Some experimental observations on the stored energy function for small values of strain invariants.
Evaluation of stored-energy function for elastomeric materials based on biaxial experiments
ENERGY STORAGE↗
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Comparison of the canonical transformation and energy functional formalisms for ab initio calculations of self-localized polarons
In materials with strong electron-phonon (e-ph) interactions, charge carriers can distort the surrounding lattice and become trapped, forming self-localized (small) polarons. We recently developed an ab initio approach based on canonical transformations to efficiently compute the formation and energetics of small polarons A different approach based on a Landau-Pekar energy functional has been proposed in the recent literature. Here in this work, we analyze and compare these two methods in detail. We show that the small polaron energy is identical in the two formalisms when using the same polaron wave function. We also show that our canonical transformation formalism can predict polaron band structures and can properly treat zero- and finite-temperature lattice vibration effects, although at present using a fixed polaron wave function. Conversely, the energy functional approach can compute the polaron wave function, but as we show here, it neglects lattice vibrations and cannot address polaron self-localization and thermal band narrowing. Taken together, this work relates two different methods developed recently to study polarons from first-principles, highlighting their merits and shortcomings and discussing them both in a unified formalism.
36 MATERIALS SCIENCE↗
The strain-energy function of a hyperelastic material in terms of the extension ratios.
Strain-energy function of hyperelastic material in terms of extension ratios, with natural rubber as example
MATERIALS SCIENCE↗
Accurate parameterization of the kinetic energy functional
The absence of a reliable formulation of kinetic energy density functional has hindered the development of orbital free density functional theory. Using the data-aided learning paradigm, we propose a simple prescription to accurately model the kinetic energy density of any system. Our method relies on a dictionary of functional forms for local and nonlocal contributions which have been proposed in the literature and the appropriate coefficients are calculated via a linear regression framework. To model the nonlocal contributions, we explore two new nonlocal functionals - a functional that captures fluctuations in electronic density and a functional that incorporates gradient information. Since, the analytical functional forms of the kernels present in these nonlocal terms are not known from theory, we propose a basis function expansion to model these seemingly difficult nonlocal quantities. This allows us to easily reconstruct kernels for any system using only a few structures. The proposed method is able to learn kinetic energy densities and total kinetic energies of molecular and periodic systems, such as H 2 , LiH, LiF and a one-dimensional chain of 8 hydrogens using data from Kohn-Sham density functional theory calculations for only a few structures. For the ease of reproduction, codes used to generate the models are provided in the supporting materials.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Investigation of the Hylleraas method for determining the potential-energy function from the phase shift
Quantum mechanical inverse problem of elastic scattering theory, discussing Hylleraas solution for determining potential energy function from phase shift
Bernstein, R. B.↗
Energy functions for regularization algorithms
Regularization techniques are widely used for inverse problem solving in computer vision such as surface reconstruction, edge detection, or optical flow estimation. Energy functions used for regularization algorithms measure how smooth a curve or surface is, and to render acceptable solutions these energies must verify certain properties such as invariance with Euclidean transformations or invariance with parameterization. The notion of smoothness energy is extended here to the notion of a differential stabilizer, and it is shown that to void the systematic underestimation of undercurvature for planar curve fitting, it is necessary that circles be the curves of maximum smoothness. A set of stabilizers is proposed that meet this condition as well as invariance with rotation and parameterization.
Delingette, H.↗
Improved unconstrained energy functional method for eigensolvers in electronic structure calculations
This paper reports on the performance of a preconditioned conjugate gradient based iterative eigensolver using an unconstrained energy functional minimization scheme. In contrast to standard implementations, this scheme avoids an explicit reorthogonalization of the trial eigenvectors and becomes an attractive alternative for the solution of very large problems. The unconstrained formulation is implemented in the first-principles materials and chemistry CP2K code, which performs electronic structure calculations based on a density functional theory approximation to the solution of the many-body Schrödinger equation. We study the convergence of the unconstrained formulation, as well as its parallel scaling, on a Cray XC40 at the National Energy Research Scientific Computing Center (NERSC). The systems we use in our studies are bulk liquid water, a supramolecular catalyst gold(III)-complex, a bilayer of MoS2-WSe2 and a divacancy point defect in silicon, with the number of atoms ranging from 2,247 to 12,288. We show that the unconstrained formulation with an appropriate preconditioner has good convergence properties and scales well to 230k cores, roughly 38% of the full machine.
