Self-consistent Field Approximation Research Articles

Extended relativistic kinetic model composed of the scalar and two vector distribution functions: Application to the spin-electron-acoustic waves

Detailed deterministic derivation of kinetic equations for relativistic plasmas is given. Focus is made on the dynamic of one-coordinate distribution functions of various tensor dimensions, but the closed set of kinetic equations is constructed of three functions: the scalar distribution function, the vector distribution function of dipole moment, and the vector distribution function of velocity (or the dipole moment in the momentum space). All two-coordinate distribution functions are discussed as well. They are presented together with their limits existing in the self-consistent field approximation. The dynamics of the small amplitude spin-electron-acoustic waves in the dense degenerate plasmas is studied within the kinetic model. This work presents the deterministic approach to the derivation and interpretation of the kinetic equations. So, no probability is introduced during the transition from the level of individual particles to the collective functions. The problem of thermalization is not considered, but we can see that the structure of kinetic equations is kept for the system before and after thermalization. Hence, the kinetic equations can be used to approach this item.

Study of nonlinear dynamic modes of native DNA via mathematical modeling methods

One of the main problems in the nonlinear dynamics of DNA is the identification of physical mechanisms responsible for the mechanical features of its behaviour. The formation and interaction of open states in the DNA structure may be one such mechanism. In this paper we propose a mathematical model of native DNA, which allows us to describe its thermodynamic and kinetic properties taking into account the collective behaviour of the ensemble of open states. The concept of a microscopic open state in DNA and its associated parameter, the displacement vector of nitrogenous bases, are introduced. By averaging the base displacement vectors over the ensemble of states, the thermodynamic variable is determined. According to the statistical model of DNA in the self-consistent field approximation, the structural parameter of the system thermalization is derived. This parameter reflects the statistical self-similarity in the behaviour of the ensemble of open states. Regularities of "criticality" for different ranges of the structural parameter are established and phenomenological representations of the free energy in the framework of the Ginzburg-Landau approach are proposed. Numerical modelling of the dynamics of native DNA is carried out for different ranges of the structural parameter, which has allowed us to establish the types of self-similar solutions and the corresponding open-state collective modes. It is shown that the latter have the nature of finite-amplitude fluctuations in the form of breahters and autosoliton modes, as well as blow-up dissipative structures. The sensitivity of the model to initial conditions is investigated, and the influence of nonlinearity inherent in the model on the modelling results is established. The limitations and prospects of using the approach proposed in this work for mathematical modelling of native DNA are discussed.

Photoionization cross sections and photoelectron angular distributions of molecules with XCHEM-2.0

