J . Phys. Chem. 1990,94, 5411-5411
5471
Electronic and Geometric Structure of the Cu, Cluster Anions ( n 5 10) Hans Akeby,* Itai Panas, Lars G. M. Pettersson, Per Siegbahn, and Ulf Wahlgren Institute of Theoretical Physics, University of Stockholm, Vanadisviigen 9, S-113 46 Stockholm, Sweden (Received: November 1, 1989: In Final Form: January 29, 1990)
The structures of the anionic Cu, clusters with n I10 have been investigated by using correlated wave functions including the effects of core (3d)-valence correlation through a model operator. The Cu core orbitals (including 3d) are described by an effective core potential (ECP). The accuracy of both the ECP description and the treatment of corevalence correlation effects is established by comparing with all-electron calculations for the smaller clusters. Recent experimental data for the electron detachment energy (EDE) and HOMO-LUMO separations is combined with the computed total energies to establish the structure of the observed C u i clusters. The EDE's are found to be strongly structure dependent, which is used to determine which of the low-lying structures is observed experimentally.
Introduction The most common theoretical approach for studying reactions on metal surfaces is to use the cluster model. In our laboratory we have over the past few years used this model to study reactions of, for example, H2, Olr and CH4 on mainly nickel clusters but also on copper clusters.' One of the main results of these studies is that cluster convergence is not straightforwardly obtained for chemisorption energies even if clusters consisting of up to 50 atoms are used. One is therefore instead restrained to try to understand the lack of cluster convergence and possibly correct for it. To understand the cluster oscillations, calculations are obviously necessary for a variety of clusters. Since the number of electrons in transition-metal clusters is very large, the accuracy of these calculations is by necessity limited. A normal ab initio treatment is only possible for clusters with few atoms, and sometimes rather crude approximations are necessary. In our studies we have thus tried to treat the nickel and copper atoms as one-electron atoms using effective core potentials (ECP's). Since the accuracy of this simplified model is not fully known, it is easy to fall into the trap of confusing the physical and chemical origin of the cluster oscillations with oscillations due to inaccuracies in the calculations. The possibility to make comparisons to accurate experimental cluster beam results is of great importance in order to directly test the accuracy of the calculations. Experiments that measure the electronic properties of transition-metal clusters such as ionization energies, electron detachment energies, and excitation energies are particularly well suited for comprisons to theoretical model calculations. In contrast to these more recent experiments, it is much more difficult to make comparisons between the measured and calculated reactivities which were previously the only available information. In order to test the accuracy of our one-electron model for the copper atom, we will in this study compare calculated results for electron detachment energies (EDEs) and HOMO-LUMO gaps for anionic copper clusters with available results from two different experiments. First, for these Cu, clusters, with n up to 10, h p o l d et al.2 have determined accurate EDEs. For some of the clusters the positions of excited electronic states were also located, giving information on the HOMO-LUMO gaps for the corresponding neutral clusters. More recently, Pettiette et aL3extended this type of study to clusters up to n = 41 including also the HOMOLUMO gap for all even clusters. As a test of the accuracy of the one-electron model for copper, we will also make comparisons to (1) Siegbahn, P.; Blomberg, M.; Panas, 1.; Wahlgren, U. Theor. Chim. Acta 1989, 75, 143. Panas, 1.; Siegbahn, P.; Wahlgren, U. In The Challenge of d andf Electrons: Theory and Computation; Salahub, D. R., Zerner, M. C., Eds.; ACS Symposium Series 394; American Chemical Society: Washington, DC,1989; Chapter 9. Panas, I.; Siegbahn, P. Chem. Phys. Lett. 1988, 153,458. SchOle, J.; Siegbahn, P.; Wahlgren, U. J . Chem. Phys. 1988, 89, 6982. Panas, 1.; Siegbahn, P.; Wahlgren, U. J. Chem. Phys. 1989,90,6791. ( 2 ) Leopold, D. G.; Ho,J.; Lineberger, W. C. J. Chem. Phys. 1987'86, 1715. (3) Pettiette, C. L.; Yang, S.H.; Craycraft, M. J.; Conceicao, J.; Laaksonen, R. T.; Chanovsky, 0.;Smalley, R. E. J . Chem. Phys. 1988.88, 5377.
more rigorous all-electron calculations for the clusters Cu, to Cug. In our study we find that the spectral properties of the clusters are very sensitive to the topological structures of the clusters. For example, for Cu8- there are two structures that differ in total binding energy by only 0.3 eV but that differ by almost 0.9 eV in their EDEs. A remarkable point in this context is that even though both these structures of Cue- can be expected to be populated under the conditions of the experiments, the experimental ground-state EDE can be given with an uncertainty of only 0.02 eV. When the accuracy of our one-electron model for the copper atom has been established, a second goal of this study is to provide new chemical information on the copper clusters. The abovementioned sensitivity of the electronic structure to the geometry is an interesting property that can be utilized in a combination of experiments and theory. If, for example, two different geometrical structures are obtained and the electron affinities or the HOMO-LUMO gaps are quite different, a comparison with the experimental values will decide which of the two geometrical structures is the actual ground state. This procedure is much more reliable than making this decision based only on the calculated total energies. In our oneelectron model of the copper atom a core polarization and core-valence correlation operator is included to mimick 3d correlation effects according to a formalism developed by Muller et al. for alkali-metal and alkaline-earth atoms.4 This type of approach has been used for small copper clusters before by Flad et and a similar approach has been used by Jeung et a1.6 However, the accuracy of this model for copper has not been tested against more rigorous calculations of 3d correlation effects before. This is therefore also a secondary goal of the present study. For this purpose comparisons will be made against the accurate theoretical investigations by Bauschlicher et al. for Cuz and Cu3?J A particularly interesting test of the model is to study the effect on the geometry of Cu, since without 3d correlation, Cu, becomes linear but with 3d correlation it becomes an almost equilateral triangle.
