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DOE OSTI2013

Generating functionals and Lagrangian partial differential equations

The main goal of this paper is to derive an alternative characterization of the multisymplectic form formula for classical field theories using the geometry of the space of boundary values. We review the concept of Type-I/II generating functionals defined on the space of boundary data of a Lagrangian field theory. On the Lagrangian side, we define an analogue of Jacobi's solution to the Hamilton–Jacobi equation for field theories, and we show that by taking variational derivatives of this functional, we obtain an isotropic submanifold of the space of Cauchy data, described by the so-called multisymplectic form formula. As an example of the latter, we show that Lorentz's reciprocity principle in electromagnetism is a particular instance of the multisymplectic form formula. We also define a Hamiltonian analogue of Jacobi's solution, and we show that this functional is a Type-II generating functional. We finish the paper by defining a similar framework of generating functions for discrete field theories, and we show that for the linear wave equation, we recover the multisymplectic conservation law of Bridges.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2013

Enantioselectivity of (321) chiral noble metal surfaces: A density functional theory study of lactate adsorption

The adsorption of the chiral molecule lactate on the intrinsically chiral noble metal surfaces Pt(321), Au(321), and Ag(321) is studied by density functional theory calculations. We use the oPBE-vdW functional which includes van der Waals forces on an ab initio level. It is shown that the molecule binds via its carboxyl and the hydroxyl oxygen atoms to the surface. The binding energy is larger on Pt(321) and Ag(321) than on Au(321). An analysis of the contributions to the binding energy of the different molecular functional groups reveals that the deprotonated carboxyl group contributes most to the binding energy, with a much smaller contribution of the hydroxyl group. The Pt(321) surface shows considerable enantioselectivity of 0.06 eV. On Au(321) and Ag(321) it is much smaller if not vanishing. The chiral selectivity of the Pt(321) surface can be explained by two factors. First, it derives from the difference in van der Waals attraction of L- and D-lactate to the surface that we trace to differences in the binding energy of the methyl group. Second, the multi-point binding pattern for lactate on the Pt(321) surface is sterically more sensitive to surface chirality and also leads to large binding energy contributions of the hydroxyl group. We also calculate the charge transfer to the molecule and the work function to gauge changes in electronic structure of the adsorbed molecule. The work function is lowered by 0.8 eV on Pt(321) with much smaller changes on Au(321) and Ag(321)

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2013

Mechanical, electronic, and transport properties of functionalized graphene monolayers from ab initio studies

We present exemplary results of extensive studies of mechanical, electronic and transport properties of covalent functionalization of graphene monolayers (GML). We report new results of ab initio studies for covalent functionalization of GML with −NH{sub 2} groups up to 12.5 % concentration. Our studies are performed in the framework of the density functional theory (DFT) and non-equilibrium Green’s function (NEGF). We discuss the stability (adsorption energy), elastic moduli, electronic structure, band gaps, and effective electron masses as a function of the density of the adsorbed molecules. We also show the conductance and current – voltage I(V) characteristics for these systems.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2014

Differentiable but exact formulation of density-functional theory

The universal density functional F of density-functional theory is a complicated and ill-behaved function of the density—in particular, F is not differentiable, making many formal manipulations more complicated. While F has been well characterized in terms of convex analysis as forming a conjugate pair (E, F) with the ground-state energy E via the Hohenberg–Kohn and Lieb variation principles, F is nondifferentiable and subdifferentiable only on a small (but dense) subset of its domain. In this article, we apply a tool from convex analysis, Moreau–Yosida regularization, to construct, for any ε > 0, pairs of conjugate functionals ({sup ε}E, {sup ε}F) that converge to (E, F) pointwise everywhere as ε → 0{sup +}, and such that {sup ε}F is (Fréchet) differentiable. For technical reasons, we limit our attention to molecular electronic systems in a finite but large box. It is noteworthy that no information is lost in the Moreau–Yosida regularization: the physical ground-state energy E(v) is exactly recoverable from the regularized ground-state energy {sup ε}E(v) in a simple way. All concepts and results pertaining to the original (E, F) pair have direct counterparts in results for ({sup ε}E, {sup ε}F). The Moreau–Yosida regularization therefore allows for an exact, differentiable formulation of density-functional theory. In particular, taking advantage of the differentiability of {sup ε}F, a rigorous formulation of Kohn–Sham theory is presented that does not suffer from the noninteracting representability problem in standard Kohn–Sham theory.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Ultra-nonlocality in density functional theory for photo-emission spectroscopy

