Compact variational wave functions for bound states in three-electron atomic systems

The variational procedure to construct compact and accurate wave functions for three-electron atoms and ions is developed. The procedure is based on the use of six-dimensional Gaussoids written in the relative four-body coordinates r ₁₂, r ₁₃, r ₂₃, r ₁₄, r ₂₄, and r ₃₄. The nonlinear parameters in each basis function have been carefully optimized. Using these variational wave functions, we have determined the energies and other bound state properties for the ground 1² S-states in a number of three-electron atoms and ions. The three-electron atomic systems considered in this work include the neutral Li atom and nine positively charged lithiumlike ions: Be⁺, B²⁺, C³⁺, ..., Na⁸⁺, and Mg⁹⁺. Our variational wave functions are used to determine the hyperfine structure splitting and field shifts for some lithium-like ions. The explicit formulas of the Q ⁻¹ expansion are derived for the total energies of these three-electron systems. The article is published in the original.     

Semi-exponential basis for highly accurate computations of three-electron atomic systems

An advanced basis set of semi-exponential radial functions is proposed and applied to highly accurate computations of three-electron atomic systems. This set is found to be significantly more effective in highly accurate, bound state computations of three-electron atomic systems, than currently used set of Hylleraas radial functions.     

Calculations of magnetic moments for three-electron atomic systems.

The first fully correlated calculations of the magnetic moment in lithium are presented. Relative to the free-electron value, the Zeeman gJ factor for the ground state lithium gJ/g(e)-1 is calculated to a computational accuracy of 200 parts in 10(9), including relativistic and radiative corrections of orders alpha2, alpha2m/M, and alpha3. The isotope shifts in gJ are predicted precisely for various isotopes. The extensions to the first excited S state of lithium and the ground state of Be+ are made.     

Criterion for the accuracy of wave functions of asymmetric atomic and molecular systems

From the condition for invariance of the energy under rotations of the Jacobi coordinates, a relationship is obtained that relates the expected values of combinations of degrees of interparticle separations in quantum-mechanical systems. This relationship, which is satisfied for atoms and molecules with like particles identically, because of the permutational symmetry, is an extremely sensitive criterion for the accuracy of the wave functions of asymmetric systems, consisting of unlike particles and having no permutational symmetry. The application of this criterion is illustrated for the example of the asymmetric ⁴Heμ^?e^? mesoatom.     

Synchronization of different chaotic systems via active radial basis functions sliding mode controller

This paper presents a new method to synchronize different chaotic systems with disturbances via an active radial basis function （RBF） sliding controller. This method incorporates the advantages of active control, neural network and sliding mode control. The main part of the controller is given based on the output of the RBF neural networks and the weights of these single layer networks are tuned on-line based on the sliding mode reaching law. Only several radial basis functions are required for this controller which takes the sliding mode variable as the only input. The proposed controller can make the synchronization error converge to zero quickly and can overcome external disturbances. Analysis of the stability for the controller is carried out based on the Lyapunov stability theorem. Finally, five examples are given to illustrate the robustness and effectiveness of the proposed synchronization control strategy.     

Configuration interaction for wave functions constructed from orthogonal many-electron group orbitals

Acta physica Academiae Scientiarum Hungaricae -     

Comparison of methods of calculating atomic radial distribution functions

Tikhonov's method of weak and uniform regularization is used to solve the Fredholm integral equation of the first kind applied to the diffraction study of the structure of disordered condensed phases. The results of the calculation of the atomic radial distribution function are compared with those using numerical Fourier transforms for the case of liquid gallium.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 62–67, February, 1982.     

Global properties of radial wave functions in Schwarzschild's space-time

Two solutions ₅(x, x _s) and ₆(x, x _s) related to the irregular singular point atx=+ of the radial wave equation in Schwarzschild's space-time are studied as functions of the independent variablex and the parameterx _s. Analytic continuations of ₅ and ₆ are derived and their relation to the flat-space case solutions is established. Explicit expressions for ₃(x, x _s) and ₄(x, x _s) (the solutions about the regular singular point atx=x _s) are given. From these expressions and the analytic continuations of ₅ and ₆ the coefficients relating linearly ₅ and ₆ with _i (i=1, 2, 3, 4) are calculated.     

