Rare-event sampling: occupation-based performance measures for parallel tempering and infinite swapping Monte Carlo methods.

In the present paper we identify a rigorous property of a number of tempering-based Monte Carlo sampling methods, including parallel tempering as well as partial and infinite swapping. Based on this property we develop a variety of performance measures for such rare-event sampling methods that are broadly applicable, informative, and straightforward to implement. We illustrate the use of these performance measures with a series of applications involving the equilibrium properties of simple Lennard-Jones clusters, applications for which the performance levels of partial and infinite swapping approaches are found to be higher than those of conventional parallel tempering.

Monte Carlo investigation of the thermodynamic properties of (H2O)n and (H2O)nH2 (n = 2-20) clusters.

The classical thermodynamic properties of water clusters modeled by the TIP4P potential with and without the inclusion of molecular hydrogen are calculated using Monte Carlo methods. The temperature-dependent heat capacity curves of the dimer and the smaller pure water clusters having ring structures show low-temperature anomalies arising from the onset of a transition from librational motion to free rotational motion. Pure water clusters having cage structures display heat capacity anomalies characteristic of "melting" phase changes. The addition of molecular hydrogen to the water clusters has little impact on the structure of the water core, but low-temperature heat capacity anomalies are observed that arise from the onset of hindered to free rotational and translational motion of the hydrogen molecule with respect to the skeletal moieties of the core. The Gibbs free energy changes associated with the growth of pure water clusters and the addition of molecular hydrogen to the water clusters are determined using a combination of state-integration methods and an application of the Gibbs-Helmholtz equation. For the temperature integration of the Gibbs-Helmholtz equation, a quadrature method is introduced that avoids numerical difficulties arising from singularities in the integrand at low temperatures. For the growth of pure water clusters, the fine structure of the enthaply and the low-temperature Gibbs free energy as a function of cluster size is rationalized using an ansatz of molecular dipole orientations. For the addition of molecular hydrogen to water clusters, the Gibbs free energy change is a virtually flat function of cluster size showing no fine structure.

Convergence characteristics of the cumulant expansion for Fourier path integrals.

The cumulant representation of the Fourier path integral method is examined to determine the asymptotic convergence characteristics of the imaginary-time density matrix with respect to the number of path variables N included. It is proved that when the cumulant expansion is truncated at order p, the asymptotic convergence rate of the density matrix behaves like N(-(2p+1)). The complex algebra associated with the proof is simplified by introducing a diagrammatic representation of the contributing terms along with an associated linked-cluster theorem. The cumulant terms at each order are expanded in a series such that the asymptotic convergence rate is maintained without the need to calculate the full cumulant at order p. Using this truncated expansion of each cumulant at order p, the numerical cost in developing Fourier path integral expressions having convergence order N(-(2p+1)) is shown to be approximately linear in the number of required potential energy evaluations making the method promising for actual numerical implementation.

Proton disorder and the dielectric constant of type II clathrate hydrates.

Computational studies are presented examining the degree of proton disorder in argon and molecular hydrogen sII clathrate hydrates. Results are presented using a variety of model potentials for the dielectric constant, the proton order parameter, and the molecular volume for the clathrate systems. The dielectric constant for the clathrate systems is found to be lower than the dielectric constant of ice in all models. The ratio of the clathrate to ice dielectric constant correlates well with the ratio of the densities, which is not the case for comparisons to the liquid, so that differences in the dielectric constants between ice and the clathrates are most likely due to differences in densities. Although the computed dielectric constant is a strong function of the model potential used, the ratio of the dielectric constant of ice to that of the clathrates is insensitive to the model potential. For the nonpolar guest molecules used in the current study, the degree proton of disorder is found to depend weakly on the identity of the guest but the dielectric constant does not appear to be sensitive to pressure or the type of guest.

The thermodynamic and ground state properties of the TIP4P water octamer.

