Jellium Model Research Articles

Optimized numerical density functional theory calculation of rotationally symmetric jellium model systems

Optimized numerical density functional theory calculation of rotationally symmetric jellium model systems

Accelerating Global Search of Large-Sized Silver Clusters Using Cluster Graph Attention Network.

Great efforts have been devoted to understanding the stability and reactivity of silver clusters, which usually depend on geometric structures, electronic configuration, and cluster size. Despite the fact that the jellium model and Wulff construction rule have successfully rationalized the stable clusters with "magic number" behavior, some experiments imply that silver clusters with 48 valence electrons also possess puzzling enhanced stability. In this work, using a recently developed deep learning technology, i.e., cluster graph attention network (CGANet), combined with a homemade comprehensive genetic algorithm (CGA) program, we searched the global minimum (GM) structures of Agn (n = 30-60) clusters with graphics processing unit acceleration, whose efficiency is about 2 orders of magnitude higher than that of the conventional density functional theory (DFT) calculations. GM structures and some representative isomers are reported at each size, revealing the competitive structural patterns based on truncated octahedra and icosahedra as well as the icosahedra-based layer-by-layer growth mode of large-sized Ag clusters. Most importantly, the size-dependent evolution behavior of structural and electronic properties of Agn (n = 30-60) clusters can successfully explain the observed stability at Ag48. Therefore, CGANet provides a powerful tool for rapidly exploring the potential energy surface of atoms with an accuracy comparable to that of DFT.

Electronic Structure, Aromaticity, and Magnetism of Minimum-Sized Regular Dodecahedral Endohedral Metallofullerenes Encapsulating Rare Earth Atoms.

A series of minimally sized regular dodecahedron-embedded metallofullerene REC20 clusters (RE = Sc, Y, La, Ce, Pr, Nd, Pm, Sm, Eu, and Gd) as basic units of nanoassembled materials with tunable magnetism and UV sensitivity have been explored using density functional theory (DFT). The contribution of the 4f orbital of the rare earth atom at the center of the C20 cage to the frontier molecular orbital of REC20 gives the REC20 cluster additional stability. The AdNDP orbitals of the four REC20 superatoms that conform to the spherical jellium model indicate that through natural population analysis and spin density diagrams, we observe a monotonic increase in the magnetic moment from Ce to Gd. This is attributed to the increased number of unpaired electrons in the 4f orbitals of lanthanide rare earth atoms. The UV-visible spectrum of REC20 clusters shows strong absorption in the mid-UV and near-UV bands. REC20 clusters encapsulating lanthanide rare earth atoms stand out for their tunable magnetism, UV sensitivity, and stability, making them potential new self-assembly materials.

Electronic shells regulate the geometric structures, stabilities and electronic properties of AlnCum (n = 10–15, m = 1–3) clusters

The properties of AlnCum (n = 10–15, m = 1–3) clusters are explored using genetic algorithm combined with density functional theory (GA-DFT). The geometric structures of Al-Cu binary clusters are spherical when the numbers of the valence electrons are around the magic number 40 predicted by Jellium model. The stabilities decrease as the number of doping Cu atom increases. The stabilities also relate to the electronic configurations. In Al13Cu-a cluster, the 3d orbitals of the central Cu atom interact strongly with the superatomic 1D orbitals and formed five bonding orbitals and five antibonding orbitals. The bonding orbitals are dominated by 3d orbitals of Cu, and antibonding orbitals are composed mainly of the superatomic 1D orbitals. The “50 valence electrons” (including 3d10 of Cu) form closed 1S21P63d101D102S21F142P6 shells, and this electronic configuration can be taken as the combination of 40-electron rule. The 3d electrons of Cu atom don’t affect the Jellium model of the cluster.

Ring Structures of Metal Atom Doped C9, C11, and C13 Clusters from Ab Initio Calculations-The Finding of a Gd@C13 Magnetic Superatom Ring.

