Terms Of Hyperspherical Coordinates Research Articles

A new approach to the quantum mechanics of atoms and small molecules

When the Schrodinger equation for a system of N particles interacting through Coulomb forces is expressed in terms of hyperspherical coordinates, it closely resembles the d-dimensional generalization of the hydrogen atom wave equation (where d = 3N). The d-dimensional hydrogenlike wave functions can be found exactly, and a set of these functions, all corresponding to the same energy (but to different charges), fulfills a potential weighted orthonormality relation. When such a set of “generalized Coulomb Sturmians” is used as a basis set for bound states, solution of the resulting secular equations yields a spectrum of parameters k0, which are proportional to the square roots of the binding energies. These parameters also govern the asymptotic behavior of the basis set appropriate for the expansion of the bound states to which they correspond.

Nonadiabatically Coupled Hyperspherical Equations Applied to the Collisional Spin Flip of Muonic Hydrogen

Nonadiabatically coupled equations in terms of hyperspherical coordinates are solved for the elastic and hyperfine transitions F→F′ of muonic hydrogen isotopes pμ, dμ, and tμ in collisions pμ(F)+p→p+μp(F′), dμ(F)+d→d+μd(F′), and tμ(F)+t→t+μt(F′), as an extension of our previous work on pμ [Igarashi et al., Phys. Rev. A 58 (1998), 1166]. Converged cross sections are obtained for collision energies 10−3 to 102 eV and for the total orbital angular momentum L=0 and 1 including the spin–spin interactions V S explicitly in the Hamiltonian, and for L≥2 neglecting V S because of the large interparticle separations. The inclusion of V S for L=0 and 1 reproduces the correct hyperfine-splitting energy Δ∈ in the separated-atom limit, and is found to be of vital importance for the calculation of the spin-flip cross sections for collision energies lower than Δ∈.

Total eigenphase description of multiparticle quantum systems.

This paper presents a formalism for studying semianalytically the development of a multiparticle quantum-mechanical system outward from its center of mass, casting its dynamics in terms of hyperspherical coordinates which emphasize collective motions of all constituents. The formalism expresses the system's devleopment through an R matix which varies with the system's hyperradius R, incorporating a greater amount of the dynamics as R grows. Eigenvalues of the R matrix are further transformed into R-dependent phases whose R variation elucidates interactions between alternative degrees of freedom. An initial application to the $^{1}$${\mathit{S}}^{\mathit{e}}$ states of helium reveals details of interactions hitherto concealed within numerical treatments.

Solution of atomic and molecular Schrödinger equation described by hyperspherical coordinates

AbstractThe Schrödinger equation for an atom or molecule is expressed in terms of hyperspherical coordinates. The eigenfunction is expanded in a series of orthonormal complete sets: Yλ,μ(Ω), eigenfunctions of generalized angular momentum scalar operator, and L, generalized Laguerre polynomials. The recurrence relation of the expansion coefficients are derived and the eigenvalues can be obtained from the secular equation. © 1993 John Wiley & Sons, Inc.

Hyperspherical formulation of the photodissociation of ozone

In a preceding paper [J. Chem. Phys. 92, 247 (1990)], we reported a preliminary three-dimensional quantum calculation of the dissociation of the ozone molecule in the Hartley band. We present here a fully converged calculation of the autocorrelation function on the excited 1B2 potential energy surface of Sheppard and Walker. The study has been reformulated in terms of hyperspherical coordinates, in order to make use of the permutation-inversion symmetry of the ozone molecule. The wave-function has been discretized on a three-dimensional hyperspherical grid. The autocorrelation function 〈φ0‖φt〉 has been computed within the time formulation, by means of the Lanczos algorithm. A calculation performed for the total angular momentum value J=0 shows a good agreement with the experimental results of Johnson and Kinsey, except for the overall intensity of the recurrence peaks. The calculated photodissociation spectrum reproduces the same oscillation pattern as observed experimentally.

Propagation in hyperspherical coordinates

TheR-matrix theory is used to discuss algorithms for the calculation of scattering amplitudes for chemical reactions. A formulation of the microscopic equations in terms of hyperspherical coordinates and the use of the hyperradius as the reaction coordinate requires a detailed specification of the multitude of available channels and particular emphasis is placed here on the treatment of the rotational manifold by means of a generator coordinate description. Examples are presented for atom-diatom systems

Numerical implementation of reactive scattering theory

Algorithms for the effective calculation of reactive scattering probabilities are developed and tested on the hydrogenic atom–diatom system described by the Siegbahn–Liu–Truhlar–Horowitz potential energy surface. A three-dimensional finite element procedure is designed from a description in terms of hyperspherical coordinates. The Wigner–Eisenbud R-matrix theory is used for a recursive procedure which admits control with limits on the hyperradial propagation inward from an asymptotic region and for a symmetry preserving transformation to arrangement channel Jacobi coordinates.