Several stochastic simulations of the TIP4P [W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey, and M. L. Klein, J. Chem. Phys. 79, 926 (1983)] water octamer are performed. Use is made of the stereographic projection path integral and the Green's function stereographic projection diffusion Monte Carlo techniques, recently developed in one of our groups. The importance sampling for the diffusion Monte Carlo algorithm is obtained by optimizing a simple wave function using variational Monte Carlo enhanced with parallel tempering to overcome quasiergodicity problems. The quantum heat capacity of the TIP4P octamer contains a pronounced melting peak at 160 K, about 50 K lower than the classical melting peak. The zero point energy of the TIP4P water octamer is 0.0348+/-0.0002 hartree. By characterizing several large samples of configurations visited by both guided and unguided diffusion walks, we determine that both the TIP4P and the SPC [H. J. C. Berendsen, J. P. Postma, W. F. von Gunsteren, and J. Hermans, (Intermolecular Forces, Reidel, 1981). p. 331] octamer have a ground state wave functions predominantly contained within the D(2d) basin of attraction. This result contrasts with the structure of the global minimum for the TIP4P potential, which is an S(4) cube. Comparisons of the thermodynamic and ground-state properties are made with the SPC octamer as well.

A stereographic projection path integral study of the coupling between the orientation and the bending degrees of freedom of water.

A Monte Carlo path integral method to study the coupling between the rotation and bending degrees of freedom for water is developed. It is demonstrated that soft internal degrees of freedom that are not stretching in nature can be mapped with stereographic projection coordinates. For water, the bending coordinate is orthogonal to the stereographic projection coordinates used to map its orientation. Methods are developed to compute the classical and quantum Jacobian terms so that the proper infinitely stiff spring constant limit is recovered in the classical limit, and so that the nonconstant nature of the Riemann Cartan curvature scalar is properly accounted in the quantum simulations. The theory is used to investigate the effects of the geometric coupling between the bending and the rotating degrees of freedom for the water monomer in an external field in the 250 to 500 K range. We detect no evidence of geometric coupling between the bending degree of freedom and the orientations.

A constant entropy increase model for the selection of parallel tempering ensembles.

The present paper explores a simple approach to the question of parallel tempering temperature selection. We argue that to optimize the performance of parallel tempering it is reasonable to require that the increase in entropy between successive temperatures be uniform over the entire ensemble. An estimate of the system's heat capacity, obtained either from experiment, a preliminary simulation, or a suitable physical model, thus provides a means for generating the desired tempering ensemble. Applications to the two-dimensional Ising problem indicate that the resulting method is effective, simple to implement, and robust with respect to its sensitivity to the quality of the underlying heat capacity model.

Pressure dependent study of the solid-solid phase change in 38-atom Lennard-Jones cluster.

Phase change phenomena in clusters are often modeled by augmenting physical interaction potentials with an external constraining potential to handle evaporation processes in finite temperature simulations. These external constraining potentials exert a pressure on the cluster. The influence of this constraining pressure on phase change phenomena in 38-atom Lennard-Jones clusters is investigated, and it is demonstrated that modest changes in the parameters of the constraining potential can lead to an order of magnitude change in the constraining pressure. At sufficiently high pressures the solid to solidlike phase change region in the 38-atom Lennard-Jones cluster is completely eliminated.

Combining smart darting with parallel tempering using Eckart space: application to Lennard-Jones clusters.

The smart-darting algorithm is a Monte Carlo based simulation method used to overcome quasiergodicity problems associated with disconnected regions of configurations space separated by high energy barriers. As originally implemented, the smart-darting method works well for clusters at low temperatures with the angular momentum restricted to zero and where there are no transitions to permutational isomers. If the rotational motion of the clusters is unrestricted or if permutational isomerization becomes important, the acceptance probability of darting moves in the original implementation of the method becomes vanishingly small. In this work the smart-darting algorithm is combined with the parallel tempering method in a manner where both rotational motion and permutational isomerization events are important. To enable the combination of parallel tempering with smart darting so that the smart-darting moves have a reasonable acceptance probability, the original algorithm is modified by using a restricted space for the smart-darting moves. The restricted space uses a body-fixed coordinate system first introduced by Eckart, and moves in this Eckart space are coupled with local moves in the full 3N-dimensional space. The modified smart-darting method is applied to the calculation of the heat capacity of a seven-atom Lennard-Jones cluster. The smart-darting moves yield significant improvement in the statistical fluctuations of the calculated heat capacity in the region of temperatures where the system isomerizes. When the modified smart-darting algorithm is combined with parallel tempering, the statistical fluctuations of the heat capacity of a seven-atom Lennard-Jones cluster using the combined method are smaller than parallel tempering when used alone.

Broken-symmetry unrestricted hybrid density functional calculations on nickel dimer and nickel hydride.

