PHYSICAL REVIEW A, VOLUME 63, 013202

Electronic excitation energies of small ZniSi clusters

Jon M. Matxain, Arantxa Irigoras, Joseph E. Fowler, and Jesus M. UgaldeKimika Fakultatea, Euskal Herriko Unibertsitatea, P.K. 1072, 20018 Donostia, Euskadi, Spain

~Received 17 March 2000; published 11 December 2000!

Improvements in the characterization of II-VI compound-based solar cells, and a recent experimental char-acterization of small clusters and nanoparticles, make the study of small II-VI clusters very interesting. In aprevious work the global minima of small ZniSi clustersi 51 –9 were characterized. In order to calculate theexcitation energies of these clusters, basically two methods are available: on the one hand, the traditionallyused configuration interaction singles~CIS! theory, and on the other hand the recently developed time depen-dent density-functional theory~TDDFT!. Calculations of the excitation energies of small ZniSi clusters,i51 –3, were performed with both methods in an attempt to find the most appropriate one. The relativisticcompact effective core potentials and shared-exponent basis set of Stevens, Krauss, Basch and Jasien~SKBJ!@Can J. Chem.70, 612 ~1992!#, systematically enlarged with extra functions, were used in this work. Theselarger basis sets are labeled according to the number of added functions. Thus, as an example, if two extraspfunctions and oned function are added, the final basis set is denoted SKBJ(2sp1d). These basis sets werecombined with both methods. In this way the most appropriate method and basis-set combination was chosen,for further excitation energy calculations on larger ZniSi clusters. It was seen in both methods that more thanone polarization function was needed. Combined with the CIS method, the smallest basis yielding good resultswas SKBJ(1sp2d3 f ), and with TDDFT SKBJ(1sp2d2 f ). In the CIS case, this basis was too large even forZn3S3. In addition to this, in the literature TDDFT was seen to provide a better description of the excitations,and therefore a TDDFT-SKBJ(1sp2d2 f ) combination was chosen for further calculations. However, due tothe fact that no experimental data are available, some results confirming the TDDFT results are necessary, inthis way ensuring that our choice is the correct one. Multireference configuration interaction calculations,combined with a triple-z double polarization~TZ2P! basis set, were carried out for ZniSi , i 51 and 2. Fori 53, a TZ basis without polarization was used; otherwise the limit of 255 on the basis function number wasexceeded. These results were clearly in agreement with the TDDFT results, and confirm our previous choice.

DOI: 10.1103/PhysRevA.63.013202 PACS number~s!: 36.40.Cg

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I. INTRODUCTION

Interest in II-VI compound semiconductors has growspectacularly in recent years due to their paramount techlogical potential. Their special semiconductor propertmake these compounds suitable for applications such astovoltaic solar cells@1–10#, optical sensitizers@11#, photo-catalysts@12,13#, or quantum devices@14#. As can be appre-ciated, such important applicability has led to extensinvestigation.

In addition to the importance of experimental researtheoretical studies are of great importance not only becaof their ability to expand our understanding, but also becaof their predictive power. Therefore, many studies of thecompounds have appeared in the literature@15–24#.

The band gaps of these compounds make them especinteresting for photovoltaic solar cells. The sun irradiatesmost of the energy within the visible range. This regionthe electromagnetic spectrum is very narrow, from 1.75 teV. Therefore, a good material for its use in photovoltsolar cells must have band gaps close to the energies gabove. In addition to this property, the solar cell must haan n-type semiconductor and ap-type semiconductor in order to obtain a potential difference between the two matals, which is necessary to produce electricity. The II-VI copound semiconductors have these properties. The bandgo from 1.45 eV for CdTe to 3.66 eV for ZnS, and all thecompounds present eitherp- or n-type semiconduction@10#.

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Therefore, these materials are ideal for solar cells.Nowadays all II-VI-based solar cells are composed

bulk systems, which basically have one absorbing bandor excitation energy. Nevertheless, the fact that clusternanoparticle characterization is becoming technologicapossible opens new possibilities. Cluster properties chaas size increases, and henceforth different-sized clusshould have different excitation energies. Combined togein a hypothetical cluster-based solar cell, the efficiency ofcell could increase considerably. In addition to this, it wobserved experimentally in CdS-CdTe solar cells that injunction there is a small region where the two cells amixed, forming CdS12xTex compounds. It is believed that inthe solar cell most of the absorption occurs in this smregion. The study of the excitation energies of small II-binary and ternary clusters appears interesting and imporat this point, and may open new ways of the high-efficiesolar cell characterization.

