Self-consistent Algorithm Research Articles

A two-step self consistent algorithm for extracting magnetic anisotropy constants from angle-dependent ferromagnetic resonance measurements

To infer anisotropy parameters of magnetic materials from experimental measurements of ferromagnetic resonance (FMR) (with data comprising resonant frequencies and corresponding resonant magnetic fields), a fitting procedure to the theoretical model is essential. The governing equation for this fitting process is the Smit–Beljers equation (SB equation), which links FMR frequencies to resonant magnetic field values and anisotropy parameters nonlinearly, and through the equilibrium magnetization angle values, which also depend on the same anisotropy parameters and magnetic field values. Obtaining an analytical expression for this equation is rarely possible. In this paper, we address the challenges posed by the Smit–Beljers equation and propose a two-step self-consistent algorithm to extract the relevant parameters efficiently.

CUDA-based focused Gaussian beams second-harmonic generation efficiency calculator

We present an object-oriented programming (OOP) CUDA-based package for fast and accurate simulation of second-harmonic generation (SHG) efficiency using focused Gaussian beams. The model includes linear as well as two-photon absorption that can ultimately lead to thermal lensing due to self-heating effects. Our approach speeds up calculations by nearly 40x (11x) without (with) temperature profiles with respect to an equivalent implementation using CPU. The package offers a valuable tool for experimental design and study of 3D field propagation in nonlinear three-wave interactions. It is useful for optimization of SHG-based experiments and mitigates undesired thermal effects, enabling improved oven designs and advanced device architectures, leading to stable, efficient high-power SHG. Program summaryProgram Title:cuSHGCPC Library link to program files:https://doi.org/10.17632/hn76s7x848.1Developer's repository link:https://github.com/alfredos84/cuSHGLicensing provisions: MITProgramming language:▪, CUDANature of problem: The problem which is solved in this work is that of second-harmonic generation (SHG) performance degradation in a nonlinear crystal with focused Gaussian beams due to thermal effects. By placing the nonlinear crystal in an oven that controls temperature, the package computes the involved electric fields along the medium. The implemented model includes the linear and nonlinear absorption which occasionally lead to self-heating effect, degrading the performance of the SHG.Solution method: The coupled differential equations for three-wave interactions, which describe the field evolution along the crystal, are solved using the well-known Split-Step Fourier method. The temperature profiles are estimated using the finite-elements method. The field evolution and thermal effects are embedded in a self-consistent algorithm that sequentially and separately solves the electromagnetic and thermal problems until the system reaches the steady state. Due to the eventual computational demand that some problems may have, we chose to implement the coupled equations in the ▪/CUDA programming language. This allows us to significantly speed up simulations, thanks to the computing power provided by a graphics processing unit (GPU) card. The output files obtained are the interacting electric fields and the temperature profile, which have to be analyzed during post-processing.

Self-consistent numerical model of mosquito dynamics with specified kinematic parameters of wing movement

Mosquito flight dynamics was considered using an advanced mathematical model with experimentally determined and biologically relevant parameters. The model was developed using a self-consistent algorithm. It described mosquito’s body and wing oscillations using mechanics equations and aerodynamic flows, simultaneously. Six equations were used for the mechanics of the mosquito (relative to the center of mass), and Navier–Stokes equations simulated aerodynamics. Numerical methods employed deformable computational mesh with specific mesh construction near the wing boundaries. Using size, shape and weight characteristics of the Culex quinquefasciatus (male) mosquito the flight dynamics was computed and analyzed. It was determined that the mosquito lift force was proportional to the square of the wing frequency. Specifically, we found a critical frequency of approximately 820 Hz that corresponded to mosquito freezing when the lift force compensated gravity. This number was consistent to experimentally determined mosquito flight characteristics. Mechanism of mosquito’s lift force generation was discussed.

Flexible DMRG-Based Framework for Anharmonic Vibrational Calculations.

