
10S. Hirako and R. Onaka, J. Phys. Soc. Jpn.51, 1255~1982!.11J. Simonetti and D. S. McClure, J. Chem. Phys.71, 793 ~1979!.12T. Koga, H. Tatewaki, and A. Thakkar, J. Chem. Phys.100, 8140
~1994!.13I. Cabria, M. T. Barriuso, J. A. Aramburu, and M. Moreno, Int.
Quantum Chem.61, 627 ~1997!.14M. T. Barriuso and M. Moreno, Phys. Rev. B26, 2271~1982!.15M. Moreno, J. A. Aramburu, and M. T. Barriuso, Phys. Lett.
87, 307 ~1982!.16J. A. Aramburu, M. Moreno, I. Cabria, M. T. Barriuso, C. Sous
C. de Graaf, and F. Illas~unpublished!.17K. Andersson, P.-A˚ . Malmqvist, B. O. Roos, A. J. Sadlej, and K
Wolinski, J. Phys. Chem.94, 5483~1990!.18K. Andersson, P.-A˚ . Malmqvist, and B. O. Roos, J. Chem. Phy
96, 1218~1992!.19K. Andersson, M. R. A. Blomberg, M. P. Fu¨lscher, G. Karlstro¨m,
R. Lindh, P.-A. Malmqvist, P. Neogra´dy, J. Olsen, B. O. RoosA. J. Sadlej, M. Schu¨tz, L. Seijo, L. Serrano-Andre´s, P. E. M.Siegbahn, and P.-O. Widmark,MOLCAS, Version 4, Universityof Lund, Lund, 1997.
20I. Cabria, M. T. Barriuso, J. A. Aramburu, and M. Moreno, Mter. Sci. Forum239–241, 171 ~1997!.
21W. Kohn, Phys. Rev. Lett.76, 3168~1996!.22E. K. U. Gross, J. F. Dobson, and M. Petersilka,Density Func-
tional Theory~Springer-Verlag, Heidelberg, 1996!.23M. E. Casida, C. Jamorski, K. C. Casida, and D. R. Salahub
Chem. Phys.108, 4439~1998!.24Z. Barandiara´n and L. Seijo, Can. J. Chem.70, 409 ~1992!.25Z. Barandiara´n and L. Seijo, J. Chem. Phys.89, 5739~1988!.26L. Seijo and Z. Barandiara´n, Computational Chemistry: Review
of Current Trends~World Scientific, Singapore, 1999! Vol. 4.27L. Seijo and Z. Barandiara´n, J. Chem. Phys.94, 8158~1991!.28B. H. Botch, T. H. Dunning Jr., and J. F. Harrison, J. Chem. Ph
75, 3466~1981!.
B
,
,
J.
.
29K. Andersson and B. O. Roos, Chem. Phys. Lett.191, 507~1992!.30B. O. Roos, in New Challenges in Computational Quantu
Chemistry, edited by R. Broer, P. J. C. Aerts, and P. S. Bag~Groningen, Netherlands, 1994!, pp. 12.
31K. Pierloot, E. Tsokos, and B. O. Roos, Chem. Phys. Lett.214,583 ~1993!.
32P. Belanzoni, E. J. Baerends, S. van Asselt, and P. B. LangeJ. Phys. Chem.99, 13 094~1995!.
33G. te Velde and E. J. Baerends, J. Comput. Phys.99, 84 ~1992!.34A. Fujimori and F. Minami, Phys. Rev. B30, 957 ~1984!.35G. J. M. Janssen and W. C. Nieuwpoort, Phys. Rev. B38, 3449
~1988!.36L. Pueyo, V. Luan˜a, M. Florez, E. Francisco, J. M. Recio, and M
Bermejo, Rev. Solid State Sci.5, 137 ~1991!.37C. de Graaf, R. Broer, and W. C. Nieuwpoort, Chem. Phys.208,
35 ~1996!.38M. Geleijns, C. de Graaf, R. Broer, and W. C. Nieuwpoort, Su
Sci. 421, 106 ~1999!.39C. Sousa, W. A. de Jong, R. Broer, and W. C. Nieuwpoort, M
Phys.92, 677 ~1997!.40H. Agren, P. S. Bagus, and B. O. Roos, Chem. Phys. Lett.82, 505
~1981!.41R. Broer and W. C. Nieuwpoort, Chem. Phys.54, 291 ~1981!.42R. Broer and W. C. Nieuwpoort, Theor. Chim. Acta73, 405
~1988!.43J. A. Aramburu, M. Moreno, and M. T. Barriuso, J. Phys.: Co
dens. Matter8, 6901~1996!.44K. Wissing, M. T. Barriuso, J. A. Aramburu, and M. Moreno,
Chem. Phys.111, 10 217~1999!.45J. A. Aramburu, M. Moreno, K. Doclo, C. Daul, and M. T. Ba
riuso, J. Chem. Phys.110, 1497~1999!.46M. T. Barriuso, J. A. Aramburu, C. Daul, and M. Moreno, Int.
Quantum Chem.61, 563 ~1999!.47I. Cabria, M. Moreno, J. A. Aramburu, M. T. Barriuso, U. Rogu
lis, and J. M. Spaeth, J. Phys.: Condens. Matter10, 6481~1998!.48B. Villacampa, R. Cases, V. M. Orera, and R. Alcala´, J. Phys.
Chem. Solids55, 263 ~1994!.49T. Ziegler, A. Rauk, and E. J. Baerends, Theor. Chim. Acta43,
261 ~1977!.50A. C. Stuckl, C. A. Daul, and H. U. Gu¨del, J. Chem. Phys.107,
4606 ~1997!.51S. Veliah, R. Pandey, and S. Marshall, Modell. Simul. Mater. S
Eng.2, 933 ~1994!.52S. Sugano and R. G. Shulmann, Phys. Rev.130, 517 ~1963!.53K. Pierloot, E. van Praet, and L. G. Vanquickenborne, J. Che
Phys.96, 4163~1992!.54J. A. Aramburu and M. Moreno, Phys. Rev. B56, 604 ~1997!.55P. W. Jacobs, J. Phys. Chem. Solids52, 35 ~1991!.
