Abstract In this paper Kalos proposed what is now known as the Green’s function quantum Monte Carlo (GFQMC) method for solving the Schrödinger equation. Kalos made the “critical” observation that the time-independent Schrödinger equation could be transformed into an integral equation for which the Green’s function is known if the boundary conditions correspond to the wavefunction vanishing at infinity. In a 3n-dimensional space R the integral form is (modified here) an equation which may be simulated by a random walk process in which particles at positions R’ are altered in weight by the V / E term and moved to new positions R obtained by sampling the Green’s function G(R’,R). As for simple diffusion, repetition leads to a distribution of particles corresponding to the lowest-energy wavefunction from which the energy E may be determined. Since time is not involved, there is no time-step error associated with the procedure. The term V / E must remain positive if mixed signs for particle weights are to be avoideda. Kalos examined the utility of the method in determining the boson ground-state energies for several three- and four-body nuclear problems involving pairwise-additive interactions with square, Gaussian, and exponential wells. The errors in the computed binding energies were estimated to be about 1%.
Abstract In this paper Kalos proposed what is now known as the Green’s function quantum Monte Carlo (GFQMC) method for solving the Schrödinger equation. Kalos made the “critical” observation that the time-independent Schrödinger equation could be transformed into an integral equation for which the Green’s function is known if the boundary conditions correspond to the wavefunction vanishing at infinity. In a 3n-dimensional space R the integral form is (modified here) an equation which may be simulated by a random walk process in which particles at positions R’ are altered in weight by the V / E term and moved to new positions R obtained by sampling the Green’s function G(R’,R). As for simple diffusion, repetition leads to a distribution of particles corresponding to the lowest-energy wavefunction from which the energy E may be determined. Since time is not involved, there is no time-step error associated with the procedure. The term V / E must remain positive if mixed signs for particle weights are to be avoideda. Kalos examined the utility of the method in determining the boson ground-state energies for several three- and four-body nuclear problems involving pairwise-additive interactions with square, Gaussian, and exponential wells. The errors in the computed binding energies were estimated to be about 1%.
We construct an eigenvalue problem by confining many-body system to a bounded domain with the boundary condition that the wave function vanishes. By changing the boundary, however, the eigenvalues of the energy can be varied continuously. The D-matrix is defined for a series of bounded problems with the same value for the ground state energy. The D-matrix is related to the S-matrix, enabling us to calculate the S-matrix at a given energy. The Schrodinger equation for the system is transformed to a diffusion equation by regarding time as imaginary. Initial ensemble, representing an approximate wave function, is evolved, through Monte Carlo simulation of random walks and branching, to the ground state ensemble. The limitations of investigation are: 1. Ingoing and outgoing channels have two fragments. 2. The interaction between the fragments is negligible outside the boundary mentioned above. 3. The particles are bosons or we know the zeros of the wave function. First we consider the scattering of a particle by a potential, which is equivalent to the two-body problem, in one dimension. Here we use the Poschl-Teller potential for which the exact solution is known. We use this case to investigate a new sampling method and study of various parameters. Next we consider three particles in one dimension. Here we take interaction to be a potential well, where at least one of the interactions is attractive so that a two-body bound state is possible.
Abstract Since the wavefunction for the ground state of the hydrogen molecule has no nodes, its energy can be determined in QMC without node location error. This paper reports such calculations for the full four-body (two-electron, two-proton) problem to obtain the energy without the use of the Born-Oppenheimer approximation or any other adiabatic approximations. The calculations were carried out with diffusion QMC and with Green’s function QMC. Importance sampling was incorporated into each of the calculations. For the diffusion calculations, extrapolation of energies obtained at several different time-step sizes was required. For the Green’s function calculations there was no time-step error, and sign changes for walkers were avoided by use of a potential energy offset and a minor approximation to eliminate positive potential energies. The energies so determined were not exact but had extremely small systematic errors. The large differences in electron and proton masses led to extensive computation requirements due to the slow equilibration and serial correlation induced by the heavier protons. The calculations were executed on one of the first massively parallel computers, a Thinking Machines CM-2 with 65,536 processors. Frequent communication among the processors was required to balance the number of walkers treated in each.
… 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.
Abstract A method for treating the nodes for small systems was described in this paper. The general procedure involves the use of both positive and negative walkers in Green’s function QMC, along with cancellation of walkers of opposite sign in close proximity. The method is illustrated for the first excited state of a particle in a one-dimensional box and for three model problems with three particles in three dimensions, interacting with each other through square-well potentials with repulsive and attractive parts. The calculations were executed with the Green’s function procedure, made convenient by availability of known Green’s functions satisfying the conditions of the sample problems. A partial cancellation of positive and negative walkers was made on the basis of overlapping Green’s functions, and the distributions of steps for the walkers involved were altered accordingly. Without cancellations, the distributions of positive and negative walkers proceed to the boson ground-state distribution. With cancellation and with control of their numbers, the positive and negative walkers tend to accumulate in separate regions and to commingle in the vicinity of a node. The arithmetic sum of the walkers yields the excited or fermion wavefunction with a node separating positive and negative regions. For the model problems the method successfully produced accuracies within a few percent of the exact energies, where known. It was noted that for larger problems, the total number of walkers required to maintain an adequate density for sufficient cancellation might be prohibitive.
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.
The Green’s function Monte Carlo method used for obtaining exact solutions to the Schrödinger equation of boson systems is generalized to treat systems of several fermions. We show that when it is possible to select eigenfunctions of the Hamiltonian based on physical symmetries, the GFMC method can be used to yield the lowest energy state of that symmetry. In particular, the lowest totally antisymmetric eigenfunction, the fermion ground state, can be obtained. Calculations on several two- and three-body model problems show the method to be computationally feasible for few-body systems.