Low-Lying S-States of Two-Electron Systems
TL;DR: In this article, hyperspherical harmonics expansion method has been applied for two-electron ions 1H− (Z = 1) to 40Ar16+ (Z= 18), negatively charged-muonium Mu− and exotic positronium ion Ps−(e+e−e−) considering purely Coulomb interaction.
Abstract: The energies of the low-lying bound S-states of some two-electron systems (treating them as three-body systems) like negatively charged hydrogen, neutral helium, positively charged-lithium, beryllium, carbon, oxygen, neon, argon and negatively charged muonium and exotic positronium ions have been calculated employing hyperspherical harmonics expansion method. The matrix elements of two-body interactions involve Raynal–Revai coefficients which are particularly essential for the numerical solution of three-body Schrődinger equation when the two-body potentials are other from Coulomb or harmonic. The technique has been applied for to two-electron ions 1H− (Z = 1) to 40Ar16+ (Z = 18), negatively charged-muonium Mu− and exotic positronium ion Ps−(e+e−e−) considering purely Coulomb interaction. The available computer facility restricted reliable calculations up to 28 partial waves (i.e. Km = 28) and energies for higher Km have been obtained by applying an extrapolation scheme suggested by Schneider.
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Dissertation•
31 Jul 2015
TL;DR: The metric space approach to quantum mechanics is a new, powerful method for deriving metrics for sets of quantum mechanical functions from conservation laws as discussed by the authors, and it has been shown that, from a standard form of conservation law, a universal method exists to generate a metric for the physical functions connected to that conservation law.
Abstract: The metric space approach to quantum mechanics is a new, powerful method for deriving metrics for sets of quantum mechanical functions from conservation laws. We develop this approach to show that, from a standard form of conservation law, a universal method exists to generate a metric for the physical functions connected to that conservation law. All of these metric spaces have an "onion-shell" geometry consisting of concentric spheres, with functions conserved to the same value lying on the same sphere. We apply this approach to generate metrics for wavefunctions, particle densities, paramagnetic current densities, and external scalar potentials. In addition, we demonstrate the extensions to our approach that ensure that the metrics for wavefunctions and paramagnetic current densities are gauge invariant.
We use our metric space approach to explore the unique relationship between ground-state wavefunctions, particle densities and paramagnetic current densities in Current Density Functional Theory (CDFT). We study how this relationship is affected by variations in the external scalar potential, pairwise electronic interaction strength, and magnetic field strength. We find that all of the metric spaces exhibit a "band structure", consisting of "bands" of points characterised by the value of the angular momentum quantum number, m. These "bands" were found to either be separated by "gaps" of forbidden distances, or be "overlapping". We also extend this analysis beyond CDFT to explore excited states.
We apply our metrics in order to gain new insight into the Hohenberg-Kohn theorem and the Kohn-Sham scheme of Density Functional Theory. For the Hohenberg-Kohn theorem, we find that the relationship between potential and wavefunction metrics, and between potential and density metrics, is monotonic and includes a linear region. Comparing Kohn-Sham quantities to many-body quantities, we find that the distance between them increases as the electron interaction dominates over the external potential.
6 citations
TL;DR: In this paper, the energies of three-body Schr\H{o}dinger equations were calculated by hyperspherical harmonics expansion method (HHEM) and the dependence of bound state energies has been checked against increasing nuclear charge Z and finally, the calculated energies have been compared with the ones of the literature.
Abstract: Energies of the low-lying bound S-states (L=0) of exotic three-body systems, consisting a nuclear core of charge +Ze (Z being atomic number of the core) and two negatively charged valence muons, have been calculated by hyperspherical harmonics expansion method (HHEM). The three-body Schr\H{o}dinger equation is solved assuming purely Coulomb interaction among the binary pairs of the three-body systems X$^{Z+}\mu^-\mu^-$ for Z=1 to 54. Convergence pattern of the energies have been checked with respect to the increasing number of partial waves $\Lambda_{max}$. For available computer facilities, calculations are feasible up to $\Lambda_{max}=28$ partial waves, however, calculation for still higher partial waves have been achieved through an appropriate extrapolation scheme. The dependence of bound state energies has been checked against increasing nuclear charge Z and finally, the calculated energies have been compared with the ones of the literature.
2 citations
TL;DR: In this paper, a hyperspherical three-body model formalism has been applied for the calculation of energies of the low-lying bound 3S-states of neutral helium and helium like Coulombic 3-body systems having nuclear charge (z) in the range 2 ≤ Z ≤ 92.