Pure C9 clusters have a linear chain structure. However, here, we report using ab initio calculations the transformation of a chain into a cyclic ring structure with the capping of Ca, Sr, and Ba atoms. Further calculations on neutral and charged clusters doped with Sc, Y, and La atoms show stabilization of a cation C9 isoelectronic cyclic ring capped with the metal (M) atom, but anion clusters doped with these trivalent atoms form a C10 like MC9 ring, which deforms to a necklace structure. Both the ring structures correspond to electronic shell closing with 20 delocalized valence electrons in a disk jellium model and have a large highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO-LUMO) gap. Calculations of IR and Raman spectra show no imaginary frequency, suggesting that the structures are stable. Addition of two C atoms to the ring structure of LaC9 leads to a capped ring structure of the cation LaC11 and an open ring structure of the LaC11 anion. Further addition of two C atoms leads to a La@C13 cation as well as Ca@C13 and Sr@C13 neutral wheel-shaped rings with endohedral doping of the M atom. These novel ring structures have a large HOMO-LUMO gap of more than 4 eV and are magic with electronic shell closing corresponding to 28 delocalized valence electrons in a disk jellium model. There is π aromaticity in this ring satisfying 4n+2 (n = 3) Hückel's rule with 14 valence electrons. Interestingly, when the dopant is a Gd atom, there is a formation of a magnetic superatom ring Gd@C13 + with 7 μB magnetic moments due to seven 4f up-spin states of Gd being fully occupied. Bonding in these novel ring structures is discussed.

Beyond Shell-Filling: Strong Enhancement of Electron Affinity of Metal Clusters through a Noninvasive Oriented External Electric Field.

Traditional electron counting rules, like the Jellium model, have long been successfully utilized in designing superhalogens by modifying clusters to have one electron less than a filled electronic shell. However, this shell-filling approach, which involves altering the intrinsic properties of the clusters, can be complex and challenging to control, especially in experiments. In this letter, we theoretically establish that the oriented external electric field (OEEF) can substantially enhance the electron affinity (EA) of diverse aluminum-based metal clusters with varying valence electron configurations, leading to the creation of superhalogen species without altering their shell arrangements. This OEEF approach offers a noninvasive alternative to traditional superatom design strategies, as it does not disrupt the clusters' geometrical structures and superatomic states. These findings contribute a vital piece to the puzzle of constructing superalkalis and superhalogens, extending beyond conventional shell-filling strategies and potentially expanding the range of applications for functional clusters.

Structural evolution and electronic properties of the La-doped germanium clusters

The geometric structure, relative stability, and electronic properties of LaGe n − (n = 3–14) are systematically studied by density functional theory, and the results are compared with the experimental literature data. It is found that the structures of 10A-I, 11A-I, and 14A-I are clearly two-parted and have the potential to design molecular chains. Combined with the average binding energy and second-order differential energy analysis, it is speculated that LaGe 9 − is a magic number cluster. The density of states and natural population analysis indicate that electrons consistently transfer from the La atom to Ge atoms. Doped La atom increases the interaction with Ge atoms and improve the stability of pure germanium clusters. Interestingly, LaGe 9 − has a high transfer charge number, and its high stability can be further explained by the spherical jellium model. The investigation into LaGe 9 − magic number clusters offer valuable insights for the exploration of novel materials with superior stability and performance.

Understanding Bonding Nature of α‐Keggin Polyoxometalates [XW12O40]n− (X = Al, Si, P, S): A Generalized Superatomic Perspective

α‐Keggin polyoxometalates (POMs) [XW12O40]n− (X = Al, Si, P, S) are widely used in batteries owing to their remarkable redox activity. However, the mechanism underlying the applications appears inconsistent with the widely accepted covalent bonding nature. Here, first‐principles calculations show that XW12 are core–shell structures composed of a shell and an XO4n− core, both are stabilized by covalent interactions. Interestingly, owing to the presence of a substantial number of electrons in W12O36 shell, the frontier molecular orbitals of XW12 are not only strongly delocalized but also exhibit superatomic properties with high‐angular momentum electrons that do not conform to the Jellium model. Detailed analysis indicates that energetically high lying filled molecular orbitals (MOs) have reached unusually high‐angular momentum characterized by quantum number K or higher, allowing for the accommodation of numerous electrons. This attribute confers strong electron acceptor ability and redox activity to XW12. Moreover, electrons added to XW12 still occupy the K orbitals and will not cause rearrangement of the MOs, thereby maintaining the stability of these structures. Our findings highlight the structure–activity relationship and provide a direction for tailor‐made POMs with specific properties at atomic level.

