Complex coordinate method

N. Moiseyev, Resonance by the Complex Coordinate Method with Hermitean Hamiltonian, Chem. Phys. Lett. 99 (1983) 364. 

N. Moiseyev, Derivations of universal exact complex absorption potentials by the generalized complex coordinate method, J. Phys. B At. Mol. Opt. Phys. 31 (7) (1998) 1431. 

Moiseyev, N. (1984). The Hermitian representation of the complex coordinate method Theory and application, in Lecture Notes in Physics, Vol. 211, ed. S. Albeverino, L.S. Ferreira, and L. Streit (Springer, Heidelberg). 

In contrast to earlier complex coordinate methods which require a specific analytic continuation of the Hamiltonian, complex basis function methods for resonances rely on the existence of a complex varla tlonal principle for complex resonance energies,... 

Following that we present calculations on resonances in a two-mathematical-dimensional (2ND) model for van der Waals systems. We will compare the complex eigenvalues obtained previonslyCl) by the complex coordinate method with those obtained from the distorted wave Born approximation (DWBA). Part of the motivation for making this comparison is to assess the accuracy of the DWBA before applying it to more realistic problems. 

Of these direct approaches the complex coordinate method is the most rigorous one. In principle it yields the exact energies of the poles of the scattering matrix, which. Ignoring the background contribution to the scattering, gives the resonance position and width. 

The complex coordinate method was applied recently by Christoffel and Bowman(l) to a model collinear hamiltonian used earlier by Bastes and Marcns(22) in qnantnm and semiclassical studies of the resonances in that system. The hamiltonian is ( >1)... 

We note that in spite of the increasing popularity of the complex coordinate method (CCM), most atomic and molecular problems... 

Due to the external electric fields the ionization threshold of an atom is lowered from the field-free value, hi fact, all states become quasi-bound, as the electron can ionize by tunneling through the potential states. This process becomes important for states close to the classical ionization energy or above. The combination of the complex coordinate method with an R-matrix quantum-defect method is discussed for parallel magnetic and... 

In the complex coordinate method the real configuration space coordinates are transformed by a complex dilatation. The Hamiltonian of the system is thus continued into the complex plane. This has the effect that, according to the boundaries of the representation, complex resonances are uncovered with square-integrable wavefunc-tions and hence the space boundary conditions remain simple. This square integrability is achieved through an additional exponentially decreasing term... 

In this article a part of the implementations of complex-coordinate methods is briefly reviewed. We consider only recent applications to free atoms (ions) or atoms interacting with an external field. Mainly recent papers of the nineties are quoted here. However, earlier review and precursory articles are also cited. 

In section 2 the most often asked physical questions concerning resonant phenomena in atoms are recollected, together with methods capable of answering them by implementation of complex coordinates. Problems connected with basis sets used within the complex-coordinate methods are described in section 3. Finally, in section 4, recent applications to atoms are listed. 

The complex-coordinate methods can be applied to atoms in external fields... 

N. Moiseyev,/. Phys. B, 31, 1431—1441 (1998). Derivations of Universal Exact Complex Absorption Potentials by the Generalized Complex Coordinate Method. 

The simplest way to add a non-adiabatic correction to the classical BO dynamics method outlined above in Section n.B is to use what is known as surface hopping. First introduced on an intuitive basis by Bjerre and Nikitin [200] and Tully and Preston [201], a number of variations have been developed [202-205], and are reviewed in [28,206]. Reference [204] also includes technical details of practical algorithms. These methods all use standard classical trajectories that use the hopping procedure to sample the different states, and so add non-adiabatic effects. A different scheme was introduced by Miller and George [207] which, although based on the same ideas, uses complex coordinates and momenta. 

The Principles Determining the Structure of Complex Ionic Crystals.—The success of the coordination method in predicting structures for brookite and topaz has led to the proposal of a set of principles governing the structure of a rather extensive class of complex ionic crystals. 

