A unified theory of free energy functionals and applications to diffusion.

SignificanceThe free energy functional is a central component of continuum dynamical models used to describe phase transitions, microstructural evolution, and pattern formation. However, despite the success of these models in many areas of physics, chemistry, and biology, the standard free energy frameworks are frequently characterized by physically opaque parameters and incorporate assumptions that are difficult to assess. Here, we introduce a mathematical formalism that provides a unifying umbrella for constructing free energy functionals. We show that Ginzburg-Landau framework is a special case of this umbrella and derive a generalization of the widely employed Cahn-Hilliard equation. More broadly, we expect the framework will also be useful for generalizing higher-order theories, establishing formal connections to microscopic physics, and coarse graining.

Computed versus experimental energy barriers in solution: Influence of the type of the density functional approximation.

Mechanistic investigations at the density functional theory level of organic and organometallic reactions in solution are now broadly accessible and routinely implemented to complement experimental investigations. The selection of an appropriate functional among the plethora of developed ones is the first challenge on the way to reliable energy barrier calculations. To provide guidelines for the choice of an initial and reliable computational level, the performances of commonly used non-empirical (PBE, PBE0, PBE0-DH) and empirical density functionals (BLYP, B3LYP, B2PLYP) were evaluated relative to experimental activation enthalpies. Most reactivity databases to assess density functional performances are primarily based on high level calculations, here a set of experimental activation enthalpies of organic and organometallic reactions performed in solution were selected from the literature. As a general trend, the non-empirical functionals outperform the empirical ones. The most accurate energy barriers are obtained with hybrid PBE0 and double-hybrid PBE0-DH density functionals, both providing similar performance. Regardless of the functional under consideration, the addition of the GD3-BJ empirical dispersion correction does not enhance the accuracy of computed energy barriers.

Application to nonlinear optical properties of the RSX-QIDH double-hybrid range-separated functional.

The effective calculation of static nonlinear optical properties requires a considerably high accuracy at a reasonable computational cost, to tackle challenging organic and inorganic systems acting as precursors and/or active layers of materials in (nano-)devices. That trade-off implies to obtain very accurate electronic energies in the presence of externally applied electric fields to consequently obtain static polarizabilities ( α i j ) and hyper-polarizabilities ( ß i j k and γ i j k l ). Density functional theory is known to provide an excellent compromise between accuracy and computational cost, which is however largely impeded for these properties without introducing range-separation techniques. We thus explore here the ability of a modern (double-hybrid and range-separated) Range-Separated eXchange Quadratic Integrand Double-Hybrid exchange-correlation functional to compete in accuracy with more costly and/or tuned methods, thanks to its robust and parameter-free nature.

The Predictive Power of Exact Constraints and Appropriate Norms in Density Functional Theory.

Ground-state Kohn-Sham density functional theory provides, in principle, the exact ground-state energy and electronic spin densities of real interacting electrons in a static external potential. In practice, the exact density functional for the exchange-correlation (xc) energy must be approximated in a computationally efficient way. About 20 mathematical properties of the exact xc functional are known. In this work, we review and discuss these known constraints on the xc energy and hole. By analyzing a sequence of increasingly sophisticated density functional approximations (DFAs), we argue that (a) the satisfaction of more exact constraints and appropriate norms makes a functional more predictive over the immense space of many-electron systems and (b) fitting to bonded systems yields an interpolative DFA that may not extrapolate well to systems unlike those in the fitting set. We discuss both how the class of well-described systems has grown along with constraint satisfaction and the possibilities for future functional development.

Study of sterically crowded alkanes: assessment of non-empirical density functionals including double-hybrid cost-effective methods.

We theoretically study the homolytic dissociation reactions of sterically crowded alkanes of increasing size, carrying three different (bulky) substituents such as tert-butyl, adamantane, and [1.1.1]propellanyl, employing a family of parameter-free functionals ranging from semi-local, to hybrid and double-hybrid models. The study is complemented with the interaction between a pair of HC(CH3)3 molecules at repulsive and attractive regions, as an example of a system composed by a pair of weakly bound sterically crowded alkanes. We also assessed the effect of incorporating reliable dispersion corrections (i.e., D4 or NL) for all the functionals assessed, as well as the use of a tailored basis set (DH-SVPD) for non-covalent interactions, which provides the best trade-off between accuracy and computational cost for a seemingly extended applications to branched or crowded systems. Overall, the PBE-QIDH/DH-SVPD and r2SCAN-QIDH/DH-SVPD methods represent an excellent compromise providing relatively low, and thus very competitive, errors at a fraction of the cost of other quantum-chemical methods in use.

