Triatomic molecule, vibration-rotation

We find it convenient to reverse the historical ordering and to stait with (neatly) exact nonrelativistic vibration-rotation Hamiltonians for triatomic molecules. From the point of view of molecular spectroscopy, the optimal Hamiltonian is that which maximally decouples from each other vibrational and rotational motions (as well different vibrational modes from one another). It is obtained by employing a molecule-bound frame that takes over the rotations of the complete molecule as much as possible. Ideally, the only remaining motion observable in this system would be displacements of the nuclei with respect to one another, that is, molecular vibrations. It is well known, however, that such a program can be realized only approximately by introducing the Eckart conditions [38]. 

An alternative form of exact nonrelativistic vibration-rotation Hamiltonian for triatomic molecules (ABC) is that used by Handy, Carter (HC), and... 

Figure 9.24 shows part of the laser Stark spectrum of the bent triatomic molecule FNO obtained with a CO infrared laser operating at 1837.430 cm All the transitions shown are Stark components of the rotational line of the Ig vibrational transition, where Vj is the N-F stretching vibration. The rotational symbolism is that for a symmetric rotor (to which FNO approximates) for which q implies that AA = 0, P implies that A/ = — 1 and the numbers indicate that K" = 7 and J" = 8 (see Section 6.2.4.2). In an electric field each J level is split into (J + 1) components (see Section 5.2.3), each specified by its value of Mj. The selection mle when the radiation is polarized perpendicular to the field (as here) is AMj = 1. Eight of the resulting Stark components are shown. 

We have discussed up to now vibrational spectra of linear and bent triatomic molecules. We address here the problem of rotational spectra and rotation-vibration interactions.3 At the level of Hamiltonians discussed up to this point we only have two contributions to rotational energies, coming from the operators C(0(3]2)) and IC(0(412))I2. The eigenvalues of these operators are... 

Jensen, P. (1983), The Nonrigid Bender Hamiltonian for Calculating the Rotation-Vibration Energy Levels of a Triatomic Molecule, Comp. Phys. Rep. 1,1. 

Sutcliffe, B. T., and Tennyson, J. (1991), A General Treatment of Vibration-Rotation Coordinates for Triatomic Molecules, Int. J. Quant. Chem. 39, 183. 

Yi, Xi-Zhang, Ding, Shi-liang and Deng, Cong-hao (1988), Lie Algebraic Approach to the Rotation-Vibrational Energy Levels for a Non-Linear Triatomic Molecule X3, Chinese J. Chem. Phys. 1, 255. 

The photofragmentation that occurs as a consequence of absorption of a photon is frequently viewed as a "half-collision" process (16)- The photon absorption prepares the molecule in assorted rovibrational states of an excited electronic pes and is followed by the half-collision event in which translational, vibrational, and rotational energy transfer may occur. It is the prediction of the corresponding product energy distributions and their correlation to features of the excited pes that is a major goal of theoretical efforts. In this section we summarize some of the quantum dynamical approaches that have been developed for polyatomic photodissociation. For ease of presentation we limit consideration to triatomic molecules and, further, follow in part the presentation of Heather and Light (17). 

In order to keep the formulation as simple as possible we confine the discussion to systems with only two degrees of freedom. The extension to more complex problems is — formally at least — straightforward. We will treat triatomic molecules ABC dissociating into products A+BC. First, we again consider in Section 3.1 the linear model, outlined in Sections 2.4 and 2.5, in which the diatomic fragment vibrates while its rotational degree of freedom is frozen. Subsequently, we treat in Section 3.2 the... 

In order to keep the expressions transparent we, once again, restrict the discussion to the dissociation of the triatomic molecule ABC into products A and BC. Furthermore, the total angular momentum is limited to J = 0. In this chapter we consider the vibration and the rotation of the fragment molecule simultaneously. The corresponding Hamilton function, i.e., the total energy as a function of all coordinates and momenta, using action-angle variables (McCurdy and Miller 1977 Smith 1986), reads... 

We consider the photofragmentation of a triatomic molecule, ABC — A + BC(j), within the model outlined in Section 3.2. The vibrational coordinate of BC is fixed and the total angular momentum is zero. According to (5.23), the classical approximation of the partial photodissociation cross section for producing BC in rotational state j is given by... 

Vibrational state distributions in direct dissociation can be described in the same way as rotational state distributions. For simplicity we consider the dissociation of the collinear triatomic molecule, ABC — A + BC, as outlined in Section 2.4. 

Mladenovic, M. and Bacic, Z. (1990). Highly excited vibration-rotation states of floppy triatomic molecules by a localized representation method The HCN/HNC molecule, J. Chem. Phys. 93, 3039-3053. 

Shapiro, M. and Balint-Kurti, G.G. (1979). A new method for the exact calculation of vibrational-rotational energy levels of triatomic molecules, J. Chem. Phys. 71, 1461-1469. 

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

Equation 2.17 is of the form A = PDP-1. The 9x9 Hessian for a triatomic molecule (three Cartesian coordinates for each atom) is decomposed by diagonalization into a P matrix whose columns are direction vectors for the vibrations whose force constants are given by the k matrix. Actually, columns 1, 2 and 3 of P and the corresponding k, k2 and k3 of k refer to translational motion of the molecule (motion of the whole molecule from one place to another in space) these three force constants are nearly zero. Columns 4, 5 and 6 of P and the corresponding k4, k5 and k6 of k refer to rotational motion about the three principal... 

Note that in a non-linear molecule, one of the vibrational modes of the linear molecule has been replaced by a rotational coordinate. As an illustration, let us consider two examples. For the stable linear triatomic molecule CO2, there are 3 x 3 — 5 = 4 internuclear coordinates, which corresponds to the vibrational degrees of freedom, namely the symmetric and antisymmetric stretch and two (degenerate) bending modes (see Appendix E). For the three atoms in the reaction D + H — H—> D — H + H, there are 3 x 3 — 6 = 3 internuclear coordinates. These coordinates can, for example, be chosen as a D H distance, the H H distance, and the I) II H angle. 

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... 

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