Second-order Gradient Expansion Research Articles

New Functional Orbital-free Within DFT for Metallic Systems

I present the continuation of a study on Laplacian Level Kinetic Energy (KE) functionals applied to metallic nanosystems. The development of novel Kinetic Energy functionals is an important topic in density functional theory (DFT). The nanoparticles are patterned using gelatin spheres of different sizes, background density and number of electrons. To reproduce the correct kinetic and potential energy density of the various nanoparticles, the use of semilocal descriptors is necessary. Need an explicit density functional expression for the kinetic energy of electrons, including the first e second functional derivative, i.e. the kinetic potential and the kinetic kernel, respectively. The exact explicit form of the non interacting kinetic energy, as a functional of the electron density, is known only for the homogeneous electron gas (HEG), i.e., the Thomas-Fermi (TF) local functional and for 1 and 2 electron systems, i.e., the von Weizsacker (VW) functional. In between these two extreme cases, different semilocal or non local approximations were developed in recent years. Most semilocal KE functionals are based on modifications of the second-order gradient expansion (GE2) or fourth-order gradient expansion (GE4). I find that the Laplacian contribute is fundamental for the description of the energy and the potential of nanoparticles. I propose a new LAP2 semilocal functional which, better than the previous ones, allows us to obtain fewer errors both of energy and potential. More details of the previous calculations can be found in my 2 previous works which will be cited in the text.

Reworking the Tao-Mo exchange-correlation functional. II. De-orbitalization.

In Paper I [H. Francisco, A. C. Cancio, and S. B. Trickey, J. Chem. Phys. 159, 214102 (2023)], 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 while essentially preserving compliance with the second-order gradient expansion. 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 functional. Here, we address de-orbitalization of that new sregTM into a pure density functional. We summarize the failures of the Mejía-Rodríguez and Trickey de-orbitalization strategy [Phys. Rev. A 96, 052512 (2017)] when used with both versions. We discuss how those failures apparently arise in the so-called z' 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 parameterized 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 cancelation 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 Tao-Perdew-Staroverov-Scuseri functional. Our work demonstrates that the intricacy of such two-indicator functionals magnifies the errors introduced by the Mejía-Rodríguez and Trickey de-orbitalization approach in ways that are extremely difficult to analyze and correct.

Gradient Expansions for the Large-Coupling Strength Limit of the Møller-Plesset Adiabatic Connection.

The adiabatic connection that has, as weak-interaction expansion, the Møller–Plesset perturbation series has been recently shown to have a large coupling-strength expansion, in terms of functionals of the Hartree–Fock density with a clear physical meaning. In this work, we accurately evaluate these density functionals and we extract second-order gradient coefficients from the data for neutral atoms, following ideas similar to the ones used in the literature for exchange, with some modifications. These new gradient expansions will be the key ingredient for performing interpolations that have already been shown to reduce dramatically MP2 errors for large noncovalent complexes. As a byproduct, our investigation of neutral atoms with large number of electrons N indicates that the second-order gradient expansion for exchange grows as N log(N) rather than as N, as often reported in the literature.

Development of novel kinetic energy functional for orbital-free density functional theory applications

The development of novel Kinetic Energy (KE) functionals is an important topic in density functional theory (DFT). In particular, this happens by means of an analysis with newly developed benchmark sets. Here, I present a study of Laplacian-level kinetic energy functionals applied to metallic nanosystems. The nanoparticles are modeled using jellium sph eres of different sizes, background densities, and number of electrons. The ability of different functionals to reproduce the correct kinetic energy density and potential of various nanoparticles is investigated and analyzed in terms of semilocal descriptors. Most semilocal KE functionals are based on modifications of the second-order gradient expansion GE2 or GE4. I find that the Laplacian contribute is fundamental for the description of the energy and the potential of nanoparticles.

