Vibrating-rotator eigenfunctions

Figure 4 shows an initial wavcpackct for a reactive scattering calculation superimposed on a potential energy surface for the Li + HF LiF + H reaction [15, 108]. The initial wavcpackct is placed in the asymptotic region of the reactant channel, where there is no force between the reactant molecules. It is constructed by first calculating the desired initial vibrational-rotational eigenfunctions of the reactant... 

Here (j)( is the t-th vibrational-rotational eigenfunction of the diatomic. Once Wf(Rr is introduced we can now also present, more explicitly, the perturbation potential, V, mentioned in Eq. (1) and (4). 

The perturbation calculation may also be described as a contact transformation. The original hamiltonian is transformed to a new effective hamiltonian which has the same eigenvalues but different eigenfunctions, to some carefully chosen order of magnitude. This contact transformation of the vibration-rotation hamiltonian was originally studied by Nielsen and co-workers. >33... 

Using the same formulation of the Hamiltonian as in Sec. VII [specifically Eqs. (67)—(70)], the two-step process makes use of five pairs of rovibrational states (specified explicitly below). The vibrational eigenstates correspond to the combined torsional and S-D asymmetric stretching modes. The rotational eigenfunctions are the parity-adapted symmetric top wave functions. Each eigenstate has additionally an Si A label denoting its symmetry with respect to inversion. Within the pairs used, the observable chiral states are composed as... 

It is always possible to express the eigenfunction corresponding to a nominal i, Vi, J electronic-vibration-rotation level as a sum of rovibronic basis functions,... 

The ability to assign a group of vibrational/rotational energy levels implies that the complete Hamiltonian for these states is well approximated by a zero-order Hamiltonian which has eigenfunctions /,( i)- The are product functions of a zero-order orthogonal basis for the molecule, or, more precisely, product functions in a natural basis representation of the molecular states, and the quantity m represents the quantum numbers defining tj>,. The wave functions are given by... 

We consider a nuclear wave function describing collisions of type A + BC(n) AC(n ) + B, where n = vj, k are the vibrational v and rotational j quantum numbers of the reagents (with k the projection of j on the reagent velocity vector of the reagents), and n = v, f, k are similarly defined for the products. The wave function is expanded in the terms of the total angular momentum eigenfunctions t X) [63], and takes the form [57-61]... 

The theoretical method, as developed before, concerns a molecule whose nuclei are fixed in a given geometry and whose wavefimctions are the eigenfunctions of the electronic Hamiltonian. Actually, the molecular structure is vibrating and rotating and the electric field is acting on the vibration itself. Thus, in a companion work, we have evaluated the vibronic corrections (5) in order to correct and to compare our results with experimental values. 

Nearly all kinetic isotope effects (KIE) have their origin in the difference of isotopic mass due to the explicit occurrence of nuclear mass in the Schrodinger equation. In the nonrelativistic Bom-Oppenheimer approximation, isotopic substitution affects only the nuclear part of the Hamiltonian and causes shifts in the rotational, vibrational, and translational eigenvalues and eigenfunctions. In general, reasonable predictions of the effects of these shifts on various kinetic processes can be made from fairly elementary considerations using simple dynamical models. 

---