We derive an exact expression for the photocurrent of photo-emission spectroscopy using time-dependent current density functional theory (TDCDFT). This expression is given as an integral over the Kohn-Sham spectral function renormalized by effective potentials that depend on the exchange-correlation kernel of current density functional theory. We analyze in detail the physical content of this expression by making a connection between the density-functional expression and the diagrammatic expansion of the photocurrent within many-body perturbation theory. We further demonstrate that the density functional expression does not provide us with information on the kinetic energy distribution of the photo-electrons. Such information can, in principle, be obtained from TDCDFT by exactly modeling the experiment in which the photocurrent is split into energy contributions by means of an external electromagnetic field outside the sample, as is done in standard detectors. We find, however, that this procedure produces very nonlocal correlations between the exchange-correlation fields in the sample and the detector.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2014

Nonequilibrium fluctuation-dissipation relations for one- and two-particle correlation functions in steady-state quantum transport

We study the non-equilibrium (NE) fluctuation-dissipation (FD) relations in the context of quantum thermoelectric transport through a two-terminal nanodevice in the steady-state. The FD relations for the one- and two-particle correlation functions are derived for a model of the central region consisting of a single electron level. Explicit expressions for the FD relations of the Green's functions (one-particle correlations) are provided. The FD relations for the current-current and charge-charge (two-particle) correlations are calculated numerically. We use self-consistent NE Green's functions calculations to treat the system in the absence and in the presence of interaction (electron-phonon) in the central region. We show that, for this model, there is no single universal FD theorem for the NE steady state. There are different FD relations for each different class of problems. We find that the FD relations for the one-particle correlation function are strongly dependent on both the NE conditions and the interactions, while the FD relations of the current-current correlation function are much less dependent on the interaction. The latter property suggests interesting applications for single-molecule and other nanoscale transport experiments.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2014

Fundamental gaps with approximate density functionals: The derivative discontinuity revealed from ensemble considerations

The fundamental gap is a central quantity in the electronic structure of matter. Unfortunately, the fundamental gap is not generally equal to the Kohn-Sham gap of density functional theory (DFT), even in principle. The two gaps differ precisely by the derivative discontinuity, namely, an abrupt change in slope of the exchange-correlation energy as a function of electron number, expected across an integer-electron point. Popular approximate functionals are thought to be devoid of a derivative discontinuity, strongly compromising their performance for prediction of spectroscopic properties. Here we show that, in fact, all exchange-correlation functionals possess a derivative discontinuity, which arises naturally from the application of ensemble considerations within DFT, without any empiricism. This derivative discontinuity can be expressed in closed form using only quantities obtained in the course of a standard DFT calculation of the neutral system. For small, finite systems, addition of this derivative discontinuity indeed results in a greatly improved prediction for the fundamental gap, even when based on the most simple approximate exchange-correlation density functional – the local density approximation (LDA). For solids, the same scheme is exact in principle, but when applied to LDA it results in a vanishing derivative discontinuity correction. This failure is shown to be directly related to the failure of LDA in predicting fundamental gaps from total energy differences in extended systems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2014

Density functional theory for carbon dioxide crystal

We present a density functional approach to describe the solid−liquid phase transition, interfacial and crystal structure, and properties of polyatomic CO{sub 2}. Unlike previous phase field crystal model or density functional theory, which are derived from the second order direct correlation function, the present density functional approach is based on the fundamental measure theory for hard-sphere repulsion in solid. More importantly, the contributions of enthalpic interactions due to the dispersive attractions and of entropic interactions arising from the molecular architecture are integrated in the density functional model. Using the theoretical model, the predicted liquid and solid densities of CO{sub 2} at equilibrium triple point are in good agreement with the experimental values. Based on the structure of crystal-liquid interfaces in different planes, the corresponding interfacial tensions are predicted. Their respective accuracies need to be tested.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