Stationary models of magnetized viscous tori around a Schwarzschild black hole

We present stationary solutions of magnetized, viscous thick accretion disks around a Schwarzschild black hole. We assume that the tori are not self-gravitating, are endowed with a toroidal magnetic field, and obey a constant angular momentum law. Our study focuses on the role of the black hole curvature in the shear viscosity tensor and in their potential combined effect on the stationary solutions. Those are built in the framework of a causality-preserving, second-order gradient expansion scheme of relativistic hydrodynamics in the Eckart frame description which gives rise to hyperbolic equations of motion. The stationary models are constructed by numerically solving the general relativistic momentum conservation equation using the method of characteristics. We place constraints in the range of validity of the second-order transport coefficients of the theory. Our results reveal that the effects of the shear viscosity and curvature are particularly noticeable only close to the cusp of the disks. The surfaces of constant pressure are affected by viscosity and curvature and the self-intersecting isocontour - the cusp - moves to smaller radii (i.e. towards the black hole horizon) as the effects become more significant. For highly magnetized disks the shift in the cusp location is smaller. Our findings might have implications on the dynamical stability of constant angular momentum tori which, in the inviscid case, are affected by the runaway instability.

Generalized gradient approximations with local parameters

We develop and demonstrate the performance of a nonseparable form of the 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 nonempirical methodology can lead to a balanced GGA description of both finite and extended systems.

Constraint-based wave vector and frequency dependent exchange-correlation kernel of the uniform electron gas

According to time-dependent density functional theory, the exact exchange-correlation kernel f$_{xc}$(n, q, $\omega$) determines not only the ground-state energy but also the excited-state energies/lifetimes and time-dependent linear density response of an electron gas of uniform density n $=$ 3/(4$\pi$r$^3_s$). Here we propose a parametrization of this function based upon the satisfaction of exact constraints. For the static ($\omega$ = 0) limit, we modify the model of Constantin and Pitarke at small wavevector q to recover the known second-order gradient expansion, plus other changes. For all frequencies $\omega$ at q $=$ 0, we use the model of Gross, Kohn, and Iwamoto. A Cauchy integral extends this model to complex $\omega$ and implies the standard Kramers-Kronig relations. A scaling relation permits closed forms for not only the imaginary but also the real part of f$_{xc}$ for real $\omega$. We then combine these ingredients by damping out the $\omega$ dependence at large q in the same way that the q dependence is damped. Away from q $=$ 0 and $\omega$ $=$ 0, the correlation contribution to the kernel becomes dominant over exchange, even at r$_s$ $=$ 4, the valence electron density of metallic sodium. The resulting correlation energy from integration over imaginary $\omega$ is essentially exact. The plasmon pole of the density response function is found by analytic continuation of f$_{xc}$ to $\omega$ just below the real axis, and the resulting plasmon lifetime first decreases from infinity and then increases as q grows from 0 toward the electron-hole continuum. A static charge-density wave is found for r$_s$ $>$ 69, and shown to be associated with softening of the plasmon mode. The exchange-only version of our static kernel confirms Overhauser's 1968 prediction that correlation enhances the charge-density wave.

Improving the Performance of Tao-Mo Non-empirical Density Functional with Broader Applicability in Quantum Chemistry and Materials Science.

A revised version of the semilocal exchange-correlation functional [Tao, J.; Mo, Y. Phys. Rev. Lett. 2016, 117, 073001] (TM) is proposed by incorporating the modifications to its correlation content obtained from the full high-density second-order gradient expansion as proposed in the case of the revised Tao-Perdew-Staroverov-Scuseria (revTPSS) [Perdew, J. P.; Ruzsinszky, A.; Csonka, G. I.; Constantin, L. A.; Sun, J. Phys. Rev. Lett. 2009, 103, 026403] functional. The present construction improves the performance of the TM functional over a wide range of quantum chemical and solid-state properties (thermochemical and structural). More specifically, the cohesive energies, jellium surface exchange-correlation energies, and real metallic surface energies are improved by preserving the accuracy of the solid-state lattice constants and bulk moduli. The present proposition is not only physically motivated but also enhances the applicability of the TM functional. New physical insights with the proper exemplification of the present modification, which is presented here, can further help in constructing more realistic non-empirical density functionals.

