Novel method for geometry optimization of molecular clusters: application to benzene clusters

A heuristic and unbiased method for searching optimal geometries of clusters of nonspherical molecules was constructed from the algorithm recently proposed for Lennard-Jones atomic clusters. In the method, global minima are searched by using three operators, interior, surface, and orientation operators. The first operator gives a perturbation on a cluster configuration by moving molecules near the center of mass of a cluster, and the second one modifies a cluster configuration by moving molecules to the most stable positions on the surface of a cluster. The moved molecules are selected by employing a contribution of the molecules to the potential energy of a cluster. The third operator randomly changes the orientations of all molecules. The proposed method was applied to benzene clusters. It was possible to find new global minima for (C6H6)11, (C6H6)14, and (C6H6)15. Global minima for (C6H6)16 to (C6H6)30 are first reported in this article.     

A Fukui function‐guided genetic algorithm. Assessment on structural prediction of Sin (n = 12–20) clusters

Theoretical studies are essential for the structural characterization of clusters, when it comes to rationalize their unique size‐dependent properties and composition. However, the rapid growth of local minima on the potential energy surface (PES), with respect to cluster size, makes the candidate identification a challenging undertaking. In this article, we introduce a hybrid strategy to explore the PES of clusters. This proposal involves the use of a biased initial population of a genetic algorithm procedure. Each individual in this population is built by assembling small fragments, according to the best matching of the Fukui function. The performance of a genetic algorithm procedure. The performance of the method is assessed on the PES exploration of medium‐sized Si_n clusters (n = 12–20). The most relevant results are: (a) the method converges at almost half of the time used by the canonical version of the GA and, (b) in all the studied cases, with the exception of Si₁₃ and Si₁₆, the method allowed to identify the global minimum (GM) and other important low‐lying structures. Additionally, the apparent deficiency of the proposal to identify the GM was corrected when a Si atom, or other low‐lying isomers, were considered to build the clusters. © 2017 Wiley Periodicals, Inc.     

An ab initio study of water clusters in gas phase and bulk aqueous media: (H2O)n,n=1–12

Geometries of several clusters of water molecules including single minimum energy structures of n‐mers (n=1–5), several hexamers and two structures of each of heptamer to decamer derived from hexamer cage and hexamer prism were optimized. One structural form of each of 11‐mer and 12‐mer were also studied. The geometry optimization calculations were performed at the RHF/6‐311G* level for all the cases and at the MP2/6‐311++G** level for some selected cases. The optimized cluster geometries were used to calculate total energies of the clusters in gas phase employing the B3LYP density functional method and the 6‐311G* basis set. Frequency analysis was carried out in all the cases to ensure that the optimized geometries corresponded to total energy minima. Zero‐point and thermal free energy corrections were applied for comparison of energies of certain hexamers. The optimized cluster geometries were used to solvate the clusters in bulk water using the polarized continuum model (PCM) of the self‐consistent reaction field (SCRF) theory, the 6‐311G* basis set, and the B3LYP density functional method. For the cases for which MP2/6‐311++G** geometry optimization was performed, solvation calculations in water were also carried out using the B3LYP density functional method, the 6‐311++G** basis set, and the PCM model of SCRF theory, besides the corresponding gas‐phase calculations. It is found that the cage form of water hexamer cluster is most stable in gas phase among the different hexamers, which is in agreement with the earlier theoretical and experimental results. Further, use of a newly defined relative population index (RPI) in terms of successive total energy differences per water molecule for different cluster sizes suggests that stabilities of trimers, hexamers, and nonamers in gas phase and those of hexamers and nonamers in bulk water would be favored while those of pentamer and decamer in both the phases would be relatively disfavored. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 90–104, 2001     

Global Minimum‐Energy Structure and Spectroscopic Properties of I2.−⋅n H2O Clusters: A Monte Carlo Simulated Annealing Study

