Abstract These two papers describe the development of a variant of diffusion QMC with importance sampling and its applications to a few simple systems. The method is based on the generalized FeynmanKac path integral formalisma and is termed the “full generalized Feynman-Kac” or FGFK method. Its major advantage, relative to other diffusion-based methods, is the elimination of branching or multiplication of walkers. Its major disadvantage is a greater complexity. The basic implementation is very nearly the same as that of standard diffusion QMC with importance sampling, and it incorporates diffusion with drift in the same way. The uses of trial functions, fixed nodes, extrapolation to zero time-step size, and other procedures except for branching are identical. However, the evaluation of mean quantities can be carried out in terms of ‘lj;2 sampling to obtain energies and other properties based on the FGFK formula. The applications described include ground and excited states of the harmonic oscillator and the helium atom and the ground state of the hydrogen molecule. For these the energies as well as a dozen other observables such as the averages of interelectron distances were determined. The results were found to be in good agreement with accepted values. A combination of FGFK with the released-node method was investigated for the case of the one-dimensional harmonic oscillator and found to be successful for that case.
Abstract These two papers describe the development of a variant of diffusion QMC with importance sampling and its applications to a few simple systems. The method is based on the generalized FeynmanKac path integral formalisma and is termed the “full generalized Feynman-Kac” or FGFK method. Its major advantage, relative to other diffusion-based methods, is the elimination of branching or multiplication of walkers. Its major disadvantage is a greater complexity. The basic implementation is very nearly the same as that of standard diffusion QMC with importance sampling, and it incorporates diffusion with drift in the same way. The uses of trial functions, fixed nodes, extrapolation to zero time-step size, and other procedures except for branching are identical. However, the evaluation of mean quantities can be carried out in terms of ‘lj;2 sampling to obtain energies and other properties based on the FGFK formula. The applications described include ground and excited states of the harmonic oscillator and the helium atom and the ground state of the hydrogen molecule. For these the energies as well as a dozen other observables such as the averages of interelectron distances were determined. The results were found to be in good agreement with accepted values. A combination of FGFK with the released-node method was investigated for the case of the one-dimensional harmonic oscillator and found to be successful for that case.
This paper presents systematic developments in the previously initiated line of research concerning a quantum Monte Carlo (QMC) method based on the use of a pure diffusion process corresponding to some reference function and a generalized Feynman–Kac path integral formalism. Not only mean values of quantum observables, but also response properties are expressed using suitable path integrals involving the diffusion measure of the reference diffusion process. Moreover, by relying on the ergodic character of this process, path integrals may be evaluated as time-averages along any sample trajectory of the process. This property is of crucial importance for the computer implementation of the method. As concerns the treatment of many-fermion systems, where the Pauli principle must be taken into account, we can use the fixed-node approximation, but we also discuss the potentially exact release-node procedure, whereby some adequate symmetry is imposed on the integrand (of the generalized Feynman–Kac formula), associated with a possibly refined choice of the reference function.
Abstract The paper describes several extensions of fixed-node diffusion in the areas of obtaining analytic derivatives of the energies of molecules, calculating expectation values for quantities other than the energy, and treating excited states. The calculation of energy derivatives is illustrated with a calculation of the first derivative of the energy of the ground state of H2 with respect to the internuclear distance, without using finite differences. With importance sampling the energy is obtained from a weighted average of local energies expressed in analytic form. These may be differentiated to obtain their derivatives with respect to any of several variables, including the internuclear distance, and their weighted averages may be used to determine the desired derivatives. The results for the H2 energy derivatives were in agreement with independent determinations. For calculation of expectation values of the energy, sampling of the (true wavefunction)(trial wavefunction) ‘1/J’l/Jr distribution is sufficient, but other quantities usually require sampling the ‘ljJ2 distribution, which is not so readily available in QMC calculations. Extrapolations of ‘!/Jf and ‘ljJ’l/Jr results, from variational and diffusion calculations, to ‘!jJ2 results are illustrated for various expectation values for the case of ground-state H2.
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 This paper introduces a successful new method for extending diffusion QMC with importance sampling to allow the direct calculation of pure ‘¢ 2 distributions in place of mixed ‘¢ 1/Jr distributions. The method removes any bias introduced with a trial function (except that due to node locations) in the calculation of observable quantities which do not commute with the Hamiltonian. It has the advantage, relative to the descendent weighting or forward walking technique,a of a controlled walker population which reduces statistical errors. The authors have named it the reptation quantum Monte Carlo (RQMC) method. Its basic object is a path or sequence of configurations called a “reptile” which may be altered by an action called “reptation,” terms adopted from related work in the polymer area. In conventional diffusion QMC with importance sampling walkers are subjected to diffusion, drift, and a local energy-based multiplication to produce a 1/J’I/Jr distribution. Without multiplication a ‘l/Jr’l/Jr distribution is produced. In RQMC multiplication is replaced by a Metropolis rejection procedure, also based on local energies, which produces a 1/J’!/J distribution of configurations in a reptile. The generation of a new reptile, by removing the tail and adding a new head, is based on diffusion and drift for proposed moves coupled with an acceptance test. Overall, the implementation is quite similar to that for variational QMC. As described, the method was tested first for the hydrogen atom, followed by the successful demonstration of a calculation for a helium fluid, 64 4He atoms in a cubic box with periodic boundary conditions. The advantages of RQMC relative to branching diffusion QMC were pointed out for treatment of clusters, films, and superfluids as well as for estimation of electronic forces.
In this paper, an importance sampling method based on the Generalized Feynman–Kac (GFK) method has been used to calculate the mean values of quantum observables from quantum correlation functions for many-body systems with the Born–Oppenheimer approximation in the nonrelativistic limit both at zero and finite temperature. Specifically, the expectation values [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] for the ground state of the lithium and beryllium and the density matrix, the partition function, the internal energy and the specific heat of a system of quantum harmonic oscillators are computed, in good agreement with the best nonrelativistic values for these quantities. Although the initial results are encouraging, more experimentation will be needed to improve the other existing numerical results beyond chemical accuracies specially for the last two properties for lithium and beryllium. Also more work needs to be done to improve the trial functions for finite temperature calculations. Although these results look promising, more work needs to be done to achieve the spectroscopic accuracy at zero temperature and to estimate the finite temperature effects from the non-Born–Oppenheimer calculations. Also more experimentation will be needed to study the convergence criteria for the inverse properties for atoms at zero temperature.