Among the several variants of quantum Monte Carlo approaches the most popular one is certainly the Diffusion Monte Carlo (DMC) method. It is based on a combination of a drifted diffusion process and a birth/death (or branching) process. In practical calculations, these two processes are realized by applying some simple stochastic rules to each member of a finite population of ``walkers''. Because of the branching term the number of walkers is not constant and a population control step is required. Unfortunately, this additional step introduces a finite error. For large enough number of walkers and for accurate enough trial wave functions this error can be made negligible. However, when trial wave functions are poor or when no trial wave function is used (e.g., in vibrational studies of floppy molecules), this is no longer true. In addition, computation of dynamical quantities are not properly defined in a presence of a branching term. A simple remedy to these various problems is to remove the birth/death step and to introduce the branching weight into the averages (to ``carry'' the weights). In principle, this method -known as the Pure diffusion Monte Carlo method (PDMC)- is quite attractive: In contrast with DMC the number of walkers is fixed, there is no bias due to the population control, the error made for a finite simulation time is under control, and the various correlation functions in time can be defined without difficulty. Unfortunately, in practice the method is unstable and works only when accurate trial wave functions are used (weights close to one). In this talk we present a method based on the use of a so-called stochastic reconfiguration step instead of branching [1,2]. Our motivation is to combine the best of both worlds: efficiency of DMC and absence of bias as in PDMC. The approach is derived within a PDMC framework (the walkers ``carry'' some weight) but the population is reconfigured using some specific rules. These rules are chosen so as to minimize as much as possible the statistical fluctuations and also to recover the PDMC and DMC methods as two well-defined limits.
I will define some measures of the error in time-evolution operators, that I think are of relevance to DMC and apply these to the errors in some approximate importance sampled real-space representations of the time-evolution operator, both with and without imposing fixed-node boundary conditions. One goal is to construct a ``cross-node" algorithm, similar to Ceperley's release-node algorithm, that employs a Fermionic guiding function in contrast to Ceperley's bosonic guiding function. This could be combined with Sorella's stochastic reconfiguration ideas could be used to control the growth of the non-fermionic components of the distribution of walkers.
We have been studying an extension of Diffusion Monte Carlo that solves the "sign problem" of fermions. The modifications to standard DMC comprise: (a) Random walkers carry signs. That is, their contributions to estimators of projected integrals required for an energy quotient are given algebraic signs. (b) Distinct guiding functions are used for walkers of different signs. (c) The Gaussians that determine diffusion for pairs of opposite walkers are correlated in such a way that the walkers drift towards each other. (d) Opposite walkers in a pair cancel when close.
Is can be shown that if the results are stable- that is, if the denominator of the energy quotient does not decay at large imaginary times, the the results are correct for fermionic systems. Experimentally, the method has been shown to be stable for periodic systems of He3 atoms, and for small systems of electrons. We will describe the method in more detail, and give the computational results for some first-row dimers.
Although quantum Monte Carlo is, in principal, an exact method for solving the Schrödinger equation, systems of Fermions still pose a challenge. Thus far all solutions to the ``sign problem'' remain inefficient (or wrong). The fixed-node approach, however, is efficient, and in many situations remains the best approach. If only we could find the exact nodes, or at least a systematic way to improve the nodes, we would, in effect, bypass the sign problem.
Despite the fundamental importance of quantum wave functions, very little is known about their nodal structure. A detailed knowledge of the topology of these high-dimensional surfaces is a necessary step for improvement of trial nodes and a systematic study has never been attempted, despite the obvious consequences for improving quantum simulations that such knowledge might generate. Such improvement would be of great benefit to quantum simulations, which could then be systematically improved within the context of the fixed-node approximation. Here we review what is known about nodes, show some recent results on simple atomic and molecular systems, and formulate some conjectures. A long term goal of this research is to develop a way to directly parametrize the nodes of accurate trial wave functions and optimize them directly.
Over the recent years quantum Monte Carlo (QMC) methods have become more and more successful in computing ground-state total energies of molecular systems. For a variety of systems including organic molecules, clusters of atoms, solids etc... the accuracy obtained by QMC is impressive. In most cases, the quality of the results is comparable or superior to that obtained with traditional techniques: DFT, MCSCF or coupled cluster (CC) methods. Unfortunately, the calculation with QMC of properties other than energy turns out to be much more difficult.
In this talk we present some recent progress toward computing efficiently observables. It is shown how the zero-variance principle responsible for the high-level of accuracy on total energies can be applied to other properties. In analogy with energy calculations, the lower the variance of some ''renormalized'' expression of the observable is, the more accurate the expectation value is. As a consequence, observables can be computed with a high accuracy at the variational level. As an important application we show how forces between atoms (gradient of the energy with respect to nuclear coordinates) can be calculated using our formalism. Finally, we discuss how this scheme can be extended to compute accurate small differences of energy.
The success of quantum Monte Carlo (QMC) methods is intimately linked to the availability of accurate trial wavefunctions. Due to the requirement of QMC for a compact representation of the trial wavefunction, only limited use has been made so far from the large variety of wavefunctions provided by other many-particle methods in quantum chemistry. On the other hand it is also difficult to transfer insight gained from QMC on the structure of the wavefunction into these methods.
We propose a Jastrow type ansatz for the wavefunction based on methods from multi-scale analysis, which might help to bridge the gap. Herein the Jastrow factor is approximated in a wavelet basis, which enables a local and adaptive representation, especially in the cusp regions. Due to the tensor product structure of the many-particle wavelet basis it is possible to apply standard methods from quantum chemistry, like coupled electron pair approximations, to specify the variational parameters of the expansion. The resulting matrix elements can be expressed in terms of one- and two-electron integrals avoiding exceedingly complicated higher dimensional integrals. These wavefunctions can be used as trial wavefunctions in diffusion Monte Carlo calculations, providing an improved flexibility for the description of electron correlations. Vice versa it is possible to expand parts of a Jastrow factor, already successfully applied in QMC calculations, in the wavelet basis and use it within other many-particle methods in order to achieve a reduction of the variational degrees of freedom and an improved convergence behaviour.
Rotational spectroscopy of molecules embedded in 4He nanodroplets offers a unique tool for probing superfluid behaviour in confined systems. The theoretical understanding of superfluidity in doped Helium clusters presently relies on permutation cycles in finite-temperature path integral simulations, hydrodynamical models, or limited knowledge of excited states. We introduce a novel ground-state estimator for the effective moment of inertia of the molecule and relate its dependence on the number of Helium atoms to current-current correlations and density profiles. The moment of inertia features a maximum for about ten Helium atoms, and for larger clusters quickly reaches its asymptotic value. Results are compared with recent measurements for HeN-OCS clusters.
Vibrational delocalisation of weakly bound complexes amongst minima of similar potential energy is of great importance in spectroscopy. This quantum effect has a bearing on the shape of clusters, such as hydrogen-bonded complexes, making their rotational spectra difficult to analyse. Unfortunately, a harmonic treatment fails to give an accurate description of the vibrational ground state needed to elucidate these spectra and a high-quality method is needed to rationalise the observations. We show that nuclear diffusion Monte Carlo (RB-DMC) is an efficient and robust technique to treat such problems, readily providing ground-state-averaged properties. Moreover, its combination with a powerful wave-function analysis tool, based on one-body density representation, gives insights into the nature of hydrogen bonding in clusters of biological interest.