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Journal ArticleDOI

Hyperspherical harmonics expansion of the ground state of the Ps - ion

01 Dec 1996-Physica Scripta (IOP Publishing)-Vol. 54, Iss: 6, pp 601-603

AbstractWe have treated the ground state of the positronium negative ion (Ps−) by a hyperspherical harmonics expansion method in which the centre of mass motion is properly accounted for. The resulting system of coupled differential equations has been solved by the renormalized Numerov method. We find that the convergence in the Binding Energy (BE) with respect to inclusion of higher hyperspherical partial waves is quite slow for this diffuse system. Using our exact numerical results up to a maximum of 28 for the hyper angular momentum quantum number (KM) in an extrapolation formula basd on the hyperspherical convergence theorems, we get the binding energy of the ground state of Ps− as 0.261 668 9 au.

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Journal ArticleDOI
01 Oct 2001-Pramana
Abstract: Hyperspherical harmonics expansion method is applied to a three-body model of two neutron halo nuclei. Convergence of the expansion has been ensured. A repulsive part is introduced in the interaction between the core and the extra-core neutron, to simulate Pauli principle. Two neutron separation energy, r.m.s. radii, correlation factor and probability density distributions have been calculated for 6He. It is found that the convergence of the two neutron separation energy is relatively slow, while other quantities reach convergence quickly.

9 citations

Journal ArticleDOI
Abstract: Ground state energies of exotic three-body atomic systems consisting two muons and a positively charged nucleus like: 1H+μ−μ−, 4He2+μ−μ−, 3He2+μ−μ−, 7Li3+μ−μ−, 6Li3+μ−μ−, 9Be4+μ−μ−, 12C6+μ−μ−, 16O8+μ−μ−, 20Ne10+μ−μ−, 28Si14+μ−μ− and 40Ar18+μ−μ− have been calculated using hyperspherical harmonics expansion (HHE) method. Calculation of matrix elements of two body interactions involved in the HHE method for a three body system is greatly simplified by expanding the bra- and ket- vector states in the hyperspherical harmonics basis states appropriate for the partition corresponding to the interacting pair. This involves the Raynal-Revai coefficients (RRC), which are the transformation coefficients between the hyperspherical harmonics bases corresponding to the two partitions. Use of these coefficients found to be very useful for the numerical solution of three-body Schrodinger equation where the two-body potentials are other than Coulomb or harmonic oscillator type. However, in this work the interaction potentials involved are purely Coulomb. The calculated energies have been compared with (i) those obtained by straight forward manner; and (ii) with those found in the literature (wherever available). The calculated binding energies agree within the computational error.

8 citations

Journal ArticleDOI
Abstract: Ground state energies of atomic three-body systems like negatively charged hydrogen, normal helium, positively charged-lithium, beryllium, carbon, oxygen, neon and negatively charged exotic- muonium and positronium atoms have been calculated adopting hyperspherical harmonics expansion method. Calculation of matrix elements of two body interactions needed in the hyperspherical harmonics expansion method for a three body system is greatly simplified by expanding the bra- and ket-vector states in the hyperspherical harmonics (HH) basis states appropriate for the partition corresponding to the interacting pair. This involves the Raynal–Revai coefficients (RRC), which are the transformation coefficients between the HH bases corresponding to the two partitions. Use of RRC become particularly essential for the numerical solution of three-body Schrődinger equation where the two-body potentials are other than Coulomb or harmonic. However in the present work the technique is used for two electron atoms 1H−(p + e − e −), D−(d + e − e −), Mu−(μ + e − e −),4He(4 He 2+ e − e −),6Li(6 Li 3+ e − e −),10Be(10 Be 4+ e − e −),12C(12 C 6+ e − e −),16O(16 O 8+ e − e −) etc. and the exotic positronium negative ion Ps −(e + e − e −) where the interactions are purely Coulomb. The relative convergence in ground state binding energy with increasing K max for 20Ne has been demonstrated as a representative case. The calculated energies at K max = 28 using RRC’s have been compared with those obtained by a straight forward manner in some representative cases to demonstrate the appropriateness of the use of RRC. The extrapolated energies have also been compared with those found in the literature. The calculated binding energies agree within the computational error.

5 citations

Journal ArticleDOI
Md. Abdul Khan1
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.

5 citations

Journal ArticleDOI
Md. Abdul Khan1
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


Additional excerpts

  • ...In hyper-spherical variables [65-66] of the i partition, three-body Schrődinger equation is [ − h̄ 2...