Making ab initio QED functional(s): Nonperturbative and photon-free effective frameworks for strong light-matter coupling.

Strong light-matter coupling provides a promising path for the control of quantum matter where the latter is routinely described from first principles. However, combining the quantized nature of light with this ab initio tool set is challenging and merely developing as the coupled light-matter Hilbert space is conceptually different and computational cost quickly becomes overwhelming. In this work, we provide a nonperturbative photon-free formulation of quantum electrodynamics (QED) in the long-wavelength limit, which is formulated solely on the matter Hilbert space and can serve as an accurate starting point for such ab initio methods. The present formulation is an extension of quantum mechanics that recovers the exact results of QED for the zero- and infinite-coupling limit and the infinite-frequency as well as the homogeneous limit, and we can constructively increase its accuracy. We show how this formulation can be used to devise approximations for quantum-electrodynamical density-functional theory (QEDFT), which in turn also allows us to extend the ansatz to the full minimal-coupling problem and to nonadiabatic situations. Finally, we provide a simple local density-type functional that takes the strong coupling to the transverse photon degrees of freedom into account and includes the correct frequency and polarization dependence. This QEDFT functional accounts for the quantized nature of light while remaining computationally simple enough to allow its application to a large range of systems. All approximations allow the seamless application to periodic systems.

Assumption-lean falsification tests of rate double-robustness of double-machine-learning estimators.

The class of doubly robust (DR) functionals studied by Rotnitzky et al. (2021) is of central importance in economics and biostatistics. It strictly includes both (i) the class of mean-square continuous functionals that can be written as an expectation of an affine functional of a conditional expectation studied by Chernozhukov et al. (2022b) and the class of functionals studied by Robins et al. (2008). The present state-of-the-art estimators for DR functionals ψ are double-machine-learning (DML) estimators (Chernozhukov et al., 2018). A DML estimator ψ^1 of ψ depends on estimates p^(x) and b^x of a pair of nuisance functions p(x) and bx, and is said to satisfy "rate double-robustness" if the Cauchy-Schwarz upper bound of its bias is o(n-1/2). Were it achievable, our scientific goal would have been to construct valid, assumption-lean (i.e. no complexity-reducing assumptions on b or p) tests of the validity of a nominal (1-α) Wald confidence interval (CI) centered at ψ^1. But this would require a test of the bias to be o(n-1/2), which can be shown not to exist. We therefore adopt the less ambitious goal of falsifying, when possible, an analyst's justification for her claim that the reported (1-α) Wald CI is valid. In many instances, an analyst justifies her claim by imposing complexity-reducing assumptions on b and p to ensure "rate double-robustness". Here we exhibit valid, assumption-lean tests of H0: "rate double-robustness holds", with non-trivial power against certain alternatives. If H0 is rejected, we will have falsified her justification. However, no assumption-lean test of H0, including ours, can be a consistent test. Thus, the failure of our test to reject is not meaningful evidence in favor of H0.

Toward a correct treatment of core properties with local hybrid functionals.

In local hybrid functionals (LHs), a local mixing function (LMF) determines the position-dependent exact-exchange admixture. We report new LHs that focus on an improvement of the LMF in the core region while retaining or partly improving upon the high accuracy in the valence region exhibited by the LH20t functional. The suggested new pt-LMFs are based on a Padé form and modify the previously used ratio between von Weizsäcker and Kohn-Sham local kinetic energies by different powers of the density to enable flexibly improved approximations to the correct high-density and iso-orbital limits relevant for the innermost core region. Using TDDFT calculations for a set of K-shell core excitations of second- and third-period systems including accurate state-of-the-art relativistic orbital corrections, the core part of the LMF is optimized, while the valence part is optimized as previously reported for test sets of atomization energies and reaction barriers (Haasler et al., J Chem Theory Comput 2020, 16, 5645). The LHs are completed by a calibration function that minimizes spurious nondynamical correlation effects caused by the gauge ambiguities of exchange-energy densities, as well as by B95c meta-GGA correlation. The resulting LH23pt functional relates to the previous LH20t functional but specifically improves upon the core region.

