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International Journal of Quantum Chemistry

Electronic energy functionals: Levy–Lieb principle within the ground state path integral quantum Monte Carlo

Luigi Delle Site, Luca M. Ghiringhelli, David M. Ceperley

Published Aug 30, 2012• 10 citations
Publisher

Abstract

Abstract We propose a theoretical/computational protocol based on the use of the Ground State Path Integral Quantum Monte Carlo for the calculation of the kinetic and Coulomb energy density for a system of N interacting electrons in an external potential. The idea is based on the derivation of the energy densities via the (N − 1)‐conditional probability density within the framework of the Levy–Lieb constrained search principle. The consequences for the development of energy functionals within the context of density functional theory are discussed. We propose also the possibility of going beyond the energy densities and extend this idea to a computational procedure where the (N − 1)‐conditional probability is an implicit functional of the electron density, independently from the external potential. In principle, such a procedure paves the way for an on‐the‐fly determination of the energy functional for any system. © 2012 Wiley Periodicals, Inc.

DOI
10.1002/qua.24321

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Abstract We propose a theoretical/computational protocol based on the use of the Ground State Path Integral Quantum Monte Carlo for the calculation of the kinetic and Coulomb energy density for a system of N interacting electrons in an external potential. The idea is based on the derivation of the energy densities via the (N − 1)‐conditional probability density within the framework of the Levy–Lieb constrained search principle. The consequences for the development of energy functionals within the context of density functional theory are discussed. We propose also the possibility of going beyond the energy densities and extend this idea to a computational procedure where the (N − 1)‐conditional probability is an implicit functional of the electron density, independently from the external potential. In principle, such a procedure paves the way for an on‐the‐fly determination of the energy functional for any system. © 2012 Wiley Periodicals, Inc.
We propose a theoretical/computational protocol based on the use of the Ground State (GS) Path Integral (PI) Quantum Monte Carlo (QMC) for the calculation of the kinetic and Coulomb energy density for a system of NNN interacting electrons in an external potential. The idea is based on the derivation of the energy densities via the N−1N-1N−1-conditional probability density within the framework of the Levy-Lieb constrained search principle. The consequences for the development of energy functionals within the context of Density Functional Theory (DFT) are discussed. We propose also the possibility of going beyond the energy densities and extend this idea to a computational procedure where the N−1N-1N−1-conditional probability is an implicit functional of the electron density, independently from the external potential. In principle, such a procedure paves the way for an {\it on-the-fly} determination of the energy functional for any system.
We consider a gas of interacting electrons in the limit of nearly uniform density and treat the one dimensional (1D), two dimensional (2D) and three dimensional (3D) cases. We focus on the determination of the correlation part of the kinetic functional by employing a Monte Carlo sampling technique of electrons in space based on an analytic derivation via the Levy-Lieb constrained search principle. Of particular interest is the question of the behaviour of the functional as one passes from 1D to 3D; according to the basic principles of Density Functional Theory (DFT) the form of the universal functional should be independent of the dimensionality. However, in practice the straightforward use of current approximate functionals in different dimensions is problematic. Here, we show that going from the 3D to the 2D case the functional form is consistent (concave function) but in 1D becomes convex; such a drastic difference is peculiar of 1D electron systems as it is for other quantities. Given the interesting behaviour of the functional, this study represents a basic first-principle approach to the problem and suggests further investigations using highly accurate (though expensive) many-electron computational techniques, such as Quantum Monte Carlo.
We present a new method for realizing the adiabatic connection approach in density functional theory, which is based on combining accurate variational quantum Monte Carlo calculations with a constrained optimization of the ground state many-body wavefunction for different values of the Coulomb coupling constant. We use the method to study an electron gas in the presence of a cosine-wave potential. For this system we present results for the exchange-correlation hole and exchange-correlation energy density, and compare our findings with those from the local density approximation and generalized gradient approximation. 1. Introduction 2. The Adiabatic connection 3. Quantum Monte Carlo realization 3.1 Variational Monte Carlo 3.2 Fixed-density variance minimization 4. Cosine-wave jellium 4.1 Many-body wavefunction 5. Results and discussion 1 Introduction Density functional theory (DFT) (1, 2) is the main computational tool for the treatment of many-body effects in solid state electroni...
It is proven that in Levy's constrained search or Lieb's ensemble constrained search, if the minimum is achieved by wave functions that satisfy the steadiness condition for the current density kernel and have continuous second derivatives, the minimizing wave functions are eigenstates of some external potential. If the excited states of this potential can only achieve stationarity and cannot achieve a minimum in the constrained search, the minimizing wave functions are ground states and the corresponding electron density is v\mathit{v}v-representable or ensemble v\mathit{v}v-representable. This is also true for the ensemble system with a varying electron number. In time-dependent density-functional theory, if the stationarity of the action integral is achieved in the constrained search by a wave function that satisfies the continuity equation for the current density kernel and has a continuous first derivative with respect to the time coordinate and continuous second derivatives with respect to the space coordinates, the stationary wave function satisfies the time-dependent Schr\"odinger equation of some time-dependent external potential and the corresponding electron density is time-dependent v\mathit{v}v-representable.
The constrained-search formulation of Levy and Lieb provides a concrete mapping from NNN-representable densities to the space of NNN-particle wave functions and explicitly defines the universal functional of density-functional theory. We numerically implement the Levy-Lieb procedure for a paradigmatic lattice system, the Hubbard dimer, using a modified variational quantum eigensolver approach. We demonstrate density variational minimization using the resulting hybrid quantum-classical scheme featuring real-time computation of the Levy-Lieb functional along the search trajectory. We further illustrate a fidelity-based quantum kernel associated with the density to pure-state embedding implied by the Levy-Lieb procedure and employ the kernel for learning observable functionals of the density. We study the kernel's ability to generalize with high accuracy through numerical experiments on the Hubbard dimer.
We derive a functional form for the energy of interacting many-particle systems from first principles. Our main contribution is that we define the effective zero-point energy of an interacting system in terms of fluctuations of momentum and the Coulomb electric field. An uncertainty equation can be derived, relating the interaction energy in terms of the kinetic energy and local charge density. The Lieb-Thirring bound is then applied to constrain the kinetic energy which takes particle (anti-)symmetry into account, and the resulting functional form applies to fermionic as well as bosonic systems; in the uniform density case, the functional form agrees with the entire range of available Quantum Monte Carlo (QMC) data of many-particle systems, to within chemical accuracy. Furthermore, the functional form shows promising agreement with QMC phase transition data found in the literature, and in the high density case, where QMC data is not as readily available for a BCC Wigner state, predicts the stability of a BCC Wigner crystal agreeing qualitatively with Unrestricted Hartree-Fock. Seen in this way, many-particle wavefunctions are minimum uncertainty states in coherence with the electromagnetic bath, where the coupling is a self-generated scalar potential.