Abstract
The notion of the nth order local energy, generated by the nth power of the Hamiltonian, has been introduced. The nth order two-particle coalescence conditions have been derived from the requirements that the nth order local energy at the coalescence point is non-singular and equal to the nth power of the Hamiltonian eigenvalue. The first condition leads to energy-independent constraints. The second one is state-specific. The analysis has been done using a radial, one-dimensional, model Hamiltonian. The model is valid in the asymptotic region of . The coalescence conditions set the relations between the expansion coefficients of the radial wave function into a power series with respect to r.
GRAPHICAL ABSTRACT
![](/cms/asset/addc2ed3-1a0a-4ea6-99a7-4c4c95977228/tmph_a_2069055_uf0001_oc.jpg)
Acknowledgments
We thank Dr Heinz-Jürgen Flad (Technische Universität München) for useful discussions.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Notes
1 It is convenient to use the radial Hamiltonian in the self-conjugate form which does not contain the first-order derivative.
2 Equation (Equation3(3)
(3) ) is meaningful if
exists, i.e. if Ψ is
-fold differentiable in its domain. As shown by Fournais et al. [Citation15], if the other electron coordinates do not coincide, then in a neighbourhood of the coalescence point Coulombic wave functions are analytic, i.e. they are differentiable an arbitrary number of times.
3 See also an early study on the coalescence conditions for non-Coulombic potentials by Silanes et al. [Citation14].
4 Note that a shift in the energy scale does not affect the eigenfunctions.
5 Explicit expressions for the continuous spectrum wave functions can be found, e.g. in the monograph by Bethe and Salpeter [Citation18]. The expansion given by Equation (Equation48(48)
(48) ) with
is the same as the one obtained from the expansion of the exact eigenfunctions.