Curvilinear bend coordinate

The majority of VTST calculations performed to date have been for atom-diatom collisions.For that kind of collision, reasonably accurate calculations of the vibrational energy levels are possible without excessive labor. For example, for a collinear minimum-energy path the vibrations orthogonal to the path consist of one stretch and a twofold degenerate bend. Use of a curvilinear bend coordinated 57,65 reduces the bend-stretch coupling, and principal anharmonicity can be included accurately in the bend by the harmonic-quartic approximation described above or by the WKB approximation. The stretch can also be treated accurately by the WKB approximation. 5 xt is also possible to estimate the effect of bend-rotational coupling,57 and in particularly... 

Figure 3 The internal bend extension coordinate for HCN is displayed pictorially for (a) the rectilinear, 5, and (b) the curvilinear R coordinate representations. (From Ref. 47.)...

Figure 3. The relation between S (the curvilinear displacement coordinate along the IRP) and the Jacobi bending angle a, is almost linear, for HCN CNH and the Murrell et al PES [54] (a) as well as for HO2 O2H and two different PES by Varandas et al [55,56] (b and c)...

From the analysis provided, it is clear that curvilinear internal coordinates are by far the best choice to represent anharmonic force fields. Then the question arises whether one could choose a best set of internal coordinates to represent the stretching and bending motions of molecular systems. 

To obtain the anharmonic terms in the potential, on the other hand, the choice of coordinates is important 130,131). The reason is that the anharmonic terms can only be obtained from a perturbation expansion on the harmonic results, and the convergence of this expansion differs considerably from one set of coordinates to another. In addition it is usually necessary to assume that some of the anharmonic interaction terms are zero and this is true only for certain classes of internal coordinates. For example, one can define an angle bend in HjO either by a rectilinear displacement of the hydrogen atoms or by a curvilinear displacement. At the harmonic level there is no difference between the two, but one can see that a rectilinear displacement introduces some stretching of the OH bonds whereas the curvilinear displacement does not. The curvilinear coordinate follows more closely the bottom of the potential well (Fig. 12) than the linear displacement and this manifests itself in rather small cubic stretch-bend interaction constants whereas these constants are larger for rectilinear coordinates. A final and important point about the choice of curvilinear coordinates is that they are geometrically defined (i.e. independent of nuclear masses) so that the resulting force constants do not depend on isotopic species. At the anharmonic level this is not true for rectilinear coordinates as it has been shown that the imposition of the Eckart conditions, that the internal coordinates shall introduce no overall translation or rotation of the body, forces them to have a small isotopic dependence 132). 

From calculations made for a number of simple molecules, it has become clear that in the cubic and quartic part of the potential written in curvilinear valence-force coordinates, the diagonal bond-stretching force constants (fm and fmr) are much larger than the bending and interaction constants. On this observation is based the simplest model potential, the anharmonic simple valence-force (SVF) model that consists of a complete harmonic potentialf with only the diagonal cubic and quartic stretching constants) and fmr added,... 

To understand the theoretical aspects of Fermi resonances, it is worthwhile to recall that in the conventional force-field method, Fermi resonance is described in terms of cubic or higher-order operators in the space of internal coordinates these internal coordinates can be either rectilinear or curvilinear, local or normal. In the previous example of interaction between v = n CH stretches and i = -I-1 CH bends, the corresponding Fermi operator can be written as... 

The equilibrium angle was obtained using a bending potential function in curvilinear coordinates. 

Corresponds to the minimum of the anharmonic bending potential function in curvilinear coordinate. 

To establish a connection between the stretch-bend potential coupling and the decay rate constant from overtone excited HC bonds in benzene, we have studied a two-mode model system describing a HC fragment of benzene. The Hamiltonian, which is expressed in internal curvilinear coordinates, includes both kinetic and potential stretch-bend coupling(13) ... 

Otherwise, Jacobi coordinates can also be used to account for isomerization reactions, e.g. HCN CNH and HO2 O2H, in which case the reaction coordinate is rather the bending angle a in Figure 2a. In addition, it is clear from Figure 3 that the connection between a and S, the curvilinear displacement along the IRP is nearly linear, for both systems, whatever the PES in the case of O2H and everywhere throughout the reaction. 

In the second-order methods we have described, the choice of coordinate system was not made explicit. Prom a quantum-chemical perspective, analytical derivatives are most conveniently computed in Cartesian (or symmetry-adapted Cartesian) coordinates. Indeed, second-order methods are not particularly sensitive to the choice of coordinate system and second-order implementations based on Cartesian coordinates usually perform quite well. As we discussed above, however, if the Hessian is to be estimated empirically, a representation in which the Hessian is diagonal, or close to diagonal, is highly desirable. This is certainly not true for Cartesian coordinates some set of internal coordinates that better resemble normal coordinates would be required. Two related choices are popular. The first choice is the internal coordinates suggested by Wilson, Decius and Cross [25], which comprise bond stretches, bond angle bends, motion of a bond relative to a plane defined by several atoms, and torsional (dihedral) motion of two planes, each defined by a triplet of atoms. Commonly, the molecular geometry is specified in Cartesian coordinates, and a linear transformation between Cartesian displacement coordinates and internal displacement coordinates is either supplied by the user or generated automatically. Less often, the (curvilinear) transformation from Cartesian coordinates to internals may be computed. The second choice is Z-matrix coordinates, popularized by a number of semiempirical... 

This expression allows for the variation of the reduced mass as a function of 0, and takes account of the fact that the coordinates of the large amplitude bending mode are curvilinear, rather than rectilinear. In the excited state, formaldehyde is a... 

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