We summarize and give perspective upon recent progress in developing non-empirical constraint-based thermal (i.e. free energy) exchange-correlation (XC) density functionals essential for accurate description of the quantum behavior of electrons in warm, dense plasmas. After delineating the critical role of ground-state functionals for zero-temperature, time-dependent DFT, we outline the underpinnings of local density approximation (LDA), generalized gradient approximation (GGA), and meta-GGA XC free-energy functionals. Two basic thermalization principles for upgrading ground-state XC functionals to successful thermal ones are emphasized. Then we turn to a long- standing challenge, assessment of the accuracy of well-founded functionals. Unlike the ground state, there are few exact results for large T and P . An exception is path integral Monte Carlo (PIMC) data for dense H/D and He plasmas. For those, we did ab-initio molecular dynamics (AIMD) simulations under selected thermodynamic conditions employing five thermal XC functionals: two approximate thermal GGAs, fully-thermal GGA, an approximate meta-GGA, and fully-thermal meta-GGA. Comparisons with the PIMC data show that functionals thermalized by augmenting a non-thermal functional with a lower-level thermal contribution are inferior to functionals with thermal XC and spatial inhomogeneity effects taken into account at the same level of refinement. We believe this and similar evidence should be convincing to the high-energy density physics community of the necessity of use of proper thermal XC functionals in simulation studies of finite-temperature quantum effects in warm, dense plasmas.

Deorbitalization of a conventional meta-generalized-gradient exchange-correlation approximation replaces its dependence upon the Kohn-Sham kinetic energy density with a dependence on the density gradient and Laplacian. In principle, that simplification should provide improved computational performance relative to the original meta-GGA form because of the shift from an orbital-dependent generalized Kohn-Sham potential to a true KS local potential. Often that prospective gain is lost because of problematic roughness in the density caused by the density Laplacian and consequent roughness in the exchange-correlation potential from the resulting higher-order spatial derivatives of the density in it. We address the problem by constructing a deorbitalizer based on the RPP deorbitalizer [Phys. Rev. Mater. 6, 083803 (2022)] with comparative smoothness of the potential along with retention of constraint satisfaction as design goals. Applied to the r²SCAN exchange-correlation functional [J. Phys. Chem. Lett. 11, 8208 (2020)], we find substantial timing improvements for solid-state calculations over both r²SCAN and its earlier deorbitalization for high precision calculations of structural properties, while improving upon the accuracy of RPP deorbitalization for both solids and molecules.

The study of spin crossover phenomena in metal complexes is of significant importance in chemistry and materials science, with implications for both theoretical advancements and practical applications. Traditionally, the analysis of electronic structure outputs in this domain often involves labor-intensive ad hoc scripting that lacks standardization and transferability. To overcome these challenges, we have developed pySCO, a library designed to automate and simplify thermodynamic analyses for this family of metal complexes, offering seamless integration with popular electronic structure codes. We feature a detailed case study on an Fe(II) metal complex to highlight the robust capabilities offered by the library and provide insights into the spin transition regimes for this material.

This roadmap presents the state-of-the-art, current challenges and near future developments anticipated in the thriving field of warm dense matter physics. Originating from strongly coupled plasma physics, high pressure physics and high energy density science, the warm dense matter physics community has recently taken a giant leap forward. This is due to spectacular developments in laser technology, diagnostic capabilities, and computer simulation techniques. Only in the last decade has it become possible to perform accurate enough simulations & experiments to truly verify theoretical results as well as to reliably design experiments based on predictions. Consequently, this roadmap discusses recent developments and contemporary challenges that are faced by theoretical methods and experimental techniques needed to create and diagnose warm dense matter. A large part of this roadmap is dedicated to specific warm dense matter systems and applications in astrophysics, inertial confinement fusion and novel material synthesis.

The application of density functional theory to materials in the warm dense matter regime has motivated the development of exchange-correlation functionals which incorporate proper, explicit temperature dependence. Previous work has yielded fully-thermal exchange-correlation free energy functionals at the local density approximation (LDA) and generalized gradient approximation (GGA) levels of refinement. Recently an additive thermal correction scheme was utilized to construct a meta-GGA exchange-correlation (XC) functional in which thermal effects are treated at the GGA level. The fTSCAN free-energy XC functional presented here includes thermal effects through the meta-GGA level in the context of the SCAN (strongly constrained and appropriately normed) ground-state functional. The fTSCAN functional provides generality while achieving similar performance to a thermal GGA functional at high temperatures, e.g. pressures within 1% of path integral Monte Carlo simulations of warm dense hydrogen, and a significant improvement over ground-state functionals. At low temperatures, fTSCAN demonstrates improvements in accuracy relative to lower-level and deorbitalized functionals, indicating that calculations using fTSCAN may be expected to perform well across experimentally relevant densities and pressures.

The relationship of the thermodynamics for non-uniform systems and density functional theory is reviewed. It is observed that the relevant universal functionals are the same in both cases under appropriate conditions for thermodynamics. The variational context arises from the convexity properties of the functionals as represented by equilibrium statistical mechanics. Within thermodynamics a choice of local density or local chemical potential for the local variable can be made. In the latter case a “dual” form for density functional theory applies (“potential functional” theory). The special case of electrons and ions is considered, and the important condition of charge neutrality for thermodynamics is noted. These general results are extended to include the interatomic pair potential as an independent variable. Its conjugate is the equilibrium pair correlation function. As an application, the construction of a classical system to represent a given quantum system in terms of effective potentials is shown to result from the Euler equations of this extended variational formulation.

Swift discovery of spin-crossover materials for their potential application in quantum information devices requires techniques which enable efficient identification of suitably spin switching candidates. To this end, we screened the Cambridge Structural Database to develop a specialized database of 1,439 materials and computed spin-switching energies from density functional theory for each material. The database was used to train an equivariant graph convolutional neural network to predict the magnitude of the spin-conversion energy. A test mean absolute error was 360 meV. For candidate identification, we equipped the system with a relevance-based classifier. This approach leads to a nearly four-fold improvement in identifying potential spin-crossover systems of interest as compared to conventional high-throughput screening.

By summarizing the constraint-based development of orbital-free free-energy density functional approximations, we provide a perspective on progress over the last 15 years, the limitations of exist- ing functionals, and the challenges awaiting resolution. We outline the chronology of the development of non-interacting and exchange-correlation free-energy orbital-free functionals and summarize the theoretical basis of existing local density approximation (LDA), second-order approximation, gen- eralized gradient approximation (GGA), and meta-GGAs. We discuss limitations and challenges such as problems with thermodynamic derivatives, free-energy nonadditivity and the closely related issue of all-electron versus valence-only local pseudo-potential performance.

De-orbitalization (replacement of orbital dependence by an explicit density functional) of a meta-generalized gradient approximation for exchange and correlation has been deemed successful if the de-orbitalized functional delivers simple error bounds comparable to those from the parent functional on standard data sets. Tacitly it has been assumed that de-orbitalization will not improve on those errors. One counter-example is known, at least on molecular data sets, the meta-GGA made very simple (MVS) functional; see Phys. Rev. A 96, 052512 (2017). On the basis of post-SCF calculations [J. Chem. Phys. 149, 144105 (2018)], it was argued that the unexpected betterment on molecules provided by that one specific de-orbitalizer does not occur in solids. Some other de-orbitalizers considered in that later work did show performance betterment of MVS on solids however. But molecules were not treated nor was the issue of ambiguous betterment pursued. We revisit the issue and show that the betterment of MVS for that particular de-orbitalizer does occur in solids when the calculations are done self-consistently and with the same computational techniques as used in other de-orbitalizations. For systems without d states or without transition metals, that betterment is improved. Imposition of second-order gradient expansion compliance as a constraint upon the de-orbitalizer refines (rather than degrades) the improvement relative to the parent MVS functional and provides insight as to why de-orbitalized MVS behaves differently from other de-orbitalized meta-GGA functionals.

We propose computationally cheap and accurate approximants to the noninteracting kinetic energy functional, T_s[n], by leveraging the simplicity and computational efficiency of the Wang-Teter functional [L.-W. Wang and M. Teter, Phys. Rev. B 45, 13196 (1992)] which depends on a single parameter, the average electron density, ρ₀. We address its limitations, which include variational instabilities and inability to approach materials with finite band gaps. We introduce three physically-motivated methods for determining ρ₀: DEN, minimizing the integral difference of the self-consistent Wang-Teter electron density from the one from Kohn-Sham DFT; KIN: minimizing the deviation between the Wang-Teter T_s[n] and the exact value from Kohn-Sham DFT; and finally ENE: minimizing the difference between the Wang- Teter and Kohn-Sham DFT total energies. The crucial result of this work is that our approaches effectively mitigate the drawbacks of the Wang-Teter functional. We provide a thorough analysis of our methods and discuss their potential for large-scale simulations and as templates for density-dependent nonlocal functionals.

We present a de-orbitalization of the recent simplified, regularized Tao-Mo exchange functional [J. Chem. Phys. 159, 214102 (2023)] that is faithful to the parent functional. That is a major gain relative to our earlier de-orbitalization which did poorly on molecular heats of formation.[J. Chem. Phys. 159, 214103 (2023)] The improvement arises from augmentation of the Mejía-Rodríguez and Trickey deorbitalization strategy (Phys. Rev. A 96, 052512 (2017)] to use a smoothed replacement for the reduced density Laplacian (conventionally denoted q) obtained from that Laplacian itself. The augmentation also rationalizes the improvement obtained from cutoff of q < 0 that was poorly understood at the time of the previous paper. The new scheme yields de-orbitalized chemical region indicators that are much closer to those from the parent, orbital-dependent functional than were obtainable from the previous de-orbitalization. It also replicates the good 3d elemental magnetization of the parent functional reasonably well.

Experimental results on two supramolecular complexes in which a Cr^III or Fe^III d-orbital single-ion magnet center is embedded between a pair of Fe^II spin-crossover moieties make those two complexes interesting as possible candidates for use in quantum information technologies. We report detailed computational results for their structure and electronic properties and use the resulting data to parametrize a spin Hamiltonian that facilitates comparison with experimental results and their interpretation. Consistent with experimental results on decoherence in [Fe(ox)₃]@[Fe₂L₃]⁺, we find it to be easy-plane type while the [Cr(ox)₃]@[Fe₂L₃]⁺ system is easy-axis type.