Computational Details For all cluster structures studied here, both anionic and neutral, the nearest-neighbor distance was maintained at that of the bulk metal (4.8304 a,,). Only topologically different clusters were investigated, and no search for optimum geometries by small changes in distances or angles has been undertaken. Some calculations on the smaller clusters were performed, however, to (4) MOller, W.; Flesch, J.; Meyer, W. J . Chem. Phys. 1984, 80, 3297. (5) Had, J.; Igel-Mann, G.; Dolg, M.; Reus, H.; Stoll, H. SurJ Sci. 1985,
163, 285.
(6) Jeung, G. H.; Malrieu, J. P.; Daudey, J. P. J . Chem. Phys. 1982, 77, 3571. (7) Langhoff, S. R.; Bauschlicher, C. W.; Walch, S. P.; Laskowski, B. C. J . Chem. Phys. 1986.85, 721 1 . (8) Bauschlicher, C. W., Jr.; Langhoff, S.R.; Taylor, P. R. J . Chem. Phys. 1988, 88, 104 1.
0022-3654/90/2094-5411%02.50/0 0 1990 American Chemical Society
Akeby et al.
5472 The Journal of Physical Chemistry, Vol. 94, No. 14, 1990 TABLE I: Cu Basis Set rod ECP Pa" Valence Basis Set exponent
ECP[Ss,3p] contraction coeff
S
S
252.440 13.160 I 2.807 06 1.120 IO 0.131 029 0.047 550
-0.005 938 0.124 873 -0.2 17 948 1 .o 1 .o
exponent
ECP[4s, 1p] contraction coeff
S
S
252.440 13.1601 2.807 06 1.12010 0.131 029 0.047 955
-0.007 135 0.130 829 -0.208 001 1 .o
coeff frozen 3s
coeff frozen 3s
~
-0.707 279 62 0.253 485 49 0.227 036 15 -0.11098540
1.o
ECP[Ss,3p] contraction coeff P - 0 . 1 I9 695 -0.267 968 0.508 310 0.588 356 1 .o
exponent P 96.078 9 23.857 6 3.675 77 1.22287 0.121 OOO
coeff frozen 3p
-0.704 352 87 0.268 230 34 0.209 120 IO -0.101 92647
1 .o
1.o
ECP[4s, 1p] contraction coeff P
exponent P
0.993 936 10 0.030 799 37
1 .o
0.121 000
ECP Parameters' ECP[SsJp] Z.a a,
A2
43.521 469 1.936591 0.916 282
0.922 677 0.477 577 0.301 283
k Is 2s
2P
9.0 Y,
12.835 83 1
Cm 1.089 877
a"
42.601 362 1.929604 0.920 259
ECP[4s,lp] Z, = 3.0 A, Ym 3.382920 12.709 189 1.557861 1.431 642
Level-Shift Parameters for the Projection Operator Part of the ECP.' Bk k Bk 3.318 328.79 3pd 3d' 40.83 1.050 35.61 3df 1 .Ooo
Cm
IO.182 894
Bl
-0.021 -0.013
a For definitions see ref 9. *The coefficients A I given in ref IO, Table I, are not correct. The calculations, both here and in ref 10, were done with the coefficients given above. CThebasis set for the s- and ptype projection operators were taken from Roos, B.; Veillard, A.; Vinot, G. Theor. Chim. Acta 1971, 20, 1, and those for the d-type projection operator were taken from Wachters, A. J. H. J. Chem. Phys. 1970,52, 1033. dThis parameter is only for the ECP[4s,lp]. 'ECP[Ss,3p] with added diffuse d function (0.30). fECP[4s,lp] with added diffuse d function (0.20).
investigate the sensitivity of the computed EDEs to variations in the interatomic distances. The Cu atom was described by a one-electron effective core potential (ECP) based on the frozen-orbital ECP method of ref 9. The interactions with the inner core orbitals (ls,2s,2p) as well as the 3d orbitals are given through the local potential and a nonlocal level-shift operator. The 3s and 3p orbitals are kept as frozen atomic orbitals and only the 4s orbital is variationally optimized. The important effects of 3d shell relaxation are included by modifying the 3d level-shift operator such as to reproduce all-electron SCF results for Cu50. To give a better description of the anionic clusters, additional ~(0.20)and p(0.38) functions were added to the valence basis set. The parameters and the basis set for the ECP (ECP[Ss,3p]) are given in Table I. A smaller ECP (ECP[4~,lpl),'~ with only a frozen 3s orbital and no diffuse functions, was used in preliminary calculations on the Cus- to Culo- clusters. Selected structures were then recalculated with the bigger ECP. The smaller ECP was tested on clusters already calculated with the bigger ECP and was found to give reliable predictions of low-lying cluster structures. All results discussed in the present investigation have been evaluated at the externally contracted CI (CCI) level" including for the effects of higher excitations. the Davidson correction 2I)*( All valence electrons were correlated and, unless otherwise stated, the wave functions were generated with the use of a single reference state. Core-valence correlation effects were included through the use of the approximate core polarization potential
(CPP) suggested by Miiller et This potential is based on a classical description of the core as a polarizable charge distribution interacting with the instantaneous fields generated by the valence electrons. The potential generated by core c on the valence electrons is thus taken proportional to the resulting field on core c and the induced dipole moment acfcassociated with this core. The additional operator to be included in the Hamiltonian and projected onto the valence space then becomes
(9) Panas, I.; Siegbahn, P. E. M.; Wahlgren, U.Chem. Phys. 1987,112, 325. (10) Mattsson, A.; Panas, I.; Siegbahn, P.; Wahlgren. U.;Akcby, H. Phys. Rev. B 1987, 36. 7389. (11) Siegbahn, P. Int. J . Quanrum Chem. 1983, 23, 1869. (12) Davidson, E. R. In The World o/Quantum Chemisrry; Daudel, R., Pullman, B., Ed.; Rcidel: Dordrecht, 1974.
