Comparative ab initio studies of small tin and lead clusters

Ann. Physik 7 (1998) 107-119 /

Annalen der Physik

0 Johann Ambrosius Barth 1998

Comparative ab initio studies of small tin and lead clusters

Bing Wang ’, L.M. Molina’, M. J. L6pez2, A. Rubio’, J. A. Alonso’, and M. J. StOtt ’ I Department of Physics, Queen’s University, Kingston, Ontario K7L 3N6, Canada

Departamento de Fisica Te6rica, Universidad de Valladolid, Valladolid 4701 1, Spain

Received 3 April 1998, accepted 21 April 1998 by ue

Abstract. We have performed ab initio total-energy pseudopotential calculations on neutral and negatively charged Sn, and Pb, (n = 3 - 10) clusters. The lowest energy structures have been deter- d e d for all clusters, and the stabilities of neutral clusters were investigated by comparing their evaporation energies and stability functions. Clusters with n = 7,lO were found to be most stable while the clusters with n = 8 and Pbs were much less stable, in agreement with features of the observed mass spectra. Calculations on Sn; and Pb; show that both atomic and electronic structures of a neutral cluster change substantially upon charging. The densities of states of Sn; clusters repro- duce the main features of the experimental photoelectron spectra. The agreement is poorer for Pb; clusters where the calculations underestimate the separation between energy levels which we think is due to the larger spin-orbit splitting in Pb, which was neglected in the calculations. We found that the differences between Sn and Pb clusters cannot be completely addressed without a more complete accounting of relativistic effects. The electron affinities of Sn, and Pb, clusters have also been calculated and the results agree fairly well with experimental values. Finally we considered Sni- and Phi- clusters and related the results to the formation of Zintl anions in liquid alkali-Sn and alkali- Pb alloys.

Keywords: Clusters; Tin; Lead

1 Introduction

In the past decade clusters of group IV elements (C, Si, Ge, Sn, Pb) have drawn much interest because of their different properties compared with the bulk phase and the change in bonding character when progressing from C to Pb. Ab initio calculations have been reported for clusters composed of up to ten atoms for C [l], Si [2, 31 and Ge [4] and numerous studies have been performed with less accurate methods [5] . However, as far as we know, calculations for Sn and Pb clusters have only been performed by Mazzone [5] using ab initio and semiempirical LCAO methods, and for Sn2, Sn3 and Pb2 by Balasubramanian [6]. Pb differs from other group IV elements by its strong metallic behavior; while Sn is of particular interest because the properties of Sn clusters are expected to be intermediate between those of Ge and Pb clusters for it is well-known that at low temperatures bulk Sn exists in the semicon- ducting diamond structure (grey tin) like Si and Ge, but transforms at 286 K into a metallic phase (white tin) with a body-centered tetragonal structure in which the atomic packing is greater and similar to that in fcc Pb. Mass abundance measure-

108 Ann. Physik 7 (1998)

ments have been performed for all group IV clusters [7-111. The mass spectra of Sn clusters do show some similarity to those of Ge clusters and some features which are present in the mass spectra of Pb clusters but not observed for Ge [lo, 111.

Photoelectron (PE) measurements, which probe the electronic structure, have also been reported for Si; and Ge; [12] as well as for Sn; and Pb; [13] clusters. The PE spectra of Sn; are similar to those of Ge; but show fewer peaks closer in energy than for Pb; suggesting that energy level spacings in Pb; clusters are greater. The energy separation between the first and second peaks in the PE spectrum of a negative cluster was originally interpreted as a measure of the gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of the corresponding neutral cluster, and such a feature as a small hump followed by a high peak with a well defined gap between them was assumed to be an indication of a closed electronic shell [12], namely, a completely-filled HOMO with a distinct HOMO-LUMO gap. However, it has been found [4, 141 that the PE spectra of negatively charged clusters cannot be simply explained in terms of neutral cluster properties and calculations of the atomic arrangement and electronic structure of negatively charged clusters are required in interpreting experimental PE spectra.