Kohn-Sham Density Functional Theory (KSDFT) is the most widely used electronic structure method in chemistry, physics, and materials science, with thousands of calculations cited annually. This ubiquity is rooted in the favorable accuracy vs cost balance of KSDFT. Nonetheless, the ambitions and expectations of researchers for use of KSDFT in predictive simulations of large, complicated molecular systems are confronted with an intrinsic computational cost-scaling challenge. Particularly evident in the context of first-principles molecular dynamics, the challenge is the high cost-scaling associated with the computation of the Kohn-Sham orbitals. Orbital-free DFT (OFDFT), as the name suggests, circumvents entirely the explicit use of those orbitals. Without them, the structural and algorithmic complexity of KSDFT simplifies dramatically and near-linear scaling with system size irrespective of system state is achievable. Thus, much larger system sizes and longer simulation time scales (compared to conventional KSDFT) become accessible; hence, new chemical phenomena and new materials can be explored. In this review, we introduce the historical contexts of OFDFT, its theoretical basis, and the challenge of realizing its promise via approximate kinetic energy density functionals (KEDFs). We review recent progress on that challenge for an array of KEDFs, such as one-point, two-point, and machine-learnt, as well as some less explored forms. We emphasize use of exact constraints and the inevitability of design choices. Then, we survey the associated numerical techniques and implemented algorithms specific to OFDFT. We conclude with an illustrative sample of applications to showcase the power of OFDFT in materials science, chemistry, and physics.

The revised, regularized Tao-Mo (rregTM) exchange-correlation density functional approximation (DFA) [J. Chem. Phys. 153, 184112 (2020); ibid. 155, 024103 (2021)] resolves the order-of-limits problem in the original TM formulation while preserving its valuable essential behaviors. Those include performance on standard thermochemistry and solid data sets that is competitive with that of the most widely explored meta-GGA DFAs (SCAN and r²SCAN) while also providing superior performance on elemental solid magnetization. Puzzlingly however, rregTM proved to be intractable for de-orbitalization via the approach of Mejía-Rodríguez and Trickey (Phys. Rev. A 96, 052512 (2017)). We report investigation that leads to diagnosis of how the regularization in rregTM of the z indicator functions (z = the ratio of the von-Weizsäcker and Kohn-Sham kinetic energy densities) leads to non-physical behavior. We propose a simpler regularization that eliminates those oddities and that can be calibrated to reproduce the good error patterns of rregTM. We denote this version as simplified, regularized Tao-Mo, sregTM. We also show that it is unnecessary to use rregTM correlation with sregTM exchange: PBE correlation is sufficient. The subsequent paper shows how sregTM enables some progress on de-orbitalization.

In the preceding paper we gave a regularization of the Tao-Mo exchange functional that removes the order-of-limits problem in the original Tao-Mo form and also eliminates the unphysical behavior introduced by an earlier regularization. The resulting simplified, regularized (sregTM) functional delivers performance on standard molecular and solid state test sets equal to that of the earlier revised, regularized Tao-Mo (rregTM) functional. Here we address de-orbitalization of sregTM into a pure density functional. We summarize the failures of the Mejía-Rodríguez and Trickey deorbitalization strategy (Phys. Rev. A 96, 052512 (2017)) when used with both versions. We discuss how those failures apparently arise in the so-called z₀ indicator function and in substitutes for the reduced density Laplacian in the parent functionals. Then we show that the sregTM functional can be de-orbitalized somewhat well with a rather peculiarly parametrized version of the previously used deorbitalizer. We discuss, briefly, a de-orbitalization that works in the sense of reproducing error patterns but that apparently succeeds by cancellation of major qualitative errors associated with the de-orbitalized indicator functions α and z, hence is not recommended. We suggest that the same issue underlies the earlier finding of comparatively mediocre performance of the de-orbitalized TPSS functional. Our work demonstrates that the intricacy of such functionals magnifies the errors introduced by the Mejía-Rodríguez and Trickey de-orbitalization approach in ways that are extremely difficult to analyze and correct.

Calculation of transition temperatures T_1/2 for thermally driven spin-crossover in condensed phases is challenging, even with sophisticated state-of-the-art density functional approximations. The first issue is the accuracy of the adiabatic crossover energy difference ΔE_HL between the low- and high-spin states of the bistable metal-organic complexes. The other is proper inclusion of entropy contributions to the Gibbs free energy from the electronic and vibrational degrees of freedom. We discuss the effects of treatments of both contributions upon the calculation of thermochemical properties for a set of twenty spin-crossover materials using a Hubbard-U correction obtained from a reference ensemble spin-state. The U values obtained from a simplest bi-molecular representation may over-correct, somewhat, the ΔE_HL values, hence give somewhat excessive reduction of the T_1/2 results with respect to their U = 0 values in the crystalline phase. We discuss the origins of the discrepancies by analyzing different sources of uncertainties. By use of a first-coordination-sphere approximation and the assumption that vibrational contributions from the outermost atoms in a metalorganic complex are similar in both low- and high-spin states, we achieve T_1/2 results with the low-cost, widely used PBE generalized gradient density functional approximation comparable to those from the more costly, more sophisticated r²SCAN meta-generalized gradient approximation. The procedure is promising for use in high-throughput materials screening because it combines rather low computational effort requirements with freedom from user manipulation of parameters.

Though calculations based on density functional theory (DFT) are used remarkably widely in chemistry, physics, materials science, and biomolecular research and though the modern form of DFT has been studied for almost 60 years, some mathematical problems remain. For context, we provide an outline of the basic structure of DFT, then pose several questions regarding both its time-independent and time-dependent forms. Progress on any of these would aid in development of better approximate functionals and in interpretation.

Effects of finite-temperature quasiparticle self-energy corrections to x-ray absorption spectra are investigated within the finite-temperature quasiparticle local density GW approximation up to temperatures T of order the Fermi temperature. To facilitate the calculations, we parametrize the quasiparticle self-energy using low-order polynomial fits. We show that temperature-driven decrease in the electron lifetime substantially broadens the spectra in the near-edge region with increasing T. However, the quasiparticle shift is most strongly modified near the onset of plasmon excitations.

High-throughput searches for spin-crossover molecules require Hubbard-U corrections to common density functional exchange-correlation (XC) approximations. However, the U_eff values obtained from linear response or based on previous studies overcorrect the spin-crossover energies. We demonstrate that employing a linearly-mixed ensemble average spin state as the reference configuration for the linear response calculation of U_eff resolves this issue. Validation on a commonly used set of spin-crossover complexes shows that these ensemble U_eff values consistently are smaller than those calculated directly on a pure spin state, irrespective of whether that be low- or high-spin. Adiabatic crossover energies using this methodology for a generalized gradient approximation XC functional are closer to the expected target energy range than with conventional U_eff values. Based on the observation that the U_eff correction is similar for different complexes that share transition metals with the same oxidation state, we devise a set of recommended averaged U_eff values for high-throughput calculations.

We report APW+lo (augmented plane wave plus local orbital) density functional theory (DFT) calculations of large molecular systems using the domain specific SIRIUS multi-functional DFT package. The APW and FLAPW (full potential linearized APW) task and data parallelism options and advanced eigen-system solver provided by SIRIUS can be exploited for performance gains in ground state Kohn-Sham calculations on large systems. This approach is distinct from our prior use of SIRIUS as a library backend to another APW+lo or FLAPW code. We benchmark the code and demonstrate performance on several magnetic molecule and metal organic framework systems. We show that the SIRIUS package in itself is capable of handling systems as large as a several hundred atoms in the unit cell without having to make technical choices that result in loss of accuracy with respect to that needed for study of magnetic systems.

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 300 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 776 entries, the paper represents a broad snapshot of DFT, anno 2022.

The deviations from linearity of the energy as a function of the number of electrons that arise with current approximations to the exchange-correlation (XC) energy functional have important consequences for the frontier eigenvalues of molecules and the corresponding valence-band maxima for solids. In this work, we present an analysis of the exact theory that allows one to infer the effects of current approximations on the highest occupied and lowest unoccupied molecular orbital eigenvalues. Then we show the importance of the asymptotic behavior of the XC potential in the generalized gradient approximation (GGA) through the functional NCAPR (nearly correct asymptotic potential revised) to determine the shift of the frontier orbital eigenvalues towards the exact values. Thereby we establish a procedure at the GGA level of refinement that allows one to make a single calculation to determine the ionization potential, the electron affinity and the hardness of molecules (and its solid counterpart, the band gap) with an accuracy equivalent to the one obtained for these properties through energy differences, a procedure which requires three calculations. For solids the accuracy achieved for the band gap lies rather close to that which is obtained through hybrid XC energy functionals, but those also demand much greater computational effort than the one required with the simple NCAPR GGA calculation.

We study intramolecular electron transfer in the single-molecule magnetic complex [Mn12O12(O2CR)16 (H2O)4] for R = -H, -CH3, -CHCl2, -C6H5, -C6H4F ligands as a mechanism for switching of the molecular dipole moment. Energetics are obtained using the density functional theory (DFT) with onsite Coulomb energy correction (DFT+U). Lattice distortions are found to be critical for localizing an extra electron on one of the easy sites on the outer ring in which localized states can be stabilized. We find that the lowest energy path for charge transfer is for the electron to go through the center via superexchange mediated tunneling. The energy barrier for such a path ranges from 0.4 meV to 54 meV depending on the ligands and the isomeric form of the complex. The electric field strength needed to move the charge from one end to the other, thus reversing the dipole moment, is 0.01 0.04 V/angstrom.

Coulomb charges confined by a harmonic potential display a rich structure at strong coupling, both classical and quantum. A simple density functional theory is reviewed showing the essential role of correlations in forming shell structure and order within the shells. An overview of previous comparisons with molecular dynamics and Monte Carlo simulations is summarized and extended. It is shown that correlations for the fluid phase (shell structure only) are well approximated by those for the uniform one-component plasma even at very strong coupling. A corresponding representation of the correlations for the ordered phase is still an open question. The confirmed success for the classical density functional theory is important for the subsequent representation of the quantum case. Here, a mapping of the quantum description onto an equivalent classical description with effective potentials allows direct application of the classical methods, both theory and simulation. This is particularly relevant at low but finite temperatures where quantum simulation methods are compromised. The special case of Coulomb charges in a harmonic trap is the simplest example of more complex systems of experimental interest where confinement and strong coupling play an essential role (e.g., quantum dots, ions in complex traps, electrons on a helium surface, dusty Yukawa plasmas, ultracold neutral plasmas).

