Rotational energy coordinates

Clodius, W. B., and Quade, C. R. (1985), Internal Coordinate Formulation for the Vibration-Rotation Energies of Polyatomic Molecules. III. Tetrahedral and Octahedral Spherical Top Molecules, /. Chem. Phys. 82, 2365. 

There are two totally symmetric ( ,) normal modes and one b2 normal mode. (The convention is to use lowercase letters for the symmetry species of the normal modes.) The symmetry species of the normal modes have been found without solving the vibrational secular equation. Moreover, since there is only one b2 normal mode, the form of this vibration must be determined from symmetry considerations together with the requirement that the vibration have no translational or rotational energy associated with it. Thus (Fig 6.1), any bent XYX molecule has a b2 normal mode with the X atoms vibrating along the X—Y bonds and the Y atom vibrating in the plane of the molecule and perpendicular to the symmetry axis. On the other hand, there are two ax symmetry coordinates and the two ax normal vibrations are linear combinations of the ax symmetry coordinates, where the coefficients are dependent on the nuclear masses and the force constants. Thus the angles between the displacement vectors of the X atoms and the X—Y bonds for the ax modes of a bent XYX molecule vary from molecule to molecule. 

Tennyson, J. (1986). The calculation of the vibration-rotation energies of triatomic molecules using scattering coordinates, Computer Physics Reports 4, 1-36. 

First, let us consider what is meant by internuclear coordinates and, in particular, how many of these coordinates are needed in order to specify the electronic energy. We consider a collection of N atomic nuclei, which in this context are considered as point particles. In the following, we will for convenience refer to any collection of nuclei and electrons as a molecule . The atomic nuclei and the electrons may form one or more stable molecules but this is of no relevance to the following argument. The internuclear coordinates are defined as coordinates that are invariant to overall translation and rotation. These coordinates can, for example, be chosen as internuclear distances and bond angles. 

The extension of the trajectory calculations to a system with any number of atoms is straightforward except for the quantization of the vibrational and rotational states of the molecules. For a molecule with three different principal moments of inertia, there does not exist a simple analytical expression for the quantized rotational energy. This is only the case for molecules with some symmetry like a spherical top molecule, where all moments of inertia are identical, and a symmetric top, where two moments of inertia are identical and different from the third. For the vibrational modes, we may use a normal coordinate analysis to determine the normal modes (see Appendix E) and quantize those as for a one-dimensional oscillator. 

Recent calculations (see Section 3.1) show that the activated complex is non-linear, that is, the average rotational energy is (3/2)ksT and Ea = Eq + (.E ib) — (E ib). %2 = 4395 cm-1 and the two vibrational frequencies associated with the activated complex are 3772 cm-1 and 296 cm-1, respectively (remember that the third vibrational degree of freedom of the non-linear triatomic molecule is the reaction coordinate which is not included in (/A). The thermal energies associated with the... 

Fig. 4. Curve of rotation of trares-diazene coordinated in complex 1(N2H2) using the BP86/RI (left) and B3LYP (right) methods (TZVP basis set). The zero point of rotational energy is fixed arbitrarily. Note that the hydrogen atoms are not aligned with the S-Fe-S axes since they always point into the direction of the sulfurs lone pairs.

This means that scalar operators are invariant with respect to rotations in coordinate or spin space. An example for a scalar operator is the Elamiltonian, i.e., the operator of the energy. 

Figure 10 A typical trajectory showing rotational excitation accompanying vibrational de-excitation (i.e. a vibration to rotational energy transfer) [71]. The top panel shows the evolution in the Z (molecule-surface distance) and r (molecular bond length) coordinates. In the lower panel, the motion is projected onto the r — 0 (molecular bond orientation) plane. Coupling of vibrations and rotations occurs because the molecule attempts to dissociate at an unfavourable bond angle.

Figure 2 The initial dissociative sticking probability for D2 on Cu(l 1 1) extracted from the state selected measurements of desorbing molecules for various vibrational (A) and rotational (B) states of the molecule [19]. Vibrational energy couples effectively to the reaction coordinate, lowering the translational energy requirement for dissociation. Rotational energy initially hinders and then promoted dissociation. Similar effects of rotational energy are predicted in the trajectory calculations shown in (C) for molecules constrained to rotate in a plane perpendicular to the surface [29].

In the following sections of this paper, we describe a new model Hamiltonian to study the vibration—inversion—rotation energy levels of ammonia. In this model the inversion motion is removed from the vibrational problem and considered with the rotational problem by allowing the molecular reference configuration to be a function of the large amplitude motion coordinate. The resulting Hamiltonian then takes a form which is very close to the standard Hamiltonian used in the study of rigid molecules and allows for a treatment of the inversion motion in a way which is very similar to the formalism developed for the study of molecules with internal rotation [see for example ]. 

Figure 28. Potential of mean force W(i) (solid curve) and the average potential energy < V( )> (dashed curve) as functions of the Tyr-35 ring rotation reaction coordinate...

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