Neural network learned Pauli potential for the advancement of orbital-free density functional theory.

The Pauli kinetic energy functional and its functional derivative, termed Pauli potential, play a crucial role in the successful implementation of orbital-free density functional theory for electronic structure calculations. However, the exact forms of these two quantities are not known. Therefore, perforce, one employs the approximate forms for the Pauli functional or Pauli potential for performing orbital-free density functional calculations. In the present study, we developed a feed-forward neural network-based representation for the Pauli potential using a 1-dimensional (1-D) model system. We expanded density in terms of basis functions, and the coefficients of the expansion were used as input to a feed-forward neural network. Using the neural network-based representation of the Pauli potential, we calculated the ground-state densities of the 1-D model system by solving the Euler equation. We calculated the Pauli kinetic energy using the neural network-based Pauli potential employing the exact relation between the Pauli kinetic energy functional and the potential. The sum of the neural network-based Pauli kinetic energy and the von Weizsäcker kinetic energy resulted in an accurate estimation of the total kinetic energy. The approach presented in this paper can be employed for the calculation of Pauli potential and Pauli kinetic energy, obviating the need for a functional derivative. The present study is an important step in the advancement of application of machine learning-based techniques toward the orbital-free density functional theory-based methods.

Structural evolution in gold nanoparticles using artificial neural network based interatomic potentials.

Relativistic effects of gold make its behavior different from other metals. Unlike silver and copper, gold does not require symmetrical structures as the stable entities. We present the evolution of gold from a cluster to a nanoparticle by considering a majority of stable structural possibilities. Here, an interatomic potential (artificial neural network), trained on quantum mechanical data comprising small to medium sized clusters, gives exceptional results for larger size clusters. We have explored the potential energy surface for "magic" number clusters 309, 561, and 923. This study reveals that these clusters are not completely symmetric, but they require a distorted symmetric core with amorphous layers of atoms over it. The amorphous geometries tend to be more stable in comparison to completely symmetric structures. The first ever gold cluster to hold an icosahedron-Au13 was identified at Au60 [S. Pande et al., J. Phys. Chem. Lett. 10, 1820 (2019)]. Through our study, we have found a plausible evolution of a symmetric core as the size of the nanoparticle increases. The stable cores were found at Au160, Au327, and Au571, which can be recognized as new magic numbers. Au923 is found to have a stable symmetric core of 147 atoms covered with layers of atoms that are not completely amorphous. This shows the preference of symmetric structures as the size of the nanoparticle increases (<3.3 nm).

A transferable artificial neural network model for atomic forces in nanoparticles.

We have designed a new method to fit the energy and atomic forces using a single artificial neural network (SANN) for any number of chemical species present in a molecular system. The traditional approach for fitting the potential energy surface for a multicomponent system using artificial neural network (ANN) is to consider n number of networks for n number of chemical species in the system. This shoots the computational cost and makes it difficult to apply to a system containing more number of species. We present a new strategy of using a SANN to compute energy and forces of a chemical system. Since atomic forces are significant for geometry optimizations and molecular dynamics simulations for any chemical system, their accurate prediction is of utmost importance. So, to predict the atomic forces, we have modified the traditional way of fitting forces from underlying energy expression. We have applied our strategy to study geometry optimizations and dynamics in gold-silver nanoalloys and thiol protected gold nanoclusters. Also, force fitting has made it possible to train smaller sized systems and extrapolate the parameters to make accurate predictions for larger systems. This proposed strategy has definitely made the mapping and fitting of atomic forces easier and can be applied to a wide variety of molecular systems.

Correlation of structure with UV-visible spectra by varying SH composition in Au-SH nanoclusters.

In the present work, we model artificial neural network (ANN) potentials for Au n (SH) m nanoclusters in the range of n = 10 to n = 38. The accuracy of ANN potentials is tested by comparing the global minimum (GM) structures of Au n (SH) m nanoclusters, at saturated amount of SH, with the earlier reported structures. The GM structures are reported for the first time for nanoclusters with compositions lower than the saturated SH composition. We calculate the probability of low energy isomers to explain the fluxional behaviour of Au n (SH) m nanoclusters at lower SH compositions. Furthermore, we try to correlate the structures of Au n (SH) m nanoclusters with UV-visible spectra based on Time-dependent density functional theory (TDDFT) calculations. The UV-visible spectral analysis reveals that significant spectroscopic variations are observed at different SH compositions. This study provides a fundamental understanding of structural changes with decreasing SH compositions and with increasing the size of the nanocluster.

c-T phase diagram and Landau free energies of (AgAu)55 nanoalloy via neural-network molecular dynamic simulations.