On the specific behavior of the work function and surface potential of an asymmetric metal-dielectric nanosandwich

We examine thin film on a dielectric substrate (vacuum/Al/SiO2) in the stabilized jellium model and the Kohn–Sham method. We investigate surface and size effects on the effective potential and the electron work function, and analyze the spatial distributions of electrons and potentials. It is found that a dielectric environment generally leads to a decrease in the work function. The effect of dielectric confinement for the electron work function of the asymmetric metal-dielectric nanosandwiches is reduced only by the surface area weighted average value of the dielectric constants. This conclusion follows from the application of the Gauss theorem for a conducting sphere with an inhomogeneous dielectric coating. The flow of electrons from the dielectric face to the vacuum one due to the contact potential difference manifests itself in the appearance of an additional dipole between the left and right face within the spatial distributions of ions. This leads to the fact that in a vacuum the electrostatic and effective potentials change sign twice, as a result of which a potential barrier appears above the vacuum level. We introduced the position of an electron conduction band in the dielectric as the input parameter in the self-consistency procedure for one of the sandwich approximations. As it turned out, the barrier height depends only on the used local or non-local approximation of the exchange-correlation energy. The nontrivial origin and behavior of the calculated effective potential on the vacuum side of the film, as well as the reasons for it, are discussed.

The potential energy surface and superatomic properties of the octahedral PtSn5 cluster

Zintl (group IV elements) clusters with a special number of valence electrons often exhibit extraordinary structures and chemical bonding patterns, and the stability of them is rationalised by the Wade-Mingos rule. However, this rule is still limited for some Zintl clusters. Here, we find that the potential energy surface of PtSn5 has a single deep funnel, which leads down to the octahedron of the global minimum. The AIMD simulations demonstrate that the octahedral PtSn5 still has a good thermal stability at 1000 K. The molecular orbitals and AdNDP analyses reveal that the 20 valence electrons from five Sn atoms of the PtSn5 fill superatomic shells resulting in an electronic configuration of 1S21P61D102S2, and the electronic structure of the Pt atom is d10, well following the magic number of the Jellium model. The result is further confirmed by the density of states and the localised orbital locator. After the CO molecule is adsorbed on the PtSn5, it prefers to be located at the Pt atom. Due to the charge-transfer effect, the C–O bond length and stretching frequency appear to change significantly. Our work opens up the study on the Jellium model for Zintl clusters.

A review of first-principles calculation methods for defects in semiconductors

Doping and defect control in semiconductors are essential prerequisites for their practical applications. First-principles calculations of defects based on density functional theory offer crucial guidance for doping and defect control. In this paper, the developments in the theoretical methods of first-principles semiconductor defect calculations are introduced. Firstly, we introduce the method of calculating the defect formation energy and finite-size errors to the formation energy caused by the supercell method. Then, we present corresponding image charge correction schemes, which include the widely used post-hoc corrections (such as Makov-Payne, Lany-Zunger, Freysoldt-Neugebauer-van de Walle schemes), the recently developed self-consistent potential correction which performs the image charge correction in the self-consistent loop for solving Kohn-Sham equations, and the self-consistent charge correction scheme which does not require an input of macroscopic dielectric constants. Further, we extend our discussion to charged defect calculations in low-dimensional semiconductors, elucidate the issue of charged defect formation energy divergence with the increase of vacuum thickness within the jellium model and introduce our theoretical model which solves this energy divergence issue by placing the ionized electrons or holes in the realistic host band-edge states instead of the virtual jellium state. Furthermore, we provide a brief overview of defect calculation correction methods due to the DFT band gap error, including the scissors operator, LDA+<i>U</i> and hybrid functionals. Finally, in order to describe the calculation of defect formation energy under illumination, we present our self-consistent two-Fermi-reservoir model, which can well predict the defect concentration and carrier concentration in the Mg doped GaN system under illumination. This work summarizes the recent developments regarding first-principles calculations of defects in semiconducting materials and low-dimensional semiconductors, under whether equilibrium conditions or non-equilibrium conditions, thus promoting further developments of doping and defect control within semiconductors.

