This thesis is composed of two parts. In part one we report results of two quantum Monte Carlo methods - variational Monte Carlo and diffusion Monte Carlo - on the potential energy curve of the helium dimer, the prototypical van der Waals system. In contrast to previous quantum Monte Carlo calculations on this system, we have employed trial wave functions of the Slater-Jastrow form and used the fixed node approximation for the fermion nodal surface. We find both methods to be in excellent agreement with the best theoretical results at short range. In addition, the diffusion Monte Carlo results give very good agreement across the whole potential energy curve, while the Slater-Jastrow wave function fails to bind the dimer at all. In part two we switch to investigations of many-body systems at finite temperature. We use the path integral representation of statistical mechanics to investigate the symmetry properties of the canonical ensemble partition function and find a representation in terms of irreducible representations of the symmetric group. We use this as a foundation to propose a novel technique for the stochastic sampling of the bosonic partition function for quantum gases. It is shown that in principle we are able to use two operators which enable us to construct a Markov chain through a graph of the irreducible representation of the symmetric group. As an illustration of this method, a test calculation of four particles in a harmonic trap is performed.
This thesis is composed of two parts. In part one we report results of two quantum Monte Carlo methods - variational Monte Carlo and diffusion Monte Carlo - on the potential energy curve of the helium dimer, the prototypical van der Waals system. In contrast to previous quantum Monte Carlo calculations on this system, we have employed trial wave functions of the Slater-Jastrow form and used the fixed node approximation for the fermion nodal surface. We find both methods to be in excellent agreement with the best theoretical results at short range. In addition, the diffusion Monte Carlo results give very good agreement across the whole potential energy curve, while the Slater-Jastrow wave function fails to bind the dimer at all. In part two we switch to investigations of many-body systems at finite temperature. We use the path integral representation of statistical mechanics to investigate the symmetry properties of the canonical ensemble partition function and find a representation in terms of irreducible representations of the symmetric group. We use this as a foundation to propose a novel technique for the stochastic sampling of the bosonic partition function for quantum gases. It is shown that in principle we are able to use two operators which enable us to construct a Markov chain through a graph of the irreducible representation of the symmetric group. As an illustration of this method, a test calculation of four particles in a harmonic trap is performed.
We report results of two quantum Monte Carlo methods -- variational Monte Carlo and diffusion Monte Carlo -- on the potential energy curve of the helium dimer. In contrast to previous quantum Monte Carlo calculations on this system, we have employed trial wave functions of the Slater-Jastrow form and used the fixed node approximation for the fermion nodal surface. We find both methods to be in excellent agreement with the best theoretical results at short range. In addition, the diffusion Monte Carlo results give very good agreement across the whole potential energy curve, while the Slater-Jastrow wave function fails to bind the dimer at all.
Trial wave function based quantum Monte Carlo is a promising family of methods for the solution of quantum mechanical Hamiltonians. These fundamentally non-perturbative methods can be used to treat bosons or fermions in weakly or strong interacting systems and in any phase. Properties computed for particles with bosonic statistics can be converged to their exact value at relatively moderate computational cost using diffusion quantum Monte Carlo. For fermions the problem is slightly more complicated. Exact methods, such as released-node diffusion quantum Monte Carlo, exist for computing unbiased fermion properties, however their computational expense increases rapidly with the Bose-Fermi energy gap and system size. It is possible to use a fixed-node version of diffusion quantum Monte Carlo, but it introduces a bias due to the nodes of the trial wave function. In this thesis we work towards reducing trial wave function bias in fixed-node calculations and then perform several benchmark studies. We begin with a pedagogical overview of the three methods used throughout the thesis, variational, diffusion, and reptation quantum Monte Carlo. Then we discuss the problem, trial wave function bias in the energy and other observables. Next we outline the Hamiltonians we are typically interested in for electronic systems and trial wave function forms used to solve them. The algorithm used to optimize the trial wave functions is presented along with special considerations for some particular cases. Next we present three studies of the pressure of the electron gas using improved estimators in variational, diffusion, and reptation quantum Monte Carlo. Benchmark results for a set of molecules are then presented for the massive multi-determinant expansion and optimization algorithm tailored to suit it. We conclude with two studies of the interacting Fermi liquid, one on the momentum properties of the electron gas in three dimensions and another on the entanglement properties of interacting Fermi liquids in two dimensions.
