Self-consistent Field Potential Research Articles

INFLUENCE OF ION VISCOSITY ON THE DISTRIBUTION OF PARAMETERS IN THE SHEATH AT THE BOUNDARY OF A STATIONARY WEAKLY IONIZED STRONGLY NONISOTHERMAL PLASMA

A stationary weakly ionized highly nonisothermal plasma is considered in the hydrodynamic approximation. Taking into account the effects of ionization, recharging, and a self-consistent field, the effect of ion viscosity on the distribution of plasma discharge parameters in the sheath was investigated. Distributions of hydrodynamicion velocity and ion density, electron density, and self-consistent field potential in the sheath were obtained.

Energy Landscapes for Electronic Structure

Orbital-optimized multiple self-consistent-field (SCF) solutions are increasingly being interpreted as mean-field approximations of diabatic or excited electronic states. However, surprisingly little is known about the topology of the electronic energy landscape from which these multiple solutions emerge. In this contribution, we extend energy landscape methods, developed for investigating molecular potential energy surfaces, to investigate and understand the structure of the electronic SCF energy surface. Using analytic gradients and Hessians, we systematically identify every real SCF minimum for the prototypical H4 molecule with the 3-21G basis set, and the index-1 saddles that connect these minima. The resulting SCF energy landscape has a double-funnel structure, with no high-energy local minima. The effect of molecular symmetry on the pathways is analyzed, and we demonstrate how the SCF energy landscape changes with the basis set, SCF potential, molecular structure, and spin state. These results provide guiding principles for the future development of algorithms to systematically identify multiple SCF solutions from an orbital optimization perspective.

Nonlocal statistical field theory of dipolar particles forming chain-like clusters

We present a nonlocal statistical field theory of a diluted solution of dipolar particles which are capable of forming chain-like clusters in accordance with the ’head-to-tail’ mechanism. As in our previous study [Yu.A. Budkov 2018 J. Phys.: Condens. Matter 30 344001], we model dipolar particles as dimers comprised of oppositely charged point-like groups, separated by fluctuating distance. For the special case of the Yukawa-type distribution function of distance between the charged groups of dipolar particles we obtain an analytical expression for the electrostatic free energy of solution within the random phase approximation. We show that an increase in the association constant leads to a decrease in the absolute value of the electrostatic free energy of solution, preventing its phase separation which is in agreement with the former computer simulations and theoretical results. We obtain a non-linear integro-differential equation for the self-consistent field potential created by the fixed external charges in a solution medium, taking into account the association of dipolar particles. As a consequence of the derived self-consistent field equation, in regime of weak electrostatic interactions, we obtain an analytical expression for the electrostatic potential of the point-like test ion, surrounded by the chain-like clusters of the dipolar particles. We show that in the mean-field approximation the association does not change the bulk dielectric permittivity of the solution, but increases the solvation radius of the point-like charge, relative to the theory of non-associating dipolar particles.

A tiered approach to Monte Carlo sampling with self-consistent field potentials

A "tiered" approach to Monte Carlo sampling of nuclear configurations is presented for ab initio, self-consistent field (SCF)-based potentials, including Hartree-Fock and density functional theory. Rather than Metropolis testing only the final SCF energy, individual cycle energies are tested in a tiered fashion, without approximation. Accordingly, rejected configurations are terminated early in the SCF procedure. The method is shown to properly obey detailed balance, and effective modifications are presented for cases in which the initial SCF guess is particularly poor. Demonstrations on simple systems are provided, including an assessment of the thermal properties of the neutral water dimer with B3LYP/6-31++G**. Cost analysis indicates a factor-of-two reduction in SCF cycles, which makes the method competitive with accelerated molecular dynamics sampling techniques, without the need for forces.

