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arXiv:1205.6503

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 May 29, 2012• 0 citations
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Abstract

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 $N$ 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 (DFT) are discussed. We propose also the possibility of going beyond the energy densities and extend this idea to a computational procedure where the $N-1$-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.

DOI
10.48550/arxiv.1205.6503

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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.
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 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.
Abstract It is difficult to imagine that density functional theory (DFT) could be useful in improving quantum Monte Carlo methods, but it is easy to imagine that QMC could be useful in improving density functional methods. Since QMC can produce accurate wavefunctions and since wavefunctions lead to all other properties one should be able, in principle, to use QMC to predict the exchangecorrelation energy functional Exc[n(r)], the key (unknown) quantity required in DFT. In practice, that is not so easy, but this paper shows how QMC can take steps in that direction by providing insights gained from comparisons of results from QMC and DFT. The authors considered the case of the strongly inhomogeneous electron gas using variational Q:tvIC to obtain wavefunctions 1/J>. and energies E[7/J>.] for the ground state of the Hamiltonian iI>- = T +AVee+V >. associated with the coupling constant A in the range 0 to 1. Optimization of a Slater-Jastrow many-body wavefunction was made by minimizing a penalty function given by a weighted sum of the variance in local energies and a sum of squares of electron density differences. The variational parameters in 7/J>. and v>. were repeatedly varied to reach convergence followed by additional calculations to obtain the local values of the exchange-correlation energies e c and the exchange-correlation holes n c from which Exe and nxc could be determined.
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...
to develop quantum Monte Carlo (QMC) approaches for excited states and for more systematic and accurate treatment of ground states of strongly correlated sytems and heavier elements. We use the auxiliary-field quantum Monte Carlo (AFQMC) framework, which is a many-body total energy method that looks like many coupled mean-field calculations. I will discuss this method’s relations and dierences with traditional density-functional and quantum chemistry methods, and with other forms of QMC. The AFQMC method uses importance-sampled random walks in the space of nonorthogonal Slater determinants to project out the many-body ground state, or excited state. The random walks are constrained by a trial wave function (TWF) to remove the so-called sign problem, giving approximate (but non-perturbative) results. With singledeterminant TWFs from Hartree-Fock or density-functional theory (DFT), this approach can be thought of as a post-DFT method with no parameter. In this form, the accuracy is typically comparable to the quantum chemistry coupled-cluster CCSD(T) method near equilibrium geometries, and better than CCSD(T) when bonds are stretched. The computational cost of AFQMC scales as N 3 with characteristic size of the system. Its accuracy can be further improved at the cost of increased computing (releasing the constraint) or better TWF. With AFQMC, we can systematically target selected orbitals to correlate. This allows many-body calculations in solids with a frozen core, with down-folded Hamiltonians, or possibly embeded in a DFT calculations to extend scales. Results from a few on-going applications will be presented, including band structures in simple solids, magnetic phases in MnO, and Cobalt adsoption on graphene.
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.