Two-particle coalescence conditions revisited

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 $r\sim 0$. 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

Keywords:

• Schrödinger equation
• two-particle coalescence
• local energy
• eigenvalue problem
• coalescence constraints

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 (3(3) $\frac{H^n{\mathrm{\Psi }}_{\nu \lambda }(r)}{{\mathrm{\Psi }}_{\nu \lambda }(r)}={\mathcal{E}}_{\nu \lambda }^{(n)}(r).n=1,2,3,\dots$(3) ) is meaningful if $H^n\mathrm{\Psi }(r)$ exists, i.e. if Ψ is $(2n)$-fold differentiable in its domain. As shown by Fournais et al. [15], 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. [14].

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 [18]. The expansion given by Equation (48(48) ${\psi }_{ϵ\lambda }(r)=F_{{\alpha }_{-1}}(r)+{\mathrm{\Delta }}_{{\alpha }_2}(r),$(48) ) with ${\mathrm{\Delta }}_{{\alpha }_2}(r)=0$ is the same as the one obtained from the expansion of the exact eigenfunctions.