Abstract: In this paper, hyperspherical three-body model formalism has been applied for the calculation of energies of the low-lying bound 3S-states of neutral helium and helium like Coulombic three-body systems having nuclear charge (z) in the range 2 ≤ Z ≤ 92. Energies of 1S-states are also calculated for those having nuclear charge in the range 14 ≤ Z ≤ 92. The calculation of the coupling potential matrix elements of the two-body potentials has been simplified by the use of Raynal–Revai Coefficients (RRC). The three-body wave function in the Schrodinger equation when expanded in terms of hyperspherical harmonics (HH), leads to an infinite set of coupled differential equation (CDE) which for practical purposes is truncated to a finite set and the truncated set of CDE’s are solved by renormalized Numerov method (RNM) to get the energy (E). The calculated energy is compared with the ones of the literature.
1 citations
Cites methods from "Low-Lying S-States of Two-Electron ..."
...The scheme of solution of the three-body Schrődinger equation in HHE approach has been described in more details in our earlier works [49-62]....
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TL;DR: In this article, a hyperspherical three-body model formalism has been applied for the calculation energies of the low-lying bound of neutral helium and helium like Coulombic three body systems having nuclear charge (Z) in the range Z=2 to Z=92.
Abstract: In this paper, hyperspherical three-body model formalism has been applied for the calculation energies of the low-lying bound $^{1,3}$S (L=0)-states of neutral helium and helium like Coulombic three-body systems having nuclear charge (Z) in the range Z=2 to Z=92. The calculation of the coupling potential matrix elements of the two-body potentials has been simplified by the introduction of Raynal-Revai Coefficients (RRC). The three-body wave function in the Schr\H{o}dinger equation when expanded in terms of hyperpherical harmonics (HH), leads to an infinite set of coupled differential equation (CDE). For practical reason the infinite set of CDE is truncated to a finite set and are solved by an exact numerical method known as renormalized Numerov method (RNM) to get the energy solution (E). The calculated energy is compared with the ones of the literature.
01 Jan 2016
TL;DR: In this article, the potential matrix element is calculated for the Coulomb interaction and the convergence of the binding energy (BE) is analyzed using the hyperspherical harmonics technique.
Abstract: Few-body Coulomb systems are discussed as examples of the hyperspherical harmonics technique. Nearly complete analytical calculation of the potential matrix element is possible for the Coulomb interaction. As a simple illustration, two-electron atoms are treated in details. There is no approximation except an upper cut-off of the HH basis, which is tested for convergence of binding energy (BE). An extrapolation formula can be obtained for the BE corresponding to the complete basis, from BE calculated with a few truncated basis functions. Hence very high precision is possible. General three-body Coulomb system with adiabatic approximation is also presented. Applications of these methods to physical systems are discussed.
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TL;DR: The 2010 self-consistent set of values of the basic constants and conversion factors of physics and chemistry recommended by the Committee on Data for Science and Technology (CODATA) for international use is presented in this article.
Abstract: This paper gives the 2010 self-consistent set of values of the basic constants and conversion factors of physics and chemistry recommended by the Committee on Data for Science and Technology (CODATA) for international use. The 2010 adjustment takes into account the data considered in the 2006 adjustment as well as the data that became available from 1 January 2007, after the closing date of that adjustment, until 31 December 2010, the closing date of the new adjustment. Further, it describes in detail the adjustment of the values of the constants, including the selection of the final set of input data based on the results of least-squares analyses. The 2010 set replaces the previously recommended 2006 CODATA set and may also be found on the World Wide Web at physics.nist.gov/constants.
2,770 citations
TL;DR: In this article, the renormalized numerov method has been generalized to bound states of the coupled-channel Schroedinger equation and a method for detecting wave function nodes is presented.
Abstract: The renormalized Numerov method, which was recently developed and applied to the one‐dimensional bound state problem [B. R. Johnson, J. Chem. Phys. 67, 4086 (1977)], has been generalized to compute bound states of the coupled‐channel Schroedinger equation. Included in this presentation is a generalization of the concept of a wavefunction node and a method for detecting these nodes. By utilizing node count information it is possible to converge to any specific eigenvalue without the need of an initial close guess and also to calculate degenerate eigenvalues and determine their degree of degeneracy. A useful interpolation formula for calculating the eigenfunctions at nongrid points is also given. Results of example calculations are presented and discussed. One of the example problems is the single center expansion calculation of the 1sσg and 2sσg states of H+2.
382 citations
TL;DR: In this paper, the authors presented a self-consistent set of values of the basic constants and conversion factors of physics and chemistry recommended by the Committee on Data for Science and Technology ~CODATA! for international use.