Variational approach to the average-atom-in-jellium and superconfigurations-in-jellium models with all electrons treated quantum-mechanically

Models of average-atom- and superconfigurations-in-plasmas with full quantum description of all electrons are important in opacity and EOS calculations, especially in case of high density plasmas in which pressure ionization phenomena can occur. However, variational formulation of models becomes difficult when quantum free electrons are considered. In our approach we use a cluster expansion of the free energy expression from which we retain the sum of the zero and of the first order terms. In the zero order the free energy is that of the homogeneous jellium of an unknown free electron density. In the first order we consider a one self-consistent-field (SCF) average atoms or ions submerged in this jellium. The SCF potential is assumed to be localized. We consider the first order atoms or ion in the whole space without imposing the neutrality of the Wigner–Seitz cell. The important part of the model is a relation between the average ion charge and the localized part of electron density that is called “ionization model”. The minimization procedure of the free energy is performed with respect to all variables except these characterizing the plasma equilibrium, i.e. except the temperature, the ion density and the atomic number. It is interesting that when the electron density is taken in the Thomas–Fermi (TF) approximation together with a specific choice of the ionization model, our approach correctly leads to the classical TF ion-in-cell average atom. It also appears that the same ionization model has to be used in the quantum case and that other ionization models lead to unphysical solutions. This observation is important with respect to the choice of correct ionization model in the case of superconfigurations-in-jellium. Due to the fully variational character of our approach the resulting expression for the thermodynamic pressure in all considered cases is simple and does not require any numerical differentiation.

Dipole and Deformation Dipole Polarization of Ions in Rock-Salt Structure Crystals

The comparative study is done on the dipole and deformation dipole polarization. With the local density approximation, the values of the dipole polarizability and the deformation dipole tensor of the ions in the rock-salt structure alkali iodide crystals are calculated. The Szigeti effective charge is obtained from the deformation dipole tensor. The crystalline potential is treated by the spherical solid model. The response to the perturbation is calculated by the self-consistent field potential. The calculated results are compared with those in other alkali halide crystals. The ease of the ionic migration is discussed by using the deformation dipole tensor.

NEXAFS experiment and multiple scattering calculations onKO2:Effects on the π resonance in the solid phase

The high-energy resolution O K-edge absorption near-edge x-ray absorption fine structure spectrum has been measured for in situ prepared potassium superoxide. The experimental data have been analyzed in detail by multiple scattering calculations using self-consistent field potentials. In particular, the so-called \ensuremath{\pi} resonance at the rising edge, which presents a double-peak structure, has been totally resolved and reproduced by the calculations. This analysis indicates that the grown material is arranged in a ${\mathrm{KO}}_{2}$ structure with an O-O distance between 1.31 and 1.34 \AA{}. Moreover, the calculation demonstrates both a complete ionic character of the bound between the ${\mathrm{O}}^{2\ensuremath{-}}$ anion and K atoms and a strong interaction between the anion and solid-state matrices.

Theory of the three-photon entangled state

Based on the multiple-scattering X alpha method, the photoabsorption spectra near the 1s threshold of CH4, NH3, H2O, and HF with ten electrons, the same as the Ne atom, are calculated. The calculated near-threshold spectra are in good agreement with the experimental spectra. With the calculated results we can provide cleat theoretical assignments for the recent highly resolved experimental spectra. In order to elucidate the attractive effect of the localized inner-shell core hole, we have adopted the ground state self-consistent field potential, the transition state model potential, and the Z + 1 model potential, respectively. The results of the Z + 1 model are in better agreement with the experimental measurements. Along the isoelectronic molecules, thr antibonding are also the initial of the corresponding Rydberg series and can be regarded as orbitals (i.e., mixed valence and Rydberg orbitals). It is interesting that the sums of the oscillator strengths of such precursor are approximately proportional to the numbers of the antibonding orbitals.

Near-threshold structures in inner-shell photoabsorption processes ofCH4,NH3,H2O, and HF

Based on the multiple-scattering $X\ensuremath{\alpha}$ method, the photoabsorption spectra near the $1s$ threshold of ${\mathrm{CH}}_{4},$ ${\mathrm{NH}}_{3},$ ${\mathrm{H}}_{2}\mathrm{O}$, and HF with ten electrons, the same as the Ne atom, are calculated. The calculated near-threshold spectra are in good agreement with the experimental spectra. With the calculated results we can provide clear theoretical assignments for the recent highly resolved experimental spectra. In order to elucidate the attractive effect of the localized inner-shell core hole, we have adopted the ground state self-consistent field potential, the transition state model potential, and the $Z+1$ model potential, respectively. The results of the $Z+1$ model are in better agreement with the experimental measurements. Along the isoelectronic molecules, the antibonding orbitals are also the initial orbitals of the corresponding Rydberg series and can be regarded as ``precursor orbitals'' (i.e., mixed valence and Rydberg orbitals). It is interesting that the sums of the oscillator strengths of such precursor orbitals are approximately proportional to the numbers of the antibonding orbitals.