For understanding the structure, dynamics, and thermal stability of (AgAu)55 nanoalloys, knowledge of the composition-temperature (c-T) phase diagram is essential due to the explicit dependence of properties on composition and temperature. Experimentally, generating the phase diagrams is very challenging, and therefore theoretical insight is necessary. We use an artificial neural network potential for (AgAu)55 nanoalloys. Predicted global minimum structures for pure gold and gold rich compositions are lower in energy compared to previous reports by density functional theory. The present work based on c-T phase diagram, surface area, surface charge, probability of isomers, and Landau free energies supports the enhancement of catalytic property of Ag-Au nanoalloys by incorporation of Ag up to 24% by composition in Au nanoparticles as found experimentally. The phase diagram shows that there is a coexistence temperature range of 70 K for Ag28Au27 compared to all other compositions. We propose the power spectrum coefficients derived from spherical harmonics as an order parameter to calculate Landau free energies.

Spherical harmonics based descriptor for neural network potentials: Structure and dynamics of Au147 nanocluster.

We propose a highly efficient method for fitting the potential energy surface of a nanocluster using a spherical harmonics based descriptor integrated with an artificial neural network. Our method achieves the accuracy of quantum mechanics and speed of empirical potentials. For large sized gold clusters (Au147), the computational time for accurate calculation of energy and forces is about 1.7 s, which is faster by several orders of magnitude compared to density functional theory (DFT). This method is used to perform the global minimum optimizations and molecular dynamics simulations for Au147, and it is found that its global minimum is not an icosahedron. The isomer that can be regarded as the global minimum is found to be 4 eV lower in energy than the icosahedron and is confirmed from DFT. The geometry of the obtained global minimum contains 105 atoms on the surface and 42 atoms in the core. A brief study on the fluxionality in Au147 is performed, and it is concluded that Au147 has a dynamic surface, thus opening a new window for studying its reaction dynamics.

Structural Evolution of Gold Clusters Aun- (n = 21-25) Revisited.

We performed a combined theoretical and experimental photoelectron spectroscopy study of the structural evolution of gold anion clusters Aun- in the size range n = 21-25, a special size range for gold anion clusters where extensive structural changes from the pyramidal structure at Au20- toward the core-shell structure at Au26- were expected to occur. Density functional theory calculations with inclusion of spin-orbit effects were employed to produce the simulated spectra for the selected low-energy isomers obtained from basin-hopping global minimum search. The comparison of these simulated spectra with reasonably well-resolved experimental photoelectron spectra resulted in the identification of the low-lying structures of the gold clusters. The fused-planar and hollow-tubular structures are found dominant in this special size range. The highly stable tetrahedral Au20 unit (viewed as the fragment of face-centered cubic (FCC) bulk gold) was found intact only in the minor isomer at n = 21, whereas hollow-tubular structures were found prevalent in the n = 22-25 range. At n = 25, the dominant structure is a hollow-tubular one with two of gold pyramids fused together, but not a core-shell one as previously believed.

Neural network potentials for dynamics and thermodynamics of gold nanoparticles.

For understanding the dynamical and thermodynamical properties of metal nanoparticles, one has to go beyond static and structural predictions of a nanoparticle. Accurate description of dynamical properties may be computationally intensive depending on the size of nanoparticle. Herein, we demonstrate the use of atomistic neural network potentials, obtained by fitting quantum mechanical data, for extensive molecular dynamics simulations of gold nanoparticles. The fitted potential was tested by performing global optimizations of size selected gold nanoparticles (Aun, 17 ≤ n ≤ 58). We performed molecular dynamics simulations in canonical (NVT) and microcanonical (NVE) ensembles on Au17, Au34, Au58 for a total simulation time of around 3 ns for each nanoparticle. Our study based on both NVT and NVE ensembles indicate that there is a dynamical coexistence of solid-like and liquid-like phases near melting transition. We estimate the probability at finite temperatures for set of isomers lying below 0.5 eV from the global minimum structure. In the case of Au17 and Au58, the properties can be estimated using global minimum structure at room temperature, while for Au34, global minimum structure is not a dominant structure even at low temperatures.

Methanol clusters (CH3OH)n: putative global minimum-energy structures from model potentials and dispersion-corrected density functional theory.

Putative global minima are reported for methanol clusters (CH3OH)n with n ≤ 15. The predictions are based on global optimization of three intermolecular potential energy models followed by local optimization and single-point energy calculations using two variants of dispersion-corrected density functional theory. Recurring structural motifs include folded and/or twisted rings, folded rings with a short branch, and stacked rings. Many of the larger structures are stabilized by weak C-H···O bonds.

Density functional theory guided Monte Carlo simulations: application to melting of Na13.