The vibrational (IR and Raman) and photoelectron spectral properties of hydrated iodine‐dimer radical‐anion clusters, I₂^.? ? n H₂O (n=1–10), are presented. Several initial guess structures are considered for each size of cluster to locate the global minimum‐energy structure by applying a Monte Carlo simulated annealing procedure including spin–orbit interaction. In the Raman spectrum, hydration reduces the intensity of the I? I stretching band but enhances the intensity of the O? H stretching band of water. Raman spectra of more highly hydrated clusters appear to be simpler than the corresponding IR spectra. Vibrational bands due to simultaneous stretching vibrations of O? H bonds in a cyclic water network are observed for I₂^.? ? n H₂O clusters with n≥3. The vertical detachment energy (VDE) profile shows stepwise saturation that indicates closing of the geometrical shell in the hydrated clusters on addition of every four water molecules. The calculated VDE of finite‐size small hydrated clusters is extrapolated to evaluate the bulk VDE value of I₂^.? in aqueous solution as 7.6 eV at the CCSD(T) level of theory. Structure and spectroscopic properties of these hydrated clusters are compared with those of hydrated clusters of Cl₂^.? and Br₂^.?.     

Global minimum geometries of acetylene clusters (HCCH)n with n ≤ 55 obtained by a heuristic method combined with geometrical perturbations

Geometry optimization of acetylene clusters (HCCH)_n in the range of n ≤ 55 was carried out with a recently proposed intermolecular potential consisting of Morse potentials, damped dispersion terms, and damped Coulomb terms. The heuristic method developed by the present author was used in the optimization: optimal geometries were searched by using geometry perturbations and subsequent local optimizations. The calculations were repeated until the global minimum was found for each cluster at least three times. The obtained results were analyzed to examine structural evolution of the clusters. The geometries of the clusters with n ≥ 25 were similar to the geometry in the cubic crystal of acetylene whereas smaller clusters take icosahedron‐based geometries. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Electronic and magnetic properties of small RhnCa (n = 1–9) clusters: A DFT study

The equilibrium geometries, relative stabilities, electronic and magnetic properties of small Rh_nCa (n = 1–9) clusters have been investigated by DFT calculations. The obtained results show that the three‐dimensional geometries are adopted for the lowest‐energy Rh_nCa clusters, and the doped Ca atom prefers locating on the surface of the cluster. Based on the analysis of the second‐order difference of energies, fragmentation energies and the HOMO‐LUMO energy gaps, we identify that the Rh₄Ca, Rh₆Ca, and Rh₈Ca clusters are relatively more stable than their neighboring clusters, and the doping of Ca enhances the chemical reactivity of the pure Rh_n clusters, suggesting that the Rh_nCa clusters can be used as nanocatalysts in many catalytic reactions. The magnetic moment for these clusters is mostly localized on the Rh atoms, and the doping Ca atom has no effect on the total magnetic moment of Rh_nCa clusters. The partial density of states, VIP, VEA, and η of these clusters in their ground‐state structures were also calculated and discussed. © 2015 Wiley Periodicals, Inc.     

Computer optimization of small atomic clusters

We have successfully identified stable configurations of both rare-gas and NaCl clusters with a new optimization procedure. An initial cluster configuration is prepared in a so-called shoot-and-stay process. Its total energy is then minimized with respect to the atomic coordinates. To prevent the system from being locked in local minima, the step size of each move is chosen as the width of the energy well at a higher level. As the system evolves, the global minimum is contained in the volume bounded by the decreasing value of step sizes. We have also carried out the optimization of NaCl clusters by the simulated annealing technique, for comparison. The results show that for such heterogeneous systems, the latter method cannot always find the global minimum, because of large energy gaps between different catchment regions in phase space.     

Search for global minimum geometries for medium sized germanium clusters: Ge12-Ge20

We have performed an unbiased search for the global minimum geometries of small-to-medium sized germanium clusters Gen(12< or =n< or =18) as well as a biased search (using seeding method) for Gen(17< or =n< or =20). We employed the basin-hopping algorithm coupled with the plane-wave pseudopotential density functional calculations. For each size, we started the unbiased search with using several structurally very different initial clusters, or we started the biased search with three different seeds. Irrespective of the initial structures of clusters we found that the obtained lowest-energy clusters of the size n=12-16 and 18 are the same. Among them, the predicted global minima of Gen(12< or =n< or =16) are identical to those reported previously [Shvartsburg et al., Phys. Rev. Lett. 83, 167 (1999)]. For n=17-20, we have identified two or three nearly isoenergetic low-lying isomers (for each size) that compete for the global minimum. Nearly all the low-lying clusters in the size range of 12< or =n< or =20 contain the tri-caped trigonal prism motif and are all prolate in geometry, in agreement with the experiment.     