In the present work we investigate the adequacy of broken-symmetry unrestricted density functional theory for constructing the potential energy curve of nickel dimer and nickel hydride, as a model for larger bare and hydrogenated nickel cluster calculations. We use three hybrid functionals: the popular B3LYP, Becke's newest optimized functional Becke98, and the simple FSLYP functional (50% Hartree-Fock and 50% Slater exchange and LYP gradient-corrected correlation functional) with two basis sets: all-electron (AE) Wachters+f basis set and Stuttgart RSC effective core potential (ECP) and basis set. We find that, overall, the best agreement with experiment, comparable to that of the high-level CASPT2, is obtained with B3LYP/AE, closely followed by Becke98/AE and Becke98/ECP. FSLYP/AE and B3LYP/ECP give slightly worse agreement with experiment, and FSLYP/ECP is the only method among the ones we studied that gives an unacceptably large error, underestimating the dissociation energy of Ni(2) by 28%, and being in the largest disagreement with the experiment and the other theoretical predictions. We also find that for Ni(2), the spin projection for the broken-symmetry unrestricted singlet states changes the ordering of the states, but the splittings are less than 10 meV. All our calculations predict a deltadelta-hole ground state for Ni(2) and delta-hole ground state for NiH. Upon spin projection of the singlet state of Ni(2), almost all of our calculations: Becke98 and FSLYP both AE and ECP and B3LYP/AE predict (1)(d(A)(x(2)-y(2)d(B)(x(2)-y(2)) or (1)(d(A)(xy) (d)(B)(xy)) ground state, which is a mixture of (1)Sigma(g) (+) and (1)Gamma(g). B3LYP/ECP predicts a (3)(d(A)(x(2)-y(2))d(B)(xy) (mixture of (3)Sigma(g) (-) and (3)Gamma(u)) ground state virtually degenerate with the (1)(d(A)(x(2)-y(2)d(B)(x)(2)-y(2)/(1)(d(A)(xy)D(B)(xy) state. The doublet delta-hole ground state of NiH predicted by all our calculations is in agreement with the experimentally predicted (2)Delta ground state. For Ni(2), all our results are consistent with the experimentally predicted ground state of 0(g) (+) (a mixture of (1)Sigma(g) (+) and (3)Sigma(g) (-)) or 0(u) (-) (a mixture of (1)Sigma(u) (-) and (3)Sigma(u) (+)).

On the encapsulation of nickel clusters by molecular nitrogen.

The structures and energetic effects of molecular nitrogen adsorbates on nickel clusters are investigated using an extended Huckel model coupled with two models of the adsorbate-nickel interaction. The potential parameters for the adsorbates are chosen to mimic experimental information about the binding strength of nitrogen on both cluster and bulk surface phases of nickel. The first model potential is a simple Lennard-Jones interaction that leads to binding sites in holes defined by sets of near-neighbor nickel atoms. The second model potential has a simple three-body form that forces the model nitrogen adsorbates to bind directly to single nickel atoms. Significant rearrangement of the core nickel structures are found in both models. A disconnectivity graph analysis of the potential energy surfaces implies that the rearrangements arise from low transition state barriers and the small differences between available isomers in the nickel core.

Phase changes in selected Lennard-Jones X13-nYn clusters.

Detailed studies of the thermodynamic properties of selected binary Lennard-Jones clusters of the type X13-nYn (where n=1, 2, 3) are presented. The total energy, heat capacity, and first derivative of the heat capacity as a function of temperature are calculated by using the classical and path integral Monte Carlo methods combined with the parallel tempering technique. A modification in the phase change phenomena from the presence of impurity atoms and quantum effects is investigated.

Taming the rugged landscape: production, reordering, and stabilization of selected cluster inherent structures in the X13-nYn system.

We present studies of the potential energy landscape of selected binary Lennard-Jones 13 atom clusters. The effect of adding selected impurity atoms to a homogeneous cluster is explored. We analyze the energy landscapes of the studied systems using disconnectivity graphs. The required inherent structures and transition states for the construction of disconnectivity graphs are found by combination of conjugate gradient and eigenvector-following methods. We show that it is possible to controllably induce new structures as well as reorder and stabilize existing structures that are characteristic of higher-lying minima. Moreover, it is shown that the selected structures can have experimentally relevant lifetimes.