The importance of being tetrahedral: the cadmium pyramids CdN; N = 4, 10, 20, 35 and 56

Using quasirelativistic density functional theory, we show the surprising stability of the tetrahedral cadmium pyramids Cd4, Cd10, Cd20, Cd35 and Cd56.

While gold clusters AuN, up to N = 13 appear to be planar,1 and those with N up to 19 have low-symmetry three-dimensional structures, Au20 was recently found to be tetrahedral.2

Actually, 20 is one of the “magic numbers” 8, 20, 40, 70 and 112 for electrons in a tetrahedron,3 stabilised by both electronic and atomic shell structure.

For the divalent Group-12 elements Zn and Cd, with two valence electrons per atom, anomalously strong abundance peaks are found for clusters of 6, 10, 18, 20, 28, 30, 32, 35, 40, 41, 46, 54, 57, 60 and 69 atoms.4,5

The peaks for CdN with N = 10, 20 and 35 are very pronounced.

While other than tetrahedral magic numbers occur, we here study the question whether the systems with the tetrahedral magic numbers are actually tetrahedral.

Strong alternative possibilities are amorphous or low-symmetry structures, which, using interatomic model potentials, have been found to be highly favoured by especially Au, Zn and Cd clusters.6–8

In an extensive pair-potential study of Zn and Cd clusters of up to 125 atoms, Doye8 found that only Zn4 and Cd4 possessed tetrahedral symmetry.

All the other potentially tetrahedral pyramids MN; N = 10, 20, 35, 56, 84 and 120, were assigned low or no-symmetry global minima.

As noted by Doye,8 the pair-potential used, being fitted to the bulk metal, is not expected to be optimal for small cluster sizes.

Here we present a reinvestigation of the smallest zinc and cadmium pyramids, using rigorous quantum chemical methodology.

The calculations were performed with the Turbomole program package, version 569. at the density functional theory (DFT) level, using the non-empirical PBE functional.10

We used Turbomole’s standard triple-zeta quality basis sets including an f-polarization function, TZVPP, and the resolution of the identity density-fitting approximation.11

The Stuttgart effective core potential12 modelled the scalar relativistic effects of cadmium.

To study the stability of the pyramids, we first optimised the molecular structures of the species using Td symmetry.

The structures of the optimised Cd-pyramids are shown in Fig. 1.

After optimisation, a frequency analysis was carried out to ensure a local minimum on the potential energy surface (PES).

Some calculated properties are tabulated in Table 1.

It can be seen that both zinc and cadmium clusters exhibit clear trends for most of the properties presented.

The energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), an important stability factor, is, as expected, seen to decrease with increasing cluster size.

For the two smallest pyramids, Zn and Cd are seen to have similar gaps, but after that, the spacings between the frontier orbitals of the Cd pyramids are larger than those of the corresponding Zn species.

The pyramidal forms always have a larger gap than the lower symmetry isomers, this difference being larger for Cd.

The reasonable HOMO–LUMO gap of especially the smaller cluster sizes does not of course mean that they are energetically the most favourable structures.

For comparison, we also reoptimised the structures of ref. 8 at DFT level, imposing the suggested symmetry.

The energy difference between the Td symmetric isomers and the more disordered isomers is denoted by ΔEGupta in Table 1.

The energy order of the Zn clusters is as expected and predicted before; the tetrahedra lie higher in energy for N > 4 and the energy difference increases with cluster size.

The cadmium clusters, on the other hand, show a markedly different behaviour.

For Cd, the tetrahedra are favoured over the low-symmetry alternatives for Cd10 and Cd20.

Furthermore; the Td-structures of Cd35 and Cd56 are only moderately destabilised with respect to the disordered structures of .ref. 8

To further establish the stability of the pyramidal Cd10 and Cd20, the low-symmetry isomers were allowed to relax to even lower symmetry.

For Cd10, this led to a change to C3v-symmetry and a stabilisation of the higher-lying, low-symmetry conformer by 7 kJ mol−1.

The symmetry of Cd20 was lowered to Cs and the energy difference fell by 16 kJ mol−1, from 50 kJ mol−1 to 34 kJ mol−1.