Global properties of radial wave functions in Schwarzschild's space-time

The radial factorR(x, x _s ) of a scalar field in Schwarzschild's space-time satisfies a second order ordinary differential equation with two regular singular points atx=0 andx=x _s and one irregular singular point atx=∞. The analytical properties of four solutions ?₁, ?₂, ?₃, and ?₄ (defined by their power series expansions aboutx=0 andx=x _s ) with respect tox _s are studied. An analytical continuation is given for each solution outside its circle of convergence. Relations to the flat-space solutions are established. Finally the coefficients relating linearly any three of these solutions are determined and studied as functions of the parameterx _s .     

High harmonic generation and the role of atomic orbital wave functions

High harmonic spectra were recorded from different rare-gas atoms under identical experimental conditions. It is shown that although each atom's spectrum is different, the differences are due almost entirely to the orbital influence in the recombination step. The amplitude of the continuum electron wave packet versus kinetic energy is derived from these data and is shown to be largely independent of the atom, in agreement with models of tunnel ionization. We compare the measurements with calculations in both the length gauge and the velocity gauge and show that the two gauges imply a different de Broglie wavelength.     

JWKB connection for radial wave functions and bound-state/free-state duality

The oscillatory and non-oscillatory regions of radial wave functions are connected by the use of spherical Bessel functions or Coulomb wave functions. The resulting formulae for the magnitude of the wave function at the origin in terms of the energy spectrum are used to exhibit the appropriate form of boundstate/free-state duality.     

Smoothed wave functions of chaotic quantum systems

It is shown that wave functions of quantum systems as ħ → 0 have an extra density near unstable periodic trajectories of the classical problem. The averaged wave function square is represented as the sum over a finite number of periodic trajectories. The contribution of each trajectory is expressed through the elements of the monodromy matrix of the trajectory. The results are compared with the numerical calculations of the wave functions for the stadium billiard.     

Relativistic recoil and spin corrections to atomic and Coulomb wave functions

A new wave equation for the relativistic two-body problem is derived and shown to be equivalent to the Breit equation and also to the “quasipotential” equation. Its unique advantage is that it provides analytic wave functions for arbitrary combinations of masses, spins, magnetic and higher moments of the two particles. Various applications are indicated, including an improvement of Dirac wave functions at small distances.     

Water vapor line width and shift calculations with accurate vibration-rotation wave functions

Calculations of H₂¹⁶O rotation-vibration line broadening and shifting due to N₂ pressure effects are performed using a semi-empirical approach. The calculations are based on impact theory modified by introducing additional parameters to extend the use of empirical data. These model parameters are determined by fitting the broadening and shifting coefficients to experimental data. The method is further developed by using anharmonic wavefunctions in the estimates of the line profiles. The main feature of the present calculation is the use of a complete set of high-accuracy vibration-rotation dipole transition moments calculated for all possible transitions using wavefunctions determined from variational nuclear motion calculations and an ab initio dipole moment surface. This approach explicitly takes into account all scattering channels induced by collisions. Results of these calculations clearly demonstrate improved agreement between observed and calculated parameters for both the line widths and the line shifts.     

Linearized augmented plane wave method utilizing the quadratic energy expansion of radial wave functions

Conclusions The developed version of the augmented plane wave method yields eigenvalues and eigenfunctions with higher accuracy than the standard LAPW, preserving its computational efficiency.The approximation of the exact radial solution by the Taylor expansion involving also the second energy derivative of the radial function, except for the first derivative, has two advantages. First, the energy dependence of the logarithmic derivatives atr=R is better described and, therefore, the method is less sensitive to the choice of the centre of expansionE ₀ or, equivalently, acceptably accurate eigenvalues are obtained for the broader energy region aroundE ₀. The other and probably more important advantage is that the approximate radial solutions are remarkably closer to the exact radial functions inside the muffin-tin sphere. This can be of use when evaluating the measurable quantities depending on the wave functions.     

Construction scheme of four-body wave functions in a hyperspherical basis

The problem of symmetrization of four-body hyperspherical functions is discussed. A construction scheme is given for the symmetrized hyperspherical asis in the (3+1)-configuration. The transformation between hyperspherical functions corresponding to the (2+2)- and (3+1)-configurations is considered.     

Application of radial basis functions to evolution equations arising in image segmentation

In this paper, radial basis functions are used to obtain the solution of evolution equations which appear in variational level set method based image segmentation. In this method, radial basis functions are used to interpolate the implicit level set function of the evolution equation with a high level of accuracy and smoothness. Then, the original initial value problem is discretized into an interpolation problem. Accordingly, the evolution equation is converted into a set of coupled ordinary differential equations, and a smooth evolution can be retained. Compared with finite difference scheme based level set approaches, the complex and costly re-initialization procedure is unnecessary. Numerical examples are also given to show the efficiency of the method.