We present a novel formulation of the vibrational density matrix renormalization group (vDMRG) algorithm tailored to strongly anharmonic molecules described by general, high-dimensional model representations of potential energy surfaces. For this purpose, we extend the vDMRG framework to support vibrational Hamiltonians expressed in the so-called n-mode second-quantization formalism. The resulting n-mode vDMRG method offers full flexibility with respect to both the functional form of the PES and the choice of the single-particle basis set. We leverage this framework to apply, for the first time, vDMRG based on an anharmonic modal basis set optimized with the vibrational self-consistent field algorithm on an on-the-fly constructed PES. We also extend the n-mode vDMRG framework to include excited-state-targeting algorithms in order to efficiently calculate anharmonic transition frequencies. We demonstrate the capabilities of our novel n-mode vDMRG framework for methyloxirane, a challenging molecule with 24 coupled vibrational modes.

The Markovian Multiagent Monte-Carlo method as a differential evolution approach to the SCF problem for restricted and unrestricted Hartree-Fock and Kohn-Sham-DFT.

In this paper we present the Markovian Multiagent Monte-Carlo Second Order Self-Consistent Field Algorithm (M3-SOSCF). This algorithm provides a highly reliable methodology for converging SCF calculations in single-reference methods using a modified differential evolution approach. Additionally, M3 is embarrassingly parallel and modular in regards to Newton-Raphson subroutines. We show that M3 is able to surpass contemporary SOSCFs in reliability, which is illustrated by a benchmark employing poor initial guesses and a second benchmark with SCF calculations which face difficulties using standard SCF algorithms. Furthermore, we analyse inherent properties of M3 and show that in addition to its robustness and efficiency, it is more user-friendly than current SOSCFs.

Breakdown of quantization in nonlinear Thouless pumping

The dynamics of solitons driven in a nonlinear Thouless pump and its connection with the system’s topology were recently explored for both weak and strong nonlinear strength. Using both a self-consistent algorithm and 4th order Runge Kutta method, this work uncovers the fate of nonlinear Thouless pumping in the regime of intermediate nonlinearity, thus establishing a fascinating crossover from the observation of nonzero and quantized pumping at weak nonlinearity to zero pumping at strong nonlinearity. We identify the presence of critical nonlinearity strength at which quantized pumping of solitons breaks down regardless of the protocol time scale. Such an obstruction to pumping quantization is attributed to the presence of self-crossing in nonlinear topological bands. By considering another type of pumping involving Bloch states, we further show how the presence of self-crossing bands also leads to breakdown of quantization, but in a completely different manner from that in the case of soliton pumping. Our results not only unveil a missing piece of physics in nonlinear Thouless pumping, but also provide a means to detect loop structures of nonlinear systems investigated in real space and momentum space.

Quantum-Classical Hybrid Systems and Ehrenfest's Theorem.

The conceptual analysis of quantum mechanics brings to light that a theory inherently consistent with observations should be able to describe both quantum and classical systems, i.e., quantum-classical hybrids. For example, the orthodox interpretation of measurements requires the transient creation of quantum-classical hybrids. Despite its limitations in defining the classical limit, Ehrenfest's theorem makes the simplest contact between quantum and classical mechanics. Here, we generalized the Ehrenfest theorem to bipartite quantum systems. To study quantum-classical hybrids, we employed a formalism based on operator-valued Wigner functions and quantum-classical brackets. We used this approach to derive the form of the Ehrenfest theorem for quantum-classical hybrids. We found that the time variation of the average energy of each component of the bipartite system is equal to the average of the symmetrized quantum dissipated power in both the quantum and the quantum-classical case. We expect that these theoretical results will be useful both to analyze quantum-classical hybrids and to develop self-consistent numerical algorithms for Ehrenfest-type simulations.

Second-Order Self-Consistent Field Algorithms: From Classical to Quantum Nuclei

This work presents a general framework for deriving exact and approximate Newton self-consistent field (SCF) orbital optimization algorithms by leveraging concepts borrowed from differential geometry. Within this framework, we extend the augmented Roothaan-Hall (ARH) algorithm to unrestricted electronic and nuclear-electronic calculations. We demonstrate that ARH yields an excellent compromise between stability and computational cost for SCF problems that are hard to converge with conventional first-order optimization strategies. In the electronic case, we show that ARH overcomes the slow convergence of orbitals in strongly correlated molecules with the example of several iron-sulfur clusters. For nuclear-electronic calculations, ARH significantly enhances the convergence already for small molecules, as demonstrated for a series of protonated water clusters.

