On the role of classical orbits in mesoscopic electronic systems

We discuss the semiclassical periodic orbit theory (POT) and some of its recent extensions which are applicable also to systems exhibiting a transition from integrable to chaotic behaviour. We apply the POT to give a semiclassical interpretation of shell effects in free metal clusters and semiconductor quantum dots in terms of classical periodic orbits. In particular, we study the ground-state deformations of large metal clusters and discuss the recently observed conductance oscillations in a circular quantum dot under the influence of an external magnetic field. We also predict the shell structure of a triangular quantum dot, using as models a triangular billiard and the Hénon-Heiles potential including weak magnetic fields.     

Semiclassical variational calculation of liquid-drop model coefficients for metal clusters

We report on semiclassical density variational calculations for spherical alkali metal clusters in the jellium model. We derive liquid-drop model expansions for total energy, ionisation potential and electron affinity and test the coefficients numerically for clusters with up toN=10⁵ atoms. From the limitN→∞, we obtain excellent agreement with surface tensions and work functions evaluated for an infinite plane metal surface.     

Thermal properties of the valence electrons in alkali metal clusters

The finite-temperature density functional approach is applied for the first time to calculate thermal properties of the valence electron system in metal clusters using the spherical jellium model. Both the canonical and the grand canonical formalism are applied and their differences are discussed. We study the temperature dependence of the total free energyF(N) (including a contribution from the ionic jellium background) for spherical neutral clusters containingN atoms. We investigate, in particular, its first and second differences, Δ₁ F =F (N ? 1) ?F (N) and Δ₂ F =F(N + 1) +F(N ? 1) ? 2F(N), and discuss their possible relevance for the understanding of the mass abundance spectra observed in cluster production experiments. We show that the typical enhancement of magic spherical-shell clusters withN=8, 20, 34, 40, 58, 92, 138, 186, 254, 338, 398, 440, 508, 612..., most of which are well established experimentally, is decreasing rather fast with increasing temperatureT and cluster sizeN. We also present electronic entropies and specific heats of spherical neutral clusters. The Koopmans theorem and related approximations for calculating Δ₁ F and Δ₂ F atT > 0 are discussed.     

Mie plasmon in polyhedral metal clusters

We study the dependence of the classical plasmon frequency on the symmetry of the metal cluster and show that all clusters with at least two three-fold axes have the same plasmon frequency as the spherical cluster, ω _p /√3. In these cases the effect of the geometry will only appear in the spill-out correction and in other quantum mechanical corrections.     

The bond nature of alkaline-earth homonuclear metal clusters investigated with pseudopotential CI method

The electronic stucture of Be_l, Mg_m and Ca_n ( 1? 7, m and n ? 5) metal clusters has been investigated by means of an ab initio pseudopotential method followed by multireference double-excitation configuration interaction (PP MRD CI). Two completely different situations arise on going from Be to heavier alkaline metals. The sp hybridization, which is effective in the Be case, completely disappears when aggregates of Mg and Ca are considered. The potential energy curves relative to Be, Mg and Ca clusters exhibit a very shallow minimum at large distances while a second deeper minimum is characteristic of some Be clusters. This inner minimum is generated by the large interaction of atomic p orbitals at an internuclear separation close to the BeBe distance present in the Be crystal. The bond in Be clusters largely depends on the geometrical arrangement which directly influences the possible sp hybridisation. Very small Be planar structures are always less favored with respect to clusters containing the tetrahedral unit as a part of the whole structure. However, this situation is changed when larger planar clusters are considered, and the highest binding energy per atom (12.0 kcal/mole) from all clusters considered in this work was found for the triplet ground state of the planar Be₇(D_6h) cluster.     

Thermal electronic properties of alkali clusters

We apply the finite-temperature Kohn-Sham method to alkali metal clusters, using the spherical jellium model and treating the valence electrons as a canonical system in the heat bath of the ions. We study the shell effects in the total free energyF(N) and the entropyS(N) for neutral clusters containingN atoms. Their strongest temperature dependence is due to the finite ground-state valueS ₀>0 of the electronic entropy for non-magic clusters. It leads to a decreasing amplitude and an increasing smear-out of the saw-tooth structure in the first difference Δ₁ F(N)=F(N?1)?F(N) with increasing temperatureT and cluster sizeN.     

Pairing effects in metal clusters

Motivated by the experimental finding of oddeven alternations in the mass spectra and ionization potentials of metal clusters, we have investigated the possibility of interpreting these results as evidence for pairing correlations among the electrons. For the present exploratory calculations, we have used a spherical BCS model which we have applied, as an example, to some Al clusters. Additional model calculations have been carried out for Na _N with 34<N<100 in order to illustrate a possible systematics.     

