A Monte Carlo algorithm for computing quantum-mechanical expectation values of coordinate operators in many-body problems is presented. The algorithm, which relies on the forward walking method, fits naturally in a Green's function Monte Carlo calculation, i.e., it does not require side walks or a bilinear sampling method. Our method evidences stability regions large enough to accurately sample unbiased pure expectation values. The proposed algorithm yields accurate results when it is applied to test problems such as the hydrogen atom and the hydrogen molecule. An excellent description of several properties of a fully many-body problem such as liquid $^{4}\mathrm{He}$ at zero temperature is achieved.
A Monte Carlo algorithm for computing quantum-mechanical expectation values of coordinate operators in many-body problems is presented. The algorithm, which relies on the forward walking method, fits naturally in a Green's function Monte Carlo calculation, i.e., it does not require side walks or a bilinear sampling method. Our method evidences stability regions large enough to accurately sample unbiased pure expectation values. The proposed algorithm yields accurate results when it is applied to test problems such as the hydrogen atom and the hydrogen molecule. An excellent description of several properties of a fully many-body problem such as liquid 4He at zero temperature is achieved.
We propose a bilinear sampling algorithm in Green’s function Monte Carlo for expectation values of operators that do not commute with the Hamiltonian and for differences between eigenvalues of different Hamiltonians. The integral representations of the Schroedinger equations are transformed into two equations whose solution has the form ψa(x)t(x, y)ψb(y), where ψa and ψb are the wavefunctions for the two related systems and t(x, y) is a kernel chosen to couple x and y. The Monte Carlo process, with random walkers on the enlarged configuration space x ⊗ y, solves these equations by generating densities whose asymptotic form is the above bilinear distribution. With such a distribution, exact Monte Carlo estimators can be obtained for the expectation values of quantum operators and for energy differences. We present results of these methods applied to several test problems, including a model integral equation, and the hydrogen atom. Key words: algorithm; bilinear sampling; energy difference; Green’s function Monte Carlo; quantum expectations; random walk. 1 I.
Despite the proven utility of quantum Monte Carlo methods in addressing the quantum many-body problem, many important observables are difficult to calculate due to the presence of large, and sometimes divergent, statistical errors. The present state of the art allows the construction of renormalized estimators which result in finite variances, but which invariably include some systematic bias. We present a simple method for calculating unbiased expectation values of local operators in the diffusion quantum Monte Carlo method which is applicable to both bare and renormalized estimators, allowing the accurate calculation of important properties such as forces.
The objective of pure-sampling quantum Monte Carlo is to calculate physical properties that are independent of the importance sampling function being employed in the calculation, save for the mismatch of its nodal hypersurface with that of the exact wave function. To achieve this objective, we report a pure-sampling algorithm that combines features of forward walking methods of pure-sampling and reptation quantum Monte Carlo (RQMC). The new algorithm accurately samples properties from the mixed and pure distributions simultaneously in runs performed at a single set of time-steps, over which extrapolation to zero time-step is performed. In a detailed comparison, we found RQMC to be less efficient. It requires different sets of time-steps to accurately determine the energy and other properties, such as the dipole moment. We implement our algorithm by systematically increasing an algorithmic parameter until the properties converge to statistically equivalent values. As a proof in principle, we calculated the fixed-node energy, static α polarizability, and other one-electron expectation values for the ground-states of LiH and water molecules. These quantities are free from importance sampling bias, population control bias, time-step bias, extrapolation-model bias, and the finite-field approximation. We found excellent agreement with the accepted values for the energy and a variety of other properties for those systems.
