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Artificial neural networks trained on 23 density functional approximations (DFAs) from multiple rungs of “Jacob's ladder” enable the prediction of where each DFA has zero curvature for chemical discovery.
Calculating charge transfer (CT) excitation energies with high accuracy and low computational cost is a challenging task. Kohn-Sham density functional theory (KS-DFT), due to its efficiency and accuracy, has achieved great success in describing ground state problems. To extend to excited state problems, our group recently demonstrated an approach with good numerical results to calculate low-lying and Rydberg excitation energies of an N-electron system from a ground state KS or generalized KS calculations of an (N-1)-electron system via its orbital energies. In the present work, we explore further the same methodology to describe CT excitations. Numerical results from this work show that performance of conventional density functional approximations (DFAs) is not as good for CT excitations as for other excitations due to the delocalization error. Applying localized orbital scaling correction (LOSC) to conventional DFAs, a recently developed method in our group to effectively reduce the delocalization error, can improve the results. Overall, the performance of this methodology is better than time dependent DFT (TDDFT) with conventional DFAs. In addition, it shows that results from LOSC-DFAs in this method reach similar accuracy to other methods, such as ΔSCF, G 0 W 0 with Bethe-Salpeter equations, particle-particle random phase approximation, and even high-level wavefunction methods like CC2. Our analysis shows that the correct 1/R trend for CT excitation can be captured from LOSC-DFA calculations, stressing that the application of DFAs with the minimal delocalization error is essential within this methodology. This work provides an efficient way to calculate CT excitation energies from ground state DFT.
Most widely used density functional approximations suffer from self-interaction error, which can be corrected using the Perdew–Zunger (PZ) self-interaction correction (SIC). We implement the recently proposed size-extensive formulation of PZ-SIC using Fermi–Löwdin Orbitals (FLOs) in real space, which is amenable to systematic convergence and large-scale parallelization. Here, we verify the new formulation within the generalized Slater scheme by computing atomization energies and ionization potentials of selected molecules and comparing to those obtained by existing FLOSIC implementations in Gaussian based codes. The results show good agreement between the two formulations, with new real-space results somewhat closer to experiment on average for the systems considered. We also obtain the ionization potentials and atomization energies by scaling down the Slater statistical average of SIC potentials. The results show that scaling down the average SIC potential improves both atomization energies and ionization potentials, bringing them closer to experiment. Finally, we verify the present formulation by calculating the barrier heights of chemical reactions in the BH6 dataset, where significant improvements are obtained relative to Gaussian based FLOSIC results.
Here, proper scaling for the mean transverse flow and Reynolds shear stress in a turbulent plane jet is determined using a scaling patch approach. By seeking an admissible scaling, a key concept in the scaling patch approach, for the mean continuity equation, a proper scale for the mean transverse flow in a turbulent plane jet is found as V ref = –δdU ctr /dx, where δ is the jet half width and dU ctr /dx is the decay rate of the mean axial velocity at the jet centerline. By seeking an admissible scaling for the mean axial momentum equation, a proper scale for the kinematic Reynolds shear stress is found as R uv,ref =U ctr V ref , which is a mix of the velocity scales in the axial and transverse directions. Approximation functions for the scaled mean transverse flow and Reynolds shear stress are developed and found to agree well with experimental and numerical data. Similarities and differences between the scales of the mean transverse flow and Reynolds shear stress in turbulent plane jets and zero-pressure-gradient turbulent boundary layer flows are clarified.
Proper scaling in turbulent planar plumes is investigated in this study using a scaling patch approach. Based on the scaled boundary conditions, a proper velocity scale for the mean axial flow is the plume centerline velocity U ref = U ctr , and a proper temperature scale for the temperature excess is Θ ref = T ctr – T ∞ , where T ctr is the plume centerline temperature and T ∞ is the ambient fluid temperature. By seeking an admissible scaling, a key concept in the scaling patch approach, for the mean continuity, mean momentum, and mean energy equations, respectively, the following is found: (1) a proper scale for the mean transverse flow is V ref = ( d δ / d x ) U ctr , where d δ / d x is the growth rate of the plume width. (2) A proper scale for the Reynolds shear stress is R vu , ref = U ctr V ref = ( d δ / d x ) U ctr 2 , a mix of the scales for the mean axial and transverse flows. (3) A proper scale for the turbulent heat flux is R v θ , ref = V ref Θ ctr , a mix of the scales for the mean transverse flow and mean temperature excess. The mean transverse flow thus plays a critical role in the scaling of turbulent planar plumes. Approximate functions are developed for the scaled mean transverse flow, Reynolds shear stress, and turbulent temperature flux, and are found to agree favorably with experimental and numerical simulation data. The integral analysis of the mean momentum equation yields a Richardson number Ri, which remains invariant in the axial direction. The Richardson number is defined as Ri = def g β Θ ctr δ t / ( U ctr V ref ) ≈ 1 / 2 , where g is the gravitational acceleration, β is the thermal expansion coefficient, and δ t is the plume half-width based on the mean temperature profile. This Richardson number arises directly from the scaling patch analysis of the mean momentum equation, including both the streamwise and transverse velocity scales.
