Because the Schrodinger equation cannot be solved analytically for systems larger than hydrogen, approximate solution methods must be applied to study systems of chemical interest. Quantum Monte Carlo is one of these methods. Quantum Monte Carlo simulations involve the diffusion of walkers through configuration space and necessitate the use of one of two imaginary time propagators, 2nd or 4th order. The choice of the propagator implemented depends on a balance between accuracy and efficiency. Obtaining accurate solutions is the primary objective; once this is accomplished, finding ways to make the simulations efficient is the next intention. Using a model system of a multidimensional harmonic oscillator, this study intends to compare and contrast both 2nd and 4th order propagators' diminished accuracy as the dimensionality of the system is increased. Two phases of the propagation are sampled to determine accuracy: a single propagation step and the asymptotic limit. To test the accuracy after one step, we start a simulation by placing walkers at the origin, moving them forward one time step in imaginary time, creating normalized histograms of their positions and comparing these with the known result for the imaginary time propagation of the delta function centered at x=0. The convergence of the simulation is determined by observing the change in distribution from the known asymptotic limit distribution. This is done by distributing the walkers in the known asymptotic distribution and observing the change in average weight upon further propagation. A converged distribution’s average weight will not change.
Because the Schrodinger equation cannot be solved analytically for systems larger than hydrogen, approximate solution methods must be applied to study systems of chemical interest. Quantum Monte Carlo is one of these methods. Quantum Monte Carlo simulations involve the diffusion of walkers through configuration space and necessitate the use of one of two imaginary time propagators, 2nd or 4th order. The choice of the propagator implemented depends on a balance between accuracy and efficiency. Obtaining accurate solutions is the primary objective; once this is accomplished, finding ways to make the simulations efficient is the next intention. Using a model system of a multidimensional harmonic oscillator, this study intends to compare and contrast both 2nd and 4th order propagators' diminished accuracy as the dimensionality of the system is increased. Two phases of the propagation are sampled to determine accuracy: a single propagation step and the asymptotic limit. To test the accuracy after one step, we start a simulation by placing walkers at the origin, moving them forward one time step in imaginary time, creating normalized histograms of their positions and comparing these with the known result for the imaginary time propagation of the delta function centered at x=0. The convergence of the simulation is determined by observing the change in distribution from the known asymptotic limit distribution. This is done by distributing the walkers in the known asymptotic distribution and observing the change in average weight upon further propagation. A converged distribution’s average weight will not change.
Direct sampling of multi-dimensional systems with quantum Monte Carlo methods allows exact account of many-body effects or particle correlations. The most straightforward approach to solve the Schrödinger equation, Diffusion Monte Carlo, has been used in several benchmark cases for other methods to pursue. Its robustness is based on direct sampling of a positive probability density for diffusion in imaginary time. It has been argued that the corresponding real time diffusion can not be realised, because the corresponding oscillating complex valued distribution can not be used to drive diffusion. Here, we demonstrate that this can be done by turning the distribution piecewise positive and normalizable, and also, by using four types of walkers. This study is a proof of concept demonstration using the well-known and transparent case: one-dimensional harmonic oscillator. Furthermore, we show that our novel method can be used to find not only the ground state but also excited states and even the time evolution of a given wave function. Considering fermionic systems, this method may turn out to be feasible for finding the wave function nodes for other approaches.
The standard method of quantum Monte Carlo for the solution of the Schrödinger equation in configuration space can be described quite generally as devising a random walk that generates-at least asymptotically-populations of random walkers whose probability density is proportional to the wave function of the system being studied. While, in principle, the energy eigenvalue of the Hamiltonian can be calculated with high accuracy, estimators of operators that do not commute the Hamiltonian cannot. Bilinear quantum Monte Carlo (BQMC) is an alternative in which the square of the wave function is sampled in a somewhat indirect way. More specifically, one uses a pair of walkers at positions x and y and introduces stochastic dynamics to sample phi(i)(x)t(x,y)phi(j)(y), where phi(i)(x) and phi(j)(y) are eigenfunctions of (possibly different) Hamiltonians, and t(x,y) is a kernel that correlates positions x and y. Using different Hamiltonians permits the accurate computation of small energy differences. We review the conceptual basis of BQMC, discuss qualitatively and analytically the problem of the fluctuations in the branching, and present partial solutions to that problem. Finally we exhibit numerical results for some model systems including harmonic oscillators and the hydrogen and helium atoms. Further research will be necessary to make this a practical and generally applicable scheme.
Quantum Monte Carlo (QMC) methods represent a powerful family of computational techniques for tackling complex quantum many-body problems and performing calculations of stationary state properties. QMC is among the most accurate and powerful approaches to the study of electronic structure, but its application is often hindered by a steep learning curve; hence it is rarely addressed in undergraduate and postgraduate classes. This tutorial is a step toward filling this gap. We offer an introduction to the diffusion Monte Carlo (DMC) method, which aims to solve the imaginary time Schrödinger equation through stochastic sampling of the configuration space. Starting from the theoretical foundations, the discussion leads naturally to the formulation of a step-by-step algorithm. To illustrate how the method works in simplified scenarios, examples such as the harmonic oscillator and the hydrogen atom are provided. The discussion extends to the fixed-node approximation, a crucial approach for addressing the fermionic sign problem in multi-electron systems. In particular, we examine the influence of trial wave function nodal surfaces on the accuracy of DMC energy by evaluating results from a non-interacting two-fermion system. Extending the method to excited states is feasible in principle, but some additional considerations are needed, supported by practical insights. By addressing the fundamental concepts from a hands-on perspective, we hope this tutorial will serve as a valuable guide for researchers and students approaching DMC for the first time.
Solving the Schrodinger equation and finding excited states for quantum mechanical many-body systems is a fundamental problem. If this problem is formulated in terms of integrals, the dimensionality of the configuration space is often too high to perform the integrations required directly. This is because the volume of configuration space increases exponentially with dimension quickly, making its complete exploration impossible. Monte Carlo methods provide a way to estimate these integrals by statistically sampling a subset of configuration space, and these methods provide 1/√ N convergence regardless of dimension. For many problems involving the quantum mechanics of molecules there exists a large time scale separation between the high-frequency internal vibrations, and low-frequency intermolecular motions. This separation motivates the rigid-body approximation which freezes internal degrees of freedom in order to study intermolecular effects. The reduction in the dimensionality of configuration space further increases the range of accessible problems. This thesis is devoted to the construction and implementation of algorithms which incorporate the rigid-body approximation into existing Monte Carlo methods for solving the quantum mechanical many-body problem. Monte Carlo estimators are constructed as averages over samples drawn from some probability distribution. If this distribution is known in closed form the samples can be generated by application of the Metropolis algorithm. In most cases the distribution is not known in closed form or even representable with a finite number of variables. If this distribution is the dominant eigenstate of some known operator then a stochastic implementation of the power method can be used to generate the required samples. For the quantum mechanical problem this operator can be taken to be the imaginary-time evolution operator and its application can be represented in terms of random walks. For rigid bodies, this method involves the implementation of rotational Brownian motion. The use of quaternion algebra to implement this rotational motion enhances simplicity, performance, and numerical stability. The method described above can be generalized to investigate excited state properties using correlation function Monte Carlo which is a Monte Carlo implementation of the Rayleigh-Ritz variational method. This generalization requires the construction of a trial subspace which is then subjected to the variational method resulting in a generalized eigenvalue problem. It is important to represent optimized trial states accurately and efficiently. For this purpose, we write trial states as functions of invariant polynomials of the interparticle distances. By reducing the number of free interparticle distances, the rigid-body approximation greatly simplifies
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.
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.