Recent Progress in Orbital-free Density Functional Theory

The following sections are included:

• Preface

• Contents

Two existing ways of deriving the Kohn-Sham equations were analyzed in great details. It turns out that both are incomplete in principle because they will lead to a paradox. It was further shown that the paradox can be resolved by introducing some arbitrary constants in the total Kohn-Sham effective potential. As a result, the functional derivative of the kinetic-energy density functional of the auxiliary non-interacting system within the Kohn-Sham method can be exactly calculated only from the highest occupied Kohn-Sham orbital , where I and are the first (lowest) ionization energy and the canonical representative of the total Kohn-Sham effective potential, respectively.

We describe a robust and practical method for the computation of implicit density functionals from electron densities based on a regularized version of the Wu-Yang method. For any given electron density that is non-interacting v-representable, our method provides a stable variational solution for calculating the non-interacting one-electron potential and the non-interacting kinetic energy. It is thus a computational approach to the challenge of constructing kinetic energy functionals. We also emphasize its application in the computation of the separate exchange and correlation components of the exchange-correlation potential.

After a short introduction to the semiclassical Thomas-Fermi (TF) method, and the gradient correction to its kinetic energy term proposed by von Weizsécker (vW), some attention is given to the differential form of the virial theorem. In one dimension, this allows the kinetic energy to be determined from the groundstate Fermion density ρ(r) generated by a given potential V(r). This is followed by a summary of the perturbative expansion of the Dirac density matrix γ(r, r′) in powers of the given three-dimensional potential V(r). This, following March and Murray, is shown, when r′ = r, to sum to the TF density-potential relation when V(r) varies only slowly in configuration space. A corresponding treatment of the kinetic energy density t(r) follows. Due to the interest in experiments in ultracold Fermion vapours, the case of harmonic confinement of independent Fermions is next summarized. In one dimension, the kinetic energy functional T[_ρ] is known, and has as building blocks the TF form ρ³(r) and the vW gradient correction term ρ′²(r)/ρ(r), even though T[ρ] is itself fully non-local in this example. For D dimensions and an arbitrary number of closed shells, the differential equation for the density is summarized, together with an explicit expression for the kinetic energy density as a functional of the Fermion ground-state density. The concept of the Pauli potential is then referred to, in relation to the functional density δT[ρ]/δρ(r) for a general potential V(r). As an example, the Pauli potential is given explicitly for the case of harmonically confined independent Fermions.

A brief section notes some significant generalizations when the potential V(r) of density functional theory is replaced by the non-local generalization V(r, r′), the most important case of this latter form being Hartree-Fock theory.

An orbital free ab initio scheme for the simulation of materials is described and applications of the scheme to liquid metal systems, the solid-liquid metal interface and to clusters are presented. The scheme is based on density functional theory and first principles pseudopotentials. The electron system is described entirely in terms of the electron density - it is free of orbitals - and the main approximation is the use of an explicit but approximate density functional for the electron kinetic energy. In addition, the absence of electron orbitals means that the pseudopotential must be local. The scheme is compared and contrasted with the popular Kohn-Sham approach in which the kinetic energy is treated exactly and accurate first principles non-local pseudopotentials are used but, at great computational cost. The error in the calculated total energy is found to be acceptable for a class of systems which includes so-called simple metals and their alloys but is not restricted to these e.g. good results have been obtained for Si and Ga. For these systems the orbital free scheme allows the simulations of much larger systems than can be treated with the full Kohn-Sham approach and long molecular dynamics runs can be achieved. The scheme is useful for investigating the static and dynamic properties of liquid systems and clusters.

In this chapter we provide an overview of the recently developed coarse-graining technique for orbital-free density functional theory that enables electronic structure calculations on multi-million atoms. The key ideas involved are: (i) a local real-space formulation of orbital-free density functional theory; (ii) a finite element discretization of the formulation; (iii) a systematic means of adaptive coarse-graining of the finite-element basis set using quasi-continuum reduction. The accuracy and effectiveness of the computational technique is demonstrated by studying the energetics of mono- and di-vacancies in multi-million atom aluminum crystals.

This paper presents the exploration of hot and dense matter by using orbital-free density-functional theory for the electrons coupled with molecular dynamics for the nuclei. Equations of state, as well as structure and transport coefficients, are computed from this formalism. By treating on an equal footing both pure elements and mixtures, the microscopic properties of mixtures are directly studied which allows the verification of standard mixing rules.

Orbital-free (OF) methods promise significant speed-up of computations based on density functional theory (DFT). In this field, the development of accurate kinetic-energy density functionals remains an open question. In this chapter we review the shell-correction method (SCM, commonly known as Strutinsky's averaging method) applied originally in nuclear physics and its more recent formulation in the context of DFT [Yannouleas and Landman, Phys. Rev. B 48, 8376 (1993)]. We demonstrate the DFT-SCM method through its earlier applications to condensed-matter finite systems, including metal clusters, fullerenes, and metal nanowires. The DFT-SCM incorporates quantum mechanical interference effects and thus offers an improvement compared to the use of Thomas-Fermi-type kinetic energy density functionals in OF-DFT.

In this chapter, we give an introduction of the finite element method for orbitalfree density functional theory. In particular, we review several efficient finite element discretization approaches to associated eigenvalue problems. We also include numerical examples for illustration.

