Coriolis matrix

The body-fixed coupled equations are thus very similar in form to those for atom-diatom systems the only differences are that the potential matrix elements here are a generalisation of the atom-diatom case and that Hmon takes a more complicated form (and may have matrix elements off-diagonal in k). As for atom-diatom systems, considerable savings in computer time may be achieved by performing helicity decoupling calculations, in which the Coriolis matrix elements off-diagonal in K are neglected. 

The operator Tang contains the cross-terms that give rise to the Coriolis coupling that mixes states with different fl (the projection of the total angular momentum quantum number J onto the intermolecular axis). This term contains first derivative operators in y. On application of Eq. (22), these operators change the matrix elements over ring according to... 

B. A. Hess Prof. Jungen, in your talk you emphasized that you don t have to calculate matrix elements of d/dQ or Coriolis coupling. My impression is that this is due to your most appropriate choice of a diabatic basis, which is generally what ab initio quantum chemists do when they want to avoid singularities in the adiabatic basis. On the other hand, the absence of explicit Coriolis coupling matrix elements is due to the transformation to a space-fixed coordinate system. 

The transition class of parameters were g (the ratio of the spin g-factor and the free proton g-factor) and H (the enhancement factor of the AKO El Nilsson matrix elements). We found g = 1.0 for the unique-parity bands and g = 0.7 for the normal-parity bands. An H of 3.0 was found to give good reproduction of the relative El and Ml rates for the Coriolis-... 

This type of distortion is characteristic of Coriolis coupling, where the matrix elements have the form,... 

In order to appreciate this point more clearly, we confine our attention to the contributions to 3Qff produced by perturbations from the spin-orbit coupling 3Q0 and the electronic Coriolis mixing 30-ot- If we represent an off-diagonal matrix element of the former by (L S) and the latter by (N L), we can describe some examples of these higher order terms, as shown in table 7.1. The third-rank terms appear only in states of quartet or higher multiplicity and the fourth-rank terms in states of quintet (or higher) multiplicity. With the important exception of transition metal compounds, the vast majority of electronic states encountered in practice have triplet multiplicity or lower. 

Because the approximation described by Eq. (3.47) fails if the inversion and vibrational wave functions are strongly mixed, the Coriolis operator defined by Eq. (5.8) cannot be treated by the numerical methods described in Sections 5.1 and 5.2. Instead of the perturbation treatment described in Section 5.1, we must use a variational approach in which the energy levels are calculated as eigenvalues of an energy matrix the off-diagonal elements of this matrix are the matrix elements of the Coriolis operator ) 2,4 ... 

Thus, for each value of / we have two 7x7 matrices, one connecting symmetric m a states with the antisymmetric states ( 1 2 + 1 4) and vice versa (Fig. 16). For special values of the rotational quantum numbers/, k, instead of a 7x7 matrix we have smaller blocks . This factorization is the analog of the factorization of the matrices describing Coriolis interactions in a C3V rigid molecule and can be used for a qualitative interpretation of the anomaly in Fjg. 17. For example, the/ =K levels in the —I component of the 1 4 level have basically the ground-state character of the rotational dependence of the inversion-splitting (Fig. 17) because they are obtained from the 1x1 block and therefore are unperturbed. 

The present stage is suitable for the introduction of the Coriolis coupling constants, f%k 63, 64). This may seem curious, but it is in accordance with the fact that these constants are appropriate only when rectilinear coordinates are involved. This will be further discussed below in relation to the vibration-rotation part of the G-matrix. Here it is convenient to give a general definition,... 

The matrix M is an inverse generalised inertia tensor while G is is an inverse metric matrix. The operator A< is associated with the Coriolis coupling and so no coordinate system can be found in which it will vanish. The operator t, is dependent on internal coordinate choice and it is possible to choose a coordinate system in which this term vanishes. 

Adiabatic states with different electronic symmetries couple not by the radial coupling of Eq. (4) but by nonadiabatic rotational (Coriolis) coupling, Hcor For X and II states of a diatomic molecule, the Coriolis coupling matrix element is given by (see Sec. Ill)... 

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

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