In this Ph.D. thesis quantum Monte Carlo methods are applied to investigate the properties of a number of ultracold quantum systems. In Chapter 1 we discuss the analytical approaches and approximations used in the subsequent Chapters; also we describe the general concepts of the two-particle scattering problem as a tool to construct Jastrow terms in trial wave functions. Chapter 2 explains in details the Quantum Monte Carlo methods employed in our calculations from the theoretical and practical points of view. In Chapter 3 we explain the Ewald summation technique, applied to a power-law 1/|r|k interaction potential, and a generic approach to obtain the Ewald terms. The obtained expressions of this analytic work are implemented into simulations of different physically relevant systems (Rydberg atoms and Yukawa particles). Chapter 5 is devoted to the modelling of a system, governed by the model potential between Rydberg atoms 1/r6. The phase diagram of the system is obtained for a relevant range of densities and temperatures, combining quantum simulations at low temperature and classical treatment at higher temperature. A special attention is paid to the classical description of this system, composed of Rydberg atoms, and its comparison to the quantum system. In Chapter 4 we present the simulation of a system with the Yukawa interaction potential. The following Chapter 6 presents the results of the Quantum Monte Carlo simulations of molecular para-hydrogen at zero and finite temperatures, performed in our Group. Conclusions are drawn in Chapter 7.
Helium systems at zero temperature are strongly correlated neutral systems which can only be described by means of quantum mechanics. The existence of the two isotopes 4He and 3He allows the comparison between the quantum effects due to different statistics, bosonic and fermionic. In this Thesis, an analysis of Helium droplets has been done. Throughout this work, both energetic and structural properties of systems with different number of particles N are obtained by means of the Quantum Monte Carlo techniques (Variational and Diffusion Monte Carlo - VMC and DMC-).
Several studies on Helium droplets have been published in the last years, most of them using the mean field theory (as the Density Functional) and variational approaches (as VMC). In the case of 4He, also exact DMC results have already been reported, actually, a great number of works have described with high accuracy the ground state of this bosonic system. The appearance of magic numbers from experimental point of view has motivated the study of 4He droplets, and one chapter of this Thesis is focused on this bosonic system. We have carried out a one-by-one energetic study in the range from 20 to 30 particles using the DMC technique to discuss the existence of magic numbers. On the other hand, a brief study about vortices is also done; using out code, one can calculate the energy of a droplet with one vortex inside (EV) and, therefore, the energy of a vortex (EV -E0).
About 3He droplets, the sign problem makes their study quite more difficult, for this reason, the number of works is considerably smaller than in the bosonic case. One aim of this thesis was to write a DMC code using both Fixed and Released Node approximations (FN and RN) able to perform high accurate calculations of energetic and structural properties of these finite fermionic systems.
Quantum chemistry is a useful tool that provides insight into the properties and behavior of chemical systems. Modern software packages have made quantum chemistry methods more easily accessible, and the continued increase in available computational resources has allowed them to be applied to larger systems at higher levels of theory. Two significant problems that the field faces are the high computational complexity of high-level methods and the shift toward parallelism in high performance computing architectures. This work examines the treatment of weakly interacting molecular systems with the fixed-node diffusion Monte Carlo (DMC) method. DMC and other quantum Monte Carlo (QMC) methods offer a possible solution to both of the aforementioned problems: they can produce near-exact results with a lower scaling (with respect to problem size) than other similarly-accurate methods, and they are inherently parallel, so there is little additional cost associated with distributing the work of a single QMC calculation across a large number of processing units. The only error in DMC that is not systematically improvable is the constraint of a fixed nodal surface of the many-particle wave function of the system being studied. There are many cases in which a single Slater determinant trial function is sufficient to obtain accurate results, but there are others in which more sophisticated multi-determinant trial functions are necessary. Furthermore, it is non-trivial to generate nodal surfaces of similar quality for isolated and interacting molecules, so cancellation of errors is not guaranteed. We examine the use of different single- and multi-determinant trial functions in DMC calculations on small chemical systems with the goal of further understanding how to construct appropriate trial functions for molecules and clusters.
Introduction In Chapters 1 to 4 we studied methods for solving the Schrödinger equation for many-electron systems. Many of the techniques described there carry over to other quantum many-particle systems, such as liquid helium, and the protons and neutrons in a nucleus. The techniques which we discussed there were, however, all of a mean-field type and therefore correlation effects could not be taken into account without introducing approximations. In this chapter, we consider more accurate techniques, which are similar to those studied in Chapter 10 and are based on using (pseudo-)random numbers – hence the name ‘Monte Carlo’ for these methods. In Chapter 10 we applied Monte Carlo techniques to classical many-particle systems; here we use these techniques for studying quantum problems involving many particles. In the next section we shall see how we can apply Monte Carlo techniques to the problem of calculating the quantum mechanical expectation value of the ground state energy. This is used in order to optimise this expectation value by adjusting a trial wave function in a variational type of approach, hence the name variational Monte Carlo (VMC). In the following section we use the similarity between the Schrödinger equation and the diffusion equation in order to calculate the properties of a collection of interacting quantum mechanical particles by simulating a classical particle diffusion process. The resulting method is called diffusion Monte Carlo (DMC).