Transformation coupled<->uncoupled

Once we have removed the terms which couple different electronic states (at least to a certain level of accuracy), we can deal with the motion in the other degrees of freedom of the molecule for each electronic state separately. The next step in the process is to consider the vibrational degree of freedom which is usually responsible for the largest energy separations within each electronic state. If we perform a suitable transformation to uncouple the different vibrational states, we obtain an effective Hamiltonian for each vibronic state. Once again, we adopt a perturbation approach. 

This is a sequence of three coupled<->uncoupled transformations, described by vector coupling or 3-j coefficients... 

The coupling coefficients (.r1yl.r2y2. ry) are introduced as members of the unitary matrix that transforms the uncoupled kets 11")ft) and r2y2), belonging to the y,th component of the irreducible representation F to the kets />) hence... 

The Rouse model, as given by the system of Eq, (21), describes the dynamics of a connected body displaying local interactions. In the Zimm model, on the other hand, the interactions among the segments are delocalized due to the inclusion of long range hydrodynamic effects. For this reason, the solution of the system of coupled equations and its transformation into normal mode coordinates are much more laborious than with the Rouse model. In order to uncouple the system of matrix equations, Zimm replaced S2U by its average over the equilibrium distribution function ... 

The point made in Eq. (3.31), namely, that the coupled, old, n action variables can be transformed to new, uncoupled, n - 1 conserved action variables is one to which we shall repeatedly return, in the quantum-algebraic context, in Chapters 4—6. Of course, we shall first discuss H0, which has n good quantum numbers, and which we shall call a Hamiltonian with a dynamical symmetry. At the next order of refinement we shall introduce coupling terms that will break the full symmetry but that will still retain some symmetry so that new, good, but fewer quantum numbers can still be exactly defined. In particular, we shall see that this can be done in a very systematic and sequential fashion, thereby establishing a hierarchy of sets of good quantum numbers, each successive set having fewer members. 

The system described is highly coupled, and although it can be solved analytically by conventional methods, such a solution requires substantial data to ensure reliable evaluation of the rate constants. Therefore, this system is transformed into an uncoupled system wherein... 

Formula (3) of the Introduction was the first example of how to use transformation matrices. The simplest case of transformation matrices is presented by formulas (10.4) and (10.5), while considering the relationships between the wave functions of coupled and uncoupled momenta. The Clebsch-Gordan coefficients served there as the corresponding transformation matrices. However, the theory of transformation matrices is most widely utilized for transformations of the wave functions and matrix elements from one coupling scheme to another (Chapters 11,12) as well as for their calculations. 

The magnitudes of < (j>(t)> versus time are shown in Fig. 4. The autocorrelation function for the positive displacement along Qx in the coupled potential (lowest curve in Fig. 4), starts out at 1 and drops to 0 over a shorter period of time than in the uncoupled potential (middle line). The Fourier transforms of these < (t) > give the spectra shown in Fig. 3. This reasoning explains why a positive displacement results in a broader progression and a negative displacement results in a narrower progression. 

Isolation — Means by which energized electrical circuits are uncoupled from each other. Two-winding transformers with primary and secondary windings are one example of isolation between circuits. In actuality, some coupling still exists in a two-winding transformer due to capacitance between the primary and the secondary windings. 

Example 7.4 Modified Graetz problem with coupled heat and mass flows The Graetz problem originally addressed heat transfer to a pure fluid without the axial conduction with various boundary conditions. However, later the Graetz problem was transformed to describe various heat and mass transfer problems, where mostly heat and mass flows are uncoupled. In drying processes, however, some researchers have considered the thermal diffusion flow of moisture caused by a temperature gradient. 

Let us now examine briefly the approach provided by the structural analysis. An examination of Eq. (5) shows that the rate of change of the amount a< of each species depends not only on at but on the amounts a, of other species as well. Thus, changes in the amount of A, during the reaction affect the amounts of species there is strong coupling between the variables in the set of Eqs. (5). It is this coupling between the variables a,-and aj that is the source of the difficulties outlined above. We shall show that a monomolecular reaction system with n species A, can be transformed, by means of appropriate mathematical operations (which involve only addition and multiplication), into a more convenient equivalent monomolecular reaction system, with n hypothetical new species Bi, which has the property that changes in the amount 6, of any species B, does not affect the amount of any other species Bj. This means that there is a set of species Bi equivalent to the set of species Ai such that the variables b, in the rate equations for the B species are completely uncoupled. 

The propagation of the solutions to the CC equations are carried out entirely in the fully-uncoupled, body-frame basis of eq. 2. At the end of the propagation, but before extraction of the S matrix, we transform the log-derivative matrix into the partially coupled basis discussed above. Here, the atomic states in the reactant arrangement are labeled by the total electronic angular momentum of the atom, ja, and... 

Examination of Eqs. 5.1.7-5.1.10 shows that the similarity transformation reduces the original set of n — 1 coupled partial differential equations to a set of n — 1 uncoupled partial differential equations in the pseudocompositions. Equation (5.1.10) for the /th pseudocomponent is... 

In this transformed Hamiltonian Hq again describes uncoupled system and bath the new element being a shift in the state energies resulting from the system-bath interactions. In addition, the interstate coupling operator is transformed to... 

In contrast, 3-j coeficients (e.g. Section 3.2.1.4) are useful in transforming between coupled abccz) and uncoupled aazbbz) basis sets. 

A transformation matrix that relates the uncoupled-spin and the coupled-spin kets can be generated by a recurrence procedure utilising only the Clebsch-Gordan coefficients. This can be applied to the uncoupled-spin interaction matrix in order to factorise the secular equation into blocks of lower dimension. 

Thus a transformation that minimizes A, the free energy of the uncoupled transformed system, Ho = Hj + H , hopefully minimizes the effects of the system-bath coupling term. If the transformation depends on some set of parameters c, this minimum is defined by the solution of the simultaneous equations... 

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