There is also interest in the relationship between the properties of small clusters and those of bulk material, not only for the elemental solids but also for mixed clusters and the corresponding alloys. The properties of solid and liquid alloys of the alkalis and group IV elements (Si, Ge, Sn and Pb) exhibit marked features at particular stoichiometries which suggest the influence of a few small, basic structural units. The structure of the equiatomic intermetallic compounds of these elements with all of the alkalis except Li have the group IV atoms in the form of separated four-atom clusters which are almost perfect regular tetrahedra. These structural units are also believed to occur in the corresponding liquid alloys [15]. Total energy calculations of clusters M4A, [16, 171 where M is Sn or Pb and A is any of the alkalis show that clusters with n = 4 are particularly stable and the lowest energy structure has the group IV atoms arranged in a tetrahedron with the alkali atoms also arranged in a surrounding tetrahedron oppositely oriented. The Sn4 and Pb4 tetrahedra are robust, persisting for n > 4, and are interpreted as Zintl anions [18], where the alkali atoms have donated electrons to form 20 electron units: Sn:- and Phi- isoelectronic with As4 and P4 molecules which in the gas phase form tetrahedral structures. However, calculations for Si [2,3] and Ge 141 suggest that four-atom clusters in the gas phase are not tetrahedra and are in fact planar structures. It is therefore of interest to investigate the structural trends of Sn, and Pb, clusters for n around 4, and for n = 4 when electrons are added to the clusters to bring the electron count closer to 20.

In this paper, we present the results of ab initio total-energy pseudopotential calculations on neutral and negatively charged Sn, and Pb, (n = 3 - 10) clusters. The lowest energy structure has been determined for each cluster. In the case of neutd clusters the stabilities were studied using calculated evaporation energies and stability functions. For the singly charged Sn; and Pb; clusters, we investigated the effect Of the extra electron and compared the calculated densities of states with expenmend PE spectra. The electron affinity has also been calculated for each of the Sn, and Pbn clusters. Through these calculations we also wish to find some clues for understand- ing the structures and properties of alkali-Sn and alkali-Pb alloys [15]. We are p d c - ulaly interested in SQ and Pb4 clusters, which may be related to Zintl ions in the liquid alloy. We therefore investigated the effect of adding extra electrons to SQ and Pb4 clusters.

R. w m g et al., Comparative ab initio studies of small tin and lead clusters 109 - 2 Computational details

f ie calculations employed the computer code developed by Scheffler and his coworkers [ 191, and treated exchange-correlation within the local density approximation (LDA), using the results of Ceperley and Alder [20] in the parametrized form given by Perdew and Zunger [XI. Inclusion of gradient corrections is expected to give mi- nor changes (see ref. [16] for a study of the NqPb cluster). The total energy func- tional was minimized with a Car-Paninello-like scheme [22]. The separable norm- conserving pseudopotentials [23] were generated for both Sn and Pb with the d-wave pseudopotential as the local component. Spin-averaged scalar relativistic pseudopo- tentids were generated and so spin-orbit splitting was not included in the cluster calculations. The effect of the spin-orbit splitting is expected to be small for structural properties. The calculations were performed in a simple cubic supercell geometry with a single k-point (T-point) for the sampling of the Brillouin zone in the calculation of the total energy and forces. The wave functions were expanded in a plane- wave basis set. The supercell lattice parameter and plane wave energy cutoff were chosen to be 30 a.u. and 10 Ry for Sn, and 34 a.u. and 8 Ry for Pb. Convergence checks were carried out for larger superlattice parameters and energy cutoffs. Differ- ences in total energy, which are more important for our purposes than the total energy itself, are already well converged to 0.001 a.u. For a given supercell of 30 a.u., we have also performed calculations with two starting configurations for a Sn3 cluster with different orientations with respect to the superlattice axis and obtained the same final equilibrium structure with the same total energy. This implies that multi- polar interactions between clusters in neighbouring cells are negligible and that our supercell is large enough to simulate the properties of isolated gas phase clusters.