Exploring the stability and aromaticity of rare earth doped tin cluster MSn16- (M = Sc, Y, La).

Rare earth elements have high chemical reactivity, and doping them into semiconductor clusters can induce novel physicochemical properties. The study of the physicochemical mechanisms of interactions between rare earth and tin atoms will enhance our understanding of rare earth functional materials from a microscopic perspective. Hence, the structure, electronic characteristics, stability, and aromaticity of endohedral cages MSn16- (M = Sc, Y, La) have been investigated using a combination of the hybrid PBE0 functional, stochastic kicking, and artificial bee colony global search technology. By comparing the simulated results with experimental photoelectron spectra, it is determined that the most stable structure of these clusters is the Frank-Kasper polyhedron. The doping of atoms has a minimal influence on density of states of the pure tin system, except for causing a widening of the energy gap. Various methods such as ab initio molecular dynamics simulations, the spherical jellium model, adaptive natural density partitioning, localized orbital locator, and electron density difference are employed to analyze the stability of these clusters. The aromaticity of the clusters is examined using iso-chemical shielding surfaces and the gauge-including magnetically induced currents. This study demonstrates that the stability and aromaticity of a tin cage can be systematically adjusted through doping.

A Density Potential Functional Approach for Mesoscopic Metal-Solution Interfaces

Mesoscopic metal-solution interfaces (MSI) are found at nanostructured electrodes in myriad electrochemical devices. Resolving the local reaction environment for electrochemical reactions occurring in mesoscopic MSIs requires a constant-potential, computationally economic method that properly treat both quantum mechanical metal electrons and almost classical electrolyte components. We have been developing a density potential functional theory (DPFT) to meet these harsh requirements [1-4]. Our DPFT method combines an orbital-free density functional theory of quantum mechanical metal electrons, a statistical field theoretic description of classical electrolyte ions and solvent, and Morse potentials describing short-range interactions between the two parts that are parameterized using first-principles based computations. In this talk, I will introduce the theoretical framework, its relationship to other similar methods including the jellium model and implicit solvation model, its parameterization and benchmark with experimental and ab initio simulation data, and recent applications to understand solvent effects on potential of zero charge, mechanoelectrochemistry of MSIs, and supported nanoparticle catalysts. The emerging opportunities, future development needs, and important caveats of the DPFT method will be discussed. Reference Huang J, Zhang Y, Li M, Groß A, Sakong S. Comparing Ab Initio Molecular Dynamics and a Semiclassical Grand Canonical Scheme for the Electric Double Layer of the Pt (111)/Water Interface. The Journal of Physical Chemistry Letters. 2023 Feb 27;14:2354-63.Huang J. Density-Potential Functional Theory of Electrochemical Double Layers: Calibration on the Ag (111)-KPF6 System and Parametric Analysis. Journal of Chemical Theory and Computation. 2023 Jan 18.Huang J, Chen S, Eikerling M. Grand-canonical model of electrochemical double layers from a hybrid density–potential functional. Journal of chemical theory and computation. 2021 Mar 31;17(4):2417-30.Huang J. Hybrid density-potential functional theory of electric double layers. Electrochimica Acta. 2021 Sep 1;389:138720. J.H. is supported by the Helmholtz Young Investigators Group Leader Program.

The s-p Nonhybrid Nature Causes Adaptive Superatomic States of Bismuth Clusters.