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Impartial assessment of spin-crossover (SCO) energy calculations is that current treatments depend on nearly artisanal skill in picking quantum mechanical approximations, and computational methods. That is incompatible with automated (work-flow-driven) screening. Cost vs. accuracy causes focus on density functional theory (DFT). By near-exhaustive study of schemes for calculating the basic molecular high-low spin energy difference, ΔE_HL, we show that presently there is no combination of a constraint-based, non-empirical density functional approximation (DFA) and a set of well-defined semi-empirical corrections to it adequate for such a protocol. Acceptable methodology must be quantum mechanically sound, yield both basic structure and property values, and accurate $\dehl$ without steering or tuning. Somewhat successful hybrid DFA calculations of ΔE_HLs are too costly for high-throughput screening of condensed phases. Lower-cost alternatives combine a generalized gradient approximation (GGA) DFA with a Hubbard-U correction (DFT + U). But we show that neither U=0 nor any currently available unsteered U calculation gives a decent ΔE_HL value for [Mn(taa)] without also degrading molecular structure or property predictions. Moving to the SCAN meta-GGA does not solve the problem. The revised-restored SCAN (r²SCAN) meta-GGA together with its deorbitalized version r²SCAN-L does give improved but not wholly satisfactory results. We also document and diagnose several non-obvious technical and procedural sensitivities and inter-code differences. In addition to being a formidable challenge to DFA development, the lack we delineate is a major impediment to progress in the development of quantum materials and spintronics.

Full-potential linearized augmented plane wave (LAPW) and APW plus local orbital (APW+lo) codes differ widely in both their user interfaces and in capabilities for calculations and analysis beyond their common central task of all-electron solution of the Kohn-Sham equations. However, that common central task opens a possible route to performance enhancement, namely to offload the basic LAPW/ATP+lo algorithms to a library optimized purely for that purpose. To explore that opportunity, we have interfaced the Exciting-Plus ("EP") LAPW/APW+lo DFT code with the highly optimized SIRIUS multi-functional DFT package. This simplest realization of the separation of concerns approach yields substantial performance over the base EP code via additional task parallelism without significant change in the EP source code or user interface. We provide benchmarks of the interfaced code against the original EP using small bulk systems, and demonstrate performance on a spin-crossover molecule and magnetic molecule that are of size and complexity at the margins of the capability of the EP code itself.

The evolution of variational Coulomb fitting from a purely practical scheme to reduce computational burden to a formal variant of Hohenberg-Kohn-Sham density functional theory (auxiliary density functional theory, ADFT) is discussed. After a summary of the historical evolution, an analysis of its connection with the Hohenberg-Kohn theorem is given, some implications for the Euler equation and for time-dependent DFT are given and some implications for the deMon2k code delineated.

The recent major modification, r²SCAN, of the SCAN (strongly constrained and appropriately normed) meta-GGA exchange-correlation functional is shown to give substantially better spin-crossover electronic energies (high spin minus low spin) on a benchmark data set than the original SCAN as well as on some Fe complexes. The deorbitalized counterpart r²SCAN-L is almost as good as SCAN and much faster in periodically bounded systems. A combination strategy for balanced treatment of molecular and periodic spin-crossover therefore is recommended.

A recent modification, r²SCAN, of the SCAN (strongly constrained and appropriately normed) meta-GGA exchange-correlation functional mostly eliminates numerical instabilities and attendant integration grid sensitivities exhibited by SCAN. Here we show that the successful deorbitalization of SCAN to SCAN-L (SCAN with density Laplacian dependence) carries over directly to yield r²SCAN-L. A major benefit is that the high iteration counts that hindered use of SCAN-L are eliminated in r²SCAN-L. It therefore is a computationally much faster meta-GGA than its orbital-dependent antecedent. Validation data for molecular heats of formation, bond lengths, and vibration frequencies (G3/99X, T96-R, T82-F test sets respectively) and on lattice constants, and cohesive energies (for 55 solids) and bulk moduli (for 40 solids) are provided. In addition, we show that the over-magnetization of bcc Fe from SCAN persists in r²SCAN but does not appear in r²SCAN-L, just as with SCAN-L.

Definitions for a local pressure in an inhomogeneous fluid are considered for both equilibrium and local equilibrium states. Thermodynamic and mechanical (hydrodynamic) contexts are reconciled. Remaining problems and uncertainties are discussed.

Specialized computational chemistry packages have permanently reshaped the landscape of chemical and materials science by providing tools to support and guide experimental efforts and for the prediction of atomistic and electronic properties. In this regard, electronic structure packages have played a special role by using first-principle-driven methodologies to model complex chemical and materials processes. Over the past few decades, the rapid development of computing technologies and the tremendous increase in computational power have offered a unique chance to study complex transformations using sophisticated and predictive many-body techniques that describe correlated behavior of electrons in molecular and condensed phase systems at different levels of theory. In enabling these simulations, novel parallel algorithms have been able to take advantage of computational resources to address the polynomial scaling of electronic structure methods. In this paper, we briefly review the NWChem computational chemistry suite, including its history, design principles, parallel tools, current capabilities, outreach, and outlook.

Using conceptually and procedurally consistent density functional theory (DFT) calculations with an advanced meta-GGA exchange-correlation functional in ab initio (Born-Oppenheimer, BO) molecular dynamics simulations, we determine the insulator-metal transition (IMT) of warm dense fluid hydrogen from 50 to 300 GPa, to be in better agreement with experiment than previous DFT predictions. Inclusion of nuclear quantum effects via path-integral molecular dynamics (PIMD) sharpens the transition and lowers its temperature relative to BOMD results. A rapid decrease in the molecular character of the system, as observed via the ionic pair correlation function, coincides with an abrupt conductivity increase, confirming a metallic transition due to molecular hydrogen dissociation that is coincident with abrupt band gap closure. Comparison of the PIMD and BOMD results clearly demonstrates an isotope effect on the IMT. Exploitation of differing methodologies for using the orbital-dependent and de-orbitalized versions of the meta-GGA enables us to quantify exchange-correlation approximation effects. Distinct from stochastic simulations, these results do not depend upon any clever but uncontrolled combination of ground-state and finite-T methodologies and should provide a reliable benchmark for further studies.

We develop and demonstrate the performance of a non-separable form of generalized gradient approximation (GGA) for exchange and correlation that includes locally varying parameters that match second-order gradient expansion behavior. Specifically, the high- and low-density limits are included to recover locally the linear response through inclusion of their dependence on the electron density. This local parametrization allows the GGA form to provide varying behavior depending on the density regime. On the basis of an extensive series of property calculations involving both molecules and solids, we show that this non-empirical methodology can lead to a balanced GGA description of both finite and extended systems.

During the past decade a number of attempts to formulate a continuum description of complex states of matter have been proposed to circumvent more cumbersome many-body and simulation methods. Typically these have been quantum systems (e.g., electrons) and the resulting phenomenologies collectively often called "quantum hydrodynamics". However, there is extensive work from the past based in non-equilibrium statistical mechanics on the microscopic origins of macroscopic continuum dynamics that has not been exploited in this context. Although formally exact, its original target was the derivation of Navier-Stokes hydrodynamics for slowly varying states in space and time. The objective here is to revisit that work for the present interest in complex quantum states - possible strong degeneracy, strong coupling, and all space-time scales. The result is an exact representation of generalized hydrodynamics suitable for introducing controlled approximations for diverse specific cases, and for critiquing existing work.

We consider a steady-state (but transient) situation in which a warm dense aggregate is a two-temperature system with equilibrium electrons at temperature T_e , ions at T_i, T_e ≠ T_i. Such states are achievable by pump-probe experiments. For warm dense H in such a two-temperature situation, we investigate nuclear quantum effects (NQEs) upon structure and thermodynamic properties, thereby delineating the limitations of ordinary ab initio molecular dynamics (AIMD). We use path integral molecular dynamics simulations driven by orbital-free density functional (OFDFT) calculations with state-of-the-art non-interacting free energy and exchange-correlation functionals for the explicit temperature de- pendence. We calibrate the OFDFT calculations against conventional (explicit orbitals) Kohn-Sham DFT. We find that when the ratio of the ionic thermal de Broglie wavelength to the mean inter-ionic distance is larger than about 0.30, the ionic radial distribution function is meaningfully affected by inclusion of NQEs. Moreover, NQEs induce a substantial increase in both the ionic and electronic pressures. This confirms the importance of NQEs for high-accuracy equation of state data on highly driven H. For T_e > 20kK, increasing T_e in the warm dense H has slight effects on the ionic radial distribution function and equation of state in the range of densities considered. In addition, compared to the thermostatted ring-polymer molecular dynamics, we confirm that the primitive path integral molecular dynamics algorithm yields overestimated electronic pressures, a consequence of the overly localized ionic description from the primitive scheme.

For orbital-free ab initio molecular dynamics, especially on systems in extreme thermodynamic conditions, we provide the first pseudo-potential-adapted generalized gradient approximation (GGA) functional for the non-interacting free energy. This is achieved by systematic finite-temperature extension of our recent LKT ground state non-interacting kinetic energy GGA functional (Phys. Rev. B 98, 041111(R) (2018)). We test the performance of the new functional first via static lattice calculations on crystalline aluminum and silicon. Then we compare deuterium equation of state results against both path-integral Monte Carlo and conventional (orbital-dependent) Kohn-Sham results. The new functional, denoted LKTF, outperforms the previous best semi-local free energy functional, VT84F (Phys. Rev. B 88, 161108(R) (2013)), and provides modestly faster simulations. We also discuss subtleties of identification of kinetic and entropic contributions to non-interacting free-energy functionals obtained by extension from ground state orbital-free kinetic energy functionals.

Two methods to calculate negative electron affinities systematically from ground-state density functional methods are presented. One makes use of the LUMO (lowest unoccupied molecular orbital) energy shift provided by approximate inclusion of derivative discontinuity in the NCAP (nearly correct asymptotic potential) non-empirical, constraint-based generalized gradient approximation (GGA) exchange functional. The other uses a second-order perturbation calculation of the derivative discontinuity based on the NCAP exchange-correlation potential. On a set of thirty-eight molecules, NCAP leads to a rather accurate description that is improved further through the perturbation correction. The results presented show the importance of the asymptotic behavior of the XC potential in the calculation of negative electron affinities as well as demonstrating the versatility of the NCAP functional.

The``regularized'' SCAN (regSCAN) XC functional was rationalized as being a numerically more stable substitute for SCAN on the basis of a small sample of solid and molecular test comparisons. We have tested regSCAN against much larger databases for both molecules and solids. We find that the seemingly minor changes from SCAN to regSCAN degrade the molecular thermochemistry accuracy of SCAN. On the positive side, regSCAN maintains the accuracy of SCAN for solids and significantly improves upon SCAN sensitivity to grid quality. In contrast, SCAN-L (de-orbitalized SCAN) retains the thermochemical balance of SCAN but also suffers the same grid sensitivity.