Theoretically, the excitation energies may be accuratcalculated at the complete active space second order pebation @25,26# ~CASPT2! level of theory. This methodmakes use of a multideterminantal wave function, whereferent determinants are generated, exciting electrons fvalence occupied orbitals to virtual valence orbitals. Theorbitals form the so-called complete active space. Thsecond-order perturbation theory is applied to this wafunction. However, the large amount of electrons in the stems under study make the use of such theory imposs

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MATXAIN, IRIGORAS, FOWLER, AND UGALDE PHYSICAL REVIEW A 63 013202

Henceforth less sophisticated methods have to be considThe Hartree-Fock ~HF!-based configuration interactiosingle~CIS! method is one of these. Single excited determnants are generated from the reference HF determinantciting one electron each time from an occupied orbital tovirtual one, and the wave function is spanned as a lincombination of the generated determinants. Neverthelessrecently developed time-dependent density-functional the~TDDFT! method was claimed to yield more accurate resu@27,28#. However, it was seen that Rydberg states arecalculated accurately@29#. In this method excitation energieare calculated from the poles of the frequency-dependentlarizability and oscillator strengths from the residues.

Our goal is to find the most proper method in ordercalculate the excitation energies of ZniSi global minima,i51 –9, characterized previously in our group at the BeckLee-Yang-Perdew~B3LYP! combined with the relativisticcompact effective core potentials and shared exponent bset of Stevens, Krauss, Basch, and Jasien~SKBJ! level oftheory @30#. For that purpose CIS and TDDFT calculatiowere performed fori 51 –3. using the SKBJ basis set,which extra sp, d, and f functions were systematicalladded in order to find the most proper basis.

Unfortunately, dealing with such compounds, which hanot been experimentally characterized, has a clear drawbthe lack of availability of experimental data. Hence it is dficult to perform a proper comparison between both methoNevertheless, according to the obtained data and the reences found in the literature, a method–basis-set combtion will be chosen. In order to dispel the uncertainties tmay arise about this choice, multireference configurationteraction~MR-CI! calculations were performed. Similarly tCASPT2, a multideterminantal wave function is generatexciting valence electrons, but instead of applying perturtion theory a configuration interaction expansion is pformed for the wave function. These results will be used aconfirmation of our previous choice.

II. METHODS

Two methods have been tested in order to find the maccurate one to calculate the excitation energies of smZniSi global minima, i 51 –3: CIS @31# method and theTDDFT method@32–34#. In order to perform the TDDFTcalculations, the hybrid Becke-style one-parameter futional method, using modified Perdew-Wang exchangePerdew-Wang 1991 correlation~MPW1PW91! @35#, wasused. This combination was seen to provide the best res@27#, along with the Becke3@36# Perdew86@37# functionalB3P86. Both CIS and TDDFT calculations were performwith the relativistic compact effective core potentials ashared-exponent basis set@38# of Stevens, Krauss, Baschand Jasien~SKBJ!. The Zn d electrons were treated as vlence electrons. In order to analyze the influence of the bset size on the calculated excitation energies, extra functwere systematically added to the SKBJ functions to a mamum of extra 3sp4d4 f functions. These functions were energy optimized using theGAMESS @39# program package athe MP2 level of theory, and the exponents are given

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Table I. These larger basis sets are labeled according tonumber of added functions. Thus, as an example, the larone is denoted SKBJ(3sp4d4 f ). CIS and TDDFT calcula-tions were carried out with theGAUSSIAN94 @40# andGAUSS-

IAN98 @41# packages.Previously, the unfortunate fact that no experimental d

are available for these compounds was mentioned. In oto obtain some ‘‘confirmation’’ values ensuring that ochoice is correct, MR-CI calculations were carried out wthe PSI@42# quantum chemistry program package developby the Schaefer group. The all electron triple-z double po-larization ~TZ2P! basis set (14s11p6d2 f /10s8p3d2 f ) forZn @43–45# and (13s10p2d1 f /6s5p2d1 f ) for S @46,47# wasused, except for Zn3S3. In the case of Zn3S3 no polarizationfunctions were used, due to the fact that the basis set limPSI, 255 functions, was exceeded.

III. RESULTS AND DISCUSSION

The results of this study were divided into three parts,hopes that in this way they will be more understandablethe reader. In Sec. III A the CIS results are analyzed, anSec. III B the TDDFT results. In both sections the basis-size effect is analyzed. Finally, in Sec. III C the MR-CI vaues are shown and compared to the previously obtainedand TDDFT results. In this way, the best behaved methobasis-set combination is chosen, which will be further usto calculate the excitation energies of larger ZniSi clusters.

A. CIS results

In this section the CIS excitation energies for ZniSi globalminima, i 51 –3, are presented. This method is combinwith the SKBJ effective core potential, and extra functioare systematically added to the SKBJ basis set in ordefind the most appropriate combination. Only the resultsZn1S1 are fully discussed, since the conclusions are simfor the rest.

In Table II the CIS excitation energies from the1Sground state of Zn1S1 are shown. For the sake of brevity, wwill focus the discussion on the calculated excitation engies with different basis sets for the1P excitation. Observe

TABLE I. Extra functions added to the SKBJ basis.