Del Ben, M↗
Stored energy function and compressibility of compressible rubberlike materials under large strain
By using new invariants in the theory of finite elasticity an expression is obtained for the stored energy function of slightly compressible materials in which the effects of the distortional change (change of shape) and of the volume change are clearly separated. The volume-related terms are expressed as a function of the third invariant, the classical compressibility, and an induced anisotropy of the effective compressibility which is due to the large deformations. After evaluating the terms, using data on pressure, volume, uniaxial strain, and fractional volume change vs strain data on natural rubber from the literature, it is shown that the volume change contribution to the total stress observed in a simple tensile experiment can be clearly separated from the distortional contribution, even at finite strains.
Peng, S. T. J.↗
Dissociation energies and potential energy functions for the ground X {sup 1}Σ{sup +} and “avoided-crossing” A {sup 1}Σ{sup +} states of NaH
A direct-potential-fit analysis of all accessible data for the A {sup 1}Σ{sup +} − X {sup 1}Σ{sup +} system of NaH and NaD is used to determine analytic potential energy functions incorporating the correct theoretically predicted long-range behaviour. These potentials represent all of the data (on average) within the experimental uncertainties and yield an improved estimate for the ground-state NaH well depth of D{sub e} = 15797.4 (±4.3) cm{sup −1}, which is ∼20 cm{sup −1} smaller than the best previous estimate. The present analysis also yields the first empirical determination of centrifugal (non-adiabatic) and potential-energy (adiabatic) Born-Oppenheimer breakdown correction functions for this system, with the latter showing that the A-state electronic isotope shift is −1.1(±0.6) cm{sup −1} going from NaH to NaD.
74 ATOMIC AND MOLECULAR PHYSICS↗
Free energy functionals for polarization fluctuations: Pekar factor revisited
The separation of slow nuclear and fast electronic polarization in problems related to electron mobility in polarizable media was considered by Pekar 70 years ago. Within dielectric continuum models, this separation leads to the Pekar factor in the free energy of solvation by the nuclear degrees of freedom. The main qualitative prediction of Pekar’s perspective is a significant, by about a factor of two, drop of the nuclear solvation free energy compared to the total (electronic plus nuclear) free energy of solvation. The Pekar factor enters the solvent reorganization energy of electron transfer reactions and is a significant mechanistic parameter accounting for the solvent effect on electron transfer. Here, we study the separation of the fast and slow polarization modes in polar molecular liquids (polarizable dipolar liquids and polarizable water force fields) without relying on the continuum approximation. We derive the nonlocal free energy functional and use atomistic numerical simulations to obtain nonlocal, reciprocal space electronic and nuclear susceptibilities. A consistent transition to the continuum limit is introduced by extrapolating the results of finite-size numerical simulation to zero wavevector. The continuum nuclear susceptibility extracted from simulations is numerically close to the Pekar factor. However, we derive a new functionality involving the static and high-frequency dielectric constants. The main distinction of our approach from the traditional theories is found for the solvation free energy due to the nuclear polarization: the anticipated significant drop of its magnitude with increasing liquid polarizability does not occur. The reorganization energy of electron transfer is either nearly constant with increasing the solvent polarizability and the corresponding high-frequency dielectric constant (polarizable dipolar liquids) or actually noticeably increases (polarizable force fields of water).
14 SOLAR ENERGY↗
Universal scaling of potential energy functions describing intermolecular interactions. II. The halide-water and alkali metal-water interactions
The scaled versions of the newly introduced [S. S. Xantheas and J. C. Werhahn, J. Chem. Phys.141, 064117 (2014)] generalized forms of some popular potential energy functions (PEFs) describing intermolecular interactions – Mie, Lennard-Jones, Morse, and Buckingham exponential-6 – have been used to fit the ab initio relaxed approach paths and fixed approach paths for the halide-water, X - (H 2 O), X = F, Cl, Br, I, and alkali metal-water, M + (H 2 O), M = Li, Na, K, Rb, Cs, interactions. The generalized forms of those PEFs have an additional parameter with respect to the original forms and produce fits to the ab initio data that are between one and two orders of magnitude better in the χ 2 than the original PEFs. They were found to describe both the long-range, minimum and repulsive wall of the respective potential energy surfaces quite accurately. Overall the 4-parameter extended Morse (eM) and generalized Buckingham exponential-6 (gBe-6) potentials were found to best fit the ab initio data for these two classes of ion-water interactions. Finally, the fitted values of the parameter of the (eM) and (gBe-6) PEFs that control the repulsive wall of the potential correlate remarkably well with the ionic radii of the halide and alkali metal ions.