The XCHEM code was introduced in 2017 [1] to provide an accurate description of electron correlation and exchange in the electronic continuum of molecules at the same level as complete or restricted active-space self-consistent field (CASSCF or RASSCF) methods. This has allowed for an accurate description of molecular photoionization in the region of Feshbach resonances, shake up processes in which ionization is accompanied by excitation of one or several of the remaining electrons, and interchannel couplings. The success of XCHEM for small molecules has led us to improve its performance in several aspects, which now allows for the description of resonant molecular photoionization in larger systems. In addition, we have incorporated the possibility to calculate photoelectron angular distributions in the laboratory and molecular frames, which are essential to interpret angularly resolved photoionization experiments. Here we show its performance in the N2 and pyrazine molecules. The new version of the code, XCHEM-2.0, is freely available at https://gitlab.com/xchem/xchem_public. Program summaryProgram Title: XCHEM-2.0CPC Library link to program files:https://doi.org/10.17632/t8tbk9gdt2.1Developer's repository link:https://gitlab.com/xchem/xchem_public/-/archive/2.0.0/xchem_public-2.0.0.tar.gzLicensing provisions: LGPLProgramming language: Fortran, pythonNature of problem: Understanding photoionization of molecules is a recurrent goal in Atomic, Molecular and Optical physics, even more nowadays with the advent of XUV and X-ray high-harmonic generation sources and free electron lasers. Photoionization can be employed to induce electron dynamics in molecules, e.g., by using ultrashort XUV or X-ray pulses, or as a probe of these dynamics by recording time-resolved photoelectron spectra. At low photoelectron energies, photoionization may result from various competing processes, such as direct ionization, autoionization of Feshbach and shape resonances, shake up, etc., each of these leaving its trace in the measured spectra. So, interpretation of these spectra can be a difficult task and theoretical simulations are often required to separate the different contributions. However, a method able to do so must be able to describe electron correlation in the electronic continuum beyond the Hartree-Fock (HF) and configuration interaction singles (CIS) approximations. XCHEM-2.0 has been designed to account for all these processes by providing an accurate description of electron correlation in the ionization continuum of molecules.Solution method: XCHEM-2.0 employs a close coupling formalism to describe the electronic continuum of molecules at the level of multiference configuration interaction (MRCI) methods in combination with a hybrid Gaussian-Bspline basis set (the so-called GABs basis). In particular, it makes use of a restricted active space self-consistent field (RASSCF) approximation to evaluate the bound electronic states of the remaining molecular cation using localized Gaussian functions. These localized Gaussians are then supplemented with a single centered Gaussian expansion to allow the electron to leave the nuclear environment. Finally, the basis set includes a set of B-spline functions, centered at the same position as the single-centered Gaussians, which reach the asymptotic region. Thanks to the last B-spline function, which does not vanish at the box boundary, XCHEM-2.0 can determine continuum states fulfilling any required asymptotic behavior. The present version of the code improves on the original version by incorporating a more efficient augmentation procedure to build Ne-electron configurations from (Ne−1) ones, reducing the space to store data, providing a more efficient removal of linear dependencies resulting from the over-completeness of the polycentric+GABS basis and a more efficient solution of the scattering equations, and including new routines to calculate photoelectron angular distributions in the laboratory and molecular frames.Additional comments including restrictions and unusual features: The proposed method focuses on RASSCF methodology, so part of electron correlation is still not included. Also, one- and two electron integrals involving exclusively Gaussian functions are done using a modified version of the OpenMOLCAS software, so the installation of this code is also required and limitation associated with the latter is applicable. The code is limited to include a few ionization channels (around 20) with medium angular momentum (limited to ℓ = 15 as in OpenMOLCAS).

Nuclear structure and Breit interaction effects in relativistic molecular electronic structure calculations

We examine the effects of nuclear structure and the self-consistent treatment of the Breit interaction on the calculation of properties associated with PT-odd interactions in TlF. We also demonstrate that 99.9999% of the Breit energy in TlF is recovered from the one-centre contributions to the interaction energy in the self-consistent field approximation. The formulation of our electronic structure code, BERTHA, facilitates the evaluation of all one-centre effects using efficient methods borrowed from relativistic atomic physics at negligible computational cost. A further analysis of this behaviour is provided for the homonuclear mean-field system, At, in the Dirac–Hartree–Fock–Breit approximation. For closed-shell systems, a formal analysis of the origin of this behaviour suggests that this computational economy may be of general applicability in relativistic molecular electronic structure theory.

Improved ground state energies of electronegative atoms using Hartree–Fock self-consistent field approximation

Improved ground state energies of electronegative atoms using Hartree–Fock self-consistent field approximation

Role of Exact Exchange in Difference Projected Double-Hybrid Density Functional Theory for Treatment of Local, Charge Transfer, and Rydberg Excitations

Difference approaches to the study of excited states have undergone a renaissance in recent years, with the development of a plethora of algorithms for locating self-consistent field approximations to excited states. Density functional theory is likely to offer the best balance of cost and accuracy for difference approaches, and yet there has been little investigation of how the parametrization of density functional approximations affects performance. In this work, we aim to explore the role of the global Hartree-Fock exchange parameter in tuning accuracy of different excitation types within the framework of the recently introduced difference projected double-hybrid density functional theory approach and contrast the performance with conventional time-dependent double-hybrid density functional theory. Difference projected double-hybrid density functional theory was demonstrated to give vertical excitation energies with average error and standard deviation with respect to multireference perturbation theory comparable to more expensive linear-response coupled cluster approaches ( J. Chem. Phys.2020, 153, 074103). However, despite benchmarking of local excitations, there has been no investigation of the methods performance for charge transfer or Rydberg excitations. In this work we report a new benchmark of charge transfer, Rydberg, and local excited state vertical excitation energies and examine how the exact Hartree-Fock exchange affects the benchmark performance to provide a deeper understanding of how projection and nonlocal correlation balance differing sources of error in the ground and excited states.