Zn Sa d d a d d

sp 0.017900 1.00 1.00 0.033139 0.21652 0.307sp 0.005967 1.00 1.00 0.01104 0.21652 0.307sp 0.001989 1.00 1.00 0.00368 0.21652 0.307d 0.3264 1.00 0.7 1.00d 0.1088 1.00 0.2333 1.00d 0.03627 1.00 0.07777 1.00d 0.01209 1.00 0.02592 1.00f 3.1109 1.00 0.55 1.00f 1.037 1.00 0.1833 1.00f 0.3457 1.00 0.0611 1.00f 0.1152 1.00 0.02037 1.00

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ELECTRONIC EXCITATION ENERGIES OF SMALL . . . PHYSICAL REVIEW A63 013202

that the basis-set size effect is similar in all states. The inence of extrasp, d andf functions is analyzed separately,procedure used along all this work.

In order to study the influence of extrasp functions, theresults of SKBJ(1d), SKBJ(1sp1d) and SKBJ(2sp1d) arecompared, which are 1.1410, 1.1564, and 1.1573 eV, restively. It is seen that while the addition of an extras andpfunctions to SKBJ(1d) changes the resulting excitation eergies significantly, mores andp additions do not. The sambehavior was observed comparing the SKBJ(1sp2d1 f ) andSKBJ(2sp2d1 f ) results, i.e., 1.1966 and 1.1975 eV.

Analyzing the influence of extrad functions, fromthe SKBJ(2sp1d), SKBJ(2sp2d), and SKBJ(2sp3d)results, 1.1573, 1.1398, and 1.1403 eV, respectively,need for at least two extrad functions is concluded. Asimilar conclusion is obtained comparing thSKBJ(1sp1d1 f ), SKBJ(1sp2d1 f ), and SKBJ(1sp3d1 f )results 1.2090, 1.1966, and 1.1974 eV, respectively. A thextrad function does not significantly change the final resu

The largest jump in theDE is produced by the addition oextra f functions. Comparing the results of SKBJ(2sp3d),SKBJ(2sp3d1 f ), SKBJ(2sp3d2 f ), SKBJ(2sp3d3 f ), andSKBJ(3sp4d4 f ), which are 1.1403, 1.1982, 1.2397, 1.255and 1.2576 eV, respectively, we conclude that at least thextra f functions are needed. From these CIS results, forZn1S1 structure the SKBJ(1sp2d3 f ) basis set appears to bthe best choice.

A similar analysis may be performed for the excitatienergies of a Zn2S2 global minimum, ofD2h symmetry and1Ag ground state, given in Table III. It may be seen th

TABLE II. CIS results for Zn1S1 . DE is the excitation energy~eV!.

1P 1S 3P 3S

SKBJ1 DE DE DE DE

- 1.1361 3.8757 0.5344 1.33471d 1.1410 3.8914 0.4932 1.40641sp1d 1.1564 3.8821 0.5142 1.42072sp1d 1.1573 3.8834 0.5154 1.42112d 1.1260 3.8970 0.4818 1.41063d 1.1328 3.9022 0.4916 1.41191sp2d 1.1389 3.8906 0.5002 1.42441sp3d 1.1394 3.8903 0.5027 1.42342sp2d 1.1398 3.8915 0.5014 1.42462sp3d 1.1403 3.8906 0.5034 1.42401 f 1.1942 3.8925 0.5401 1.41221sp1 f 1.2090 3.8827 0.5603 1.42711sp2d1 f 1.1966 3.8934 0.5493 1.43041sp3d1 f 1.1974 3.8931 0.5516 1.43072sp1 f 1.2099 3.8839 0.5614 1.42762sp2d1 f 1.1975 3.8941 0.5504 1.43072sp3d1 f 1.1982 3.8934 0.5512 1.43132sp3d2 f 1.2397 3.8949 0.5991 1.49082sp3d3 f 1.2557 3.8980 0.6165 1.50833sp4d4 f 1.2576 3.8985 0.6186 1.5107

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again SKBJ(1sp2d3 f ) seems to be the smallest proper bsis.

The CIS excitation energies for Zn3S3 global minimum,of D3h symmetry and1A18 ground state, are given in TablIV. There it can be viewed that the only SKBJ basis wenlarged by a maximum of twof functions. This was due tothe fact that the addition of a third function required too lara disk space. Hence it was not possible to perform an ansis as in the previous cases.

These results show the importance of including polarition functions in the basis set. Although it is too large evfor Zn3S3, the SKBJ(1sp2d3 f ) basis set has been seen tothe smallest proper basis to be combined to a CIS statorder to calculate reliable excitation energies. Neverthelthis combination cannot be used for larger clusters, atherefore does not seem to be a practical choice.

B. TDDFT results

In this section the calculated TDDFT excitation energfor ZniSi global minima,i 51 –3, are shown. As in the CIScase, the SKBJ effective core potential is systematicallylarged in order to find the most proper basis set. Similarlythe CIS case, only the Zn1S1 case is fully discussed. In TablV, the TDDFT excitation energies of Zn1S1 are shown.

It may be observed that the addition of extra functionsthe SKBJ basis changes significantly the final result. Asthe CIS case, we follow a step by step method. First

TABLE III. CIS results for Zn2S2 . DE is the excitation energy~eV!.

