Complex rotation

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 scale parameter A in (10.4.7) is used to adjust the spatial range of the basis functions to the spatial extension of the wave functions of the model atom. 

The dots shown in the ten panels of Fig. 10.8 indicate the positions of bound states and resonances in the complex energy plane. The horizontal 

For n — oo the real parts of the resonances of an m series converge toward the ionization threshold energy Em = — l/2m. Therefore, as shown in Fig. 10.8, the scaled energy of the resonances converges toward Y — m for highly excited members of the m series. The fine horizontal lines in the panels correspond to the energy levels predicted by the analytical 

This formula is an empirical fit to the real parts of the resonance energies. It is derived in Section 10.4.2. 

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Moiseyev N, Certain P R and Weinhold F 1978 Resonance properties of complex-rotated Hamiltonians Molec. Phys. 36 1613... 

In this proposed process, p-hydride elimination first yields a putative hydride olefin rc-complex. Rotation of the -coordinated olefin moiety about its co-ordination axis, followed by reinsertion produces a secondary carbon unit and therefore a branching point. Consecutive repetitions of this process allows the metal center to migrate down the polymer chain, thus producing longer chain branches. Chain termination occurs via monomer assisted p-hydrogen elimination, either in a fully concerted fashion as illustrated in Figure 2b or in a multistep associative mechanism as implicated by Johnson1 et al. 

Since products of the electrode process are quickly transported out of the vicinity of the electrode disk, use of the rotating disk electrode complements the more complex rotating ring disk electrode (RRDE) [32]. Here, redox active products can be detected at the ring electrode, which is held at a separately controlled potential. 

The haphazard rotational motions of molecules or one or more segments of a molecule. This diffusional process strongly influences the mutual orientation of molecules (particularly large ones) as they encounter each other and proceed to form complexes. Rotational diffusion can be characterized by one or more relaxation times, t, describing the motion of a molecule or segment of volume, V, in a medium of viscosity, 17, as shown in the following equation ... 

A Fourier series is an example of an orthogonal polynomial, meaning that the individual terms which it comprises are independent of each other. It should be possible, therefore, to dissect a complex rotational energy profile into a series of N-fold components, and interpret each of these components independent of all others. For example, the onefold term (the difference between syn and anti conformers) in /7-butane probably reflects the crowding of methyl groups. 

An alternative approach to the calculation of MCD is suggested by its definition in terms of the complex rotation. Rather than... 

Proteins with cyclic or dihedral symmetry are particularly common. More complex rotational symmetries are possible, but only a few are regularly encountered. One example is icosahedral symmetry. An icosahedron is a regular 12-cornered polyhedron having 20 equilateral triangular faces (Fig. 4-24c). Each face can... 

It is well known that in n-body complexes rotational transitions of the order n may occur [400]. However, we will assume here that the interaction forces are pairwise additive and isotropic so that rotations and translations are uncorrelated. In this case, at most double transitions occur [400] and the correlation function of the total dipole moment can be written as... 

S. A. Rice Prof. Shapiro, is your method extendable to spectra that include resonances, either those that generate L2 states after complex rotation, or possibly other resonances If the answer is yes, I infer that we can put to rest the issue of the existence of quantum chaos. 

The more complex rotations and vibrations of polyatomic molecules are subject to the same principles, and distribution laws of the same kind apply, as will be shown in the following section. 

The l3C NMR spectrum in CDC13 at ambient temperature displays two Mo—CO (<5 222.2 and 228.79 ppm) and two characteristic high-field coordinated C=C (<5 58.64 and 71.91 ppm) resonances, indicating that the solid state structure of the Mo-chelate complex is maintained in solution. Conformational rigidity is generally observed in many -complexes. Rotational barriers about the metal-olefin axes and conformational preferences in these complexes have been estimated using extended Huckel-type calculations283. 

At each cf the wavelengths, the observed rotational distributions were determined by successively stripping the higher rotational distribution out of the total distribution. It is clear from the results in this table that complex rotational distributions are observed at all wavelengths. 

The situation is different when the rotational correlation time is shorter than zs and therefore dominates the overall correlation time rc in the dipolar term. In this case, the relative importance of the dipolar term is decreased by a factor xs/xr, which can be as large as 102 to 103 for small complexes with rotational correlation times of 10 10 to 10-n s and electronic relaxation times of 10 8 to 10-9 s - for example, Cu2+, Mn2+, and V02+. In macromolecular complexes, rotational correlation times are much larger, and situations of this type do not occur. 

The key geometric features of the optimized Pd-Cl bridging structures were found to nicely match those determined experimentally, and the presence of rather strong Cl-B interactions was substantiated by Natural Bond Orbital (NBO) analyses. For both complexes, rotation around the P— Pd bond allowed the location of another minimum devoid of any Cl-B interaction (B-pendant form, coordination mode E) (Figure 27). The two forms lie very close in energy (AG298 = 3-5 kcal), in agreement with the coexistence of the two coordination modes in solution. 

N. Moiseyev, P.R. Certain, F. Weinhold, Resonance Properties of Complex-Rotated Hamiltonians, Mol. Phys. 36 (1978) 1613. 

P. Krylstedt, M. Rittby, N. Elander, E. Brandas, A Complex Rotated Approach to Resonant Electron Scattering on Atoms in a Static Exchange Plus Polarization Formulation, J. Phys. B At. Mol. Phys. 20 (1987) 1295. 

M. Rittby, N. Elander, E.J. Brandas, Complex Rotated Titchmarsh-Weyl Theory A Review and Some Recent Results, Int. J. Quant. Chem. S17 (1983) 117. 

A technique for direct computations of the eigenvalues Er —zT/2 of H(6 = 0) with the outgoing-wave boundary condition is reviewed in detail in a chapter in Part I of this two-volume special issue of Advances in Quantum Chemistry on Unstable States in the Continuous Spectra [27]. Determination of the wavefunction of Eq. (2) with a real eigenvalue EQ using a judiciously chosen real, square-integrable basis set, followed by diagonalization of a complex Hamiltonian matrix for the whole Hq + H constructed in terms of basis functions of complex-rotated coordinates, is shown to be quite useful. 

For more accurate estimation, complex-rotation calculations [80] and coupled-channel calculations [81, 82] with the correct relativistic and radiative splitting AE taken into account reveal that, for H Feshbach series below the H(n = 2) threshold, the number vmax of resonances for the 1Se series is 4 (and the same for the system e+H) [81], vmax = 3 for the 1P° series [80,82], and vmax = 4 for the 3P° series [82], The relativistic effects also mix different LS states. This affects the resonance positions only slightly, but the components... 

A. Igarashi, I. Shimamura, Stable complex-rotation eigenvalues that correspond to no full resonances in scattering Examples in positron scattering by the helium ion, Phys. Rev. A 70 (2004) 012706. 

Complex rotation can be usefully applied also to the case of the interaction of an atom with a time-dependent perturbation. With the Floquet formalism by Shirley [41], it was shown that, for a time-periodic field, the dressed states of the combined atom-field system can be characterized non-perturbatively by the eigenstates of a time-independent, infinite-dimensional matrix. The combination of the Floquet approach with complex rotation, proposed by Chu, Reinhardt, and coworkers [37, 42, 43], permits to account for the field-induced coupling to the continuum in an efficient way. As in the time-independent case, this results in complex eigenvalues (this time to the Floquet Hamiltonian matrix) and again the imaginary parts give the transition rate to the continuum. This combination has since then been successfully used to examine various strong field phenomena a review can be found in Ref. [44]. 

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