Exact conditions have long been used to guide the construction of density functional approximations. However, hundreds of empirical-based approximations tailored for chemistry are in use, of which many neglect these conditions in their design. We analyze well-known conditions and revive several obscure ones. Two crucial distinctions are drawn: that between necessary and sufficient conditions and that between all electronic densities and the subset of realistic Coulombic ground states. Simple search algorithms find that many empirical approximations satisfy many exact conditions for realistic densities and non-empirical approximations satisfy even more conditions than those enforced in their construction. The role of exact conditions in developing approximations is revisited.
To accurately describe the energetics of transition metal systems, density functional approximations (DFAs) must provide a balanced description of s- and d- electrons. One measure of this is the sd transfer error, which has previously been defined as E ( 3 d n − 1 4 s 1 ) − E ( 3 d n − 2 4 s 2 ) . Theoretical concerns have been raised about this definition due to its evaluation of excited-state energies using ground-state DFAs. A more serious concern appears to be strong correlation in the 4s 2 configuration. Here, we define a ground-state measure of the sd energy imbalance, based on the errors of s- and d-electron second ionization energies of the 3d atoms, that effectively circumvents the aforementioned problems. We find an improved performance as we move from the local spin density approximation (LSDA) to the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) to the regularized and restored Strongly Constrained and Appropriately Normed (r 2 SCAN) meta-GGA for first-row transition metal atoms. However, we find large (∼2 eV) ground-state sd energy imbalances when applying a Perdew–Zunger 1981 self-interaction correction. This is attributed to an “energy penalty” associated with the noded 3d orbitals. A local scaling of the self-interaction correction to LSDA results in a balance of s- and d-errors.
In this work, an improved version of the CarMa code is presented, called CarMa-D, for the analysis of resistive wall modes (RWMs) in fusion devices, simultaneously considering the effects of volumetric three-dimensional conducting structures, plasma dynamics, toroidal rotation or drift-kinetic damping. Unlike static CarMa, CarMa-D does not rely on the simplifying assumptions such as neglecting the plasma mass, toroidal rotation and kinetic damping. The new coupling strategy is based on matrix-based Padé rational functions approximation of plasma a response. The arising mathematical model is formally equivalent to the original CarMa model, but with a higher number of degrees of freedom to model the dynamics of the plasma. CarMa-D overcomes the main limitations of the original CarMa, in particular: (i) the massless assumption for the plasma is removed, allowing modeling of global modes growing both on ideal kink time scales and in the typical RWM growth rate regime, with a suitable treatment of the model; (ii) the effects of toroidal plasma flow and drift kinetic damping can be included into the new model, providing a powerful tool to study macroscopic phenomena where both plasma dynamics and 3-D conducting structures play important roles.
We report a systematic and accurate approach for deriving the bulk free energy surface (FES), a function of temperature, polarization, and strain, from the first-principles density functional theory (DFT) of proper ferroelectrics. The core of our approach is the metadynamics algorithm that extracts the polarization dependence of the FES from all-atom molecular dynamics simulations without an a priori ansatz. The rest of the FES is derived from the metadynamics trajectories that span the relevant phase space. We demonstrate our approach in the case of lead titanate. The errors across the phase transition, due to DFT numerics, all-atom molecular dynamics, and free energy evaluation by enhanced sampling, can be systematically controlled and are of the order of 1 meV/atom. The accuracy of the resulting ab initio FES is only limited by the adopted functional approximation of DFT.
Since the discovery of heavy-fermion superconductivity in CeCu 2 Si 2 , the material has attracted great interest, particularly with regard to the nature of the superconducting pairing and its mechanism. Consequently, it is essential to better understand the electronic Fermi surface topology and its role in strong antiferromagnetic fluctuations. The standard density functional theory method is insufficient to model the interplay of strong on-site Coulomb repulsion in localized 4𝑓 electrons and their hybridization with itinerant ligand-orbital electrons. We have performed electronic ground-state calculations on CeCu 2 Si 2 using the Gutzwiller wave function approximation. The Gutzwiller approximation captures the quasiparticle band renormalization from the strong on-site Coulomb repulsion. We have performed an analysis of this effect on the electronic structure and the Fermi surface topology by varying the interaction strength and taking into account the crystal-field splitting. Using the de Haas-van Alphen effect, the extremal Fermi surface cross-sectional areas were calculated to quantify the effects of quasiparticle mass renormalization on the Fermi surface. Our results confirm the presence of two Fermi surface sheets corresponding to the heavy (488𝑚 𝑒 ) and light (4.35𝑚 𝑒 ) quasiparticles when the crystal-field splitting is accounted for on equal footing with the electronic correlations. This method gives the best agreement with experimental measurements as well as the renormalized band method.
