E. Curotto

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.

Thermodynamic properties of ammonia clusters (NH3)n n=2–11: Comparing classical and quantum simulation results for hydrogen bonded species

Classical and quantum simulations of ammonia clusters in the dimer through the hendecamer range are performed using the stereographic projection path integral. Employing the most recent polarizable potential to describe intermolecular interactions, energetic and structural data obtained with our simulations provide support for a more fluxional or flexible nature at low temperature of the ammonia dimer, pentamer, and hexamer than in the other investigated species. The octamer and the hendecamer display a relatively strong melting peak in the classical heat capacity and a less intense but significant melting peak in the quantum heat capacity. The latter are shifted to lower temperature (roughly 15 and 40 K lower, respectively) by the quantum effects. The features present in both classical and quantum constant volume heat capacity are interpreted as an indication of melting even in the octamer case, where a large energy gap is present between its global minimum and second most stable species. We develop a first order finite difference algorithm to integrate the geodesic equations in the inertia ellipsoid generated by n rigid nonlinear bodies mapped with stereographic projections. We use the technique to optimize configurations and to explore the potential surface of the hendecamer.

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.

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.

Quantum simulations of the hydrogen molecule on ammonia clusters

Mixed ammonia-hydrogen molecule clusters [H2-(NH3)n] have been studied with the aim of exploring the quantitative importance of the H2 quantum motion in defining their structure and energetics. Minimum energy structures have been obtained employing genetic algorithm-based optimization methods in conjunction with accurate pair potentials for NH3-NH3 and H2-NH3. These include both a full 5D potential and a spherically averaged reduced surface mimicking the presence of a para-H2. All the putative global minima for n ≥ 7 are characterized by H2 being adsorbed onto a rhomboidal ammonia tetramer motif formed by two double donor and two double acceptor ammonia molecules. In a few cases, the choice of specific rhombus seems to be directed by the vicinity of an ammonia ad-molecule. Diffusion Monte Carlo simulations on a subset of the species obtained highlighted important quantum effects in defining the H2 surface distribution, often resulting in populating rhomboidal sites different from the global minimum one, and showing a compelling correlation between local geometrical features and the relative stability of surface H2. Clathrate-like species have also been studied and suggested to be metastable over a broad range of conditions if formed.

On the convergence of diffusion Monte Carlo in non-Euclidean spaces. II. Diffusion with sources and sinks

We test the second order Milstein method adapted to simulate diffusion in general compact Riemann manifolds on a number of systems characterized by nonconfining potential energy surfaces of increasing complexity. For the 2-sphere and more complex spaces derived from it, we compare the Milstein method with a number of other first and second order approaches. In each case tested, we find evidence that demonstrate the versatility and relative ease of implementation of the Milstein method derived in Part I.

On the convergence of diffusion Monte Carlo in non-Euclidean spaces. I. Free diffusion

We develop a set of diffusion Monte Carlo algorithms for general compactly supported Riemannian manifolds that converge weakly to second order with respect to the time step. The approaches are designed to work for cases that include non-orthogonal coordinate systems, nonuniform metric tensors, manifold boundaries, and multiply connected spaces. The methods do not require specially designed coordinate charts and can in principle work with atlases of charts. Several numerical tests for free diffusion in compactly supported Riemannian manifolds are carried out for spaces relevant to the chemical physics community. These include the circle, the 2-sphere, and the ellipsoid of inertia mapped with traditional angles. In all cases, we observe second order convergence, and in the case of the sphere, we gain insight into the function of the advection term that is generated by the curved nature of the space.

