Complex coordinate rotation method

As with Ps , there is only one bound state of Ps2 but there exist Rydberg series of autodissociating states arising from the attractive interaction between one of the positrons and the residual Ps- (or between one of the electrons and the charge conjugate of Ps ). The positions and widths of several of these states were determined by Ho (1989) using the complex coordinate rotation method. To date Ps2 has not been observed in the laboratory. 

J. Simons, The complex coordinate rotation method and exterior scaling A simple example, Int. J. Quant. Chem. 14 (1980) 113. 

When time resolution is required, then it is time-dependent probabilities rather than rates (= probability per unit of time) that have to be computed. For the case of the static field, the SSEA has not been applied yet. However, for the one-electron hydrogen atom, the problem of the time-resolved evolution of its ground and its first excited states was tackled a few years ago, first by Geltman [25], who used an expansion method (the TDSE turns into a set of coupled equations), with a box-normalized discretized continuum, and then, more extensively, by Durand and Paidarova [26], who approached the problem in terms of Eq. (6c) and the complex coordinate rotation method. In addition to their numerical results, the publications [25, 26] contain interpretations and critical remarks on aspects of the dynamics. 

Figure 33 Pictorial illustration of the Balslev-Combes theorem and the complex coordinate rotation method. Horizontal and vertical axes represent the real and imaginary parts of the complex energy, respectively. Application of the complex-scaling transformation X xe rotates the continuum hy an angle of —26 in the complex plane, leaving the resonances exposed as discrete states with square-integrahle wave functions and complex energies. Bound states remain on the real axis. Adapted with permission from Ref. 187 copyright 2013 American Institute of Physics.

Y.K. Ho, The method of complex coordinate rotation and its applications to atomic collision processes, Phys. Rep. 99 (1983) 1. 

We present a new effective numerical method to compute resonances of simple but non-integrable quantum systems, based on a combination of complex coordinate rotations with the finite element and the discrete variable method. By using model potentials we were able to compute atomic data for alkali systems. As an example we show some results for the radial Stark and the Stark effect and compare our values with recent published ones. 

To document the accuracy of our method we compared our results for the radial Stark effect obtained by complex coordinate rotation with results recently published [14]. 

The complex coordinate rotation (CCR) or complex scaling method (5,6,10,19) is directly based on the ABCS theory (1-3), therefore Reinhardt (5) also called it the direct approach. A complex rotated Hamiltonian, H 0), is obtained from the electron Hamiltonian of the atom, H, by replacing the radial coordinates r by re, where 0 is a real parameter. The eigenproblem of this non-Hermitian operator is solved variationally in a basis of square-integrable functions. The matrix representation of H ) is obtained by simple scaling of matrices T and V representing the kinetic and Coulomb potential part of the unrotated Hamiltonian H,... 

Unlike the above mentioned methods, another Floquet-theorem-based approach, the many-electron many-photon theory (MEMPT) of Mercouris and Nicolaides (71,72) does not involve complex rotated Hamiltonians. The complex coordinate rotation is used only to regularize that part of the wave functions which describes unbound electrons (see the CESE method). This allows efficient description of bound or quasi-bound states, involved in a problem under consideration, by MCHF solutions and therefore enables ab initio application to many-electron systems (71,72,83-87). 

The computational use of complex scaling of coordinates in the Hamiltonian is normally called the "complex coordinate rotation" (CCR) method. A brief reference to it is given in Sections 3.3 and 5.1, with references to related review articles. 

The underlying idea behind the complex coordinate rotation (CCR) method " that is suggested by the Balslev-Combes theorem is a complex scaling of the Cartesian coordinates in the Hamiltonian operator, each by the same complex phase factor x xe. This transformation defines a new, complex-scaled Hamiltonian, H H 0). In one dimension (for simplicity), the complex-scaled Hamiltonian is... 

The method of complex rotation, introduced by Aguilar and Combes (1971) and Balslev and Combes (1971) is reviewed, e.g., by Junker (1982), Reinhardt (1982), Ho (1983) and Buchleitner et al. (1994). It is a convenient and powerful tool for the calculation of positions and widths of bound states and resonances. Introducing the complex coordinates... 

The complex stabilization method of Junker (7), although it was introduced in a different way, gives practically the same computational prescription as the CESE method, as far as the way of using complex coordinates is considered. Another approach of this type, resembling the CESE method as well as the complex stabilization method, is the saddle-point complex-rotation technique of Chung and Davis (29). These methods provide cleair physical insight into the resonance wave function. They differ in the way the localized paxt of the wave function is expanded in basis sets and how it is optimized. 

Recently Lindroth (30) has introduced the concept of complex coordinates into nonvariational methods. The idea of Lindroth is to produce a basis set of one-particle complex rotated functions as solutions of the one-particle complex-rotated problem, and then to apply this basis within a bound-state method for a many-particle system. In this way the complex rotation has been combined with the many-body perturbation theory (30) and with the coupled cluster method (31). Apart from the fact that the resulting energy is complex and its real and imaginary part can be interpreted as the position and width of a resonance under consideration, the use of complex coordinates has the advantage that singularities caused by the degeneracy on the real axis between the resonance and the adjacent continuum are not present in the complex energy plane. 

If the condition 0 < ( 7 — jS) < tt is fulfilled, the second exponential factor in the last form of exp [iknR] goes to zero as p = R — oo. The channel function then behaves as that of a bound state. It is also important to note that this complex transformation of the coordinate does not affect the decreasing asymptotic behavior and the square integrability of a bound sfafe wavefunc-tion. This means that any method available for bound sfafe calculafions can be used for resonance calculations. A variant of the complex rotation method consists in transforming the reaction coordinate only after some value, say Rq. The form given to the coordinate is then Rq + [R — Ro)exp(k). This procedure is called exterior scaling [42,43]. 

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