Vector Wave Functions

We define the vector spherical harmonics of left- and right-handed type as 

Since Imn and Vmn are linear combinations of nimn and Umn, we deduce that the system of vector spherical harmonics of left- and right-handed type is also orthogonal and complete in 

The independent solutions to the vector wave equations can be constructed as [215] 

The superscript T stands for the regular vector spherical wave functions while the superscript 3 stands for the radiating vector spherical wave functions. It is useful to note that for n = m = 0, we have mI q = = 0. Mi, , 

The vector spherical wave functions can be expressed in terms of vector spherical harmonics as follows  

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To solve the single-particle problem it is convenient to introduce a new representation, where the coefficients ca in the expansion (1) are the components of a vector wave function (we assume here that all states a are numerated by integers)... 

As mentioned earlier, the resolvent is a tool allowing one to formally write down the solution of an eigenvalue/eigenvector problem. It is also useful for developing perturbation expansions, which, as we saw previously, require somewhat tedious work when done in terms of vectors (wave functions). 

The minute particles, which a solid consists of, have the extraordinary quantum features. However, there is a gap between quantum theory on the one hand and engineering on the other hand. Even the principal notions and terms are different. The quantum physics operates with such notions as electron, nucleus, atom, energy, the electronic band structure, wave vector, wave function, Fermi surface, phonon, and so on. The objects in the engineering material science are crystal lattice, microstructure, grain size, alloy, strength, strain, wear properties, robustness, creep, fatigue, and so on. 

The concept of GMM and the details of computation were presented by Yu-Lin Xu in a series of papers (e.g. Xu 1995,1997,1998a). The main idea is to express the scattering field of the particle j as an infinite series of spherical vector wave functions (SVWF) (i.e. spherical harmonics for vector fields) that are defined for... 

O. Cruzan, Translational addition theorems for spherical vector wave functions. Q. Appl. Math. 20 (1), 33 0 (1962)... 

The spherical vector wave functions (SVWF) are the general solution of the vectorial Helmholtz differential equation in spherical coordinates (Xu 1995) ... 

For a system of Af particles with o = f3i = 7 = 0, I = 1,2,. ..,AA, the transformations of the vector spherical vector wave functions involve only the addition theorem under coordinate translations, i.e.,... 

The regular and radiating spherical vector wave functions can be expressed as integrals over vector spherical harmonics [26]... 

The following theorems state the completeness and linear independence of the system of regular and radiating spherical vector wave functions on two enclosing surfaces. 

K.A. Aydin, A. Hizal, On the completeness of the spherical vector wave functions, J. Math. Anal. Appl. 117, 428 (1986)... 

A. Bostrom, G. Kristensson, S. Strom, Transformation properties of plane, spherical and cylindrical scalar and vector wave functions, in Field Representations and Introduction to Scattering, ed. by V.V. Varadan, A. Lakhtakia, V.K. Varadan (North Holland, Amsterdam, 1991) pp. 165-210... 

As mentioned in the introduction, the simplest way of approximately accounting for the geomehic or topological effects of a conical intersection incorporates a phase factor in the nuclear wave function. In this section, we shall consider some specific situations where this approach is used and furthermore give the vector potential that can be derived from the phase factor. 

Single surface calculations with a vector potential in the adiabatic representation and two surface calculations in the diabatic representation with or without shifting the conical intersection from the origin are performed using Cartesian coordinates. As in the asymptotic region the two coordinates of the model represent a translational and a vibrational mode, respectively, the initial wave function for the ground state can be represented as. 

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

One starts with the Hamiltonian for a molecule H r, R) written out in terms of the electronic coordinates (r) and the nuclear displacement coordinates (R, this being a vector whose dimensionality is three times the number of nuclei) and containing the interaction potential V(r, R). Then, following the BO scheme, one can write the combined wave function [ (r, R) as a sum of an infinite number of terms... 

In an Abelian theory [for which I (r, R) in Eq. (90) is a scalar rather than a vector function, Al=l], the introduction of a gauge field g(R) means premultiplication of the wave function x(R) by exp(igR), where g(R) is a scalar. This allows the definition of a gauge -vector potential, in natural units... 

In the -electronic-state adiabatic representation involving real electronic wave functions, the skew-symmetiic first-derivative coupling vector mahix... 

When constructing more general molecular wave functions there are several concepts that need to be defined. The concept of geometry is inhoduced to mean a (time-dependent) point in the generalized phase space for the total number of centers used to describe the END wave function. The notations R and P are used for the position and conjugate momenta vectors, such that... 

The END equations are integrated to yield the time evolution of the wave function parameters for reactive processes from an initial state of the system. The solution is propagated until such a time that the system has clearly reached the final products. Then, the evolved state vector may be projected against a number of different possible final product states to yield coiresponding transition probability amplitudes. Details of the END dynamics can be depicted and cross-section cross-sections and rate coefficients calculated. 

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