Higher-order gradient corrections for exchange-correlation functionals

In the study of the ground‐state binding energy of atoms, relatively more attention has been paid to the gradient expansion of the kinetic energy term of the Thomas–Fermi–Dirac model than to the gradient expansion of the exchange term of this model. Recently Shih and Shih, Murphy, and Wang have focused on this problem and calculated the first gradient expansion correction to the Dirac exchange term by making use of electron densities constructed from Hartree–Fock wave functions. In previous work, aimed to introduce the shell structure of atoms via a variational procedure using the energy density functional formalism, electron densities have been obtained for the Na atom within the Thomas–Fermi–Dirac model with and without the Weizsäcker and Hodges gradient expansion corrections to the kinetic energy term. In this paper we make use of the respective electron densities and calculate the first gradient expansion corrections to the Dirac exchange term. The results show that, for the Na atom, the magnitude of this correction is about 1% of the magnitude of the total binding energy.

A gradient expansion for the total kinetic energy was derived. We imposed the condition that the densities should be a slowly varying function. It is applicable up to the fourth order in ▿n. The leading terms are the same as those of Thomas-Fermi and Weizsacker. The fourth-order term is in good agreement with the nonlinear response part of Hodges correction. A small term,Y[n], has been added to the derived gradient expansion. A new formula for the total kinetic energy with full nonlinear response is achieved as a result. The same method is applied to the exchange energy functional, where the well-known gradient expansion has been found but it differs only in the numerical coefficients. Divergent terms have been found in both expansions.

The topological derivative provides the sensitivity of a given shape functional with respect to an infinitesimal domain perturbation. Classically, this derivative comes from the second term of the topological asymptotic expansion, dealing only with infinitesimal perturbations. Therefore, as a natural extension of this concept, we can consider higher-order terms in the expansion. In particular, the next one we recognize as the second-order topological derivative, which allows us to deal with perturbations of finite sizes. This term depends explicitly on higher-order gradients of the solution associated to the non-perturbed problem and also implicitly through the solution of an auxiliary variational problem. In this article, we calculate the explicit as well as implicit terms of the second-order topological asymptotic expansion for the total potential energy associated to the Laplace equation in the two-dimensional domain. The domain perturbation is done by the insertion of a small inclusion with a thermal conductivity coefficent value different from the bulk material. Finally, we present some numerical experiments showing the influence of the second-order term in the topological asymptotic expansion for several values of the thermal conductivity coefficent of the inclusion.

The pertinence of four-order gradient corrections to the exchange-only energy functional ${\mathit{E}}_{\mathit{x}}$[n] for producing accurate exchange potentials ${\mathit{v}}_{\mathit{x}}$(r) is examined utilizing the exact ${\mathit{v}}_{\mathit{x}}$(r)'s obtained from the optimized-potential model (OPM) for spherical atoms and jellium spheres (whose inner regions have slowly varying densities for which the gradient expansion should be valid). It is found that the fourth-order contributions containing ${\mathrm{\ensuremath{\nabla}}}^{2}$n represent important ingredients of ${\mathit{E}}_{\mathit{x}}$[n] which should be included in gradient-based nonlocal extensions of the local-density approximation. In accordance with this observation the ${\mathit{v}}_{\mathit{x}}$(r)'s resulting from the fourth-order gradient expansion are closer to the exact ${\mathit{v}}_{\mathit{x}}^{\mathrm{OPM}}$(r)'s than those from generalized gradient approximations which do not contain ${\mathrm{\ensuremath{\nabla}}}^{2}$n contributions.

A systematic procedure for refining gradient corrections in Kohn–Sham exchange-correlation functionals is presented. The procedure is based on least-squares fitting to accurate thermochemical data. In this first application of the method, we use the G2 test set of Pople and co-workers to generate what we believe to be an optimum GGA/exact-exchange density-functional theory (i.e., generalized gradient approximation with mixing of exactly computed exchange).

