Exterior complex scaling

B. Simon, The Definition of Molecular Resonance Curves by the Method of Exterior Complex Scaling, Phys. Lett. A71 (1979) 211. 

An alternative to uniform complex scaling (used in the studies mentioned so far) is exterior complex scaling [45,46], where the scaling of the coordinates starts at some finite distance from the origin. This approach has been used extensively on electron scattering problems in the absence of resonances, for example, in Refs. [47-50], for a review see Ref. [51], as well as to obtain resonance parameters in connection with electronic autoionization [52,53]. 

In practical implementations of complex scaling, the Hamiltonian is regularly discretized in finite space, for example, in a box of radius R. This yields a discrete pseudo-continuum with energies that fulfill Ek e w for Z = 0 and approaches it with increasing k and R for Z 0. If exterior complex scaling is made in such a finite box, Eq. (15) is adjusted to... 

The matrix elements in both BRe and Bim will, for both uniform and exterior complex scaling, be built from terms which, when the matrix representation of the rotated operator is constructed, are multiplied with complex constants. This will make the matrices BRe and BIm complex, but they will still be symmetric and antisymmetric, respectively, with respect to transposition, i.e.,... 

For transparency, we now use exterior complex scaling [Eq. (9)]. With Z = 2 the lowest resonance of -symmetry is known to be —0.78 a.u., see, for example, Ref. [15]. The bound part is dominated by 2s2 and the outgoing component is lses. Projection of the resonance wavefunction onto Is should then, for large enough r, gives an outgoing Coulomb wave... 

T.N. Rescigno, W. Vanrose, D.A. Homer, F. Martin, C.W. McCurdy, First principles study of double photoionization of H2 using exterior complex scaling, J. Electron Spectros. Relat. Phenom. 161 (2007) 85. 

C.W. McCurdy, M. Baertschy, T.N. Rescigno, Solving the three-body Coulomb breakup problem using exterior complex scaling, J. Phys. B At. Mol. Opt. Phys. 37 (2004) R137. 

N. Elander, E. Yarevsky, Exterior complex scaling method applied to doubly excited states of helium, Phys. Rev. A. 57 (1998) 3119. 

A. Scrinzi, Infinite-range exterior complex scaling as a perfect absorber in time-dependent problems, Phys. Rev. A 81 (5) (2010) 053845. 

A. Scrinzi, N. Elander, A finite element implementation of exterior complex scaling for the accurate determination of resonance energies, 98 (5) (1993) 3866. 

Figure 4 Complex scaling in three different forms the original Uniform complex scaling of Balslev and Combes. The Exterior complex scaling proposed by Simon and the Smooth exterior complex scaling studied by Helffer are all discussed in the text.

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. 

From Eqs. (12 and 13), to computational methods such as the CESE, the exterior complex scaling and the complex absorption potential methods... 

Corresponding to the above idea are methods that implement the regularization technique of "exterior complex scaling" (ECS), which was introduced to atomic physics in Ref. [92b]—see also Refs. [95,96], and which is discussed in Section 4.3. 

C.A. Nicolaides, H.J. Gotsis, M. Chrysos, Y. Komninos, Resonances and exterior complex scaling, Chem. Phys. Lett. 168 (1990) 570. 

Exterior Complex Scaling Finite Element Calculations Based on an Adiabatic and a Strictly Diabatic Basis, J. Mol. Spectrosc. 214 (2002) 103-110. 

In this section, we apply the Floquet-TDDFT approach with exterior complex scaling to the calculation of multiphoton detachment of Li in monochromatic linearly polarized laser field [28,64]. When applying a complex scaling transformation, a... 

N. Rom, E. Engdahl, and N. Moiseyev,/. Chem. Phys., 93, 3413—3419 (1990). Tunneling Rates in Bound Systems Using Smooth Exterior Complex Scaling within the Framework of the Finite Basis Set Approximation. 

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