Partial averaging and the centroid virial estimator for stereographic projection path-integral simulations in curved spaces

We develop and test three different partial averaging theories for the stereographic projection path integral in curved spaces. Additionally, we adapt and test the centroid virial estimator for the kinetic energy in curved spaces. We tested both a confining as well as a nonconfining potential for the particle in a ring. All three partial averaging theories are demonstrated to converge linearly in the asymptotic region with k(-2)max, the number of Fourier coefficients. We use an error estimator to determine the optimal parameters for an extrapolation to infinite kmax. We verify that the centroid virial estimator (derived from the primitive DeWitt path-integral formula) converges to the kinetic energy for both confining and nonconfining potentials.     

Stereographic projection path-integral simulations of (HF)n clusters

We perform several quantum canonical ensemble simulations of (HF)(n) clusters. The HF stretches are rigid, and the stereographic projection path-integral method is employed for the simulation in the resulting curved configuration space. We make use of the reweighted random series techniques to accelerate the convergence of the path-integral simulation with respect to the number of path coefficients. We develop and test estimators for the total energy and heat capacity based on a finite difference approach for non-Euclidean spaces. The quantum effects at temperatures below 400 K are substantial for all sizes. We observe interesting thermodynamic behaviors in the quantum simulations of the octamer and the heptamer.     

Function extended spaces

Extended function spaces defined over the real field are defined as vector spaces made by the Cartesian product of a real Euclidian space and a real function space. This construct is related to the Holographic Electronic Density Theorem and to the stereographic projection of quantum chemically related and well behaved functions in general. It permits to establish the basis for the Holographic General Function Theorem.     

Stereographic projection diffusion monte carlo (SPDMC) algorithms for molecular condensed matter

We develop and test three algorithms for diffusion Monte Carlo simulations in non-Euclidean manifolds. The methods are based on the construction of the "velocity" distribution by rejection techniques and are capable of functioning in a broad class of non-Euclidean spaces generated by holonomic constraints. The formulation of the propagator for non-Euclidean manifolds avoids the use of Lagrange multipliers; it is derived instead from the Feynman quantization in manifolds proposed by DeWitt. The manifolds are mapped onto Rd by using stereographic projection coordinates. Numerical tests are conducted for the particle in a ring of unit radius subjected to a sinusoidal potential, for the electron in the field of an infinitely massive proton, and for a water molecule modeled as an asymmetric top subjected to an external field.     

Quantum Monte Carlo simulations of selected ammonia clusters (n = 2-5): isotope effects on the ground state of typical hydrogen bonded systems

Variational Monte Carlo, diffusion Monte Carlo, and stereographic projection path integral simulations are performed on eight selected species from the (NH(3))(n), (ND(3))(n), (NH(2)D)(n), and (NH(3))(n-1)(ND(3)) clusters. Each monomer is treated as a rigid body with the rotation spaces mapped by the stereographic projection coordinates. We compare the energy obtained from path integral simulations at several low temperatures with those obtained by diffusion Monte Carlo, for two dimers, and we find that at 4 K, the fully deuterated dimer energy is in excellent agreement with the ground state energy of the same. The ground state wavefunction for the (NH(3))(2-5) clusters is predominantly localized in the global minimum of the potential energy. In all simulations of mixed isotopic substitutions, we find that the heavier isotope is almost exclusively the participant in the hydrogen bond.     

Stereographic projections path integral for inertia ellipsoids: applications to Arn-HF clusters

The DeWitt formula for inertia ellipsoids mapped by stereographic projection coordinates is developed. We discover that by remapping the quaternion parameter space with stereographic projections, considerable simplification of the differential geometry for the inertia ellipsoid with spherical symmetry takes place. The metric tensor is diagonal and contains only one independent element in that case. We find no difficulties testing and implementing the DeWitt formula for the inertia ellipsoids of asymmetric tops mapped by stereographic projections. The path integral algorithm for the treatment of Rm x S2 manifolds based on a mixture of Cartesian and stereographic projection coordinates is tested for small Arn-HF clusters in the n = 2 to n = 5 range. In particular, we determine the quantum effects of the red shift and the isomerization patterns at finite temperatures. Our findings are consistent with previously reported computations and experimental data for small Arn-HF clusters.     