Total and orbital density-based analyses of molecules revealing long-range interaction regions.

Total and orbital electron densities of molecules are explored for the effect of the long-range correction (LC) for density functional theory (DFT) exchange functionals by comparing to the effect of the ab initio coupled cluster singles and doubles (CCSD) method. Calculating the LC effect on the total electron densities shows that the LC stabilizes the electrons around the long-range interaction regions of kinetic energy density, which are assumed to be electrons other than free electrons and self-interacting electrons, while the CCSD method stabilizes the electrons in the long-range interaction regions in the vertical molecular planes. As a more precise test, the LC effect on orbital densities are compared to the CCSD effect on Dyson orbital densities. Surprisingly, these effects are similar for the unoccupied orbitals, indicating that the LC covers the effects required to reproduce the CCSD Dyson unoccupied orbitals. For exploring the discrepancies between these effects on the occupied orbitals, the photoionization cross sections are calculated as a direct test for the shapes of the HOMOs to investigate the differences between these effects on the occupied orbitals. Consequently, the LC clearly produces the canonical HOMOs close to the CCSD Dyson and experimental ones, except for the HOMO of benzene molecule that mixes with the HOMO - 1 for the CCSD Dyson orbitals. This indicates that the orbital analyses using the photoionization cross sections are available as a direct test for the quality of DFT functionals.

Hybrid Functional and Plane Waves based Ab Initio Molecular Dynamics Study of the Aqueous Fe2+ /Fe3+ Redox Reaction.

Kohn-Sham density functional theory and plane wave basis set based ab initio molecular dynamics (AIMD) simulation is a powerful tool for studying complex reactions in solutions, such as electron transfer (ET) reactions involving Fe2+ /Fe3+ ions in water. In most cases, such simulations are performed using density functionals at the level of Generalized Gradient Approximation (GGA). The challenge in modelling ET reactions is the poor quality of GGA functionals in predicting properties of such open-shell systems due to the inevitable self-interaction error (SIE). While hybrid functionals can minimize SIE, standard plane-wave based AIMD at that level of theory is typically 150 times slower than GGA for systems containing ∼100 atoms. Among several approaches reported to speed-up AIMD simulations with hybrid functionals, the noise-stabilized MD (NSMD) procedure, together with the use of localized orbitals to compute the required exchange integrals, is an attractive option. In this work, we demonstrate the application of the NSMD approach for studying the Fe2+ /Fe3+ redox reaction in water. It is shown here that long AIMD trajectories at the level of hybrid density functionals can be obtained using this approach. Redox properties of the aqueous Fe2+ /Fe3+ system computed from these simulations are compared with the available experimental data for validation.

Combined measures of mimetic fidelity explain imperfect mimicry in a brood parasite-host system.

The persistence of imperfect mimicry in nature presents a challenge to mimicry theory. Some hypotheses for the existence of imperfect mimicry make differing predictions depending on how mimetic fidelity is measured. Here, we measure mimetic fidelity in a brood parasite-host system using both trait-based and response-based measures of mimetic fidelity. Cuckoo finches Anomalospiza imberbis lay imperfectly mimetic eggs that lack the fine scribbling characteristic of eggs of the tawny-flanked prinia Prinia subflava, a common host species. A trait-based discriminant analysis based on Minkowski functionals-that use geometric and topological morphometric methods related to egg pattern shape and coverage-reflects this consistent difference between host and parasite eggs. These methods could be applied to quantify other phenotypes including stripes and waved patterns. Furthermore, by painting scribbles onto cuckoo finch eggs and testing their rate of rejection compared to control eggs (i.e. a response-based approach to quantify mimetic fidelity), we show that prinias do not discriminate between eggs based on the absence of scribbles. Overall, our results support relaxed selection on cuckoo finches to mimic scribbles, since prinias do not respond differently to eggs with and without scribbles, despite the existence of this consistent trait difference.

A stochastic parabolic model of MEMS driven by fractional Brownian motion.

In this paper, we study a stochastic parabolic problem that emerges in the modeling and control of an electrically actuated MEMS (micro-electro-mechanical system) device. The dynamics under consideration are driven by an one dimensional fractional Brownian motion with Hurst index [Formula: see text]. We derive conditions under which the resulting SPDE has a global in time solution, and we provide analytic estimates for certain statistics of interest, such as quenching times and the corresponding quenching probabilities. Our results demonstrate the non-trivial impact of the fractional noise on the dynamics of the system. Given the significance of MEMS devices in biomedical applications, such as drug delivery and diagnostics, our results provide valuable insights into the reliability of these devices in the presence of positively correlated noise.