We report a joint experimental and theoretical study on the stability and reactivity of Bin+ (n = 5-33) clusters. The alternating odd-even effect on the reaction rates of Bin+ clusters with NO is observed, and Bi7+ finds the most inertness. First-principles calculation results reveal that the lowest energy structures of Bi6-9+ exhibit quasi-spherical geometry pertaining to the jellium shell model; however, but the Bin+ (n ≥ 10) clusters adopt assembly structures. The prominent stability of Bi7+ is associated with its highly symmetric structure and superatomic states with a magic number of 34e closed shell. For the first time, we demonstrate that the unique s-p nonhybrid feature in bismuth rationalizes the stability of Bi6-9+ clusters within the jellium model, by filling the 6s electrons into the superatomic orbitals (forming "s-band"). Interestingly, the stability of 18e "s-band" coincides with the compact structure for Bin+ at n≤9 but assembly structures for n ≥10, showing an accommodation of the s electrons to the geometric structure. The atomic p-orbitals also allow to form superatomic orbitals at higher energy levels, contributing to the preferable structures of tridentate binding units. We illustrate the s-p nonhybrid nature accommodates the structure and superatomic states of bismuth clusters.

Structures, Electronic, and Magnetic Properties of CoKn (n = 2-12) Clusters: A Particle Swarm Optimization Prediction Jointed with First-Principles Investigation.

Transition-metal-doped clusters have long been attracting great attention due to their unique geometries and interesting physical and/or chemical properties. In this paper, the geometries of the lowest- and lower-energy CoKn (n = 2-12) clusters have been screened out using particle swarm optimization and first principles relaxation. The results show that except for CoK2 the other CoKn (n = 3-12) clusters are all three-dimensional structures, and CoK7 is the transition structure from which the lowest energy structures are cobalt atom-centered cage-like structures. The stability, the electronic structures, and the magnetic properties of CoKn clusters (n = 2-12) clusters are further investigated using the first principles method. The results show that the medium-sized clusters whose geometries are cage-like structures are more stable than smaller-sized clusters. The electronic configuration of CoKn clusters could be described as 1S1P1D according to the spherical jellium model. The main components of petal-shaped D molecular orbitals are Co-d and K-s states or Co-d and Co-s states, and the main components of sphere-like S molecular orbitals or spindle-like P molecular orbitals are K-s states or Co-s states. Co atoms give the main contribution to the total magnetic moments, and K atoms can either enhance or attenuate the total magnetic moments. CoKn (n = 5-8) clusters have relatively large magnetic moments, which has a relation to the strong Co-K bond and the large amount of charge transfer. CoK4 could be a magnetic superatom with a large magnetic moment of 5 μB.

Out-of-equilibrium dynamics of repulsive ranked diffusions: The expanding crystal.

We study the nonequilibrium Langevin dynamics of N particles in one dimension with Coulomb repulsive linear interactions. This is a dynamical version of the so-called jellium model (without confinement) also known as ranked diffusion. Using a mapping to the Lieb-Liniger model of quantum bosons, we obtain an exact formula for the joint distribution of the positions of the N particles at time t, all starting from the origin. A saddle-point analysis shows that the system converges at long time to a linearly expanding crystal. Properly rescaled, this dynamical state resembles the equilibrium crystal in a time-dependent effective quadratic potential. This analogy allows us to study the fluctuations around the perfect crystal, which, to leading order, are Gaussian. There are however deviations from this Gaussian behavior, which embody long-range correlations of purely dynamical origin, characterized by the higher-order cumulants of, e.g., the gaps between the particles, which we calculate exactly. We complement these results using a recent approach by one of us in terms of a noisy Burgers equation. In the large-N limit, the mean density of the gas can be obtained at any time from the solution of a deterministic viscous Burgers equation. This approach provides a quantitative description of the dense regime at shorter times. Our predictions are in good agreement with numerical simulations for finite and large N.

