Complex scaling rotation

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

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.,... 

The Bom-Oppenheimer Hamiltonian is not dilatation analytic if only the electronic co-ordinates are scaled since relS — R vanishes for a continuous range of 0 3 r = R and r.R = cos(0) /80/. The conceptual and numerical difficulties associated with complex scaling of both the nuclear and electronic coordinates and solution of the combined problem with a plethora of rotational, vibrational and electronic thresholds have prompted the formulation of (see ref. 22) two different schemes by McCurdy and Rescigno /81 / and Moiseyev and Corcoran... 

Fig.3a. Trajectory for the nearest scattering root (denoted by o in Fig. 1) as a function of the real scale factor p. The starting value is p = 0.65 at top left, and the increment Ap is 0.025. The rotation angle is fixed at a = 0.38. Note that the slope of the line corresponds to the value of a expected from the complex scaling theorem (AIm[E]/A Re[E] = tan 2a).

Fig. 3. Important valence orbitals of some metal fragments. The energy scale markings are eV. (Reprinted, with permission, from Ethylene Complexes, Bonding, Rotational Barriers, and Conformational Preferences, Albright, T. A. et al. J. Am. Chem. Soc. 101, 3802, Fig. 1, copyright, 1979, by the American Chemical Society)...

Fig. 10.7. Bound states and resonances in the complex plane with axes = 3ie(S) and E O = Qm(E). Complex scaling with rotation angle 0 rotates the continuum cuts into the negative imaginary energy plane (angle 20) thus exposing the resonances. (Adapted from Bliimel and Reinhardt (1992).)...

One of the most powerful tools to study resonances is complex scaling techniques (see Ref. 157 and references therein). In complex scaling the coordinate x of the Hamiltonian was rotated into the complex plane that is, H(x) > ll(xe 2). For resonances that have 0 .v tan-1 [Im(/i <,s )/Re ( (" ,v))] < < ) the wave functions of both the bound and resonance states are represented by square-integrable functions and can be expanded in standard L2 basis functions. 

Of the many approaches developped for the treatment of resonances, none has been in recent years so extensively used as the method of complex rotation or complex scaling (for reviews see 1-3). The appealing aspects of the method are ... 

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,... 

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. 

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... 

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.

To move up the scale of complexity one now needs to consider the energetics o rotation about each bond. The simplest approach is to assume that each bond can be treatec independently 2md that the total energy of the chain is the sum of the individual torsiona energies for each bond. However, this particular model has some serious shortcoming arising from the assumption of independence. 

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