1B2g1B1u

1Au3B2g

3B1u3Au

SKBJ1 DE DE DE DE DE DE

1d 3.3668 4.0397 4.1394 3.1271 3.2631 3.5061sp1d 3.3627 4.0299 4.1329 3.1262 3.2659 3.5081sp2d 3.3604 4.0823 4.1339 3.1059 3.3158 3.5091sp3d 3.3645 4.0843 4.1361 3.1072 3.3169 3.5092sp1d 3.3650 4.0240 4.1349 3.1276 3.2667 3.5082sp2d 3.3613 4.0825 4.1346 3.1061 3.3158 3.5102sp3d 3.3644 4.0842 4.1366 3.1062 3.3163 3.5081d1 f 3.4117 4.0401 4.1816 3.1713 3.2613 3.5101d2 f 3.4362 4.0701 4.2035 3.1942 3.2923 3.5321d3 f 3.4495 4.0932 4.2145 3.2058 3.3101 3.5391sp1d1 f 3.4064 4.0227 4.1742 3.1692 3.2643 3.5131sp1d2 f 3.4299 4.0533 4.1971 3.1915 3.2964 3.5361sp1d3 f 3.4443 4.0792 4.2061 3.2046 3.3145 3.5442sp1d1 f 3.4086 4.0239 4.1762 3.1705 3.2652 3.5132sp1d2 f 3.4320 4.0548 4.1988 3.1929 3.2976 3.5372sp1d3 f 3.4461 4.0804 4.2072 3.2060 3.3157 3.5451sp2d2 f 3.4383 4.0976 4.2081 3.1827 3.3294 3.5382d3 f 3.4460 4.1101 4.2193 3.1875 3.3296 3.5351sp2d3 f 3.4491 4.1070 4.2178 3.1929 3.3383 3.5441sp2d1 f 3.4098 4.0770 4.1808 3.1544 3.3083 3.5163d3 f 3.4605 4.1204 4.2233 3.2019 3.3427 3.5461sp3d3 f 3.4540 4.1117 4.2212 3.1960 3.3421 3.5462sp3d3 f 3.4541 4.1120 4.2216 3.1954 3.3419 3.546

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MATXAIN, IRIGORAS, FOWLER, AND UGALDE PHYSICAL REVIEW A 63 013202

influence of extrasp functions is studied, then the influencof extrad functions and finally the influence of extraf func-tions.

An extrasp function seems not to be very important fthe 1P state, as the SKBJ(1d) basis yields an excitationenergy of 0.7633 eV and the SKBJ(1sp1d) basis one of0.7687 eV. Nevertheless, for the1S state, note that theSKBJ(1d) energy is 3.9933 eV and the SKBJ(1sp1d) en-ergy is 3.9782 eV. The influence of a second extrasp func-tion, on the other hand, is small in all the cases. For instafor the 1P state, SKBJ(1sp1d) and SKBJ(2sp1d) energiesalmost yield the sameDE: 0.7687 and 0.7685 eV, respetively. One extrasp function seems a good choice at thpoint. At least two extrad functions are needed, as followfrom the results of SKBJ(1sp1d), SKBJ(1sp2d), andSKBJ(1sp3d) energies, which are 0.7687, 0.7414, a0.7393 eV, respectively. The results obtained by adding

TABLE IV. CIS results for Zn3S3 . DE is the excitation energy~eV!.

1E8 1E9 3E8 3E9SKBJ1 DE DE DE DE

1d 2.8602 2.8676 2.4687 2.57392d 2.8689 2.8438 2.4559 2.54263d 2.8857 2.8674 2.4725 2.56431sp1d 2.8693 2.8816 2.4808 2.58772sp1d 2.8702 2.8834 2.4817 2.58921sp2d 2.8835 2.8625 2.4727 2.56012sp2d 2.8840 2.8633 2.4732 2.56071d1 f 2.9132 2.9387 2.5206 2.64391d2 f 2.9735 2.9988 2.4559 2.54261d3 f

TABLE V. TDDFT results for Zn1S1 . DE is the excitation en-ergy ~eV!.

1P 1S 3P 3S

SKBJ1 DE DE DE DE

1d 0.7633 3.9933 20.3156 1.84121sp1d 0.7687 3.9782 20.3068 1.84572sp1d 0.7685 3.9777 20.3068 1.84481sp2d 0.7414 3.9808 20.3244 1.82962sp2d 0.7416 3.9803 20.3243 1.82921sp3d 0.7393 3.9775 20.3247 1.82262sp3d 0.7394 3.9770 20.3247 1.82261d1 f 0.8051 3.9795 20.3188 1.82041d2 f 0.8203 3.9646 20.3095 1.84381sp1d1 f 0.8094 3.9649 20.3098 1.82531sp2d1 f 0.7844 3.9678 20.3302 1.80921sp1d2 f 0.8243 3.9506 20.2997 1.85101sp2d2 f 0.8061 3.9560 20.3159 1.83842sp3d3 f 0.8135 3.9537 20.3130 1.84093sp4d4 f 0.8135 3.9536 20.3128 1.8400

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and three extrad functions do not differ significantly. Twoextraf functions have been seen to be necessary as well.the 1P state we now focus on the SKBJ(1sp2d),SKBJ(1sp2d1 f ), and SKBJ(1sp2d2 f ) excitation energies0.7414, 0.7844, and 0.8061 eV, respectively. This last vastill diverges considerably from the one with only one exf function. However, comparing it to the SKBJ(2sp3d3 f )energies, which is 0.8135 eV, it is seen that a third one isnecessary. This value is also found for a larger basis set.difference between SKBJ(1sp2d2 f ) and SKBJ(2sp3d3 f )energies is small enough that we can choose this first basa proper one to combine with TDDFT, in this way savinCPU usage. It should also be pointed out that a negaexcitation energy is calculated for the3P state.

