Cartesian coordinates polyatomic molecules

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). 

A polyatomic molecule, such as a sugar, may be regarded as a system of masses joined by bonds having spring-like properties. The vibration of each of the masses (atoms) can be resolved into components parallel to the x, y, and z axis of a Cartesian system of coordinates. This means that each atom has three degrees of freedom, and a system of N nuclei has 3 N... 

The kinetic and potential energies of a polyatomic molecule can be expressed in terms of Cartesian coordinates (Ax, Ay, Az) or internal coordinates such as increments of bond length (Ar) and bond angles (Aa). In the former case, 3N coordinates are required for an iV-atom molecule. Figure 1-44 shows the nine Cartesian coordinates of the H20 molecule. Since the number of normal vibrations is 3 (3 x 3 — 6), this set of Cartesian coordinates includes six extra coordinates. On the other hand, only three coordinates (A/-, Ar2 andAa) shown in Fig. 1-44 are necessary to express the energies in terms of internal... 

The equations depend essentially on six coordinates in the Cartesian space, and it includes a sixfold integral. This integral is the one that prevents the theory from applications to polyatomic molecules. It is the interaction-site model and the RISM approximation proposed by Chandler and Andersen [16] that enabled one to solve the equations. The idea behind the model is to project the functions onto the one-dimensional space along the distance between the interaction sites, usually placed on the center of atoms, by taking the statistical average over the angular coordinates of the molecules with fixation of the separation between a pair of interaction site. 

For the general treatment of the vibrations of a polyatomic molecule we choose a system of Cartesian displacement coordinates, with the origin for each nucleus at its equilibrium position,... 

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 ... 

For polyatomic molecules the situation is more complicated, and the vibrations are characteristic not of individual pairs of atoms but of the molecule as a whole. If it contains N atoms, their positions in space are describable by ZN Cartesian coordinates which may be written qp. These may conveniently be measured from... 

For polyatomic molecules the electronic potential is a function of more than one internuclear distance. It cannot be graphically represented in the plane of V(r) and r, but requires a space of higher dimensionality. For a molecule of N atoms, 3N Cartesian coordinates are required to specify the position of each atom. The electronic potential energy is independent of location in space and depends only on the relative positions of the atoms, which for any pair x and y is given by the vector Thus, only N— vector distances, or 3N—3 Cartesian coordinates, are required, the discarded three Cartesian coordinates being those that locate the (unnecessary) position of the molecule in space. 

The positions of all N nuclei in a polyatomic molecule may be specified using the 3N Cartesian coordinates < 2,..., < 3jv In terms of these, the nuclear kinetic energy is given by... 

We pointed out in the preceding section that a realistic potential energy function may not be easily expressible in Cartesian coordinates, but may be written more naturally in terms of 3JV generalized coordinates related to the mass-weighted coordinates by a linear transformation (6.40). In fact, only 3N — 6 such coordinates are required to fully specify the potential (3N — 5 in linear molecules) 2K is not sensitive to the center-of-mass position or molecular orientation in space, and a polyatomic molecule exhibits only 3N — 6 (3N — 5) independent bond lengths and bond angles. Such a truncated set of 3N — 6 (iN — 5) generalized coordinates is called an internal coordinate basis, and is commonly denoted S. To illustrate how an internal coordinate basis may be used to evaluate normal modes, we consider the bent H2O molecule in Fig. 6.3. The three internal coordinates are conveniently chosen to be... 

Let Ui represents a displacement of the ith cartesian coordinate from its equilibrium value i = 1,2, ZN), where N is the number of nuclei in a molecule, and qi = y/MiUi are the generalized coordinate (M is the mass of the atom associated with the ith coordinate) and its derivative with respect to time q = Pi. In the harmonic approximation the classical vibrational Hamiltonian of a polyatomic molecule becomes... 

For an N-atom molecule there are 3N - 6 internal degrees of freedom, so there must be 3N - 6 independent vibrations and one needs as many independent coordinates for their description. Chemical intuition works in terms of chemical bonds, so that the most intuitive way of representing molecular vibrations is in terms of the variations in bond lengths, bond angles, and torsion angles. These are internal coordinates, [ i]. Cartesian coordinates are independent by definition, but finding a set of truly independent internal coordinates for a large polyatomic molecule is all but a trivial matter [1],... 

The calculation of Cartesian coordinates for a molecule distorted by a specified amount along one or some of the curvilinear internal coordinates is not trivial for polyatomic molecules, owing to the nonlinear nature of the transformation between the two coordinate systems (see Section 2.6). 

Polyatomic molecules have more than one vibrational frequency. The number can be calculated from the following. One atom in the molecule can move independently in three directions, the x, y, and z directions in a Cartesian coordinate system. Therefore, in a molecule with n atoms, the n atoms have 3n independent ways they can move. The center of mass of the molecule can move in three independent directions, x, y, and z. A nonlinear molecule can rotate in three independent ways about the x, y, and z axes, which pass through the center of mass. A linear molecule has one less degree of rotational freedom since rotation about its own axis does not displace any atoms. These translations of the center of mass and rotations can be performed with a rigid molecule and do not change its shape or size. Substracting these motions, there remain 3n — 6 degrees of freedom of internal motion for nonlinear molecules and 3n —5 for linear molecules. These... 

For collisions of an atom B with a polyatomic molecule A, the reactive cross section may be determined as a function of relative velocity v cx, and either the vibrational and rotational quantum numbers n, /, and or temperature Ta of A. For the latter the quantum numbers are chosen from their thermal Boltzmann distributions [i.e., Eqs. [38], [39], [55]-[58]]. These two different samplings of A s vibrational-rotational states give the cross sections Or = ar v.,, r A,jA,RA) and a,. = ar(i rei T4), Eq. [62], respectively. To choose random initial conditions for an ensemble of A + B collisions the vibrational and rotational quantum numbers of A are first transformed to Cartesian coordinates and momenta as described above (in Eqs. [32]-[48]) and [59]. The following steps are then performed to choose random initial conditions for the A - - B collision ... 

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