Comparison of approximate quantum simulation methods applied to normal liquid helium at 4 K

The Feynman-Kleinert linearized path integral molecular dynamics (FK-LPI), ring polymer molecular dynamics (RPMD), and centroid molecular dynamics (CMD) methods are applied to the simulation of normal liquid helium. Comparisons of the simulation results at the T = 4 K and rho = 0.01873 A-3 state point are presented. The calculated quantum correlation functions for the three methods show significant differences, both in the short time and in the intermediate regions of the spectrum. Our simulation results are also compared to the recent results of other approximate quantum simulation methods. We find that FK-LPI qualitatively agrees with other approximate quantum simulation results while CMD and RPMD predict a qualitatively different impulsive rebound in the velocity autocorrelation function. Frequency space analysis reveals that RPMD exhibits a broad high-frequency tail similar to that from quantum mode coupling theory and numerical analytic continuation approaches, while FK-LPI provides a somewhat more rapid decay at high frequency than any of these three methods. CMD manifests a high-frequency component that is greatly reduced compared with the other methods.     

Importance sampling for quantum Monte Carlo in manifolds: addressing the time scale problem in simulations of molecular aggregates

Several importance sampling strategies are developed and tested for stereographic projection diffusion Monte Carlo in manifolds. We test a family of one parameter trial wavefunctions for variational Monte Carlo in stereographically projected manifolds which can be used to produce importance sampling. We use the double well potential in one dimensional Euclidean space to study systematically sampling issues for diffusion Monte Carlo. We find that diffusion Monte Carlo with importance sampling in manifolds is orders of magnitude more efficient compared to unguided diffusion Monte Carlo. Additionally, diffusion Monte Carlo with importance sampling in manifolds can overcome problems with nonconfining potentials and can suppress quasiergodicity effectively. We obtain the ground state energy and the wavefunction for the Stokmayer trimer.     

Parameter space minimization methods: applications to Lennard-Jones-dipole-dipole clusters

The morphology of the uniform Lennard-Jones-dipole-dipole cluster with 13 centers (LJDD)13 is investigated over a relatively wide range of values of the dipole moment. We introduce and compare several necessary modifications of the basin-hopping algorithm for global optimization to improve its efficiency. We develop a general algorithm for T=0 Brownian dynamics in curved spaces, and a graph theoretical approach necessary for the elimination of dissociated states. We find that the (LJDD)13 cluster has icosahedral symmetry for small to moderate values of the dipole moment. As the dipole moment increases, however, its morphology shifts to an hexagonal antiprism, and eventually to a ring.     

Replica sub-permutation method for molecular dynamics and monte carlo simulations

We propose an improvement of the replica-exchange and replica-permutation methods, which we call the replica sub-permutation method (RSPM). Instead of considering all permutations, this method uses a new algorithm referred to as sub-permutation to perform parameter transition. The RSPM succeeds in reducing the number of combinations between replicas and parameters without the loss of sampling efficiency. For comparison, we applied the replica sub-permutation, replica-permutation, and replica-exchange methods to a β-hairpin mini protein, chignolin, in explicit water. We calculated the transition ratio and number of tunneling events in the parameter space, the number of folding–unfolding events, the autocorrelation function, and the autocorrelation time as measures of sampling efficiency. The results indicate that among the three methods, the proposed RSPM is the most efficient in both parameter and conformational spaces. © 2019 Wiley Periodicals, Inc.     

Stereographic projection path integral simulations of (HCl)n clusters (n=2-5): evidence of quantum induced melting in small hydrogen bonded networks

We investigate the quantum thermodynamic properties of small (HCl)(n) clusters using stereographic projection path integral simulations. The HCl stretches are rigid, the orientations are mapped with stereographic projection coordinates, and we make use of the reweighted random series techniques to obtain cubic convergence with respect to the number of path coefficients. Path integral simulations are converged at and above 10 K for the pentamer and above 15 K for the dimer and the trimer. None of the systems display a melting feature in the classical limit. We find an evidence of quantum induced melting between 15 and 45 K.     