Recent investigations have found that the strongly constrained and appropriately normed (SCAN) meta-GGA exchange-correlation functional significantly over-magnetizes elemental Fe, Co, and Ni solids. For the paradigmatic case, bcc Fe, the error relative to experiment is ≳ 20%. Comparative analysis of magnetization results from SCAN and its deorbitalized counterpart, SCAN-L, leads to identification of the source of the discrepancy. It is not from the difference between Kohn-Sham (SCAN-L) and generalized Kohn-Sham (SCAN) procedures. The key is the iso-orbital indicator ? (the ratio of the local Pauli and Thomas-Fermi kinetic energy densities). Its deorbitalized counter- part, ?_L , has more dispersion in both spin channels with respect to magnetization in an approximate region between 0.6 Bohr and 1.2 Bohr around an Fe nucleus. The overall effect is that the SCAN switching function evaluated with ?_L reduces the energetic disadvantage of the down channel with respect to up compared to the original ?, which in turn reduces the magnetization. This identifies the cause of the SCAN magnetization error as insensitivity of the SCAN switching function to ? values in the approximate range 0.5⪅ ?⪅ 0.8 and an oversensitivity for ? ⪆ 0.8.

Renewed interest in the homogeneous electron gas (HEG) has been stimulated by recent accurate simulations of it over a wide range of densities and temperatures. Those data, combined with known theoretical limits, have led to analytical representations of the free energy. Such a representation is, at least in principle, the complete HEG equation of state. The initial objective here is to establish that the two best representations [``corrKSDT'', Phys. Rev. Lett. 112, 076403 (2014), Phys. Rev. Lett. 120, 07640 (2018), and ``GDB'' Phys. Rev. Lett. 119, 135001 (2017)] of the simulation data and constraints are effectively the same in both functional form and accuracy of representation. The second objective is to disclose and delineate a significant difficulty. Despite their expected accuracy for the free energy, the underlying functional form is not adequate for derived thermodynamic properties. As an example, the specific heats obtained from the representations exhibit anomalies suggesting the need first for additional simulation data in critical regimes, then for refined fitting functions. The existing representations are, however, almost certainly adequate for applications based on the free energy alone (e.g., density functional theory for warm dense matter). The third objective is to show that, despite their inability to provide a complete thermodynamic description of the HEG, the best analytical representations do provide a fully adequate exchange-correlation local density approximation for free energy density functional calculations.

We develop and validate a non-empirical generalized gradient approximation (GGA) exchange (X) density functional which performs as well as the SCAN meta-GGA on standard thermochemistry tests. Additionally, the new functional (NCAP) yields Kohn-Sham eigenvalues that are useful approximations of the density functional theory (DFT) ionization potential theorem values by inclusion of a systematic derivative discontinuity shift of the X potential. NCAP also enables time- dependent DFT (TD-DFT) calculations of good quality polarizabilities, hyper-polarizabilities, and one-fermion excited states without modification (calculated or ad hoc) of the long-range behavior of the exchange potential or other patches. NCAP is constructed by reconsidering the imposition of the asymptotic correctness of the X potential (-1/r) as a constraint. Inclusion of derivative discontinuity and approximate integer self-interaction correction treatments along with first-principles determination of the effective second-order gradient expansion coefficient yields a major advance over our earlier correct asymptotic potential functional [CAP; J. Chem. Phys. 142, 054105 (2015)]. The new functional reduces a spurious bump in the CAP atomic exchange potential and moves it to distances irrelevantly far from the nucleus (outside the tail of essentially all practical basis functions). It therefore has nearly correct atomic exchange-potential behavior out to rather large finite distances r from the nucleus but eventually goes as -c/r with an estimated value for the constant c of around 0.3 , so as to achieve other important properties of exact DFT exchange within the restrictions of the GGA form. We illustrate the results with the Ne atom optimized effective potentials and with standard molecular benchmark test data sets for thermochemical, structural and response properties.

A procedure for removing explicit orbital dependence from meta-generalized-gradient approximation (mGGA) exchange-correlation functionals by converting them into Laplacian-dependent functionals recently was developed by us and shown to be successful in molecules. It uses an approximate kinetic energy density functional (KEDF) parametrized to Kohn-Sham results (not experimental data) on a small training set. Here we present extensive validation calculations on periodic solids that demonstrate that the same deorbitalization with the same parametrization also is successful for those extended systems. Because of the number of stringent constraints used in its construction and its recent prominence, our focus is on the SCAN meta-GGA. Coded in VASP, the deorbitalized version, SCAN-L, can be as much as a factor of three faster than original SCAN, a potentially significant gain for large-scale ab initio molecular dynamics.

A simple, unconventional nonempirical, constraint-based orbital-free generalized gradient approximation (GGA) noninteracting kinetic energy density functional is presented along with illustrative applications. The innovation is adaptation of constraint-based construction to the essential properties of pseudodensities from the pseudopotentials that are essential in plane-wave-basis ab initio molecular dynamics. This contrasts with constraining to the qualitatively different Kato-cusp-condition densities. The single parameter in the proposed functional is calibrated by satisfying Pauli potential positivity constraints for pseudoatom densities. In static lattice tests on simple metals and semiconductors, the LKT (for the authors' initials) functional outperforms the previous best constraint-based GGA functional, VT84F [Phys. Rev. B 88, 161108(R) (2013)], is generally superior to a recently proposed meta-GGA, is reasonably competitive with parametrized two-point functionals, and is substantially faster.

The density linear response function for an inhomogeneous system of electrons in equilibrium with an array of fixed ions is considered. Two routes to its evaluation for extreme conditions (e.g., warm dense matter) are considered. The first is from a recently developed short-time kinetic equation; the second is from time-dependent density functional theory (tdDFT). The result from the latter approach agrees with that from kinetic theory in the "adiabatic approximation", providing support and contextual clarity for each. Both provide a connection to the phenomenological Kubo-Greenwood method for calculating transport properties.

The potential for predictive density functional calculations of matter under extreme conditions depends crucially upon having an exchange-correlation (XC) free energy functional accurate over a wide range of state conditions. Unlike the ground-state case, no such functional exists. We remedy that with systematic construction of a generalized gradient approximation XC free-energy functional based on rigorous constraints, including the free energy gradient expansion. The new functional provides the correct temperature dependence in the slowly varying regime and the correct zero-T, high-T, and homogeneous electron gas limits. Kohn-Sham calculations for hot electrons in a static fcc Aluminum lattice demonstrate the combined magnitude of thermal and gradient effects handled by this functional. Its accuracy in the warm dense matter regime is attested by excellent agreement of the calculated deuterium equation of state with reference path integral Monte Carlo results at intermediate and elevated T and by low density Al calculations over a wide T range.

The calculation of dynamical properties for matter under extreme conditions is a challenging task. The popular Kubo-Greenwood model exploits elements from equilibrium density-functional theory (DFT) that allow a detailed treatment of electron correlations, but its origin is largely phenomenological; traditional kinetic theories have a more secure foundation but are limited to weak ion-electron interactions. The objective here is to show how a combination of the two evolves naturally from the short-time limit for the generator of the effective single-electron dynamics governing time correlation functions without such limitations. This provides a theoretical context for the current DFT-related approach, the Kubo-Greenwood model, while showing the nature of its corrections. The method is to calculate the short-time dynamics in the single-electron subspace for a given configuration of the ions. This differs from the usual kinetic theory approach in which an average over the ions is performed as well. In this way the effective ion-electron interaction includes strong Coulomb coupling and is shown to be determined from DFT. The correlation functions have the form of the random-phase approximation for an inhomogeneous system but with renormalized ion-electron and electron-electron potentials. The dynamic structure function, density response function, and electrical conductivity are calculated as examples. The static local field corrections in the dielectric function are identified in this way. The current analysis is limited to semiclassical electrons (quantum statistical potentials), so important quantum conditions are excluded. However, a quantization of the kinetic theory is identified for broader application while awaiting its detailed derivation.

Kinetic energy density functionals (KEDFs) are central to orbital-free density functional theory. Limitations on the spatial derivative dependencies of KEDFs have been claimed from differential virial theorems. We point out a central defect in the argument: the relationships are not true for an arbitrary density but hold only for the minimizing density and corresponding chemical potential. Contrary to the claims therefore, the relationships are not constraints and provide no independent information about the spatial derivative dependencies of approximate KEDFs. A simple argument also shows that validity for arbitrary v-representable densities is not restored by appeal to the density-potential bijection.

We explore the simplification of widely used meta-generalized-gradient approximation (mGGA) exchange-correlation functionals to the Laplacian level of refinement by use of approximate kinetic energy density functionals (KEDFs). Such deorbitalization is motivated by the prospect of reducing computational cost while recovering a strictly Kohn-Sham local potential framework (rather than the usual generalized Kohn-Sham treatment of mGGAs). A KEDF that has been rather successful in solid simulations proves to be inadequate for deorbitalization but we produce other forms which, with parametrization to Kohn-Sham results (not experimental data) on a small training set, yield rather good results on standard molecular test sets when used to deorbitalize the meta-GGA made very simple, TPSS, and SCAN functionals. We also study the difference between high-fidelity and best-performing deorbitalizations and discuss possible implications for use in ab initio molecular dynamics simulations of complicated condensed phase systems.

Currently, the most common method to calculate transport properties for materials under extreme conditions is based on the phenomenological Kubo-Greenwood method. The results of an inquiry into the justification and context of that model are summarized here. Specifically, the basis for its connection to equilibrium DFT and the assumption of static ions are discussed briefly.

In high magnetic field calculations, anisotropic Gaussian type orbital (AGTO) basis functions are capable of reconciling the competing demands of the spherically symmetric Coulombic interaction and cylindrical magnetic (B field) confinement. However, the best available a priori procedure for composing highly accurate AGTO sets for atoms in a strong B field [Phys. Rev. A 90, 022504 (2014)] yields very large basis sets. Their size is problematical for use in any calculation with unfavorable computational cost scaling. Here we provide an alternative constructive procedure. It is based upon analysis of the underlying physics of atoms in B fields that allows identification of several principles for the construction of AGTO basis sets. Aided by numerical optimization and parameter fitting, followed by fine tuning of fitting parameters, we devise formulae for generating accurate AGTO basis sets in an arbitrary B field. For the hydrogen iso-electronic sequence, a set depends on B field strength, nuclear charge, and upon orbital quantum numbers. For multi-electron systems, the basis set formulae also include adjustment to account for orbital occupations. Tests of the new basis sets for atoms H through C (1 ≤ Z ≤ 6), and ions Li⁺, Be⁺, and B⁺, in a wide B field range (0 ≤ B ≤ 2000 a.u.), show an accuracy better than a few μH for single-electron systems, and a few hundredths to a few mHs for multi-electron atoms. The relative errors are similar for different atoms and ions in a large B field range, from a few to a couple of tens of millionths, thereby confirming rather uniform accuracy across the nuclear charge Z and B field strength values. Residual basis set errors are two to three orders of magnitude smaller than the electronic correlation energies in muti-electron atoms, a signal of the usefulness of the new AGTO basis sets in correlated wavefunction or density functional calculations for atomic and molecular systems in an external strong B field.

