The quantum Monte Carlo (QMC) technique is an extremely powerful method to treat many-body systems. Usually the quantum Monte Carlo method has been applied in cases where the interaction potential has a simple analytic form, like the 1/r Coulomb potential. However, in a complicated environment as in a semiconductor heterostructure, the evaluation of the interaction itself becomes a nontrivial problem. Obtaining the potential from any grid-based finite-difference method for every walker and every step is infeasible. We demonstrate an alternative approach of solving the Poisson equation by a classical Monte Carlo calculation within the overall quantum Monte Carlo scheme. We have developed a modified "walk on spheres" algorithm using Green's function techniques, which can efficiently account for the interaction energy of walker configurations, typical of quantum Monte Carlo algorithms. This stochastically obtained potential can be easily incorporated with variational, diffusion, and other Monte Carlo techniques. We demonstrate the validity of this method by studying a simple problem, the polarization of a helium atom in the electric field of an infinite capacitor.
The quantum Monte Carlo (QMC) technique is an extremely powerful method to treat many-body systems. Usually the quantum Monte Carlo method has been applied in cases where the interaction potential has a simple analytic form, like the 1/r Coulomb potential. However, in a complicated environment as in a semiconductor heterostructure, the evaluation of the interaction itself becomes a nontrivial problem. Obtaining the potential from any grid-based finite-difference method for every walker and every step is infeasible. We demonstrate an alternative approach of solving the Poisson equation by a classical Monte Carlo calculation within the overall quantum Monte Carlo scheme. We have developed a modified "walk on spheres" algorithm using Green's function techniques, which can efficiently account for the interaction energy of walker configurations, typical of quantum Monte Carlo algorithms. This stochastically obtained potential can be easily incorporated with variational, diffusion, and other Monte Carlo techniques. We demonstrate the validity of this method by studying a simple problem, the polarization of a helium atom in the electric field of an infinite capacitor.
Quantum Monte Carlo (QMC) is an extremely powerful method to treat many-body systems. Usually quantum Monte Carlo has been applied in cases where the interaction potential has a simple analytic form, like the 1/r Coulomb potential. However, in a complicated environment as in a semiconductor heterostructure, the evaluation of the interaction itself becomes a non-trivial problem. Obtaining the potential from any grid-based finite-difference method, for every walker and every step is unfeasible. We demonstrate an alternative approach of solving the Poisson equation by a classical Monte Carlo within the overall quantum Monte Carlo scheme. We have developed a modified ''Walk On Spheres'' algorithm using Green's function techniques, which can efficiently account for the interaction energy of walker configurations, typical of quantum Monte Carlo algorithms. This stochastically obtained potential can be easily incorporated within popular quantum Monte Carlo techniques like variational Monte Carlo (VMC) or diffusion Monte Carlo (DMC). We demonstrate the validity of this method by studying a simple problem, the polarization of a helium atom in the electric field of an infinite capacitor.
Introduction In Chapters 1 to 4 we studied methods for solving the Schrödinger equation for many-electron systems. Many of the techniques described there carry over to other quantum many-particle systems, such as liquid helium, and the protons and neutrons in a nucleus. The techniques which we discussed there were, however, all of a mean-field type and therefore correlation effects could not be taken into account without introducing approximations. In this chapter, we consider more accurate techniques, which are similar to those studied in Chapter 10 and are based on using (pseudo-)random numbers – hence the name ‘Monte Carlo’ for these methods. In Chapter 10 we applied Monte Carlo techniques to classical many-particle systems; here we use these techniques for studying quantum problems involving many particles. In the next section we shall see how we can apply Monte Carlo techniques to the problem of calculating the quantum mechanical expectation value of the ground state energy. This is used in order to optimise this expectation value by adjusting a trial wave function in a variational type of approach, hence the name variational Monte Carlo (VMC). In the following section we use the similarity between the Schrödinger equation and the diffusion equation in order to calculate the properties of a collection of interacting quantum mechanical particles by simulating a classical particle diffusion process. The resulting method is called diffusion Monte Carlo (DMC).
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.
In this brief article, I introduce the various types of quantum Monte Carlo (QMC) methods, in particular, those that are applicable to systems in extreme regimes of temperature and pressure.We give references to longer articles where detailed discussion of applications and algorithms appear. MOTIVATIONOne does QMC for the same reason as one does classical simulations; there is no other method able to treat exactly the quantum many-body problem aside from the direct simulation method where electrons and ions are directly represented as particles, instead of as a "fluid" as is done in mean-field based methods.However, quantum systems are more difficult than classical systems because one does not know the distribution to be sampled, it must be solved for.In fact, it is not known today which quantum problems can be "solved" with simulation on a classical computer in a reasonable amount of computer time.One knows that certain systems, such as most quantum many-body systems in 1D and most bosonic systems are amenable to solution with Monte Carlo methods, however, the "sign problem" prevents making the same statement for systems with electrons in 3D.Some limitations or approximations are needed in practice.On the other hand, in contrast to simulation of classical systems, one does know the Hamiltonian exactly: namely charged particles interacting with a Coulomb potential.Given sufficient computer resources, the results can be of quite high quality and for systems where there is little reliable experimental data.For this reason, QMC methods, though more expensive, are useful to benchmark and validate results from other methods.The two main applications that we will discuss in this review are the "electronic structure problem"; computing the energy of the an interacting systems of electrons and fixed ions, and the problem of quantum effects of the nuclei. RANDOM WALK AND MARKOV CHAINSMost QMC algorithms are based on random walks, known mathematically as Markov chains.This is a general class of algorithms, introduced in the famous work by Metropolis, the Rosenbluths and the Tellers (1953) for sampling any probability distribution.Let us denote the desired distribution as Π(S), where the variable "S" represents the state of the walk, for example the coordinates of all of the particles.In a random walk, one invents some moving rules (transition probabilities) for changing the state of the system so that the probability of being in state S n+1 in the (n+1) step, given that the random walk was S n is given by T(S n ; S n+1 ).By
Quantum Monte Carlo (QMC) is one of the most promising methods for solving quantum many-body problems. QMC combines known analytical properties of wave functions, results from other methods, and stochastic techniques, into a powerful tool for investigation of the electronic structure of real materials. Furthermore, the method has a favorable scaling ( O[N3] in the number of electrons), perfect scalability on parallel architectures, and can be applied to a wide range of systems. Our work has focused both on expanding the capabilities of QMC for calculation of quantities beyond energies, such as interatomic forces, and on application of the method to challenging problems in nanostructure materials research. For the calculation of interatomic forces we have implemented a finite difference correlated sampling method. The correlated sampling enables us to avoid the problem of statistical noise in evaluating the energy differences and allows us to obtain forces with an average 1% error. We have analyzed the effect of the method's main approximations on the estimate of the forces, both by theoretical arguments and by performing a variety of tests on small systems. Furthermore, we have used this newly developed method for estimating equilibrium geometries in a problem where other methods have failed. QMC enabled us to treat the electron correlation effects with high accuracy and to provide accurate predictions in a number of interesting applications. In particular, for carbon rings of C4N+2 stoichiometry we have resolved a long-standing problem regarding the competition between the aromaticity and second order Jahn-Teller effects. We have found that the dominant mechanism at small and intermediate sizes, which stabilizes the bond angle and bond length alternated geometries, is the secondorder Jahn-Teller effect while the aromatic isomer is always found to be a transition state. These high accuracy calculations which have involved multi-reference wave functions, extensive variational optimizations, and the use of massively parallel platforms demonstrate clearly the unique capabilities of the QMC approach.