The parameters a, and pe in the CPP operator were taken from Pettersson et al.I3 and were not reoptimized for the present study. The core dipole polarizability, a, (6.428 a&, was obtained in ab
tcpp= - f/2C(a,f,Z - 2fC.l.c,O+ f,O.l.c,O) C
where F: is the induced dipole moment of core c and f: is the expectation value of the field at core c. The latter two terms give corrections for the actual induced core dipole moments as computed with the basis set used in the calculations. In the ECP calculations in the present work these two correction terms vanish. The field, f,, is the resulting field on core c from all the valence electrons and surrounding cores; Le., f, = f: + fen, where f: = x,f:(i) is summed over all electrons i. A cutoff function is used on the valence charge density to remove contributions to the field from electronic charge inside the spherically symmetric core. Following ref 4, this is taken as
= (1 - exP(-(rd/P,)2))2 such that the field on core c from the valence electrons is given by C(PCJcA
(13) Pettersson, L. G.M.; Akcby, H. To be published.
The Journal of Physical Chemistry, Vol. 94, No. 14, 1990 5473
Structure of Cu, Cluster Anions TABLE 11: Calibration Calculations. Comparison of Computed EDE’s (eV) Using AU-Electron (AE) rad ECP Valence Dcscriptionr and with Cu(3d) Correlation Illeluded Y ACEkctron @ or Using CPP Operator (SCF Results in Parentheses, All Otber Results at CCI+Q Level)
cluster CUIcui cu,cus-
Electron Detachment Energy no CPP symmetry AE’ ECP AE 1.01 0.45 0.66 D,h D,h D3r D2h
0.67 1.97 1.16
c,
c4n D3h
Cue-
oh
0.32 1.38 0.91 0.74 (0.23) (0.57) (0.74) (0.55)
0.44 1.51 0.90 0.80 (0.38) (0.72) (0.84) (0.81)
with CPP ECP AE 0.65 0.59 1.69 1.16 1.22 (1.15) (1.43) (1.41) (1.48)
0.89 0.66 1.93 1.23 1.23 (1.25) (1.56) (1.42) (1.72)
HOMO-LUMO Separation no CPP cluster Cu, Cus cu6
----.
symmetry Du (IA, 3BI,) C, (zAl *BI) C, ( B ‘A2) D3h (’AI’ ‘A,”) oh (3T2g ‘E,)
ECP 0.51 (1.17) (0.08) (0.1 1) (0.95)
AE
with CPP AE ECP
0.61 (1.54) (0.55) (0.00) (0.25) (0.74) (0.80) 0.63 (1.43) (0.22)
0.82 (1.93) (0.41) (0.48) (0.55)
aResults including explicit correlation of the 3d shell from ref 8.
initio calculations on the Cu+ core, and the cutoff parameter pE (1.472) was determined so as to reproduce the experimental ionization energy of Cu(%). It should be noted that including core-valence correlation effects gives, apart from quantitatively better results, also qualitatively different results for the optimum geometries. For instance, neutral Cu3 without the CPP operator included is linear, while the core (3d)-valence correlation effects give a larger energy decrease for the triangular, D3h,structure. At the all-electron level the D3,,structure then becomes the lowest, in agreement with more elaborate calculation^.^
Calibration Calculations The calculations on the larger clusters utilize two basic computational approximations which will be discussed separately in this section. We have separately investigated the ECP approximation for the smaller clusters (16 atoms) by performing the corresponding all-electron calculations. In the correlation treatment all valence electrons are always included, but the important effects stemming from corevalence correlation have also to be considered. The formalism used in the present work to describe these effects will be compared with all-electron calculations using large basis sets and including explicit correlation of the Cu 3d shell performed by Bauschlicher et aL7 The all-electron calibration calculations were performed by use of the DZC h i s set suggested by Tatewaki and Huzinaga.14 This was extended with the same s and p functions as used in the ECP[Ss,3p] and with an additional diffuse d function. For the smaller clusters ( n 1 4) both SCF and CCI calculations were performed, while for Cus and Cu6 the calculations were done at the SCF level only. This difference in the treatment affects the comparison of the magnitude of the effect of the CPP operator between the clusters, since at the SCF level normally larger effects are obtained than when the valence shell is correlated (see below). The results of these calibration calculations are presented in Table 11, where both the computed EDE’s and the HOMOLUMO separations are shown for the different clusters. The effects of including the CPP operator are given separately. Starting by comparing the ECP results with the corresponding all-electron results, we find a discrepancy of the order of 0.10-0.20 eV in the computed EDE. This result is fairly independent of the size of the clusters and also of whether or not the CPP operator has been included. The EDE computed using the ECP description