Generalization of the Hartree-Fock approach to collision processes

The conventional Hartree and Hartree-Fock approaches for bound states are generalized to treat atomic collision processes. All the single-particle orbitals, for both bound and scattering states, are determined simultaneously by requiring full self-consistency. This generalization is achieved by introducing two Ans\"atze: (a) the weak asymptotic boundary condition, which maintains the correct scattering energy and target orbitals with correct number of nodes, and (b) square integrable amputated scattering functions to generate self-consistent field (SCF) potentials for the target orbitals. The exact initial target and final-state asymptotic wave functions are not required and thus need not be specified a priori, as they are determined simultaneously by the SCF iterations. To check the asymptotic behavior of the solution, the theory is applied to elastic electron-hydrogen scattering at low energies. The solution is found to be stable and the weak asymptotic condition is sufficient to produce the correct scattering amplitudes. The SCF potential for the target orbital shows the strong penetration by the projectile electron during the collision, but the exchange term tends to restore the original form. Potential applicabilities of this extension are discussed, including the treatment of ionization and shake-off processes.

Real‐space multigrid solution of electrostatics problems and the Kohn–Sham equations

A multigrid method for numerically solving electrostatics and quantum chemical problems in real space is discussed. Multigrid techniques are used to solve both the linear Poisson equation and the nonlinear Kohn–Sham and Poisson–Boltzmann equations. The electrostatic potential, Laplacian, charge densities (electrons and nuclei), Kohn–Sham DFT orbitals, and the self-consistent field potential are all represented discretely on the Cartesian grid. High-order finite differences are utilized to obtain physically reasonable results on modestly sized grids. The method is summarized and numerical results for all-electron atomic and molecular structure are presented. The strengths and weaknesses of the method are discussed with suggested directions for future developments, including a new high-order conservative differencing scheme for accurate composite grid computations which preserves the linear scaling property of the multigrid method. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 477–486, 1997

Dynamical and Nondynamical Correlation

The variation of correlation energies with bond distances of various first row diatomic molecules has been studied. Self-consistent field and complete active space self-consistent field potential curves of these molecules have been calculated. Exact potential energy curves are constructed from experimental data using the Rydberg−Klein−Rees method. With appropriate definitions, the dynamical and nondynamical correlation energies are obtained and the variation of these with bond distance is calculated. Two definitions of nondynamical correlation are examined. Classifying the angular correlation as dynamical seems to be a better way to partition the correlation energy. The correlation functionals of density functional theory, VWN, LYP, and P86, are also evaluated and compared with the ab initio dynamical correlation energies. LYP appears to give the closest agreement with the dynamical correlation energy.

Dilatation transformation and sum rules for general potentials including self-consistent field potentials

The eigenfunctions and energies of general dilated Hamiltonians are expanded in powers of the dilatation parameter. These expansions, augmented by stationarity and stability conditions, are used to derive exact sum rules for bound and resonance states. Particular attention is paid to Hamiltonians with potentials which depend on external parameters, such as the nuclear coordinates in molecules, and to self-consistent potentials. The sum rules can be employed in practical computations to improve the quality of the results and may also serve in analyzing the results from approximate calculations.

Analytical expressions of potentials and wave functions for atoms and ions

A method to calculate the self-consistent field (SCF) potential and wave functions for atoms and ions in analytical forms is described. The numerical SCF potential for atoms and ions is fitted to an analytical function by the use of the nonlinear least-squares method. The SCF wave functions are obtained numerically by solving the Schrödinger equation with the analytical SCF potential. The analytical wave functions are calculated with the nonlinear least-squares fitting technique. The obtained results are tested by comparing energy eigenvalues and radiative transition probabilities with those from numerical calculations.

