Rotations/rotation matrix elements

A value of 0 = 0° corresponds to a pure ground state, and 6 = 90° to a pure 3,2 ground state. Since the d orbital rotation matrix elements are different for the d and d -y orbitals, this will lead to a variation of the local g tensor of the Fe" site with the mixing angle d ... 

The electron capture processes are driven by non-adiabatic couplings between molecular states. All the non-zero radial and rotational eoupling matrix elements have therefore been evaluated from ab initio wavefunctions. 

The rotational coupling matrix elements between Z-IT and fl-A states have been evaluated analytically by use of the L. and L. operators. 

Fig. 3. Rotational coupling matrix element between the ll (N (3p) + He (ls)) state and the states of single-electron capture.

On the basis of these formulae one can convert measurements of area, which equals the integral in the latter formula, under spectral lines into values of coefficients in a selected radial function for electric dipolar moment for a polar diatomic molecular species. Just such an exercise resulted in the formula for that radial function [129] of HCl in formula 82, combining in this case other data for expectation values (0,7 p(v) 0,7) from measurements of the Stark effect as mentioned above. For applications involving these vibration-rotational matrix elements in emission spectra, the Einstein coefficients for spontaneous emission conform to this relation. 

Although transformation of coefficients pj into coefficients qj is readily practicable, the resulting values for CO adopt unwieldy magnitudes. Chackerian and Tipping [141] fitted a function of the latter form from experimental and theoretical (computations of molecular electronic structure) information in judicious combination, according to which they calculated vibration-rotational matrix elements for transitions in bands 5-0 and 6-0 fitting the latter values with formula 84 yielded the values of quantities presented above. Rational functions, such as those in formulae 92 - 94 or others, transcend the spirit of Dunham s approach because their construction incorporates physical knowledge of a quantity that is superfluous for invocation of a mere truncated polynomial. 

Purely rotational matrix elements. For v = v = 0 and j = f = 0, with the choice of 5 given above, we thus have... 

The last equation shows that the statistical tensors obey well-defined rotational properties which are similar to those of the angular momentum functions, except for the complex-conjugated rotation matrix elements which describe the inverse rotation x", y", z" - x, y, z, and that no physical significance is attached to the coefficients k and k (they are convenient summation indices only). 

Similarly, one obtains (in order to make the reference axis more explicit, the angles (O, 0, x) in the rotation matrix elements of equ. (8.108c) have been replaced by ((P,x))... 

The latter can be done analytically by employing the standard expressions for the integrals of three rotation matrix elements M respectively three spherical harmonics Yjq (Edmonds I974 ch.4). Without explicitly quoting the result for the angular integral we note the following selection rules ... 

If the matrix A is symmetric, Hermitian, or unitary, then there is a system of 3 x 3 rotation matrices R (and their inverses R x) which will rotate the matrix elements Ay so that the only nonzero elements will appear on the diagonal this is known as a similarity transformation or as a principal-axis transformation or diagonalization ... 

The rotation matrix elements reduce to spherical harmonics when j is an integer and m or n is zero ... 

With our choice of phases, the symmetric top wave function is related to the corresponding rotation matrix element by... 

This means that S is invariant under rotations. The matrix elements of S are easily obtained using Eq. (34). Thus... 

Radial and rotational coupling matrix elements have been calculated in both the triplet and singlet manifolds. The radial coupling between all pairs of states of the same symmetry have been calculated by means of the finite difference technique ... 

The vibration-rotation interaction is the effect arising from coupling terms between angular and vibrational momenta as well as from the dependence of the rotational G-matrix elements (the /u-tensor) on the internal coordinates. The importance of this effect may to some extent be reduced provided an appropriate axis convention is used. The axis convention is the set of rules defining the orientation of the molecular axes, eg, g = x,y, z, relative to an arbitrary configuration as given by the position vectors, Ra, a. = 1, 2,... N. These rules can be expressed in three relations between the rag components, similar to the center of mass conditions(2.4). We shall refer to these relations as the axial constraints . Usually Eckart-condi-tions39 are imposed, but other possibilities may be considered. 

P indicates that the components are those for the principal axis system (PAS) of the tensor. The terms o(QpL(f)) are Wigner rotation matrix elements. They are functions of the set of Euler angles, Qpf (f), which relates the PAS of the chemical shift to the laboratory frame. Due to MAS, these angles are time dependent. A full treatment of the orientation dependence of the chemical shift requires the transformation between several different reference frames. 

There is no coupling between the two manifolds (coupling matrix element within each manifold is... 

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