Self-consistent Method Research Articles

A multi-scale diffusional-mechanically coupled model for super-elastic NiTi shape memory alloy wires in hydrogen-rich environment

The functional devices made by NiTi shape memory alloys (SMAs) are often serviced in hydrogen (H)-rich environment. Recently experimental results showed that the martensite transformation (MT) of NiTi SMA changed strongly after charging H, which originates from the H diffusion in the material and an additional transformation resistance caused by the absorbed H atoms. In this work, a multi-scale diffusional-mechanically coupled constitutive model is constructed to describe the super-elastic deformation of NiTi SMA wires in H-rich environment. In the grain scale, a crystal plasticity-based constitutive model is developed within the framework of irreversible thermodynamics. Four deformation mechanisms, i.e., elasticity, MT, transformation-induced plasticity (TRIP) and H expansion are taken into account. Since the H atoms can be trapped by the lattice defects, the total H concentration is further decomposed into two parts, i.e., the lattice hydrogen concentration and the trapped one. The generalized thermodynamic forces of MT and TRIP are derived from the Gibbs free energy and Clausius-Duhem inequality. The evolution of H concentration field is derived by combining the chemical potential, diffusion balance equation and the Fick's law. In the polycrystalline aggregate scale, in order to measure the interaction among the grains and obtain the overall response of the polycrystalline aggregate, a diffusional-mechanically coupled self-consistent homogenization method is developed. Meanwhile, the finite volume method is employed to calculate the H concentration in each grain. An assumption of uniform stress field in the macroscopic scale is adopted to achieve the scale transition from the polycrystalline aggregate to the whole wire. To validate the prediction capability of the proposed model, the predictions are compared with the experiments. Moreover, the influences of loading rate on the deformation of the wires in the process of in-situ charging H are predicted and discussed.

Application of the reference interaction site model self-consistent field method based on the Dirac-Hartree–Fock wave function to a chemical reaction

The reference interaction site model self-consistent field (RISM-SCF) method is a combined method of the electronic structure theory of molecules and the integral equation theory of molecular liquids. The RISM-SCF method based on the Dirac-Hartree-Fock wave function, recently proposed, is applied to a chemical reaction, specifically, a Menshutkin reaction in aqueous solution. The Helmholtz energy profile along the reaction coordinate is calculated and the characteristics of the reaction are discussed based on energy component analysis.

A Charge Conserving Exponential Predictor Corrector FEMPIC Formulation for Relativistic Particle Simulations

The state of art of charge conserving electromagnetic finite element particle-in-cell (EM-FEMPIC) has grown by leaps and bounds in the past few years. These advances have primarily been achieved for leap-frog time stepping schemes for Maxwell solvers, in large part, due to the method strictly following the proper space for representing fields, charges, and measuring currents. Unfortunately, leap-frog-based solvers (and their other incarnations) are only conditionally stable. Recent advances have made it possible to construct EM-FEMPIC methods built around unconditionally stable time-stepping methods while conserving charge. Together with the use of a quasi-Helmholtz decomposition, these methods were both unconditionally stable and satisfied Gauss’ laws to machine precision. However, this architecture was developed for systems with explicit particle integrators where fields and velocities were off by a time step. While completely self-consistent methods exist in the literature, they follow the classic rubric: collect a system of first-order differential equations (Maxwell and Newton equations) and use an integrator to solve the combined system. These methods suffer from the same side effect as earlier—they are conditionally stable. Here, we propose a different approach; we pair an unconditionally stable Maxwell solver to an exponential predictor corrector (PC) method for Newton’s equations. As we will show via numerical experiments, the proposed method conserves energy within a particle-in-cell (PIC) scheme, has an unconditionally stable electromagnetic (EM) solve, solves Newton’s equations to much higher accuracy than a traditional Boris solver, and conserves charge to machine precision. We further demonstrate benefits compared with other polynomial methods to solve Newton’s equations, like the well-known Boris push.

