Angular Triatomic Molecules

Let us now illustrate the discretization process using the vibration of a triatomic molecule (ABC) as an example. The nuclear Hamiltonian with zero total angular momentum (J = 0) can be conveniently written in the Jacobi coordinates (h = 1 thereafter) ... 

Triatomic molecules may be linear or bent (i.e. V-shaped or angular). The shape adopted by any particular molecule is that which is consistent with the minimization of its total energy. 

The structural sequences in Fig. 8.5 change between the s-valent, p-valent and sp-valent cases because the shape parameter, s, is very sensitive to the angular character of the orbitals. We have already seen in Fig. 4,13 for the triatomic molecule AH2 that the three-atom contributions to the fourth moment are dependent on the bond angle. It follows from eqs (7.14)—(7.17) that hopping along a three-atom path and back again amongst sp-valent orbitals will lead to the fourth-moment contribution... 

There are several important effects associated with degenerate vibrational levels. We begin by considering the doubly degenerate v2a and v2b vibrational modes of a linear triatomic molecule. In these modes, the three atoms each vibrate in a plane perpendicular to the molecular axis, the vibrations being either in the xz or the yz plane, where the z axis is the molecular axis (Fig. 6.2). Classically, if both these modes are excited, the vibrations may give rise to vibrational angular momentum about the internuclear axis for example, if v2a and v2b are of equal amplitude and differ in phase by 90°, then the resultant motion of each nucleus is a circle about the molecular axis, as shown in Fig. 6.8 (see Problem 6.18). 

A triatomic molecule is never called triangular or planar, even though it always is both it always should be called either linear or angular. In either case, the three atoms will lie in the same plane, because any three noncollinear points always determine a plane. 

As the second example of a triatomic molecule, we shall consider the bonding in the water molecule. Water in its ground state has an angular configuration in which the H—O—H angle is about 105°. 

The bending motion of the precursor molecule results in a so-called dynamic axis switching effect (53). This effect is connected with a small deviation of the diatomic axis from the equilibrium axis of the bound triatomic molecule. It turns out that this effect is important for large values of j and makes a noticeable contribution to the partitioning of j into diatomic angular momentum and the relative angular momentum of the atom about the fragment. 

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... 

In this chapter we discuss only the scalar aspect of rotational excitation, i.e., the forces which promote rotational excitation and how they show up in the final state distributions. The simple model of a triatomic molecule with total angular momentum J = 0, outlined in Section 3.2, is adequate for this purpose without concealing the main dynamical effects with too many indices and angular momentum coupling elements. The vector properties and some more involved topics will be discussed in Chapter 11. 

The theory outlined above can be used to calculate the exact bound-state energies and wavefunctions for any triatomic molecule and for any value J of the total angular momentum quantum number. We can solve the set of coupled equations (11.7) subject to the boundary conditions Xjfi (R Jp) —> 0 in the limits R —> 0 and R — oo (Shapiro and Balint-Kurti 1979). Alternatively we may expand the radial wavefunctions in a suitable set of one-dimensional oscillator wavefunctions ipm(R),... 

Balint-Kurti, G.G. and Shapiro, M. (1981). Photofragmentation of triatomic molecules. Theory of angular and state distribution of product fragments, Chem. Phys. 61, 137-155. 

Morse, M.D. and Freed, K.F. (1981). Rotational and angular distributions from photodissociations. III. Effects of dynamic axis switching in linear triatomic molecules, J. Chem. Phys. 74, 4395-4417. 

Figure 2.5-2 Stretching vibrations of a symmetric angular triatomic molecule Y-Y-Y, depending on the bond angle a, Y = C /i =/2 = 5 N cm r. symmetric (in-phase), Ua antisymmetric (out-of-phase vibration), the trace - - . shows the frequency of the uncoupled oscillators.

Having tested our formalism and code on the two coulomb interacting systems antiprotonic helium and doubly excited states in normal helium we were ready to attack a model of predissociating triatomic molecule. We choose the NelCl van der Waals complex as our triatomic test molecule since a number of more and more accurate studies had been performed on zero-angular momentum levels of this system[38, 39, 40, 41, 42]. 

Previous fully quantum mechanical studies of predissociation phenomena in triatomic molecules do not, to our knowledge, use a Hamiltonian that has a non-zero total angular momentum. Tennyson et al[43, 44, 45, 46, 47, 48, 49, 50, 51] solve the same equations as we do but have not yet, to our knowledge, treated any predissociation problems. The adiabatic rotation approximation method of Carter and Bowman[52] plus a complex C2 modification have, on the other hand, been used to compute rovibrational energies and widths in the HCO[53, 54] and HOCl[55, 56, 57] molecules. This method is based upon the the Wilson and Howard[58], Darling and Dennison[59] and Watson[60] formalism. It is less transparent but the exact formalism in refs.[58, 59, 60] is equivalent to the one presented here and in ref [43]. While both we and Tennyson et al[43] include the exact Hamiltonian in our formalism the latter authors 152] use an approximate method which they have analysed and motivated. 

In an indirect reaction [2] A + BC —t B-A-C —t AB + C or AC + B. In a first step, the A atom inserts into the BC diatom forming an ABC complex. Two new bonds (AB and AC) are formed while the BC bond is broken. Then the complex dissociates with a breaking of one of these two bonds. This reaction mechanism is called insertion. In contrast with abstraction reactions, all three bonds in the triatomic molecule ABC participate actively in the reaction. Two bonds are formed teni] )orarily while only one exists for the reactants and products. Thus, the potential energy surface involves a very deep well (several eV) which correspond to a stable ABC molecnle or radical. When the lifetime of the ABC molecule is larger than its rotational period, angular distributions of the products are symetric with a backward/forward peak and the population of rovibrational states of the products presents a statistical character. 

---