Orbital-free molecular dynamics simulations of melting in Na8 and Na20: Melting in steps

Orbital-Free Density Functional Theory: An Attractive Electronic Structure Method for Large-Scale First-Principles Simulations

Kohn-Sham Density Functional Theory (KSDFT) is the most widely used electronic structure method in chemistry, physics, and materials science, with thousands of calculations cited annually. This ubiquity is rooted in the favorable accuracy vs cost balance of KSDFT. Nonetheless, the ambitions and expectations of researchers for use of KSDFT in predictive simulations of large, complicated molecular systems are confronted with an intrinsic computational cost-scaling challenge. Particularly evident in the context of first-principles molecular dynamics, the challenge is the high cost-scaling associated with the computation of the Kohn-Sham orbitals. Orbital-free DFT (OFDFT), as the name suggests, circumvents entirely the explicit use of those orbitals. Without them, the structural and algorithmic complexity of KSDFT simplifies dramatically and near-linear scaling with system size irrespective of system state is achievable. Thus, much larger system sizes and longer simulation time scales (compared to conventional KSDFT) become accessible; hence, new chemical phenomena and new materials can be explored. In this review, we introduce the historical contexts of OFDFT, its theoretical basis, and the challenge of realizing its promise via approximate kinetic energy density functionals (KEDFs). We review recent progress on that challenge for an array of KEDFs, such as one-point, two-point, and machine-learnt, as well as some less explored forms. We emphasize use of exact constraints and the inevitability of design choices. Then, we survey the associated numerical techniques and implemented algorithms specific to OFDFT. We conclude with an illustrative sample of applications to showcase the power of OFDFT in materials science, chemistry, and physics.

Melting and Freezing of Metal Clusters

Recent developments allow heat capacities to be measured for size-selected clusters isolated in the gas phase. For clusters with tens to hundreds of atoms, the heat capacities determined as a function of temperature usually have a single peak attributed to a melting transition. The melting temperatures and latent heats show large size-dependent fluctuations. In some cases, the melting temperatures change by hundreds of degrees with the addition of a single atom. Theory has played a critical role in understanding the origin of the size-dependent fluctuations, and in understanding the properties of the liquid-like and solid-like states. In some cases, the heat capacities have extra features (an additional peak or a dip) that reveal a more complex behavior than simple melting. In this article we provide a description of the methods used to measure the heat capacities and provide an overview of the experimental and theoretical results obtained for sodium and aluminum clusters.

Metal clusters with hidden ground states: Melting and structural transitions in Al115(+), Al116(+), and Al117(+)

Heat capacities measured as a function of temperature for Al(115)(+), Al(116)(+), and Al(117)(+) show two well-resolved peaks, at around 450 and 600 K. After being annealed to 523 K (a temperature between the two peaks) or to 773 K (well above both peaks), the high temperature peak remains unchanged but the low temperature peak disappears. After considering the possible explanations, the low temperature peak is attributed to a structural transition and the high temperature peak to the melting of the higher enthalpy structure generated by the structural transition. The annealing results show that the liquid clusters freeze exclusively into the higher enthalpy structure and that the lower enthalpy structure is not accessible from the higher enthalpy one on the timescale of the experiments. We suggest that the low enthalpy structure observed before annealing results from epitaxy, where the smaller clusters act as a nucleus and follow a growth pattern that provides access to the low enthalpy structure. The solid-to-solid transition that leads to the low temperature peak in the heat capacity does not occur under equilibrium but requires a superheated solid.

Metal clusters that freeze into high energy geometries

Heat capacities measured for isolated aluminum clusters show peaks due to melting. For some clusters with around 60 and 80 atoms there is a dip in the heat capacities at a slightly lower temperature than the peak. The dips have been attributed to structural transitions. Here we report studies where the clusters are annealed before the heat capacity is measured. The dips disappear for some clusters, but in many cases they persist, even when the clusters are annealed to well above their melting temperature. This indicates that the dips do not result from badly formed clusters generated during cluster growth, as originally suggested. We develop a simple kinetic model of melting and freezing in a system consisting of one liquidlike and two solidlike states with different melting temperatures and latent heats. Using this model we are able to reproduce the experimental results including the dependence on the annealing conditions. The dips result from freezing into a high energy geometry and then annealing into the thermodynamically preferred solid. The thermodynamically preferred solid has the higher freezing temperature. However, the liquid can bypass freezing into the thermodynamically preferred solid (at high cooling rates) if the higher energy geometry has a larger freezing rate.