Functional differentiability in time-dependent quantum mechanics

In this work, we investigate the functional differentiability of the time-dependent many-body wave function and of derived quantities with respect to time-dependent potentials. For properly chosen Banach spaces of potentials and wave functions, Fréchet differentiability is proven. From this follows an estimate for the difference of two solutions to the time-dependent Schrödinger equation that evolve under the influence of different potentials. Such results can be applied directly to the one-particle density and to bounded operators, and present a rigorous formulation of non-equilibrium linear-response theory where the usual Lehmann representation of the linear-response kernel is not valid. Further, the Fréchet differentiability of the wave function provides a new route towards proving basic properties of time-dependent density-functional theory.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Too big, too small, or just right? A benchmark assessment of density functional theory for predicting the spatial extent of the electron density of small chemical systems

Multipole moments are the first-order responses of the energy to spatial derivatives of the electric field strength. The quality of density functional theory prediction of molecular multipole moments thus characterizes errors in modeling the electron density itself, as well as the performance in describing molecules interacting with external electric fields. However, only the lowest non-zero moment is translationally invariant, making the higher-order moments origin-dependent. Therefore, instead of using the 3 × 3 quadrupole moment matrix, we utilize the translationally invariant 3 × 3 matrix of second cumulants (or spatial variances) of the electron density as the quantity of interest (denoted by K). The principal components of K are the square of the spatial extent of the electron density along each axis. A benchmark dataset of the principal components of K for 100 small molecules at the coupled cluster singles and doubles with perturbative triples at the complete basis set limit is developed, resulting in 213 independent K components. The performance of 47 popular and recent density functionals is assessed against this Var213 dataset. Several functionals, especially double hybrids, and also SCAN and SCAN0 predict reliable second cumulants, although some modern, empirically parameterized functionals yield more disappointing performance. The H, Li, and Be atoms, in particular, are challenging for nearly all methods, indicating that future functional development could benefit from the inclusion of their density information in training or testing protocols.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

DMC-ICE13 : Ambient and high pressure polymorphs of ice from diffusion Monte Carlo and density functional theory

Ice is one of the most important and interesting molecular crystals, exhibiting a rich and evolving phase diagram. Recent discoveries mean that there are now 20 distinct polymorphs; a structural diversity that arises from a delicate interplay of hydrogen bonding and van der Waals dispersion forces. This wealth of structures provides a stern test of electronic structure theories, with Density Functional Theory (DFT) often not able to accurately characterize the relative energies of the various ice polymorphs. Thanks to recent advances that enable the accurate and efficient treatment of molecular crystals with Diffusion Monte Carlo (DMC), we present here the DMC-ICE13 dataset; a dataset of lattice energies of 13 ice polymorphs. This dataset encompasses the full structural complexity found in the ambient and high-pressure molecular ice polymorphs, and when experimental reference energies are available, our DMC results deliver sub-chemical accuracy. Using this dataset, we then perform an extensive benchmark of a broad range of DFT functionals. Of the functionals considered, revPBE-D3 and RSCAN reproduce reference absolute lattice energies with the smallest error, while optB86b-vdW and SCAN+rVV10 have the best performance on the relative lattice energies. Our results suggest that a single functional achieving reliable performance for all phases is still missing, and that care is needed in the selection of the most appropriate functional for the desired application. The insights obtained here may also be relevant to liquid water and other hydrogen-bonded and dispersion-bonded molecular crystals.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Importance of imposing gauge invariance in time-dependent density functional theory calculations with meta-generalized gradient approximations

It has been known for more than a decade that the gauge variance of the kinetic energy density τ leads to additional terms in the magnetic orbital rotation Hessian used in linear-response time-dependent density functional theory (TDDFT), affecting excitation energies obtained with τ-dependent exchange–correlation functionals. While previous investigations found that a correction scheme based on the paramagnetic current density has a small effect on benchmark results, we report more pronounced effects here, in particular, for the popular M06-2X functional and for some other meta-generalized gradient approximations (mGGAs). In the first part of this communication, this is shown by a reassessment of a set of five Ni(II) complexes for which a previous benchmark study that did not impose gauge invariance has found surprisingly large errors for excitation energies obtained with M06-2X. These errors are more than halved by restoring gauge invariance. The variable importance of imposing gauge invariance for different mGGA-based functionals can be rationalized by the derivative of the mGGA exchange energy integrand with respect to τ. In the second part, a large set of valence excitations in small main-group molecules is analyzed. For M06-2X, several selected n → π* and π→π$^{*}_{⊥}$ excitations are heavily gauge-dependent with average changes of –0.17 and –0.28 eV, respectively, while π→π$^{*}_{∥}$ excitations are marginally affected (–0.04 eV). Similar patterns, but of the opposite signs, are found for SCAN0. Here, the results suggest that reevaluation of previous gauge variant TDDFT results based on M06-2X and other mGGA functionals is warranted.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2013