Shea, Rees & Wood (1994) apply null models and an analytical argument to test our suggestion (Silvertown et al. 1993) that correlations among elasticities reflect evolutionary trade-offs between plant life history variables. We welcome such scrutiny and accept their analytical proof that trade-offs between matrix elements do not necessarily alter the elasticities of the elements and therefore trade-offs cannot be detected in this way. Clearly, our original suggestion was simplistic, but we mantain that the relationships between elasticities are worth further investigation. Shea et al. (1994) found complex, curvilinear relationships between elasticities derived from artificially constructed matrices. We argue that these patterns were dependent upon two assumptions of the null models; allowing X to vary and using an exponential function in the trade-off between reproduction & survival. While it is true that X varied between species in our original dataset, the ideal model designed to test the relationship between elasticities at equilibrium should constrain X= 1. In order to test whether Shea, Rees & Wood's results were a consequence of the exponential function in their model and of not constraining X to a fixed value, we repeated their exercise using the smallest figures of their Appendix I with further decreases in some of the elements to yield X = 1. We then simulated a trade-off between fecundity (f) and survival in the last category following their stated rules (a15 = 0.6f; a25 = 0.4f) and found that keeping X constant required the linear funciim a15 = 3.14-4.209a55 ad a25 = 1.711-2.293a.55 Similarly, elasticity analysis on matrices determined in this way showed that the relationships between composite elasticities E1-E6 were all linear, not curvilinear or 'complex' as in Shea etal.'s model. The reason for this linear behaviour is that, in a linear matrix model, a decrease in the probability of survival of reproductive individuals must be compensated by a proportional increase in their fecundity. This shows that the patterns Shea et al. found were model-dependent and that they were a consequence of their unrealistic assumption of an exponential trade-off function which changes X systematically from a low value when survival is low but fecundity is high, to a high value when the reverse is true. When we simulated trade-offs between elements in different columns of the projection matrix (e.g. early vs. late survival) complex patterns of the kind found by Shea et al. were obtained. This reflects the interdependence of matrix elements. The point we wish to draw from this is that, although in their simplest form elasticities cannot be interpreted in terms of trade-offs, disentangling direct from indirect effects may still allow the use of elasticities to draw conclusions on possible trade-offs. Indeed, P. van Tienderen (unpublished manuscript) has explicitly derived formulas to separate the direct from the indirect effects in what he terms 'integrated elasticities' and 'integrated sensitivities'. The integrated elasticity (and the integrated sensitivity) of a matrix element is calculated using a linear combination of the slope relating that particular element to each other element, weighted by their respective elasticities (or sensitivities). Determining the relationships among matrix elements involves measuring trade-offs. According to this view, sensitivities and elasticities still have a role to play in the interpretation of trade-offs among life cycle components.

Coupled tensorial forms of the second-order effective Hamiltonian for open-subshell atoms in jj-coupling

A numerically exact Monte Carlo scheme for calculation of open quantum system dynamics is proposed and implemented. The method consists of a Monte Carlo summation of a perturbation expansion in terms of trajectories in Liouville phase-space with respect to the coupling between the excited states of the molecule. The trajectories are weighted by a complex decoherence factor based on the second-order cumulant expansion of the environmental evolution. The method can be used with an arbitrary environment characterized by a general correlation function and arbitrary coupling strength. It is formally exact for harmonic environments, and it can be used with arbitrary temperature. Time evolution of an optically excited Frenkel exciton dimer representing a molecular exciton interacting with a charge transfer state is calculated by the proposed method. We calculate the evolution of the optical coherence elements of the density matrix and linear absorption spectrum, and compare them with the predictions of standard simulation methods.