Distribution of interatomic distances in large metallic clusters

Spherically averaged pseudopotential (SAPS) calculations have been done for Mg _n clusters, withn up to 250 within the framework of density functional theory. The electronic structure is computed resorting to the Thomas-Fermi-Dirac-Weizsäcker (TFDW) approximation for the kinetic energy. The equilibrium geometries have been obtained by minimizing the total cluster energy with respect to the atomic positions using the steepest-descent method. The ground state geometries obtained in this way are formed by spherical atomic shells, the number of them increasing with cluster size, up to a number of four for the biggest sizes considered here. An analysis of the distribution of the interatomic distances shows that the more internal is the shell, the more contracted are the interatomic distances. This effect diminishes progressively with increasing cluster size. For the purpose of comparison, similar calculations have been done with Cs _n clusters in the same size range, allowing us to reproduce previous results obtained using a more elaborated density functional technique (Kohn-Sham method). The inhomogeneous contraction of interatomic distances then appears as a general fact for simple metallic clusters and not only for alkaline ones.     

Structures and electronic properties of the small rubidium‐doped silicon RbSin (n = 1–12) clusters

The geometries, relative stabilities, and electronic properties of small rubidium‐doped silicon clusters RbSi_n (n = 1–12) have been systematically investigated using the density functional theory at the B3LYP/GENECP level. The optimized structures show that lowest‐energy isomers of RbSi_n are similar with the ground state isomers of pure Si_n clusters and prefer the three‐dimensional for n = 3–12. The relative stabilities of RbSi_n clusters have been analyzed on the averaged binding energy, fragmentation energy, second‐order energy difference, and highest occupied molecular orbital‐lowest unoccupied molecular orbital energy gap. The calculated results indicate that the doping of Rb atom enhances the chemical activity of Si_n frame and the magic number is RbSi₂. The Mulliken population analysis reveals that the charges in the corresponding RbSi_n clusters transfer from the Rb atom to Si atoms. The partial density of states and chemical hardness are also discussed. © 2014 Wiley Periodicals, Inc.     

A detailed investigation on the global minimum structures of mixed rare‐gas clusters: Geometry,energetics, and site occupancy

We performed a global minimum search of mixed rare‐gas clusters by applying an evolutionary algorithm (EA), which was recently proposed for binary atomic systems (Marques and Pereira, Chem. Phys. Lett. 2010, 485, 211). Before being applied to the potentials used in this work, the EA was further tested against results previously reported for the Ar_NXe_38?N clusters and several new putative global minima were discovered. We employed either simple Lennard‐Jones (LJ) potentials or more realistic functions to describe pair interactions in Ar_NKr_38?N, Ar_NXe_38?N, and Kr_NXe_38?N clusters. The long‐range tail of the pair‐potentials shows some influence on the energetic features and shape of the structure of clusters. In turn, core–shell type structures are mostly observed for global minima of the binary rare‐gas clusters, for both accurate and LJ potentials. However, the long‐range tail of the potential may have influence on the type of atoms that segregate on the surface or form the core of the cluster. While relevant differences for the preferential site occupancy occur between the two potentials for Ar_NKr_38?N (for N > 21), the type of atoms that segregate on the surface for Ar_NXe_38?N and Kr_NXe_38?N clusters is unaffected by the accuracy of the long‐range part of the interaction in almost all cases. Moreover, the global minimum search for model‐potentials in binary systems reveals that the surface‐site occupancy is mainly determined by the combination of two parameters: the size ratio of the two types of particles forming the cluster and the minimum‐energy ratio corresponding to the pair‐interactions between unlike atoms. © 2012 Wiley Periodicals, Inc.     