The roles of the large exchange dimensionless gradient limit, \(s\rightarrow\infty\) with \(s = \frac{\vert \nabla n(r)\vert}{2k_f(r)n(r)}\), \(k_f = (3\pi^2 n(r))^{1/3}\), and of dispersion interactions computed by Grimme's scheme in the context of solids are considered. Two families of recently developed generalized gradient approximation exchange functionals in combination with a suitably calibrated dispersion contribution are studied. Furthermore, the effects of changing the correlation functional or including exact exchange in the calculations also are explored. The results indicate that the large exchange dimensionless gradient limit has a small influence and that the most important contribution for a better description of the structure and energetics of porous materials is dispersion. The functional that provides best overall agreement with the experimental stability trend of a large set of pure silica zeolites is an exchange functional (denoted lsRPBE) based on the modified version of PBE, the exchange functional RPBE, corrected to satisfy the large exchange dimensionless limit, combined with PBE correlation and including a calibrated Grimme dispersion contribution. It outperforms any of the functionals that include exact exchange which were tested. Remarkably, the simple local density approximation does almost as well.

As the foundation for a new computational implementation, we survey the calculation of the complex electrical conductivity tensor based on the Kubo- Greenwood (KG) formalism (J. Phys. Soc. Jpn. 12, 570 (1957); Proc. Phys. Soc. 71, 585 (1958)), with emphasis on derivations and technical aspects per- tinent to use of projector augmented wave datasets with plane wave basis sets (Phys. Rev. B 50, 17953 (1994)). New analytical results and a full implemen- tation of the KG approach in an open-source Fortran 90 post-processing code for use with Quantum Espresso (J. Phys. Cond. Matt. 21, 395502 (2009)) are presented. Named KGEC ([K]ubo [G]reenwood [E]lectronic [C]onductivity), the code calculates the full complex conductivity tensor (not just the average trace). It supports use of either the original KG formula or the popular one approximated in terms of a Dirac delta function. It provides both Gaussian and Lorentzian representations of the Dirac delta function (though the Lorentzian is preferable on basic grounds). KGEC provides decomposition of the con- ductivity into intra- and inter-band contributions as well as degenerate state contributions. It calculates the dc conductivity tensor directly. It is MPI par- allelized over k-points, bands, and plane waves, with an option to recover the plane wave processes for their use in band parallelization as well. It is designed to provide rapid convergence with respect to k-point density. Examples of its use are given.

Density Functional Theory relies on universal functionals characteristic of a given system. Those functionals in general are different for the electron gas and for jellium (electron gas with uniform background). However, jellium is frequently used to construct approximate functionals for the electron gas (e.g., local density approximation, gradient expansions). The precise relationship of the exact functionals for the two systems is addressed here. In particular, it is shown that the exchange - correlation functionals for the inhomogeneous electron gas and inhomogeneous jellium are the same. This justifies theoretical and quantum Monte Carlo simulation studies of jellium to guide the construction of functionals for the electron gas. Related issues of the thermodynamic limit are noted as well.

This comment has no Abstract.

The direct random phase approximation (RPA) and RPA with second-order screened exchange (SOSEX) have been implemented with complex orbitals as a basis for treating open-shell atoms. Both RPA and RPA+SOSEX are natural implicit current density functionals because the paramagnetic current density implicitly is included through the use of complex orbitals. We confirm that inclusion of the SOSEX correction improves the total energy accuracy substantially compared to RPA, especially for smaller-Z atoms. Computational complexity makes post self-consistent-field (post-SCF) evaluation of RPA-type expressions commonplace, so orbital basis origins and properties become important. Sizable differences are found in correlation energies, total atomic energies, and ionization energies for RPA-type functionals evaluated in the post-SCF fashion with orbital sets obtained from different schemes. Reference orbitals from Kohn-Sham calculations with semi-local functionals are more suitable for RPA+SOSEX to generate accurate total energies, but reference orbitals from exact exchange (non-local) yield essentially energetically degenerate open-shell atom ground states. RPA+SOSEX correlation combined with exact exchange calculated from a hybrid reference orbital set (half the exchange calculated from exact-exchange orbitals, the other half of the exchange from orbitals optimized for the Perdew-Burke-Ernzerhof (PBE) exchange functional) gives the best overall performance. Numerical results show that the RPA-like functional with SOSEX correction can be used as a practical implicit current density functional when current effects should be included.

To demonstrate that there are specific temperature effects in the description of static and dynamic polar- izabilities which arise from generalized gradient approximation exchange-correlation functionals that obey distinctive asymptotic constraints, we present calculations for a test set of small molecules, at the experimental geometry, at the optimized ground-state geometry, and at the Born-Oppenheimer molecular dynamics geometries that arise from simulating a temperature of 300 K. The results indicate that a functional with the correct asymptotic potential (CAP) provides a better description at room tem- perature than does a GGA functional with an exponentially decaying exchange potential such as PBE.

A pair of families of generalized gradient approximation (GGA) exchange functionals is presented. The aim is to simplify the PW91 enhancement factor considerably, yet retain its shape, not degrade its performance and, within those confines, improve on it. The functionals are constructed non-empirically by taking as kernels the PBE and RPBE analytic forms and adding a Gaussian tail to comply with the asymptotic reduced-gradient constraint from non-uniform scaling. Standard heats of formation are considerably improved by the functionals compared to PBE. Globally, the new functionals exhibit much better balance in predicting thermodynamic and kinetic properties than any competing non-empirical GGA.

Finding the crystalline phase transitions along the zero-temperature equation of state (?cold curve?) of elemental solids has been a triumph of density functional theory even in the simple local-density approximation. The present work shows how quantitatively significant improve- ment on molecules in a generalized gradient approximation functional can result in unexpected ambiguity in the cold curve. The specific cause is obscure but some possibilities are discussed.

A non-empirical global hybrid exchange-correlation energy functional which leads to an exchange potential with correct asymptotic behavior is presented. The exchange functional combines one-fourth of exact exchange with three-fourths of the correct asymptotic potential (CAP) generalized gradient approximation functional. It is combined with the Perdew-Burke-Ernzerhof correlation energy with a slightly modified parameterization so as to cancel the gradient terms of CAP exchange with that of correlation, in the limit of slowly varying density. The resulting global hybrid functional, called CAP0, gives heats of formation, ionization potentials, electron affinities, proton affinities, binding energies of weakly interacting systems, barrier heights for hydrogen and non-hydrogen transfer reactions, bond distances, and harmonic frequencies on standard test sets that are competitive with results from other long-range-corrected, Coulomb-attenuated, or global hybrid functionals. In fact, they are generally superior to or competitive with CAM-PBE0 and, except for heats of formation, with CAM-B3LYP as well. Advantageously, the Rydberg excitation energies from CAP0 are superior to those of other global hybrids and of the long-range-corrected hybrids. They are similar to those from CAM-B3LYP and modestly inferior to the CAM-PBE0 errors. For the valence excitations, we did not find substantial differences for all the hybrid functionals considered, while the oscillator strengths from CAP0 are better to those of other global hybrids and comparable to those obtained with long-range-corrected and Coulomb-attenuated hybrids.

Approximations to the many-fermion free energy density functional that include the Thomas-Fermi (TF) form for the non-interacting part lead to singular densities for singular external potentials (e.g. attractive Coulomb). This limitation of the TF approximation is addressed here by a formal map of the exact Euler equation for the density onto an equivalent TF form characterized by a modified Kohn-Sham potential. It is shown to be a "regularized" version of the Kohn-Sham potential, tempered by convolution with a finite-temperature response function. The resulting density is non-singular, with the equilibrium properties obtained from the total free energy functional evaluated at this density. This new representation is formally exact. Approximate expressions for the regularized potential are given to leading order in a non-locality parameter and the limiting behavior at high and low temperatures is described. The non-interacting part of the free energy in this approximation is the usual Thomas-Fermi functional. These results generalize and extend to finite temperatures the ground-state regularization by Parr and Ghosh (Proc. Nat. Acad. Sci. 83, 3577 (1986)) and by Pratt, Hoffman, and Harris (J. Chem. Phys. 92, 1818 (1988)) and formally systematize the finite-temperature regularization given by the latter authors.

A quantum system of N Coulomb charges confined within a harmonic trap is considered over a wide range of densities and temperatures. A recently described construction of an equivalent classical system is applied in order to exploit the rather complete classical description of harmonic confinement via liquid state theory. Here, the effects of quantum mechanics on that representation are described with attention focused on the origin and nature of shell structure. The analysis extends from the classical strong Coulomb coupling conditions of dusty plasmas to the opposite limit of low temperatures and large densities characteristic of "warm, dense matter".

The effects of an explicit temperature dependence in the exchange correlation (XC) free-energy functional upon calculated properties of matter in the warm dense regime are investigated. The comparison is between the Karasiev-Sjostrom-Dufty-Trickey (KSDT) finite-temperature local-density approximation (TLDA) XC functional [Karasiev et al., Phys. Rev. Lett. 112, 076403 (2014)] parametrized from restricted path-integral Monte Carlo data on the homogeneous electron gas (HEG) and the conventional Monte Carlo parametrization ground-state LDA XC [Perdew-Zunger (PZ)] functional evaluated with T-dependent densities. Both Kohn-Sham (KS) and orbital-free density-functional theories are used, depending upon computational resource demands. Compared to the PZ functional, the KSDT functional generally lowers the dc electrical conductivity of low-density Al, yielding improved agreement with experiment. The greatest lowering is about 15% for T = 15 kK. Correspondingly, the KS band structure of low-density fcc Al from the KSDT functional exhibits a clear increase in interband separation above the Fermi level compared to the PZ bands. In some density-temperature regimes, the deuterium equations of state obtained from the two XC functionals exhibit pressure differences as large as 4% and a 6% range of differences. However, the hydrogen principal Hugoniot is insensitive to the explicit XC T dependence because of cancellation between the energy and pressure-volume work difference terms in the Rankine-Hugoniot equation. Finally, the temperature at which the HEG becomes unstable is T≥7200 K for the T-dependent XC, a result that the ground-state XC underestimates by about 1000 K.