Theoretical Calculation of Differential Cross Section for e+-Ar and e--Ar Elastic Scattering

The differential cross sections for e--Ar and e+-Ar elastic scattering are calculated by relativistic partial wave expansion method. In the processes an incident particle (electron or positron) is treated as moving relativistically in an effective central potential. For electron-atom scattering, the effective potential is calculated by relativistic self-consistent-field (SCF) method with the Slater local exchange approximation. For the positron scattering, the effective potential is the electrostatic potential determined by the ground-state electron charge density calculated by the SCF method. For atomic target polarization due to the incident charge particle(electron or positron), the effect is treated approximately by adding a static polarization potential to the SCF potentials. The calculated results show that the polarization effect is very important for the low energy positron-atom scattering.

Topological analysis of the charge density response of a Ni4 cluster to a probe H2 molecule

Local density self-consistent field (SCF) discrete variationalXα calculations are performed on a Ni4 tetrahedron interacting with a probe H2 molecule in special geometries. Optimized basis functions generated from the spherically averaged SCF potential are used. Topological charge-density analyses and binding energy calculations are used to study a portion of the energy surface for the approach of the H2 molecule toward the Ni4 tetrahedron. The effect of the H2 molecule on Ni-Ni, Ni-H bonds and changes in the H-H covalent bond are investigated with the help of the ∇ρ field and various data at its critical points. The qualitative relationship between these data and the calculated binding energies is exploited.

Ground-state potential curves forAl2andAl26+in the repulsive region

Self-consistent-field (SCF) calculations have been performed for the short-range interaction between two aluminum atoms. The ground-state potential curve (/sup 3/..sigma../sup -//sub g/..-->../sup 5/..sigma../sup -//sub u/..-->../sup 3/..sigma../sup +//sub u/..-->../sup 1/..sigma../sup +//sub g/ ..-->../sup 3/Pi/sub g/) is presented for internuclear separations between R/sub e/ and 0.1 a.u. (repulsive energies up to approx.10/sup 3/ a.u.). Basis sets consist of scaled even-tempered Slater orbitals of double-zeta quality, augmented by diffuse functions, with the addition of united or semiunited atom-basis sets centered on the bond. The SCF potentials are compared with Thomas-Fermi theory and with the electron-gas calculations of Wilson, Haggmark, and Biersack. A kink in the SCF screening function is found near R=2 a.u. This feature is believed related to changes in the ground-state configuration that occur in this region. Additional calculations (SCF and linear combination of atomic orbitals were performed on Al/sub 2//ts/sup 6 +// to determine individually the core overlap and the valence-electron contributions to the interatomic potential. The core-overlap interaction is compared with Gordon-Kim electron-gas calculations. Schematic calculations were also performed on the Al-Al/sup 3 +/ system to estimate the interaction between an atom and an energetic ion.

Vertical electronic spectrum of CS molecule

Results from non-empirical self-consistent field (SCF) and configuration interaction (CI) calculations on the ground state and low-lying valence and Rydberg states of CS molecule are reported. SCF potential energy curves obtained in the region 2.50 bohr to 3.70 bohr are given. Configuration interaction calculations at a CS distance 2.90 bohr show that the resulting term series including the Ĩ 3Σ −(π → π *) agree well with the experimental ordering of terms. The reassignment of the third triplet by Robbe and Schamps to 3Δ(π → π *) instead of 3Π, as originally proposed by Field and Bergeman, is ascertained. The transition 39138 cm −1 observed by Field and Bergeman is excluded to be X̃ 1Σ* → d 3Δ.

Studies in Perturbation Theory. V. Some Aspects on the Exact Self-Consistent Field Theory

The independent-particle model in the theory of many-particle systems is studied by means of the self-consistent-field (SCF) idea. After a review of the characteristic features of the Hartree and Hartree-Fock schemes, the extension of the SCF method developed by Brueckner is further refined by introducing the exact reaction operator containing all correlation effects. This operator is here simply defined by means of the partitioning technique, and, if the SCF potentials are derived from this operator, one obtains a formalism which is completely analogous to the Hartree scheme but which still renders the exact energy and the exact wave function. An elementary derivation of the linked-cluster theorem is given, and finally the inclusion of various symmetry properties is discussed.