Semilocal properties of the Pauli kinetic potential

The logarithmic singularity of the Lindhard linear response function plays an important role in various phenomena, such as the long-range Friedel oscillations and Kohn anomaly in phonon dispersion. Such a weak singularity cannot be captured by the known gradient expansion of the kinetic energy (KE), but it can be somewhat mimicked by the second-order gradient singularity expansion (GSE2) developed in this work. We show that the GSE2 Pauli KE potential of atoms, computed with the Kohn-Sham density, is remarkably accurate, being the best possible approximation provided by any second-order KE gradient expansion. Next, we study the utility of GSE2 for orbital-free density functional theory, and we prove that the GSE2-based KE functionals give an important and systematic improvement over other popular KE functionals.

Generalized Gradient Approximation Exchange Energy Functional with Near-Best Semilocal Performance.

We develop and validate a nonempirical generalized gradient approximation (GGA) exchange (X) density functional that performs as well as the SCAN (strongly constrained and appropriately normed) meta-GGA on standard thermochemistry tests. Additionally, the new functional (NCAP, nearly correct asymptotic potential) 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. 2015 , 142 , 054105 ]. 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.

Simple Modifications of the SCAN Meta-Generalized Gradient Approximation Functional.

We analyzed various possibilities to improve upon the SCAN meta-generalized gradient approximation density functional obeying all known properties of the exact functional that can be satisfied at this level of approximation. We examined the necessity of locally satisfying a strongly tightened lower bound for the exchange energy density in single-orbital regions, the nature of the error cancellation between the exchange and correlation parts in two-electron regions, and the effect of the fourth-order term in the gradient expansion of the correlation energy density. We have concluded that the functional can be modified to separately reproduce the exchange and correlation energies of the helium atom by locally releasing the strongly tightened lower bound for the exchange energy density in single-orbital regions, but this leads to an unbalanced improvement in the single-orbital electron densities. Therefore, we decided to keep the FX ≤ 1.174 exact condition for any single-orbital density, where FX is the exchange enhancement factor. However, we observed a general improvement in the single-orbital electron densities by revising the correlation functional form to follow the second-order gradient expansion in a wider range. Our new revSCAN functional provides more-accurate atomization energies for the systems with multireference character, compared to the SCAN functional. The nonlocal VV10 dispersion-corrected revSCAN functional yields more-accurate noncovalent interaction energies than the VV10-corrected SCAN functional. Furthermore, its global hybrid version with 25% of exact exchange, called revSCAN0, generally performs better than the similar SCAN0 for reaction barrier heights. Here, we also analyzed the possibility of the construction of a local hybrid from the SCAN exchange and a specific locally bounded nonconventional exact exchange energy density. We predict compatibility problems since this nonconventional exact exchange energy density does not really obey the strongly tightened lower bound for the exchange energy density in single-orbital regions.

Visualization and analysis of the Kohn-Sham kinetic energy density and its orbital-free description in molecules.

We visualize the Kohn-Sham kinetic energy density (KED) and the ingredients--the electron density, its gradient, and Laplacian--used to construct orbital-free models of it, for the AE6 test set of molecules. These are compared to related quantities used in metaGGA's, to characterize two important limits--the gradient expansion and the localized-electron limit typified by the covalent bond. We find the second-order gradient expansion of the KED to be a surprisingly successful predictor of the exact KED, particularly at low densities where this approximation fails for exchange. This contradicts the conjointness conjecture that the optimal enhancement factors for orbital-free kinetic and exchange energy functionals are closely similar in form. In addition we find significant problems with a recent metaGGA-level orbital-free KED, especially for regions of strong electron localization. We define an orbital-free description of electron localization and a revised metaGGA that improves upon atomization energies significantly.

Improved analytical representation of combinations of Fermi–Dirac integrals for finite-temperature density functional calculations

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.