Structural evolution of subnano platinum clusters

We present a theoretical study of the structural evolution of small minimum energy platinum clusters, using density functional theory (DFT). Three growth pathways were identified. At the subnanoscale, clusters with triangular packing are energetically most favorable. At a cluster size of approximately n = 19, a structural transition from triangular clusters to icosahedral clusters occurs. A less energetically favorable transition from triangular clusters to fcc‐like clusters takes place at around n = 38. Ionization potentials, electron affinities, and magnetic moments of the triangular clusters were also calculated. Understanding the structures and properties will facilitate studies of the chemical reactivity of Pt nanoclusters toward small molecules. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Optimization of bimetallic Cu–Au and Ag–Au clusters by using a modified adaptive immune optimization algorithm

A modified adaptive immune optimization algorithm (AIOA) is designed for optimization of Cu–Au and Ag–Au bimetallic clusters with Gupta potential. Compared with homoatom clusters, there are homotopic isomers in bimetallic cluster, so atom exchange operation is presented in the modified AIOA. The efficiency of the algorithm is tested by optimization of Cu_nAu_38‐n (0 ≤ n ≤ 38). Results show that all the structures with the putative global minimal energies are successfully located. In the optimization of Ag_nAu_55‐n (0 ≤ n ≤ 55) bimetallic clusters, all the structures with the reported minimal energies are obtained, and 36 structures with even lower potential energies are found. On the other hand, with the optimized structures of Cu_nAu_55‐n, it is shown that all 55‐atom Cu–Au bimetallic clusters are Mackay icosahedra except for Au₅₅, which is a face‐centered cubic (fcc)‐like structure; Cu₅₅, Cu₁₂Au₄₃, and Cu₁Au₅₄ have two‐shell Mackay icosahedral geometries with I_h point group symmetry. © 2009 Wiley Periodicals, Inc. J Comput Chem 2009     

N2O in small para‐hydrogen clusters: Structures and energetics

We present the minimum‐energy structures and energetics of clusters of the linear N₂O molecule with small numbers of para‐hydrogen molecules with pairwise additive potentials. Interaction energies of (p‐H₂)–N₂O and (p‐H₂)–(p‐H₂) complexes were calculated by averaging the corresponding full‐dimensional potentials over the H₂ angular coordinates. The averaged (p‐H₂)–N₂O potential has three minima corresponding to the T‐shaped and the linear (p‐H₂)–ONN and (p‐H₂)–NNO structures. Optimization of the minimum‐energy structures was performed using a Genetic Algorithm. It was found that p‐H₂ molecules fill three solvation rings around the N₂O axis, each of them containing up to five p‐H₂ molecules, followed by accumulation of two p‐H₂ molecules at the oxygen and nitrogen ends. The first solvation shell is completed at N = 17. The calculated chemical potential oscillates with cluster size up to the completed first solvation shell. These results are consistent with the available experimental measurements. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

Theoretical study on the adsorption of aromatic compounds on platinum clusters

B3 LYP hybrid functional with LACVP* pseudopotential was applied for the optimization of geometries of complexes resulting from interaction of benzene, pyridine, naphthalene, and quinoline with Pt_n (n = 4, 7) clusters. For benzene‐containing complexes, the most stable form corresponds to a bridge adsorption, with benzene undergoing considerable geometric distortions, assuming a boat‐like conformation. C? H bonds are bended upward from the plane of the cluster. C? C bonds stretch, especially when they form π‐complexes with low coordinated Pt atoms. Some arrangements for pyridine complexes involving the N atom of the organic moiety undergo further distortions, apparently preserving a formal C? N π bond. Except for that distortion, the behavior of any heteroaromatic complex is similar to that of benzene in the same arrangement. The quinoline–Pt₇ complex can suitably be used for simulation of the cinchonidine (CD) anchorage over Pt. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Theoretical prediction for the structures of gas phase lithium oxide clusters: (Li2O)n (n = 1–8)

We apply genetic algorithm combining directly with density functional method to search the potential energy surface of lithium‐oxide clusters (Li₂O)_n up to n = 8. In (Li₂O)_n (n = 1–8) clusters, the planar structures are found to be global minimum up to n = 2, and the global minimum structures are all three‐dimensional at n ≥ 3. At n ≥ 4, the tetrahedral unit (TU) is found in most of the stable structures. In the TU, the central Li is bonded with four O atoms in sp³ interactions, which leads to unusual charge transformation, and the probability of the central Li participating in the bonding is higher by adaptive natural density partitioning analysis, so the central Li is in particularly low positive charge. At large cluster size, distortion of structures is viewed, which breaks the symmetry and may make energy higher. The global minimum structures of (Li₂O)₂, (Li₂O)₆, and (Li₂O)₇ clusters are the most stable magic numbers, where the first one is planar and the later both have stable structural units of tetrahedral and C_4v. © 2012 Wiley Periodicals, Inc.     