Of course, one might think that the proper basis for Zn1S1

is not for larger clusters. The results for Zn2S2, are given inTable VI. Looking at this table, one may observe that tsmallest basis yielding good values is the SKBJ(1sp2d2 f )energy. The Zn3S3 data, given in Table VII, also show thSKBJ(1sp2d2 f ) basis set as the smallest one yielding goresults. These calculations were performed withC2v symme-try and an 1A1 ground state, for better comparison wiMR-CI results.

It seems logical then to choose this SKBJ(1sp2d2 f ) ba-sis set for further calculations. It is seen, as in the CIS cathat the inclusion of more than one polarization functionthe basis set is indispensable. For TDDFT, two polarizatfunctions are enough, and three are needed in the CIS c

C. MR-CI results

We have already obtained results with both CIS aTDDFT methods, and found a proper basis to combine weach one. However, the lack of availability of experimendata makes one uncertain about the accuracy of these c

TABLE VI. TDDFT results for Zn2S2 . DE is the excitationenergy~eV!.

1B2g1B1u

1B3u3B2g

3B1u3Au

DE DE DE DE DE DE

1d 2.0627 2.8697 2.9279 1.8958 2.2998 2.5971sp1d 2.0651 2.8654 2.9274 1.8997 2.3031 2.5992sp1d 2.0655 2.8654 2.9275 1.8999 2.3030 2.5991sp2d 2.0597 2.8898 2.9081 1.8863 2.3287 2.5862sp2d 2.0596 2.8891 2.9079 1.8863 2.3278 2.5861sp3d 2.0612 2.8877 2.9086 1.8878 2.3264 2.5882sp3d 2.0611 2.8876 2.9086 1.8873 2.3260 2.5881d1 f 2.1036 2.8674 2.9662 1.9363 2.2936 2.6281d2 f 2.1215 2.8845 2.9817 1.9540 2.31091sp1d1 f 2.1048 2.8633 2.9647 1.9391 2.2973 2.6281sp1d2 f 2.1225 2.8813 2.9801 1.9566 2.3159 2.6461sp2d1 f 2.1035 2.8845 2.9482 1.9300 2.3182 2.6171sp2d2 f 2.1251 2.8978 2.9669 1.9521 2.3295 2.6382sp3d3 f 2.1296 2.8928 2.9691 1.9559 2.3226 2.6443sp4d4 f 2.1332 2.8963 2.9723 1.9594 2.3259 2.646

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ELECTRONIC EXCITATION ENERGIES OF SMALL . . . PHYSICAL REVIEW A63 013202

lations. In order to check the suitability of thCIS-SKBJ(1sp2d3 f ) or TDDFT-SKBJ(1sp2d2 f ) combi-nations, MR-CI/TZ2P calculations were performed, andobtained results were used as reference values.

The MR-CI calculations were carried out with the Pprogram package, as mentioned earlier. This program muse of the molecular point group and the corresponding iducible representations. Nevertheless, one of the drawbis that the largest available symmetry isD2h . Therefore, inthe cases of Zn1S1 and Zn3S3, belonging to theC`v andD3h

cases, respectively, lower symmetries must be used.Zn1S1

1S states,C2v symmetry was used, and for the1PstateC2 symmetry, as will be explained further. Zn3S3 cal-culations were carried out withC2v symmetry, as in theTDDFT calculations. Two electron excitations were allowin the MR part, and in the configuration interaction as weresulting in final configuration interaction singles doub~triples and quadrupoles!, CISD~TQ!, calculations.

In Table VIII MR-CI results are shown for Zn1S1. The

TABLE VII. TDDFT results for Zn3S3 . DE is the excitationenergy~eV!.

