Cartesian coordinates atom + diatom

The simple harmonic motion of a diatomic molecule was treated in Chapter 1, and will not be repeated here. Instead, emphasis is placed on polyatomic molecules whose electronic energy s dependence on the 3N Cartesian coordinates of its N atoms can be written (approximately) in terms of a Taylor series expansion about a stable local minimum. We therefore assume that the molecule of interest exists in an electronic state for which the geometry being considered is stable (i.e., not subject to spontaneous geometrical distortion). 

The second common type of operationally defined structure is the so-called substitution or rt structure.10 The structural parameter is said to be an rs parameter whenever it has been obtained from Cartesian coordinates calculated from changes in moments of inertia that occur on isotopic substitution at the atoms involved by using Kraitchman s equations.9 In contrast to r0 structures, rs structures are very nearly isotopically consistent. Nonetheless, isotope effects can cause difficulties as discussed by Schwendeman. Watson12 has recently shown that to first-order in perturbation theory a moment of inertia calculated entirely from substitution coordinates is approximately the average of the effective and equilibrium moments of inertia. However, this relation does not extend to the structural parameters themselves, except for a diatomic molecule or a very few special cases of polyatomics. In fact, one drawback of rs structures is their lack of a well-defined relation to other types of structural parameters in spite of the well-defined way in which they are determined. It is occasionally stated in the literature that r, parameters approximate re parameters, but this cannot be true in general. For example, for a linear molecule Watson12 has shown that to first order ... 

This particular choice of coordinates seems eminently reasonable for a diatom-diatom collision. On the other hand, let us say that a reaction occurs and the products are a triatom and an atom (see Fig. 2). The Cartesian coordinates are now as appropriate as before, whereas the Jacobi coordinates are rather less useful for visualization purposes. However it must be emphasized that the trajectory can still be integrated in these conditions. (In general, you will see in the remainder of this book that the choice of coordinates is crucial in quantum mechanical calculations it is far less important in classical mechanics.) In general, it is worthwhile using Jacobi coordinates for three- and four-atom systems. 

Molecules containing N atoms require 3N Cartesian coordinates to specify the locations of all the nuclei. For convenience, we may use three coordinates to locate the center of mass of the molecule, and three more coordinates to orient a chosen axis passing through the molecule. Only two are needed to describe the orientation if the molecu-le is diatomic. The remaining 3N - 6 coordinates are used to. sped.fy the relative posit 0.ns of the atorns. The potential energy V (r )... 

The preceding presentation describes how the collision impact parameter and the relative translational energy are sampled to calculate reaction cross sections and rate constants. In the following, Monte Carlo sampling of the reactant s Cartesian coordinates and momenta is described for atom + diatom collisions and polyatomic + polyatomic collisions. Initial energies are chosen for the reactants, which corresponds to quantum mechanical vibrational-rotational energy levels. This is the quasi-classical model [2-4]. 

Cartesian coordinates xt and x2 are chosen for the two atoms of the diatomic (identified as 1 and 2) with the origin at the center of mass. 

Choosing initial Cartesian coordinates for the polyatomic reactants follows the procedures outlined above for an atom + diatom collision and for normal-mode sampling. If the cross section is calculated as a function of the rotational quantum numbers J and K, the components of the angular momentum are found from... 

For a bound stationary state, the quantum-mechanical virial theorem states that 2(r) = Hi qiidV/dqi)), where the sum is over the Cartesian coordinates of all the particles. If F is a homogeneous function of degree n, then 2(7 ) = n(V). For a diatomic molecule, the virial theorem becomes = - u - R dU/dR) and (V) = 2U + R dU/dR), where U R) is the potential-energy function for nuclear motion. The virial theorem shows that at 7 , ( F) of a diatomic molecule is less than the total ( V) of the separated atoms, and ( T i) is greater than the total (Tei) of the separated atoms. 

The simplest model for a diatomic molecule consists of two atoms of mass m and m2 connected by a rigid, massless spring of length r, which has the value Yq at equilibrium (Figure 1). If the z-axis is taken to lie along the internuclear line, then the Cartesian coordinates of the two atoms, referred to the center of... 

The potential energy E(x, X2) is defined by taking the CN diatom fixed on the x -axis with the zero point in the center of mass, and taking pure (x, X2) Cartesian coordinates to describe the migration of the H-atom around the CN kernel. 

Although conceptually straightforward, the use of Cartesian coordinates and momenta is not convenient when describing the motion of a diatomic molecule. This is because these degrees of freedom do not change independently, since the chemical bond between the two atoms constrains their motion. 

The center of mass for a diatomic molecule is found, using the Cartesian coordinates of the atoms, at... 

The angular velocity is then added using the relationship rt = to x r(, where r, is the vectorial distance of the atom from the center of mass. With the diatom placed along the x- axis and = coz, the resulting Cartesian velocities are only for the -coordinate and equal... 

It is of interest to note that the identical result (2.38) is obtained for the librations by a treatment using cartesian displacement coordinates for the atoms of a diatomic molecule (Suzuki, 1970). Although we do not face the problems of separability of coordinates in the kinetic energy in this case, the assiunption of harmonicity of the potential energy presumably causes the same inaccuracies. In other words, it is immaterial which coordinates are used but in terms of Eulerian angle displacement coordinates the harmonic approximation is not confined to the potential energy but also appears in the kinetic energy. 

These analogies fail to explain the coarse structure observed in diatomic electronic spectra under low resolution (Chapter 4), because the additional nuclear coordinates introduced by the second atom in a diatomic molecule AB create new internal modes that have no counterpart in atoms. The nuclear positions relative to an arbitrary origin fixed in space can be specified using the Cartesian vectors and Rg (Fig. 3.1). They may be equivalently described in terms of the coordinates... 

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