As an initial test of the generated pseudopotentials we studied the Sn dimer, Pb4, bulk Sn in the diamond structure (grey tin) and bulk Pb in the fcc structure since comparison with experiment is possible in these cases. The bulk calculations were performed with 64 k-points for the Brillouin-zone integration. The results are given in Table 1. The calculated bond length of Sn;? and lattice constants of the solids are

Table 1 Test of the Sn and Pb pseudopotentials for Sn2, Pb4, bulk Sn in the diamond structure and bulk Pb in the fcc structure. The binding and cohesive energies were obtained by using as a reference the energy of a pseudoatom with the inclusion of spin-polarization. The numbers in parenthe- ses are experimental values

Snz Pb4 bulk Sn bulk Pb

Bond length (A) 2.79

Lattice constant (A) 6.62 4.86 (2.75)a

(6.49)b (4.95)b Binding energy (eV) 2.01

Cohesive energy (eV/atom) 0.99 3.44 2.10 (1.91)'

(1 .06)d (3.14)' (2.03)'

a Ref. [24] Ref. [25] Ref. [26] Ref. [27] Ref. [28]

b

d

Ann. Physik 7 (1998) 110

in good agreement with experimental values. The binding and cohesive energies have been calculated by taking as reference a spin-polarized free pseudoatom, and are also in good agreement with experiment. However, because of our good results for bond length and lattice constants, we think that total energy differences will be much less affected by spin-polarization effects and we have not included such effects in deter- mining the equilibrium structures of clusters. In addition, our results for bond length and binding energy of Sn2 are closer to the experimental values than the LCAO results of Mazzone [5].

We have also performed calculations for charged clusters, which require the addition of an artificial uniform background charge to maintain charge neutrality in a supercell calculation. This introduces additional spurious interactions within and between supercells, and corrections to the total energy need to be applied to account for these effects. We used a spherical approximation for the charge distribution of the cluster in making these corrections. Although the charging corrections affect the calculated electron affinities, tests conducted on Sn: using supercells of different sizes indicate that they do not seem to influence the lowest energy structure of a negatively charged cluster.

3 Results and discussion

3.1 Neutral Sn, and Pb, clusters

3.1.1 Lowest energy structures

To find the lowest energy structure for each cluster, several reasonable starting configurations were selected and relaxed to the equilibrium structure. Starting configurations for a (n + 1)-atom cluster were obtained from those of n or (n - 1)-atom clusters by adding one or two atoms at possible bonding sites. Although this procedure does not assure, in general, that all low-lying local minimum energy structures have been explored, we are confident that the lowest energy structure has been ascertained for each cluster.

We found the same lowest energy structure for each of Sn, and Pb, (n = 3 - 10) clusters with the same n and this is illustrated in Fig. 1. Only for n = 8 do the structures differ slightly, as we discuss below. For n = 3, the structure is an isosceles triangle, which for Sn3 has an apex angle of 83" and a shortest bond length of 2.71A, in excellent agreement with the result of relativistic configuration-interaction calculations [29]; the corresponding values for Pb3 are 80" and 2.74A. For n = 4, a planar rhombus was found. Triangular and rhombic structures were also obtained for n = 3 and n = 4 respectively, by Mazzone [5]. Starting with n = 5, the lowest energy structures for Sn, and Pb, are three-dimensional; this is the same as for Si [2, 31 and Ge [41 clusters but in contrast to carbon for which the small clusters are linear or mono- cyclic [l]. The structures for n = 5, 6 and 7 are respectively trigonal, tetragonal and pentagonal bipyramids, each compressed along the symmetry axis, indicating at least a weak bonding between the two apex atoms. Very similar lowest energy structures have also been found for Sin [2, 31 and Ge, [4] with n = 3 - 7. The structure for Sn8 and pb8 was found to be a distorted pentagonal bipyramid with the eighth atom Capping a lateral edge of one of the pyramids. In the case of Pb8 the distortion of the edge-capped bipyramid is slightly larger, giving a structure with almost D2d symme-