A Novel Liquid Water Content Retrieval Method Based on Mass Absorption for Single-Wavelength Cloud Radar

Low-level clouds (LLC), mainly composed of liquid water droplets, cool the climate system by strongly reflecting solar radiation back to space, and thus play an important role in the Earth energy budget. However, the LLC properties and their radiative effects are poorly represented in climate models, leading to the largest source of uncertainty in the climate prediction. Liquid water content (LWC) is a key property of LLC determining cloud extinction characteristics and is a fundamental parameter in the radiative transfer model. To improve the understanding of LWC properties, algorithms have been proposed to retrieve LWC based on millimeter-wavelength radar. However, the traditional retrieval relies on pre-constructed empirical relationship between reflectivity and LWC and have noticeable limitations. Particularly, the retrieval uncertainty is strongly depended on the assumed particle size distribution, the existence of drizzle particle; and on the accuracy of reflectivity measurement. In this study, we develop a new self-consistent algorithm to retrieve LWC by constraining radar reflectivity factor and attenuation in the whole liquid cloud layer. A relationship between the radar measured reflectivity, LWC, and the intrinsic reflectivity is first constructed based on the radiative transfer theory under Rayleigh scattering regime. A nonlinear least-square regression technique is then applied to derive the optimal parameters in the retrieval equations to obtain the LWC. Comparison with the microwave radiometer (MWR) derived liquid water path (LWP) indicates that our proposed method retrieves more accurate LWC products than that from the traditional empirical algorithms.

Phase diagram of graphene in the presence of spin-orbit coupling and electron-electron interaction

We theoretically investigate in this work the interplay between the intrinsic spin-orbit coupling (SOC) and electron-electron interaction on the magnetism in the graphene honeycomb lattice. Particularly, a phase diagram is here explored and studied by Kane-Mele-Hubbard (KMH) model combined with mean-field theory and self-consistent algorithm. Results show that the graphene undergoes a phase transition from the gapless semi-metal to topological band insulator (TBI) at a finite SOC and weak Coulomb energy U, and another transition from the TBI to antiferromagnetic ordered insulator at stronger Coulomb energy U observed as well. Moreover, our calculations also point out the prior developing orientation of magnetic moments in the in-plane direction driven by the SOC rather than that in the out-of-plane direction without the SOC.

An Adaptive Planewave Method for Electronic Structure Calculations

We propose an adaptive planewave method for eigenvalue problems in electronic structure calculations. The method combines a priori convergence rates and accurate a posteriori error estimates into an effective way of updating the energy cut-off for planewave discretizations, for both linear and nonlinear eigenvalue problems. The method is error controllable for linear eigenvalue problems in the sense that for a given required accuracy, an energy cut-off for which the solution matches the target accuracy can be reached efficiently. Further, the method is particularly promising for nonlinear eigenvalue problems in electronic structure calculations as it shall reduce the cost of early iterations in self-consistent algorithms. We present some numerical experiments for both linear and nonlinear eigenvalue problems. In particular, we provide electronic structure calculations for some insulator and metallic systems simulated with Kohn--Sham density functional theory (DFT) and the projector augmented wave (PAW) method, illustrating the efficiency and potential of the algorithm.

Iterative subspace algorithms for finite-temperature solution of Dyson equation.

One-particle Green's functions obtained from the self-consistent solution of the Dyson equation can be employed in the evaluation of spectroscopic and thermodynamic properties for both molecules and solids. However, typical acceleration techniques used in the traditional quantum chemistry self-consistent algorithms cannot be easily deployed for the Green's function methods because of a non-convex grand potential functional and a non-idempotent density matrix. Moreover, the optimization problem can become more challenging due to the inclusion of correlation effects, changing chemical potential, and fluctuations of the number of particles. In this paper, we study acceleration techniques to target the self-consistent solution of the Dyson equation directly. We use the direct inversion in the iterative subspace (DIIS), the least-squared commutator in the iterative subspace (LCIIS), and the Krylov space accelerated inexact Newton method (KAIN). We observe that the definition of the residual has a significant impact on the convergence of the iterative procedure. Based on the Dyson equation, we generalize the concept of the commutator residual used in DIIS and LCIIS and compare it with the difference residual used in DIIS and KAIN. The commutator residuals outperform the difference residuals for all considered molecular and solid systems within both GW and GF2. For a number of bond-breaking problems, we found that an easily obtained high-temperature solution with effectively suppressed correlations is a very effective starting point for reaching convergence of the problematic low-temperature solutions through a sequential reduction of temperature during calculations.

