Browsing by Author "Prof. Dean Lee, Committee Member"

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    Generalized Pairing Wave Functions and Nodal Properties for Electronic Structure Quantum Monte Carlo
    (2007-04-06) Bajdich, Michal; Prof. Lubos Mitas, Committee Chair; Prof. Dean Lee, Committee Member; Prof. Christopher Roland, Committee Member; Prof. Jerry L. Whitten, Committee Member
    The quantum Monte Carlo (QMC) is one of the most promising many-body electronic structure approaches. It employs stochastic techniques for solving the stationary Schr" odinger equation and for evaluation of expectation values. The key advantage of QMC is its capability to use the explicitly correlated wave functions, which allow the study of many-body effects beyond the reach of mean-field methods. The most important limit on QMC accuracy is the fixed-node approximation, which comes from necessity to circumvent the fermion sign problem. The size of resulting fixed-node errors depends on the quality of the nodes (the subset of position space where the wave function vanishes) of a used wave function. In this dissertation, we analyze the nodal properties of the existing fermionic wave functions and offer new types of variational wave functions with improved nodal structure. In the first part of this dissertation, we study the fermion nodes for spin-polarized states of a few-electron ions and molecules with $s$, $p$, $d$ and $f$ one-particle orbitals. We find exact nodes for some cases of two electron atomic and molecular states and also the first exact node for the three-electron atomic system in $ˆ4S(pˆ3)$ state using appropriate coordinate maps and wave function symmetries. We analyze the cases of nodes for larger number of electrons in the Hartree-Fock approximation and for some cases we find transformations for projecting the high-dimensional nodal manifolds into 3D space. The nodal topologies and other properties are studied using these projections. Finally, for two specific cases of spin-unpolarized states, we show how correlations reduce the nodal structure to only two maximal nodal cells. In the second part, we investigate several types of trial wave functions with pairing orbitals and their nodal properties in the fixed-node quantum Monte Carlo. Using a set of first row atoms and molecules we find that the wave functions in the form of single Pfaffian provide very consistent and systematic behavior in recovering the correlation energies on the level of 95%. In order to get beyond this limit we explore the possibilities of expanding the wave function in linear combinations of Pfaffians. We observe that molecular systems require much larger expansions than atomic systems and that the linear combinations of a few Pfaffians lead to rather small gains in correlation energy. Further, we test the wave function based on fully-antisymmetrized product of independent pair orbitals. Despite its seemingly large variational potential, we do not observe significant gains in correlation energy. Finally, we combine these developments with the recently proposed inhomogeneous backflow transformations.
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    Non-Trivial Vacuum Solutions of Low Dimensional Scalar Field Theories in the Oscillator Representation Method.
    (2005-08-13) Shalaby, Abouzeid Mohammed; Prof. Chueng-Ryong Ji, Committee Chair; Prof. Dean Lee, Committee Member; Prof. Lung O. Chung, Committee Member; Prof. Thomas Schaefer, Committee Member
    This dissertation presents a study of self-organizing nature of quantum field theories. In particular, we study the low-dimensional scalar field theories using a nonpperturbative approach known as the Oscillator Representation (OR) method which appears to be simpler than other well-known nonp-perturbative approaches such as the Hartree Approximation (HA) and the Gaussian Effective Potential (GEP) method. The key idea of the OR method is to make canonical transformations between the original particle theory and the quasi-particles theory and find non-trivial vacuum solutions which satisfy the self-consistency conditions required by a desired form of the quasi-particles effective Hamiltonian. In the low-dimensional scalar field theories, we find the duality property between the original particle theory and the quasi-particles theory, i.e. the original nonp-perturbative strong interaction theory is equivalent to the weakly interacting quasi-particles theory. We present explicit examples of the duality which allows the conversion of the original nonp-perturbative strong interaction problem into a weekly interacting quasi-particles problem that can be solved by the usual perturbative analysis. We also make a link between the OR method and the Effective Action approach with a specific canonical transformation of field shifts. However, we point out a difficulty of these nonp-perturbative methods (OR, HA, GEP) in describing the order of the phase transition near the critical coupling region where the nontrivial vacuum condensations occur. For the case of OR method, we present a detailed prescription how one can overcome this difficulty using the Borel summation technique and the Kleinert algorithm. In the example of (4)1+1 theory, we show the recovery of correct second order phase transition with this improvement. The OR applications to the 2+1 dimensional scalar field theories of $phiˆ4$ and $phiˆ6$ interactions are also detailed with the description of the renormalization procedure for the different levels of loop calculations such as involving one-loop, normal-ordering and two-loop regularizations. Various numerical results of nontrivial vacuum solutions, including their energy densities, quasi-particles mass spectra and classical effective potentials, are discussed along with their symmetry properties.