is never larger than that computed at the all-electron level. A reasonable estimate of the EDEs from the ECP calculations would then be to increase the computed value by 0.10 eV and assigning an uncertainty of fO.10 eV to this value. The nearest-neighbor distance was maintained at that of the bulk in all clusters studied. The effects of this approximation on the EDEs was investigated through additional calculations performed on Cu2 and Cu3, where the largest differences between bulk and optimized geometries may be expected. Computing the EDE of CUI at the computed bond distance (4.29 a,J of neutral Cuz from ref 8 affects the EDE by 0.2 eV. For linear and triangular Cu, the EDE’s are similarly affected by 0.05 and 0.10 eV, respectively, from using the geometries of the neutral systems as reported in ref 8. The differences in nearest-neighbor distance here were 0.3 Q. For the larger clusters the effects are expected to be smaller, and the computed EDEs are thus expected to be only somewhat affected by the neglect of geometry optimization in the different clusters. Furthermore, the optimal structures of the Cu; clusters are found to be very similar to the fully optimized structures of Li; clusters reported in ref 15. We thus feel reasonably confident that the symmetry contraints imposed on the structures have not resulted in the exclusion of important nonsymmetric structures. The effects of the CPP operator compared with explicitly correlating the 3d shell are obtained by comparison (Table 11, column 1) with the results of ref 7. The largest discrepancy is obtained for the smallest system, the Cu atom. The difference in the computed EDEs or electron affinity in this case is 0.12 eV, whereas for Cuz and Cu3 very good agreement is obtained. The effects of the CPP operator is to increase the EDE by 0 . 2 4 4 eV for Cu2to Cu4, while the larger clusters show much larger effects. This latter result, however, is largely an effect of the SCF treatment used in the comparative calculations performed for the larger clusters. Correlation of the valence electrons is expected to reduce the effects of the CPP operator also for those clusters. As examples, we mention that the computed effects at the SCF level were 0.47 eV (Cu,), 0.43 eV (Cu,), 0.81 eV (Cu,), and 0.53 eV (Cu,). This should be compared with the 0.23-, 0.22-, 0.42-, and 0.43-eV CPP effects obtained at the correlated level (Table 11) for the corresponding clusters. The origin of this correlation effect is that atomic valence correlation tends to reduce the electric fields felt by the core electrons. Finally, for the neutral Cu3cluster we obtain the Dmkstructure 0.08 eV lower than the equilateral D3,, triangular structure when the CPP operator is not included. The effect of the CPP operator then is to lower the D3h structure by 0.1 1 eV relative to the linear one. Thus, at the CPP level the correct structure is obtained, such that the qualitative effect of 3d correlation is correctly reproduced by this approximate operator also in this case. In conclusion, we find very good agreement between our all-electron CCI+Q+CPP results and the results of the large calculations of ref 7. Turning now to the calculation of HOMO-LUMO separations, the ECP description gives a somewhat larger uncertainty than in the computation of the EDE’s. The differences are now of the order of f0.20 eV, which is increased to b0.30 eV when also the CPP operator is included. This will then be the uncertainty we will have to ascribe to the computed HOMO-LUMO separations of the larger clusters. As it turns out, the actual differences between the experimental results and the calculations are in fact less than this uncertainty. Apparently there is thus a fortuitous cancellation of errors.
Cluster Geometries and Electronic Spectra In the preceding section we have shown that the crude approximation to treat the copper atom as a one-electron system works reasonably well. The modeling of 3d correlation effects by a polarization operator also gives results in reasonable agreement with previous calculations on Cu, to Cu3 in which the 3d electrons were correlated.’J Compared with all-electron calculations, the errors in our ECP calculations are usually less ~
(14) Tatewaki, H.; Huzinaga, S. J . Chem. Phys. 1979, 71, 4339.
(15) Boustani, I.; Kouteckg, J. J . Chem. Phys. 1988, 88, 5657.
5474
Akeby et al.
The Journal of Physical Chemistry, Vol. 94, No. 14, 1990 EDE(eVI
TABLE Ilk Results (in eV) for Vertical EDE .#1HOMO-LUMO Scpontion (HIS) of the Ground-StateCu, Anionic Clusters. AU Results at CCI + Q Level Using the ECPfWp]
cluster CUICUT
cu,cuqcu< cu6cu,cu*cug
svmmetrv
experiment' EDE' HLS
Cb Cb
1.235 0.89 2.35-2.55 1.45 1.97 1.97 2.15 1.58
CS
2.40"
0.h
Dpb D2h
Ca CS
0.80 >0.60 0.31
>0.20 1.36d
this work EDE" HLS 0.75 0.69 1.79 1.32 1.63 1.76 2.03 1.40 2.17 1.96
0.63 1.76 0.16
" 1
1.46
0.88 CS 2.04 0.65e CUld 'Reference 2. bCalculated as vertical IP of anionic clusters. 'To the computed EDE's a 0.1-eV correction for the ECP approximation has been added (see text). The estimated uncertainty in the EDEs is f0.10 eV and that in the HLS's is 10.30 eV. dReference 3. *Reference 2; 0.14 eV.
