Hyperspherical Approach Research Articles

Highly excited states for the helium atom in the hyperspherical adiabatic approach

The hyperspherical adiabatic approach is used to obtain the highly excited series and of the helium atom. The introduction of appropriate asymptotic conditions at large values of the hyperspherical radius results in a stable algorithm that allows the calculation of the full atomic spectrum with precision of a few parts per million. Comparison with the variational calculations available in the literature shows that the accuracy of the results improves with increasing principal quantum number. We present the energies up to n = 31 which is the typical value used in multiphoton excitation experiments.

Quantum Reaction Dynamics of Heavy−Light−Heavy Systems: Reduction of the Number of Potential Curves and Transitions at Avoided Crossings

Two decoupling procedures are proposed within the framework of the hyperspherical coordinate approach in order to reduce the number of states in the close-coupling calculations and to clarify the reaction mechanisms. Sharply avoided adiabatic states are diabatically connected and relabeled without any diabatic coupling there. This is named diabatic decoupling and is useful for decoupling two manifolds of states belonging to different categories. Furthermore, the number of states is reduced in such a way that only a limited number of adjacent adiabatic states are taken into account in the close-coupling calculations for each relevant state. This is called adiabatic decoupling scheme. These reductions of the number of states enable us to analyze reactions in terms of nonadiabatic transitions at avoided crossings among the small number of adiabatic potential curves. The method is applied to O(3P) + HCl → OH + Cl and Cl + HCl → HCl + Cl. The idea of vibrationally nonadiabatic transitions at avoided crossings together with the concept of potential ridge introduced in our previous paper can untangle the congested potential curves and clarify reaction mechanisms.

Hyperspherical adiabatic approach for excitons bound to ionized donors in semiconductors

Binding energies for excitons trapped by impurities in ZnSe and CdS are calculated within the hyperspherical adiabatic approach. Nonadiabatic couplings are included in the radial equations, leading to energies lower than the variational values available in the literature. Resonant states similar to autoionizing lines in atoms are predicted to lie above the first electron-impurity ionization threshold.

Hyperquantization algorithm. I. Theory for triatomic systems

In this paper we present the theoretical concepts and methodology of the hyperquantization algorithm for the three body quantum mechanical problem. Within the framework of the hyperspherical approach to reaction dynamics, we use angular momentum algebra (or its generalization, e.g., including Hahn coefficients which are orthonormal polynomials on a set of grid points which span the interaction region) to compute matrix elements of the Hamiltonian operator parametrically in the hyperradius. The particularly advantageous aspects of the method proposed here is that no integrals are required and the construction of the kinetic energy matrix is simple and universal: salient features are the block tridiagonal structure of the Hamiltonian matrix and a number of symmetry properties. The extremely sparse structure is a further advantage for the diagonalization required to evaluate the adiabatic hyperspherical states as a function of the hyperradius. Numerical implementation is illustrated in the following paper by a specific example.

Quantum mechanical elucidation of reaction mechanisms of heavy-light-heavy systems: Role of potential ridge

A new idea to elucidate quantum reaction dynamics of heavy-light-heavy (HLH) systems is proposed on the basis of the hyperspherical elliptic coordinate approach. This coordinate system has a big advantage of nicely expressing good vibrational adiabaticity in the HLH systems. Taking this advantage, the concept of potential ridge is introduced, for the first time, in three-dimensional reactions. The potential ridge is proved to be very useful to extract some important avoided crossings which dominate the reaction dynamics. In fact, qualitative features of the reaction dynamics can be interpreted in terms of nonadiabatic transitions at those important avoided crossings near the potential ridge. Examples are: (i) onset of reaction for a specified initial rotational state, and (ii) major reactive transition for a specified initial rotational state. Avoided crossings to the left of the potential ridge are also useful to interpret certain aspects of reactions accompanying vibrational transitions. The new idea mentioned above is applied to a typical HLH reaction O(3P)+HCl→OH+Cl with the use of two types of potential energy surfaces.

Subthreshold resonances in few-neutron systems

Three- and four-neutron systems are studied within the framework of the hyperspherical approach with a local S-wave nn-potential. Possible bound and resonant states of these systems are sought as zeros of three- and four-body Jost functions in the complex momentum plane. It is found that zeros closest to the origin correspond to subthreshold (nnn) and (nnnn) resonant states. The positions of these zeros turned out to be sensitive to the choice of the nn-potential. For the Malfliet - Tjon potential they are (MeV) and (MeV). Movement of the zeros with an artificial increase of the potential strength also shows an extreme sensitivity to the choice of potential. Thus, to generate and bound states, the Yukawa potential needs to be multiplied by 2.67 and 2.32 respectively, while for the Malfliet - Tjon potential the required multiplicative factors are 4.04 and 3.59.

