Recently, Gisin introduced a Heisenberg picture of the quantum-state diffusion model and considered correlation functions in this framework. It is shown here that these correlation functions in general do not coincide with the usual functions, in contrast with what one might have expected. Furthermore it is pointed out how conventional correlation functions can be calculated using the quantum-state diffusion model. For higher-order correlation functions this approach may offer advantages compared with the Monte Carlo wavefunction method.
Recently, Gisin introduced a Heisenberg picture of the quantum-state diffusion model and considered correlation functions in this framework. It is shown here that these correlation functions in general do not coincide with the usual functions, in contrast with what one might have expected. Furthermore it is pointed out how conventional correlation functions can be calculated using the quantum-state diffusion model. For higher-order correlation functions this approach may offer advantages compared with the Monte Carlo wavefunction method.
An alternative approach to conventional quantum chemistry techniques for molecular studies is described.This approach uses the quantum Monte Carlo method, which was developed and used primarily in nuclear and condensed-matter physics.In this approach the many-body Schrodinger equation is re-interpreted as a diffusion equation.Simulation of the appropriate random walk process allows one to calculate expectation values of molecular properties.In principle these expectation values can be calculated exactly, subject only to statistical errors (which may be made arbitrarily small).In preliminary work on small molecules (H 2 , LiH, Li 2 , and H 2 0), we use a simple but accurate approximation to ease the treatment of Fermi statistics.In that approximation, the calculated total energy remains an upper bound to the true energy.How good the bound is, as well as the magnitude of the statistical error, depends on the quality of an "importance function" which guides the diffusion into more probable regions of phase space.With relatively simple importance functions, we obtain from 75-100% of the correlation energy of the above-mentioned molecules.
Quantum confinement in 2D materials strongly enhances electronic correlation effects. Therefore, predicting the properties of these unique materials, with both a high level of accuracy and computational efficiency, without relying on adjustable parameters or functionals, remains an outstanding theoretical challenge. The majority of theoretical studies are based on the approximations of density functional theory (DFT). The reliability of DFT predictions are heavily dependent on the choice of an approximated exchange-correlation functional. Here, we estimate the magnitude of impact of correlation on the total energy for the quintessential 2D material, graphene, by performing and comparing state-of-the-art selected CI and quantum Monte Carlo extrapolated calculations for a single unit cell at the Γ point. We demonstrate that Self-Healing Diffusion Monte Carlo (SHDMC) obtains a very compact, but high-quality wavefunction for this system that lacks the strong basis set dependence displayed by state of the art quantum chemistry methods. The SHDMC wavefunction is of higher quality compared to that obtained from sCI, in the same orbital basis, while being ∼ 1000 times smaller in terms of determinant count compared to sCI. We also demonstrate that extrapolating SHDMC results to the infinite determinant limit compares extremely well with complete basis set extrapolated sCI. Our work paves the way for future validation of SHDMC applied to challenging 2D materials.
In computing the spectra of quantum mechanical systems one encounters the Fourier transforms of time correlation functions, as given by the quantum regression theorem for systems described by master equations. Quantum state diffusion (QSD) gives a useful method of solving these problems by unravelling the master equation into stochastic trajectories; but there is no generally accepted definition of a time correlation function for a single QSD trajectory. In this paper we show how QSD can be used to calculate these spectra directly; by formally solving the equations which arise, we arrive at a natural definition for a two-time correlation function in QSD, which depends explicitly on the stochastic noise of the particular trajectory, and which agrees in the mean with the ensemble average definition of correlation functions.
In computing the spectra of quantum mechanical systems one encounters the Fourier transforms of time correlation functions, as given by the quantum regression theorem for systems described by master equations. Quantum state diffusion (QSD) gives a useful method of solving these problems by unravelling the master equation into stochastic trajectories; but there is no generally accepted definition of a time correlation function for a single QSD trajectory. In this paper we show how QSD can be used to calculate these spectra directly; by formally solving the equations which arise, we arrive at a natural definition for a two-time correlation function in QSD, which depends explicitly on the stochastic noise of the particular trajectory, and which agrees in the mean with the ensemble average definition of correlation functions.
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.