Abstract This work extended the simple diffusion QMC method of solving the Schrödinger equation for a wavefunction ψ; to that of solving for a correction to a trial wavefunction ψo such that ψ= ψo+ . The method may be applied in the manner of successive corrections to obtain very high accuracies. As described, the method is limited to solutions for systems with fixed nodes. The method was illustrated for the cases of the particle-in-a-box and the hydrogen atom. For the latter, successive corrections improved the accuracy in energy by about a factor of ten with each iteration. For higher dimensional systems, difficulties in the fitting procedure may be expected to limit the applicability of the method.
Abstract This work extended the simple diffusion QMC method of solving the Schrödinger equation for a wavefunction ψ; to that of solving for a correction to a trial wavefunction ψo such that ψ= ψo+ . The method may be applied in the manner of successive corrections to obtain very high accuracies. As described, the method is limited to solutions for systems with fixed nodes. The method was illustrated for the cases of the particle-in-a-box and the hydrogen atom. For the latter, successive corrections improved the accuracy in energy by about a factor of ten with each iteration. For higher dimensional systems, difficulties in the fitting procedure may be expected to limit the applicability of the method.
Abstract In an earlier papera a method for the direct calculation of the difference between a true wavefunction and a trial wavefunction was described for simple diffusion calculations. This paper reports further increases in accuracy obtained by a combination of this difference method with importance sampling. The combination allows the calculation of corrections to the product ΨΨT of true and trial wavefunctions, rather than to the wavefunction Ψ itself. Using a new function g = ΨΨT- Ψ2T, the original diffusion equation with drift and local energies is altered by replacement of ΨΨT with g and by addition of a distributed source term. Since walkers fed by sampling the source term may be positive or negative, cancellation is required, but both sets of walkers proceed to the same distribution and age-based cancellation may be used. Node locations are not changed and remain at their original locations as specified by the trial function. The corrected wavefunction and the energy obtained are the same as for a fixed-node diffusion calculation using the same trial function for importance sampling. The method is useful in reducing statistical error, and because the statistical error occurs only in the difference terms, it is most effective when the trial wavefunction is already reasonably accurate. In test calculations for the hydrogen atom and the helium atom, both nodeless, the statistical error in the QMC calculations was greatly reduced. Thus, for hydrogen a trial function with an expectation value of -0.4998 hartrees led to a QMC energy of -0.499995(4) hartrees.
Abstract This little paper describes two procedures for achieving higher speeds in fixed-node diffusion QMC calculations with importance sampling. The first is an iterative procedure for reducing timestep error, thereby allowing larger and fewer time steps to obtain a given accuracy. The movement of a walker is divided into two parts: a diffusion step selected at random from an appropriate distribution and a drift step calculated from the drift velocity at the initial location. These give a temporary step to a new location for which the drift velocity is again determined. The average of the two velocities together with the original diffusion step is used to determine a permanent step. In tests, the procedure was found to reduce time-step error by a factor of 5 to 10. The second procedure makes use of two trial functions for importance sampling, the first a simple function for walker motion and the second a more complex and more accurate function for determining energies. These allow the more computationally expensive function to be used at large intervals for samples having a lower serial correlation.
Abstract In this paper the combination of importance sampling° with diffusion QMC was shown to be remarkably effective in improving the accuracy of electonic structure calculations for molecular systems. The key to success was the use of a simple, fairly accurate trial wavefunction ΨT for which the function itself, its derivatives and the local energy Eioc may be easily computed. This, used in combination with the diffusion-with-drift expression for walkers corresponding to the product f = ΨΨT according to allows the energy to be evaluated as an average for all walkers of every step after reaching a fluctuating steady state. Since there is a time-step error associated with the simulation of diffusion, drift, and multiplication, varying the time-step and an extrapolation to zero step size are required. The result for H3+ in its triangular configuration of side length 1.6500 bohr was -1.3439 ± 0.0002 hartrees, a value well below the lowest variational energy levels at the time and having an error bar straddling the best modern value of -1.343 835 hartrees. The use of importance sampling along with higher computer speeds resulted in an uncertainty about a factor of 60 lower than that of the previous calculation.b
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.
The diffusion quantum Monte Carlo method (DMC) provides a numeric solution of the time dependent Schrödinger equation by means of a random walk process. To fulfil the pauli principle the nodes of a so called trial wavefunction, obtained with standard ab initio quantum chemistry methods like HF, DFT, or CASSCF, are considered as additional boundary conditions. In case of such nodes coincide with those of the exact wavefunction DMC yields the exact solution of the Schrödinger equation. Generally, this is not true and one ends up with an systematic bias, referred to as the node location error. Theoretical chemistry aims the prediction of reaction, excitation, or activation energies, i.e. energy differences. Therefore, the node location error has to be seen under the light of efficient error cancellation. One can distinguish dynamic and non-dynamic electron correlation. The first one accounts for the instantaneous, short distance interaction among the electrons. The non-dynamic correlation addresses the interaction between different states of the same symmetry. Both kinds of correlation have a significant impact on the nodal structure and therefore govern the node location error. In this thesis the isomerization of bicylo[1.1.0]butane has been studied by exploiting the DMC method. The underlying reaction mechanism poses a great challenge to quantum chemistry because it comprises species with different amounts of non-dynamical correlation. Indeed, very sophisticated methods like for example CCSD(T) fail in giving a proper description. For the DMC method, the construction of the trial function is of crucial importance. Hence, the latter is generated by means of CASSCF calculations. With regard to efficient error cancellation it is most reasonable to take only those CSFs into account which contribute significantly to the proper description of the non-dynamic correlation. Since the DMC method is variational for the fermionic ground state, the incorporation of such a CSFs leads to a decrease of the energy compared with the output of a computation just based on the reference determinant. In the present case, only up to three CSFs very necessary. As a consequence, the direct optimization of the DMC energy with respect to the CASSCF coefficients is possible. By means of this construction scheme, dynamic and non-dynamic electron correlation effects are accounted for in equal measure for every molecule comprising the reaction path. Hence, the error cancellation is excellent. The obtained reaction, and activation energies are in very good agreement with experimental data and completely renormalized coupled cluster calculations. The presented CSF selection scheme yields very short linear combinations of CSFs and therefore should be applicable even for larger organic molecules. As it has already pointed out, quantum chemistry aims the prediction of energy differences rather than absolute values. A very efficient approach to calculate energy differences within a Monte Carlo framework is correlated sampling. The main idea behind this procedure is to describe the two different systems whose energy difference is to be calculated by means of the same random walk process. The resulting statistical correlation gives rise to a reduction of the statistical uncertainty of the energy difference. As far as we know, until now no implementation of the correlated sampling formalism into the DMC method has been reported in the literature. To implement correlated sampling in the DMC method a mathematical formalism based on the theory of stochastic differential equations has been devised. Based on this formalism an appropriate algorithm has been implemented for test systems like the particle in a one dimensional box, and the harmonic oszillator. Correlated sampling is very well suited for the computation of small energy differences with considerable small statistical uncertainties. A possible application would be the calculation of the weak interaction between DNA base pairs. For this purpose the corresponding formalism has to be implemented in our quantum Monte Carlo program Amolqc. In doing so, a major task is the implementation of routines which account for the propagation with respect to a positive guide function what has already been finished in the course of this work.