B. Wang et al., Comparative ab initio studies of small tin and lead clusters 111 /

A

n n

Fig. 1 Lowest energy structures for Sn, (n = 3-10) clusters. The structures for Pb, are the same, except for Pb8, in which case the distortion of the bipyramid is a little larger, giving a structure with almost D2d symmetry.

try. Our result for n = 8 is different from the lowest energy structure of both Sig [2, 31 and Geg [4] which is a distorted transcapped octahedron. We also performed calculations on this alternative structure for Sng and Pbg, and found that this has a higher energy than the edge-capped distorted pentagonal bipyramid. There is substantial disagreement over the lowest energy structure for the nine-atom clusters of Si and Ge, where different calculations give different results. The Car-Parrinello molecular- dynamics calculations performed by Ballone et al. [2] suggest a strongly distorted tricapped octahedron as the lowest energy structure for Si9, which is similar to that given by the Hartree-Fock calculations of Raghavachari and Rohlfing [3]. However, the calculations of Chelikowsky and coworkers [3, 41 based on Langevin molecular dynamics and the high-order finite-difference pseudopotential method indicate that the lowest energy structure for Si9 and Ge9 is a distorted pentagonal bipyramid with the eighth and ninth atoms capping two adjacent faces in the vertical direction. We have considered these structures for Sn9 and Pb9, and both were found to be higher in energy than another isomer, a horizontally bicapped distorted pentagonal bipyramid (having C2,, symmetry) as shown in Fig. 1. The results for Snlo and Pblo clusters are consistent with those for Silo [2, 31 and Gel0 [4]. All calculations suggest a tetracapped trigonal prism as the lowest energy structure.

The calculated bond lengths of Sn, are smaller, although surprisingly close to those of Pb,. We attribute this to the omission in our calculations of spin-orbit split- h g , which is well known to be particularly important for Pb and should be included, at least for Pb, if all the differences in the structure and properties between Sn and

112 Ann. Physik 7 (1998)

Pb clusters are to be addressed. However, relativistic effects were incorporated into our pseudopotentials in a spin-averaged form and contribute to some extent to the differences between Sn and Pb. For example, we found that the sp splitting in Pb, clusters is typically 1.3 eV larger than that of Sn,. Such a difference also occurs in Pb and Sn atoms as well as in Pb and Sn solids [30].

3.1.2 Stability

A measure of the stability of a cluster is the evaporation energy, E,,, defined as the minimum energy required to remove an atom from the cluster. For a n-atom cluster it is given by

E,,(n) = E(n - 1) + E(1) - E(n) , (1)

where E(n) is the lowest energy of a n-atom cluster and E( 1) is the total energy of a single atom. An alternative measure is the stability function, which is related to the second derivative of the total energy with respect to the number of atoms in the cluster, The stability function of a n-atom cluster, A2(n), is defined as

A,(n) = E(n - 1) + E(n + 1) - 2E(n). (2)

Large values of evaporation energy and stability function of a cluster imply that it is very stable and can be expected to have a higher abundance in the mass spectrum than neighboring clusters containing one more or one less atom. The experimental procedure to measure the abundance population in a cluster beam is to ionize the neutral clusters by electron impact or photoionization before mass selection and counting, but ionization is believed to induce extensive fragmentation, which makes the correspondence between a mass spectral peak and a “magic” neutral-cluster size somehow uncertain. Even so, under carefully chosen photoionization conditions, ionization-induced fragmentation and other unwanted effects can be avoided or limited, and it is possible to obtain a mass spectrum which provides a reasonably accurate measure of the neutral-cluster abundance [ 111. Comparison of the calculated results with such a mass spectrum is still qualitative since no dynamic study of the cluster formation and fragmentation has been performed in this work.