A state-specific partially internally contracted multireference coupled cluster approach

A state-specific partially internally contracted multireference coupled cluster approach is presented for general complete active spaces with arbitrary number of active electrons. The dominant dynamical correlation is included via an exponential parametrization of internally contracted cluster operators ( ?T) which excite electrons from a multideterminantal reference function. The remaining dynamical correlation and relaxation effects are included via a diagonalization of the transformed Hamiltonian ?? =e(- ?T)H?e( ?T) in the multireference configuration interaction singles space in an uncontracted fashion. A new set of residual equations for determining the internally contracted cluster amplitudes is proposed. The second quantized matrix elements of ?? , expressed using the extended normal ordering of Kutzelnigg and Mukherjee, are used as the residual equations without projection onto the excited configurations. These residual equations, referred to as the many-body residuals, do not have any near-singularity and thus, should allow one to solve all the amplitudes without discarding any. There are some relatively minor remaining convergence issues that may arise from an attempt to solve all the amplitudes and an initial analysis is provided in this paper. Applications to the bond-stretching potential energy surfaces for N(2), CO, and the low-lying electronic states of C(2) indicate clear improvements of the results using the many-body residuals over the conventional projected residual equations.     

Analytical calculation of the solvent-accessible surface area and its nuclear gradients by stereographic projection: A general approach for molecules,polymers, nanotubes,helices, and surfaces

In this article, we explore an alternative to the analytical Gauss–Bonnet approach for computing the solvent-accessible surface area (SASA) and its nuclear gradients. These two key quantities are required to evaluate the nonelectrostatic contribution to the solvation energy and its nuclear gradients in implicit solvation models. We extend a previously proposed analytical approach for finite systems based on the stereographic projection technique to infinite periodic systems such as polymers, nanotubes, helices, or surfaces and detail its implementation in the Crystal code. We provide the full derivation of the SASA nuclear gradients, and introduce an iterative perturbation scheme of the atomic coordinates to stabilize the gradients calculation for certain difficult symmetric systems. An excellent agreement of computed SASA with reference analytical values is found for finite systems, while the SASA size-extensivity is verified for infinite periodic systems. In addition, correctness of the analytical gradients is confirmed by the excellent agreement obtained with numerical gradients and by the translational invariance achieved, both for finite and infinite periodic systems. Overall therefore, the stereographic projection approach appears as a general, simple, and efficient technique to compute the key quantities required for the calculation of the nonelectrostatic contribution to the solvation energy and its nuclear gradients in implicit solvation models applicable to both finite and infinite periodic systems.     

On the short-time limit of ring polymer molecular dynamics

We examine the short-time accuracy of a class of approximate quantum dynamical techniques that includes the centroid molecular dynamics (CMD) and ring polymer molecular dynamics (RPMD) methods. Both of these methods are based on the path integral molecular dynamics (PIMD) technique for calculating the exact static equilibrium properties of quantum mechanical systems. For Kubo-transformed real-time correlation functions involving operators that are linear functions of positions or momenta, the RPMD and (adiabatic) CMD approximations differ only in the choice of the artificial mass matrix of the system of ring polymer beads that is employed in PIMD. The obvious ansatz for a general method of this type is therefore to regard the elements of the PIMD (or Parrinello-Rahman) mass matrix as an adjustable set of parameters that can be chosen to improve the accuracy of the resulting approximation. We show here that this ansatz leads uniquely to the RPMD approximation when the criterion that is used to select the mass matrix is the short-time accuracy of the Kubo-transformed correlation function. In particular, we show that the leading error in the RPMD position autocorrelation function is O(t(8)) and the error in the velocity autocorrelation function is O(t(6)), for a general anharmonic potential. The corresponding errors in the CMD approximation are O(t(6)) and O(t(4)), respectively.     