Werhahn, Jasper C.↗
A Bayesian analysis of nuclear deformation properties with Skyrme energy functionals
In spite of numerous scientific and practical applications, there is still no comprehensive theoretical description of the nuclear fission process based solely on protons, neutrons and their interactions. Furthermore, the most advanced simulations of fission are currently carried out within nuclear density functional theory (DFT). In spite of being fully quantum-mechanical and rooted in the theory of nuclear forces, DFT still depends on a dozen or so parameters characterizing the energy functional. Calibrating these parameters on experimental data results in uncertainties that must be quantified for applications. This task is very challenging because of the high computational cost of DFT calculations for fission. In this paper, we use Gaussian processes to build emulators of DFT models in order to quantify and propagate statistical uncertainties of theoretical predictions for a range of nuclear deformations relevant to describing the fission process.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Temperature-dependent vibrational spectra and structure of liquid water from classical and quantum simulations with the MB-pol potential energy function
The structure of liquid water as a function of temperature is explored through the modeling of infrared and Raman spectra along with structural order parameters calculated from classical and quantum molecular dynamics simulations with the MB-pol many-body potential energy function. The magnitude of nuclear quantum effects is also monitored by comparing the vibrational spectra obtained from classical and centroid molecular dynamics, both in intensities and peak positions. The observed changes in spectral activities are shown to reflect changes in the underlying structure of the hydrogen-bond network and are found to be particularly sensitive to many-body effects in the representation of the electrostatic interactions. As a whole, good consensus is found with the experimental spectra, which provides further evidence for the accuracy of MB-pol in predicting the properties of water.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Universal scaling of potential energy functions describing intermolecular interactions. I. Foundations and scalable forms of new generalized Mie, Lennard-Jones, Morse, and Buckingham exponential-6 potentials
Based on the formulation of the analytical expression of the potential V(r) describing intermolecular interactions in terms of the dimensionless variables r*=r/rm and !*=V/!, where rm is the separation at the minimum and ! the well depth, we propose more generalized scalable forms for the commonly used Lennard-Jones, Mie, Morse and Buckingham exponential-6 potential energy functions (PEFs). These new generalized forms have an additional parameter from and revert to the original ones for some choice of that parameter. In this respect, the original forms can be considered as special cases of the more general forms that are introduced. We also propose a scalable, but nonrevertible to the original one, 4-parameter extended Morse potential.
Xantheas, Sotiris S.↗
HIGH-RESOLUTION ROTATIONAL SPECTRUM, DUNHAM COEFFICIENTS, AND POTENTIAL ENERGY FUNCTION OF NaCl
We report laboratory spectroscopy for the first time of the J = 1–0 and J = 2–1 lines of Na{sup 35}Cl and Na{sup 37}Cl in several vibrational states. The hyperfine structure has been resolved in both transitions for all vibrational levels, which permit us to predict with high accuracy the hyperfine splitting of the rotational transitions of the two isotopologues at higher frequencies. The new data have been merged with all previous works at microwave, millimeter, and infrared wavelengths and fitted to a series of mass-independent Dunham parameters and to a potential energy function. The obtained parameters have been used to compute a new dipole moment function, from which the dipole moment for infrared transitions up to Δ v = 8 has been derived. Frequency and intensity predictions are provided for all rovibrational transitions up to J = 150 and v = 8, from which the ALMA data of evolved stars can be modeled and interpreted.