Inferring structural and dynamical properties of gene networks from data with deep learning.

The reconstruction of gene regulatory networks (GRNs) from data is vital in systems biology. Although different approaches have been proposed to infer causality from data, some challenges remain, such as how to accurately infer the direction and type of interactions, how to deal with complex network involving multiple feedbacks, as well as how to infer causality between variables from real-world data, especially single cell data. Here, we tackle these problems by deep neural networks (DNNs). The underlying regulatory network for different systems (gene regulations, ecology, diseases, development) can be successfully reconstructed from trained DNN models. We show that DNN is superior to existing approaches including Boolean network, Random Forest and partial cross mapping for network inference. Further, by interrogating the ensemble DNN model trained from single cell data from dynamical system perspective, we are able to unravel complex cell fate dynamics during preimplantation development. We also propose a data-driven approach to quantify the energy landscape for gene regulatory systems, by combining DNN with the partial self-consistent mean field approximation (PSCA) approach. We anticipate the proposed method can be applied to other fields to decipher the underlying dynamical mechanisms of systems from data.

Particle-hole symmetry broken solutions in graphene nanoribbons: A multi-orbital, mean-field perspective

Mean-field theories have since long predicted edge magnetism in graphene nanoribbons, where the order parameter is given by the local magnetization. However, signatures of edge magnetism appears elusive in the experiments, suggesting another class of solutions. We employ a self-consistent mean field approximation within a multi-orbital tight-binding model and obtain particle-hole symmetry broken solutions, where the local filling plays the role of the order parameter. Unlike the magnetic edge solutions, these are topologically non-trivial and show zero local magnetization. A small and a large doping regime are studied, and a free energy minimum for finite hole doping is encountered, which may serve as an explanation for the absence of experimental evidence for magnetic edge states in zigzag graphene nanoribbons. The electronic interaction may increase the finite \(d\)-orbital occupation, which leads to a change of the effective Coulomb interaction of the dominant \(p_z\)-orbitals. Our findings indicate that the non-magnetic solution for finite hole doping becomes energetically preferred, compared to the magnetic phases at half-filling, once thermal fluctuations or unintentional doping from the substrate are considered. This result persists even in the presence of the \(d\)-orbitals and the Coulomb interaction therein.

Hybrid stars and the QCD phase transition with an NJL-type model

In this paper, we introduce a self-consistent mean field approximation to study the QCD phase transition and the structure of hybrid stars within the framework of NJL model. In our practice, a phenomenological parameter $\alpha$ is introduced, which reflects the weights of "direct" channel and "exchange" channel under a finite chemical potential. The mass-radius relation is obtained by solving the Tolman-Oppenheimer-Volkoff equation using a crossover equation of state (EOS). We calculate the density distribution in a two solar-mass hybrid star to show the effects of different parameters. We also calculate the tidal Love number $k_2$ and the deformability $(\Lambda)$. It is found that the stiffness of the EOS increases with $\alpha $, which allows us to obtain a hybrid star with a maximum mass of 2.40 solar-mass through our model. The observation of over 2.06 solar-mass neutron stars may indicates that the chiral transition may be a crossover on the whole $T-\mu$ plane.