than 0.3 eV for the EDE's and the HOMO-LUMO gaps. It should, however, be remembered that the all-electron calculations for Cu, to Cu3 have errors of 0.3-0.5 eV for spectral energies in comparison with experiments. These errors are most probably due to missing 3d correlation effects. The main errors in our results should thus, in one way or another, be related to the amount of valence electron penetration into the 3d region. It is therefore to be expected that the results should be more accurate as the clusters get larger since the valence electrons will be more and more delocalized and thus penetrate less into the 3d regions. This expectation is also borne out by the present calculations. In this section we will make direct comparisons between our calculated values and experiment for copper clusters with up to 10 atoms. The first conclusion that can be drawn from the results collected in Table 111 and Figure 1 is that the model calculations are in quite good agreement with experiment for the clusters with four atoms and more. For these clusters the deviation compared with experiment for both the EDEs and the HOMO-LUMO gaps, with one exception, is 0.2 1 eV and smaller, which must be considered a very satisfactory result. In the assignments, most of the structures selected as ground states are those that also have the lowest energy. In some cases a comparison to the experimental EDEs and HOMO-LUMO separations was also important for making the final structural assignments. Since in particular the EDE is strongly structure dependent, we feel confident that the structures reported here are most likely the ones observed. Before studying the individual assignments in detail, it is instructive to compare the properties of different size clusters and in particular the effects of the CPP operator on these (Table IV). The structures used in the table are those obtained from the calculations as corresponding to the experimentally observed clusters (see below). The binding energy computed per atom increases rapidly up to 1.3 eV for Cu,- with a break at Cus- and then smaller subsequent increases for the larger clusters. Since the neutral Cu8 cluster corresponds to an especially stable closed shell structure in the superatom picture, the decreased per atom binding energy of Cus- compared with Cu7- is not unexpected.
2
3
-Experimental
5
4
6
. t..Ali-Electron
I -1-
8
9 IO No of Atoms
ECP+ CPP + C C I
Figure 1. Computed and experimented EDEs for Cu, cluster anions (n I IO). (-) is experimental; (---) is calculated with ECP+CPP+CI(v);
(.-) at the all-electron+CPP+Cf(v) level (the EDE's for CU,and CUg in this graph are estimated by correcting the SCF + CPP results, obtained at the all-electron level, with the valence correlation energics from the ECP calculations on these clusters).
The contribution from core-valence correlation to the binding energy is large and of the order of 0.5 eV/atom for the larger clusters. Thus, in computing atomization energies of clusters or metals, this would be an important effect to take into account. In the same way, the CPP contributions to the EDEs are substantial and are contributing about 30% of the total EDE, always improving the comparison with experiment, while the HOMOLUMO separations are much less affected by these effects. For cluster binding energies and EDEs the core (3d)-valence correlation must thus be included for accurate results. Hitherto, in previous studies, when calculations have been used to decide on the preferred geometry for a cluster, the only basis for these decisions has been the lowest total energy. Since it is common that there are many different geometries which have very similar total energies, the result based on this procedure can be uncertain. The advantage of being able to make comparisons also to the experimentally known EDE's and HOMO-LUMO gaps, which are strongly structure sensitive, is well illustrated by the results for the C u c cluster (see Table V). The experimental EDE for this cluster is 1.58 eV and the HOMO-LUMO gap is 1.36 eV. The topological structure with the lowest total energy is the Djdstructure in Figure 3. However, for this structure the calculated HOMO-LUMO gap is only 0.18 eV and the EDE is as high as 2.16 eV. It is very rare that a calculated electron affinity (or EDE) will become too high compared with experiment for any molecule, and for the Cu, to Cu3 systems the calculated values were all substantiallylower than the experimental values. An error in the computed HOMO-LUMO gap of as much as 1.18 eV (0.18 eV compared to 1.36 eV) would also make this cluster unique among all the studied copper clusters. The other clusters show errors of only 0.1-0.4 eV. We can therefore safely conclude that the structure that we calculate to have the lowest total energy cannot be the ground state of Cus-. The C , structure, shown in Figure 2, has a total energy that is only 0.30 eV higher than the D3,, structure. Our calculations are not accurate enough to conclude that in reality there could not be a reversal of stability. For the C , structure the EDE is 1.30 eV, which in comparison with the experimental value 1.58 eV shows the expected deviation.
TABLE I V Computed (CCI+Q+CPP) C r d S h t e AtomiutIoll Eecrgy (EB)!Binding Energy per Atom ( E J o ) , Vertical Eketrm Detachment Energy (EDE), and the HOMO-LUMO Separation (HLS), AU in eV, for Anionic Cu, Clusters. Effects of Core-Valence Correlation (ACPP) Given for Each Property. Calculatiom Use the ECPfSs3pl Description cluster symmetry EB E dn ACPPf n EDE ACPP HLS ACPP 0.49 0.87 0.94 1.08 1.20 1.32 1.29 1.38 1.41
0.15 0.20 0.30 0.34 0.45 0.46 0.44 0.51 0.54
0.65 0.59 1.69 I .22 1.53 1.66 1.93 1.30 2.07 1.86
0.18 0.27 0.3 1 0.48 0.59 0.66 0.62 0.45 0.63 0.59
0.63 1.78 0.16
0.22 0.03
1.46
0.20
0.88
0.10
The Journal of Physical Chemistry, Vol. 94, No. 14, 1990 5475
Structure of Cu, Cluster Anions
n
TABLE V: COwith Crousd-sCrtc Adolie CURClusters for All Computed (CCI+Q+CPP) Clusters. Mffereaca in A t o m i P t h Enem A E b Vertierl Ekctroa b ~ g (EDE). y md the HOMO-LUMO Separation (HLS) for the CNeutral Clusters, All in eV. Calcuhtion8 Use tbe Ecqs13p) Dcseripth. Values in Parentheses Indicate CAS+MR-CCI
cluster
symmetry
cu; cu,cu,-
cus-
cu,-
cue cu,- c; ilA?' C,-B (
C,-C (lA')d CU~O-C, ('A')' Czh (2B,)d
AER
EDE
0.00 0.59 0.00 0.19 0.25 0.43 (0.38) 0.51 0.00 0.25 0.27 0.30 0.70 0.83 0.88 0.00 0.16 0.24 0.46 0.00 0.04 0.13 0.00 -0.30 0.55
1.69 1.16 1.22 1.39 1.62 1.35 (1.39) 1.60 1.53 1.65 1.75 1.94 2.20 1.31 2.84 1.64 (1.66) 1.90 0.96 0.96 1.93 1.31 1.55 1.30 2.16 1.19 2.07 2.16 1.96 1.86 2.05 (1.94)
0.00 0.13 0.42
0.00 0.43
HLS tu, -
1.46 0.18 1.76
0.88 0.32 (0.44)
h
LJ - c,
U tu,-
C2Y
c, c u m - c, Figure 2. Cluster structures, predicted by considering the lowest total energy, the EDE of the anion, and the HOMCbLUMO gap for each cluster (see text). cue -
c2v
O3d
CUB
- Td
U
0.16 0.70 (0.22) 1.65 1.82
Linear. bEquilateral triangle. 'Figure 2. dFigure 3. 'Square. /Tetrahedron. #Trigonal bipyramid. Pyramid. 'C, with corner base atoms bent down 1 5 O . jTwo planar equilateral triangles with a common top. k T ~ twisted o equilateral triangles with a common top. 'Octahedron. "Pentagonal pyramid. "Planar structure like CU, C, with one atom added on top. OPentagonal bipyramid. PTetrahedral with one atom on tou of three faces.