Abstract: This paper gives the 1998 self-consistent set of values of the basic constants and conversion factors of physics and chemistry recommended by the Committee on Data for Science and Technology ~CODATA! for international use. Further, it describes in detail the adjustment of the values of the subset of constants on which the complete 1998 set of recommended values is based. The 1998 set replaces its immediate predecessor recommended by CODATA in 1986. The new adjustment, which takes into account all of the data available through 31 December 1998, is a significant advance over its 1986 counterpart. The standard uncertainties ~i.e., estimated standard deviations ! of the new recommended values are in most cases about 1/5 to 1/12 and in some cases 1/160 times the standard uncertainties of the corresponding 1986 values. Moreover, in almost all cases the absolute values of the differences between the 1998 values and the corresponding 1986 values are less than twice the standard uncertainties of the 1986 values. The new set of recommended values is available on the World Wide Web at physics.nist.gov/ constants. ©1999 American Institute of Physics and American Chemical Society. @S0047-2689 ~00!00301-9#
377 citations
01 Jan 2006
TL;DR: In this paper, the authors present a mathematical method for estimating the density matrix of an atom atoms in a multiconfiguration atoms and demonstrate the effect of different density matrices on different properties of the atom.
Abstract: Units and Constants- Part A Mathematical Methods: Angular Momentum Theory- Group Theory for Atomic Shells- Dynamical Groups- Perturbation Theory- Second Quantization- Density Matrices- Computational Techniques- Hydrogenic Wave Functions- Part B Atoms: Atomic Spectroscopy- High Precision Calculations for Helium- Atomic Multipoles- Atoms in Strong Fields- Rydberg Atoms- Rydberg Atoms in Strong Static Fields- Hyperfine Structure- Precision Oscillator Strength and Lifetime Measurements- Ion Beam Spectroscopy- Line Shapes and Radiation Transfer- Thomas - Fermi and Other Density-Functional Theories- Atomic Structure: Multiconfiguration Hartree - Fock Theories- Relativistic Atomic Structure- Many-Body Theory of Atomic Structure and Processes- Photoionization of Atoms- Autoionization- Green's Functions of Field Theory- Quantum Electrodynamics- Tests of Fundamental Physics- Parity Nonconserving Effects in Atoms- Atomic Clocks and Constraints on Variations of Fundamental Constants- Molecular Structure- Molecular Symmetry and Dynamics- Radiative Transition Probabilities- Molecular Photodissociation- Time-Resolved Molecular Dynamics- Nonreactive Scattering- Gas Phase Reactions- Gas Phase Ionic Reactions- Clusters- Infrared Spectroscopy- Laser Spectroscopy in the Submillimeter and Far-Infrared Region- Spectroscopic Techniques: Lasers- Spectroscopic Techniques: Cavity-Enhanced Methods- Spectroscopic Techniques: Ultraviolet- Part C Scattering Theory: Elastic Scattering: Classical, Quantal, and Semiclassical- Orientation and Alignment in Atomic and Molecular Collisions- Electron-Atom, Electron-Ion, and Electron-Molecule Collisions- Positron Collisions- Adiabatic and Diabatic Collision Processes at Low Energies- Ion -Atom and Atom - Atom Collisions- Ion - Atom Charge Transfer Reactions at Low Energies- Continuum Distorted-Wave and Wannier Methods- Ionization in High Energy Ion - Atom Collisions- Electron - Ion and Ion - Ion Recombination- Dielectronic Recombination- Rydberg Collisions: Binary Encounter, Born and Impulse Approximations- Mass Transfer at High Energies: Thomas Peak- Classical Trajectory and Monte Carlo Techniques- Collisional Broadening of Spectral Lines- Part D Scattering Experiments: Photodetachment- Photon - Atom Interactions: Low Energy- Photon - Atom Interactions: Intermediate Energies- Electron - Atom and Electron - Molecule Collisions- Ion - Atom Scattering Experiments: Low Energy- Ion - Atom Collisions:High Energy- Reactive Scattering- Ion - Molecule Reactions- Part E Quantum Optics: Light - Matter Interaction- Absorption and Gain Spectra- Laser Principles- Types of Lasers- Nonlinear Optics- Coherent Transients- Multiphoton and Strong-Field Processes- Cooling and Trapping- Quantum Degenerate Gases: Bose - Einstein Condensation- De Broglie Optics- Quantized Field Effects- Entangled Atoms and Fields: Cavity QED- Quantum Optical Tests of the Foundations of Physics- Quantum Information- Part F Applications: Applications of Atomic and Molecular Physics to Astrophysics- Comets- Aeronomy- Applications of Atomic and Molecular Physics to Global Change- Atoms in Dense Plasmas- Conduction of Electricity in Gases- Applications to Combustion- Surface Physics- Interface with Nuclear Physics- Charged-Particle - Matter Interactions- Radiation Physics- About the Authors- Subject Index
370 citations
TL;DR: The authors provides key ideas, basic techniques and data, including recent advances in all related fields, including AMOP and AMOP-based techniques and their applications, for AMOP research.
Abstract: This indespensable resource on AMOP contains key ideas, basic techniques and data, including recent advances in all related fields
300 citations