The method of complex rotation

Complex rotation of the Hamiltonian, H He, is usually performed by a direct transformation of the radial variable either uniformly, 

With the scaling in Eq. (8), the one-particle hydrogen-like Schrodinger equation without external fields transforms as 

At first sight, one might think that complex scaling is just a variable transformation. Since one does not expect a variable transformation to change 

The requirement that all eigenvectors, both the proper and the generalized ones, remain finite as r - oo 

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 

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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... 

There are also important advances concerning the theoretical description of Rydberg atoms in strong radiation fields. Buchleitner et d. (1995) report on fully fiedged three-dimensional computations of the microwave ionization problem. They use the method of complex rotation discussed in Section 10.4.1, adapted to the computation of the resonances of the Floquet operator. The computed ionization probabilities are in good overall agreement with existing experimental data. 

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 ... 

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

Microwave studies in molecular beams are usually limited to studying the ground vibrational state of the complex. For complexes made up of two molecules (as opposed to atoms), the intennolecular vibrations are usually of relatively low amplitude (though there are some notable exceptions to this, such as the ammonia dimer). Under these circumstances, the methods of classical microwave spectroscopy can be used to detennine the stmcture of the complex. The principal quantities obtained from a microwave spectmm are the rotational constants of the complex, which are conventionally designated A, B and C in decreasing order of magnitude there is one rotational constant 5 for a linear complex, two constants (A and B or B and C) for a complex that is a symmetric top and tliree constants (A, B and C) for an... 

Abernathy and Sharp (130,145) treated the intermediate regime, when the reorientation of the paramagnetic species is in-between the slow- and fast-rotations limits. They applied the spin-dynamics method, described in Section VI, to the case of outer-sphere relaxation and interpreted NMRD profiles for non-aqueous solvents in the presence of complexes of Ni(II) (S = 1) and Mn(III) (S = 2). 

Since the early times of stereochemistry, the phenomena related to chirality ( dis-symetrie moleculaire, as originally stated by Pasteur) have been treated or referred to as enantiomericaUy pure compounds. For a long time the measurement of specific rotations has been the only tool to evaluate the enantiomer distribution of an enantioimpure sample hence the expressions optical purity and optical antipodes. The usefulness of chiral assistance (natural products, circularly polarized light, etc.) for the preparation of optically active compounds, by either resolution or asymmetric synthesis, has been recognized by Pasteur, Le Bel, and van t Hoff. The first chiral auxiliaries selected for asymmetric synthesis were alkaloids such as quinine or some terpenes. Natural products with several asymmetric centers are usually enantiopure or close to 100% ee. With the necessity to devise new routes to enantiopure compounds, many simple or complex auxiliaries have been prepared from natural products or from resolved materials. Often the authors tried to get the highest enantiomeric excess values possible for the chiral auxiliaries before using them for asymmetric reactions. When a chiral reagent or catalyst could not be prepared enantiomericaUy pure, the enantiomeric excess (ee) of the product was assumed to be a minimum value or was corrected by the ee of the chiral auxiliary. The experimental data measured by polarimetry or spectroscopic methods are conveniently expressed by enantiomeric excess and enantiomeric... 

Even in modern quality control laboratories you will find a number of traditional methods for the identification of single flavour compounds, for example the estimation of optical rotation, refractive index, density and melting point, since these methods are generally accepted, effective and less time-consuming. Especially for the purpose of fast identification checks of more complex systems, spectroscopic methods, above all infrared (IR) and near-IR spectroscopy, are gaining more and more importance. 

The w parameter is conveniently tabulated as a function of t. It may be noted that the correction to the semi-classical expression is largest when the vibrational frequencies are spread over a wide range. The method of Whitten and Rabinovitch [8] has been widely used and has provided a valuable service to the kinetics community. An expression for the density of states is obtained upon differentiation of eqn. (12). Their method also encompasses the case where there is an internal rotation. For the transition complex at low energies, W( e + ) is conveniently found by a straight count. 

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