Computational NMR of carbohydrates: 1. Glucopyranoses.

A high-level calculation of 1 H and 13 C NMR chemical shifts of α- and ß-d-glucopyranoses is carried out at the DFT level with taking into account their conformational composition to reveal the most effective computational protocols. A number of dedicated DFT functionals in combination with Jensen's pcS-n (n = 0-4) family of basis sets were applied to evaluate the most reliable combination. It was found that BHandHLYP/pcS-2 provided the most accurate and reliable computational protocol. Based on the performed calculations, the established computational protocol is generally recommended for the calculation of 1 H and 13 C NMR chemical shifts of a wide series of carbohydrates.

Recent advances in the liquid-phase 6,7 Li nuclear magnetic resonance.

The present review is focused on experimental methods and structural applications, including computational aspects, of classical lithium liquid-phase nuclear magnetic resonance (NMR). It consists of four parts covering accordingly a brief overview, early experimental reports (papers of up to about 2015) and more recent (papers appearing in the interim of about 2015 until 2022) results, together with very few but highly prospective computational results.

Computational 19 F NMR of trifluoromethyl derivatives of alkenes, pyrimidines, and indenes.

The 19 F NMR chemical shifts of 13 trifluoromethyl derivatives of alkenes, pyrimidines, and indenes were calculated at the DFT level using the BhandHLYP, BHandH, PBE, PBE0, O3LYP, B3LYP, KT2, and KT3 functionals in combination with the pcS-2 basis set. Best result was documented for the BHandHLYP functional: The mean absolute error (MAE) of 0.66 ppm for the scaled values was achieved for the range of about 20 ppm. Solvent, vibrational, and relativistic corrections were found to be rather small, especially when taken in combination, generally demonstrating a slight decrease in the difference between calculated and experimental fluorine chemical shifts. As a measure of the practical importance of these compounds, one should recall that the growing number of life science products that contain trifluoromethyl groups provides a continuing driving force for the development of an effective methodology that enables both regio- and stereoselective introduction of trifluoromethyl groups into both aliphatic and aromatic systems.

Comparison of the Performance of Density Functional Methods for the Description of Spin States and Binding Energies of Porphyrins.

This work analyzes the performance of 250 electronic structure theory methods (including 240 density functional approximations) for the description of spin states and the binding properties of iron, manganese, and cobalt porphyrins. The assessment employs the Por21 database of high-level computational data (CASPT2 reference energies taken from the literature). Results show that current approximations fail to achieve the "chemical accuracy" target of 1.0 kcal/mol by a long margin. The best-performing methods achieve a mean unsigned error (MUE) <15.0 kcal/mol, but the errors are at least twice as large for most methods. Semilocal functionals and global hybrid functionals with a low percentage of exact exchange are found to be the least problematic for spin states and binding energies, in agreement with the general knowledge in transition metal computational chemistry. Approximations with high percentages of exact exchange (including range-separated and double-hybrid functionals) can lead to catastrophic failures. More modern approximations usually perform better than older functionals. An accurate statistical analysis of the results also casts doubts on some of the reference energies calculated using multireference methods. Suggestions and general guidelines for users are provided in the conclusions. These results hopefully stimulate advances for both the wave function and the density functional side of electronic structure calculations.

Global Existence and Fixed-Time Synchronization of a Hyperchaotic Financial System Governed by Semi-Linear Parabolic Partial Differential Equations Equipped with the Homogeneous Neumann Boundary Condition.

Chaotic nonlinear dynamical systems, in which the generated time series exhibit high entropy values, have been extensively used and play essential roles in tracking accurately the complex fluctuations of the real-world financial markets. We are concerned with a system of semi-linear parabolic partial differential equations supplemented by the homogeneous Neumann boundary condition, which governs a financial system comprising the labor force, the stock, the money, and the production sub-blocks distributed in a certain line segment or planar region. The system derived by removing the terms involved with partial derivatives with respect to space variables from our concerned system was demonstrated to be hyperchaotic. We firstly prove, via Galerkin's method and establishing a priori inequalities, that the initial-boundary value problem for the concerned partial differential equations is globally well posed in Hadamard's sense. Secondly, we design controls for the response system to our concerned financial system, prove under some additional conditions that our concerned system and its controlled response system achieve drive-response fixed-time synchronization, and provide an estimate on the settling time. Several modified energy functionals (i.e., Lyapunov functionals) are constructed to demonstrate the global well-posedness and the fixed-time synchronizability. Finally, we perform several numerical simulations to validate our synchronization theoretical results.

