Ensemble Density-Functional Theory forAb InitioMolecular Dynamics of Metals and Finite-Temperature Insulators

Abstract

A new method is presented for performing first-principles molecular-dynamics simulations of systems with variable occupancies. We adopt a matrix representation for the one-particle statistical operator Γ, to introduce a "projected" free energy functional G that depends on the Kohn-Sham orbitals only and that is invariant under their unitary transformations. The Liouville equation [Γ,Ĥ] = 0 is always satisfied, guaranteeing a very efficient and stable variational minimization algorithm that can be extended to non-conventional entropic formulations or fictitious thermal distributions.In recent years, the range of problems that can be studied with quantitative accuracy using the methods of computational solid state physics has expanded dramatically. It is now possible to calculate many materials properties with an accuracy that is often comparable to that of experiments. This degree of confidence is based on the fundamental quantum-mechanical treatment offered by density-functional theory (DFT) [1], coupled with the availability of increasingly powerful computers and with the development of algorithms tuned towards optimal performance [2].The application of these methods and techniques to metallic systems has nonetheless encountered several difficulties, that have made progress slower than for the case of semiconductors and insulators. The discontinuous variation of the orbital occupancies across the Brillouin zone (BZ) makes the occupation numbers rather ill-conditioned variables, and the self-consistent solution of the screening problem can suffer from several instabilities. The absence of a gap in the energy spectrum and the requirement of an exact diagonalization for the Hamiltonian matrix everywhere in the BZ (in order to assign the occupation numbers) introduce "slow frequencies" in the evolution of the orbitals towards the ground state and preclude the straightforward extension to metals of algorithms which performed well for insulators. Smearing the Fermi surface with a finite electronic temperature [3] allows for an improved BZ sampling, but only partially alleviates the problems alluded to above.In this Letter, we introduce a new approach which solves many of these problems in a natural way, and which provides a general and efficient framework for obtaining the ground state of a Kohn-Sham Hamiltonian at a finite electronic temperature. The typical context is the Mermin formulation for the Fermi-Dirac statistics [4], but the method also applies when generalized entropic functionals are introduced [5], as it is often the case for metallic systems. Other applications include DFT studies of insulators or semiconductors with thermally excited states [6], and fractional quantum Hall states [7]. The language of ensemble-DFT [8] is used, and an orbitalbased variational algorithm for the minimization to the ground state is developed and implemented. Dramatic improvements are obtained in the convergence of the energies and especially of the Hellmann-Feynman forces.Within ensemble-DFT, the Helmholtz free energy fun...

Absolute proton hydration free energy, surface potential of water, and redox potential of the hydrogen electrode from first principles: QM/MM MD free-energy simulations of sodium and potassium hydration

“…For the CPMD/BOMD calculations, error sources include the shortcomings of DFT methods in terms of electron correlation (including dispersion), affecting both bulk water properties [121][122][123][124][125][126][127][128][129][130] and ion-water interactions, 66,67,105,131 the small system sizes considered (resulting in a high weight for the approximate finite-size correction term), the very limited sampling times, the ambiguity in defining the appropriate reference potential within DFT water, 21,112,[132][133][134][135][136] and the numerical problems (see, e.g., footnotes 42 and 45 in Ref. 111) associated with creating a species or injecting electrons when applying computational alchemy in a QM framework [137][138][139][140][141][142][143] (see, however, Refs. 114 and 144-149 for possible grandcanonical alternatives).…”

Extended Lagrangian free energy molecular dynamics

“…14,15,60,64 In this paper, first principles free energy molecular dynamics schemes were developed based on extended Lagrangian Born-Oppenheimer molecular dynamics. 3,4,10,34 The formulation was given both for density matrix and plane wave representations.…”

Preconditioned iterative minimization for linear-scaling electronic structure calculations

“…It is worth noting that all of the occupation numbers f n for the model system have been set to unity. This amounts to an additional occupancy preconditioning, first introduced by Gillan 36 in the context of metallic systems and then by Marzari et al 29 in the general framework of ensemble density-funtional theory.…”

“…We have adopted the Einstein summation convention for all repeated Greek suffixes, and continue to do so from here on. It is both convenient and physically meaningful to perform the minimization of the energy functional in two nested loops, as in the ensemble density-functional method of Marzari et al: 29 In the inner loop we minimize the energy with respect to the elements of the density kernel K ␣␤ using one of a number of methods [30][31][32] to impose the constraint that the ground state density matrix be idempotent and give the correct number of electrons; in the outer loop we optimize the localized functions ␣ (r) with respect to their coefficients c ␣ in the basis D (r). 16 …”