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References
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Journal ArticleDOI
Abstract: A new method is developed for solving the wave equation for two-electron atoms. The wave function is expanded into a triple orthogonal set in three perimetric coordinates. From the wave equation one obtains an explicit recursion relation for the coefficients in the expansion, and the vanishing of the determinant of these coefficients provides the condition for the energy eigenvalues and for the eigenvectors. The determinant was solved on WEIZAC for $Z=1 \mathrm{to} 10$, using an iteration method. Since the elements of the determinant are integers, and only an average of about 20 per row are nonvanishing, it has been possible to go to an order of 214 before exceeding the capacity of the fast memory of WEIZAC. The nonrelativistic energy eigenvalues obtained for the ground state are lower than any previously published for all $Z$ from 1 to 10. In the case of helium, our nonrelativistic energy value is accurate to within 0.01 ${\mathrm{cm}}^{\ensuremath{-}1}$ and is 0.40 ${\mathrm{cm}}^{\ensuremath{-}1}$ lower than the value computed by Kinoshita. From the wave functions obtained, the mass-polarization and the relativistic corrections were evaluated for $Z=1 \mathrm{to} 10$. Using the values of the Lamb shift computed by Kabir, Salpeter, and Sucher, we obtain an ionization potential for helium of 198 310.67 ${\mathrm{cm}}^{\ensuremath{-}1}$ as against Herzberg's value of 198 ${310.8}_{2}$\ifmmode\pm\else\textpm\fi{}0.15 ${\mathrm{cm}}^{\ensuremath{-}1}$. Comparison is also made with the available experimental data for the other values of $Z$. By the use of our magnetic tape storage, the accuracy of the nonrelativistic energy value for helium could be pushed to about 0.001 ${\mathrm{cm}}^{\ensuremath{-}1}$, should future improvements in the experimental values and in the computed radiative corrections warrant it.

703 citations

Journal ArticleDOI
Abstract: The method described previously for the solution of the wave equation of two-electron atoms has been applied to the $1^{1}S$ and $2^{3}S$ states of helium, with the purpose of attaining an accuracy of 0.001 ${\mathrm{cm}}^{\ensuremath{-}1}$ in the nonrelativistic energy values. For the $1^{1}S$ state we have extended our previous calculations by solving determinants of orders 252, 444, 715, and 1078, the last yielding an energy value of -2.903724375 atomic units, with an estimated error of the order of 1 in the last figure. Applying the mass-polarization and relativistic corrections derived from the new wave functions, we obtain a value for the ionization energy of 198 312.0258 ${\mathrm{cm}}^{\ensuremath{-}1}$, as against the value of 198 312.011 ${\mathrm{cm}}^{\ensuremath{-}1}$ derived previously from the solution of a determinant of order 210. With a Lamb shift correction of -1.339, due to Kabir, Salpeter, and Sucher, this leads to a theoretical value for the ionization energy of 198 310.687 ${\mathrm{cm}}^{\ensuremath{-}1}$, compared with Herzberg's experimental value of 198 ${310.8}_{2}$\ifmmode\pm\else\textpm\fi{}0.15 ${\mathrm{cm}}^{\ensuremath{-}1}$.For the $2^{3}S$ state we have solved determinants of orders 125, 252, 444, and 715, the last giving an energy value of -2.17522937822 a.u., with an estimated error of the order of 1 in the last figure. This corresponds to a nonrelativistic ionization energy of 38 453.1292 ${\mathrm{cm}}^{\ensuremath{-}1}$. The mass-polarization and relativistic corrections bring it up to 38 454.8273 ${\mathrm{cm}}^{\ensuremath{-}1}$. Using the value of 74.9 ry obtained by Dalgarno and Kingston for the Lamb-shift excitation energy ${K}_{0}$, we get a Lamb-shift correction to the ionization energy of the $2^{3}S$ state of -0.16 ${\mathrm{cm}}^{\ensuremath{-}1}$. The resulting theoretical value of 38 454.66 ${\mathrm{cm}}^{\ensuremath{-}1}$ for the ionization potential is to be compared with the experimental value, which Herzberg estimates to be 38 454.73\ifmmode\pm\else\textpm\fi{}0.05 ${\mathrm{cm}}^{\ensuremath{-}1}$. The electron density at the nucleus $D(0)$ comes out 33.18416, as against a value of 33.18388\ifmmode\pm\else\textpm\fi{}0.00023 which Novick and Commins deduced from the hyperfine splitting. We have also determined expectation values of several positive and negative powers of the three mutual distances, which enter in the expressions for the polarizability and for various sum rules.

440 citations

Journal ArticleDOI
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.

364 citations

Journal ArticleDOI
Abstract: The general formalism of scattering theory is extended to cover reactions in which more than two particles exist in the entrance or exit channels, and the behaviour of the scattering matrix in this general case is discussed at arbitrary energies, in particular near thresholds. The energy dependence of observable quantities such as cross-sections and polarizations is given, and the occurrence and origin of the Wigner cusps and their counterpart in the general case are clearly seen; such singularities do not arise from the behaviour of the eigenphase-shifts. Coulomb effects are not considered in the general case.

295 citations

Journal ArticleDOI
Abstract: A variational calculation for the ground state of two-electron atoms is carried out with a function containing the nonconventional terms $\mathrm{ln}({r}_{1}+{r}_{2})$, ${[\mathrm{ln}({r}_{1}+{r}_{2})]}^{2}$, and ${({{r}_{1}}^{2}+{{r}_{2}}^{2})}^{\frac{1}{2}}$. The convergence of the energy eigenvalues is very good, lending support to the existence of the logarithmic terms in the exact solution of the wave equation.

275 citations