Hydrodynamic tensor density functional theory with correct susceptibility

In a previous work the authors developed a family of orbital-free tensor equations for the density functional theory [J. Chem. Phys. 124, 024105 (2006)]. The theory is a combination of the coupled hydrodynamic moment equation hierarchy with a cumulant truncation of the one-body electron density matrix. A basic ingredient in the theory is how to truncate the series of equation of motion for the moments. In the original work the authors assumed that the cumulants vanish above a certain order (N). Here the authors show how to modify this assumption to obtain the correct susceptibilities. This is done for N=3, a level above the previous study. At the desired truncation level a few relevant terms are added, which, with the right combination of coefficients, lead to excellent agreement with the Kohn-Sham Lindhard susceptibilities for an uninteracting system. The approach is also powerful away from linear response, as demonstrated in a nonperturbative study of a jellium with a repulsive core, where excellent matching with Kohn-Sham simulations is obtained, while the Thomas-Fermi and von Weiszacker methods show significant deviations. In addition, time-dependent linear response studies at the new N=3 level demonstrate the author's previous assertion that as the order of the theory is increased new additional transverse sound modes appear mimicking the random phase approximation transverse dispersion region.

Ion calorimetry: Using mass spectrometry to measure melting points

Calorimetry measurements have been used to probe the melting of aluminum cluster cations with 63 to 83 atoms. Heat capacities were determined as a function of temperature (from 150 to 1050 K) for size-selected cluster ions using an approach based on multicollision-induced dissociation. The experimental method is described in detail and the assumptions are critically evaluated. Most of the aluminum clusters in the size range examined here show a distinct peak in their heat capacities that is attributed to a melting transition (the peak is due to the latent heat). The melting temperatures are below the bulk melting point and show enormous fluctuations as a function of cluster size. Some clusters (for example, n = 64, 68, and 69) do not show peaks in their heat capacities. This behavior is probably due to the clusters having a disordered solid-like phase, so that melting occurs without a latent heat.

Orbital-free tensor density functional theory

We propose a family of time-dependent orbital-free density-based theories that go beyond the usual current-density description of electrons or other particles. The theories deal with physical quantities that characterize the one-particle density matrix and consequently the kinetics of the particles. We analyze the first two theories in the family. The "lowest-order" theory is quantum hydrodynamics. The second one yields not only the longitudinal plasmon collective excitations, but also the transverse phonon modes that are associated with elementary excitations in Fermi liquids. The theories should make it feasible to do large orbital-free simulations of time-dependent and stationary systems.

First-principles investigation of finite-temperature behavior in small sodium clusters

A systematic and detailed investigation of the finite-temperature behavior of small sodium clusters, Na(n), in the size range of n=8-50 are carried out. The simulations are performed using density-functional molecular dynamics with ultrasoft pseudopotentials. A number of thermodynamic indicators such as specific heat, caloric curve, root-mean-square bond-length fluctuation, deviation energy, etc., are calculated for each of the clusters. Size dependence of these indicators reveals several interesting features. The smallest clusters with n=8 and 10 do not show any signature of melting transition. With the increase in size, broad peak in the specific heat is developed, which alternately for larger clusters evolves into a sharper one, indicating a solidlike to liquidlike transition. The melting temperatures show an irregular pattern similar to the experimentally observed one for larger clusters [Schmidt et al., Nature (London) 393, 238 (1998)]. The present calculations also reveal a remarkable size-sensitive effect in the size range of n=40-55. While Na(40) and Na(55) show well-developed peaks in the specific-heat curve, Na(50) cluster exhibits a rather broad peak, indicating a poorly defined melting transition. Such a feature has been experimentally observed for gallium and aluminum clusters [Breaux et al., J. Am. Chem. Soc. 126, 8628 (2004); Breaux et al., Phys. Rev. Lett. 94, 173401 (2005)].

Competing Thermal Activation Mechanisms in the Meltinglike Transition of NaN (N = 135−147) Clusters

The meltinglike transition in unsupported icosahedral Na(N)() clusters, with N = 135-147, has been studied by isokinetic molecular dynamics simulations based on an orbital-free version of density functional theory. A maximum in the melting temperature, T(m), is obtained for Na141, while the latent heat, deltaE, and entropy of melting, deltaS, are maximal for Na147. These observations are in close agreement with calorimetric experiments on N clusters. The size evolution of deltaS is rationalized by the emergence of important premelting effects associated with the diffusive motion of atomic vacancies at the cluster surface. The precise location of the maximum in T(m) is explained in terms of two different thermally activated structural instability mechanisms which trigger the meltinglike transition in the size ranges N = 135-141 and N = 141-147, respectively.

