Gradient Expansion Approximation Research Articles

Density functional theory of material design: fundamentals and applications—II

Abstract This is the second and the final part of the review on density functional theory (DFT), referred to as DFT-II. In the first review, DFT-I, we have discussed wavefunction-based methods, their complexity, and basics of density functional theory. In DFT-II, we focus on fundamentals of DFT and their implications for the betterment of the theory. We start our presentation with the exact DFT results followed by the concept of exchange-correlation (xc) or Fermi-Coulomb hole and its relationship with xc energy functional. We also provide the exact conditions for the xc-hole, xc-energy and xc-potential along with their physical interpretation. Next, we describe the extension of DFT for non-integer number of electrons, the piecewise linearity of total energy and discontinuity of chemical potential at integer particle numbers, and derivative discontinuity of the xc potential, which has consequences on fundamental gap of solids. After that, we present how one obtains more accurate xc energy functionals by going beyond the LDA. We discuss the gradient expansion approximation (GEA), generalized gradient approximation (GGA), and hybrid functional approaches to designing better xc energy functionals that give accurate total energies. However, these functionals fail to predict properties like the ionization potential and the band gap. Thus, we next describe different methods of modelling these potentials and results of their application for calculation of the band gaps of different solids to highlight accuracy of different xc potentials. Finally, we conclude with a glimpse on orbital-free density functional theory and the machine learning approach.

Leading Correction to the Local Density Approximation for Exchange in Large-Z Atoms.

The large-Z asymptotic expansion of atomic energies has been useful in determining exact conditions for corrections to the local density approximation in density functional theory. The correction for exchange is fit well with a leading ZlnZ term, and we find its coefficient numerically. The gradient expansion approximation also has such a term, but with a smaller coefficient. Analytic results in the limit of vanishing interaction with hydrogenic orbitals (a Bohr atom) lead to the conjecture that the coefficients are precisely 2.7 times larger than their gradient expansion counterparts, yielding an analytic expression for the exchange-energy correction which is accurate to ∼5% for all Z.

Threshold for primordial black holes: Dependence on the shape of the cosmological perturbations

Primordial black holes may have formed in the radiative era of the early Universe from the collapse of large enough amplitude perturbations of the metric. These correspond to non linear energy density perturbations characterized by an amplitude larger than a certain threshold, measured when the perturbations reenter the cosmological horizon. The process of primordial black hole formation is studied here within spherical symmetry, using the gradient expansion approximation in the long wavelength limit, where the pressure gradients are small, and the initial perturbations are functions only of a time-independent curvature profile. In this regime it is possible to understand how the threshold for primordial black hole formation depends on the shape of the initial energy density profile, clarifying the relation between local and averaged measures of the perturbation amplitude. Although there is no universal threshold for primordial black hole formation, the averaged mass excess of the perturbation depends on the amplitude of the energy density peak, and it is possible to formulate a well-defined criterion to establish when a cosmological perturbation is able to form a black hole in terms of one of these two key quantities. This gives understanding of how the abundance of primordial black holes depends on the shape of the the inflationary power spectrum of cosmological perturbations.

Towards an accurate description of perovskite ferroelectrics: exchange and correlation effects

Using the van der Waals density functional with C09 exchange (vdW-DF-C09), which has been applied to describing a wide range of dispersion-bound systems, we explore the physical properties of prototypical ABO3 bulk ferroelectric oxides. Surprisingly, vdW-DF-C09 provides a superior description of experimental values for lattice constants, polarization and bulk moduli, exhibiting similar accuracy to the modified Perdew-Burke-Erzenhoff functional which was designed specifically for bulk solids (PBEsol). The relative performance of vdW-DF-C09 is strongly linked to the form of the exchange enhancement factor which, like PBEsol, tends to behave like the gradient expansion approximation for small reduced gradients. These results suggest the general-purpose nature of the class of vdW-DF functionals, with particular consequences for predicting material functionality across dense and sparse matter regimes.

