Frame transformation

This definition of the quadrupole is obviously dependent upon the orientation of the chargi distribution within the coordinate frame. Transformation of the axes can lead to alternativi definitions that may be more informative. Thus the quadrupole moment is commonl defined as follows ... 

The above density operator needs to be transformed to the rotating frame before calculating the evolution of the density operator by the PIP2. It can be done by a frame transformation of... 

A three-dimensional simulation method was used to simulate this extrusion process and others presented in this book. For this method, an FDM technique was used to solve the momentum equations Eqs. 7.43 to 7.45. The channel geometry used for this method was essentially identical to that of the unwound channel. That is, the width of the channel at the screw root was smaller than that at the barrel wall as forced by geometric constraints provided by Fig. 7.1. The Lagrangian reference frame transformation was used for all calculations, and thermal effects were included. The thermal effects were based on screw rotation. This three-dimensional simulation method was previously proven to predict accurately the simulation of pressures, temperatures, and rates for extruders of different diameters, screw designs, and resin types. 

The Kf matrix is related to the conventional K matrix of MQDT by a frame transformation from parabolic to spherical co-ordinates the K matrix is then related by a further frame transformation to the quantum defects [43, 45], The first term in Eq. (4) gives a contribution to the photoionization intensity borrowed from the bound-state spectrum. The dlyom term represents direct photoionization, and the overall expression allows Fano-type interference between these terms. In Eq. (5) A is a phase shift in the parabolic rep-... 

II. Frame Transformations and Bound States [II. High Orbital Angular Momentum States... 

A central feature of molecular quantum defect theory is the use of frame transformations. These provide an elegant way of treating the breakdown of the Bom-Oppenheimer approximation that occurs systematically once an electron is excited into a high Rydberg state. 

Notice that in our calculation we have both closed and open ionization channels. This means that the quantum defect/frame transformation approach appears to work very well both below and above the critical region for which n - 100,..., 1000.1 would find it surprising if the approach failed in that region. 

W. H. Miller I believe that the reason the multichannel quantum defect theory (MCQDT) works well is that it assumes the ordinary Bom-Oppenheimer approximation (i.e., that the electron follows the molecular vibrational and rotational motion adiabatically) in the region close to the molecule, but not so in the region far from the molecule (where the electron moves more slowly than molecular vibration and rotation). The frame transformation provides the transition between... 

Ch. Jungen All I can say at this point is that the quantum defect/frame transformation approach appears to work for CaF around n = 14. We have chosen CaF, which is so highly polar, in order to ascertain this, and this is also the reason why we have made ab initio calculations in order to compare experiment and theory. I suppose that the dipole field is averaged out by the rotational motion, and thus one can get away with the customary frame transformation approach. 

In principle, the evolution in the laboratory frame differs from that in the rotation frame. Transformation to/from the rotation frame at the beginning and the end of the time interval introduces corrections to the evolution operator or order cv/B. This is however a negligible boundary contribution. Indeed, for a sufficiently long time interval St X 1 /o the phase shift due to the Lamb shift, of order c2St/B, is much larger (but still small, as long as SI, C B/c2). 

This coupling potential is smooth everywhere, which allows numerical calculations with high precision. There is no nonadiabatic coupling since the basis functions [0< )( 2C) are independent of p in each sector. The solution I Wf/o, 2C) is connected smoothly, in principle, from sector to sector by a unitary frame transformation from the /th set of channels to the (/ + l)st set [97-99]. The coordinate system is transformed from the hyperspherical to the Jacobi coordinates at some large p, beyond which the conventional close-coupling equations are employed for determining the asymptotic form of the wavefunction appropriate for the scattering boundary condition [100]. 

Chang, E.S. and Fano, U. (1972). Theory of electron-molecule collisions by frame transformation, Phys. Rev. A 6, 173-185. 

The elements of the reference frame transformation matrix are given by Eqs. 

SFCCCC Calculations of Ar-HD vdW Complex (20). The potential surface for Ar-HD can be obtained from the 3(6,8) potential of Ar-H2 by performing the asymmetric Isotope frame transformation (33) to a coordinate system based on the centre-of-mass of HD. This transformation Introduces Legendre terms of odd order Into the potential expansion, so that the diagonal vlbratlonally averaged Ar-HD potential, for example, has the form... 

The nuclear spin-interaction tensor is most readily expressed in its principal axis frame where only the M = 0, +2 terms are non-zero (and only the M = 0 term is non-zero for axial symmetry cases). It can then be expressed in the rotating frame as required for Eq. (8) by performing the frame transformation from principal axis frame to rotating frame ... 

In summary, then, all relaxation processes can ultimately be described as some linear combination of spectral density functions of the form shown in Eq. (11). We have here only considered explicitly the case of longitudinal or spin lattice relaxation in the laboratory frame (the so-called spin-lattice relaxation in the rotating frame being a different process), but a similar case can be made for transverse relaxation, relaxation processes in the rotating frame and crossrelaxation processes. The spectral densities involved in each case are J f( Mo> ) where co is the frequency of rotating frame transformation required to remove the stationary part of the total spin Hamiltonian in each case. This will be the Larmor frequency, co0, for any relaxation process taking place in the laboratory frame. For relaxation processes taking place... 

The idea is to incorporate some of the very important developments of MQDT made in atomic physics. An ambitious programme (not addressed here) is to develop and extend MQDT to molecular species by including full rovibronic structures for simple molecules within the framework of an extended theory by using frame transformations. Instead, we describe a far more restricted phenomenological approach, akin to MQDT for atoms, and applicable only to high n states of polyatomic species in which the rotational and most of the vibrational structure has collapsed. We can then compare the Lu-Fano graphs directly with those of corresponding atoms, and discuss both similarities and differences. 

The basic concepts of MQDT axe unfamiliar to most molecular spectro-scopists. The crucial ideas of frame transformation and phase shift are briefly explained in the next two paragraphs. 

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