Hydrous and nominally anhydrous minerals are a fundamental class of solids of enormous significance to geophysics. They are the water carriers in the deep geological water cycle and impact structural, elastic, plastic, and thermodynamic properties and phase relations in Earth's forming aggregates (rocks). They play a critical role in the geochemical and geophysical processes that shape the planet. Their complexity has prevented predictive calculations of their properties, but progress in materials simulations ushered by machine-learning potentials is transforming this state of affairs. Here, we adopt a hybrid approach that combines deep learning potentials (DPs) with the strongly constrained and appropriately normed meta-generalized gradient approximation functional to simulate a prototypical hydrous system. We illustrate the success of this approach to simulate δ − AlOOH ( δ ) , a phase capable of transporting water down to near the core-mantle boundary of the Earth ( ∼ 2900 km depth and ∼ 135 GPa ) in subducting slabs. A high-throughput sampling of phase space using molecular dynamics simulations with DPs sheds light on the hydrogen-bond behavior and proton diffusion at geophysical conditions. These simulations provide a pathway for a deeper understanding of these crucial components that shape Earth's internal state. Published by the American Physical Society 2024
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Here, we present an approach for compressing volumetric scalar fields using implicit neural representations. Our approach represents a scalar field as a learned function, wherein a neural network maps a point in the domain to an output scalar value. By setting the number of weights of the neural network to be smaller than the input size, we achieve compressed representations of scalar fields, thus framing compression as a type of function approximation. Combined with carefully quantizing network weights, we show that this approach yields highly compact representations that outperform state-of-the-art volume compression approaches. The conceptual simplicity of our approach enables a number of benefits, such as support for time-varying scalar fields, optimizing to preserve spatial gradients, and random-access field evaluation. We study the impact of network design choices on compression performance, highlighting how simple network architectures are effective for a broad range of volumes.
LBC encompasses the source code and data to reproduce and extend results for a manuscript that compares demand responsive control schemes for multi-zone buildings. It includes examples of model predictive control (MPC), value function approximation via CVXPYLAYERS, and differentiable predictive control (DPC). The goal of the study is to evaluate state-of-the-art controllers and establish the efficacy (if any) of learning-based approaches that leverage deep neural networks in one way or another.
The need to approximate functions is ubiquitous in science, either due to empirical constraints or high computational cost of accessing the function. In high-energy physics, the precise computation of the scattering cross-section of a process requires the evaluation of computationally intensive integrals. A wide variety of methods in machine learning have been used to tackle this problem, but often the motivation of using one method over another is lacking. Comparing these methods is typically highly dependent on the problem at hand, so we specify to the case where we can evaluate the function a large number of times, after which quick and accurate evaluation can take place. We consider four interpolation and three machine learning techniques and compare their performance on three toy functions, the four-point scalar Passarino-Veltman D_0 D 0 function, and the two-loop self-energy master integral M. We find that in low dimensions (d = 3), traditional interpolation techniques like the Radial Basis Function perform very well, but in higher dimensions (d=5, 6, 9) we find that multi-layer perceptrons (a.k.a neural networks) do not suffer as much from the curse of dimensionality and provide the fastest and most accurate predictions.
This report demonstrates universal differential equations (UDEs) as an approach to bridge the gap between ordinary differential equations (ODE) models and agent-based models (ABMs). Using UDE models as surrogates for ABMs allows us to preserve the foundational ODE that represents global disease dynamics while coupling it with a neural network model to approximate functions for the local behaviors of the ABM.
It is well known that, under plastic deformation, dislocations are not only created but also move through the crystal, and their mobility is impeded by their interaction with the crystal structure. At high stress and temperature, this “drag” is dominated by phonon wind, i.e., phonons scattering off dislocations. Employing the semi-isotropic approach discussed in detail in a previous paper (J. Phys. Chem. Solids 2019, 124, 24–35), we discuss here the approximate functional dependence of dislocation drag B on dislocation velocity in various regimes between a few percent of transverse sound speed c T and c T (where c T is the effective average transverse sound speed of the polycrystal). In doing so, we find an effective functional form for dislocation drag B ( v ) for different slip systems and dislocation characters at fixed (room) temperature and low pressure.
The Neutron Star Interior Composition Explorer collaboration recently published a joint estimate of the mass and the radius of PSR J0030+0451, derived via X-ray pulse-profile modeling. Raaijmakers et al. explored the implications of this measurement for the dense matter equation of state (EOS) using two parameterizations of the high-density EOS: a piecewise-polytropic model, and a model based on the speed of sound in neutron stars (NSs). In this work we obtain further constraints on the EOS following this approach, but we also include information about the tidal deformability of NSs from the gravitational wave signal of the compact binary merger GW170817. Further, we compare the constraints on the EOS to those set by the recent measurement of a 2.14 Me pulsar, included as a likelihood function approximated by a Gaussian, and find a small increase in information gain. To show the flexibility of our method, we also explore the possibility that GW170817 was a NS–black hole merger, which yields weaker constraints on the EOS.