Understanding the helicoidal damage behavior of bio-inspired laminates by conducting multiscale concurrent simulation and experimental analysis

Bio-inspired helicoidal structures exhibit excellent fracture toughness. A multiscale concurrent simulation of the damage propagation process is critical for understanding the fracture behavior of such structures with pre-existing cracks. However, such an analysis is extremely challenging owing to computational complexity. In this study, the in-plane critical stress intensity factor and energy release rate of six different helicoidal carbon-fiber-reinforced plastic structures are measured experimentally by conducting a single-edge-notch bending test. The damage distribution of the different structures is examined by performing computed tomography scanning. Several multiscale concurrent analyses are conducted by using the self-consistent clustering analysis method to study the specific damage propagation process and evaluate the fracture toughness contribution of different layers. The results indicate that the pitch angle significantly impacts the fracture toughness, whereas the stacking mode changes the crack propagation orientation. When the pitch angle is appropriately adjusted, the fiber damage length is enhanced, resulting in a significant increase in the fracture toughness. The interaction between fiber damage, matrix damage, and hybrid damage during the failure process is interpreted using the dissipated energy ratios of the fiber damage layer, transition layer, and matrix damage layer.

Target State Optimized Density Functional Theory for Electronic Excited and Diabatic States.

A flexible self-consistent field method, called target state optimization (TSO), is presented for exploring electronic excited configurations and localized diabatic states. The key idea is to partition molecular orbitals into different subspaces according to the excitation or localization pattern for a target state. Because of the orbital-subspace constraint, orbitals belonging to different subspaces do not mix. Furthermore, the determinant wave function for such excited or diabatic configurations can be variationally optimized as a ground state procedure, unlike conventional ΔSCF methods, without the possibility of collapsing back to the ground state or other lower-energy configurations. The TSO method can be applied both in Hartree-Fock theory and in Kohn-Sham density functional theory (DFT). The density projection procedure and the working equations for implementing the TSO method are described along with several illustrative applications. For valence excited states of organic compounds, it was found that the computed excitation energies from TSO-DFT and time-dependent density functional theory (TD-DFT) are of similar quality with average errors of 0.5 and 0.4 eV, respectively. For core excitation, doubly excited states and charge-transfer states, the performance of TSO-DFT is clearly superior to that from conventional TD-DFT calculations. It is shown that variationally optimized charge-localized diabatic states can be defined using TSO-DFT in energy decomposition analysis to gain both qualitative and quantitative insights on intermolecular interactions. Alternatively, the variational diabatic states may be used in molecular dynamics simulation of charge transfer processes. The TSO method can also be used to define basis states in multistate density functional theory for excited states through nonorthogonal state interaction calculations. The software implementing TSO-DFT can be accessed from the authors.

Elastic scattering of electrons by water: An ab initio study

In this work we devise a theoretical and computational method to compute the elastic scattering of electrons from a non-spherical potential, such as in the case of molecules and molecular aggregates. Its main feature is represented by the ability of calculating accurate wave functions for continuum states of polycentric systems via the solution of the Lippmann-Schwinger equation, including both the correlation effects and multi-scattering interference terms, typically neglected in widely used approaches, such as the Mott theory. Within this framework, we calculate the purely elastic scattering matrix elements. As a test case, we apply our scheme to the modelling of electron-water elastic scattering. The Dirac-Hartree-Fock self-consistent field method is used to determine the non-spherical molecular potential projected on a functional space spanned by Gaussian basis set. By adding a number of multi-centric radially-arranged s-type Gaussian functions, whose exponents are system-dependent and optimized to reproduce the properties of the continuum electron wave function in different energy regions, we are able to achieve unprecedented access to the description of the low energy range of the spectrum (0.001 < E < 10 eV) up to keV, finding a good agreement with experimental data and previous theoretical results. To show the potential of our approach, we also compute the total elastic scattering cross section of electrons impinging on clusters of water molecules and zundel cation. Our method can be extended to deal with inelastic scattering events and heavy-charged particles.

Extended mean-field homogenization of unidirectional piezoelectric nanocomposites with generalized Gurtin-Murdoch interfaces

This paper presents for the first time an extended Mori-Tanaka approach aimed at identifying the little-explored piezoelectric response of unidirectional nanoporous composites with energetic surfaces. The interface is simulated using the generalized Gurtin-Murdoch coherent interface model considering nonvanishing in-plane surface stress and surface electric displacement. The analytical solutions for Eshelby’s inhomogeneity problems are obtained by solving four systems of equations, generated from the known macroscopic electromechanical loading conditions. The dilute concentration tensors for the fiber and energetic surface are evaluated exactly for coupled electromechanical fields, which are then utilized in the extended multiphysics Mori-Tanaka homogenization scheme. The reliability and accuracy of the extended theory are demonstrated through an extensive comparison with the predictions of the composite cylinder assemblage (CCA), the generalized self-consistent method (GSCM), as well as the finite-element (FE) homogenization technique. New results generated in this work demonstrate that, except for the transverse shear modulus, the extended Mori-Tanaka, CCA, and FE provide indistinguishable results under axisymmetric, axial shear, and electric loading, even at high volume fractions and small pore sizes where the surface piezoelectric effects are significant.