Atomic ionization by electron impact: A hyperspherical approach

An {ital ab initio} calculation, following a hyperspherical partial-wave method [J. N. Das, Aust. J. Phys. {bold 47}, 743 (1994)], satisfactorily represents the recent experimental triple differential cross-section results of R{umlt o}der {ital et al.} [Phys. Rev. A {bold 53}, 225 (1996)] for electron hydrogen atom ionization collisions at low energies. The qualitative agreement with their theoretical results is also good. Moreover, our results are in absolute units and in some instances our results are in better agreement with the experiments compared to their theoretical results. Here it is also exhibited that the hyperspherical partial wave method is a promising formalism for studying atomic ionization problems. {copyright} {ital 1997} {ital The American Physical Society}

A Monte Carlo (N,V,T) study of the stability of charged interfaces: A simulation on a hypersphere

We have used an exact expression of the Coulombic interactions derived on a hypersphere of an Euclidian space of dimension four to determine the swelling behavior of two infinite charged plates neutralized by exchangeable counterions. Monte Carlo simulations in the (N,V,T) ensemble allows for a derivation of short-ranged hard core repulsions and long-ranged electrostatic forces, which are the two components of the interionic forces in the context of the primitive model. Comparison with numerical results obtained by a classical Euclidian method illustrates the efficiency of the hyperspherical approach, especially at strong coupling between the charged particles, i.e., for divalent counterions and small plate separation.

Pair-Rydberg description of doubly excited states: Diabatic evolution of correlation patterns

The standard adiabatic hyperspherical approach to the description of doubly excited two-electron systems is adapted to the special case of ridge states. The two electrons are treated as a pair throughout the whole range of the adiabatic variable R. Diabatic corrections to the adiabatic potential curves are determined that are similar to the diabatic-by-sector method. The evolution of the correlation patterns (correlation diagram) is investigated and the most appropriate sets of coordinates and corresponding quantum numbers are identified for three different regions of the adiabatic variable.

Hyperspherical approach to the calculation of few-body atomic resonances

Direct solution of the Schr\odinger equation for the doubly excited resonant $^{1}\mathrm{S}^{\mathrm{e}}$ state of the helium atom is obtained with the help of the complex rotation method (which reduces the resonance problem to that of bound states with complex energy) and the correlation function hyperspherical harmonic (CFHH) method. In the CFHH method the solution is a product of a correlation function and a smooth factor expanded into hyperspherical harmonic functions. Given a proper correlation function, chosen from physical considerations, the method generates resonant wave functions, accurate in the whole range of interparticle distances. Since the method is nonvariational no stabilization procedure is needed. The calculated energy and width are shown to be strictly independent of the angle of complex rotation, hence there is no need for the angle optimization procedure as well. The results are compared with variational and other precise computations.

The local characteristics of the bound states of muonic molecules

The sticking probabilitiesΩ Jv,G- andγ-factors for all bound states of mesic molecules ppµ, pdµ, ptµ, ddµ, dtµ, and ttµ withJ=0 andv=0, 1 have been calculated in the adiabatic hyperspherical approach (AHSA).

Reduced adiabatic hyperspherical basis in the Coulomb three-body bound state problem

A new version of the adiabatic hyperspherical approach (AHSA) is suggested which has significant advantages for the calculation of three-body states with total angular momentumJ> 0. The binding energies of all bound states of mesic molecules with normal parity are calculated by the suggested method. Comparison with results of variational calculations and the fast convergence of the method confirm its high efficiency.

Hyperspherical approach to three-electron atomic systems.

The hyperspherical coordinates approach to three-electron atomic systems is further refined by taking into account the symmetry of the electrons analytically. Spins are treated explicitly for both doublet and quartet cases to reduce the three-electron Schr\odinger equation to a set of coupled two-dimensional partial differential equations in a compact, symmetric form ready for numerical calculations. It is shown that the resulting equations can be accurately solved using B splines. By adopting the adiabatic approximation, the hyperspherical potential curves for singly, doubly, and triply excited states for the $^{2}$${\mathit{P}}^{\mathit{o}}$ symmetry of Li are obtained. \textcopyright{} 1996 The American Physical Society.

Muonic three-body Coulomb systems in the hyperspherical approach.

