Triatomic symmetry coordinates

Figure 7.23 Symmetry coordinates of a bent symmetric triatomic molecule. After Ref [75].

T is a rotational angle, which determines the spatial orientation of the adiabatic electronic functions v / and )/ . In triatomic molecules, this orientation follows directly from symmetry considerations. So, for example, in a II state one of the elecbonic wave functions has its maximum in the molecular plane and the other one is perpendicular to it. If a treatment of the R-T effect is carried out employing the space-fixed coordinate system, the angle t appearing in Eqs. (53)... 

Figure 6 shows the relation between the several coordinate systems and the molecular conformations for a constant perimeter of the triatomic molecule. Also shown in this figure is the locus of the possible symmetry point groups (we consider the symmetry associated with the group operations applicable to identical nuclei) unless stated otherwise, the symmetry group will be Cs. 

Figure Al.2.6. Anharmonic stretch normal modes of a S5mimetric triatomic. The plot is similar to figure Al.2.5. except the normal modes are now anharmonic and can be curvilinear in the bond displacement coordinates 2. The antis5mimetric stretch is curved, but the S5mimetric stretch is linear because of symmetry.

These nine irreducible representations correspond to the nine motional degrees of freedom of the triatomic water molecule. To obtain the symmetry of the genuine vibrations, the irreducible representations of the translational and rotational motion have to be separated. This can be done using some considerations described in Chapter 4. The translational motion always belongs to those irreducible representations where the three coordinates, x, y, and z, belong. 

As discussed in Section 8.2.1, when nonadiabatic couplings cannot be neglected, the BO approximation is not reliable and coupled electronic states must be considered simultaneously with their interactions. For small systems, several full-dimensional approaches based on the vibronic or spin-rovibronic wavefunctions and taking into account simultaneously at least two electronic states have been developed [2, 100-104]. To quote some examples, the full vibronic Hamiltonians have been derived and employed for linear tetra-atomic molecules showing Renner-Teller interactions [103] or CXaY-like molecules of Csv symmetry showing Jahn-Teller interactions [104]. In the following, we will present the computational approaches based on the full rovibronic Carter-Handy Hamiltonian [100], developed for triatomic molecules and expressed in internal coordinates, which allows us to take into account up to three interacting electronic states [2, 100, 101]. 

A novel effect of SSB in linear molecules was revealed recently broken cyhn-drical symmetry [47]. The common understanding is that the bending of linear molecules is independent of the directions of the bending, as aU directions are equivalent for the free molecule. In cylindrical coordinates p, q>, and z the two normal coordinates of the bending of a triatomics are = pcosq> and = psintp, so the APES in the space of these two coordinates is expected to be a surface of rotation dependent on p only. 

Fig. 2.1. A triatomic molecule, say water. Left the overall molecular motion involves displacement of all atoms in a combination of translation, rotation, and vibration. On the right, the arrows sketch the displacements of the nuclei along normal coordinates, symmetry-adapted in terms of the molecular symmetry, represented by a mirror plane whose trace is denoted by m. In all normal coordinates, also the oxygen atom is slightly displaced so as to ensure that there is not net translational or rotational motion.

Beryllium hydride, BeH2, is the simplest triatomic molecule. This substance has apparently not been shown experimentally to exist, but calculations indicate that it should be a bound linear molecule with two equal Be-H bond distances. We assume this equilibrium conformation and apply the Bom-Oppenheimer approximation. We place the molecule in a Cartesian coordinate system with the Be atom at the origin and with the H atoms on the z axis, as shown in Figure 21.1. We denote one of the H atoms by Ha and the other by Hb. The molecule possesses the same symmetry operations as hJ E, i, ah, Cooz, and infinitely many cr and C2 operations perpendicular to the bond axis. 

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