Reentrant superconductivity in a hybrid heterostructure with a high transparency barrier

Within the framework of the self-consistent effective field approximation of the time-dependent perturbation theory, an influence of the electron tunneling on the spontaneously induced order parameters in a normal metal–superconductor hybrid structure is considered. For a normal-metal model, which does not take into account electron-electron scattering, as well as electron-phonon coupling, a critical barrier transparency, corresponding to the disappearance of superconductivity in the ground state, was obtained. The presence of incoherent excitations leads to a complex relationship between the effects of ordering, thermal fluctuations, and tunneling. Near the critical barrier transparency, this can stabilize a superconducting state in the certain temperature intervals. As a result, a reentrant superconductivity phenomenon was observed. The studied spectral properties of the hybrid structure reflect the existence of both coherent and incoherent elementary excitations.

The behaviors of the solutions of the quark gap equation in the NJL model with the self-consistent mean field approximation

In this paper, we use the self-consistent mean field approximation to study the Quantum Chromodynamics (QCD) phase transition. In the self-consistent mean field approximation of the Nambu–Jona-Lasinio (NJL) model, a parameter [Formula: see text] is introduced, which reflects the weight of “direct” channel and the “exchange” channel and needs to be determined by experiments (as mentioned in a recent work [T. Zhao, W. Zheng, F. Wang, C.-M. Li, Y. Yan, Y.-F. Huang and H.-S. Zong, Phys. Rev. D 100, 043018 (2019)], the results with [Formula: see text] are in good agreement with astronomical observation data on the latest binary neutron star merging. This indicates that the contribution of “exchange” channel should be considered, and [Formula: see text] is a possible choice). By comparing the results with different parameter [Formula: see text]’s ([Formula: see text], [Formula: see text] and [Formula: see text]), we study the influence of “exchange” channel on the behavior of the solutions of the quark gap equation and the critical point of chiral phase transition. Our results show that the second-order chiral phase turns to the crossover from the chiral limit to the non-chiral limit around [Formula: see text] in the case of [Formula: see text]. The difference of the quark mass with different [Formula: see text]’s mainly occurs in the intermediate temperatures for the different fixed chemical potentials. At zero temperature and the chemical potential [Formula: see text] there will be two solutions (including a meta-stable solution) of gap equation with [Formula: see text], and as [Formula: see text] increases it will be only one solution left (the meta-stable solution will disappear until [Formula: see text]). Besides, the discrepancy of the critical temperature (above which the pseudo-Wigner solution and negative Nambu solution will disappear) in the three cases of [Formula: see text] will become large when the chemical potential increases.

Superconducting fluctuations and giant negative magnetoresistance in a gate-voltage tuned two-dimensional electron system with strong spin-orbit impurity scattering

We present a quantitative theory of the gate-voltage tuned superconductor-to-insulator transition (SIT) observed experimentally in the 2D electron system created in the (111) interface between crystalline ${\mathrm{Sr}\mathrm{Ti}\mathrm{O}}_{3}$ and ${\mathrm{La}\mathrm{Al}\mathrm{O}}_{3}$. Considering two fundamental opposing effects of Cooper-pair fluctuations---the critical conductivity enhancement, known as paraconductivity, and its suppression associated with the loss of unpaired electrons due to Cooper-pairs formation---we generalize the standard thermal fluctuations theory to include interaction between fluctuations within the self-consistent field approximation and quantum tunneling between mesoscopic superconducting puddles within a phenomenological approach. Relying on the quantitative agreement found between our theory and a large body of experimental sheet-resistance data, we conclude that spin-orbit scatterings, via significant enhancement of the interaction between fluctuations, strongly enhance the sheet resistance peak at high fields and reveal anomalous metallic behavior at low fields, due to mixing of relatively heavy electron bands with a light electron band near a Lifshitz point. The large enhancement of the interaction between fluctuations at high fields, where the sheet resistance is strongly amplified, is shown to result in localization of Cooper-pair fluctuations within mesoscopic puddles.