cu,
c 3v
0.61 (0.63) 0.78 0.72 0.43 (0.14) 0.57 (0.20) 1.78 0.40 0.32 1.56 1.16
tu,-
Also the calculated HOMO-LUMO gap of 1.46 eV compares favorably with the experimental value 1.36 eV. The Td structure of Cu8- shown in Figure 3 has a calculated EDE of 1.19 eV and a HOMO-LUMO gap of 1.76 eV. None of these values is unreasonably far away from the experimental results, but both of them have much larger errors than those for the C, structure. Also, the total energy for the Td structure is 0.88 eV higher than , structure. An error, due to our model, of this that for the C
cu, - cs- 8
tu,-
cs-c
"C%
v
- c 2h
Figure 3. Other investigated cluster structures. See also Table V.
magnitude, which would reverse the stabilities of these two clusters, seems unlikely. We therefore conclude that the most plausible ground-state structure of Cus- is the C , structure. The tug- cluster is another example where the result for the EDE is more conclusive than the computed total energy. In the calculations four structures were obtained within 0.5 eV with the C, structure as the lowest. The difference to the oh structure is only 0.16 eV, however, and the C, structure is only an additional 0.08 eV higher. Comparing with the experimental EDE value of 1.97 eV and the HOMO-LUMO separation of 0.31 eV, we may rule out the C, structure as a candidate for the ground state. The computed values (0.96 and 1.65 eV, respectively) for this structure differ from experiment by much more than the uncertainty in the calculations and this structure may thus safely be ruled out. The values for the higher lying D3hstate are very similar to those of the C,, structure, and the D?hastructuremay then be ruled out for the same reason. The remaining C, and Oh structures both have computed EDEs and HOMO-LUMO separations in good agreement with experiment. From the calibration calculations (Table 11) we know that, using the ECP in this case, we underestimate the EDE by 0.24 eV compared with the all-electron calculation. Furthermore, the oh (*TI,) state is Jahn-Teller unstable while the corresponding neutral structure is not and allowing the anion to distort should then make the EDE still larger. With these additional considerations we then obtain an EDE for this structure that is higher than experiment, and also this structure has to be ruled out. We thus assign the C,structure as the one observed experimentally. It should again be pointed out that, from the computed energies alone, no assignment could have been made. Having many different low-lying geometrical structures available with sometimes large variations in EDE's, it would seem possible that peaks assigned to the next ionization level, Le., the HOMO level, might actually be a hot band originating from a different geometrical structure of the anionic cluster. A careful study of Table V reveals, however, that among the clusters larger than Cus only cu6- exhibits this possibility. We thus cannot exclude that the HOMO level in this case is not in fact a detachment from the oh structure, which should be very close in energy and have an EDE differing by close to 0.3 eV from that of the C, structure. The Culo- cluster is interesting in that it is the only cluster for which there is a clear discrepancy in the results from the two experiments. In the first study, Leopold et al. obtained a HOMO-LUMO splitting of only 0.14 eV, while Pettiette et al., who were able to obtain spectral information for higher energies, concluded that the HOMO-LUMO gap should be 0.65 eV. This deviation was not commented on in the experimental work. For the EDE there is hardly any deviation between the results of the two groups of 1.99 and 2.05 eV, respectively. In our work we find
5476 The Journal of Physical Chemistry, Vol. 94, No. 14, 1990
the C, structure lowest in energy with the next structure C, 0.4 eV higher in energy. The former cluster has a HOMO-LUMO gap of 0.88 eV, which is thus in reasonable agreement with the result by Pettiette et al. However, the other structure has a HOMO-LUMO gap of only 0.32 eV, which is not too far away from the result found by Leopold et ai. Since both of these structures have EDEs in reasonable agreement with experiment at 1.86 and 2.05 eV, respectively, it is tempting to suggest that both these structures are actually seen in the experiments. However, on the basis of our calculated total energies, we suggest that the C,structure is the ground-state geometry. Cus- is the cluster for which the optimal geometry was hardest to select. Seven topologically different structures were investigated, falling within an energy range of 0.9 eV. Of these the linear D,,, and the DM structures, composed of two equilateral twisted triangles with a common top, were ruled out since their computed EDEs became much larger than the experimental value. This is also the case for the C&-Bstructure if corrections for the ECP approximation are made according to the calibration calculations. For