Hierarchical chlorine-doped rutile TiO2 spherical clusters of nanorods: Large-scale synthesis and high photocatalytic activity

In this study, we report the synthesis of hierarchical chlorine-doped rutile TiO₂ spherical clusters of nanorods photocatalyst on a large scale via a soft interface approach. This catalyst showed much higher photocatalytic activity than the famous commercial titania (Degussa P25) under visible light (λ>420 nm). The resulting sample was characterized by X-ray diffraction (XRD), scanning electron microscopy (SEM), transmission electron microscopy (TEM), high-resolution TEM (HRTEM), nitrogen adsorption, X-ray photoelectron spectroscopy (XPS), UV-vis diffuse reflectance spectroscopy, ¹H solid magic-angle spinning nuclear magnetic resonance (MAS-NMR) and photoluminescence spectroscopy. On the basis of characterization results, we found that the doping of chlorine resulted in red shift of absorption and higher surface acidity as well as crystal defects in the photocatalyst, which were the reasons for high photocatalytic activity of chlorine-doped TiO₂ under visible light (λ>420 nm). These hierarchical chlorine-doped rutile TiO₂ spherical clusters of nanorods are very attractive in the fields of environmental pollutants removal and solar cell because of their easy separation and high activity.     

Triaxial shapes of sodium clusters

A modified Nilsson-Clemenger model is combined with Strutinsky's shell correction method. For spherical clusters, the model potential is fitted to the single-particle spectra obtained from selfconsistent Kohn-Sham calculations. The deformation energy surfaces of sodium clusters with sizes of up toN=270 atoms are calculated for a combination of triaxial, quadrupole and hexadecapole deformations. The ground state shapes and energies are determined by simultaneous minimization with respect to the three shape parameters. A significant fraction of the clusters is predicted to be triaxial. The deviations from the axial shape do not generate any systematic odd-even staggering of the binding energies.     

Structure and bonding in [M6X8] units of nonmetallic transition metal cluster compounds

A simple quantum mechanical method, based on the Wolfsberg-Helmholtz approximation, has been applied to some nonmetallic4d and5d transition metal cluster compounds containing M₆X₈ units. Our aim was to calculate bonding energies ofM₆ clusters as a function of electronic configuration and symmetry. Energy sequences of clusters containing elements adjacent to each other in the periodic table are discussed, in particular, niobium and molybdenum as well as tungsten and rhenium. Compared with experimentally well-characterized compounds, the computional results show good qualitative agreement. So the occurrence ofM₆ clusters withO_h or lower symmetry can be explained by electronic effects.     

Atomic structure and collective excitations of medium size NanKm clusters

Density functional theory is used to study the atomic and electronic structure of Na_nK_m clusters with up to seventy atoms. The simplifying approximation has been made of replacing the external potential of the ionic background by its spherical average about the cluster centre in the iterative process of solving the Kohn-Sham equations for each geometry tested. The search for the equilibrium geometry is performed by employing steepest descent and simulated annealing techniques. We have found segregation of K to the surface and when the cluster is large enough, a neat stratification of K and Na shells. Those effects (segregation and stratification) do not perturb the electronic magic numbers well known for pure alkali metal clusters. Our results for the atomic structure are rather similar to those reported earlier for Na_nCs_n clusters. We have also studied in a selected case, Na₂₀Cs₂₀, the dependence of the collective electronic excitation spectrum on the segregation and other geometric characteristics of the cluster.     

Front Cover: High-level ab initio evidence of bipyramidal Cu5 clusters as fluxional Jahn-Teller molecules (ChemPhysChem 19/2023)

Novel highly selective synthesis techniques have enable the production of atomically precise monodisperse metal clusters (AMCs) of subnanometer size. These AMCs exhibit ‘molecule-like’ structures that have distinct physical and chemical properties, significantly different from those of nanoparticles and bulk material. In this work, we study copper pentamer Cu₅ clusters as model AMCs by applying both density functional theory (DFT) and high-level (wave-function-based) ab initio methods, including those which are capable of accounting for the multi-state multi-reference character of the wavefunction at the conical intersection (CI) between different electronic states and augmenting the electronic basis set till achieving well-converged energy values and structures. After assessing the accuracy of a high-level multi-multireference ab initio protocol for the well-known Cu₃ case, we apply it to demonstrate that bypiramidal Cu₅ clusters are distorted Jahn-Teller (JT) molecules. The method is further used to evaluate the accuracy of single-reference approaches, finding that the coupled cluster singles and doubles and perturbative triples CCSD(T) method delivers the results closer to our ab initio predictions and that dispersion-corrected DFT can outperform the CCSD method. Finally, we discuss how JT effects and, more generally, conical intersections, are intimately connected to the fluxionality of AMCs, giving them a ‘floppy’ character that ultimately facilitates their interaction with environmental molecules and thus enhances their functioning as catalysts.     