The calculated evaporation energies and stability functions for Sn, and Pb, clusters are plotted as a function of the cluster size in Fig. 2. First we note the similarity between the results for Sn and Pb, in particular both eight-atom clusters are at minima of E,, and A2 suggesting that they are less stable than neighboring clusters, while the seven-atom clusters have large values of both indicators implying that they are particularly stable. This is consistent with the experimental mass spectra for Sn, and Pbn (Fig. 7 in Ref. [ll]), where a high abundance peak was found at n = 7 and a large drop at n = 8. This feature can be explained by the drastic reduction of the HOMO- LUMO gap between n = 7 and n = 8, which is a consequence of the loss of symmetry. The calculated E, and 8 2 of Pb, also show a large drop between n = 4 and n = 5. This is again consistent with the mass spectrum of Pb, [Ill, where Pbs was found to be a local abundance minimum. The mass spectra of Sn, and Pb, exhibit a high peak at n = 10 as well. We have not attempted any calculations of Sn, clusters for n > 10, but we have studied Pb, clusters up to n = 14 [16]. The calculated EV and A2 for Pblo and Pbll are included in Fig. 2. It can be seen that a large drop in

B. wmg et al., Comparative ab initio studies of small tin and lead clusters 113 I

4.5

0.5

Fig. 2 Calculated evaporation energy, E,, and stability function,

atoms in the cluster.

-1.5

Number of atoms in the cluster

E,, and A2 also occurs between n = 10 and n = 11 while Pblo has fairly large values of both quantities, suggesting that Pblo is more stable than the neighboring clusters. Considering the similarity between Sn and Pb, we expect that the calculated results would be the same for Snlo and Snll.

It is straightforward to obtain from the calculated total energies the lowest energy required to fragment a n-atom cluster into a (n - m)-atom and m-atom cluster. We found that the Sn, clusters with n 5 8 prefer monomer evaporation, which is the same as the calculated result for Sin [31] and Ge, [4] clusters. Sn9 tends to dissociate into Sn7 and Sn2 (which again shows the high stability of Sn7), while Snlo favors the fragmentation into Sn6 and Sn4. These fragmentation channels are the same as for Si9 and Silo [31] but different from the results for Ge [4]. There has been some experimental work on the fragmentation branching ratios of Si [8] and Ge [9] clusters, but to our knowledge, no such experimental results have been reported for Sn clusters.

3.2 Negatively charged Sn, and Pb, clusters

3.2.1 Lowest energy structures The lowest energy structure of a neutral cluster and several low-lying isomers were used as starting configurations for the corresponding negative cluster. The calculations show that the extra electron induces significant changes in the atomic structure

114 Ann. Physik 7 (1998)

or in the bond lengths for all clusters except for the four-atom cluster, where there are only slight modifications of the bond lengths and bond angles. We found, in general, that the lowest energy structure of a cluster becomes more spherical upon charging. This structural change has also been observed in calculations for Si; clusters