LED pumped Raman laser: Towards the design of an on-chip all-silicon laser

Design of an on-chip silicon Raman laser is proposed based on a slotted photonic crystal nanocavity. The slot has been considered to be filled with the silicon nanocrystal material having ultra-high Raman gain coefficient, which has further been fortified by the tight spatio-temporal optical confinement inside the nanocavity. Consequently, the threshold power as required for the lasing is miniaturized substantially, and a light emitting diode (LED) suffices for optical pumping. The LED is amenable for integration on the same platform as of the silicon nanocrystal embedded slotted photonic crystal nanocavity exploiting the efficient electroluminescence of the silicon nanocrystal material. The coupled mode equations for stimulated Raman scattering interactions, which govern the evolution of the pump and the Stokes photons, have been solved using a novel self-consistent algorithm to analyze the characteristics of the proposed laser. A lasing threshold-power, as small as, 5 μW has been achieved. Moreover, small-signal frequency response of the proposed LED pumped laser has also been evaluated in steady-state condition, and a modulation bandwidth as high as ∼1.6 GHz has been demonstrated which is almost two orders of magnitude higher than that of the conventional Raman lasers.

A variational framework for computing Wannier functions using dictionary learning

This article introduces a data-driven variational framework for computing Wannier functions by using basis pursuit to transfer information from a general feature dictionary. General features are learned by applying dictionary learning to a dataset of Wannier functions. Our approach displays a systematically controllable energy-localization trade-off, an objective functional allowing for the use of fast numerical solvers, and is capable of being used in highly-efficient self-consistent algorithms.

Reduced density matrix sampling: Self-consistent embedding and multiscale electronic structure on current generation quantum computers

We investigate fully self-consistent multiscale quantum-classical algorithms on current generation superconducting quantum computers, in a unified approach to tackle the correlated electronic structure of large systems in both quantum chemistry and condensed matter physics. In both of these contexts, a strongly correlated quantum region of the extended system is isolated and self-consistently coupled to its environment via the sampling of reduced density matrices. We analyze the viability of current generation quantum devices to provide the required fidelity of these objects for a robust and efficient optimization of this subspace. We show that with a simple error mitigation strategy and optimization of compact tensor product bases to minimize the number of terms to sample, these self-consistent algorithms are indeed highly robust, even in the presence of significant noises on quantum hardware. Furthermore, we demonstrate the use of these density matrices for the sampling of non-energetic properties, including dipole moments and Fermi liquid parameters in condensed phase systems, achieving a reliable accuracy with sparse sampling. It appears that uncertainties derived from the iterative optimization of these subspaces is smaller than variances in the energy for a single subspace optimization with current quantum hardware. This boosts the prospect for routine self-consistency to improve the choice of correlated subspaces in hybrid quantum-classical approaches to electronic structure for large systems in this multiscale fashion.

Implementation of Relativistic Coupled Cluster Theory for Massively Parallel GPU-Accelerated Computing Architectures.

In this paper, we report reimplementation of the core algorithms of relativistic coupled cluster theory aimed at modern heterogeneous high-performance computational infrastructures. The code is designed for parallel execution on many compute nodes with optional GPU coprocessing, accomplished via the new ExaTENSOR back end. The resulting ExaCorr module is primarily intended for calculations of molecules with one or more heavy elements, as relativistic effects on the electronic structure are included from the outset. In the current work, we thereby focus on exact two-component methods and demonstrate the accuracy and performance of the software. The module can be used as a stand-alone program requiring a set of molecular orbital coefficients as the starting point, but it is also interfaced to the DIRAC program that can be used to generate these. We therefore also briefly discuss an improvement of the parallel computing aspects of the relativistic self-consistent field algorithm of the DIRAC program.

Method for Calculating Excited Electronic States Using Density Functionals and Direct Orbital Optimization with Real Space Grid or Plane-Wave Basis Set.

