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-coordinatm ethod is greater than erg(k")~f or 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 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.

An Investigation on the He−(1s2s2 2S) Resonance in Debye Plasmas

“…The uncertainty in our results lies on the sixth decimal place. Table 2 also includes the results obtained by using some of the most accurate calculations [15,16,21,22,27]. Comparison indicates that our computed resonance energy is close to the value reported by others, but resonance width differs considerably.…”

“…in He − (1s2s 2 2 S) for µ = 0. (a) Method of complex coordinate rotation [21]; (b) R-matrix method [22]; (c) complex stabilization method [27]; (d) experimental results [15]; (e): experimental results [16].…”

“…In particular, the aim of our study is to reveal the plasma screening effects on the resonance parameters. In vacuum, a great deal of experimental and theoretical investigations on 1s2s 2 2 S resonance in e − − He scattering has appeared in the literature [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] (and further references therein). In fact, such a resonance was the first resonance observed in electron-atom scattering [11].…”

Complex coordinate calculations of resonances in N‐electron atoms

“…Then the solution for the bound states of hydrogenic systems will be discussed followed by a discussion of the asymptotic form of the wave function for a resonance [4].…”

“…Here we have defined Xa as that complex non-Hermitian Hamiltonian obtained from the Hermitian Hamiltonian X by the transformation r + p exp (icy) (4) Although there is only one radial coordinate, it is convenient to define p as in Eq. (4).…”