For triatomic molecules

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... 

In this case, the situation is essentially equivalent to that for triatomics molecules. (We shall always assume that Ur > uc the fommlas for the opposite case, Ur < uc, are obtained from those to be derived by interchanging simply... 

Figure 3.1 Bond and intrinsic variables for triatomic molecules (a) and the Euler angles characterizing the orientation in space of the molecule (b).

Figure 4.16 Linear-bent correlation diagram for triatomic molecules.

Recoupling coefficients are important in computing matrix elements of operators. Consider, for example, the C operators defined, for triatomic molecules, in Eq. (4.68). For three bonds (tetratomic molecules) one has... 

Jensen, P. (1988), A New Morse Oscillator-Rigid Bender Internal Dynamics (MORBID) Hamiltonian for Triatomic Molecules, J. Mol. Sped. 128,478. 

Sorbie, K. S., and Murrell, J. N. (1975), Analytical Potentials for Triatomic Molecules from Spectroscopic Data, Mol. Phys. 29, 1387. 

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

In this book we present an algebraic approach to molecular vibrotation spectroscopy. We discuss the underlying algebraic techniques and illustrate their application. We develop the approach from its very beginning so as to enable newcomers to enter the field. Also provided are enough details and concrete examples to serve as a reference for the expert. We seek not only to introduce the spirit and techniques of the approach but also to demonstrate its quantitative application. For this reason a compilation of results for triatomic molecules (both linear and nonlinear) is provided. (See Appendix C.)... 

In the present work, we must carry out transformations of the dipole moment functions analogous to those descrihed for triatomic molecules in Refs. [18,19]. Our approach to this problem is completely different from that made in Refs. [18,19]. We do not transform analytical expressions for the body-fixed dipole moment components (/Zy, fiy, fi ). Instead we obtain, at each calculated ab initio point, discrete values of the dipole moment components fi, fiy, fif) in the xyz axis system, and we fit parameterized, analytical functions of our chosen vibrational coordinates (see below) through these values. This approach has the disadvantage that we must carry out a separate fitting for each isotopomer of a molecule Different isotopomers with the same geometrical structure have different xyz axis systems (because the Eckart and Sayvetz conditions depend on the nuclear masses) and therefore different dipole moment components (/Z, fiy, fij. We resort to the approach of transforming the dipole moment at each ab initio point because the direct transformation of analytical expressions for the body-fixed dipole moment components (/Zy, fiyi, fi i) is not practicable for a four-atomic molecule. The fact that the four-atomic molecule has six vibrational coordinates causes a huge increase in the complexity of the transformations relative to that encountered for the triatomic molecules (with three vibrational coordinates) treated in Refs. [18,19]. 

The promise of the early work on Be and He has recently been confirmed in the work of Nakatsuji and Mazziotti, which started to appear in 2001. This work showed that the lower bound method combined with second-order approximations yields accurate information for atoms and molecules. Nakatsuji and his co-workers [12] did a series of computational experiments where accuracies of between four and five figures were typically achieved. More precisely, they reported the correlation energy as a percentage of the exact correlation energy for a variety of atoms and molecules. They found these percentages ranged between 100% and 110% for atoms and diatomic molecules, and between 110% and 120% for triatomic molecules since these percentages are for lower bounds they never go below 100%. 

More complicated behaviors are expected for triatomic molecules (i.e., for three-body problems). In general, the analysis is facilitated by the fact that... 

The statistical prior distribution predicts for triatomic molecules a more resonant behavior of energy-transfer processes than in the diatomic cases. 

Figure 27. Experimental energy-transfer spectrum for Na + C02 and +N20 quenching and comparison with prior distribution for triatomic molecule.

Some examples are given in Figs E.1.1 and E.1.2, for triatomic molecules. A linear triatomic molecule has four (3x3 — 5) vibrational modes two bond-stretching modes and two (degenerate) bending modes. Fig. E.1.2 shows one of the four normal modes... 

Figure 2. Some relevant coordinate systems for triatomic molecules (R, R2, R3), (r, R, 0,). and (Kj.Rj.Jtn).

In this section we apply the model presented in Section 3 to the XS of triatomic molecules such as O3, 2/ CO2. In line with Heller [5], Schinke, on pages 115-116 of his book [6], proposed a 2D version of the simple reflection model but without the curvature and quantum effects taken into account in Sections 2 and 3. For triatomic molecules the quantities corresponding to Vq/P and V / or r (see Formula (12)) defined in Section 2 and corresponding to t (see Formula (27)) defined... 

For triatomic molecules, the contribution of hot bands cannot be expressed as a function of energy alone (see (5)) and therefore cannot be expressed in a compact analytic formula like Formula (C.3). However, for rigid triatomic molecules like CO2, NO2, SO2, O3 and N2O, the contribution of hot bands is weak at room temperature (and below) because hco kT for all normal mode frequencies. Note that the width of the contribution to the Abs. XS associated with each excited vibrational level (hot bands) is proportional to the slope of the upper FES along the normal mode of the ground electronic corresponding to each excited (thermally populated) vibrational level. This fact explains why numerical models (e.g. using ground state normal coordinates) are able to calculate the Abs. XS. These calculations are of Frank-Condon type. 

E. Garcia, A. Lagana, A new bond-order functional form for triatomic-molecules a fit of the BETH potential-energy. Mot Phys. 56 (3) (1985) 629-639. 

Purely quantum studies of the fully coupled anharmonic (and sometimes nonrigid) rovibrational state densities have also been obtained with a variety of methods. The simplest to implement are spectroscopic perturbation theory based studies [121, 122, 124]. Related semiclassical perturbation treatments have been described by Miller and coworkers [172-174]. Vibrational self-consistent field (SCF) plus configuration interaction (Cl) calculations [175, 176] provide another useful alternative, for which interesting illustrative results have been presented by Christoffel and Bowman for the H + CO2 reaction [123] and by Isaacson for the H2 + OH reaction [121]. The MULTIMODE code provides a general procedure for implementing such SCF-CI calculations [177]. Numerous studies of the state densities for triatomic molecules have also been presented. 

H. Wei and T. Carrington, Jr., An exact Eckart-embedded kinetic energy operator in Radau coordinates for triatomic molecules. Chem. Phys. Lett. 287, 289—300 (1998). 

TABLE 6. Equilibrium rotational and structural parameters for triatomic molecules ... 

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