Self-Consistent Study of GaAs/AlGaAs Quantum Wells with Modulated Doping

In this work, the characterization and analysis of the physics of a GaAs quantum well with AlGaAs barriers were carried out, according to an interior doped layer. An analysis of the probability density, the energy spectrum, and the electronic density was performed using the self-consistent method to solve the Schrödinger, Poisson, and charge-neutrality equations. Based on the characterizations, the system response to geometric changes in the well width and to non-geometric changes, such as the position and with of the doped layer as well as the donor density, were reviewed. All second-order differential equations were solved using the finite difference method. Finally, with the obtained wave functions and energies, the optical absorption coefficient and the electromagnetically induced transparency between the first three confined states were calculated. The results showed the possibility of tuning the optical absorption coefficient and the electromagnetically induced transparency via changes to the system geometry and the doped-layer characteristics.

Nonlocal corrections to dynamical mean-field theory from the two-particle self-consistent method

Theoretical methods that are accurate for both short-distance observables and long-wavelength collective modes are still being developed for the Hubbard model. Here, we benchmark an approach that combines dynamical mean-field theory (DMFT) observables with the two-particle self-consistent theory (TPSC). This offers a way to include non-local correlations in DMFT while also improving TPSC. The benchmarks are published diagrammatic quantum Monte Carlo results for the two-dimensional square lattice Hubbard model with nearest-neighbor hopping. This method (TPSC+DMFT) is relevant for weak to intermediate interaction, satisfies the local Pauli principle and allows us to compute a spin susceptibility that satisfies the Mermin-Wagner theorem. The DMFT double occupancy determines the spin and charge vertices through local spin and charge sum rules. The TPSC self-energy is also improved by replacing its local part with the local DMFT self-energy. With this method, we find improvements for both spin and charge fluctuations and for the self-energy. We also find that the accuracy check developed for TPSC is a good predictor of deviations from benchmarks for this model. TPSC+DMFT can be used in regimes where quantum Monte Carlo is inaccessible. In addition, this method paves the way to multi-band generalizations of TPSC that could be used in advanced electronic structure codes that include DMFT.

Phase diagram of twisted bilayer graphene at filling factor ν=±3

We study the correlated insulating phases of twisted bilayer graphene (TBG) in the absence of lattice strain at integer filling $\ensuremath{\nu}=\ifmmode\pm\else\textpm\fi{}3$. Using the self-consistent Hartree-Fock method on a particle-hole symmetric model and allowing translation symmetry breaking terms, we obtain the phase diagram with respect to the ratio of $AA$ interlayer hopping $({w}_{0})$ and $AB$ interlayer hopping $({w}_{1})$. When the interlayer hopping ratio is close to the chiral limit (${w}_{0}/{w}_{1}\ensuremath{\lesssim}0.5$), a quantum anomalous Hall state with Chern number ${\ensuremath{\nu}}_{c}=\ifmmode\pm\else\textpm\fi{}1$ can be observed consistent with previous studies. Around the realistic value ${w}_{0}/{w}_{1}\ensuremath{\approx}0.8$, we find a spin and valley polarized, translation symmetry breaking, state with ${C}_{2z}T$ symmetry, a charge gap and a doubling of the moir\'e unit cell, dubbed the ${C}_{2z}T$ stripe phase. The real-space total charge distribution of this ${C}_{2z}T$ stripe phase in the flat band limit does not have modulation between different moir\'e unit cells, although the charge density in each layer is modulated, and the translation symmetry is strongly broken. Other symmetries, including ${C}_{2z}, {C}_{2x}$, particle-hole symmetry $P$, and the topology of the ${C}_{2z}T$ stripe phase, are also discussed in detail. We observed braiding and annihilation of the Dirac nodes by continuously turning on the order parameter to its fully self-consistent value, and provide a detailed explanation of the mechanism for the charge gap opening despite preserving ${C}_{2z}T$ and valley $\mathrm{U}(1)$ symmetries. In the transition region between the quantum anomalous Hall phase and the ${C}_{2z}T$ stripe phase, we find an additional competing state with comparable energy corresponding to a phase with a tripling of the moir\'e unit cell.