For the clusters with n = 3 , 9 and 10, the lowest energy structure is completely changed by the extra electron. The lowest energy structure of Sn, and Pb, is an equilaterial triangle (Fig. 3) while the lowest energy structure of the corresponding neutral clusters, an isosceles triangle, is no longer even a local minimum for the negative clusters. Similarly, the lowest energy structure for Snp and Pb, is a tricapped trigonal prism (Fig. 3), which is also not a local minimum for the corresponding neutral clusters. However, in the case of the ten-atom clusters a local minimum structure of the neutral cluster, a bicapped tetragonal antiprism, was found to be the lowest energy structure for Sn, and Pb,, and the lowest energy structure for the neutral clusters (see Fig. 1) becomes a local minimum structure for Snlo and Pb,. The change in the lowest energy structure from low symmetry to high symmetry upon charging can be related to features of the energy levels in the different structures. In the neutral case, the high-symmetry structure has a partially filled HOMO while the low-symmetry structure has a fully occupied HOMO and a large HOMO-LUMO gap. Thus when an extra electron is added, the high-symmetry structure will be energetically more favorable. Pb; has the same lowest energy structure (with almost D2d symmetry) as the corresponding neutral species. Charging Sn8 distorts a little more the edge-capped pentagonal bipyramid characteristic of the neutral cluster, yielding the same structure with almost D2d symmetry as for Pb8 and Pb; (Fig. 3). For the clusters with n=5, 6 and 7, the extra electron does not change the symmetry of the lowest energy structure, but the bond lengths are substantially modified with the relative change typically ranging from 2% to 17%. A common feature is the elongation along the symmetry axis and inward contraction on the middle plane, which makes the lowest energy structure more spherical. This may be interpreted as a mutual repulsion between the two apex atoms, arising from the fact that the LUMO orbitals of the neutral cluster are predominantly located in the region of the apex atoms, as also dis- cussed by Chelikowsky [4] for Ge.

An interesting feature of the ten-atom cluster reveals a difference in behaviour between Sn and Pb. In both neutral clusters the tetracapped trigonal prism is lower in

~ 4 1 .

B. wmg et al., Comparative ab initio studies of small tin and lead clusters 115 //

than the bicapped tetragonal antiprism but only by 0.18 eV for Pblo compared \Nib 0.71 eV for Snlo. For the charged clusters the tetracapped trigonal prism is the higher in energy of the two structures, by 0.53 eV for Pb,S but only by 0.19 eV for Snlo. This means that the bicapped tetragonal antiprism, which is the more symmetri- cal and close-packed one of the two structures, is more favored by Pb. Thus in Pblo and Pb,, we see a hint of the greater tendency towards close-packing of Pb which becomes significant in fcc bulk Pb.

Compared with Ge; clusters [4], the major differences occur in the clusters with n = 3 and 8. For Ge;, the lowest energy structure remains an isosceles triangle. For Ge,, a vertically bicapped octahedron was obtained as the lowest energy structure. In our calculations for Sn; and Pb;, this structure was found to be higher in energy than the edge-capped pentagonal bipyramid. In summary, some structural modifications are generally observed in the clusters of group IV elements upon charging.

3.2.2 Electronic structure m e extra electron also induces significant changes in the electronic structure, the main effects being the raising of the energy levels by about 1-2 eV and the reduction of the HOMO-LUMO gaps from those in the neutral clusters by about 0.5-1.5 eV. An exception is the four-atom cluster, where the HOMO-LUMO gap of the neutral cluster is widened slightly upon charging. To compare our results with the experimental photoelectron spectra [ 131, we have broadened the discrete energy eigenvalue spectrum into a continuous density of states (DOS) using a Gaussian lineshape of width 0.05 eV. The calculated DOS for each of the Sn; and Pb; clusters are shown in Fig. 4. Our DOS reproduces the main features in the photoelectron spectra of Sn; clusters. For SnT and Sn;, where our calculations indicate that the symmetry of the lowest energy structure is not changed upon charging, both the photoelectron spectrum and calculated DOS exhibit a small hump followed by a larger peak with a distinct gap between them, which is taken to be an indication that the corresponding neutral cluster has a closed-shell structure, the small hump arising from the extra electron occupying the LUMO of the neutral cluster. According to our calculations Sns and Sn7 are also closed-shell clusters, but unfortunately this cannot be confirmed by the experiment because the measured photoelectron spectra provide no informa- tion about energy levels deeper than -3.0 eV. For clusters with n = 3, 9 and 10, the photoelectron spectra or calculated DOS alone cannot tell us whether the corresponding neutral clusters have a closed shell or not since their lowest energy structures are changed upon charging. The situation of Sng is subtle. The calculated DOS seems to indicate that Sng is closed shell, but the photoelectron peaks are broad and do not show a gap. In general, the calculated DOS peaks of Sn; clusters are well resolved in comparison with the experimental spectra. We attribute this to the isomerization of Sn; clusters and thermal broadening [14]. While we calculate the ground-state properties at the temperature T=O, the temperature of the clusters produced in the experiment may be high and several isomers may coexist. For Sng, we have plotted the DOS of both the lowest energy structure and the isomer with an energy just 0.22 eV higher than the lowest energy. The combined DOS is in better agreement with the photoelectron spectrum.

