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arXiv:2411.00310 · The Journal of Chemical Physics

Pair approximating the action for molecular rotations in path integral Monte Carlo

Muhammad Shaeer Moeed, Tobias Serwatka, Pierre–Nicholas Roy

Published Jan 9, 2025• 3 citations
arXivHTMLPDF

Abstract

Typical path integral Monte Carlo approaches use the primitive approximation to compute the probability density for a given path. In this work, we develop the pair discrete variable representation (pair-DVR) approach to study molecular rotations. The pair propagator, which was initially introduced to study superfluidity in condensed helium, is naturally well-suited for systems interacting with a pairwise potential. Consequently, paths sampled using the pair action tend to be closer to the exact paths (compared to primitive Trotter paths) for such systems leading to convergence with less imaginary time steps. Our approach relies on using the pair factorization approach in conjunction with a discretized path integral ground state paradigm to study a chain of planar rotors interacting with a pairwise dipole interaction. We first use the Wigner-Kirkwood density expansion to analyze the asymptotics of the pair propagator in imaginary time. Then, we exhibit the utility of the pair factorization scheme via convergence studies comparing the pair and primitive propagators. Finally, we compute energetic and structural properties of this system including the correlation function and Binder ratio as functions of the coupling strength to examine the behavior of the pair-DVR method near criticality. The density matrix renormalization group results are used for benchmarking throughout.

DOI
10.1063/5.0246327

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Typical path integral Monte Carlo approaches use the primitive approximation to compute the probability density for a given path. In this work, we develop the pair discrete variable representation (pair-DVR) approach to study molecular rotations. The pair propagator, which was initially introduced to study superfluidity in condensed helium, is naturally well-suited for systems interacting with a pairwise potential. Consequently, paths sampled using the pair action tend to be closer to the exact paths (compared to primitive Trotter paths) for such systems leading to convergence with less imaginary time steps. Our approach relies on using the pair factorization approach in conjunction with a discretized path integral ground state paradigm to study a chain of planar rotors interacting with a pairwise dipole interaction. We first use the Wigner-Kirkwood density expansion to analyze the asymptotics of the pair propagator in imaginary time. Then, we exhibit the utility of the pair factorization scheme via convergence studies comparing the pair and primitive propagators. Finally, we compute energetic and structural properties of this system including the correlation function and Binder ratio as functions of the coupling strength to examine the behavior of the pair-DVR method near criticality. The density matrix renormalization group results are used for benchmarking throughout.
We introduce a path integral Monte Carlo approach that uses the angular momentum representation for the description of interacting rotor systems. Such a choice of representation allows the calculation of momentum properties without having to break the paths. The discrete nature of the momentum basis also allows the use of rejection-free Gibbs sampling techniques. To illustrate the method, we study the collective behavior of N confined planar rotors with dipole-dipole interactions, a system known to exhibit a quantum phase transition from a disordered to an ordered state at zero temperature. Ground state properties are obtained using the path integral ground state method. We propose a Bond-Hamiltonian decomposition for the high temperature density matrix factorization of the imaginary time propagator. We show that cluster-loop type moves are necessary to overcome ergodicity issues and to achieve efficient Markov chain updates. Ground state energies and angular momentum properties are computed and compared with density matrix renormalization group benchmark results. In particular, the derivative of the kinetic energy with respect to the interaction strength estimator is presented as a successful order parameter for the detection of the quantum phase transition.
In this work, we propose a path integral Monte Carlo approach based on discretized continuous degrees of freedom and rejection-free Gibbs sampling. The ground state properties of a chain of planar rotors with dipole-dipole interactions are used to illustrate the approach. Energetic and structural properties are computed and compared to exact diagonalization and numerical matrix multiplication for N ≤ 3 to assess the systematic Trotter factorization error convergence. For larger chains with up to N = 100 rotors, Density Matrix Renormalization Group calculations are used as a benchmark. We show that using Gibbs sampling is advantageous compared to traditional Metropolis-Hastings rejection importance sampling. Indeed, Gibbs sampling leads to lower variance and correlation in the computed observables.
This thesis describes several novel approaches in quantum molecular dynamics for obtaining properties of molecular systems in different regimes. We investigate ground state properties of chains of linear rotors with dipole–dipole interactions via the density matrix renormalization group (DMRG), by deriving the appropriate form of the interaction operator and implementing it in ITensor. This provides us with further evidence of a quantum phase transition in this system. We also improve the sampling of Gaussian mixture distributions for finite temperature path integral Monte Carlo (PIMC) of vibronic Hamiltonians. To do this, we replace random sampling by quasi-random sampling, and improve sampling distributions by optimizing their parameters. Finally, we introduce estimators and integrators for constrained free energy simulations in path integral molecular dynamics (PIMD). This method is applied to the study of a water dimer, for which we obtain a quantum potential of mean force.
We present a numerical implementation of the quantum chemistry density matrix renormalization group (DMRG) using the hybrid discrete variable representation (DVR)/Gaussian basis set. The z -axis of real space is discretized by a DVR basis set and each transversal plane is described by the eigenstates of the transversal core Hamiltonian, represented in a set of primitive Gaussian basis functions. Such a hybrid basis can reduce the computation of two-electron repulsion integrals. One main advantage of using the local DVR basis sets over finite-difference methods for real space discretization is that the kinetic energy operator matrix elements can be computed exactly. This method is first applied to a one-dimensional pseudo-hydrogen chain under screened Coulomb potential and then to a realistic hydrogen chain, whereby the DMRG/DVR results are comparable in accuracy to the full configuration interaction results.
We present a numerical implementation of the density matrix renormalization group (DMRG) using the discrete variable representation (DVR) basis set. One main advantage of using the local DVR basis sets is that the computations of one-electron integral and two-electron repulsion integrals are drastically simplified. For comparison, we further implemented DVR complete active space configuration interaction (CASCI) using canonical molecular orbitals. These methods are applied to a one-dimensional pseudo-hydrogen chain under screened Coulomb potential. The DMRG ground state energy agrees with CASCI up to 0.1 mEh with a very small number of bond dimensions.
Abstract The focus of Chapter 12 is the Feynman path integral formulations of quantum mechanics and quantum statistical mechanics. The connection between real-time quantum propagator and the density matrix is established via the Wick rotation, and a Trotter expansion is used to derive discrete path integral expressions for both. The continuous limit of the discrete path integral is derived to yield the functional integral formulation of the path integral. The construction of estimators and their averages for computing quantum observables is discussed. Next, the path integral is extended to incorporate Boltzmann, Fermi-Dirac, and Bose-Einstein statistics, and a useful recursive formula for the partition function is derived that accelerates convergence by building up N-body partition functions from those of smaller systems. Quantum free energies are discussed, and algorithms for the evaluation of path integrals using molecular dynamics or Monte Carlo are presented. The chapter concludes with a discussion of higher-order factorizations and ring-polymer contraction methods to reduce the computational overhead of path-integral calculations