Anomalous size dependence in the melting temperatures of free sodium clusters: an explanation for the calorimetry experiments

The meltinglike transition in unsupported Na(N) clusters (N = 55, 92, 147, 181, 189, 215, 249, 271, 281 and 299) is studied by first-principles isokinetic molecular dynamics simulations. The irregular size dependence of the melting temperatures Tm observed in the calorimetry experiments of Schmidt et al. [Nature (London) 393, 238 (1998)] is quantitatively reproduced. We demonstrate that structural effects alone can explain all broad features of experimental observations. Specifically, maxima in Tm(N) correlate with high surface stability and with structural features such as a high compactness degree.

First principles local pseudopotential for silver: Towards orbital-free density-functional theory for transition metals

Orbital-free density-functional theory (OF-DFT) with modern kinetic-energy density functionals (KEDFs) is a linear scaling technique that accurately describes nearly-free-electron-like (main group) metals. In an attempt towards extending OF-DFT to transition metals, here we consider whether OF-DFT can be used effectively to study Ag, a metal with a localized d shell. OF-DFT has two approximations: use of a KEDF and local pseudopotentials (LPSs). This paper reports construction of a reasonably accurate LPS for Ag by means of inversion of the Kohn-Sham (KS) DFT equations in a bulk crystal environment. The accuracy of this LPS is determined within KS-DFT (where the exact noninteracting kinetic energy is employed) by comparing its predictions of bulk properties to those obtained from a conventional (orbital-based) nonlocal pseudopotential (NLPS). We find that the static bulk properties of fcc and hcp Ag predicted within KS-DFT using this LPS compare fairly well to those predicted by an NLPS. With the transferability of the LPS established, we then use this LPS in OF-DFT, where several approximate KEDFs were tested. We find that a combination of the Thomas-Fermi (T(TF)) and von Weizsacker (T(vW)) functionals (T(vW)+0.4T(TF)) produces better densities than those from the linear-response-based Wang-Teter KEDF. However, the equations of state obtained from both KEDFs in OF-DFT contain unacceptably large errors. The lack of accurate KEDFs remains the final barrier to extending OF-DFT to treat transition metals.

Melting, premelting, and structural transitions in size-selected aluminum clusters with around 55 atoms

Heat capacities have been determined for unsupported aluminum clusters, Al49(+) - Al63(+), from 150 to 1050 K. Peaks in the heat capacities due to melting occur between 450 and 650 K (well below the bulk melting point of 933 K). The peaks for Al+51 and Al+52 are bimodal, suggesting the presence of a premelting transition where the surface of the clusters melts around 100 K before the core. For clusters with n > 55 the melting temperatures suddenly drop, and there is a dip in the heat capacities due to a transition between two solid forms before the clusters melt.

Improving the orbital-free density functional theory description of covalent materials

The essential challenge in orbital-free density functional theory (OF-DFT) is to construct accurate kinetic energy density functionals (KEDFs) with general applicability (i.e., transferability). During the last decade, several linear-response (LR)-based KEDFs have been proposed. Among them, the Wang-Govind-Carter (WGC) KEDF, containing a density-dependent response kernel, is one of the most accurate that still affords a linear scaling algorithm. For nearly-free-electron-like metals such as Al and its alloys, OF-DFT employing the WGC KEDF produces bulk properties in good agreement with orbital-based Kohn-Sham (KS) DFT predictions. However, when OF-DFT, using the WGC KEDF combined with a recently proposed bulk-derived local pseudopotential (BLPS), was applied to semiconducting and metallic phases of Si, problems arose with convergence of the self-consistent density and energy, leading to poor results. Here we provide evidence that the convergence problem is very likely caused by the use of a truncated Taylor series expansion of the WGC response kernel. Moreover, we show that a defect in the ansatz for the first-order reduced density matrix underlying the LR KEDFs limits the accuracy of these KEDFs. By optimizing the two free parameters involved in the WGC KEDF, the two-body Fermi wave vector mixing parameter gamma and the reference density rho* used in the Taylor expansion, OF-DFT calculations with the BLPS can achieve semiquantitative results for nine phases of bulk silicon. These new parameters are recommended whenever the WGC KEDF is used to study nonmetallic systems.