Our calculated DOS of Pb; clusters is rather similar to that of Sn; clusters. Differ- ences can only be seen in the clusters with n=3, 9 and 10. The DOS peak in Snp and the second peak in Sn?, which involve nearly-degenerate energy levels, split into

Ann. Physik 7 (1998) 116

I

' -1.0 -2.0 -3.0 0.0 -1.0 -2.0 -3.0 Enew (eV) Energy (ev) Binding energy (eV)

Fig. 4 The calculated density of states (DOS) (left-hand side panel) for Sn; and Pb; (n=3-10) clusters in comparison with the experimental photoelectron spectra obtained at a photon energy of 3.68 eV (right-hand side panel) [13]. For Sng, the Calculated DOS of the lowest energy structure is represented by a solid curve while the dotted curve is the DOS of the second lowest energy isomer.

two peaks in Pb; and Pby respectively. Also, the degeneracy of the HOMO in Snh is lifted in Pb,, resulting in two separate DOS peaks. However, the calculated DOS does not show all the differences in the photoelectron spectra between Sn; and Pb, clusters. The energy separation between the 1st and 2nd peaks in the calculated DOS of Pb; clusters is generally smaller than that in the photoelectron spectra. These could be the consequences of our neglect of spin-orbit splitting in the calculations.

The adiabatic electron afinities (EA) of the neutral tin and lead clusters were esti- mated using total energy differences, that is,

EA(n) = E ( n ) - E-(n) , (3)

B. wmg et al., Comparative ab initio studies of small tin and lead clusters 117

%ble 2 The calculated electron affinities for Sn, and Pb, (n=2-10) clusters. The numbers in pa- ="theses are experimental values (from Ref. [ 131) (unit: eV)

- - n 2 3 4 5 6 7 8 9 10 - Sn, 2.2 2.3 2.0 2.3 2.3 2.3 2.4 2.6 2.5

(1.7) (1.8) (1.7) (2.2) (1.8) (1.8) (2.0) (1.8) (1.9) 2.08 1.80 2.34 2.01 2.14 2.26 2.24 2.41 (1.70) (1.55) (2.35) (2.05) (1.95) (2.15) (2.45) (2.55)

Pbn

where E-(n) is the total energy of a singly charged n-atom cluster. The charging corrections were applied to the total energies of the charged clusters and ranged from 4.4 eV to -1.0 eV. The calculated affinities, given in Table 2, compare fairly well with the experimental values. For Snz, our result is also in good agreement with an accurately measured value, 1.96 eV, given by Ho et al. [32]. However, we notice that the calculated results for Sn clusters are consistently larger than the experimental values [13]. The experimental values were obtained from the threshold energy of the photoelectron spectra of Sn; and Pb;, and are generally taken to be estimates of the vertical EA as it is assumed that the geometry of a cluster is unchanged upon photodetachment. For Sn, they are only rough estimates of vertical EA's due to the flat- ness of the thresholds, particularly for Snp and Snro. On the other hand, calculated vertical photodetachment energies would be larger than the adiabatic ones tabulated, thereby increasing the discrepancy with the measured threshold energies. Also, per- haps a more accurate form of the charging correction [33] needs to be incorporated into the total energy expression in order to obtain better results of EA.