the Da structure, which is the planar structure corresponding to the DM form, the computed EDE is 0.66 eV smaller than the experimental value 1.97 eV. In comparison to the errors obtained for the other clusters, in particular the larger ones, this error seems too large and this structure is therefore ruled out. The three remaining structures gave EDE's in at least reasonable agreement with experiments. The lowest energy was obtained for a parallel trapezoidal C, form. For this form the computed EDE is 1.53 eV and the HOMO-LUMO gap 1.78 eV, compared to the experimental values 1.97 and > O h 0 eV, respectively. The next lowest energy was obtained for a trigonal bipyramidal D3,, structure, which had an energy 0.25 eV higher than that of the C, structure. The computed EDE for the D3* structure is 1.65 eV and the HOMO-LUMO gap 0.40 eV. The final structure investigated was the pyramidal C, structure. The total energy of this form is very close to that of the D3kform, the computed EDE is 1.75 eV, which also compares well with experiment, and the HOMOLUMO gap is 0.32 eV. It is unfortunate that the HOMO-LUMO gap is not known more accurately experimentally since such a value would easily distinguish between the C, structure on the one hand and the D3*and C, structures on the other. At present, mainly for the fact that the total energy is lowest for the C , structure, this geometry appears most likely for the ground state. As a secondary argument, it could be added that the other two clusters actually have HOMO-LUMO gaps which fall below the lower limit set by the experiment, but not by a very significant amount. Since the error of 0.44 eV for the EDE obtained for the most optimal structure, the C, form, is the largest computed error for all the clusters with more than three atoms, the study of the Cu,cluster was carried a bit further. First, extending the basis set did not change the computed EDE significantly. Second, minor geometry distortions from the parallel trapezoidal form were investigated. The latter effects were inspired by the fact that Boustani and KouteckpI5 actually found the corresponding structure of Li< to be slightly distorted. In fact, bending the comer atoms of the base of the Cus- parallel trapezoid structure (similar to ref 15) down by 15' gave a much better value for the EDE of 1.94 eV. However, the total energy went up by nearly 0.3 eV, so this structure was also abandoned. This investigation, never, structure theless, gave the information that the EDE for the C is extremely sensitive to details of the geometry, and the fairly large difference between the computed and experimental EDE's is in light of this easier to understand. For Cud- and Cu9- the calculations cannot with certainty determine which of the lowest structures is observed. The structures selected as the ground states for these clusters were then simply the ones with lowest total energies, but this choice can be debated. Finally, the assignment of the ground-state geometry for Cu,- was unproblematic. A large number of neutral Cu, structures corresponding to the anionic structures were also studied. The computed energy differences between structures and electronic excitation energies used
Akeby et al. TABLE VI: Compdsom with Groaal-State CU, clr6crs for AU Colqrtcd (CCI+Q+CPP) ckstcnr. Dldicnnecs ia Atomization Energy AEb and HOMO-LUMO Separation (HIS)All in eV. CUse tbe ECqSs3el Description. V h in Parentheses AW CAS+MR-CCI Results cluster symmetry HLS 0.00 0.06 0.00 (-0.02) 0.61 0.36 1.14 0.56 (0.55) 0.99 (0.69) 0.65 1.37 0.89 (0.88) 1.46 (1.46)
0.61 (0.63) 0.78 0.43 (0.14) 0.72 0.57 (0.20)
0.00 1.78 0.37 0.76 0.48 0.80 0.60 0.70 2.26 1.37 (1.33) 2.65 2.19 0.00 1.65 0.23 2.05 0.44 (0.40) 0.60 (0.56) 0.87 1.57 (1.09) 0.00 0.32 0.58 0.00 1.93 0.69 2.14 1.13 2.89 1.25 1.43 0.00 0.22 0.31 0.00 0.88 0.63 (0.51) 0.95
1.78 0.40 0.32 1.56 1.16 1.65 1.82 0.16 (0.16) 0.70 (0.22)
1.93 1.46 1.76 0.18
0.88 0.32 (0.44)
to compute the HOMO-LUMO separations are reported in Table VI. Summary and Discussion
We have shown that a crude treatment of the copper atom as a one-electron system can give cluster results that are in quantitative agreement with experiment for both EDE's and HOMO-LUMO gaps. For Cu; to CU,Qthe discrepancy between the calculated results and experiment is less than 0.21 eV with one exception. This exception is Cu5-, for which the EDE is in error by 0.44 eV. The main part of this error does not necessarily come from the model potential description of copper, but could also be due to the assumption of perfect C+ symmetry. The calculations show that only a small distortion improves the EDE value substantially. In order to obtain quantitative agreement with experiment, the modeling of 3d correlation effects has been absolutely essential. The effect of the core polarization operator is most important for and the EDEs and ranges from 0.45 to 0.70 eV for Cu4- to CU,,,leads to improved results in all cases.