Proton Reaction Path in Base Pairs of DNA Molecule According to the Complete Active Space Self-Consistent Field Method

The double proton transfer reaction paths in AT and CG base pairs of DNA molecule are calculated in the Complete Active Space Self-Consistent Field method and compared with the same paths in Density Functional Theory with B3LYP approximation approach. We found that an essential increase of an activation energy, which significantly reduces the probability of spontaneous mutations in DNA via double proton transfer. There exist two transition points on the singlet potential energy surface divided by a flat region for GC base pair. The applicability of various quantum-chemical methods for description of double proton transfer reactions was discussed.

Light–matter quantum interface with continuous pump and probe

Spin-polarized atomic ensembles probed by light based on the Faraday interaction are a versatile platform for numerous applications in quantum metrology and quantum information processing. Here we consider an ensemble of Alkali atoms that are continuously optically pumped and probed. Due to the collective scattering of photons at large optical depth, the steady state of atoms does not correspond to an uncorrelated tensor-product state, as is usually assumed. We introduce a self-consistent method to approximate the steady state including the pair correlations, taking into account the multilevel structure of atoms. We find and characterize regimes of Raman lasing, akin to the model of a superradiant laser. We determine the spectrum of the collectively scattered photons, which also characterizes the coherence time of the collective spin excitations on top of the stationary correlated mean-field state, as relevant for applications in metrology and quantum information.

Exchange correction for allowed β decay

We investigate the exchange effect between the final atom's bound electrons and those emitted in the allowed $\ensuremath{\beta}$ decay of the initial nucleus. The electron wave functions are obtained with the Dirac-Hartree-Fock-Slater self-consistent method, and we ensure the orthogonality between the continuum and bound electron states, in the potential of the final atom, by modifying the last iteration of the self-consistent method. We show that orthogonality plays an essential role in calculating the exchange correction. After imposing the orthogonality, we found considerable differences in magnitude and energy dependence compared to previous results. We argue that our findings can solve the mismatch between the previous predictions and experimental measurements in the low-energy region of the $\ensuremath{\beta}$ spectrum. First, we calculate the exchange effect for the low-energy $\ensuremath{\beta}$ transitions in $^{14}\mathrm{C}$, $^{45}\mathrm{Ca}$, $^{63}\mathrm{Ni}$, and $^{241}\mathrm{Pu}$, recently investigated in the literature. Next, we compute the total exchange correction for a large number of $\ensuremath{\beta}$ emitters with $Z$ from 1 to 102. From the systematic study, we found that for ultralow energy, i.e., 5 eV, the $Z$ dependence of the total exchange effect is affected by ${s}_{1/2}$ and ${p}_{1/2}$ orbitals closure. We also show that the contributions from orbitals higher than the $2{s}_{1/2}$ orbital are essential for correctly calculating the total effect, especially for low energies and heavy $\ensuremath{\beta}$ emitters. Finally, we provide an analytical expression of the total exchange correction for each atomic number for easy implementation in experimental investigations.

Calculations of positron annihilation lifetime in solid based on finite difference and conjugate-gradient methods

We employ a real-space grids technique to calculate positron annihilation lifetimes with pseudopotentials. This method is based on the two-component of density functional theory (TCDFT), the Hamiltonian operator of the Kohn-Sham equation is discretized on a point grid by the finite difference method (FDM), the positron eigenstates are searched by the conjugate-gradient method (CG) and the positron Kohn-Sham equation is solved by a self-consistent iteration method. We show that the numerical results under this scheme for bulk and monovacancy of positron annihilation lifetimes are in reasonable agreement with the available experimental data. Furthermore, all the results of our calculations demonstrate the accuracy and efficiency of the method from different aspects.

Toward an efficient f-in-core/f-in-valence switchable description for DFTB calculations of Ce 4f states in ceria.