Within the framework of density functional theory, a study of approximations to the enhancement factor of the non- interacting kinetic energy functional T_s[ρ] has been presented. For this purpose, the model of Liu and Parr [Liu and Parr, Phys Rev A 55, 1792 (1997)] based on a series expansion of T_s[ρ] involving powers of the density was employed. Application to 34 atoms, at the Hartree‐Fock level has shown that the enhancement factors present peaks that are in excellent agreement with those of the exact ones and give an accurate description of the shell structure of these atoms. The application of Z-dependent expansions to represent some of the terms of these approximations for neutral atoms and for positive and negative ions, which allows T_s[ρ] to be cast in a very simple form, is also explored. Indications are given as to how these functionals may be applied to molecules and clusters.

A previous analysis of scaling, bounds, and inequalities for the non-interacting functionals of thermal density functional theory is extended to the full interacting functionals. The results are obtained from analysis of the related functionals from the equilibrium statistical mechanics of thermodynamics for an inhomogeneous system. Their extension to the functionals of density functional theory is described.

We suggest a more nuanced view of the merit and utility of generalized gradient approximations (GGAs) for the non-interacting kinetic energy than the critique of Xia and Carter (X-C). Specifically, the multiple-valuedness of the Pauli term enhancement factor (denoted G[n] by X-C), with respect to the inhomogeneity variable s can be excluded by enforcement of a bound on the Kohn-Sham KE to achieve universality of the functional along with enforcement of proper large-s behavior. This is physically sensible in that the excluded G values occur for s values that occur only at very low densities. The behavior is exacerbated by peculiarities of pseudo-densities. The VT84F GGA, constructed with these constraints, does not have the numerical instability in our older PBE2 functional analyzed by X-C.

The thermodynamics for a system with given temperature, density, and volume is described by the Canonical ensemble. The thermodynamics for a corresponding system with the same temperature, volume, and average density is described by the Grand Canonical ensemble. In general a chosen thermodynamic potential (e.g., free energy) is different in the two cases. Their relationship is considered here as a function of the system size. Exact expressions relating the fundamental potential for each (free energy and pressure, respectively) are identified for arbitrary system size. A formal asymptotic analysis for large system size gives the expected equivalence, but without any characterization of the intermediate size dependence. More detailed evaluation is provided for the simple case of a homogeneous, non-interacting Fermi gas. In this case, the origin of size dependence arises from only two length scales, the average inter-particle distance and quantum length scale (thermal deBroglie or Fermi length). The free energies per particle calculated from each ensemble are compared for particle numbers $2\le N\le 64$ for a range of temperatures above and below the Fermi temperature. The relevance of these results for applications of density functional theory is discussed briefly.

F.E. Harris has been a significant partner in our work on orbital-free density functional approximations for use in ab initio molecular dynamics. Here we mention briefly the essential progress on single-point functionals since our original paper (2006). Then we focus on the advantages and limitations of generalized gradient approximation (GGA) non-interacting kinetic-energy functionals. We reconsider the constraints provided by near-origin conditions in atomic-like systems and their relationship to regularized versus physical external potentials. Then we seek the best empirical GGA for the non-interacting KE for a modest-sized set of molecules with a well-defined near-origin behavior of their densities. The search is motivated by a desire for insight into GGA limitations and for a target for constraint-based development.

A recent description of an exact map for the equilibrium structure and thermodynamics of a quantum system onto a corresponding classical system is summarized. Approximate implementations are constructed by pinning exact limits (ideal gas, weak coupling), and illustrated by calculation of pair correlations for the uniform electron gas and shell structure for harmonically conned charges. A wide range of temperatures and densities are addressed in each case. For the electron gas, comparisons are made to recent path integral Monte Carlo simulations (PIMC) showing good agreement. Finally, the relevance for orbital free density functional theory for conditions of warm, dense matter is discussed brie y.

Smooth, highly accurate analytical representations of Fermi-Dirac (FD) integral combinations important in free-energy density functional calculations are presented. Specific forms include those that occur in the local density approximation (LDA), generalized gradient approximation (GGA), and fourth-order gradient expansion of the non-interacting free energy as well as in the LDA and second-order gradient expansion for exchange. By construction, all the representations and their derivatives of any order are continuous on the full domains of their independent variables. The same type of technique provides an analytical representation of the function inverse to the FD integral of order 1/2. It plays an important role in physical problems related to the electron gas at finite temperature. From direct evaluation, the quality of these improved representations is shown to be substantially superior to existing ones, many of which were developed before the era of large-scale computation or early in the era.

A new non-empirical exchange energy functional of the generalized gradient approximation type which gives an exchange potential with the correct asymptotic behavior is developed and explored. In combination with the Perdew-Burke-Ernzerhof correlation energy functional, the new CAP-PBE (CAP stands for correct asymptotic potential) exchange-correlation functional gives heats of formation, ionization potentials, electron affinities, proton affinities, binding energies of weakly interacting systems, barrier heights for hydrogen and non-hydrogen transfer reactions, bond distances, and harmonic frequencies on standard test sets that are fully competitive with those obtained from other GGA-type functionals that do not have the correct asymptotic exchange potential behavior. Distinct from them, the new functional provides important improvements in quantities dependent upon response functions, e.g., static and dynamic polarizabilities and hyperpolarizabilities. When CAP is combined with the Lee-Yang-Parr correlation energy (CAP-LYP) one finds roughly equivalent results. Consideration of the computed dynamical polarizabilities in the context of the broad spectrum of other properties considered tips the balance to the non-empirical CAP-PBE combination. Intriguingly, these improvements arise primarily from improvements in the highest occupied and lowest unoccupied molecular orbitals, and not from shifts in the associated eigenvalues. Those eigenvalues do not change dramatically with respect to eigenvalues from other GGA-type functionals that do not provide the correct asymptotic behavior of the potential.

This communication comments on the work of A.M. Frolov and D.M. Wardlaw in Phys. Rev. A 78, 042506 (2008), and specifically on its treatment of the expansion of the total nonrelativistic energies of small atomic systems in inverse powers of the nuclear charge.

Implementation of orbital-free free-energy functionals in the Profess code and the coupling of Profess with the Quantum Espresso code are described. The combination enables orbital-free DFT to drive ab initio molecular dynamics simulations on the same footing (algorithms, thermostats, convergence parameters, etc.) as for Kohn-Sham (KS) DFT. All the non-interacting free-energy func- tionals implemented are single-point: the local density approximation (LDA; also known as finite-T Thomas-Fermi, ftTF), the second-order gradient approximation (SGA or finite-T gradient-corrected TF), and our recently introduced finite-T generalized gradient approximations (ftGGA). Elimination of the KS orbital bottleneck via orbital-free methodology enables high-T simulations on ordinary computers, whereas those simulations would be costly or even prohibitively time-consuming for KS molecular dynamics (MD) on very high-performance computer systems. Example MD simulations on H over a temperature range 2, 000 K <= T <= 4, 000, 000 K are reported, with timings on small clusters (16-128 cores) and even laptops. With respect to KS-driven calculations, the orbital-free calculations are between a few times through a few hundreds of times faster.

An accurate analytical parametrization for the exchange-correlation free energy of the homogeneous electron gas, including interpolation for partial spin-polarization, is derived via thermodynamic analysis of recent restricted path integral Monte-Carlo (RPIMC) data. This parametrization constitutes the local spin density approximation (LSDA) for the exchange-correlation functional in density functional theory. The new finite-temperature LSDA reproduces the RPIMC data well, satisfies the correct high-density and low- and high-T asymptotic limits, and is well-behaved beyond the range of the RPIMC data, suggestive of broad utility.

State-of-the-art treatment of nuclei and electrons in materials uses ab initio molecular dynamics for nuclear motion driven by Born-Oppenheimer forces from the electrons. Almost universally, those forces are calculated from density functional theory in the Kohn-Sham form. The computational costs of the conventional KS implementation scale at least as the cube of the number of electrons. This is a formidable barrier to complex system simulations with long MD runs on department-scale machines, since the DFT force calculation dominates the per step cost. The difficulty arises from the explicit dependence of the non-interacting kinetic energy on the KS non-interacting orbitals. The cost scaling worsens with use of explicitly orbital-dependent exchange-correlation functionals are used. The alternative approach, use of DFT in its basic form, dates to Thomas-Fermi-Dirac theory. The challenge is to have sufficiently accurate orbital-free expressions for the KS kinetic energy and exchange-correlation functionals. We discuss progress on these tasks via constraint-based methods, with emphasis on developments since the Sept. 2010 ``New Approaches to Many-Electron Theory'' meeting.

Reliable, tractable computational characterization of warm dense matter is a challenging task because of the wide range of important aggregation states and effective interactions involved. Contemporary best practice is to do ab initio molecular dynamics on the ion constituents with the forces from the electronic population provided by density functional calculations. Issues with that approach include the lack of reliable approximate density functionals and the computational bottleneck intrinsic to Kohn-Sham calculations. Our research is aimed at both problems, via the so-called orbital-free approach to density functional theory. After a sketch of the relevant properties of warm dense matter to motivate our research, we give a survey of our results for constraint-based non-interacting free energy functionals and exchange-correlation free-energy functionals. That survey includes comparisons with novel finite-temperature Hartree-Fock calculations and also presents progress on both pertinent exact results and matters of computational technique.

For a wide range of magnetic fields, $0 \le B \le 2000$ a.u., we present a systematic comparative study of the performance of different types of density functional approximations in light atoms ($2 \le Z \le 6$). Local, generalized gradient approximation (GGA; semi-local), and meta-GGA ground state exchange-correlation (XC) functionals are compared on an equal footing with exact-exchange, Hartree-Fock (HF), and current-density-functional-theory (CDFT) approximations. Comparison also is made with published quantum Monte Carlo data. Though all approximations give qualitatively reasonable results, the exchange energies from local and GGA functionals are too negative for large $B$. Results from the PBE ground-state GGA and TPSS ground state meta-GGA functionals are very close. Because of confinement, self-interaction error in such functionals is more severe at large $B$ than at $B=0$, hence self-interaction correction is crucial. Exact-exchange combined with the TPSS correlation functional results in a self-interaction-free (XC) functional, from which we obtain atomic energies of comparable accuracy to those from correlated wavefunction methods. Specifically for the B and C atoms, we provide the best beyond-HF energies in a wide range of $B$ fields. Fully self-consistent CDFT calculations were done with the Vignale-Rasolt-Geldart (VRG) functional in conjunction with the PW92 XC functional. Current effects turn out to be small, and the vorticity % ${\boldsymbol \nu}({\mathbf r})$, variable in the VRG functional diverges in some low-density regions. This part of the study suggests that non-local, self-interaction-free functionals may be better than local approximations as a starting point for CDFT functional construction and that some basic variable other than the vorticity could be helpful in making CDFT calculations practical.