A Cationic Metal Glue Strategy for Expanding Paramagnetic Hetero-Multinuclear Metal-Oxo Clusters within Polyoxometalate Ligands

Precise structural design of large hetero-multinuclear metal-oxo clusters is crucial for controlling their large spin ground states and multielectron redox properties for application as a single-molecule magnet (SMM), molecular magnetic refrigeration, and efficient redox catalyst. However, it is difficult to synthesize large hetero-multinuclear metal oxo clusters as designed because the final structures are unpredictable when employing conventional one-step condensation reaction of metal cations and ligands. Herein, we report a “cationic metal glue strategy” for increasing the size and nuclearity of hetero-multinuclear metal-oxo clusters by using lacunary-type anionic molecular metal oxides (polyoxometalates, POMs) as rigid multidentate ligands. The employed method enabled the synthesis of {(FeMn₄)Mn₂Ln₂(FeMn₄)} oxo clusters (Ln=Gd, Tb, Dy, and Lu), which are the largest among previously reported paramagnetic hetero-multinuclear metal-oxo clusters in POMs and showed unique SMM properties. These clusters were synthesized by conjugating {FeMn₄} oxo clusters with Mn and Ln cations as glues in a predictable way, indicating that the “cationic metal glue strategy” would be a powerful tool to construct desired large hetero-multinuclear metal clusters precisely and effectively.     

Fission of ionized alkali metal clusters

We study the symmetric fragmentation of ionized alkali clusters within a liquid-drop type model. The interplay of surface and Coulomb interactions leads to a stability condition against small deformations which depends on the ratioZ ²/N. For systems which are stable against small-amplitude oscillations we consider the possibility of large-amplitude modes eventually leading to fission and give in terms of the same quantity an estimate of the potential barrier for this fission channel.     

The response of metal clusters toq- andL-dependent external fields

We have calculated the static polarizability and mean excitation energy of metal clusters submitted toq-andL-dependent external fields ofj _L (qr)Y _L0(Ω) type. Use has been made of an Extended Random-Phase Approximation which includes exchange and correlation effects within a local model, and of the spherical jellium model to describe the neutralizing positive background.     

Investigation of the phonon band gap effect on Raman-active optical phonons in SrMoO4 crystal

The phonon dispersions of SrMoO₄ crystal are calculated using the lattice dynamical calculations approach. Spontaneous Raman spectra in the SrMoO₄ were measured in the temperature range from 10 K to 295 K, and the temperature dependence of the linewidth of the B_g (95 cm⁻¹) and A_g (888 cm⁻¹) Raman modes was analyzed using the lattice dynamical perturbative approach. We found that different behaviors of these two modes in the case of temperature broadening could be attributed to the large energy band gap in the phonon spectrum resulting in different anharmonic interactions. The calculated temperature dependence of the linewidth of A_g (888 cm⁻¹) mode was well accounted for the experimental one by including both down-conversion by the cubic term and the dephasing by quartic term. The dephasing processes are increased only at high temperatures and the effect of dephasing is related to the size of a large phonon band gap.     

Structural Evolution and Superatoms in Molybdenum Atom Stabilized Boron Clusters: MoB<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> (<Emphasis Type="Italic">n </Emphasis>= 10–24)

Doping transition metal atom is known as an effective approach to stabilize an atomic cluster and modify its structure and electronic properties. We herein report the effect of molybdenum doping on the structural evolution of medium-sized boron clusters. The lowest-energy structures of MoB_n (n?=?10, 12, 14, 16, 18, 20, 22, 24) clusters are globally searched using genetic algorithm combined with density functional theory calculations. We found that Mo doping has significantly affected the grow behaviors of B_n clusters, leading to a structural evolution from bowl-like to tubular and finally endohedral cage. The size-dependent binding energy, HOMO–LUMO gap, vertical ionization potential and vertical electron affinity show that MoB₁₂, MoB₂₂ and MoB₂₄ clusters have relatively higher stability and enhanced chemical inertness. More interestingly, the endohedral MoB₂₂ cage is identified as an elegant superatom, which satisfies 18-electron closed shell configuration well.