We present a density functional theory (DFT) based Monte Carlo simulation method in which a simple energy function gets fitted on-the-fly to DFT energies and gradients. The fitness of the energy function gets tested periodically using the classical importance function technique [R. Iftimie, D. Salahub, D. Wei, and J. Schofield, J. Chem. Phys. 113, 4852 (2000)]. The function is updated to fit the DFT energies and gradients of the most recent structures visited whenever it fails to achieve a preset accuracy. In this way, we effectively break down the problem of fitting the entire potential energy surface (PES) into many easier problems, which are to fit small local regions of the PES. We used the scaled Morse potential empirical function to guide a DFT Monte Carlo simulation of Na(13) at various temperatures. The use of empirical function guide produced a computational speed-up of about 7 in our test system without affecting the quality of the results.

Using swarm intelligence for finding transition states and reaction paths.

We describe an algorithm that explores potential energy surfaces (PES) and finds approximate reaction paths and transition states. A few (≈6) evolving atomic configurations ("climbers") start near a local minimum M1 of the PES. The climbers seek a shallow ascent, low energy, path toward a saddle point S12, cross over to another valley of the PES, and climb down to a new minimum M2 that was not known beforehand. Climbers use both energy and energy derivatives to make individual decisions, and they use relative fitness to make team-based decisions. In sufficiently long runs, they keep exploring and may go through a sequence M1-S12-M2-S23-M3 ... of minima and saddle points without revisiting any of the critical points. We report results on eight small test systems that highlight advantages and disadvantages of the method. We also investigated the PES of Li(8), Al(7)(+), Ag(7), and Ag(2)NH(3) to illustrate potential applications of this new method.

CO chemisorption on the surfaces of the golden cages.

We report a joint experimental and theoretical study of CO chemisorption on the golden cages. We find that the Au(17)(-) cage is highly robust and retains its cage structure in Au(17)(CO)(-). On the other hand, the Au(16)(-) cage is transformed to a structure similar to Au(17)(-) upon the adsorption of CO. Au(18)(-) is known to consist of two nearly degenerate structures, i.e., a cage and a pyramidal isomer, which coexist in the cluster beam. However, upon CO chemisorption only the cage isomer is observed while the pyramidal isomer no longer exists due to its less favorable interaction with CO, compared to the cage isomer. We find that inclusion of the spin-orbit effects is critical in yielding simulated spectra in quantitative agreement with the experimental data and providing unequivocal structural information and molecular insights into the chemical interactions between CO and the golden cages.

Structural transition of gold nanoclusters: from the golden cage to the golden pyramid.

How nanoclusters transform from one structural type to another as a function of size is a critical issue in cluster science. Here we report a study of the structural transition from the golden cage Au(16)(-) to the pyramidal Au(20)(-). We obtained distinct experimental evidence that the cage-to-pyramid crossover occurs at Au(18)(-), for which the cage and pyramidal isomers are nearly degenerate and coexist experimentally. The two isomers are observed and identified by their different interactions with O(2) and Ar. The cage isomer is observed to be more reactive with O(2) and can be preferentially "titrated" from the cluster beam, whereas the pyramidal isomer has slightly stronger interactions with Ar and is favored in the Au(18)Ar(x)(-) van der Waals complexes. The current study allows the detailed structural evolution and growth routes from the hollow cage to the compact pyramid to be understood and provides information about the structure-function relationship of the Au(18)(-) cluster.

Search for lowest-energy structure of Zintl dianion Si(12)(2-), Ge(12)(2-), and Sn(12)(2-).

We perform an unbiased search for the lowest-energy structures of Zintl dianions (Si(12)(2-), Ge(12)(2-), and Sn(12)(2-)), by using the basin-hopping (BH) global optimization method combined with density functional theory geometric optimization. High-level ab initio calculation at the coupled-cluster level is used to determine relative stabilities and energy ranking among competitive low-lying isomers of the dianions obtained from the BH search. For Si(12)(2-), all BH searches (based on independent initial structures) lead to the same lowest-energy structure Si(12a)(2-), a tricapped trigonal prism (TTP) with C(s) group symmetry. Coupled-cluster calculation, however, suggests that another TTP isomer of Si(12c)(2-) is nearly isoenergetic with Si(12a)(2-). For Sn(12)(2-), all BH searches lead to the icosahedral structure I(h)-Sn(12a)(2-), i.e., the stannaspherene. For Ge(12)(2-), however, most BH searches lead to the TTP-containing Ge(12b)(2-), while a few BH searches lead to the empty-cage icosahedral structure I(h)-Ge(12a)(2-) (named as germaniaspherene). High-level ab initio calculation indicates that I(h)-Ge(12a)(2-) and TTP-containing Ge(12b)(2-) are almost isoenergetic and, thus, both may be considered as candidates for the lowest-energy structure at 0 K. Ge(12a)(2-) has a much larger energy gap (2.04 eV) between highest occupied molecular orbital and lowest unoccupied molecular orbital than Ge(12b)(2-) (1.29 eV), while Ge(12b)(2-) has a lower free energy than I(h)-Ge(12a)(2-) at elevated temperature (>980 K). The TTP-containing Si(12a)(2-) and Ge(12b)(2-) exhibit large negative nuclear independent chemical shift (NICS) value (approximately -44) at the center of TTP, indicating aromatic character. In contrast, germaniaspherene I(h)-Ge(12a)(2-) and stannaspherene I(h)-Sn(12a)(2-) exhibit modest positive NICS values, approximately 12 and 3, respectively, at the center of the empty cage, indicating weakly antiaromatic character.