1B11A2

1B13B1

3A23B1

DE DE DE DE DE DE

1d 3.6054 3.6056 4.3346 3.4464 3.4467 4.0451sp1d 3.5996 3.5997 4.3279 3.4406 3.4408 4.0382sp1d 3.5995 3.5997 4.3277 3.4403 3.4406 4.0381sp2d 3.5502 3.5503 4.2747 3.3859 3.3861 3.9862sp2d 3.5503 3.5504 4.2748 3.3859 3.3861 3.9872sp3d 3.5535 3.5536 4.2774 3.3866 3.3868 3.9861d1 f 3.6409 3.6410 4.3688 3.4813 3.4815 4.0791d2 f 3.6423 3.6424 4.3705 3.4818 3.4820 4.0801sp1d1 f 3.6338 3.6339 4.3610 3.4742 3.4744 4.0711sp1d2 f 3.6343 3.6345 4.3618 3.4740 3.4742 4.0721sp2d1 f 3.5873 3.5874 4.3099 3.4225 3.4227 4.0211sp2d2 f 3.5969 3.5970 4.3205 3.4320 3.4022 4.0322sp3d3 f 3.6017 3.6018 4.3273 3.4339 3.4341 4.035

TABLE VIII. MR-CI results for different Zn1S1 electronicstates. NMR indicates the number of reference determinants,NCI the number of determinants used in the CI part of the calction. The obtained MR-CI energy is in hartrees, and the MR-CIS, and TDDFT excitation energies are in eV.

1S (g.s.) 1P (e.s.) 1S (e.s.)

NMR 461 524 461NCI 217013 221666 217013E ~MR-CI! 22175.41804222175.39027422175.273990DE(MR-CI) 0.7562 3.9221DE@CIS/SKBJ(3sp4d4 f )]

1.2576 3.8985

DE@TDDFT/SKBJ(1sp2d2 f )]

0.8061 3.9560

01320

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energy of the1S states, for both ground and excited statwere calculated with 461 determinants in the valence reence, and 217013 determinants in the CISD step, withC2vsymmetry. The1P excited-state energy was calculated w524 determinants as reference for the 221666 determinof the CISD calculation, withC2 symmetry. The obtainedMR-CI excitation energies for the1P and 1S states are0.7562 and 3.9221 eV, respectively. These results areagreement with the TDDFT/SKBJ(1sp2d2 f ) energies, i.e.,0.8061 and 3.9560 eV.

The excitation energy of the1P state was calculated withC2v symmetry at the beginning. The obtainedDE was closeto that obtained with the CIS method: 1.13 eV. WithC2vsymmetry there were two degenerate statesB1 andB2 that,due to the fact that they belong to different irreducible reresentations, do not mix. That is, this excitation is treateda single excitation. If this is correct, calculations withC2symmetry should be similar. We have seen that they areIf the calculation is performed withC2 symmetry, bothB1andB2 states inC2v belong to theB irreducible representation. In this way they mix, and the excitation is correctdescribed as a double excitation. The CIS method is not ato describe these excitations, since only single excitationstaken into account. On the other hand, the TDDFT methdescribes these excitations correctly. At this point the usethe TDDFT method seems logical in further calculations.

In Table IX the obtained MR-CI results for Zn2S2 aregiven, and are compared to the CIS and TDDFT results.1Ag state energy was calculated using 905 determinantsreference, and the CISD calculation made use of 281determinants. The energy of1B2g was calculated with 1598determinants in the reference level and 773112 determinin the CISD level, similarly to the1B1u state, where therewas 1596 determinants in the reference level and 77202the CISD level. The calculated excitation energies againin agreement with the TDDFT/SKBJ(1sp2d2 f ) method.The DE(MR-CI) 1B2g excitation energy, 2.211 eV, is verclose to that obtained at the TDDFT/SKBJ(1sp2d2 f ) level,which is 2.1251 eV. However, the1B1u excitation energy isnot so accurately calculated: 3.284 eV at the MR-CI levand 2.8978 at the TDDFT level. Again the CIS resultgreater: 4.1070 eV.

d-,

TABLE IX. MR-CI results for different Zn2S2 electronic states.NMR indicates the number of reference determinants, and NCI thenumber of determinants used in the CI part of the calculation. Tobtained MR-CI energy is in hartree, and the MR-CI, CIS, aTDDFT excitation energies are in eV.

1Ag1B2g

1B1u

NMR 905 1598 1596NCI 281382 773112 772023E ~MR-CI! 24350.81484724350.73362824350.692521DE(MR-CI) 2.211 3.284DE@CIS/SKBJ

(1sp2d3 f )]3.4491 4.1070

DE@TDDFT/SKBJ(1sp2d2 f )]

2.1251 2.8978

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eR-CIow the

MATXAIN, IRIGORAS, FOWLER, AND UGALDE PHYSICAL REVIEW A 63 013202

TABLE X. MR-CI results for different Zn3S3 electronic states. NMR indicates the number of referencdeterminants, and NCI the number of determinants used in the CI part of the calculation. The obtained Menergy is hartree, and the MR-CI,CIS, and TDDFT excitation energies are in eV. In parentheses we shstates corresponding to the CISD3h calculations.