A new recursive procedure is reported for the evaluation of certain three-body integrals involving expo- nentially correlated atomic orbitals. The procedure is more rapidly convergent than those reported earlier. The for- mulas are relevant to ab initio electronic-structure com- putations on three- and four-body systems. They also illustrate techniques that are useful in the evaluation of summations involving binomial coefficients.

We report a purely nonempirical generalized gradient approximation for the noninteracting free energy functional of orbital-free density functional theory obtained via a novel constraint-based parametrization scheme. We use that functional to provide forces for finite-temperature molecular dynamics simulations in the warm dense matter (WDM) regime and demonstrate good-to-excellent agreement with reference Kohn-Sham calculations under WDM conditions at a minuscule fraction of the computational cost of corresponding orbital-based simulations.

We calculate the free energy of the quantum uniform electron gas for temperatures from near zero to 100 times the Fermi energy, approaching the classical limit. An extension of the Vashista-Singwi theory to finite temperatures and self-consistent compressibility sum rule is presented. Comparisons are made to other local field correction methods, as well as recent quantum Monte Carlo simulation and classical map based results. Accurate fits to the exchange-correlation free energy from both theory and simulation are given for future practical applications.

A simple, practical model for computing the equilibrium thermodynamics and structure of the uniform electron gas (jellium) by classical strong coupling methods is proposed. Conditions addressed are those of interest for recent studies of warm dense matter: solid densities and temperatures from zero to plama states. An effective pair potential and coupling constant are introduced, incorporating the ideal gas, low density, and weak coupling quantum limits. The resulting parameter-free, analytic model is illustrated by the calculation of the pair correlation function via strong coupling classical liquid state theory. The results compare favorably with the first finite temperature restricted path integral Monte Carlo simulations reported recently.

A quantum system at equilibrium is represented by a corresponding classical system, chosen to reproduce thermodynamic and structural properties. The motivation is to allow application of classical strong-coupling theories and molecular dynamics simulation to quantum systems at strong coupling. The correspondence is made at the level of the grand-canonical ensembles for the two systems. An effective temperature, local chemical potential, and pair potential are introduced to define the corresponding classical system. These are determined formally by requiring the equivalence of the grand potentials and their functional derivatives. Practical inversions of these formal definitions are indicated via the integral equations for densities and pair correlation functions of classical liquid theory. Application to the ideal Fermi gas is demonstrated, and the weak-coupling form for the pair potential is given. In a companion paper two applications are described: the thermodynamics and structure of uniform jellium over a range of temperatures and densities and the shell structure of harmonically bound charges.

In the preceding paper, the structure and thermodynamics of a given quantum system was represented by a corresponding classical system having an effective temperature, local chemical potential, and pair potential. Here, that formal correspondence is implemented approximately for applications to two quantum systems. The first is the electron gas (jellium) over a range of temperatures and densities. The second is an investigation of quantum effects on shell structure for charges confined by a harmonic potential.

Summary: This paper gives a way to modify the PBE exchange-correlation functional to satisfy the relevant large reduced-density-gradient limit \( s \rightarrow \infty \) and reconsiders the distribution of \( s \) values important for generalized gradient approximation functionals.

With rare exceptions, explicit particle number dependence (Ne-dependence) in approximate density functionals is viewed as a serious deficiency because of apparent size-consistency issues. In contrast, there are multiple manifestations of explicit Ne-dependence in density functional bounds (including the Gázquez-Robles kinetic energy bound), constraints, and approximations. We argue that these constitute inescapable motivation for exploring Ne-dependent approximate functionals. Doing so would be consistent with a mostly-ignored result of Lieb about properties of the universal functional.

Con contadas excepciones, en la teoría de funcionales de la densidad aproximada la dependencia explícita en el número de partículas se visualiza como una seria deficiencia debido a problemas aparentes asociados con la consistencia en tamaño. En contraste, existen múltiples manifestaciones de cotas (como la cota de Gázquez-Robles de la energía cinética), restricciones y aproximaciones de los funcionales de la densidad las cuales tienen una dependencia explícita en el número de partículas. En este trabajo discutimos que lo anterior es una motivación ineludible para explorar la búsqueda y construcciín de funcionales aproximados con dependencia explícita en el número de partículas. Hacerlo será consistente con un resultado debido a Lieb, y muy frecuentemente ignorado, respecto a la estructura del funcional universal.

In this work, based on the experimental electron density, we present the approximate spatial distributions of the Pauli potential, one of the key quantities in the orbital-free density functional, for three crystalline systems: diamond, cubic boron nitride, and magnesium diboride. Our aim is to reveal a link between the Pauli potential and the orbital-free picture of chemical bond. We also expand the theoretical framework by developing the concept of the Pauli charge density. We find that both these quantities reproduce the electronic shell structure in the atomic core regions, while in the bonding region they reveal the different features for different bonding types, thereby distinguishing between ionic and covalent bond and also identifying the distinction between polar and nonpolar covalent bonds. Therefore, the Pauli potential and its associated charge density can be used as the orbital-free descriptors of chemical bond in the crystalline systems.

We develop a framework for orbital-free generalized gradient approximations (GGAs) for the non-interacting free energy density and its components (kinetic energy, entropy) based upon analysis of the corresponding gradient expansion. From that we obtain a new finite-temperature GGA (ftGGA) pair. We discuss implementation of the finite-temperature Thomas-Fermi, second-order gradient expansion, and our new ftGGA free energy functionals in an orbital-free density functional theory (OF-DFT) code, including the construction and validation of required local pseudopotentials. Then we compare results of self-consistent OF-DFT calculations on hydrogen using those non-interacting free energy functionals (in combination with the zero-temperature local density approximation (LDA) for exchange-correlation) with results from conventional finite-temperature Kohn-Sham calculations and the same LDA. As an aid to implementation, we provide analytical expressions for the temperature-dependent scaling factors involved.

We compare the behavior of the finite-temperature Hartree-Fock model with that of thermal density functional theory using both ground-state and temperature-dependent approximate exchange functionals. The test system is bcc Li in the temperature-density regime of warm dense matter (WDM). In this exchange-only case, there are significant qualitative differences among the three approaches. Those differences may be important for Born-Oppenheimer molecular dynamics studies of WDM with ground-state approximate density functionals and thermal occupancies. Such calculations require well-characterized, reliable pseudopotentials over a demanding range of temperatures and densities. For that, we evaluate and validate pseudopotential techniques for the high-density regime. We compare pseudopotential and all-electron results for small Li clusters of local bcc symmetry and bond-lengths appropriate to high density bulk Li. We determine the density range over which both standard projector augmented wave (PAW) and norm-conserving pseudopotentials are reliable. Then we construct PAW data sets with a small cutoff radius which are valid for lithium densities up to at least 80 g/cm\(^3\) for both the local density approximation and the generalized gradient approximation for exchange-correlation.

We report the performance of a non-empirical meta-GGA that comes from converting our simple VT{84} 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. Its validation 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{84} are substantially better but reaction barrier heights are considerably worse. This suggests that additional constraints and better correlation functionals are needed.

New local "hybrid" functionals proposed by V. V. Karasiev in [J. Chem. Phys. 118, 8567 (2003)] are benchmarked against nonlocal hybrid functionals. Their performance is tested on the total and high occupied orbital energies, as well as the electric moments of selected diatomic molecules. The new functionals, along with the Hartree-Fock and non-hybrid functionals, are employed for finite-difference calculations, which are basis-independent. Basis set errors in the total energy and electric moments are calculated for the 6-311G, 6-311G++G(3df,3pd) and AUG-cc-pVnZ (n=3,4,6) basis sets used in conjunction with the Hartree-Fock and conventional density functional methods. A comparison between the results of the finite-difference local "hybrid" and basis set nonlocal hybrid functional shows that total energies of local and nonlocal hybrid functionals agree to within the basis set error. Discrepancies for multipole moments are larger in magnitude when compared to the basis set errors, but still reasonably small (smaller than errors produced by the 6-311G basis set). Thus, we recommend using the new local "hybrid" functionals whenever the accuracy is expected to be sufficient, because they require a solution of just differential Kohn-Sham equations, instead of integro-differential ones in the case of hybrid functionals.

Finding a proper local measure of chemical hardness has been a long-standing aim of density functional theory. The traditional approach to defining a local hardness index, by the derivative of the chemical potential \( \mu \) with respect to the electron density \( n(\vec{r}) \) subject to the constraint of a fixed external potential \( v(\vec{r}) \), has raised several questions, and its chemical applicability has proved to be limited. Here, we point out that the only actual possibility to obtain a local hardness measure in the traditional approach emerges if the external potential constraint is dropped; consequently, utilizing the ambiguity of a restricted chemical potential derivative is not an option to gain alternative definitions of local hardness. At the same time, however, the arising local hardness concept turns out to be fatally undermined by its inherent connection with the asymptotic value of the second derivative of the universal density functional. The only other local hardness concept one may deduce from the traditional approach, \( \delta\mu[n]/\delta n(\vec{r}) |_{v(\vec{r})} \) , is the one that gives a constant value, the global hardness itself, throughout an electron system in its ground state. Consequently, the traditional approach is in principle incapable of delivering a local hardness indicator. The parallel case of defining a local version of the chemical potential itself is also outlined, arriving at similar conclusions.

The performance of eight generalized gradient approximation exchange-correlation (xc) functionals is assessed by a series of scalar relativistic all-electron calculations on octahedral palladium model clusters Pd\( _n \) with n = 13, 19, 38, 55, 79, 147 and the analogous clusters Au\( _n \) (for n up to 79). For these model systems we determined the cohesive energies and average bond lengths of the optimized octahedral structures. We extrapolate these values to the bulk limits and compare with the corresponding experimental values. While the well-established functionals BP, PBE, and PW91 are the most accurate at predicting energies, the more recent forms PBEsol, VMTsol, and VT{84}sol significantly improve the accuracy of geometries. The observed trends are largely similar for both Pd and Au. In the same spirit, we also studied the scalability of the ionization potentials and electron affinities of the Pd clusters, and extrapolated those quantities to estimates of the work function. Overall, the xc functionals can be classified into four distinct groups according to the accuracy of the computed parameters. These results allow a judicious selection of xc approximations for treating transition metal clusters.