Stochastic Simulations of Clusters : Quantum Methods in Flat and Curved Spaces

FUNDAMENTALS FORTRAN Essentials Introduction What Is FORTRAN? FORTRAN Basics Data Types The IMPLICIT Statement Initialization and Assignment Order of Operations Column Position Rules A Typical Chemistry Problem Solved with FORTRAN Free Format I/O The FORTRAN Code for the Tertiary Mixtures Problem Conditional Execution Loops Intrinsic Functions User-Defined Functions Subroutines Numerical Derivatives The Extended Trapezoid Rule to Evaluate Integrals Basics of Classical Dynamics Introduction Some Important Variables of Classical Physics The Lagrangian and the Hamiltonian in Euclidean Spaces The Least Action Principle and the Equations of Motion The Two-Body Problem with Central Potential Isotropic Potentials and the Two-Body Problem The Rigid Rotor Numerical Integration Methods Hamiltons Equations and Symplectic Integrators The Potential Energy Surface Dissipative Systems The Fourier Transform and the Position Autocorrelation Function Basics of Stochastic Computations Introduction Continuous Random Variables and Their Distributions The Classical Statistical Mechanics of a Single Particle The Monoatomic Ideal Gas The Equipartition Theorem Basics of Stochastic Computation Probability Distributions Minimizing V by Trial and Error The Metropolis Algorithm Parallel Tempering A Random Number Generator Vector Spaces, Groups, and Algebras Introduction A Few Useful Definitions Groups Number Fields Vector Spaces Algebras The Exponential Mapping of Lie Algebras The Determinant of a n x n Matrix and the Levi-Civita Symbol Scalar Product, Outer Product, and Vector Space Mapping Rotations in Euclidean Space Complex Field Extensions Dirac Bra-Ket Notation Eigensystems The Connection between Diagonalization and Lie Algebras Symplectic Lie Algebras and Groups Lie Groups as Solutions of Differential Equations Split Symplectic Integrators Supermatrices and Superalgebras Matrix Quantum Mechanics Introduction The Failures of Classical Physics Spectroscopy The Heat Capacity of Solids at Low Temperature The Photoelectric Effect Black Body Radiator The Beginning of the Quantum Revolution Modern Quantum Theory and Schrodingers Equation Matrix Quantum Mechanics The Monodimensional Hamiltonian in a Simple Hilbert Space Numerical Solution Issues in Vector Spaces The Harmonic Oscillator in Hilbert Space A Simple Discrete Variable Representation (DVR) Accelerating the Convergence of the Simple DVR Elements of Sparse Matrix Technology The Gram-Schmidt Process The Krylov Space The Row Representation of a Sparse Matrix The Lanczos Algorithm Orbital Angular Momentum and the Spherical Harmonics Complete Sets of Commuting Observables The Addition of Angular Momentum Vectors Computation of the Vector Coupling Coefficients Matrix Elements of Anisotropic Potentials in the Angular Momentum Basis The Physical Rigid Dipole in a Constant Electric Field Time Evolution in Quantum Mechanics Introduction The Time-Dependent Schrodinger Equation Wavepackets, Measurements, and Time Propagation of Wavepackets The Time Evolution Operator The Dyson Series and the Time-Ordered Exponential Representation The Magnus Expansion The Trotter Factorization The time_evolution_operator Program Feynmans Path Integral Quantum Monte Carlo A Variational Monte Carlo Method for Importance Sampling Diffusion Monte Carlo (IS-DMC) IS-DMC with Drift Greens Function Diffusion Monte Carlo The Path Integral in Euclidean Spaces Introduction The Harmonic Oscillator Classical Canonical Average Energy and Heat Capacity Quantum Canonical Average Energy and Heat Capacity The Path Integral in Rd The Canonical Fourier Path Integral The Reweighted Fourier-Wiener Path Integral ATOMIC CLUSTERS Characterization of the Potential of Ar7 Introduction Cartesian Coordinates of Atomic Clusters Rotations and Translations The Center of Mass The Inertia Tensor The Structural Comparison Algorithm Gradients and Hessians of Multidimensional Potentials The Lennard-Jones Potential V(LJ) The Gradient of V(LJ) Brownian Dynamics at 0 K Basin Hopping The Genetic Algorithm The Hessian Matrix Normal Mode Analysis Transition States with the Cerjan-Miller Algorithm Optical Activity Classical and Quantum Simulations of Ar7 Introduction Simulation Techniques: Parallel Tempering Revisited Thermodynamic Properties of a Cluster with n Atoms The Program parallel_tempering_r3n.f The Variational Ground State Energy Diffusion Monte Carlo (DMC) of Atomic Clusters Path Integral Simulations of Ar7 Characterization Techniques: The Lindemann Index Characterization Techniques: Bond Orientational Parameters Characterization Techniques: Structural Comparison Appendices METHODS IN CURVED SPACES Introduction to Differential Geometry Introduction Coordinate Changes: Einsteins Sum Convention and the Metric Tensor Contravariant Tensors Gradients as 1-Forms Tensors of Higher Ranks The Metric Tensor of a Space Integration on Manifolds Stereographic Projections Dynamics in Manifolds The Hessian Metric The Christofell Connections and the Geodesic Equations The Laplace-Beltrami Operator The Riemann-Cartan Curvature Scalar The Two-Body Problem Revisited Stereographic Projections for the Two-Body Problem The Rigid Rotor and the Infinitely Stiff Spring Constant Limit Relative Coordinates for the Three-Body Problem The Rigid-Body Problem and the Body Fixed Frame Stereographic Projections for the Ellipsoid of Inertia The Spherical Top The Riemann Curvature Scalar for a Spherical Top Coefficients and the Curvature for Spherical Tops with Stereographic Projection Coordinates (SPCs) The Riemann Curvature Scalar for a Symmetric Nonspherical Top A Split Operator for Symplectic Integrators in Curved Manifolds The Verlet Algorithm for Manifolds Simulations in Curved Manifolds Introduction The Invariance of the Phase Space Volume Variational Ground States DMC in Manifolds The Path Integral in Space-Like Curved Manifolds The Virial Estimator for the Total Energy Angular Momentum Theory Solution for a Particle in S2 Variational Ground State for S2 DMC in S2 Stereographic Projection Path Integral in S2 Higher Dimensional Tops The Free Particle in a Ring The Particle in a Ring Subject to Smooth Potentials APPLICATIONS TO MOLECULAR SYSTEMS Clusters of Rigid Tops Introduction The Stockmayer Model The Map for R3n (S2)n The Gradient of the Lennard-Jones Dipole-Dipole (LJDD) Potential Beyond the Stockmayer Model for Rigid Linear Tops The Hessian Metric Tensor on R3n (S2)n Reweighted Random Series Action for Clusters of Linear Rigid Tops The Local Energy Estimator for Clusters of Linear Rigid Tops Clusters of Rigid Nonlinear Tops Coordinate Transformations for R3n n The Hessian Metric Tensor for R3n n Local Energy and Action for R3n n Concluding Remarks Bibliography Index