The one-electron equation for orbitals embedded in frozen electron density (Eqs. 20-21 in [Wesolowski and Warshel, J. Phys. Chem, 97 (1993) 8050]) in its exact and approximated version is solved for an analytically solvable model system. The system is used to discuss the role of the embedding potential in preventing the collapse of a variationally obtained electron density onto the nucleus in the case when the frozen density is chosen to be that of the innermost shell. The approximated potential obtained from the second-order gradient expansion for the kinetic energy prevents such a collapse almost perfectly but this results from partial compensation of flaws of its components. It is also shown that that the quality of a semi-local approximation to the kinetic-energy functional, a quantity needed in orbital-free methods, is not related to the quality of the non-additive kinetic energy potential - a key component of the effective embedding potential in one-electron equations for embedded orbitals.

Current applications of frozen-density embedding (FDE)—or more generally subsystem density-functional theory (DFT) schemes—are limited to subsystems that are not connected by covalent bonds. This restriction is due to the insufficiencies of the available approximate kinetic-energy functionals, which are used to calculate the contribution of the nonadditive kinetic energy to the effective embedding potential. In this Chapter, we discuss two different approaches to overcome these limitations and to extend the applicability of the FDE scheme to subsystems connected by covalent bonds. First, we outline possibilities to improve the currently available approximations applied for the kinetic-energy component of the embedding potential. Second, we show how a generalized three-partition FDE scheme can be employed to circumvent the problems in the approximate kinetic-energy functionals by introducing capping groups, thus allowing for a subsystem DFT treatment of proteins.

Electronic spectra of molecules in condensed phase can be sensitive to a variety of factors arising from interactions with the surrounding solvent or more general environments including protein matrices, membranes, crystal environments or surfaces. If many chromophores are present, exciton-like interactions may lead to additional changes in the absorption properties. Orbital-free embedding calculations and related subsystem-density-functional theory approaches offer a convenient way of describing these effects in a consistent manner while exploiting the subsystem structure of the total system to keep the computational effort managable. The diabatic picture inherent in these approaches bridges the gap between exciton-like model theories and quantum chemical calculations for systems of coupled chromophores.

A principal difference is revealed between the exact and approximate frozendensity embedding (FDE) theory with respect to the incoming frozen density ρf. In the exact FDE the resultant total density does not depend on ρf, which has to belong to a restricted set of admissible densities. The FDE “freeze-and-thaw” (FT) procedure cannot alter the outcome of the exact FDE. On the other hand, in approximate FDE Kohn-Sham equations the effective “external” potential could be self-consistently adjusted to a given ρf, which offers a greater flexibility at the approximate level. In this case FT is characterized as a self-consistent procedure, which at a given iteration searches for a stationary point of the FT functional specified with the solution from the previous iteration. A condition is obtained for the FT saturation point. A generalization of the frozen-core procedure is presented in the context of frozen-orbital embedding theory.

By combining rigorous mathematical conditions and a computational approach for sampling the electronic configurations in space, we build a procedure for the design of kinetic functionals for electrons which is internally consistent. We then apply it to the test case of the almost uniform interacting electron gas and derive the corresponding kinetic functional. Interestingly the functional form obtained has strong similarities with the Shannon Entropy within the Theory of Information.

The relationship between the kinetic energy and the Fisher information is discussed. It is argued that the sum of the one-electron Fisher information, called modified Fisher information, is the original Fisher information plus a sum of differences of quantum and classical variances. A generalization of the Stam's inequality is presented. It is shown how to derive the Euler equation of the density functional theory from the principle of minimum Fisher information. Making use of the differential virial theorem of Nagy and March, a first-order differential equation for the functional derivative of the kinetic energy functional is derived for spherically symmetric systems.

Density functional theory, which employs the one-electron density as the fundamental variable, is extensively employed for the calculation of atomic and molecular properties. Achieving greater chemical accuracy or computational efficiency is highly desirable and this has motivated attempts to construct improved kineticenergy functionals. We hope that our results will ultimately aid in this endeavor but our aim in this chapter is conceptual rather than practical. We employ a dequantization procedure to obtain an exact expression for the kinetic energy of an N-electron system as the sum of an N-electron classical kinetic energy and an N-electron purely quantum kinetic energy arising from the quantum fluctuations that turn the classical momentum into the quantum momentum as in Nelson's stochastic approach to quantum mechanics. We show that the N-electron purely quantum kinetic energy can be written as the sum of the Weizsäcker term, which is directly proportional to the Fisher information, and a purely quantum kinetic correlation term. We further show that the Weizsäcker term results from local quantum fluctuations while the purely quantum kinetic correlation term results from nonlocal quantum fluctuations. The N-electron classical kinetic energy is seen to be explicitly dependent on the phase of the N-electron wave function and this has implications for the development of accurate orbital-free kinetic-energy functionals. For stationary one-electron systems we show that there is a direct connection between the classical kinetic energy and the angular momentum. For nonstationary one-electron systems we show that the classical kinetic energy is complementary to the purely quantum kinetic energy which is directly proportional to the Fisher information. In this sense, the classical kinetic energy plays a role analogous to that of the Shannon information.

Approximations to the non-interacting kinetic energy T_s[ρ], which take the form of semilocal analytic expressions are collected. They are grouped according to the quantities on which they explicitly depend. Additionally, the approximations for quantities related to T_s[ρ] (kinetic potential and non-additive kinetic energy), for which the analytic expressions for the “parent” approximation for the functional T_s[ρ] are unknown, are also given.

The following sections are included:

• Author Index

• Subject Index