Experimental determination of magnetic landau parameters in ErRh 4B 4

M1-matrix elements and F-spin symmetry in nuclei

Yield calculation for delayed gamma-ray neutron activation analysis

An improved microwave spectrograph has been developed which permits measurements of electric field intensity to one part in ten thousand. The instrument is employed to make precise Stark effect measurements. A method is also given for measuring the polarizability of polar molecules in a given rotational state. The transition $J=1\ensuremath{\rightarrow}2$, $\ensuremath{\Delta}M=0$ for the ground vibrational state of carbonyl sulfide (OCS) was studied and the electric dipole moment was determined to be (0.7124\ifmmode\pm\else\textpm\fi{}0.0002) Debye units. In addition the polarizability anisotropy (${\ensuremath{\alpha}}_{\mathrm{zz}}\ensuremath{-}{\ensuremath{\alpha}}_{\mathrm{xx}}$) was measured and found to be (2.4\ifmmode\pm\else\textpm\fi{}3.0)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}24}$ ${\mathrm{cm}}^{3}$.Effects caused by the fourth-order term in the perturbation expansion for the Stark Effect were observed and found to be in good agreement with theoretical predictions.

Machine learning (ML) of kinetic energy functionals (KEFs), in particular kinetic energy density (KED) functionals, is a promising way to construct KEFs for orbital-free density functional theory (DFT). Neural networks and kernel methods including Gaussian process regression (GPR) have been used to learn Kohn-Sham (KS) KED from density-based descriptors derived from KS DFT calculations. The descriptors are typically expressed as functions of different powers and derivatives of the electron density. This can generate large and extremely unevenly distributed datasets, which complicates effective application of ML techniques. Very uneven data distributions require many training datapoints, can cause overfitting, and can ultimately lower the quality of an ML KED model. We show that one can produce more accurate ML models from fewer data by working with smoothed density-dependent variables and KED. Smoothing palliates the issue of very uneven data distributions and associated difficulties of sampling while retaining enough spatial structure necessary for working within the paradigm of KEDF. We use GPR as a function of smoothed terms of the fourth order gradient expansion and KS effective potential and obtain accurate and stable (with respect to different random choices of training points) kinetic energy models for Al, Mg, and Si simultaneously from as few as 2000 samples (about 0.3% of the total KS DFT data). In particular, accuracies on the order of 1% in a measure of the quality of energy-volume dependence B'=EV0-ΔV-2EV0+E(V0+ΔV)ΔV/V02 (where V0 is the equilibrium volume and ΔV is a deviation from it) are obtained simultaneously for all three materials.

We present a machine-learned (ML) model of kinetic energy for orbital-free density functional theory (OF-DFT) suitable for bulk light weight metals and compounds made of group III–V elements. The functional is machine-learned with Gaussian process regression (GPR) from data computed with Kohn-Sham DFT with plane wave bases and local pseudopotentials. The dataset includes multiple phases of unary, binary, and ternary compounds containing Li, Al, Mg, Si, As, Ga, Sb, Na, Sn, P, and In. A total of 433 materials were used for training, and 18 strained structures were used for each material. Averaged (over the unit cell) kinetic energy density is fitted as a function of averaged terms of the 4th order gradient expansion and the product of the density and effective potential. The kinetic energy predicted by the model allows reproducing energy-volume curves around equilibrium geometry with good accuracy. We show that the GPR model beats linear and polynomial regressions. We also find that unary compounds sample a wider region of the descriptor space than binary and ternary compounds, and it is therefore important to include them in the training set; a GPR model trained on a small number of unary compounds is able to extrapolate relatively well to binary and ternary compounds but not vice versa.

An extension of the Helfand theory of equilibrium properties of interfaces between different polymers is presented. The corrections to the interface width due to higher-order terms in gradient expansion of the free energy of the system are shown to be negligible in comparison with «end» corrections due to the connectivity of blocks. It is also shown that the latter corrections that are specific for block copolymers are typically much larger than end corrections for a system of immiscible homopolymers. The effect of fluctuations of concentration profiles is also analyzed. It is shown that the effect results in an appreciable increase of the apparent interface width