CCR method

Most of the actual realizations of the CCR method diagonalize the matrix H 0). However, there are some versions of the CCR method which do not diagonalize H(0), but some other matrices related to it. Sommerfeld et al. (20) diagonalizes the matrix e H 0) = e T + V, so that the complex arithmetic is connected only with the matrix T, which is very sparse in comparison to... 

V. The resulting eigenvalues are multiplied by e to obtain the eigenvalues of H 0). In the Hermitian representation of the CCR method, introduced by Moiseyev (21) and modified by Bylicki (22), the eigenproblem of a Hermitian operator defined in terms of the Hermitian and non-Hermitian parts of H 0) is solved no complex arithmetic is involved. However, a disadvantage of this method is that it is addressed to a single resonance, whose approximate energy has to be known in advance. 

From a technical point of view, in the OCR method the electric and square magnetic terms of the Hamiltonian need to be scaled by factor e and e , respectively. From a mathematical point of view, the ABCS theory does not apply to this case. The application of the CCR method to the Stark problem has been justified by Herbst and Simon (48). 

Unlike the CCR method, the complex stabilization technique of Junker... 

The P spectrum of He" has also been treated by Bylicki (121) using an r, correlated basis set within the CCR method. Apart from the resonances reported by Themelis and Nicolaides (46), sixteen other doubly excited Feshbach as well as shape resonances have been discovered in the energy region up to the ls7p P° helium threshold. 

Triply-excited states of He" ion consitute a challenging subject for investigation demanding taking into account electron correlation effects and ein infinite number of open channels of autoionization. Application of a basis set of r, -correlated functions within the CCR method (119,120) gave accurate results for the positions and widths of the 2s 2p P , 2s2p P, P, ... 

The CCR method has been applied to exotic atoms. Among those, the resonances of negative positronium ion have been mostly investigated by Ho (147) and Ho and Bhatia (104-107). Positron-atom scattering resonances have been considered by Ho (108,148). Hu and Bhatia (149) have studied resonances in muonic systems. 

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. 

In spite of its conceptual simplicity, the CCR method, i.e., the repeated direct diagonalization of on a single function space of functions and... 

Furthermore, and most important, the CCR method is not suitable for the solution of the MEP, just like the direct diagonalization of H(r) on a single set of basis functions is not a practical method for solving the Schrodinger equation for even the ground states of polyelectronic atoms or molecules (More discussion is given in Sections 7 and 8.)... 

Furthermore, the CESE approach was proven convenient and very efficient when Tif dependent basis sets are used. Specifically, Bednarz and Bylicki [117] published results of their computations on the He 2s resonance, where they followed the CESE procedures with explicitly correlated basis functions. In explaining their work, these authors pointed out that "... one should realize that the p- dependent basis functions are not capable of repairing the slow radial convergence of the CCR method."... 

As is well-known, the two-electron system e -F H has been attracting theoretical attention for decades, with a large number of reports on the identification and nature of its resonances. Hence, it is possible to compare the CESE results of Ref. [123,124] with those published by other groups, who applied large scale R-matrix methods [130-132], or the CCR method [133-136], or specially improved close-coupling methods [137], or implementation of Feshbach s formalism. The related references are cited in Refs. [123,124] and below. 

In order to support the above, it suffices to recall here only a subset of the results that were published in Refs. [123,124]. To this purpose, 1 chose the resonances of H in the energy region below the n = 3 threshold, for the ipo, ijy ip symmetries. The CESE results [123] are listed in Tables 4.2 and 4.3, together with those obtained from large scale computations using the R-matrix method [130-132], or the CCR method [133-136]. 

Let me repeat the above argument, using the hydrogenic Hamiltonian with complex coordinates, which is the hallmark of the CCR method. According to mathematical analysis [107, 108], the poles of the hydrogenic H(rd ) are real and correspond to the discrete spectrum below threshold. As 6 increases from zero, these poles must remain at their initial positions on the real energy axis. 

In analogy to the field-free problem, the "Stark CCR" and the "Floquet CCR" methods have serious limitations with respect to the MEP. The way out was proposed in the late 1980s, when ideas and the general methodology of the SSA for the field-free resonances that solves state-specific CESEs were adapted so as to achieve the nonperturbative solution of problems of interaction of strong dc-and ac- electric fields and static magnetic fields with atomic (molecular) ground or excited states in terms of non-Hermifian formulafions, e.g.. Refs. [103,179-190]. 

Table 4-18 Correlation of temperature and residual weight data from TGA and CCR methods.

The Ohio State University proposed a method to combine coal gasification with a carbonation/calcination reaction (CCR) process to produce hydrogen [27]. CCR is also a sorption-enhanced process. In contrast to the ZECA process, the CCR method uses CaO to promote WGS reaction as follows ... 

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... 

This idea is readily extended to the Born-Oppenheimer electronic Hamiltonian by noting that x -r xe implies that interparticle coordinates should be scaled asr re . For 0 0, the operator H 6) is non-Hermitian and therefore admits complex eigenvalues. In its simplest form, the CCR method consists of determining these eigenvalues. 

It is emphasized in Ref. 187 that very flexible basis sets are required to deal with the finite-basis 0-dependence of the complex eigenvalues in CCR methods, and in particular to converge the resonance widths. However, the term very flexible is used in comparison to standard basis sets for valence anions, and in fact good results are obtained for auto-ionizing resonances of He, H , and Be using the aug-cc-pVTZ-l-[3s3p] basis,which includes three even-tempered diffuse s and p shells. This is not much different from the basis sets recommended here for proper description of loosely-bound electrons in general gas-phase calculations. 

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