A formulation of the quantum-mechanical many-body problem in terms of hyperspherical coordinates

Hyperspherical coordinates are used to construct a matrix representation of a general N-particle Hamiltonian in the case where the interaction is electrostatic. The Yukawa interaction can be treated similarly, as is shown in an appendix. The basis functions used to construct the matrix representation of H are mononomials inthe 3N coordinates of the particles, multiplied by functions of the hyperradius. Methods for transforming from this representation to a symmetry-adapted representation are also discussed.

Hyperspherical coordinate representation of potential surfaces of large molecules

In a previous study an energy criterion has been applied for the simplification of the enumeration problem of various conformers of biomolecules. In the present note, the above energy criterion is combined with a spatial criterion that in general leads to a reduction in the size of those domains of biomolecule potential surfaces within which the search for chemically important conformers is warranted. The above model and the conformational domains of primary interest can be formulated in a hyperspherical representation. Advantages and disadvantages of representing biomolecule potential surfaces in terms of hyperspherical coordinates are discussed.

Classification and supermultiplet structure of doubly excited states

Based upon the analysis of electron correlations in hyperspherical coordinates for arbitrary $L$, $S$, and $\ensuremath{\pi}$ states, a classification scheme for all doubly excited states of two-electron atoms is presented. A new set of internal correlation quantum numbers, $K$, $T$, and $A$, are introduced. Here ($K$,$T$) describe angular correlations and $A=\ifmmode\pm\else\textpm\fi{}1,0$ describes radial correlations. These quantum numbers are used to label the first-order wave functions which are approximated as $\ensuremath{\Psi}={F}_{\ensuremath{\mu}}^{n}(R){\ensuremath{\Phi}}_{\ensuremath{\mu}}(R;\ensuremath{\Omega})$ in terms of hyperspherical coordinates. The channel index $\ensuremath{\mu}$ is $\ensuremath{\mu}\ensuremath{\equiv}|{(K,T)}_{N}^{A}^{2S+1}L^{\ensuremath{\pi}}〉$, where $N$ is the dissociation limit of the channel. Rules for constructing the correlation diagram for channel potential ${U}_{\ensuremath{\mu}}(R)$ and the labeling of each channel are discussed. By comparing the rotation-averaged surface charge densities, it is shown that channels which have identical ${(K,T)}_{N}^{+,\ensuremath{-}}$ have isomorphic correlation patterns, irrespective of the overall $L$, $S$, and $\ensuremath{\pi}$. Such isomorphism is shown to be the underlying origin of the supermultiplet structure of intrashell doubly excited states. In fact, it is shown that such supermultiplet structures actually extend to all states which have $A=+1 or \ensuremath{-}1$. A new Grotrian diagram for energy levels grouped according to ${(K,T)}^{+}$ and ${(K,T)}^{\ensuremath{-}}$ displays rotor-like structure. Such diagrams can easily reveal missing or misclassified levels. It is also shown that all $A=0$ states are similar to singly excited states where for a given ${(K,T)}_{N}^{0}$, the triplet state always has a slightly lower energy than the singlet state. Approximate selection rules for photoabsorption and for $e$-H and $e$-${\mathrm{He}}^{+}$ scatterings are discussed.

Hyperspherical representations of triatomic energy surfaces

Examples of triatomic electronic energy surfaces are presented in terms of hyperspherical coordinates using a mapping introduced by Mead. An approximate electronic ground state energy is determined from the Hubbard model for the H3 system and graphs are prepared for different values of the hyperspherical radius.

Use of computer simulation to determine the triplet distribution function in dense fluids

In this paper a new method is described for determining the triplet distribution function g 3 from computer simulations of dense fluids. The method involves expressing g 3 in terms of hyperspherical coordinates (ϱ,θ,φ) and then expanding g 3 in spherical harmonics of the angular variables θ and φ. We first test the convergence of the spherical harmonic expansion by applying the method to the superposition approximation for g 3. We then show how the expansion coefficients may be obtained from an analysis of computer simulation data. Lastly, we report new results for g 3 obtained by applying the method to data from a molecular dynamics simultion of the Lennard-Jones fluid.

Supermultiplets in doubly excited atoms

Take an excited hydrogen atom, add a second excited electron, and you change the calculation of energy levels from an easy problem to a formidable task. In such doubly excited atoms, the dominance of electron correlations considerably alters the single‐particle shell structure. Alternative formulations of the electronic states in terms of hyperspherical coordinates have greatly elucidated the angular and radial correlations. That progress was recently boosted by the insight from dynamical group theory: A supermultiplet classification gives striking order to the energy spectrum and predicts which electron states become mixed. This supermultiplet classification visually depicts the doubly excited atom as a linear triatomic molecule having the electrons across the nucleus from one another and exhibiting collective rotational and bending motions. This new view is compatible with that of the hyperspherical approach. All the varying perspectives were discussed last month at the meeting of the APS Division of Electron and Atomic Physics in New York.