Ground State Energies of Helium-Like Ions Using a Simple Parameter-Free Matrix Method

This study aims to use hydrogenic orbitals within an analytic and numeric parameter-free truncated-matrix method to solve the projected Schrödinger equation of some Helium-like ions (3 ≤ Z ≤ 10). We also derived a new analytical expression of the ion ground state energies, which was simple and accurate and improved the accuracy of the analytic calculation, numerically using Mathematica. The standard matrix method was applied, where the wave function of the ions was expanded in a finite number of eigenvectors comprising hydrogenic orbitals. The Hamiltonian of the systems was calculated using the wave function and diagonalized to obtain their ground state energies. The results showed that a simple analytic expression of the ground state energies of He-like ions was successfully derived. Although the analytic expression was derived without involving any variational parameter, it was reasonably accurate with a 0.12% error for Ne8+ ion. From this method, the accuracy of the analytic energies was also numerically improved to 0.10% error for Ne8+ ion. The results clearly showed that the energies obtained using this method were more accurate than the hydrogenic perturbation theory and the uncertainty principle-variational approach. In addition, for Z > 4, our results were more accurate than those from the geometrical model.

Bethe ansatz on a quantum computer?

We consider the feasibility of studying the anisotropic Heisenberg quantum spin chain with the Variational Quantum Eigensolver (VQE) algorithm, by treating Bethe states as variational states, and Bethe roots as variational parameters. For short chains, we construct exact one-magnon trial states that are functions of the variational parameter, and implement the VQE calculations in Qiskit. However, exact multi-magnon trial states appear to be out out of reach.

Hartree–Fock implementation using a Laguerre-based wave function for the ground state and correlation energies of two-electron atoms

An implementation of the Hartree–Fock (HF) method using a Laguerre-based wave function is described and used to accurately study the ground state of two-electron atoms in the fixed nucleus approximation, and by comparison with fully correlated (FC) energies, used to determine accurate electron correlation energies. A variational parameter A is included in the wave function and is shown to rapidly increase the convergence of the energy. The one-electron integrals are solved by series solution and an analytical form is found for the two-electron integrals. This methodology is used to produce accurate wave functions, energies and expectation values for the helium isoelectronic sequence, including at low nuclear charge just prior to electron detachment. Additionally, the critical nuclear charge for binding two electrons within the HF approach is calculated and determined to be Z HF C =1.031 177 528. This article is part of the theme issue ‘Modern theoretical chemistry’.

Two-Parameter Harmonic Oscillator Expansion Method for Calculating Non-Relativistic Ground State Energy of Coulomb Three-Particle Systems with Two Identical Particles

The variational method in oscillator representation with individual parameters for each Jacobi coordinate is applied to the non-relativistic calculation of the ground state energy of a number of three-particle Coulomb systems, consisting of two identical particles and a different one. The accuracy and convergence rate of the calculations in the constructed oscillator basis are studied up to a total of 28 oscillator quanta. The results are compared with those of the traditional approach using only one such nonlinear variational parameter. The method with individual parameters for Jacobi coordinates is found to possess a number of advantages as compared to the traditional approach.

Application of optimization method to the x4 model in the Tsallis nonextensive statistics

We study the effects of the environment described by the Tsallis nonextensive statistics on physical quantities using an optimization method in the case of small deviation from the Boltzmann–Gibbs statistics. The x4 model is used and the density operator is restricted to be a Gaussian form. The variational parameter is the frequency Ω of a harmonic oscillator in the optimization method. We obtain an approximate expression of free energy and of the expectation value of βmΩ2 x2/2, where β is the inverse of the temperature T and m is the mass. The optimized frequency is estimated numerically. The expectation value of βmΩ2 x2/2 and the derivative of the energy with respect to the temperature [Formula: see text] are also calculated numerically. The effects of the Tsallis nonextensive statistic in the case of small deviation from the Boltzmann–Gibbs statistics are found: (1) the frequency modulation, (2) the variation of the expectation value of βm Ω2 x2/2 and (3) the variation of [Formula: see text].

A Coons Patch Spanning a Finite Number of Curves Tested for Variationally Minimizing Its Area

In surface modeling a surface frequently encountered is a Coons patch that is defined only for a boundary composed offouranalytical curves. In this paper we extend the range of applicability of a Coons patch by telling how to write it for a boundary composed of an arbitrary number of boundary curves. We partition the curves in a clear and natural way into four groups and then join all the curves in each group intooneanalytic curve by using representations of the unit step function including one that isfully analytic. Having a well-parameterized surface, we do some calculations on it that are motivated by differential geometry but give a better optimized and possibly more smooth surface. For this, we use an ansatz consisting of the original surface plus a variational parameter multiplying the numerator part of its mean curvature function and minimize with the respect to it the rms mean curvature and decrease the area of the surface we generate. We do a complete numerical implementation for a boundary composed of five straight lines, that can model a string breaking, and get about 0.82 percent decrease of the area. Given the demonstrated ability of our optimization algorithm to reduce area by as much as 23 percent for a spanning surface not close of being a minimal surface, this much smaller fractional decrease suggests that the Coons patch we have been able to write is already close of being a minimal surface.

Assimilation of Satellite Cloud Data into the GMAO Finite-Volume Data Assimilation System Using a Parameter Estimation Method. Part I: Motivation and Algorithm Description

General circulation models are unable to resolve subgrid-scale moisture variability and associated cloudiness and so must parameterize grid-scale cloud properties. This typically involves various empirical assumptions and a failure to capture the full range (synoptic, geographic, diurnal) of the subgrid-scale variability. A variational parameter estimation technique is employed to adjust empirical model cloud parameters in both space and time, in order to better represent assimilated International Satellite Cloud Climatology Project (ISCCP) cloud fraction and optical depth and Special Sensor Microwave Imager (SSM/I) liquid water path. The value of these adjustments is verified by much improved cloud radiative forcing and persistent improvement in cloud fraction forecasts.