QCD phase diagram at finite isospin and baryon chemical potentials with the self-consistent mean field approximation * *Supported by National Natural Science Foundation of China (12075117, 11535005, 11775118, 11690030, 11905104) and National Major state Basic Research and Development of China (2016YEF0129300)

The self-consistent mean field approximation of the two-flavor NJL model, with a free parameter to reflect the competition between the "direct" channel and the "exchange" channel, is employed to study the QCD phase structure at finite isospin chemical potential , finite baryon chemical potential and finite temperature T, and especially to study the location of the QCD critical point. Our results show that in order to match the corresponding lattice results of isospin density and energy density, the contributions of the "exchange" channel need to be considered in the framework of the NJL model, and a weighting factor should be taken. It is also found that for fixed isospin chemical potentials, the lower temperature of the phase transition is obtained with increasing in the plane, and the largest difference of the phase transition temperature with different 's appears at . At the temperature of the QCD critical end point (CEP) decreases with increasing , while the critical baryon chemical potential increases. At high isospin chemical potential ( MeV), the temperature of the QCD tricritical point (TCP) increases with increasing , and in the low temperature regions the system will transition from the pion superfluidity phase to the normal phase as increases. At low density, the critical temperature of the QCD phase transition with different 's rapidly increases with at the beginning, and then increases smoothly around MeV. In the high baryon density region, the increase of the isospin chemical potential will raise the critical baryon chemical potential of the phase transition.

Efficient implementation of isotropic cubic response functions for two-photon absorption cross sections within the self-consistent field approximation.

Within the self-consistent field approximation, computationally tractable expressions for the isotropic second-order hyperpolarizability have been derived and implemented for the calculation of two-photon absorption cross sections. The novel tensor average formulation presented in this work allows for the evaluation of isotropic damped cubic response functions using only ∼3.3% (one-photon off-resonance regions) and ∼10% (one-photon resonance regions) of the number of auxiliary Fock matrices required when explicitly calculating all the needed individual tensor components. Numerical examples of the two-photon absorption cross section in the one-photon off-resonance and resonance regions are provided for alanine-tryptophan and 2,5-dibromo-1,4-bis(2-(4-diphenylaminophenyl)vinyl)-benzene. Furthermore, a benchmark set of 22 additional small- and medium-sized organic molecules is considered. In all these calculations, a quantitative assessment is made of the reduced and approximate forms of the cubic response function in the one-photon off-resonance regions and results demonstrate a relative error of less than ∼5% when using the reduced expression as compared to the full form of the isotropic cubic response function.

Quantum hydrodynamics description of spin-1 Bose–Einstein condensates

We develop a novel model of the magnetized spin-1 Bose–Einstein condensate (BEC) of neutral atoms, using the method of many-particle quantum hydrodynamic (QHD) and propose an original derivation of the system of continual equations. We consider bosons with a spin-spin interaction and a short range interaction in the first order in the interaction radius, on the of basis of the self-consistent field approximation of the QHD equations. We demonstrate that the dynamics of the fluid velocity and magnetization is determined by a nontrivial modification of the Euler and Landau–Lifshitz equation, and show that a nontrivial modification of the spin density evolution equation contains the spin torque effect that arises from the self-interactions between spins of the bosons. The properties of the dispersion spectrum of collective excitations are described. We obtain the new contribution of the self-interaction of spins in the spin wave spectrum together with the influence of an external magnetic field and spin-spin interactions between polarized particles. The anisotropic spin wave instability is predicted.

On the stability of two-flavor and three-flavor quark matter in quark stars within the framework of NJL model

Following our recently proposed self-consistent mean field approximation approach, we have done some researches on the chiral phase transition of strong interaction matter within the framework of Nambu-Jona-Lasinio (NJL) model. The chiral susceptibility and equation of state (EOS) are computed in this work for both two-flavor and three-flavor quark matter for contrast. The Pauli–Villars scheme, which can preserve gauge invariance, is used in this paper. Moreover, whether the three-flavor quark matter is more stable than the two-flavor quark matter or not in quark stars is discussed in this work. In our model, when the bag constant are the same, the two-flavor quark matter has a higher pressure than the three-flavor quark matter, which is different from what Witten proposed in his pioneering work.