J. Phys. Chem. 1990,94, 5411-5482 It is clear that with the successful modeling of the copper atoms as one-electron systems it is not too surprising to find that the results for the copper clusters are often qualitatively similar to results for alkali-metal clusters. This fact has been noted several times in both experimentaland theoretical papers. The most direct comparisons with the present copper clusters can be made with the corresponding anionic lithium clusters, which have been studied theoretically by Boustani and Ko~teckp.'~ Apart from the similarities it can be noted that the smaller Li, clusters up to n = 5 were all found to be linear. This could be a real difference between lithium and copper clusters but could also be an effect of different treatments of core correlation. In our study we have noted that nonlinear structures were strongly stabilized in comparison to linear structures after core correlation effects were added and such effects were not included for Li; in ref 15. In this study we have shown that the present simplified treatment of transition-metal atoms can give useful and reliable information about cluster properties. In terms of reactivities copper clusters are not the most interesting metal clusters. For example, no copper cluster is reactive toward Hl, while for nickel clusters the opposite is true and all clusters are reactive. Iron and cobalt clusters are more interesting in that the reactivities vary consid-
5477
erably from cluster to cluster. In a recent paper' we have suggested that the oscillations in reactivity for these clusters are due to a tendency for iron and cobalt to mix in d"s2 atoms among the normal d*'s atoms and thereby cause some of the clusters to have a closed valence shell character. The closed-shell character will lead to a drastically reduced reactivity as compared to the corresponding nickel clusters. It seems reasonable to expect that the formation of closed valence shells for iron and cobalt clusters should be seen in some spectral properties, such as very low electron affinities for these clusters. The same type of experimental information as has been obtained for the copper clusters in terms of EDE's and HOMO-LUMO gaps would thus be even more interesting if it could be obtained for iron and cobalt clusters. It is likely that such information together with theoretical calculations could definitely establish the origin of the oscillationsin reactivity that were observed already several years ago and for which conflicting explanations have been proposed. R d ~ t r yNO. C U ~107865-35-0; , Cuj-, 108658-50-0; C U ~108658, 51-1; C U ~108658-52-2; , C U ~108658-53-3; , CUT,108658-54-4; C U ~ , 108658-55-5; Cb-, 108658-56-6; CU~O-, 108658-57-7; CUI,12190-70-4; CU,, 66771-03-7; Cud, 65357-62-2; CU,,66771-05-9; C U ~681 , 12-39-0; CUT,81 543-63-7; C U ~62863-28-9; , Cup, 81 543-64-8; CUI&127398-94-1.
The Restrlcted Active Space Self-Consistent-Field Method, Implemented with a Split Graph Unitary Group Approach Per-brke Malmqvist,* Alistair Rendell2 and Bjorn 0. Roos Department of Theoretical Chemistry, Chemical Centre, P.O.B. 124, S-221 00 Lund, Sweden (Received: November 1 , 1989; In Final Form: February 20, 1990)
An MCSCF method based on a restricted active space (RAS) type wave function has been implemented. The RAS concept is an extension of the complete active space (CAS) formalism, where the active orbitals are partitioned into three subspaces: RASI, which contains up to a given maximum number of holes; RAS2, where all possible distributions of electrons are allowed; and RAS3, which contains up to a given maximum number of electrons. A typical example of a RAS wave function is all single, double, etc. excitations with a CAS reference space. Spin-adapted configurationsare used as the basis for the MC expansion. Vector coupling coefficients are generated by using a split graph variant of the unitary group approach, with the result that very large MC expansions (up to around 1@) can be treated, still keeping the number of coefficients explicitly computed small enough to be kept in central storage. The largest MCSCF calculation that has so far been performed with the new program is for a CAS wave function containing 716418 CSF's. The MCSCF optimization is performed using an approximate form of the super-CI method. A quasi-Newton update method is used as a convergence accelerator and results in much improved convergence in most applications.
Introduction The complete active space (CAS) SCF method'+ has during the past 10 years been used in a large number of applications covering many areas of applied quantum chemistry.$ The pop ularity of the CASSCF approach is mostly due to its simple structure, where the only effort from the user is to select an adequate set of active orbitals in order to defme the wave function. In most applications this is a straightforward procedure, even if there exist cases where a deeper insight into the nature of the wave function is necessary before a proper choice can be made. The major drawback of the CASSCF approach is that the number of configuration state functions (CSFs) increases steeply with the number of active orbitals. For example, distributing 12 electrons among 12 active orbitals yields 226 5 12 singlet CSF's, if no reduction due to spatial symmetry is assumed. Adding one more active orbital increases the number to 736 164. This strong dependence on the size of the MC expansion on the number of active orbitals makes it difficult to perform CASSCF calculations Present address: NASA Am- Research Center, Computational Chemistry, Moffett Field, CA 94035.
with more than around 12 active orbitals, at least in cases where the number of active electrons is about the same as the number of active orbitals. In most applications this limit does not pose a problem, but there are situations where it would be desirable to be able to increase the number of correlating orbitals in the MCSCF wave function. Normally dynamic correlation effects are included in the calculation in a subsequent step, where a multireference (MR) CI calculation is performed using selected CSF's from the CASSCF wave function as reference configurations. The CASSCF calculation is then restricted to include only static correlation effects, and is used to determine the orbitals. (1) Roos, B. 0.;Taylor, P. R.;Siegbahn, P. E. M. Chem. Phys. 1980,48, 151. ( 2 ) Siegbahn, P. E. M.; Almlbf, J.; Heiberg, A.; Roos, B. 0. J . Chem. Phys. 1981, 74, 2384. (3) R.oos, B. 0. Inr. J. Quantum Chem. Symp. 1980, 14, 175. (4) Siegbahn,P. E. M.; Heiberg. A.; Roos, B. 0.;Levy,B. Phys. Scr. 1980, 21, 323.
(5) Roos, B. 0. In Ab Initio Methods in Quantum Chemistry, Part I&
Adv. Chem. Phys.. LXIX;Lawley, K.P., Ed.;Wiley: Chichester, U.K.,1987.
0 1990 American Chemical Society