The cadmium pyramids have large planar surfaces.

As the surface atoms have fewer bonding partners than atoms in a more contracted, amorphous structure, higher energies could be expected.

On the other hand, in solid Zn and Cd, the c/a ratio is anomalously large.

The ultimate reason for this trend does not seem to be well established.

A given atom prefers to have six nearest neighbours in a hexagonal lattice, and six others, further away in the two neighbouring planes.

As seen from Fig. 1, this is exactly what the present tetrahedra strive to do, visibly buckling outwards.

Also, as noted before,8 in order to increase their coordination number, the surface atoms tend to contract their bond lengths.

The shortest bonds in the pyramids are indeed found between surface atoms, the bond length systematically decreasing with pyramid size, as seen in Table 1.

Especially striking is the stability of the tetrahedral Cd20.

Here we have to point out that no rigorous conformer search for Cd20 has been performed, so only an upper limit for the energy difference between full and lesser order has been obtained.

A priori, nothing prevents a conformer even lower in energy than the pyramid from existing.

We believe this to be unlikely, however, at least at the level of theory used in this work.

Comparing Cd20 with the Au20 cluster of Li et al2. reveals many similarities; the most stable structure of both seems to be a tetrahedral pyramid, they both fulfil a “magic number” electron-shell closing rule for their suggested potentials, their HOMO–LUMO gap is nearly identical and both have previously been assigned disordered minima.

Why are the small Cd pyramids so stable?

The stability goes against the paradigm of Soler et al.,6 stating that the tendency for amorphisation should increase downwards (and to the right) in the periodic table.

One difference between Zn and Cd is that relativistic effects are more important for cadmium.

To check this, we performed a single point all-electron, non-relativistic calculation on Cd20, using the relativistic structures.

With relativistic effects omitted, the energy difference in favour of the pyramidal form decreases from 34 kJ mol−1 to only 10 kJ mol−1.

This finding is similar to what has been noted for gold clusters.

For gold, relativistic effects stabilise planar structures over 3D ones.13,14

The bond lengths seem to be indifferent to the inclusion of relativistic effects.

The bonds of the all-electron structure of Td-Cd20 are slightly shorter, but the difference is less than 1 pm.

In any case, the underlying reason for the stability of the Cd pyramids has to be a many-body effect, not accounted for by empirical potentials.

One possibly decisive effect is the nonmetal to metal transition threshold.

Zn clusters have been found to show stronger metallicity than Cd clusters,15 which again turn metallic at smaller N than Hg clusters.

In fact, the next pyramid in the Cd series, Cd84, is at least 126 kJ mol−1 higher in energy than the no-symmetry isomer of .ref. 8

A transition from moderate to large energy difference between order and disorder is then seen between N = 10–20 and N = 56–84 for the Zn and Cd pyramids, respectively.

We also briefly considered the tetrahedral HgN pyramids.

They have much longer bond lengths than the Cd pyramids, and are best characterized as van der Waals systems.

A further point is thermal accessibility.

For N = 56, the energy difference of 28 kJ mol−1 corresponds to a mean vibrational energy 3(N − 6)kT/2 or 0.37 kJ mol−1 mode−1, or a temperature of 45 K. Thus, at higher temperatures, both forms may be accessible.

This of course also applies in the cases where the pyramidal forms are more stable.

If the low-symmetry structures have higher entropies than the tetrahedral ones, they will be favoured further at finite temperatures by their free energy G = HTS.

Concluding, we find that the CdN clusters with N = 4, 10 and 20 indeed seem to prefer tetrahedral symmetries, while those with N = 35 and 56 have low-lying tetrahedral minima, only slightly above the alternatives.

The ultimate driving force behind this remains unknown but could be related to the structure of the bulk metal, whose lattice parameters a resemble the present surface bond lengths.

Among the numerous disordered Cd clusters, it is pleasing to find these beautiful exceptions.

We duly note, however, that an exhaustive scan of the potential energy surface has not been performed; at present, this is still computationally unfeasible.