A Computational Study of 3‐D Helium Clusters

Physical and thermodynamic properties have been calculated and analyzed for the best and optimized geometries of the 3‐D clusters with N = 3 to N = 10 atoms and unit cells of three types of crystalline systems using ab initio RHF/6–31G^** method. Dependence of the lattice binding energy on the cluster parameter, R, has been studied. Similar behavior observed for the binding energies for all clusters shows that probabilities of their existence in the condensed phase are more or less the same. In the next step, thermodynamic properties have been calculated and analyzed for He₂₇ 3‐D helium clusters with simple cubic, body centered cubic (bcc), trigonal and hexagonal (hcp) configurations. The results show that the hexagonal cluster is more favored over other clusters. It is found that these clusters are electronically stable over a limited range of the values for the lattice parameter. Δ_fH is constant in this stability region and thus the Δ_fG exactly follows the variations of TΔ_fS. Surface effects have been investigated by comparing the square and hexagonal He₉ 2‐D lattices with the cubic and hexagonal He₂₇ 3‐D lattices, respectively. The lattice parameters, densities and molar volumes calculated for the clusters with hcp and bcc configurations have satisfactory agreement with the available experimental values. Properties of the He₁₃, He₃₄ and He₁₀₄ hcp clusters have also been calculated and analyzed.     

Density functional study of structural and electronic properties of AlnN (1 ≤ n ≤ 12) clusters

Low‐lying equilibrium geometric structures of Al_nN (n = 1–12) clusters obtained by an all‐electron linear combination of atomic orbital approach, within spin‐polarized density functional theory, are reported. The binding energy, dissociation energy, and stability of these clusters are studied within the local spin density approximation (LSDA) and the three‐parameter hybrid generalized gradient approximation (GGA) due to Becke–Lee–Yang–Parr (B3LYP). Ionization potentials, electron affinities, hardness, and static dipole polarizabilities are calculated for the ground‐state structures within the GGA. It is observed that symmetric structures with the nitrogen atom occupying the internal position are lowest‐energy geometries. Generalized gradient approximation extends bond lengths as compared with the LSDA lengths. The odd–even oscillations in the dissociation energy, the second differences in energy, the highest occupied molecular orbital–lowest unoccupied molecular orbital (HOMO–LUMO) gaps, the ionization potential, the electron affinity, and the hardness are more pronounced within the GGA. The stability analysis based on the energies clearly shows the Al₇N cluster to be endowed with special stability. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Benzene clusters in a supersonic beam

A supersonic beam is employed to produce benzene clusters (C₆H₆) _n up ton=40. Mass analysis is achieved after two-photon ionization in a reflectron mass spectrometer. Photon energy is chosen so that the internal energy of the cluster ions is less than 700 meV and a slow decay on the µs time scale is observed. By an energy analysis with the reflecting field it is found that the elimination of one neutral benzene monomer is the favoured dissociation process of the cluster ions. Information about the dissociation pathways of the cluster ions is essential if one is to obtain neutral cluster abundances from the ion mass spectrum. Furthermore an experimental method is presented to obtain pure intermediate state (S ₁←S₀) spectra of selected clusters without interferences from the other clusters present in the molecular beam. This method is based on the observation of the metastable decay of the corresponding cluster ion. When the metastable signal is recorded as a function of photon energy it reflects theS ₁←S ₀ intermediate state spectrum. Spectra are presented for the benzene dimer, trimer, tetramer and pentamer.     

Quantum chemistry of quantum size‐effects in semiconductors: Small clusters electronic structure calculations

Ab initio RHF SCF calculations are used for some small clusters M_xX_y, where M=Cd, Ag; X=S, I; and x, y≤7. Variation of electronic structure with size for some clusters with the bulklike tetrahedral coordination and with the lower symmetry allows one to predict their possible geometries which are compared with experimental data on the existence of the clusters. The chemical‐bonding factor (the chemical nature of bounded atoms, coordination number for metal and nonmetal atoms, hybridization, etc.) is of more importance for properties of the clusters than is the familiar quantum confinement effect of semiconductor clusters. The essential difference in regularities of small cluster formation is analyzed for CdS‐ and AgI‐based structures. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 337–341, 1999