Almost Surely Exponential Convergence Analysis of Time Delayed Uncertain Cellular Neural Networks Driven by Liu Process via Lyapunov-Krasovskii Functional Approach.

As with probability theory, uncertainty theory has been developed, in recent years, to portray indeterminacy phenomena in various application scenarios. We are concerned, in this paper, with the convergence property of state trajectories to equilibrium states (or fixed points) of time delayed uncertain cellular neural networks driven by the Liu process. By applying the classical Banach's fixed-point theorem, we prove, under certain conditions, that the delayed uncertain cellular neural networks, concerned in this paper, have unique equilibrium states (or fixed points). By carefully designing a certain Lyapunov-Krasovskii functional, we provide a convergence criterion, for state trajectories of our concerned uncertain cellular neural networks, based on our developed Lyapunov-Krasovskii functional. We demonstrate under our proposed convergence criterion that the existing equilibrium states (or fixed points) are exponentially stable almost surely, or equivalently that state trajectories converge exponentially to equilibrium states (or fixed points) almost surely. We also provide an example to illustrate graphically and numerically that our theoretical results are all valid. There seem to be rare results concerning the stability of equilibrium states (or fixed points) of neural networks driven by uncertain processes, and our study in this paper would provide some new research clues in this direction. The conservatism of the main criterion obtained in this paper is reduced by introducing quite general positive definite matrices in our designed Lyapunov-Krasovskii functional.

Improving the scaling and performance of multiple time stepping-based molecular dynamics with hybrid density functionals.

Density functionals at the level of the generalized gradient approximation (GGA) and a plane-wave basis set are widely used today to perform ab initio molecular dynamics (AIMD) simulations. Going up in the ladder of accuracy of density functionals from GGA (second rung) to hybrid density functionals (fourth rung) is much desired pertaining to the accuracy of the latter in describing structure, dynamics, and energetics of molecular and condensed matter systems. On the other hand, hybrid density functional based AIMD simulations are about two orders of magnitude slower than GGA based AIMD for systems containing ~100 atoms using ~100 compute cores. Two methods, namely MTACE and s-MTACE, based on a multiple time step integrator and adaptively compressed exchange operator formalism are able to provide a speed-up of about 7-9 in performing hybrid density functional based AIMD. In this work, we report an implementation of these methods using a task-group based parallelization within the CPMD program package, with the intention to take advantage of the large number of compute cores available on modern high-performance computing platforms. We present here the boost in performance achieved through this algorithm. This work also identifies the computational bottleneck in the s-MTACE method and proposes a way to overcome it.

Benchmarking of Density Functionals for the Description of Optical Properties of Newly Synthesized π-Conjugated TADF Blue Emitters.

Computational modeling of the optical characteristics of organic molecules with potential for thermally activated delayed fluorescence (TADF) may assist markedly the development of more efficient emitting materials for organic light-emitting diodes. Recent theoretical studies in this area employ mostly methods from density functional theory (DFT). In order to obtain accurate predictions within this approach, the choice of a proper functional is crucial. In the current study, we focus on testing the performance of a set of DFT functionals for estimation of the excitation and emission energy and the excited singlet-triplet energy gap of three newly synthesized compounds with capacity for TADF. The emitters are designed specifically to enable charge transfer by π-electron conjugation, at the same time possessing high-energy excited triplet states. The functionals chosen for testing are from various groups ranging from gradient-corrected through global hybrids to range-separated ones. The results show that the monitored optical properties are especially sensitive to how the long-range part of the exchange energy is treated within the functional. The accurate functional should also be able to provide well balanced distribution of the π-electrons among the molecular fragments. Global hybrids with moderate (less than 0.4) share of exact exchange (B3LYP, PBE0) and the meta-GGA HSE06 are outlined as the best performing methods for the systems under study. They can predict all important optical parameters correctly, both qualitatively and quantitatively.