Hyperspherical harmonics as Sturmian orbitals in momentum space: A systematic approach to the few-body Coulomb problem (original) (raw)
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Chemical Physics, 1997
Hydrogenoid orbitals, i.e. the solutions to the Schriidinger equation for a central Coulomb field, are considered in mathematical dimensions d = 2 and d > 3 different from the physical case, d = 3. Extending known results for d = 3, Sturmian basis sets in configuration (or direct) spacecorresponding to variable separation in parabolic coordinates -are introduced as alternatives to the ordinary ones in spherical coordinates: extensions of Fock ste:L%graphic projections allow us to establish the relationships between the corresponding momentum (or reciprocal) space orbitals and the alternative fomrs of hyperspherical harmonics. Properties of the latter and multi-dimensional Fourier integral transforms are exploited to obtain the matrix elements connecting the alternative basis sets explicitly in terms of Wigner's rotation matrix elements for d = 2 and generalized vector coupling (or Hahn) coefficients for d > 3. The use of these orbitals as complete and orthonormal expansion basis sets for atomic and molecular problems is briefly commented.
The relationship between alternative separable solutions of the Coulomb problem both in configuration and in momentum space is exploited in order to obtain Sturmian orbitals of use as expansion basis sets in atomic and molecular problems. The usual spherical basis is obtained by separation in polar coordinates. The mathematical properties are explored for a type of basis set for quantum mechanical problems with axial symmetry, examples being diatomic molecules, or atoms under the influence of a uniform electric field. Because of its appropriateness for treatment of the Stark effect in atomic physics, this alternative basis set is called Stark basis. The Stark basis corresponds to separation in parabolic coordinates in configuration space and in cylindrical coordinates in momentum space. Fock's projection onto the surface of a sphere in the four dimensional hyperspace allows us to establish the connections of the momentum space wave functions with hyperspherical harmonics. As an ...
Hyperspherical harmonics for polyatomic systems: basis set for collective motions
Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta), 2004
As a continuation of our previous work for the construction of expansion basis sets for the quantum mechanical treatment of N -body problems of interest for intermolecular and intramolecular and reactive dynamics of polyatomic molecules and clusters, we develop here a group-theoretical procedure which allows us to obtain explicitly the hyperspherical harmonics for the description of the collective motions. The coordinates involved are related to the invariants of the N -body system, which are referred to the moments of inertia. Although this work is limited to the case where both external and internal (kinematic) rotations are zero, and the example of N ¼ 4 is explicitly worked out, this method, which gives hyperspherical harmonics as linear combinations of ordinary spherical harmonics, can be extended to cover the general N -body case.
Sturmian bases for two-electron systems in hyperspherical coordinates
Journal of Physics B: Atomic, Molecular and Optical Physics, 2016
We give a detailed account of an ab initio spectral approach for the calculation of energy spectra of two active electron atoms in a system of hyperspherical coordinates. In this system of coordinates, the Hamiltonian has the same structure as the one of atomic hydrogen with the Coulomb potential expressed in terms of a hyperradius and the nuclear charge replaced by an angle dependent effective charge. The simplest spectral approach consists in expanding the hyperangular wave function in a basis of hyperspherical harmonics. This expansion however, is known to be very slowly converging. Instead, we introduce new hyperangular sturmian functions. These functions do not have an analytical expression but they treat the first term of the multipole expansion of the electron-electron interaction potential, namely the radial electron correlation, exactly. The properties of these new functions are discussed in detail. For the basis functions of the hyperradius, several choices are possible. In the present case, we use Coulomb sturmian functions of half integer angular momentum. We show that, in the case of H − , the accuracy of the energy and the width of the resonance states obtained through a single diagonalization of the Hamiltonian, is comparable to the values given by state-of-the-art methods while using a much smaller basis set. In addition, we show that precise values of the electric-dipole oscillator strengths for S → P transitions in helium are obtained thereby confirming the accuracy of the bound state wave functions generated with the present method.
Hyperspherical harmonics for polyatomic systems: Basis set for kinematic rotations
International Journal of Quantum Chemistry, 2002
As a continuation of our previous work for the construction of expansion basis sets for the quantum mechanical treatment of N -body problems of interest for intermolecular and intramolecular and reactive dynamics of polyatomic molecules and clusters, we develop here a group-theoretical procedure which allows us to obtain explicitly the hyperspherical harmonics for the description of the collective motions. The coordinates involved are related to the invariants of the N -body system, which are referred to the moments of inertia. Although this work is limited to the case where both external and internal (kinematic) rotations are zero, and the example of N ¼ 4 is explicitly worked out, this method, which gives hyperspherical harmonics as linear combinations of ordinary spherical harmonics, can be extended to cover the general N -body case.
Theoretical Chemistry Accounts, 2007
The hyperspherical method is a widely used and successful approach for the quantum treatment of elementary chemical processes. It has been mostly applied to three-atomic systems, and current progress is here outlined concerning the basic theoretical framework for the extension to four-body bound state and reactive scattering problems. Although most applications only exploit the advantages of the hyperspherical coordinate systems for the formulation of the few-body problem, the full power of the technique implies representations explicitly involving quantum hyperangular momentum operators as dynamical quantities and hyperspherical harmonics as basis functions. In terms of discrete analogues of these harmonics one has a universal representation for the kinetic energy and a diagonal representation for the potential (hyperquantization algorithm). Very recently, advances have been made on the use of the approach in classical dynamics, provided that a hyperspherical formulation is given based on "classical" definitions of the hyperangular momenta and related quantities. The aim of the present paper is to offer a retrospective and prospective view of the hyperspherical methods both in quantum and classical dynamics. Specifically, regarding the general quantum hyperspherical approaches for three-and four-body systems, we first focus on the basis set issue, and then we present developments on the classical formulation that has led to applications involving the implementations of hyperspherical techniques for classical molecular dynamics simulations of simple nanoaggregates.
Matrix elements of potentials in the correlation-function hyperspherical-harmonic method
Physical Review A, 1990
Matrix elements of two-body potentials and correlation functions between threeand four-body hyperspherical states, including their velocity-dependent parts, are calculated analytically for any value of the total orbital angular momentum. The resulting formulas contain explicitly written functions of the radial variable, and the Raynal-Revai coefficients. The latter are expressible through finite sums of 3-j and 9-j symbols. The formulas allow precise and fast evaluation of matrix elements of the effective potential in the correlation-function hyperspherical-harmonic method for atomic, molecular, and nuclear threeand four-body problems. The generalization to any number of particles is straightforward.
New methods for old Coulomb few-body problems
International Journal of Quantum Chemistry, 2004
Coulomb problems involving three or four particles can advantageously be described using wavefunctions that explicitly involve all the interparticle distances. In what is known as the Hylleraas method, a set of Slater-type orbitals for the electrons of an atom is augmented by linear (or in some cases higher powers) of the interelectron separation(s). An alternative approach, of particular value when all the particles have comparable masses, is to use a wavefunction containing exponentials in all the interparticle distances. This "exponential ansatz" is straightforward to implement when there are three particles but has only recently been implemented for four-body systems. Integral evaluation is reviewed for both the exponential ansatz and the Hylleraas method, the treatment of general angular symmetry is discussed, and kinetic-energy matrix elements are examined and related to simpler integrals. Some illustrative results for the positronium molecule (e ϩ e Ϫ e ϩ e Ϫ ) are included.