Muonic three-body bound states and resonances are treated within a hyperspherical adiabatic expansion scheme. A new method for determining the basis functions of this expansion is developed: decomposing these functions into Faddeev-type components, an equivalent treatment of all two-body contributions, and thus the correct asymptotics, are guaranteed. This approach is characterized by its high symmetry and a considerable reduction of the numerical effort. Using partial wave and B-spline expansions for the components, wave functions and energies of the dt\ensuremath{\mu} and ${\mathit{d}}^{3}$He\ensuremath{\mu} molecules are calculated in extreme and uncoupled adiabatic approximation. For dt\ensuremath{\mu} good agreement with alternative calculations, which are based on a much higher number of expansion functions, is found, and the results for the ${\mathit{d}}^{3}$He\ensuremath{\mu} system are rather close to variational calculations. \textcopyright{} 1996 The American Physical Society.

A consistent study of the the low energy baryon spectrum and the nucleon-nucleon interaction within the chiral quark model

By solving the Schrödinger equation for the three-quark system in the hyperspherical harmonic approach, we have studied the low energy part of the nucleon and Δ spectra using a quark-quark interaction which reproduces the nucleon-nucleon phenomenology. The quark-quark hamiltonian considered includes, besides the usual one-gluon exchange, pion and sigma exchanges generated by the chiral symmetry breaking. The baryonic spectrum obtained is reasonable and the resulting wave function gives consistency to the ansatz used in the two baryon system.

Quantum dynamics of the Mu+H2(HD,D2) and H+MuH(MuD) reactions

Quantum mechanically accurate calculations are carried out for the following reactions involving muonium atom (Mu) using the hyperspherical coordinate approach: Mu+H2→MuH+H, Mu+D2→MuD+D, Mu+HD→MuH(MuD)+D(H), H+MuH→MuH+H, and H+MuD ↔MuH+D. The initial vibrational state is restricted to the ground state (vi=0) and the collision energies considered are up to ∼1.2 eV. The various aspects of the dynamics, such as the isotope effects, the initial rotational state (ji) dependence, and the final rotational state (jf) distribution are analyzed for a wide range of ji and jf. Some of the isotope effects can be interpreted in terms of the variations in reaction barrier and endothermicity. The following two intriguing features are also found: (1) strong enhancement of reaction by initial rotational excitation, and (2) oscillation of integral cross section as a function of collision energy in the case of the Mu-transfer reactions.

HSTERM — A program to calculate potential curves and radial matrix elements for two-electron systems within the hyperspherical adiabatic approach

A FORTRAN 77 program is presented which calculates potential curves and matrix elements of radial coupling for two-electron systems using the hyperspherical coordinate method. The adiabatic and diabatic-by-sector close-coupling approaches are considered. The program calculates also the overlap matrices on borders of all sectors which are necessary for integration of close-coupling hyperradial equations within the sector-diabatic approach. It performs also the computation of the angular part of dipole amplitudes (in the length and acceleration forms) for dipole transitions between two given atomic states. The convergence and accuracy of the computational scheme elaborated are studied in details. Radial matrix elements computed by the HSTERM program can be used for the solution of the bound state and scattering problems for two-electron systems in both the adiabatic and diabatic-by-sector close-coupling approaches.

Hyperspherical adiabatic approach for the helium atom.

The nonadiabatic ground state for the helium atom is obtained with the hyperspherical adiabatic approach. Potential curves, nonadiabatic couplings, and channel functions are calculated by a numerically exact procedure based on the analytical expansion of the channel functions. The coupled radial equations are solved by standard techniques. The convergence of the procedure is investigated as nonadiabatic couplings are systematically introduced. The inclusion of 11 potential curves and channel functions gives a ground-state energy that differs from the best variational calculation by 0.1 parts per million.

Quantum dynamics of the Mu + H 2 and Mu + D 2 reactions

Quantum mechanically accurate calculations are carried out for the Mu + H 2 and Mu + D 2 reactions using the hyperspherical coordinate approach. The results for the rate constant are in good agreement with experiment. Comparisons are also made with other more approximate calculations. The effect of initial rotational excitation and final rotational state distribution are also analyzed.

Semiclassical hyperspherical matrix elements for helium doubly excited states

A classical description of the two-electron atom, analogous to the quantum adiabatic hyperspherical channel approach, is presented. The classical problems, analogous to the quantum eigenvalue problem for the grand angular momentum operator, and the separated dynamical systems defined by each of the other constants of the motion of the non-interacting system are solved by using the Hamilton-Jacobi method. Some matrix elements of the Coulomb interaction terms of the Hamiltonian for doubly excited helium atoms using the Heisenberg correspondence principle are calculated.