Hermitian one-particle density matrix through a semiclassical gradient expansion

We carry out the semiclassical expansion of the one-particle density matrix up to the second order in We use the method of Grammaticos and Voros based on the Wigner transform of operators. We show that the resulting density matrix is Hermitian and idempotent in contrast with the well-known result of the semiclassical Kirzhnits expansion. Our density matrix leads to the same particle density and kinetic energy density as in the literature, and it satisfies the consistency criterion of the Euler equation. The derived Hermitian density matrix clarifies the ambiguity in the usefulness of gradient expansion approximations and might reignite the development of density functionals with semiclassical methods.

Van der Waals density functional made accurate

I propose a van der Waals density functional (vdW-DF) that improves upon the description of energetics and geometries of molecules, solids, and adsorption systems over the original vdW-DF. The functional is based on the nonlocal correlation for the second version of the vdW-DF [Lee et al., Phys. Rev. B 82, 081101(R) (2010)] and an exchange functional that recovers the second-order gradient expansion approximation in the slowly varying limit, while reproducing the large density gradient behavior proposed by Becke [J. Chem. Phys. 85, 7184 (1986)]. A systematic assessment of the proposed functional is presented, which demonstrates the applicability of the proposed vdW-DF to a wide range of systems.

A new meta-GGA exchange functional based on an improved constraint-based GGA

We report the performance of a non-empirical meta-GGA that comes from converting our simple VT{8,4} GGA. That GGA satisfies the large dimensionless reduced gradient limit, obeys the Lieb–Oxford bound, and reduces to the exact second-order gradient expansion approximation in the slowly varying limit. Validation studies of meta-VT{8,4} for several properties using well-known test sets shows a modest improvement with respect to revTPSS. Compared with the heavily parameterized M06-L, the heats of formation of meta-VT{8,4} are substantially better but reaction barrier heights are considerably worse. This suggests the opportunity for additional constraints and the need for better correlation functionals.

Beating in electronic transport through quantum dot based devices

Electronic transport through a two-level system driven by external electric field and coupled to (magnetic or nonmagnetic) electron reservoirs is considered theoretically. The basic transport characteristics such as charge and spin current and tunnel magnetoresistance (TMR) are calculated in the weak-coupling approximation by the use of rate equation connected with Green's function formalism and slave-boson approach. The time-dependent phenomenon is considered in the gradient expansion approximation. The results show that coherent beating pattern can be observed both in current and TMR. The proposed system consisting of two quantum dots attached to external leads, in which the dots' levels can be tuned independently, can be realized experimentally to test this well-known physical phenomenon. Finally, we also indicate possible practical applications of such device.

Kaluza-Klein two-brane-worlds cosmology at low energy

We study two $(4+n)$-dimensional branes embedded in $(5+n)$-dimensional spacetime. Using the gradient expansion approximation, we find that the effective theory is described by $(4+n)$-dimensional scalar-tensor gravity with a specific coupling function. Based on this theory we investigate the Kaluza-Klein two-brane-worlds cosmology at low energy, in both the static and the nonstatic internal dimensions. In the case of the static internal dimensions, the effective gravitational constant in the induced Friedmann equation depends on the equations of state of the brane matter, and the dark radiation term naturally appears. In the nonstatic case we take a relation between the external and internal scale factors of the form $b(t)={a}^{\ensuremath{\gamma}}(t)$ in which the brane world evolves with two scale factors. In this case, the induced Friedmann equation on the brane is modified in the effective gravitational constant and the term proportional to ${a}^{\ensuremath{-}4\ensuremath{\beta}}$. For dark radiation, we find $\ensuremath{\gamma}=\ensuremath{-}2/(1+n)$. Finally, we discuss the issue of conformal frames which naturally arises with scalar-tensor theories. We find that the static internal dimensions in the Jordan frame may become nonstatic in the Einstein frame.

Van der Waals density functional: An appropriate exchange functional

In this Rapid Communication, an exchange functional which is compatible with the nonlocal Rutgers-Chalmers correlation functional [van der Waals density functional (vdW-DF)] is presented. This functional, when employed with vdW-DF, demonstrates remarkable improvements on intermolecular separation distances while further improving the accuracy of vdW-DF interaction energies. The key to the success of this three-parameter functional is its reduction in short-range exchange repulsion through matching to the gradient expansion approximation in the slowly varying/high-density limit while recovering the large reduced gradient, $s$, limit set in the revised Perdew-Burke-Ernzerhof (revPBE) exchange functional. This augmented exchange functional could be a solution to long-standing issues of vdW-DF lending to further applicability of density-functional theory to the study of relatively large, dispersion bound (van der Waals) complexes.

