Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Monte-Carlo solution of Schrödinger’s equation

R C Grimm, R. G. Storer

Published Jun 18, 2007• 0 citations
Publisher

Abstract

Abstract With this paper Grimm and Storer introduced importance sampling to the field of quantum Monte Carlo. They had earlier used numerical quadrature for estimating the integrals required in evaluating energies in a diffusion-like path integral iteration scheme for solving the Schrödinger equation.0 This was extended to larger numbers of particles and higher dimensions with use of Monte Carlo methods. In conjunction with this they recognized they could take advantage of prior knowledge of the general form of a wavefunction to obtain more accurate estimates of the eigenvalue Eo .

DOI
10.1093/oso/9780195310108.003.0007

Related papers

Quantum Monte Carlo methods

Joseph Marie Thijssen

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).

Cambridge University Press eBooks•Mar 22, 2007•26 citations

Applications of quantum Monte Carlo methods in condensed systems

Jindřich Kolorenč, Luboš Mitáš

Quantum Monte Carlo methods represent a powerful and broadly applicable computational tool for finding very accurate solutions of the stationary Schrödinger equation for atoms, molecules, solids and a variety of model systems. The algorithms are intrinsically parallel and are able to take full advantage of present-day high-performance computing systems. This review paper concentrates on the fixed-node/fixed-phase diffusion Monte Carlo method with emphasis on its applications to the electronic structure of solids and other extended many-particle systems.

arXiv:1010.4992•Reports on Progress in Physics•Jan 21, 2011•208 citations

Bilinear diffusion quantum Monte Carlo methods

F. Arias de Saavedra, Malvin H. Kalos

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.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Feb 27, 2003•7 citations

Projector quantum Monte Carlo

Richard M. Martin, Lucia Reining, David M. Ceperley

… as suggested by Fermi, the time-independent Schrödinger equation … can be interpreted as describing the behavior of a system of particles each of which performs a random walk, i.e., diffuses isotropically and at the same time is subject to multiplication, which is determined by the value of the point function V. N. Metropolis and S. Ulam, 1949 Summary In the projector quantum Monte Carlo method, one uses a function of the hamiltonian to sample a distribution proportional to the exact ground-state wavefunction, and thereby computes exact matrix elements of it. An importance sampling transformation makes the algorithm much more efficient. In this chapter we introduce and develop the diffusion Monte Carlo method, which involves drifting, branching random walks. For any excited state, including any system with more than two electrons, one encounters the sign problem, limiting the direct application of these algorithms for most fermion systems. Instead, by using approximate fixed-node or fixed-phase boundary conditions, one can achieve efficiency similar to variational Monte Carlo. We also discuss the application of the projector method in a basis of Slater determinants. In this chapter, we discuss a different quantum Monte Carlo method, projector Monte Carlo (PMC). This general method was first suggested by Fermi [1049]; see the quote at the start of this chapter by two of the inventors of the Monte Carlo method. An implementation of PMC was tried out in the early days of computing [1050]. Advances in methodology, in particular importance sampling, resulted in a significant large-scale application: the exact calculation of the ground-state properties of 256 hard-sphere bosons by Kalos, Levesque, and Verlet [1051] in 1974. Calculations for electronic systems and the fixednode approximation were introduced by Anderson [1052, 1053]. One of the most important projector MC algorithms, the diffusion Monte Carlo algorithm with importance sampling for fermions, was used to compute the correlation energy of homogeneous electron gas by Ceperley and Alder [109] in 1980; the resulting HEG correlation energy was crucial in the development of density functional calculations. Types and properties of projectors In this method, a many-body projector G ( R , R ) = Ĝ is repeatedly applied to filter out the exact many-body ground state from an initial state; the operation of the projector is carried out with a random walk, hence the name of this class of methods.

Cambridge University Press eBooks•May 31, 2016•0 citations

EXTENSIONS FOR THE RIGID-BODY QUANTUM MONTE CARLO METHOD

Chris Smith

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

•Jan 1, 2010•0 citations

Iterative Monte Carlo for quantum dynamics

Vikram Jadhao, Nancy Makri

We present a fully quantum mechanical methodology for calculating complex-time correlation functions by evaluating the discretized path integral expression iteratively on a grid selected by a Monte Carlo procedure. Both the grid points and the summations performed in each iteration utilize importance sampling, leading to favorable scaling with the number of particles, while the stepwise evaluation of the integrals circumvents the exponential growth of statistical error with time.

The Journal of Chemical Physics•Oct 28, 2008•58 citations
Abstract With this paper Grimm and Storer introduced importance sampling to the field of quantum Monte Carlo. They had earlier used numerical quadrature for estimating the integrals required in evaluating energies in a diffusion-like path integral iteration scheme for solving the Schrödinger equation.0 This was extended to larger numbers of particles and higher dimensions with use of Monte Carlo methods. In conjunction with this they recognized they could take advantage of prior knowledge of the general form of a wavefunction to obtain more accurate estimates of the eigenvalue Eo .
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).
Quantum Monte Carlo methods represent a powerful and broadly applicable computational tool for finding very accurate solutions of the stationary Schrödinger equation for atoms, molecules, solids and a variety of model systems. The algorithms are intrinsically parallel and are able to take full advantage of present-day high-performance computing systems. This review paper concentrates on the fixed-node/fixed-phase diffusion Monte Carlo method with emphasis on its applications to the electronic structure of solids and other extended many-particle systems.
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.
… as suggested by Fermi, the time-independent Schrödinger equation … can be interpreted as describing the behavior of a system of particles each of which performs a random walk, i.e., diffuses isotropically and at the same time is subject to multiplication, which is determined by the value of the point function V. N. Metropolis and S. Ulam, 1949 Summary In the projector quantum Monte Carlo method, one uses a function of the hamiltonian to sample a distribution proportional to the exact ground-state wavefunction, and thereby computes exact matrix elements of it. An importance sampling transformation makes the algorithm much more efficient. In this chapter we introduce and develop the diffusion Monte Carlo method, which involves drifting, branching random walks. For any excited state, including any system with more than two electrons, one encounters the sign problem, limiting the direct application of these algorithms for most fermion systems. Instead, by using approximate fixed-node or fixed-phase boundary conditions, one can achieve efficiency similar to variational Monte Carlo. We also discuss the application of the projector method in a basis of Slater determinants. In this chapter, we discuss a different quantum Monte Carlo method, projector Monte Carlo (PMC). This general method was first suggested by Fermi [1049]; see the quote at the start of this chapter by two of the inventors of the Monte Carlo method. An implementation of PMC was tried out in the early days of computing [1050]. Advances in methodology, in particular importance sampling, resulted in a significant large-scale application: the exact calculation of the ground-state properties of 256 hard-sphere bosons by Kalos, Levesque, and Verlet [1051] in 1974. Calculations for electronic systems and the fixednode approximation were introduced by Anderson [1052, 1053]. One of the most important projector MC algorithms, the diffusion Monte Carlo algorithm with importance sampling for fermions, was used to compute the correlation energy of homogeneous electron gas by Ceperley and Alder [109] in 1980; the resulting HEG correlation energy was crucial in the development of density functional calculations. Types and properties of projectors In this method, a many-body projector G ( R , R ) = Ĝ is repeatedly applied to filter out the exact many-body ground state from an initial state; the operation of the projector is carried out with a random walk, hence the name of this class of methods.
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