The objective of pure-sampling quantum Monte Carlo is to calculate physical properties that are independent of the importance sampling function being employed in the calculation, save for the mismatch of its nodal hypersurface with that of the exact wave function. To achieve this objective, we describe a pure-sampling algorithm that combines features of forward-walking methods of pure-sampling and reptation quantum Monte Carlo. The importance sampling is performed by using a single-determinant basis set composed of Slater-type orbitals. We implement our algorithm by systematically increasing an algorithmic parameter until the properties sampled from the electron distributions converge to statistically equivalent values, extrapolated in the limit of zero time-step. In doing so, we are able to unambiguously determine the values for the ground-state fixed-node energies and one-electron properties of various molecules. These quantities are free from importance sampling bias, population control bias, time-step bias, extrapolation-model bias, and the finite-field approximation. We applied our algorithm to the ground-states of lithium hydride, water and ethylene molecules, and found excellent agreement with the accepted literature values for the energy and a variety of other properties for those systems. Some of our one-electron properties of ethylene had not been calculated before at any level of theory. In a detailed comparison, we found reptation quantum Monte Carlo, our closest competitor, to be less efficient by at least a factor of two. It requires different sets of time-steps to accurately determine the ground-state energy and one-electron properties, whereas our algorithm can achieve the same objective by using a single set of time-step values.
We present a general formulation of the Green's function Monte Carlo method in imaginary-time quantum Monte Carlo which employs exact propagators. This algorithm has no time-step errors and is obtained by minimal modifications of the time-independent Green's function Monte Carlo method. We describe how the method can be applied to the many-body Schrödinger equation, lattice Hamiltonians, and simple field theories. Our modification of the Green's function Monte Carlo algorithm is applied to the ground state of liquid 4He. We calculate the zero-temperature imaginary-time diffusion constant and relate that to the effective mass of a mass-four "impurity" atom in liquid 4He.
Quantum Monte Carlo methods are a powerful suite of techniques for solving the quantum many-body problem. By using random numbers to stochastically sample quantum properties, QMC methods are capable of studying low-temperature quantum systems well beyond the reach of conventional deterministic techniques. QMC techniques have likewise been indispensible tools for augmenting our current knowledge of superfluidity and superconductivity. In this thesis, I present two new quantum Monte Carlo techniques, the Monte Carlo Power Method and Bose-Fermi Auxiliary-Field Quantum Monte Carlo, and apply previously developed Path Integral Monte Carlo methods to explore two new phases of quantum hard spheres and hydrogen. I lay the foundation for a subsequent description of my research by first reviewing the physics of quantum liquids in Chapter One and the mathematics behind Quantum Monte Carlo algorithms in Chapter Two. I then discuss the Monte Carlo Power Method, a stochastic way of computing the first several extremal eigenvalues of a matrix too memory-intensive to be stored and therefore diagonalized. As an illustration of the technique, I demonstrate how it can be used to determine the second eigenvalues of the transition matrices of several popular Monte Carlo algorithms. This information may be used to quantify how rapidly a Monte Carlo algorithm is converging to the equilibrium probability distribution it is sampling. I next present the Bose-Fermi Auxiliary-Field Quantum Monte Carlo algorithm. This algorithm generalizes the well-known Auxiliary-Field Quantum Monte Carlo algorithm for fermions to bosons and Bose-Fermi mixtures. Despite some shortcomings, the Bose-Fermi Auxiliary-Field Quantum Monte Carlo algorithm represents the first exact technique capable of studying Bose-Fermi mixtures of any size in any dimension. In Chapter Six, I describe a new Constant Stress Path Integral Monte Carlo algorithm for the study of quantum mechanical systems under high pressures. While the eventual hope is to apply this algorithm to the exploration of yet unidentified high-pressure, low-temperature phases of hydrogen, I employ this algorithm to determine whether or not quantum hard spheres can form a low-temperature bcc solid if exchange is not taken into account. In the final chapter of this thesis, I use Path Integral Monte Carlo once again to explore whether glassy para-hydrogen exhibits superfluidity. Physicists have long searched for ways to coax hydrogen into becoming a superfluid. I present evidence that, while glassy hydrogen does not crystallize at the temperatures at which hydrogen might become a superfluid, it nevertheless does not exhibit superfluidity. This is because the average binding energy per p-H2 molecule poses a severe barrier to exchange regardless of whether the system is crystalline. All in all, this work extends the reach of Quantum Monte Carlo methods to new systems and brings the power of existing methods to bear on new problems.