Specific heat and Lindemann-like parameter of metallic clusters: Mono- and polyvalent metals

The Brownian-type molecular dynamics simulation is revisited and applied to study the thermal and geometric properties of four mono- and two polyvalent metallic clusters. For the thermal property, we report the specific heat at constant volume CV and study the solid-liquid-like transition by scrutinizing its characteristic. For the geometric property, we calculate the root mean square relative bond-length fluctuation delta as a function of increasing temperature. The thermal change in delta reflects the movement of atoms and hence is a relevant parameter in understanding the phase transition in clusters. The simulated results for the CV of alkali and aluminum clusters whose ground state structures exhibit icosahedral symmetry generally show one phase transition. In contrast, the tetravalent lead is quite often seen to exhibit two phase transitions, a premelting process followed by a progressive melting. In connection with the premelting scenario, it is found here that those (magic number) clusters identified to be of lesser stability (among other stable ones) according to the second energy difference are clusters showing a greater possibility of undergoing premelting process. This energy criterion applies to aluminum clusters nAl=28 and 38. To delve further into the thermal behavior of clusters, we have analyzed also the thermal variation of deltaT and attempted to correlate it with CV(T). It turns out that the premelting (if exist) and melting temperatures of the smaller size clusters (n less, similar 50) extracted from CV do not always agree quantitatively with that deduced from delta.

On the premelting features in sodium clusters

Melting in Na(n) clusters described with an empirical embedded-atom potential has been reexamined in the size range 55</=n</=147 with a special attention at sizes close to 130. Contrary to previous findings, premelting effects are also present at such medium sizes, and they turn out to be even stronger than the melting process itself for Na(133) or Na(135). These results indicate that the empirical potential is qualitatively inadequate to model sodium clusters.

Molecular dynamics implementation in MSINDO: Study of silicon clusters

Born-Oppenheimer molecular dynamics is implemented in the semiempirical self-consistent field molecular orbital method MSINDO. The method is employed for the investigation of the structure and dynamics of silicon clusters of various sizes. The reliability of the present parameterization for silicon compounds is demonstrated by a comparison of the results of simulated annealing and of density functional calculations of Si(n) clusters (n = 5-7). The melting behavior of the Si(7) cluster is investigated and the MSINDO results are compared to previous high-level calculations. The efficiency of the present approach for the treatment of large systems is demonstrated by an extensive simulated annealing study of the Si(45) and Si(60) clusters. New Si(45) and Si(60) structures are found and evaluated. The relative stability of various energy minimum structures is compared with density functional calculations and available literature data.

Influence of energy and entropy on the melting of sodium clusters

Energetic and entropic influences on the melting temperatures of size selected sodium clusters are experimentally separated. It is shown that the energetic difference between solid and liquid is the leading influence for the still puzzling features in the size dependence of sodium melting points. Additionally, this energy difference decreases towards smaller cluster sizes and causes steplike melting phase transitions to vanish. The entropy difference between solid and liquid has been found to be strongly correlated with the energy and causes a pronounced damping of the energetic influences.

Exchange monte carlo for molecular simulations with monoelectronic hamiltonians

We introduce a general Monte Carlo scheme for achieving atomistic simulations with monoelectronic Hamiltonians including the thermalization of both nuclear and electronic degrees of freedom. The kinetic Monte Carlo algorithm is used to obtain the exact occupation numbers of the electronic levels at canonical equilibrium, and comparison is made with Fermi-Dirac statistics in infinite and finite systems. The effects of a nonzero electronic temperature on the thermodynamic properties of liquid silver and sodium clusters are presented.

Thermal expansion in small metal clusters and its impact on the electric polarizability

The thermal expansion coefficients of Na(N) clusters with 8</=N</=40, and Al7, Al-13, and Al-14 clusters are obtained from ab initio Born-Oppenheimer local-density-approximation molecular dynamics. Thermal expansion of small metal clusters is considerably larger than that in the bulk and is size dependent. We demonstrate that the average static electric dipole polarizability of Na clusters depends linearly on the mean interatomic distance and only to a minor extent on the detailed ionic configuration when the overall shape of the electron density is enforced by electronic shell effects. Taking thermal expansion into account brings theoretical and experimental polarizabilities into quantitative agreement.