Statistics of general functions of a Gaussian field-application to non-Gaussianity from preheating

We provide a general formula for calculating correlators of arbitrary function of a Gaussian field. This work extends the standard leading-order approximation based on the δN formalism to the case where truncation of the δN at some low order does not yield the correct answer. As an application of this formula, we investigate 2, 3 and 4-point functions of the primordial curvature perturbation generated in the massless preheating model by approximating the mapping between the curvature perturbation and the Gaussian field as a sum of the many spiky normal distribution functions as suggested by lattice calculations. We also discuss observational consequences of this case and show that trispectrum would be a key observable to search signature of preheating in the CMB map. It is found the forms of the curvature correlation functions for any δN, at the leading order in the correlator of the Gaussian field, coincide with the standard local type ones. Within this approximation, it is also found that the standard formula for the non-linearity parameters given by the product of the derivatives of the e-folding number still holds after we replace the bare e-folding number appearing in the original δN expansion with the one smoothed in the field space with a Gaussian window function.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Highly accurate and constrained density functional obtained with differentiable programming

Using an end-to-end differentiable implementation of the Kohn-Sham self-consistent field equations, we obtain a highly accurate neural network–based exchange and correlation (XC) functional of the electronic density. The functional is optimized using information on both energy and density while exact constraints are enforced through an appropriate neural network architecture. Here we evaluate our model against different families of XC approximations and show that at the meta-GGA level our functional exhibits unprecedented accuracy for both energy and density predictions. For nonempirical functionals, there is a strong linear correlation between energy and density errors. We use this correlation to define an XC functional quality metric that includes both energy and density errors, leading to an improved way to rank different approximations.

36 MATERIALS SCIENCE↗
DOE OSTI2018

Configurational forces in electronic structure calculations using Kohn-Sham density functional theory

Here we derive the expressions for configurational forces in Kohn-Sham density functional theory, which correspond to the generalized variational force computed as the derivative of the Kohn-Sham energy functional with respect to the position of a material point x. These configurational forces that result from the inner variations of the Kohn-Sham energy functional provide a unified framework to compute atomic forces as well as stress tensor for geometry optimization. Importantly, owing to the variational nature of the formulation, these configurational forces inherently account for the Pulay corrections. The formulation presented in this work treats both pseudopotential and all-electron calculations in a single framework, and employs a local variational real-space formulation of Kohn-Sham density functional theory (DFT) expressed in terms of the nonorthogonal wave functions that is amenable to reduced-order scaling techniques. We demonstrate the accuracy and performance of the proposed configurational force approach on benchmark all-electron and pseudopotential calculations conducted using higher-order finite-element discretization. To this end, we examine the rates of convergence of the finite-element discretization in the computed forces and stresses for various materials systems, and, further, verify the accuracy from finite differencing the energy. Wherever applicable, we also compare the forces and stresses with those obtained from Kohn-Sham DFT calculations employing plane-wave basis (pseudopotential calculations) and Gaussian basis (all-electron calculations). Finally, we verify the accuracy of the forces on large materials systems involving a metallic aluminum nanocluster containing 666 atoms and an alkane chain containing 902 atoms, where the Kohn-Sham electronic ground state is computed using a reduced-order scaling subspace projection technique

36 MATERIALS SCIENCE↗
DOE OSTI2017

Climate, soil and plant functional types as drivers of global fine-root trait variation