Generalized Gradient Approximation Correlation Energy Functionals Based on the Uniform Electron Gas with Gap Model

We studied uniform electron gas with a gap model in the context of density functional theory. On the basis of this analysis, we constructed two local gap models that are used in generalized gradient approximation (GGA) correlation functionals that satisfy numerous exact constraints for correlation energy. The first one, named GAPc, fulfills the full second-order correlation gradient expansion at any density regime and is very accurate for jellium surfaces, comparable to state-of-the-art GGAs for atomic systems and molecular systems, and is well compatible with known semilocal exchanges. The second functional, named GAPloc, satisfies the same exact conditions, except that the second-order gradient expansion is sacrificed for a better behavior under the Thomas-Fermi scaling and a more realistic correlation energy density of the helium atom. The GAPloc functional displays a high accuracy for atomic correlation energies, still preserving a reasonable behavior for jellium surfaces. Moreover, it shows a higher compatibility with the Hartree-Fock exchange than other semilocal correlation functionals. This feature is explained in terms of the real-space analysis of the GAPloc correlation energy.

Van der Waals density functional made accurate

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

Density functional theory and Bader's atoms-in-molecules theory: towards a vivid dialogue

Density functional theory and Bader's atoms-in-molecules theory share the same primary ingredient: the electron density, which is the fundamental physical observable in quantum chemistry. In this paper, we elaborate on the decomposition of the Kohn-Sham molecular energy in terms of Bader's partition, discussing how Pendás' Interacting Quantum Atoms framework could be adapted to a DFT context. Besides, another bridge between these two theories is built through a general formalism able to generate new local descriptors from any second-order density gradient expansion. These approaches are then applied to two classes of intramolecular bonds: between two electronegative atoms and intramolecular hydrogen bonds, illustrating how fruitful a dialog between both theories can be.

Relevance of coordinate and particle-number scaling in density-functional theory

We discuss a $\beta$-dependent family of electronic density scalings of the form $n_\lambda(\R)=\lambda^{3\beta+1}\; n(\lambda^\beta \R)$ in the context of density functional theory. In particular, we consider the following special cases: the Thomas-Fermi scaling ($\beta=1/3$ and $\lambda \gg 1$), which is crucial for the semiclassical theory of neutral atoms; the uniform-electron-gas scaling ($\beta=-1/3$ and $\lambda\gg 1$), that is important in the semiclassical theory of metallic clusters; the homogeneous density scaling ($\beta=0$) which can be related to the self-interaction problem in density functional theory when $\lambda \leq 1$; the fractional scaling ($\beta=1$ and $\lambda\leq 1$), that is important for atom and molecule fragmentation; and the strong-correlation scaling ($\beta=-1$ and $\lambda \gg 1$) that is important to describe the strong correlation limit. The results of our work provide evidence for the importance of this family of scalings in semiclassical and quantum theory of electronic systems, and indicate that these scaling properties must be considered as important constraints in the construction of new approximate density functionals. We also show, using the uniform-electron-gas scaling, that the curvature energy of metallic clusters is related to the second-order gradient expansion of kinetic and exchange-correlation energies.

Generalized-gradient-approximation noninteracting free-energy functionals for orbital-free density functional calculations

We develop a framework for orbital-free generalized gradient approximations (GGAs) for the noninteracting 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 noninteracting 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.

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

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

A meta-GGA Made Free of the Order of Limits Anomaly.

We have improved the revised Tao-Perdew-Staroverov-Scuseria (revTPSS) meta-generalized gradient approximation (GGA) in order to remove the order of limits anomaly in its exchange energy. The revTPSS meta-GGA recovers the second-order gradient expansion for a wide range of densities and therefore provides excellent atomization energies and lattice constants. For other properties of materials, however, even the revTPSS does not give the desired accuracy. The revTPSS does not perform as well as expected for the energy differences between different geometries for the same molecular formula and for the related nonbarrier height chemical reaction energies. The same order of limits problem might lead to inaccurate energy differences between different crystal structures and to inaccurate cohesive energies of insulating solids. Here we show a possible way to remove the order of limits anomaly with a weighted difference of the revTPSS exchange between the slowly varying and iso-orbitals (one- or two-electron) limits. We show that the new regularized (regTPSS) gives atomization energies comparable to revTPSS and preserves the accurate lattice constants as well. For other properties, the regTPSS gives at least the same performance as the revTPSS or TPSS meta-GGAs.