Imposition of the constraint that, for the hydrogen atom, the exchange energy cancels the Coulomb repulsion energy yields a non-empirical re-parameterization of the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) exchange-correlation energy functional, and of the related PBE hybrid (PBE0). The re-parameterization, which leads to an increase of the gradient contribution to the exchange energy with respect to the original PBE functional, is tested through the calculation of heats of formation, ionization potentials, electron affinities, proton affinities, binding energies of weakly interacting systems, barrier heights for hydrogen and non-hydrogen transfer reactions, bond distances and harmonic frequencies, for some well known test sets designed to validate energy functionals. The results for the re-parameterized PBE GGA, called PBEmol, show a substantial improvement over the original PBE in the prediction of the heats of formation, while retaining the quality of the original PBE functional for description of all the other properties considered. The results for the hybrids indicate that, although the PBE0 functional provides a rather good description of these properties, the predictions of the re-parameterized functional, called PBEmol\( \beta \)0, are, except in the case of the ionization potentials, modestly better. Also, the results for PBEmol\( \beta \)0 are better than B3LYP, except for the ionization potentials and harmonic frequencies. The re-parameterization for the pure GGA (PBEmol) differs from that for the hybrid (PBEmol\( \beta \)0), illustrating that improvement at the GGA level of complexity does not necessarily provide the best GGA for use in a hybrid.

A quantum system at equilibrium is represented by a corresponding classical system, chosen to reproduce the thermodynamic and structural properties. The objective is to develop a means for exploiting strong coupling classical methods (e.g., MD, integral equations, DFT) to describe quantum systems. The classical system has an effective temperature, local chemical potential, and pair interaction that are defined by requiring equivalence of the grand potential and its functional derivatives with respect to the external and pair potentials for the classical and quantum systems. Practical inversion of this mapping for the classical properties is effected via the hypernetted chain approximation, leading to representations as functionals of the quantum pair correlation function. As an illustration, the parameters of the classical system are determined approximately such that ideal gas and weak coupling RPA limits are preserved.

Though there is fevered effort on orbital-dependent approximate exchange-correlation functionals, generalized gradient approximations, especially the Perdew-Burke-Ernzerhof form, remain the overwhelming choice in calculations. A simple GGA exchange functional ("VMT", J. Chem. Phys. 130, 244103 (2009)) was developed that improves substantially over PBE in energetics (on a typical test set) while being almost as simple in form. The improvement came from constraining the exchange enhancement factor to be below the Lieb-Oxford bound for all but one value of the exchange dimensionless gradient, \(s\), and to go to the uniform electron gas limit at both \(s = 0\) and \(s \to \infty \). Here we discuss the issue of asymptotic constraints for GGAs and show that imposition of the large \(s\) constraint, \( \displaystyle \lim_{s \to \infty} s^{1/2} F_{xc}(n,s) \lt \infty \), where \(F_{xc}(n,s)\) is the enhancement factor and \(n\) is the electron density, upon the VMT exchange functional yields modest but non-trivial further improvement. The resulting exchange functional, denoted VT{8,4}, is only slightly more complicated than VMT and easy to program. Additional improvement is obtained by combining VT{8,4} or VMT exchange with the LYP correlation functional. Extensive computational results on several data sets are provided as verification of the overall performance gains of both versions.

Solving the Euler equation which corresponds to the energy minimum of a density functional expressed in orbital-free form involves related but distinct computational challenges. One is the choice between all-electron and pseudo-potential calculations and, if the latter, construction of the pseudo-potential. Another is the stability, speed, and accuracy of solution algorithms. Underlying both is the fundamental issue of satisfactory quality of the approximate functionals (kinetic energy and exchange-correlation). We address both computational issues and illustrate them by some comparative performance testing of our recently developed modified-conjoint generalized gradient approximation kinetic energy functionals. Comparisons are given for atoms, diatomic molecules, and some simple solids.

Many-electron systems at substantial finite temperatures and densities present a major challenge to density functional theory. Very little is known about the free-energy behavior over the temperature range of interest, for example, in the study of warm dense matter. As a result, it is difficult to assess the validity of proposed approximate free-energy density functionals. Here we address, at least in part, this need for detailed results on well-characterized systems for purposes of testing and calibration of proposed approximate functionals. We present results on a comparatively simple, well-defined, but computationally feasible model, namely thermally occupied Hartree-Fock states for eight one-electron atoms at arbitrary positions in a hard-walled box. We discuss the main technical tasks (defining a suitable basis and evaluation of the required matrix elements) and discuss the physics which emerges from the calculations.

Finite temperature density functional theory requires representations for the internal energy, entropy, and free energy as functionals of the local density field. A central formal difficulty for an orbital-free representation is construction of the corresponding functionals for non-interacting particles in an arbitrary external potential. That problem is posed here in the context of the equilibrium statistical mechanics of an inhomogeneous system. The density functionals are defined and shown to be equal to the extremal state for a functional of the reduced one-particle statistical operators. Convexity of the latter functionals implies a class of general inequalities. First, it is shown that the familiar von Weizsäcker lower bound for zero temperature functionals applies at finite temperature as well. An upper bound is obtained in terms of a single-particle statistical operator corresponding to the Thomas-Fermi approximation. Next, the behavior of the density functionals under coordinate scaling is obtained. The inequalities are exploited to obtain a class of upper and lower bounds at constant temperature, and a complementary class at constant density. The utility of such constraints and their relationship to corresponding results at zero temperature are discussed.

Recently several variants of a new orbital-free density functional for the total \(N_e\)-electron kinetic energy (KE) have been proposed. These are based on a systematically constructed (\(N_e-1\))-electron conditional probability function and Monte Carlo evaluation of the associated conditional expectation of the KE operator in the case of the homogeneous electron gas. Because the resulting functionals depend on \(n \;\ln \; n\) (\(n\) = the electron number density), they have been interpreted as being the leading term in a Shannon information power expression for the non-von Weizsäcker part of the a total KE. We show that these functionals violate known positivity constraints, are inconsistent with known results for the correlation energy of the homogeneous electron gas, and that the Shannon information power interpretation also violates known constraints. We consider both the full KE and Kohn-Sham KE cases. Possible corrections and extensions are considered, including a possible new form for parametrization.

We present an analysis and extension of our constraint-based approach to orbital-free (OF) kinetic-energy (KE) density functionals intended for the calculation of quantum-mechanical forces in multiscale molecular-dynamics simulations. Suitability for realistic system simulations requires that the OF-KE functional yield accurate forces on the nuclei yet be computationally simple. We therefore require that the functionals be based on density-functional theory constraints, be local, be dependent at most upon a small number of parameters fitted to a training set of limited size, and be applicable beyond the scope of the training set. Our previous "modified-conjoint" generalized-gradient-type functionals were constrained to producing a positive-definite Pauli potential. Though distinctly better than several published generalized-gradient-approximation-type functionals in that they gave semiquantitative agreement with Born-Oppenheimer forces from full Kohn-Sham results, those modified-conjoint functionals suffer from unphysical singularities at the nuclei. Here we show how to remove such singularities by introducing higher-order density derivatives and analyze the consequences. We give a simple illustration of such a functional and a few tests of it.

Effective, explicitly density-dependent (i.e. orbital-free) approximations to the Kohn-Sham kinetic energy functional \(T_s\) remain elusive. In the course of developing such functionals in collaboration with Frank Harris, various issues have arisen that have gone unaddressed. We consider four of them here: positivity of the KS kinetic energy density, supposed requirements on its functional dependence, the use of cutoffs to insure positivity, and the role of the Coulomb virial theorem in the asymptotic analysis of \(T_s\).

We propose a different way to satisfy both gradient expansion limiting behavior and the Lieb–Oxford bound in a generalized gradient approximation exchange functional by extension of the Perdew–Burke–Ernzerhof (PBE) form. Motivation includes early and recent exploration of modified values for the gradient expansion coefficient in the PBE exchange-correlation functional (cf. the PBEsol functional) and earlier experience with a numerical cutoff for large-s \( (s \propto | \nabla n |/ n^{4/3}) \) in a version of the deMon molecular code. For either the original PBE or the PBEsol choice of the gradient coefficient, we find improved performance from using an s-dependent (spatially varying) satisfaction of the Lieb–Oxford bound which quenches to uniform electron gas behavior at large s. The mean absolute deviations (MADs) in atomization energies for a widely used test set of 20 small molecules are reduced by about 22% relative to PBE and PBEsol. For these small molecules, the bond length MADs are essentially unchanged.

Rapid calculation of Born-Oppenheimer (B-O) forces is essential for driving the so-called quantum region of a multi-scale molecular dynamics simulation. The success of density functional theory (DFT) with modern exchange-correlation approximations makes DFT an appealing choice for this role. But conventional Kohn-Sham DFT, even with various linear-scaling implementations, really is not fast enough to meet the challenge of complicated chemo-mechanical phenomena (e.g. stress-induced cracking in the presence of a solvent). Moreover, those schemes involve approximations that are difficult to check practically or to validate formally. A popular alternative, Car-Parrinello dynamics, does not guarantee motion on the B-O surface. Another approach, orbital-free DFT, is appealing but has proven difficult to implement because of the challenge of constructing reliable orbital-free (OF) approximations to the kinetic energy (KE) functional. To be maximally useful for multi-scale simulations, an OF-KE functional must be local (i.e. one-point). This requirement eliminates the two-point functionals designed to have proper linear-response behavior in the weakly inhomogeneous limit. In the face of these difficulties, we demonstrate that there is a way forward. By requiring only that the approximate functional deliver high-quality forces, by exploiting the "conjointness" hypothesis of Lee, Lee, and Parr, by enforcing a basic positivity constraint, and by parameterizing to a carefully selected, small set of molecules we are able to generate a KE functional that does a good job of describing various \( \rm{H_q Si_m O_n} \) clusters as well as CO (providing encouraging evidence of transferability). In addition to that positive result, we discuss several major negative results. First is definitive proof that the conjointness hypothesis is not correct, but nevertheless is useful. The second is the failure of a considerable variety of published KE functionals of the generalized gradient approximation type. Those functionals yield no minimum on the energy surface and give completely incorrect forces. In all cases, the problem can be traced to incorrect behavior of the functionals near the nuclei. Third, the seemingly obvious strategy of direct numerical fitting of OF-KE functional parameters to reproduce the energy surface of selected molecules is unsuccessful. The functionals that result are completely untransferable.