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Journal of Physics Condensed Matter

Real-space finite-difference approach for multi-body systems: path-integral renormalization group method and direct energy minimization method

Akira Sasaki, Masashi Kojo, Kikuji Hirose, Hidekazu Goto

Published Oct 13, 2011• 3 citations
Publisher

Abstract

The path-integral renormalization group and direct energy minimization method of practical first-principles electronic structure calculations for multi-body systems within the framework of the real-space finite-difference scheme are introduced. These two methods can handle higher dimensional systems with consideration of the correlation effect. Furthermore, they can be easily extended to the multicomponent quantum systems which contain more than two kinds of quantum particles. The key to the present methods is employing linear combinations of nonorthogonal Slater determinants (SDs) as multi-body wavefunctions. As one of the noticeable results, the same accuracy as the variational Monte Carlo method is achieved with a few SDs. This enables us to study the entire ground state consisting of electrons and nuclei without the need to use the Born-Oppenheimer approximation. Recent activities on methodological developments aiming towards practical calculations such as the implementation of auxiliary field for Coulombic interaction, the treatment of the kinetic operator in imaginary-time evolutions, the time-saving double-grid technique for bare-Coulomb atomic potentials and the optimization scheme for minimizing the total-energy functional are also introduced. As test examples, the total energy of the hydrogen molecule, the atomic configuration of the methylene and the electronic structures of two-dimensional quantum dots are calculated, and the accuracy, availability and possibility of the present methods are demonstrated.