Quantum hydrodynamic model for noble metal nanoplasmonics

The quantum hydrodynamic model (QHDM) has become a versatile and efficient tool for plasmonics at nanometer and even subnanometer length scales, but the theory is principally applicable only to simple metals. For the most common plasmonic materials, i.e., noble metals, QHDM has not been duly justified to describe their optical responses. In particular, when noble metal structures interface with dielectric materials, electron spillover plays an important role. Although a background optical permittivity is usually invoked in Amp\`ere's law to encompass the $d$-band electrons' effect on optical responses in a direct manner, their indirect effects, e.g., acting via conduction electrons, still remain elusive. Here we identify these indirect effects and develop a near-field potential ${U}_{\mathrm{aff}}$ refined QHDM (U-QHDM), in which the $d$-band electrons' effects on the conduction electrons are recognized and the electrostatic properties of dielectric are incorporated. We report the detailed procedures of calibrating U-QHDM. Material-independent model parameters are fitted with density functional theory (DFT) calculations for simple metals. Last, the calibrated U-QHDM is benchmarked against DFT and time-dependent DFT within stabilized jellium model for Ag, Au, and Al nanostructures interfacing with dielectrics. We expect U-QHDM will be a valuable asset for the rapidly developing field of extreme nanophotonics.

Planar σ-Aromaticity in Ga-Doped Au Clusters

Recently, the first example of Au-Ga clusters is synthesized and characterized, which can be described by the jellium model as a superatom with 8 valence electrons that come from the joint contribution of Au and Ga atoms, opening a whole new field for further research. Here, the structure features and stability of one Ga-doped Au cluster with magic number electrons (6 and 8) are analyzed in detail. Moreover, the valence electron fillings and chemical bonding of them are also further explored. It is found that Au3Ga and Au5Ga clusters present planar configurations, and they have higher stability than that of neighbor clusters. The AIMD simulations show that these two clusters still have a good thermal stability at 500 K. The molecular orbital analyses show that the Au3Ga and Au5Ga have three and one typical delocalization orbital throughout the whole planar spaces, respectively, following the planar σ-aromaticity rule. The ELF and LOL analyses are further performed, and the results are consistent with the molecular orbital analyses. The NICSzz-scan curves confirm that the Au3Ga is more aromatic than the Au5Ga, and the reason is that the former has more delocalized electrons than the latter. Our work opens up aromaticity studies in the Au-Ga clusters.

An exact formula for the variance of linear statistics in the one-dimensional jellium model

We consider the jellium model of N particles on a line confined in an external harmonic potential and with a pairwise one-dimensional Coulomb repulsion of strength α > 0. Using a Coulomb gas method, we study the statistics of where f(x), in principle, is an arbitrary smooth function. While the mean of s is easy to compute, the variance is nontrivial due to the long-range Coulomb interactions. In this paper we demonstrate that the fluctuations around this mean are Gaussian with a variance for large N. In this paper, we provide an exact compact formula for the constant . In addition, we also calculate the full large deviation function characterizing the tails of the full distribution for several different examples of f(x). Our analytical predictions are confirmed by numerical simulations.

Effect of Group-10 Element M (Ni, Pd, Pt) on Electronic Structure of Icosahedral M@Au12 Cores of MAu24L18 (L = Alkynyl, Thiolate)

Heterometal doping into chemically modified gold clusters is a straightforward method to modulate their electronic structure. Although examples exist of alloy clusters containing a series of heterometeal dopants from different groups of the periodic table, it is rare for a complete set of alloy clusters to contain dopants from the same group. In this work, we newly synthesize Ni-doped gold clusters with NiAu24L18 [L = ArFC≡C– (ArF = 3,5-(CF3)2C6H3), PhC2H4S–] by using hydrogen-containing phosphine-protected NiAu clusters as precursors. The synthesis of the Ni-doped counterpart allows us to compare the geometric and electronic structures of prototypical alloy clusters with a M@Au12 core (M = Ni, Pd, Pt). Single-crystal X-ray diffraction analysis indicates that due to the small atomic radius of Ni, NiAu24L18 has a slightly contracted core than clusters with M = Pd or Pt. Voltammetry and optical absorption spectroscopy suggest that the modified jellium model cannot solely explain the period-dependent doping effect because the Ni@Au12 core contracts. MAu24L18 clusters (L = PhC2H4S–) are paramagnetized by single electron reduction, and their magnetic properties are investigated by using electron spin resonance spectroscopy. This work thus provides a guide for designing gold clusters by heterometal doping within a given group of the periodic table.