Performance of Kinetic Energy Functionals for Interaction Energies in a Subsystem Formulation of Density Functional Theory.

We have tested the performance of a large set of kinetic energy density functionals of the local density approximation (LDA), the gradient expansion approximation (GEA), and the generalized gradient approximation (GGA) for the calculation of interaction energies within a subsystem approach to density functional theory. Our results have been obtained with a new implementation of interaction energies for frozen-density embedding into the Amsterdam Density Functional program. We present data for a representative sample of 39 intermolecular complexes and 15 transition metal coordination compounds with interaction energies spanning the range from -1 to -783 kcal/mol. This is the first time that kinetic energy functionals have been tested for such strong interaction energies as the ligand-metal bonds in the investigated coordination compounds. We confirm earlier work that GGA functionals offer an improvement over the LDA and are particularly well suited for weak interactions like hydrogen bonds. We do, however, not find a particular reason to prefer any of the GGA functionals over another. Functionals derived from the GEA in general perform worse for all of the weaker interactions and cannot be recommended. An unexpectedly good performance is found for the coordination compounds, in particular with the GEA-derived functionals. However, the presently available kinetic energy functionals cannot be applied in cases in which a density redistribution between the subsystems leads to strongly overlapping subsystem electron densities.

Regularized Gradient Expansion for Atoms, Molecules, and Solids.

A new, regularized gradient expansion (RGE) approximation density functional (i.e., a generalized gradient approximation or GGA that recovers the second-order gradient expansion for exchange in the slowly varying limit) was designed in an attempt to obtain good solid-state and molecular properties at the same time from a single GGA. We assess the performance of this functional for molecular atomization energies, solid lattice constants, and jellium surface energies. We compare the performance of this functional to the modified Perdew-Burke-Ernzerhof generalized gradient approximation (PBEsol GGA), the original PBE GGA, and the Tao-Perdew-Staroverov-Scuseria (TPSS) meta-GGA.

Kinetic energy density study of confined noninteracting fermions: The importance of the angular momentum

In this work we present a study of the kinetic energy density of an arbitrary number of noninteracting fermions confined in a Coulombic potential. We present the results for the total kinetic energy and the kinetic energy density using three simple but representative functionals: Thomas-Fermi, von Weizsaker, and the second order gradient expansion approximation. In order to assess their quality, the deviation of the kinetic energy density of every functional is measured through a parameter that reflects the quality of the functionals in a better way than their relative errors (for that reason, we call it quality factor). Trying to understand the performance of the different functionals we study degenerate states with the same energy but different angular momenta and densities. Therefore, the functionals' ability to capture the effects of each density corresponding to the different angular momenta is discussed, as well as the trends of these contributions.

Dynamics of scalar-tensor cosmology from a Randall-Sundrum two-brane model

We consider a Randall-Sundrum two-brane cosmological model in the low energy gradient expansion approximation by Kanno and Soda. It is a scalar-tensor theory with a specific coupling function and a specific potential. Upon introducing the Friedmann-Lema\^{\i}tre-Robertson-WalkerFLRW metric and perfect fluid matter on both branes in the Jordan frame, the effective dynamical equation for the $A$-brane (our Universe) scale factor decouples from the scalar field and $B$-brane matter leaving only a nonvanishing integration constant (the dark radiation term). We find exact solutions for the $A$-brane scale factor for four types of matter: cosmological constant, radiation, dust, and cosmological constant plus radiation. We perform a complementary analysis of the dynamics of the scalar field (radion) using phase space methods and examine convergence towards the limit of general relativity. In particular, we find that radion stabilizes at a certain finite value for suitable negative matter densities on the $B$-brane. Observational constraints from solar system experiments (PPN) and primordial nucleosynthesis (BBN) are also briefly discussed.

