Complex-coordinate method. Structure of the wave function

Abstract

The functional forms of the wave functions for the bound states of hydrogenic systems and for free particles as described by the complex-coordinate method are presented. Writing the wave function for an electron resonance as the sum of a "boundlike" or "$Q$-space" part and a "scatteringlike" or "$P$-space" part, we suggest functional forms or bases for these two parts based on the solutions of the hydrogenic and freeparticle systems. We present an argument suggesting that when the rotation angle in the complex-coordinate method is greater than $\left|\arg \right(k_r\left)\right|$ for the electron resonance, this method is identical to calculations based on a Siegert resonance. This assumed structure of the wave function should yield a rate of convergence similar to other methods. The advantages of this method are that the basis functions are all square integrable, a single calculation yields both the position and width, only a solution of a straightforward eigenvalue problem is required, arbitrarily accurate target states are easily incorporated, and polarization terms can be explicitly included. Variational calculations for the position and width of the lowest ${}_{}{}^2{}_{}{}^{}S$ resonance in the negative helium ion are reported using trial wave functions containing 39, 43, 55, 59, and 67 configurations. These wave functions contain 8, 8, 20, 24, and 32 "$P$-space" configurations, respectively. Values of 19.387 eV and 12.1 meV are obtained for the position and width, respectively, of the resonance. One also finds that inclusion of free-particle-like basis functions improves the representation of the scattering states.