The crystal structures in Chapters 5 and 6 were determined by x-ray diffraction, and the papers illustrate Pauling s approach to this experimental technique, including his most notable methodological contributions—the coordination method (SP 42) and the stochastic method (SP 47). In its day, SP 47 was a tour de force in the determination of a complex crystal structure. SP 46 contains Pauling s famous discovery of two quite different crystal structures giving the same x-ray diffraction pattern, which violated the then-current conventional wisdom in x-ray crystallography. 

Asymmetric hydrosilylation can be extended to 1,3-diynes for the synthesis of optically active allenes, which are of great importance in organic synthesis, and few synthetic methods are known for their asymmetric synthesis with chiral catalysts. Catalytic asymmetric hydrosilylation of butadiynes provides a possible way to optically allenes, though the selectivity and scope of this reaction are relatively low. A chiral rhodium complex coordinated with (2S,4S)-PPM turned out to be the best catalyst for the asymmetric hydrosilylation of butadiyne to give an allene of 22% ee (Scheme 3-20) [59]. 

Deng, L., Ziegler, T., 1997, Theoretical Study of the Oxidation of Alcohols to Aldehyde by d° Transition-Metal-Oxo Complexes Combined Approach Based on Density Functional Theory and die Intrinsic Reaction Coordinate Method , Organometallics, 16, 716. 

Distibines fall lowest in the spectrochemical series P2 > As2 > St>2,1529 a result which corresponds with the electronic spectra data for the monodentate series PEt3, AsEt3, SbEt3.1530 Mossbauer spectroscopy is a useful method for platinum stibine complexes. Coordination of the stibine causes a large increase in isomer shift and a decrease in quadrupole coupling constant in the 121Sb spectrum, as expected for the formation of a donor bond.1531... 

Supramolecular or coordinative [120] complexation A method of preparing dendrimers with central metal complex entity accomplished by Kawa and Frechet consists in the complexation of a metal cation with suitably functionalised den-dron ligands. Self-assembly of three Frechet-type dendrons, each with a carboxy-... 

As with Ps , there is only one bound state of Ps2 but there exist Rydberg series of autodissociating states arising from the attractive interaction between one of the positrons and the residual Ps- (or between one of the electrons and the charge conjugate of Ps ). The positions and widths of several of these states were determined by Ho (1989) using the complex coordinate rotation method. To date Ps2 has not been observed in the laboratory. 

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

Y.K. Ho, The method of complex coordinate rotation and its applications to atomic collision processes, Phys. Rep. 99 (1983) 1. 

While molecular assembly has proven to be effective for a photoelectric conversion system, coordination reactions are possibly a simple approach for connecting such functional molecules, as presented in the previous section. We applied the stepwise coordination method to prepare a photoelectric conversion system. Since the molecular wire exhibits redox conduction through the wire,11,13 efficient photo-electron transport through the redox sites in the wire is also expected. In this section, we demonstrate the fabrication of a photoelectric conversion system using ITO electrodes modified with M(tpy)2 (M = Co, Fe, Zn) complex wires with a terminal porphyrin moiety as a photosensitizer. The behavior of photo-electron transfer from porphyrin to ITO through the molecular wire was investigated by changing the metal element in the M(tpy)2 moieties.14... 

FIGURE 11. Chemical structure of the ligands used in this chapter, and stepwise coordination methods for the preparation of modified ITO electrodes (i) immobilization of tpy-BzA, (ii) complexation with a metal ion, and (iii) complexation with tpy-ZnTPP. (Reprinted with permission from Ref. 14.)... 

This section described the fabrication of ITO electrodes modified with porphyrin-terminated M(tpy)2 complex wires by the stepwise coordination method, and it is demonstrated that the electronic nature of the molecular wire is critical to the photoelectron transfer from the porphyrin to ITO. These results suggest that the new facile fabrication method of molecular assemblies is effective in the construction of photoelectron transfer systems. The system could be upgraded by extending the wire length, embedding the redox potential step in the wire, increasing the photoreceptors in the wire, and/or incorporating donors and acceptors. 

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