DOI
10.1088/0953-8984/23/43/434001

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The path-integral renormalization group and direct energy minimization method of practical first-principles electronic structure calculations for multi-body systems within the framework of the real-space finite-difference scheme are introduced. These two methods can handle higher dimensional systems with consideration of the correlation effect. Furthermore, they can be easily extended to the multicomponent quantum systems which contain more than two kinds of quantum particles. The key to the present methods is employing linear combinations of nonorthogonal Slater determinants (SDs) as multi-body wavefunctions. As one of the noticeable results, the same accuracy as the variational Monte Carlo method is achieved with a few SDs. This enables us to study the entire ground state consisting of electrons and nuclei without the need to use the Born-Oppenheimer approximation. Recent activities on methodological developments aiming towards practical calculations such as the implementation of auxiliary field for Coulombic interaction, the treatment of the kinetic operator in imaginary-time evolutions, the time-saving double-grid technique for bare-Coulomb atomic potentials and the optimization scheme for minimizing the total-energy functional are also introduced. As test examples, the total energy of the hydrogen molecule, the atomic configuration of the methylene and the electronic structures of two-dimensional quantum dots are calculated, and the accuracy, availability and possibility of the present methods are demonstrated.
An approach for obtaining the ground state of Coulombic many-body systems is presented. This approach is based on the path-integral renormalization-group method with nonorthogonal Slater determinants, is free of the negative sign problem, and can handle higher dimensional systems with consideration of the correlation effect. Furthermore, it can be easily extended to the multicomponent quantum systems that contain more than two kinds of quantum particles. According to our results obtained with the present approach, it achieves the same accuracy as the variational Monte Carlo method with a few Slater determinants and enables us to study the entire ground state consisting of electrons and nuclei without the need to use the Born-Oppenheimer approximation.
A new efficient numerical algorithm for interacting fermion systems is proposed and examined in detail. The ground state is expressed approximately by a linear combination of numerically chosen basis states in a truncated Hilbert space. Two procedures lead to a better approximation. The first is a numerical renormalization, which optimizes the chosen basis and projects onto the ground state within the fixed dimension, LLL, of the Hilbert space. The second is an increase of the dimension of the truncated Hilbert space, which enables the linear combination to converge to a better approximation. The extrapolation L→∞L\to\inftyL→∞ after the convergence removes the approximation error systematically. This algorithm does not suffer from the negative sign problem and can be applied to systems in any spatial dimension and arbitrary lattice structure. The efficiency is tested and the implementation explained for two-dimensional Hubbard models where Slater determinants are employed as chosen basis. Our results with less than 400 chosen basis indicate good accuracy within the errorbar of the best available results as those of the quantum Monte Carlo for energy and other physical quantities.
In calculating energy spectra, unlimited accuracy is only obtained if N-body wavefunctions contain N-body correlations except in special cases. These wave-functions are however not practical both for analytic and computational considerations. To this extent, the possiblity of utilizing correlations less than N is investigated. These investigations are based on the hypothesis whereby in D-dimensions, correlations greater than (D + 1)-body correlations are considerably less important. To accomplish these investigations, we compare the quality Q of the wavefunctions, pertaining to two-, three- and four-body correlations, in various dimensions. The trial wavefunctions used in our Monte Carlo method are linear combinations of elementary basis functions that consist of non-linear variational parameters. This method entails a linear optimization nested in a non-linear one. Once these trial wavefunctions are optimized we employ them as basis functions in a correlation function Monte Carlo which reduces systematically the variational bias of the energy estimates. This method is used to calculate ground and excited state energies of small bosonic van der Waal clusters using wavefunctions with two-, three- and four-body correlations. In calculating energy spectra in different dimensions, it was observed that for N atoms there exists an inter-dimensional degeneracy in the energy spectrum for (N − 1)-dimensions and (N + 1)-dimensions. The finite difference method is also used to calculate 'exact' energy spectra of two-body clusters, and these results are used as a benchmark to compare results obtained by our Monte Carlo method. We also describe a method of obtaining better statistics from quantum Monte Carlo estimators based on the assumption that these estimators are afflicted with distributions that resemble the lognormal probability distribution.
We introduce the Nuclear-Electronic All-Particle Density Matrix Renormalization Group (NEAP-DMRG) method for solving the time-independent Schrödinger equation simultaneously for electrons and other quantum species. In contrast to the already existing multicomponent approaches, in this work, we construct from the outset a multi-reference trial wave function with stochastically optimized non-orthogonal Gaussian orbitals. By iterative refining of the Gaussians’ positions and widths, we obtain a compact multi-reference expansion for the multicomponent wave function. We extend the DMRG algorithm to multicomponent wave functions to take into account inter- and intra-species correlation effects. The efficient parameterization of the total wave function as a matrix product state allows NEAP-DMRG to accurately approximate the full configuration interaction energies of molecular systems with more than three nuclei and 12 particles in total, which is currently a major challenge for other multicomponent approaches. We present the NEAP-DMRG results for two few-body systems, i.e., H2 and H3+, and one larger system, namely, BH3.
A numerical algorithm for studying strongly correlated electron systems is proposed. The groundstate wavefunction is projected out after numerical renormalization procedure in the path integral formalism. The wavefunction is expressed from the optimized linear combination of retained states in the truncated Hilbert space with numerically chosen basis. This algorithm does not suffer from the negative sign problem and can be applied to any type of Hamiltonian in any dimension. The efficiency is tested in examples of the Hubbard model where the basis of Slater determinants is numerically optimized. We show results on fast convergence and accuracy achieved with small number of retained states.
In the eigen-functional theoretical framework, exact/accurate treatment of high-dimensional strongly correlated electron systems may be realized. The two key steps of this method are that under the path integral formulation we first change a D-dimensional quantum many-particle system into an (D+1)-dimensional (time-dependent) effective "single-particle" problem, then by solving the eigen-equation of the propagator operator of the particles, we can obtain the ground state energy functional and action, respectively. Under the eigen-functional theory, the problems of quantum many-particle systems end in to solve the equation of phase fields that are completely determined by the electron interaction. In practice this equation of the phase fields can be numerically solved for large lattice sites. After replacing the real time by the imaginary time, this method can also be applied for finite temperature cases.