3.2.3 Snj- and Pbz- clusters

Sw and Pb4 are of particular interest because of their possible implications for the structure of solid and liquid alkali-Sn and alkali-Pb alloys [15]. Both Sn4 and Pb4 have a planar rhombus as the lowest energy structure. We found that the tetrahedral isomer, which is seen in the equiatomic solid alloys with the alkalis except for Li and believed also to be present in the liquid alloys, is much higher in energy for the gas phase clusters (1.2 eV for Sn4 and 1.05 eV for Pb4) and is not even a local minimum. When an extra electron is added, the planar rhombus remains the lowest energy structure of Sn, and Pb, clusters, but the energy difference between the tetrahedron and planar rhombus is reduced to 0.6 eV for S n i and 0.5 eV for Pb;. When a second electron is added we found that the lowest energy structure of Sni- and Phi- becomes a distorted tetrahedron which can be obtained by folding the planar rhombus along its shorter diagonal. This change in shape of the lowest energy structure as electrons are added to Sn4 and Pb4 is similar to the change in the arrangement of the four group IV atoms as alkali atoms are added to Sn4 and Pb4 [16, 171. For example, the lowest energy structure of NanSn4 with n = 1 is obtained by adding the Na atom above the slightly distorted rhombus of Sn atoms, but another isomer which can be viewed as a distorted Sn4 tetrahedron with one face capped by the Na atom is almost indistinguishable in energy. The lowest energy structure for n = 2 is clearly a tetrahedron of Sn atoms, although somewhat distorted, with the Na atoms capping two of the faces. The compact SQ tetrahedron persists as more Na atoms are added with the Na atoms first completing the capping of the Sn4 faces and, when this is accom-

118 Ann. Phvsik 7 (1998)

plished, capping the edges. The Sn4 and Pb4 tetrahedra seem to be very robust structures when there are additional electrons available to occupy the bonding levels of the tetrahedron. Four extra electrons will fill these levels but evidently addition of two electrons is enough for a tetrahedral structure for the group IV atoms to be energetically favored. Whereas in the case of the mixed alkali-group IV clusters the alkali atoms provide a cheap source of extra electrons, in the calculations reported here extra electrons are actually added to the cluster which becomes charged. We were un- able to find stable structures when more than two extra electrons are added to the clusters, presumably because of the coulomb energy cost of the charging.

4 Conclusions

In summary, we have performed ab initio total-energy pseudopotential calculations for neutral and negatively charged Sn, and Pb, clusters. The lowest energy structure has been determined for each cluster. A striking result is the similarity of the lowest energy structures of the Sn and Pb clusters, not only in morphology but also in the bond lengths. Since relativistic effects were included albeit only at the spin-averaged level we suggest that the experimentally observed differences between Sn and Pb clusters are mainly due to spin-orbit splitting which we have omitted and is much larger in Pb than in Sn. For neutral Sn, and Pb, clusters, we found that the clusters with n=7, 10 are particularly stable while those with n=8 and Pbs are relatively less stable, in agreement with the experimental mass spectra. For Sn; and Pb; clusters, we investigated the effect of the extra electron and found that it induces significant changes in both the atomic and electronic structures. The calculated DOS reproduces the main features of the experimental photoelectron spectra of Sn; clusters but does not display all the differences in the photoelectron spectra between Sn; and Pb; clusters possibly due to the omission of spin-orbit splitting in our calculations. We also calculated the electron affinities of Sn, and Pb, clusters and the results compare fairly well with experimental values. Calculations of Snip and Phi- clusters indicate that two more electrons change the lowest energy structure of SQ and Pb4 from a planar rhombus to a distorted tetrahedron, a result which must be related to the formation of Zintl ions in alkali-Sn and alkali-Pb alloys.

The work was supported by the National Sciences and Engineering Research Council of Canada, DGES of Spain (Grant PB95-0720-C02-01 and postdoctoral and graduate fellowships to M.J.L. and L.M.M. respectively) and NATO (Grant CGR.961128). B.W. and M.J.S. acknowledge the hospitality of the University of Valladolid, and M.J.S. is grateful to Fundaci6n BBV (Programa CAtedra) for support.

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Comparative ab initio studies of small tin and lead clusters

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