Infinite swapping in curved spaces

We develop an extension of the infinite swapping and partial infinite swapping techniques [N. Plattner, J. D. Doll, P. Dupuis, H. Wang, Y. Liu, and J. E. Gubernatis, J. Chem. Phys. 135, 134111 (2011)] to curved spaces. Furthermore, we test the performance of infinite swapping and partial infinite swapping in a series of flat spaces characterized by the same potential energy surface model. We develop a second order variational algorithm for general curved spaces without the extended Lagrangian formalism to include holonomic constraints. We test the new methods by carrying out NVT classical ensemble simulations on a set of multidimensional toroids mapped by stereographic projections and characterized by a potential energy surface built from a linear combination of decoupled double wells shaped purposely to create rare events over a range of temperatures.

Quest for Inexpensive Hydrogen Isotopic Fractionation: Do We Need 2D Quantum Confining in Porous Materials or Are Rough Surfaces Enough? The Case of Ammonia Nanoclusters

We study the adsorption energetics and quantum properties of the molecular hydrogen isotopes H2, D2, and T2 onto the surface of rigid ammonia nanoclusters with quantum simulations and accurate model potential energy surfaces (PES). A highly efficient diffusion Monte Carlo (DMC) algorithm for rigid rotors allowed us to accurately define zero-point adsorption energies for the three isotopes, as well as the degree of translational and rotational delocalization that each affords on the surface. From the data emerges that the quantum adsorption energy (Eads) of T2 can be up to twice the one of H2 at 0 K, suggesting the possibility of exploiting some form of solid ammonia to selectivity separate hydrogen isotopes at low temperatures (≃20 K). This is discussed by focusing on the structural motif that may be more effective for the task. The analysis of the contributions to Eads, however, surprisingly indicates that the average kinetic energy (Ekin) and rotation energy (Erotkin) of T2 can also be, respectively, 2 times and 20 times higher than those of H2; this finding markedly deviates from what is predicted for hydrogen molecules inside carbon nanotubes (CNT) or metallic-organic frameworks (MOF), where Ekin and Erotkin is higher for H2 due to the unavoidable effects of confinement and hindrance to its rotational motion. The rationale for these differences is provided by the geometrical distributions for the rigid rotors, which reveal an increasingly stronger coupling between rotational and translational degrees of freedom upon increasing the isotopic mass. This effect has never been observed before on adsorbing surfaces (e.g., graphite) and is induced by a strongly anisotropic and anharmonic bowl-like potential experienced by the rotors.