A computational protocol is developed for efficient studies of partially reduced redox-active oxides using the self-consistent charge density functional tight-binding method. The protocol is demonstrated for ceria, which is a prototypical reducible oxide material. The underlying idea is to achieve a consistent (and harmonized) set of Slater-Koster (SK) tables with connected repulsive potentials that enable switching on and off the in-valence description of the Ce 4f states without serious loss of accuracy in structure and energetics. The implicit treatment of the Ce 4f states, with the use of f-in-core SK-tables, is found to lead to a significant decrease in computational time. More importantly, it allows for explicit control of the oxidation states of individual Ce atoms. This makes it possible to "freeze" the electronic configuration, thereby allowing the exploration of the energetics for various meta-stable configurations. We anticipate that the outlined strategy can help to shed light on the interplay between the size, shape, and redox activity for nanoceria and other related materials.

Coulson–Fischer Wave Function on Self-consistent Field and Perturbation Correction

We propose a computational method to obtain Coulson–Fischer (CF) wave functions using the conventional Gaussian-type orbital function as the basis sets and the self-consistent field method. Because the obtained wave functions are similar to unrestricted Hartree–Fock functions, the electron correlation energies can be estimated by the perturbation correction using our CF wave function. Comparing the corrected total energy with the result via the full configuration interaction confirms the usability of our method.

Electron and ion spectroscopy of azobenzene in the valence and core shells.

Azobenzene is a prototype and a building block of a class of molecules of extreme technological interest as molecular photo-switches. We present a joint experimental and theoretical study of its response to irradiation with light across the UV to x-ray spectrum. The study of valence and inner shell photo-ionization and excitation processes combined with measurement of valence photoelectron-photoion coincidence and mass spectra across the core thresholds provides a detailed insight into the site- and state-selected photo-induced processes. Photo-ionization and excitation measurements are interpreted via the multi-configurational restricted active space self-consistent field method corrected by second order perturbation theory. Using static modeling, we demonstrate that the carbon and nitrogen K edges of azobenzene are suitable candidates for exploring its photoinduced dynamics thanks to the transient signals appearing in background-free regions of the NEXAFS and XPS.

Electric-field-tunable thermal conductivity in anti-ferroelectric materials

Based on density functional theory (DFT) calculations, we evaluate the thermal conductivities of LiBeP in both the ferroelectric (FE) and anti-ferroelectric (AFE) phases by using the self-consistent phonon method (SCP) combined with the compressive sensing (CS) approach. It is observed that the thermal conductivities can be tailored by more than 50% at room temperature upon the FE-AFE phase transitions. Detailed analysis on the phonon modes and scattering space reveals that there are substantial anharmonic effects in LiBeP. The significant change of the thermal conductivity makes it a promising candidate for thermal management materials with tunable thermal conductivities by electric fields. Our work provides a proof-of-concept example for electric-field controllable thermal management, which is promising for future applications.

A Distributed Self-Consistent Control Method for Electric Vehicles to Coordinate Low-Carbon Transportation and Energy

Dear Editor, In this letter, a distributed self-consistent control method to coordinate low-carbon transportation and energy is proposed to address the efficient utilization of regional transportation energy and renewable energy. Specifically, taking into account the coordinated development of transportation, power grids, and renewable energy, transportation energy self-consistent, including instant self-consistent rate and power self-consistent rate, is introduced to the distributed control, which considers electric vehicle (EV) charging and discharging willingness, and integrates the multiple demands of transportation, power grids, and renewable energy. The simulation results show that the proposed method can effectively increase the instant self-consistent rate and improve the renewable energy accommodation.

The temperature dependence on the coherence time of a quantum pseudodot qubit

In this work, the energies and eigenfunctions of ground state and first-excited states (GFES) of a strongly coupled polaron in a quantum pseudo-dot (QPD) were studied by using variational method of Pekar type (VMPT). A single qubit can be realized in this two-level quantum system. Then, we calculated the coherence time of a QPD qubit by employing the Fermi Golden Rule. The temperature effects on the coherence time are taken into account by using the quantum statistics theory (QST) and self-consistent calculation method. According to the obtained results, it is found that the coherence time increases with decreasing temperature. Also, this time is a decaying function of the chemical potential of the two-dimensional electron gas and the zero point of the PHP.