Complex-rotation method

K.T. Chung, B.F. Davis, Saddle-point complex-rotation method for resonances, Phys. Rev. A 26 (1982) 3278. 

Calculations by this method have shown remarkable Insensitivity to the nonlinear parameters of the complex part of the trial function. For the 2s autolonlzlng state, for which Equation 14 Is the trial function, Chung and Davis (33) obtained Ej. 57.8483 eV and r 0.12468 eV which compare well with the experimental (34) values of Ej. 57.8210.04 eV and T 0.13810.15 eV. The saddle point complex-rotation method Is strictly speaking another variant of CCI, but It demonstrates the premium In accuracy and efficiency to be gained from a well chosen trial function. In this case one In which the Feshbach Q-space (resonance) part of the trial wave-function Is optimized. 

An interesting problem of behaviour of the resonance levels along isoelec-tronic sequences can be investigated by the Z-dependent perturbation theory. Manning and Sanders (32,33) combined the complex rotation method with the Z-dependent perturbation theory. Expansion of the complex eigenvalue corresponding to the resonance as a power series in simultaneously yields values of both the resonance position and width for all members of an isoelectronic sequence. 

Actucdly, in the opinion of the author, the R-matrix complex-rotation method, presented above, should be considered as an implementation of the idea proposed by Nicolaides and Beck (37) and then by Simon (38), and known as the exterior complex-scaling. This interpretation is natural due to the fact that the R-matrix method and the exterior complex-scaling share the idea of dividing the configuration space into internal and external parts. 

Rescigno and McKoy (63) have presented a prescription for computing photoabsorption cross sections by the complex rotation method. The cross section for absorption of photons, fiw, by an atom or molecule in an initial state of energy Eg, can be computed as... 

Methods involving ri -correlated trial functions have also been developed for three-electron bound systems (117). One of them, the superposition of correlated configuration method of Woznicki (118), has been recently combined with the complex rotation method and succesfully applied to He autoionizing resonances (22,119-121). 

In this context, the idea of discrete numerical basis sets, introduced by Sa-lomonson and Oster (129) for the bound-state problem and combined with the complex-rotation method by Lindroth (30), is very interesting. One-particle basis functions are defined on a discrete grid inside a spherical box containing the system under cosideration. The functions are evaluated by diagonalizing the discretized one-particle complex-rotated Hamiltonian. Such basis sets are then used to compute autoionizing state parameters by means of bound-state methods (30,31,66). 

Ho also computed series of high-lying Feshbach-type resonances converging to the TV = 3,..,9 hydrogen threshols (133). Several tens of resonances were reported. Chen (134) applied a basis set of B-spline functions within the saddle-point complex rotation method to compute parameters of twelve and low lying Feshbach resonances. 

Manning and Sanders (33) used the Z-dependent perturbation theory combined with the complex rotation method to calculate the resonance position and width for the 2s2p autoionizing states of all members of the helium isoelectronic sequence. 

Energies and widths of fifteen resonances lying between the Li+ 2 P and 2 S thresholds have been computed by Wu and Chung (143) using the saddle-point complex-rotation method. 

Gou and Chung (127) used the saddle-point complex-rotation method to investigate triply excited resonances of Be" " and C " ". They computed the energies and the total widths of seven resonances. They also considered partial widths in one open channel approximation. 

Rydberg states of Ba and Sr in an external magnetic field have been considered by Halley, Delande, and Taylor (35), by means of the R-matrix complex rotation method. Seipp and Taylor (36) used the same method for the Stark and Stark-Zeeman problem of Rydberg states of Na. Themelis and Nicolaides (96) investigated the ls 2s 2p 3s 5, 3p 4s 5, and 3d bound states of Na. They used the CESE method to compute tunneling rates and scalar and tensor polarizabilities and hyperpolarizabilities. Medikeri, Nair, and Mishra (145,146) considered shape resonances in Be", Mg" and Ca" in two-particle-one-hole-Tamm-Dancoff approximation. Photodetachment rate for Cl" described by one-electron model was computed by Yao and Chu (73)... 

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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