Theoretical study of the effect of the Coulomb blockade and Kondo resonance in the density of states using self-consistent field approximation

In a recent paper, we theoretically investigated the density of states of the composite channel–contact system in the Coulomb and Kondo regimes using the self-consistent field approximation. There are the main experimental observations of vibration features in the Coulomb blockade [H. Park et al., Nature (London) 407, 57 (2000)] and Kondo [L. H. Yu et al., Phys. Rev. Lett. 93, 266802 (2004)] regimes. In the Kondo regime, our results show that one peak at E=μ can be observed in the density of states at low temperatures (0.0026 eV ≤ kBT ≤ 0.0000026 eV). Also, the real part of ∑3 has one minimum peak at E=−μ and the real par of ∑2 has one maximum peak at E=−μ for 0.01 ≤ μ ≤ 0.07 in the Kondo regime at low temperatures.

Nonlinear dielectric relaxation of polar molecules with linear reaction dynamics in self-consistent field approximation

The nonlinear dielectric relaxation of polar molecules with annihilation reaction in the self-consistent field is studied according to the reaction–diffusion equation. The general nonlinear responses of three kinds of reaction systems, the noninteracting system, the polarizable system and the interacting system, are evaluated via the perturbation theory. In these three systems, the response is strongly dependent on the reaction dynamics due to the variation of the total number of polar molecules in the chemical reaction process. In particular, the analytic equations of nonlinear polarization and loss tangent in the noninteracting system and the polarizable system are also derived in the time-harmonic electric field. Diagrams of the distribution functions are presented showing the effect of the electric field, the rate constant, the anisotropic polarizability and the interaction between molecules.

Dynamics and coupling of fractional-order models of the motor cortex and central pattern generators

Objective. Fractional calculus plays a key role in the analysis of neural dynamics. In particular, fractional calculus has been recently exploited for analyzing complex biological systems and capturing intrinsic phenomena. Also, artificial neural networks have been shown to have complex neuronal dynamics and characteristics that can be modeled by fractional calculus. Moreover, for a neural microcircuit placed on the spinal cord, fractional calculus can be employed to model the central pattern generator (CPG). However, the relation between the CPG and the motor cortex is still unclear. Approach. In this paper, fractional-order models of the CPG and the motor cortex are built on the Van der Pol oscillator and the neural mass model (NMM), respectively. A self-consistent mean field approximation is used to construct the potential landscape of the Van der Pol oscillator. This landscape provides a useful tool to observe the 3D dynamics of the oscillator. To infer the relation of the motor cortex and CPG, the coupling model between the fractional-order Van der Pol oscillator and the NMM is built. As well, the influence of the coupling parameters on the CPG and the motor cortex is assessed. Main results. Fractional-order NMM and coupling model of the motor cortex and the CPG are first established. The potential landscape is used to show 3D probabilistic evolution of the Van der Pol oscillator states. Detailed observations of the evolution of the system states can be made with fractional calculus. In particular, fractional calculus enables the observation of the creation of stable modes and switching between them. Significance. The results confirm that the motor cortex and CPG have associated modes or states that can be switched based on changes in the fractional order and the time delay. Fractional calculus and the potential landscape are helpful methods for better understanding of the working principles of locomotion systems.

Self-consistent mean field approximation and application in three-flavor NJL model

In this study, we apply a self-consistent mean field approximation of the three-flavor Nambu–Jona-Lasinio (NJL) model and compare it with the two-flavor NJL model. The self-consistent mean field approximation introduces a new parameter, , that cannot be fixed in advance by the mean field approach itself. Due to the lack of experimental data, the parameter, , is undetermined. Hence, it is regarded as a free parameter and its influence on the chiral phase transition of strong interaction matter is studied based on this self-consistent mean field approximation. affects numerous properties of the chiral phase transitions, such as the position of the phase transition point and the order of phase transition. Additionally, increasing will decrease the number densities of different quarks and increase the chemical potential at which the number density of the strange quark is non-zero. Finally, we observed that affects the equation of state (EOS) of the quark matter, and the sound velocity can be calculated to determine the stiffness of the EOS, which provides a good basis for studying the neutron star mass-radius relationship.