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Journal ArticleDOI

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

TL;DR: In this article, the ground state of the positronium negative ion (Ps−) was treated by a hyperspherical harmonics expansion method in which the center of mass motion was properly accounted for.
Abstract: We 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
Md. Abdul Khan1
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


Additional excerpts

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

    [...]

Journal ArticleDOI
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.
Book ChapterDOI
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.
References
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Journal ArticleDOI
TL;DR: In this article, a new method was developed for solving the wave equation for two-electron atoms, and the wave function was expanded into a triple orthogonal set in three perimetric coordinates.
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.

716 citations

Journal ArticleDOI
TL;DR: In this article, the mass-polarization and relativistic corrections derived from the new wave functions were applied to the wave equation of two-electron atoms to obtain a value for the ionization energy of 198 312.
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.

449 citations

Journal ArticleDOI
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

Journal ArticleDOI
TL;DR: In this paper, 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.

301 citations

Journal ArticleDOI
TL;DR: In this paper, a variational calculation for the ground state of two-electron atoms is carried out with a function containing the nonconventional terms (i.e., logarithmic terms) and the convergence of the energy eigenvalues is very good.
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.

282 citations