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Back | Show only these items: | The importance of being tetrahedral: the cadmium pyramids CdN; N = 4, 10, 20, 35 and 56

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj1

Using quasirelativistic density functional theory, we show the surprising stability of the tetrahedral cadmium pyramids Cd_4, Cd₁₀, Cd₂₀, Cd₃₅ and Cd₅₆.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con1

While gold clusters Au_N, 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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac1

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj2

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac5

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac6

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac7

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac8

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj3

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

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met1

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod2

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

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met2

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs1

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

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met3

Some calculated properties are tabulated in Table 1.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs2

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res1

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res3

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res4

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res5

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs3

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res6

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res7

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res8

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac9

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

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met4

For Cd₁₀, this led to a change to C_3v-symmetry and a stabilisation of the higher-lying, low-symmetry conformer by 7 kJ mol⁻¹.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res9

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res10

The cadmium pyramids have large planar surfaces.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res11

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

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp1

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac10

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

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot1

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac11

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res12

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac12

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res13

Especially striking is the stability of the tetrahedral Cd₂₀.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot2

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res14

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

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

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp2

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac13

Why are the small Cd pyramids so stable?

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot3

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac14

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod5

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

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met5

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res15

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res16

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac15

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac16

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res17

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.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp3

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

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp4

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac17

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}

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac18

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res18

We also briefly considered the tetrahedral Hg_N pyramids.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj4

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res19

A further point is thermal accessibility.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot4

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res20

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res21

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 = H − TS.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

Concluding, we find that the Cd_N 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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con2

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con3

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res22

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con4