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Generalized spheroidal wave equation for real and complex valued parameters. An algorithm based on the analytic derivatives for the eigenvalues

Mykhaylo V. Khoma

Abstract

This paper presents a new approach for the computation of eigenvalues of the generalized spheroidal wave equations. The novelty of the present method is in the use of the analytical derivatives of the eigenvalues to minimize losses in accuracy. The derivatives are constructed in the form of three-term recurrent relations within the method of continued fractions associated with the corresponding spheroidal wave equation. Very accurate results for the eigenvalues are obtained for a wide range of the parameters of the problem. As an illustrative example, the electronic energies and the separation constants are computed for various electronic states and geometries of selected ($\rm{H}_2^{+}$, $\rm{HeH}^{2+}$, and $\rm{BH}^{5+}$) quasimolecular systems. The computations for high lying ${}^{2}Σ$ electronic states of $\rm{H}_2^{+}$ up to very large internuclear separations ($ \leq 1.7 \times 10^5$ au) are presented. Also presented the computations for the eigenvalues of the generalized spheroidal wave equations with complex valued parameters. The agreement between the obtained results and the results of other authors is discussed.

Generalized spheroidal wave equation for real and complex valued parameters. An algorithm based on the analytic derivatives for the eigenvalues

Abstract

This paper presents a new approach for the computation of eigenvalues of the generalized spheroidal wave equations. The novelty of the present method is in the use of the analytical derivatives of the eigenvalues to minimize losses in accuracy. The derivatives are constructed in the form of three-term recurrent relations within the method of continued fractions associated with the corresponding spheroidal wave equation. Very accurate results for the eigenvalues are obtained for a wide range of the parameters of the problem. As an illustrative example, the electronic energies and the separation constants are computed for various electronic states and geometries of selected (, , and ) quasimolecular systems. The computations for high lying electronic states of up to very large internuclear separations ( au) are presented. Also presented the computations for the eigenvalues of the generalized spheroidal wave equations with complex valued parameters. The agreement between the obtained results and the results of other authors is discussed.

Paper Structure

This paper contains 13 sections, 55 equations, 4 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. The GSWE and the two-Coulomb-center problem
  4. The algorithm to compute the eigenvalues
  5. Derivation of the equations for $\bf dE/dR$, $\bf dp/dR$, and $\bf d\lambda/dR$
  6. Details of numerical computations and examples
  7. Hydrogen molecular ion $\mathrm{H}_2^{+}$
  8. The case of small $R$
  9. The case of large $R$
  10. Exact Demkov's solution
  11. Continuum spectrum of the two-Coulomb-center problem
  12. SWE and GSWE with complex valued parameters
  13. Concluding remarks

Figures (4)

  • Figure 1: PECs for selected highly excited states of $\rm{H}_2^{+}$ (see also Table \ref{['tab:HighE-Rmin']}): 1, $(q,k)=(189,28)$; 2, $(q,k)=(201,30)$; 3, $(q,k)=(237,11)$; 4, $(q,k)=(233,17)$; 5, $(q,k)=(265,2)$; 6, $(q,k)=(283,0)$.
  • Figure 2: Ridge of avoided crossings shown for states $(k,q,m)=(0,q,0)$, $q=162,164,...,214$.
  • Figure 3: The same as in Fig. \ref{['fig:ridge1']} but for the states $q=322,324,...,362$.
  • Figure 4: The values $A_{qm} = \lambda_{qm} + c^2$ as a function of the parameter $b$ at $c=1$. The numbers in the square brackets $[qm]$ denotes the quantum numbers $q$ and $m$.