Bending coordinates

The complete vibration-rotation Flamiltonian for acetylene-like tetraatomic molecules has been derived by Handy et al. by hand [155] and using a computer algebra program [156]. (Note that in both of the mentioned papers there are some minor errors, see also [144,157,158]). Handy uses as bending coordinates... [Pg.518]

First, let us note that the adiabatic potentials and V [Eq. (67)], even in the lowest order (harmonic) approximation, depend on the difference of the angles 4>j- and t >c this is an essential difference with respect to triatomics where the adiabatic potentials depend only on the radial bending coordinate p. The foims of the functions V, Vt, and Vc are determined by the adiabatic potentials via the following relations... [Pg.524]

On the basis of tbe above analyses, it follows that them is no need to compute multidimensional potential surfaces if one wishes to handle the R-T effect in the framework of the model proposed. In spite of that, such computations were carried out in [152] in order to demonstrate the reliability of the model for handling the R-T effect and to estimate the range in which it can safely be applied in its lowest order (quadratic) approximation. The 3D potential surfaces involving the variation of the bending coordinates pi, p2 and the relative azimuth angle 7 = 4 2 1 were computed for both component of the state. [Pg.527]

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... [Pg.533]

We introduce the dimensionless bending coordinates qr = t/XrPr anti qc = tAcPc ith Xt = (kT -r) = PrOir, Xc = sJ kcPc) = Pc nc. where cor and fOc are the harmonic frequencies for pure trans- and cis-bending vibrations, respectively. After integrating over 0, we obtain the effective Hamiltonian H = Ho + H, which is employed in the perturbative handling of the R-T effect and the spin-orbit coupling. Its zeroth-order pait is of the foim... [Pg.534]

The surface in Fig. 12 demonstrates that there is little coupling between the C—F translation coordinate and the bending coordinate of the complex. Stated another way, the time scale for intramolecular vibrational redistribution between these coordinates is slow compared to the time scale for breaking the C—F bond. These conclusions are not obvious upon examination of the minimum energy path shown in Fig. 11, and indeed such diagrams, while generally instructive, can lead to improper conclusions because they hide the multidimensional nature of the true PFS. A central assumption of statistical product distribution theories... [Pg.248]

Fig. 5.15 Schematic representation of the normal modes of the Fe(ni)-azide complex with the largest iron composition factors. The individual displacements of the Fe nucleus are depicted by a blue arrow. All vibrations except for V4 are characterized by a significant involvement of bond stretching and bending coordinates (red arrows and archlines), hi such a case, the length of the arrows and archlines roughly indicate the relative amplitude of bond stretching and bending, respectively. Internal coordinates vibrating in antiphase are denoted by inward and outward arrows respectively (taken from [63])...

Atomic units (me = 1, qe = 1, h = 1) are used throughout this chapter.] The coefficients T, T2, and To are assumed to be in general analytical functions of the bending coordinate p. The term Tz represent the operator describing the rotation of the molecule around the (principal) axis z corresponding to the smallest moment inertia—this axis coincides at the linear nuclear arrangement with the molecular axis. Now Tz can be written in the form... [Pg.587]

Figure 3.80 Angular dependences of (a) the second-order stabilization energy - A gem(2) and (b) the geminal Fock-matrix element F%

In a very recent publication [1], we have presented a new model for the rotation-vibration motion of pyramidal XY3 molecules, based on the Hougen-Bunker-Johns (henceforth HBJ) approach [2] (see also Chapter 15, in particular Section 15.2, of Ref. [3]). In this model, inversion is treated as a large-amplitude motion in the HBJ sense, while the other vibrations are assumed to be of small amplitude they are described by linearized stretching and bending coordinates. The rotation-vibration Schrddinger equation is solved variationally to determine rotation-vibration energies. The reader is referred to Ref. [1] for a complete description of the theoretical and computational details. [Pg.210]

Fig. 4. Cr(CO)s excited state relaxation dynamics comparison of semi-classical trajectory surface hopping (left), and MCTDH wave packet dynamics (right). Trajectory shows molecule passing through TBP Jahn-Teller geometry within 130 fs, then oscillating in SP potential well afterward. Wave packet dynamics plotted for the Si and S0 adiabatic states in the space the symmetric and asymmetric CCrC bending coordinates.

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