A direct orbital optimization method is presented for density functional calculations of excited electronic states using either a real space grid or a plane-wave basis set. The method is variational, provides atomic forces in the excited states, and can be applied to Kohn-Sham (KS) functionals as well as orbital-density-dependent (ODD) functionals including explicit self-interaction correction. The implementation for KS functionals involves two nested loops: (1) An inner loop for finding a stationary point in a subspace spanned by the occupied and a few virtual orbitals corresponding to the excited state; (2) an outer loop for minimizing the energy in a tangential direction in the space of the orbitals. For ODD functionals, a third loop is used to find the unitary transformation that minimizes the energy functional among occupied orbitals only. Combined with the maximum overlap method, the algorithm converges in challenging cases where conventional self-consistent field algorithms tend to fail. The benchmark tests presented include two charge-transfer excitations in nitrobenzene and an excitation of CO to degenerate π* orbitals where the importance of complex orbitals is illustrated. The application of this method to several metal-to-ligand charge-transfer and metal-centered excited states of an FeII photosensitizer complex is described, and the results are compared to reported experimental estimates. This method is also used to study the effect of the Perdew-Zunger self-interaction correction on valence and Rydberg excited states of several molecules, both singlet and triplet states, and the performance compared to semilocal and hybrid functionals.

Interacting conformal Carrollian theories: Cues from electrodynamics

We construct the free Lagrangian of the magnetic sector of Carrollian electrodynamics. The construction relies on Helmholtz integrability condition for differential equations in a self-consistent algorithm, working hand in hand with imposing invariance under infinite dimensional Conformal Carroll algebra. It requires inclusion of new fields in the dynamics and the system is free of gauge redundancies. We next add interaction (quartic) terms to the free Lagrangian, strictly constrained by conformal invariance and Carrollian symmetry. The dynamical realization of the non-semi-simple infinite dimensional symmetry algebra at the level of charge algebra is exact and free from central terms.

SCERPA Simulation of Clocked Molecular Field-Coupling Nanocomputing

Among all the possible technologies proposed for post-CMOS computing, molecular field-coupled nanocomputing (FCN) is one of the most promising technologies. The information propagation relies on electrostatic interactions among single molecules, overcoming the need for electron transport, significantly reducing energy dissipation. The expected working frequency is very high, and high throughput may be achieved by introducing an efficient pipeline of information propagation. The pipeline could be realized by adding an external clock signal that controls the propagation of data and makes the transmission adiabatic. In this article, we extend the Self-Consistent Electrostatic Potential Algorithm (SCERPA), previously introduced to analyze molecular circuits with a uniform clock field, to clocked molecular devices. The single-molecule is analyzed by ab initio calculations and modeled as an electronic device. Several clocked devices have been partitioned into clock zones and analyzed: the binary wire, the bus, the inverter, and the majority voter. The proposed modification of SCERPA enables linking the functional behavior of the clocked devices to molecular physics, becoming a possible tool for the eventual physical design verification of emerging FCN devices. The algorithm provides some first quantitative results that highlight the clocked propagation characteristics and provide significant feedback for the future implementation of molecular FCN circuits.

Precise PEM fuel cell parameter extraction based on a self-consistent model and SCCSA optimization algorithm

Mathematical modeling of a polymer electrolyte membrane fuel cell (PEMFC) is widely used for investigating its performance. The development of a precise model is crucial for achieving a consistent and accurate simulation of PEMFC performance. Although many studies have been conducted to address the simulation of the characteristics of PEMFC by identifying the uncertain model parameters, they overwhelmingly suffer from a physical flaw, which invalidates their reported results. The aims of the present study are twofold: (1) to highlight the critical inconsistency in the previous research works, and (2) to develop an accurate, self-consistent model, which prevents yielding physically flawed results that are frequently seen in the open literature. The newly proposed model is then used in a coalition with a recent optimization algorithm called hybrid sine-cosine crow search algorithm (SCCSA). Using existing experimental data, the accuracy and efficiency of the proposed formulation are evaluated. Our results indicate good agreement between experimental and computed data for all the test cases, which in return demonstrates the decisive accuracy and usefulness of the developed model. A comparison between the obtained fitness values of this study and the ones reported in the literature shows the superiority of the proposed model. Furthermore, the capability of SCCSA algorithm is examined by comparing the obtained results with those of other meta-heuristic optimization algorithms. Performance comparison across SCCSA and other algorithms, such as particle swarm optimization (PSO), proves the adequate capability of this method to find the optimal fitness and its high convergence speed.