Hyperspherical harmonics as Sturmian orbitals in momentum space: A systematic approach to the few-body Coulomb problem

Aquilanti, Vincenz̊o; Cavalli, Simonetta; Coletti, Cecilia; Domênico, Daniela Di; Grossi, Gaia

doi:10.1080/01442350110075926

Cited by 56 publications

(38 citation statements)

References 97 publications

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“…(20) allows one to find the global multidimensional extent of hydro-gen-like state (n, l, {}) by means of the momentum Shannon entropy (58) according to Eq. (17), where the angular part S͑Y͒ is the entropy of hyperspherical harmonics previously discussed, and the radial part is given by…”

Section: The Shannon Entropy In Momentum Spacementioning

confidence: 99%

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Information theory of D-dimensional hydrogenic systems: Application to circular and Rydberg states

Dehesa

López-Rosa

Martínez–Finkelshtein

et al. 2009

Int. J. Quantum Chem.

151

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ABSTRACT:The analytic information theory of quantum systems includes the exact determination of their spatial extension or multidimensional spreading in both position and momentum spaces by means of the familiar variance and its generalization, the power and logarithmic moments, and, more appropriately, the Shannon entropy and the Fisher information. These complementary uncertainty measures have a global or local character, respectively, because they are power-like (variance, moments), logarithmic (Shannon) and gradient (Fisher) functionals of the corresponding probability distribution. Here we explicitly discuss all these spreading measures (and their associated uncertainty relations) in both position and momentum for the main prototype in D-dimensional physics, the hydrogenic system, directly in terms of the dimensionality and the hyperquantum numbers which characterize the involved states. Then, we analyze in detail such measures for s-states, circular states (i.e., single-electron states of highest angular momenta allowed within an electronic manifold characterized by a given principal hyperquantum number), and Rydberg states (i.e., states with large radial hyperquantum numbers n).

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Section: The Shannon Entropy In Momentum Spacementioning

confidence: 99%

“…Moreover, in quantum chemistry the D-dimensional hydrogenic wavefunctions play a relevant role because they have been used in both position and momentum spaces as complete orthonormal sets for many body problems, such as three-body Coulomb systems (hydrogen molecular ion, helium atom, etc.) and in Sturmian approaches [16,17].…”

Section: Introductionmentioning

confidence: 99%

Information theory of D-dimensional hydrogenic systems: Application to circular and Rydberg states

Dehesa

López-Rosa

Martínez–Finkelshtein

et al. 2009

Int. J. Quantum Chem.

151

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“…18, we also discussed the use of combinations of complex-valued Wigner D-functions 33,34 to obtain a complete orthonormal set of real-valued hyperspherical harmonics R MM Ј ͑␣ , ␤ , ␥͒. 35 It was shown that the potential energy can be expanded in terms of such functions as follows: …”

Section: Hyperspherical Harmonics Expansionmentioning

confidence: 99%

A quantum chemical study of H2S2: Intramolecular torsional mode and intermolecular interactions with rare gases

Maciel

Barreto

Palazzetti

et al. 2008

The Journal of Chemical Physics

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The structural and energetic properties of the H 2 S 2 molecule have been studied using density functional theory, second-order Møller-Plesset method, and coupled cluster theory with several basis sets. In order to extend previous work on intra-and intermolecular dynamics of the chirality changing modes for H 2 O 2 and its derivatives, our focus has been on the torsion around the S-S bond, along with an extensive characterization of the intermolecular potentials of H 2 S 2 with the rare gases ͑He, Ne, Ar, and Kr͒. Use is made of previously defined coordinates and expansion formulas for the potentials which allow for a faithful representation of geometrical and symmetry properties of these systems that involve the interaction of an atom with a floppy molecule. The potential energy surfaces obtained in this work are useful for classical and quantum mechanical simulations of molecular collisions responsible for chirality changing processes of possible interest in the modeling of prebiotic phenomena.

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“…Extension of such an algorithm to more than three bodies requires advances in harmonic analysis in the direction indicated in this work. Also, the search for harmonics that must be adapted to the continuous and discrete symmetries of a given problem as presented here runs in parallel with investigations on the use of hyperspherical harmonics as generalized Sturmian orbitals in quantum chemistry (see 49–52). Interestingly, prescriptions due to the D 2 group symmetry, leading to “jumps” by two in harmonics labels (such as the k projection quantum number in Section 3.3) have been already encountered in a recent investigation 53, 54, which also includes a recurrence algorithm for the explicit computation.…”

Section: Discussionmentioning

confidence: 99%

Hyperspherical harmonics for polyatomic systems: Basis set for kinematic rotations

Aquilanti

Beddoni

Lombardi

et al. 2002

Int J of Quantum Chemistry

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ABSTRACT:In a symmetrical hyperspherical framework, the internal coordinates for the treatment of N-body systems are conveniently broken up into kinematic invariants and kinematic rotations. Kinematic rotations describe motions that leave unaltered the moments of the inertia of the N-body system and perform the permutation of particles. This article considers the corresponding expansions of the wave function in terms of hyperspherical harmonics giving explicit examples for the four-body case, for which the space of kinematic rotations (the "kinetic cube") is the space SO(3)/V 4 and then the related eigenfunctions will provide a basis on such manifold, as well as be symmetrical with respect to the exchange of identical particles (if any). V 4 is also denoted as D 2 . The eigenfunctions are obtained studying the action of projection operators for V 4 on Wigner D-functions. When n of the particles are identical, the exchange symmetry can be obtained using the projection operators for the S n group. This eigenfunction expansion basis set for kinematic rotations can be also of interest for the mapping of the potential energy surfaces.

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Hyperspherical harmonics as Sturmian orbitals in momentum space: A systematic approach to the few-body Coulomb problem

Cited by 56 publications

References 97 publications

Information theory of D-dimensional hydrogenic systems: Application to circular and Rydberg states

Information theory of D-dimensional hydrogenic systems: Application to circular and Rydberg states

A quantum chemical study of H2S2: Intramolecular torsional mode and intermolecular interactions with rare gases

Hyperspherical harmonics for polyatomic systems: Basis set for kinematic rotations

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