79 ASTRONOMY AND ASTROPHYSICS↗
Accuracy of first-principles interatomic interactions and predictions of ferroelectric phase transitions in perovskite oxides: Energy functional and effective Hamiltonian
While first-principles density functional theory (DFT)-based models have been effective in capturing the physics of ferroelectric phase transitions in BaTiO 3 , PbTiO 3 , and KNbO 3 , quantitative estimates of the transition temperatures (T C ) have errors that are believed to originate from the errors in estimating lattice constants obtained within the local density approximation (LDA) and generalized gradient approximation (GGA) of DFT. The recently developed strongly constrained and appropriately normed (SCAN) meta-GGA functional has been shown to be quite accurate in the estimation of lattice constants. Here, we present a quantitative analysis of the estimates of ferroelectric ground-state properties of eight perovskite oxides and transition temperatures of BaTiO 3 , PbTiO 3 , and KNbO 3 obtained with molecular dynamics simulations using an effective Hamiltonian derived from the SCAN meta-GGA-based DFT. Relative to LDA, we find an improvement in the estimates of T C , which arises from the changes in the calculated strain-phonon, anharmonic coupling constants, and strength of ferroelectric instabilities, i.e., frequencies of the soft modes. We also assess the errors in T C originating from approximately integrating out the high-energy phonons during construction of the model Hamiltonian through estimates of the effects of fourth-order couplings between the soft mode and higher-energy modes of BaTiO 3 , PbTiO 3 , and KNbO 3 . We find that inclusion of these anharmonic couplings results in deeper double-well energy functions of ferroelectric distortions and further improvement in the estimates of transition temperatures. Consistently improved estimates of lattice constants and transition temperatures with the SCAN meta-GGA calculations augur well for their use in simulations of superlattices or heterostructures of perovskite oxides, in which the effects of lattice matching are critical.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Efficient classical density-functional theories of rigid-molecular fluids and a simplified free energy functional for liquid water
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catalysis (heterogeneous)↗
Interfacial Properties and Design of Functional Energy Materials
The vital importance of energy to society continues to demand a relentless pursuit of energy responsive materials that can bridge fundamental chemical structures at the molecular level and achieve improved functionality, such as efficient energy conversion/storage/transmission, over multiple length scales. This demand can potentially be realized by harnessing the power of self-assembly a spontaneous process where molecules or much larger entities form ordered aggregates as a consequence of predominately non-covalent (weak) interactions. Self-assembly is the key to bottom-up design of molecular devices, because the nearly atomic-level control is very difficult to realize in a top-down, e.g., lithographic approach. However, while function (e.g., charge mobility) in simple systems such as single crystals can often be predicted, predicting the function of the great variety of self-assembled molecular architectures is complicated by the lack of understanding and control over nanoscale interactions, mesoscale architectures, and macroscale (long-range) order. To establish a foundation toward delivering practical solutions, it is critical to develop an understanding of the chemical and physical mechanisms responsible for the self-assembly of molecular and hybrid materials on various substrates. Typically molecular self-assembly involves poorly understood non-covalent intermolecular and substrate-molecule interactions compounded by local and/or collective influences from the substrate atomic lattice (symmetry and/or topological features) and electronic structure. Thus, progress towards unraveling the underlying physicochemical processes that control the structure and macroscopic physical, mechanical, electrical, and transport properties of materials increasingly requires tight integration of theory, modeling and simulation with precision synthesis, advanced experimental characterization, and device measurements. In this mode, theory and simulation can greatly accelerate the process of materials discovery by providing atomic level understanding of physicochemical phenomena and for making predictions of trends. In particular, this approach can provide understanding, prediction and exploration of new materials and conditions before they are realized in the lab, to illuminate connections between experimental observations, and help identify new materials for targeted synthesis. Toward this end, Density Functional Theory (DFT) can provide a suitable computational framework for investigating the inter- and intramolecular bonding, molecular conformation, charge and spin configurations that are intrinsic to self-assembly of molecules on substrates. This Account highlights recent advances in using an integrated approach based on DFT and scanning probe microscopy [STM(s), AFM] to study/develop electronic materials formed from the self-assembly of molecules into supramolecular or polymeric architectures on substrates. Here it is the interplay between molecular interactions and surface electrons that is used to control the final architecture and subsequent bulk properties of the two-dimensional patterns/assemblies. Indeed a rich variety of functional energy materials become possible.
Sumpter, Bobby G↗