Approximating the kinetic energy functional T s[ρ]: lessons from four-electron systems

It is shown that for pairs of electron densities (ρα and ) obtained from mixing orbital densities in a spin-compensated four-electron system, the kinetic energy functional of the non-interacting reference system (T s[ρ]) satisfies the general inequality . This condition is discussed in the context of the gradient expansion approximation to T s[ρ] and its possible use in variational orbital-free calculations. In particular, it is shown that the second-order term of the analytic form given by von Weizsäcker violates this inequality for the considered pairs.

Quantum Monte Carlo investigations of density functional theory of the strongly inhomogeneous electron gas

We use a variational quantum Monte Carlo realization of the adiabatic connection technique to calculate the most relevant quantities in Hohenberg-Kohn-Sham density functional theory for several strongly inhomogeneous electron-gas systems. Results for the coupling-constant dependence of the exchange-correlation energy, the pair-correlation function, the exchange-correlation hole, and the exchange and correlation energy densities are presented. Comparisons are made with the interaction strength interpolation ~ISI! approximation, the local density approximation ~LDA!, the gradient expansion approximation ~GEA!, the generalized gradient approximation ~GGA!, and the weighted density approximation ~WDA!. The coupling-constant dependence of the exchange-correlation energy is accurately described by an ISI model that incorporates information on the strong-interaction limit. Unlike either the LDA or GEA, the WDA is successful in describing the nonlocal structure of the exchange-correlation hole. The LDA errors in the exchange-correlation energy density show a remarkable correlation with the Laplacian of the density. The GGA worsens the error in the integrated exchange-correlation energy as the inhomogeneity of the systems increases. This failure is shared by current meta-GGA functionals and is shown to be caused by the inability of these functionals to describe the LDA overestimation ~in absolute value! of the exchange energy density around density maxima. It is suggested that this effect could be taken into account by including Laplacian terms in semilocal density functionals.

Introduction of the explicit long-range nonlocality as an alternative to the gradient expansion approximation for the kinetic-energy functional

The approximate nonempirical kinetic-energy functional proposed by Tal and Bader is analyzed for polyatomic systems. The performance of this functional and the functionals derived from the gradient expansion approximation truncated to zeroth, second, and fourth order is investigated for a testing set of 68 neutral and charged molecules. It is shown that the Tal–Bader functional, despite the simplicity of the idea behind its construction, leads to significantly better total kinetic energies than the gradient expansion approximation functionals. The local behavior of the kinetic-energy density derived from the Tal–Bader functional is also discussed.

Nonequilibrium time evolution of the spectral function in quantum field theory

Transport or kinetic equations are often derived assuming a quasi-particle (on-shell) representation of the spectral function. We investigate this assumption using a three-loop approximation of the 2PI effective action in real time, without a gradient expansion or on-shell approximation. For a scalar field in 1+1 dimensions the nonlinear evolution, including the integration over memory kernels, can be solved numerically. We find that a spectral function approximately described by a nonzero width emerges dynamically. During the nonequilibrium time evolution the Wigner transformed spectral function is slowly varying, even in presence of strong qualitative changes in the effective particle distribution. These results may be used to make further analytical progress towards a quantum Boltzmann equation including off-shell effects and a nonzero width.

The role of the kinetic energy density in approximations to the exchange energy

Local and gradient corrected functionals for the non-interacting kinetic energy (Ts) are applied to the atomization process. These functionals and a newly developed generalized gradient approximation (GGA) for Ts are not accurate enough for thermochemistry. The limited accuracy of the gradient expansion approximation for Ts is shown to be related to the remaining errors in existing GGAs for the exchange energy Ex. Various new approximation schemes for Ex are developed which attempt to overcome the problems arising from approximations to Ts.

Density-gradient expressions for the correlation-energy of many-electron systems

Based on gradient expansion approximations to an earlier proposed method [Chem. Phys. Lett. 175 (1990) 641; J. Mol. Struct. (Theochem) 287 (1993) 11] we presently report a new derivation of correlation functionals. Their application to atoms and molecules is discussed. A relation between the correlation energy and the expectation value of r 2 is derived.