Melting in Large Sodium Clusters: An Orbital-Free Molecular Dynamics Study

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.

Machine Learning Approaches toward Orbital-free Density Functional Theory: Simultaneous Training on the Kinetic Energy Density Functional and Its Functional Derivative

Orbital-free approaches might offer a way to boost the applicability of density functional theory by orders of magnitude in system size. An important ingredient for this endeavor is the kinetic energy density functional. Snyder et al. [Phys. Rev. Lett.2012, 108, 25300223004593] presented a machine learning approximation for this functional achieving chemical accuracy on a one-dimensional model system. However, a poor performance with respect to the functional derivative, a crucial element in iterative energy minimization procedures, enforced the application of a computationally expensive projection method. In this work we circumvent this issue by including the functional derivative into the training of various machine learning models. Besides kernel ridge regression, the original method of choice, we also test the performance of convolutional neural network techniques borrowed from the field of image recognition.

Excess thermal energy and latent heat in nanocluster collisional growth

Nanoclusters can form and grow by nanocluster-monomer collisions (condensation) and nanocluster-nanocluster collisions (coagulation). During growth, product nanoclusters have elevated thermal energies due to potential and thermal energy exchange following a collision. Even though nanocluster collisional heating may be significant and strongly size dependent, no prior theory describes this phenomenon for collisions of finite-size clusters. We derive a model to describe the excess thermal energy of collisional growth, defined as the kinetic energy increase in the product cluster, and latent heat of collisional growth, defined as the heat released to the background upon thermalization of the nonequilibrium cluster. Both quantities are composed of a temperature-independent term related to potential energy minimum differences and a size- and temperature-dependent term, which hinges upon heat capacity and energy partitioning. Example calculations using gold nanoclusters demonstrate that collisional heating can be important and strongly size dependent, particularly for reactive collisions involving nanoclusters composed of 14-20 atoms. Excessive latent heat release may have considerable implications in cluster formation and growth.

Investigation of Melting Dynamics of Hafnium Clusters

Melting dynamics of hafnium clusters are investigated using a novel approach based on the idea of the chemical similarity index. Ground state configurations of small hafnium clusters are first derived using Basin-Hopping and Genetic Algorithm in the parallel tempering mode, employing the COMB potential in the energy calculator. These assumed ground state structures are verified by using the Low Lying Structures (LLS) method. The melting process is carried out either by using the direct heating method or prolonged simulated annealing. The melting point is identified by a caloric curve. However, it is found that the global similarity index is much more superior in locating premelting and total melting points of hafnium clusters.

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.

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.

Second-Order Phase Transitions in Amorphous Gallium Clusters

Ion mobility and calorimetry measurements have been used to probe the nature of the phase transitions in gallium clusters with 29-55 atoms. While most clusters appear to undergo a first-order transition between solidlike and liquidlike phases, a few show the signature of melting without a significant latent heat. These transitions appear to be the finite size analogue of a second-order phase transition, and they presumably occur for some cluster sizes because their solidlike phase is amorphous.

Single impurity effect on the melting of nanoclusters

We show by molecular dynamics simulations that the melting temperature of clusters can be tuned by selective doping. In fact, a single Ni or Cu impurity in Ag icosahedral clusters considerably increases the melting temperature even for sizes of more than a hundred atoms. The upward shift is correlated to the strain relaxation induced by a small central impurity in icosahedral clusters.

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 of sodium clusters: where do the magic numbers come from?

Melting temperatures of Na clusters show size-dependent fluctuations that have resisted interpretation so far. Here we discuss that these temperatures, in fact, cannot be expected to exhibit an easily understandable behavior. The energy and entropy differences between the liquid and the solid clusters turn out to be much more relevant parameters. They exhibit pronounced maxima that correlate well with geometrical shell closings, demonstrating the importance of geometric structure for the melting process. Icosahedral symmetry dominates, a conclusion corroborated by new photoelectron spectra measured on cold cluster anions. In the vicinity of the geometrical shell closings the measured entropy change upon melting is in good agreement with a simple combinatorial model.

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.