Ecosystem functioning relies heavily on below-ground processes, which are largely regulated by plant fine-roots and their functional traits. However, our knowledge of fine-root trait distribution relies to date on local- and regional-scale studies with limited numbers of species, growth forms and environmental variation. We compiled a world-wide fine-root trait dataset, featuring 1115 species from contrasting climatic areas, phylogeny and growth forms to test a series of hypotheses pertaining to the influence of plant functional types, soil and climate variables, and the degree of manipulation of plant growing conditions on species fine-root trait variation. Most particularly, we tested the competing hypotheses that fine-root traits typical of faster return on investment would be most strongly associated with conditions of limiting versus favourable soil resource availability. We accounted for both data source and species phylogenetic relatedness. We demonstrate that: (i) Climate conditions promoting soil fertility relate negatively to fine-root traits favouring fast soil resource acquisition, with a particularly strong positive effect of temperature on fine-root diameter and negative effect on specific root length (SRL), and a negative effect of rainfall on root nitrogen concentration; (ii) Soil bulk density strongly influences species fine-root morphology, by favouring thicker, denser fine-roots; (iii) Fine-roots from herbaceous species are on average finer and have higher SRL than those of woody species, and N 2 -fixing capacity positively relates to root nitrogen; and (iv) Plants growing in pots have higher SRL than those grown in the field. Synthesis. This study reveals both the large variation in fine-root traits encountered globally and the relevance of several key plant functional types and soil and climate variables for explaining a substantial part of this variation. Climate, particularly temperature, and plant functional types were the two strongest predictors of fine-root trait variation. High trait variation occurred at local scales, suggesting that wide-ranging below-ground resource economics strategies are viable within most climatic areas and soil conditions.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Soil enzymes as indicators of soil function: A step toward greater realism in microbial ecological modeling

Soil carbon (C) and nitrogen (N) cycles and their complex responses to environmental changes have received increasing attention. However, large uncertainties in model predictions remain, partially due to the lack of explicit representation and parameterization of microbial processes. One great challenge is to effectively integrate rich microbial functional traits into ecosystem modeling for better predictions. Here, using soil enzymes as indicators of soil function, we developed a competitive dynamic enzyme allocation scheme and detailed enzyme-mediated soil inorganic N processes in the Microbial-ENzyme Decomposition (MEND) model. We conducted a rigorous calibration and validation of MEND with diverse soil C-N fluxes, microbial C:N ratios, and functional gene abundances from a 12-year CO 2 × N grassland experiment (BioCON) in Minnesota, USA. In addition to accurately simulating soil CO 2 fluxes and multiple N variables, the model correctly predicted microbial C:N ratios and their negative response to enriched N supply. Model validation further showed that, compared to the changes in simulated enzyme concentrations and decomposition rates, the changes in simulated activities of eight C-N-associated enzymes were better explained by the measured gene abundances in responses to elevated atmospheric CO 2 concentration. In conclusion, our results demonstrated that using enzymes as indicators of soil function and validating model predictions with functional gene abundances in ecosystem modeling can provide a basis for testing hypotheses about microbially mediated biogeochemical processes in response to environmental changes. Further development and applications of the modeling framework presented here will enable microbial ecologists to address ecosystem-level questions beyond empirical observations, toward more predictive understanding, an ultimate goal of microbial ecology.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2018

Variation in hydroclimate sustains tropical forest biomass and promotes functional diversity

The fate of tropical forests under climate change is unclear as a result, in part, of the uncertainty in projected changes in precipitation and in the ability of vegetation models to capture the effects of drought-induced mortality on aboveground biomass (AGB). In this work, we evaluated the ability of a terrestrial biosphere model with demography and hydrodynamics (Ecosystem Demography, ED2-hydro) to simulate AGB and mortality of four tropical tree plant functional types (PFTs) that operate along light- and water-use axes. Model predictions were compared with observations of canopy trees at Barro Colorado Island (BCI), Panama. We then assessed the implications of eight hypothetical precipitation scenarios, including increased annual precipitation, reduced inter-annual variation, El Niño-related droughts and drier wet or dry seasons, on AGB and functional diversity of the model forest. When forced with observed meteorology, ED2-hydro predictions capture multiple BCI benchmarks. ED2-hydro predicts that AGB will be sustained under lower rainfall via shifts in the functional composition of the forest, except under the drier dry-season scenario. Furthermore these results support the hypothesis that inter-annual variation in mean and seasonal precipitation promotes the coexistence of functionally diverse PFTs because of the relative differences in mortality rates. If the hydroclimate becomes chronically drier or wetter, functional evenness related to drought tolerance may decline.

54 ENVIRONMENTAL SCIENCES↗