Investigation of the possibility of geometrical electrophoresis

We investigate the possibility of geometrical electrophoresis, which is based on nanofabrication techniques. (GEE) utilizes geometrical effects during electrophoresis, which are generated by physical interactions between walls and a macromolecule confined in spaces smaller than the Flory radius. When a polymer is injected into a small space, confinement energy is usually required. However, the confinement energy form depends on the geometry of the space. In the case of electrophoresis, the electric field itself changes depending on the geometry. Using a nanofabricated quartz chip with a curved channel, we investigated electrophoretic behavior of high molecular weight DNA based on the curvature effect.     

Quantum monte carlo methods for constrained systems

The torsional ground state for ethane, the torsional, rotational, and mixed torsional and rotational ground state of propane are computed with a version of diffusion Monte Carlo adapted to handle the geometric complexity of curved spaces such as the Ramachandra space. The quantum NVT ensemble average for the mixed torsional and rotational degrees of freedom of propane is computed, using a version of Monte Carlo path integral, also adapted to handle curved spaces. These three problems are selected to demonstrate the generality and the applicability of the approaches described. The spaces of coordinates can be best constructed from the parameters of continuous Lie groups, and alternative methods based on vector spaces, where extended Lagrangian terms would be too cumbersome to implement. We note that the geometric coupling between the torsions and the rotations of propane produces a substantial effect on the ground state energy of propane, and that the quantum effects on the energy of propane are quite large even well above room temperature. © 2014 Wiley Periodicals, Inc.     

Interactions between nanocolloidal particles in polymer solutions: effect of attractive interactions

We present a density-functional theory study of nanoparticle interactions in a concentrated polymer solution. The polymers are modeled as freely jointed tangent chains; all nonbonded interactions between polymer segments and nanoparticles are described by Lennard-Jones potentials. We test several recently proposed methods of treating attractive interactions within the density-functional theory framework by comparing theoretical results with recent simulation data. We find that the simple van der Waals approach provides the most accurate results for the polymer-mediated potential of mean force between two dilute nanoparticles. We employ this approach to study nanoparticle interactions as a function of nanoparticle-segment interaction strength and polymer solution density and temperature.     

Comparative study of cluster Ag17Cu2 by instantaneous normal mode analysis and by isothermal Brownian-type molecular dynamics simulation

We perform isothermal Brownian-type molecular dynamics simulations to obtain the velocity autocorrelation function and its time Fourier-transformed power spectral density for the metallic cluster Ag(17)Cu(2). The temperature dependences of these dynamical quantities from T = 0 to 1500 K were examined and across this temperature range the cluster melting temperature T(m), which we define to be the principal maximum position of the specific heat is determined. The instantaneous normal mode analysis is then used to dissect the cluster dynamics by calculating the vibrational instantaneous normal mode density of states and hence its frequency integrated value I(j) which is an ensemble average of all vibrational projection operators for the jth atom in the cluster. In addition to comparing the results with simulation data, we look more closely at the entities I(j) of all atoms using the point group symmetry and diagnose their temperature variations. We find that I(j) exhibit features that may be used to deduce T(m), which turns out to agree very well with those inferred from the power spectral density and specific heat.     

Applications of molecular dynamics simulations coupled with Harris functional approximation to argon

We have demonstrated molecular dynamics simulations using a combination of the classical molecular dynamics with density functional theory for argon clusters. Three different molecular dynamics schemes, which differ in their treatment of the potential energy and forces, have been carried out. The first uses a Lennard-Jones potential. In the second, the potential is computed using the Harris functional, and in the third, a combination of Lennard-Jones and Harris functional potentials is used. In addition to direct examination of the trajectories, the velocity autocorrelation function and its power spectrum have been computed to demonstrate the agreement between these three methods. The present studies show that a scheme that used a combination of model potentials and density functional theory provides a very useful tool for the dynamics simulation of systems that contain some fragments in which the analytical model potentials are not available. © 1994 John Wiley & Sons, Inc.