Ab initio calculation of carbon clusters. II. Relative stabilities of fullerene and nonfullerene C24.

Chemical stabilities of six low-energy isomers of C24 derived from global-minimum search are investigated. The six isomers include one classical fullerene (isomer 1) whose cage is composed of only five- and six-membered rings (56-MRs), three nonclassical fullerene structures whose cages contain at least one four-membered ring (4-MR), one plate, and one monocyclic ring. Chemical and electronic properties of the six C24 isomers are calculated based on a density-functional theory method (hybrid PBE1PBE functional and cc-pVTZ basis set). The properties include the nucleus-independent chemical shifts (NICS), singlet-triplet splitting, electron affinity, ionization potential, and gap between the highest occupied molecular orbital and the lowest unoccupied molecular orbital (HOMO-LUMO) gap. The calculation suggests that the neutral isomer 2, a nonclassical fullerene with two 4-MRs, may be more chemically stable than the classical fullerene (isomer 1). Analyses of molecular orbital NICS show that the incorporations of 4-MRs into the cage considerably reduce paratropic contributions from HOMO, HOMO-1, and HOMO-2, which are mainly responsible for the sign change in NICS from positive for isomer 1 (42) to negative (-19) for isomer 2, although C24 clusters satisfy neither 4N+2 nor 2(N+1)2 aromaticity rule. Anion photoelectron spectra of four cage isomers, one plate, one monocyclic ring, and one tadpole isomer, as well as three bicyclic ring isomers are calculated. The simulated photoelectron spectra of mono- and bicyclic rings (with C1 symmetry) appear to match the measured HOMO-LUMO gap (between the first and second band in the experimental spectra) [S. Yang et al., Chem. Phys. Lett. 144, 431 (1988)]. Nevertheless, the nonclassical fullerene isomers 3 and 4 apparently also match the measured vertical detachment energy (2.90 eV) reasonably well. These results suggest possible coexistence of nonclassical fullerene isomers with the mono- and bicyclic ring isomers of C24(-) under the experimental conditions.

Structures and relative stability of neutral gold clusters: Aun (n=15-19).

We performed a global-minimum search for low-lying neutral clusters (Au(n)) in the size range of n=15-19 by means of basin-hopping method coupled with density functional theory calculation. Leading candidates for the lowest-energy clusters are identified, including four for Au(15), two for Au(16), three for Au(17), five for Au(18), and one for Au(19). For Au(15) and Au(16) we find that the shell-like flat-cage structures dominate the population of low-lying clusters, while for Au(17) and Au(18) spherical-like hollow-cage structures dominate the low-lying population. The transition from flat-cage to hollow-cage structure is at Au(17) for neutral gold clusters, in contrast to the anion counterparts for which the structural transition is at Au(16) (-) [S. Bulusu et al., Proc. Natl. Acad. Sci. U.S.A. 103, 8362 (2006)]. Moreover, the structural transition from hollow-cage to pyramidal structure occurs at Au(19). The lowest-energy hollow-cage structure of Au(17) (with C(2v) point-group symmetry) shows distinct stability, either in neutral or in anionic form. The distinct stability of the hollow-cage Au(17) calls for the possibility of synthesizing highly stable core/shell bimetallic clusters M@Au(17) (M=group I metal elements).

Lowest-energy structures of water clusters (H2O)11 and (H2O)13.

We employed a four-step searching/screening approach to determine best candidates for the global minima of (H2O)11 and (H2O)13. This approach can be useful when there exist a large number of low-lying and near-isoenergetic isomers, many of which have the same oxygen-skeleton structure. On the two new candidates for the global minimum of (H2O)11, one isomer can be viewed as placing the 11th molecule onto the side of the global minimum of (H2O)10 and the other can be viewed as removing the 12th molecule from the middle layer of the global minimum of (H2O)12. The three leading lowest-energy clusters of (H2O)13 can all be built starting from the global minimum of (H2O)12, with the difference being in the location of the 13th water molecule.