1A11B1 (1E8) 1B1 (1E9)

NMR 121 115 115NCI 141337 191195 191195E ~MR-CI! 26526.144962 26525.983187 26525.953076DE(MR-CI) 4.4046 5.2245DE@CIS/SKBJ(1d2 f )# 2.9735 2.9988DE@TDDFT/SKBJ(1sp2d2 f )# 3.5969 4.3205

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In Table X the Zn3S3 MR-CI results may be seen. Thsymmetry of Zn3S3 was reduced fromD3h to C2v , as ex-plained above. The1E8 state found in the CISD3h calcula-tions splits into the near degenerate1B1 and 1A2 statesfound in theC2v calculations. The remaining1B1 state cor-responds to the1E9 state. No polarization functions weradded to the TZ basis set. This was due to the fact thatlimit of 255 functions was exceeded. One may expect tthe obtained results will not be very accurate, but nevertless it could be very useful to compare qualitatively with tCIS and TDDFT results. It may be seen that the MR resuare a bit far from the TDDFT results, but are even furthfrom the CIS results. In addition to this, the energy diffeence between the two1B1 excited states are very similar iboth MR and TDDFT method, 0.8199 and 0.7236 eV,spectively, which compare well and lends further supporthe TDDFT calculations.

These results show that TDDFT method is more approate than the CIS method in order to perform this typecalculation. It better describes both single and double etations. Therefore, for further calculations of the excitatienergies of ZniSi clusters, the TDDFT/SKBJ(1sp2d2 f )method is chosen.

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IV. CONCLUSIONS

It was seen that both CIS and TDDFT methods need bsets with more than one polarization functions. TSKBJ(1sp2d2 f ) basis set was seen to be the smallest byielding converged results with the TDDFT method, andSKBJ(1sp2d3 f ) to be the smallest basis required in the Cmethod. Nevertheless, this basis was too large to be ueven for Zn3S3, so it appeared to be a bad choice for furthcalculations.

The MR-CI results used as reference show the clearvantage of the TDDFT method. It better describes the twfold degenerate1P excitation in Zn1S1 than the CIS methodand also appears to be superior for the rest of the stuexcitation in all the clusters investigated. Therefore, for fther calculations the TDDFT/SKBJ(1sp2d2 f ) combinationis recommended.

ACKNOWLEDGMENTS

This research was funded by Euskal Herriko Unibertatea Grant No. UPV 203.215-G50/98. J.E.F. would likethank Eusko Jaurlaritza~the Basque Government! for fund-ing.

ter.

ells

s.

n-

@1# A. Kampmann and D. Lincot, J. Electroanal. Chem.418, 73~1996!.

@2# Y. Y. Loginov, K. Durose, H. M. Al-Allak, S. A. Galloway, S.Oktik, A. W. Brinkman, H. Richter, and D. Bonnet, J. CrysGrowth 161, 159 ~1996!.

@3# A. Niemegeers and M. Burgelman, J. Appl. Phys.81, 2881~1997!.

@4# K. Li, A. T. S. Wee, J. Lin, K. L. Tan, L. Zhou, S. F. Y. Li, ZC. Feng, H. C. Chou, S. Kamra, and A. Rohatgi, J. Mater. S8, 125 ~1997!.

@5# K. Omura, A. Hanahusa, T. Arita, H. Higuchi, T. Aramoto, TNishio, S. Sibutani, S. Kumazawa, M. Murozono, Y. Yabuchi, and H. Takakura~unpublished!.

@6# C. Ferekides and J. Britt, Sol. Energy Mater. Sol. Cells35, 255~1994!.

@7# H. C. Chou, A. Rohatgi, N. M. Jokerst, S. Kamra, S. R. StoS. L. Lowrie, R. K. Ahrenkiel, and D. H. Levi, Mater. ChemPhys.43, 178 ~1996!.

i.

,

@8# S. Naseem, D. Nazir, R. Mumtaz, and K. Hussain, J. MaSci. Technol.12, 89 ~1996!.

@9# J. Touukova, D. Kindl, and J. Tousek, Thin Solid Films293,272 ~1997!.

@10# T. L. Chu and S. S. Chu, Solid-State Electron.38, 533~1995!.@11# P. J. Sebastian and M. Ocampo, Sol. Energy Mater. Sol. C

44, 1 ~1996!.@12# A. J. Hoffman, G. Mills, H. Yee, and M. R. Hoffmann, J. Phy

Chem.96, 5546~1992!.@13# S. Kuwabata, K. Nishida, R. Tsuda, H. Inoue, and H. Yo

eyama, J. Electrochem. Soc.141, 1498~1994!.@14# E. Corcoran, Sci. Am.263, 74 ~1990!.@15# J. L. Martins and N. Troullier, Phys. Rev. B43, 2213~1991!.@16# Y. N. Xu and W. Y. Ching, Phys. Rev. B48, 4335~1993!.@17# J. Muilu and T. A. Pakkanen, Surf. Sci.364, 439 ~1996!.@18# J. Muilu and T. A. Pakkanen, Phys. Rev. B49, 11185~1994!.@19# P. Schroer, P. Kruger, and J. Pollmann, Phys. Rev. B47, 6971

~1993!.

2-6

o-

d

.

ys

C.

A

J

em

,

n.

.

A.t-

.A.

la-Y.S.y,C.

M.m-m,-

C..R.,, P..le,,

. J.al7,

ys.

J.

