Coriolis matrix elements

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 ... 

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... 

Coriolis interactions [63A11, 84Gor] are caused by the coupling of the total angular momentum Jg and the vibrational angular momentum pg. The interaction matrix element between two interacting states v = (u, 0 ) and o = (o + 1, o -1) may be written... 

The operators (p) through. Jfj (p) are defined by Belov et al. [80 Bel] and p is the coordinate which describes the inversion motion. The terms containing (p), 2 (p) through (p) and 2/ ip) represent the harmonic, Coriolis, and anharmonic contributions, respectively, to the Ak = 3n interactions. The matrix elements of the Hamiltonian 3 are calculated in the basis of symmetrized inversion-rotation wavefunctions ... 

Although for planar geometries these two states, being of different symmetry, do not intersect (since the nonadiabatic matrix elements are zero for out of plane vibrational modes) the planar Cs symmetry is destroyed, and the non adiabaticity, which now becomes non zero, can lead to the deexcitation of OH(A 5 ). For triatomics the planar symmetry is conserved and the mixing of different spatial symmetry states can occur through the Coriolis interaction part of the Hamiltonian. Such a behaviour was found in the N2 + Na system [18], and can also be present for CO + OH. 

Coriolis interaction between a fundamental and a harmonic or combination band as well as between two harmonic or combination bands may also occur. The corresponding matrix elements may be found in [82Cha]. 

Coriolis interaction parameters (they are defined by the interaction matrix element = (6 (1+ 46 Ka and similar for Wse) ... 

Coriolis interaction exists not only in the 07=2, v = vg= and vg=2 triad, but is present between all vibrationalfy excited states listed above a- and 6-type between the states of different symmetry and c-type if the states are of the same symmetry. The 06=1, 05=1, and 04=1 fundamental states and the doubly excited states 07=2 and 09=2 are connected by Fermi interactions. The fitting parameters entering the off-diagonal matrix elements (w, 0 ) of the Hamiltonian are given by J =Jb i/c, [24,5]+ =+45+54). 

Equation (13) is the most important relation involved in the transformation to the new variables. Given the Coriolis coupling elements j i(s) = B(s), equation (13) is an integral equation for the unitary matrix U(s). By differentiating with respect to s, it is converted into the differential equation... 

Non-adiabatic transitions are induced by off-diagonal matrix elements of die nuclear kinetic operator on the electronic wavefunctions. In the case of OCS, a rotational (Coriolis) coupling is essential. If we assume the total angular momentum J to be zero, a rotational coupling term is expressed as follows ... 

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