ELECTRONIC EXCITATION ENERGIES OF SMALL . . . PHYSICAL REVIEW A63 013202

@20# P. Schroer, P. Kruger, and J. Pollmann, Phys. Rev. B48,18 264~1993!.

@21# P. Schroer, P. Kruger, and J. Pollmann, Phys. Rev. B49,17 092~1994!.

@22# D. Vogel, P. Kruger, and J. Pollmann, Phys. Rev. B52, 14 316~1995!.

@23# J. Pollmann, P. Kruger, M. Rohlfing, M. Sabisch, and D. Vgel, Appl. Surf. Sci.104Õ105, 1 ~1996!.

@24# D. Vogel, P. Kruger, and J. Pollmann, Phys. Rev. B54, 5495~1996!.

@25# K. Andersson, P.-A˚ . Malmqvist, B. O. Roos, A. J. Sadlej, anK. Wolinski, J. Phys. Chem.94, 5483~1990!.

@26# K. Andersson, P.-A˚ . Malmqvist, and B. O. Roos, J. ChemPhys.96, 1218~1992!.

@27# S. Hirata and M. Head-Gordon, Chem. Phys. Lett.302, 375~1999!.

@28# K. B. Wiberg, R. E. Stratmann, and M. J. Frisch, Chem. PhLett. 297, 60 ~1998!.

@29# J. C. Shieh, J. L. Chang, J. C. Wu, R. Li, A. M. Mebel, N.Handy, and Y. T. Chen, J. Chem. Phys.112, 7384~2000!.

@30# J. M. Matxain, J. E. Fowler, and J. M. Ugalde, Phys. Rev.61, 053201~2000!.

@31# J. B. Foresman, M. Head-Gordon, J. A. Pople, and M.Frisch, J. Phys. Chem.96, 135 ~1992!.

@32# R. E. Stratmann, G. E. Scuseria, and M. J. Frisch, J. ChPhys.109, 8218~1998!.

@33# R. Bauernschmitt and R. Ahlrichs, Chem. Phys. Lett.256, 454~1996!.

@34# M. E. Casida, C. Jamorski, K. C. Casida, and D. R. SalahubChem. Phys.108, 4439~1998!.

@35# C. Adamo and V. Barone, Chem. Phys. Lett.274, 242 ~1997!.@36# A. D. Becke, J. Chem. Phys.98, 5648~1993!.@37# J. P. Perdew, Phys. Rev. B33, 8822~1986!.@38# W. J. Stevens, M. Krauss, H. Basch, and P. G. Jasien, Ca

Chem.70, 612 ~1992!.@39# M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M

01320

.

.

.

J.

J.

S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K.Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Mongomery, J. Comput. Chem.14, 1347~1993!, GAMESS Package.

@40# M. J. Frisch, G. W. Trucks, H. B. Schlegel, P. M. W. Gill, BG. Johnson, M. A. Robb, J. R. Cheeseman, T. Keith, G.Petersson, J. A. Montgomery, K. Raghavachari, M. A. Alham, V. G. Zakrzewski, J. V. Ortiz, J. B. Foresman, C.Peng, P. Y. Ayala, W. Chen, M. W. Wong, J. L. Andres, E.Replogle, R. Gomperts, R. L. Martin, D. J. Fox, J. S. BinkleD. J. Defrees, J. Baker, J. P. Stewart, M. Head-Gordon,Gonzalez, and J. A. Pople,GAUSSIAN94 B.2. ~Gaussian, Inc.,Pittsburgh, PA, 1995!.

@41# M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria,A. Robb, J. R. Cheeseman, V. G. Zakrzewski, J. A. Montgoery, R. E. Stratmann, B. J. Burant, S. Dapprich, J. M. MillaA. D. Daniels, K. N. Kudin, M. C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Mennucci,Pomelli, C. Adamo, S. Clifford, J. Ciolowski, J. V. Ortiz, B.BStefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi,Gomperts, R. L. Martins, D. J. Fox, T. Keith, M. A. Al-LahamC. Y. Peng, A. Nanayakkara, C. Gonzalez, M. ChallacombeM. W. Gill, B. G. Johnson, W. Chen, M. W. Wong, J. LAndres, M. Head-Gordon, E. S. Replogle, and J. A. PopGAUSSIAN98 ~Revision A5! ~Gaussian, Inc., Pittsburgh, PA1998!.

@42# J. C. Stephens, G. Vacek, C. D. Sherrill, J. T. Fermann, TVan Huis, and T. D. Crawford, Center for ComputationQuantum Chemistry University of Georgia, Athens, GA, 199PSI 2.0.8. program package.

@43# A. J. H. Wachters, J. Chem. Phys.52, 1033~1970!.@44# P. J. Hay, J. Chem. Phys.66, 4377~1977!.@45# D. M. Hood, R. M. Pitzer, and H. F. Schaefer, J. Chem. Ph

71, 705 ~1979!.@46# A. D. McLean and G. S. Chandler, J. Chem. Phys.72, 5639

~1980!.@47# R. Krishnan, J. S. Binkley, R. Seeger, and J. A. Pople,

Chem. Phys.72, 650 ~1980!.

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