Molecules polyatomic, linear

Rotation (Rigid molecule approximation) Linear polyatomic or diatomic molecules... 

Figure 1. Translation, rotation, and vibration of a diatomic molecule. Every molecule has three translational degrees of freedom corresponding to motion of the center of mass of the molecule in the three Cartesian directions (left side). Diatomic and linear molecules also have two rotational degrees of freedom, about rotational axes perpendicular to the bond (center). Non-linear molecules have three rotational degrees of freedom. Vibrations involve no net momentum or angular momentum, instead corresponding to distortions of the internal structure of the molecule (right side). Diatomic molecules have one vibration, polyatomic linear molecules have 3V-5 vibrations, and nonlinear molecules have 3V-6 vibrations. Equilibrium stable isotope fractionations are driven mainly by the effects of isotopic substitution on vibrational frequencies.

In Section 5.1, we noted that to a good approximation the nuclear motion of a polyatomic molecule can be separated into translational, vibrational, and rotational motions. If the molecule has N nuclei, then the nuclear wave function is a function of 3/V coordinates. The translational wave function depends on the three coordinates of the molecular center of mass in a space-fixed coordinate system. For a nonlinear molecule, the rotational wave function depends on the three Eulerian angles 9, principal axes a, b, and c with respect to a nonrotating set of axes with origin at the center of mass. For a linear molecule, the rotational quantum number K must be zero, and the wave function (5.68) is a function of 6 and only only two angles are needed to specify the orientation of a linear molecule. Thus the vibrational wave function will depend on 3N — 5 or 3N — 6 coordinates, according to whether the molecule is linear or nonlinear we say there are 3N — 5 or 3N — 6 vibrational degrees of freedom. 

ROTATION (Rigid Molecule Approximations) Linear Polyatomic or Diatomic Molecules ... 

First, we describe briefly the calculation of the absorption spectrum for bound-bound transitions. In order to keep the presentation as clear as possible we consider the simplest polyatomic molecule, a linear triatom ABC as illustrated in Figure 2.1. The motion of the three atoms is confined to a straight line overall rotation and bending vibration are not taken into account. This simple model serves to define the Jacobi coordinates, which we will later use to describe dissociation processes, and to elucidate the differences between bound-bound and bound-free transitions. We consider an electronic transition from the electronic ground state (k = 0) to an excited electronic state (k = 1) whose potential is also binding (see the lower part of Figure 2.2 the case of a repulsive upper state follows in Section 2.5). The superscripts nu and el will be omitted in what follows. Furthermore, the labels k used to distinguish the electronic states are retained only if necessary. 

Calculations on triatomic molecules, because of their theoretical and experimental significance, will be mentioned below in some detail. Herzberg s Polyatomics [6] gives diagrams of the correlation of orbitals between large and small internuclear distances in linear AH2 molecules and in linear ABa molecules [6]. Walsh, in a classic series of papers [97], gave correlations of molecular orbitals between linear and bent AH2 molecules and linear and... 

It may occur to the reader that it is always possible to bring the orbitals together in such a way that the overlap is positive. For example, in Fig. 5.8g, h if negative overlap is obtained, one need only invert one of the atoms to achieve positive overlap. This is true for diatomic molecules or even for polyatomic linear molecules. However, when we come to cyclic compounds, we no longer have the freedom arbitrarily to invert atoms to obtain proper overlap matches. One example will suffice to illustrate this. 

According to the data of analysis of many adsorption systems, the first term in Equation 9 corresponding to the second order appears only v hen considering adsorption of relatively small molecules. They include molecules of linear shape, such as the diatomic gases, carbon dioxide, carbon monoxide, etc. Experimentally realizable orders, n, are integers from 3 to 6 in the general case. With larger polyatomic molecules, no adsorption space remains in the zeolite voids for final adsorption under the effect of dispersion forces. Then Equation 9 retains only the second term, and Uon is expressed by Equation 12. 

Whereas monatomic molecules can only possess translational thermal energy, two additional kinds of motions become possible in polyatomic molecules. A linear molecule has an axis that defines two perpendicular directions in which rotations can occur each represents an additional degree of freedom, so the two together contribute a total of 1/2 R to the heat capacity. For a non-linear molecule, rotations are possible along all three directions of space, so these molecules have a rotational heat capacity of 3/2 R. Finally, the individual atoms within a molecule can move relative to each other, producing a vibrational motion. A molecule consisting of N atoms can vibrate in 3N-6 different ways or modes1. For mechanical reasons that we cannot go into here, each vibrational mode contributes R (rather than 1/2 R) to the total heat capacity. 

Similarly, most terms cancel in the calculation of the ratio of rotational partition functions. For diatomic molecules and linear polyatomic molecules, this ratio is given by fjl ... 

Suppose that the dissociation products are a polyatomic linear molecule and an atom (e.g., BrC C- + Br), and that the available energy is well above the vibrational energies of the BrC=C- fragment. Then the vibrational density of states must be included in the PED. If the upper Br atom spin-orbit state is ignored, the PED for a... 

In the particular case of a molecules polyatomic (with N atoms) having structure linear (all the atoms constituents are willing co-linear), the number of actual oscillation of the molecule is equal to (3N - 5). In this case is deleted of the total number of 3N possible directions of displacement 3. Degrees of freedom correlated with translation no deformation of the entire molecule and only 2 degrees of freedom corresponding to rotation molecule around two mutually orthogonal axis and perpendicular to the longitudinal axis of the molecule. 

Data for diatomic diamagnetic molecules are contained in subvolmne 11/29A, and polyatomic linear molecnles are dealt with in subvolume II/29B. Data on paramagnetic species will be contained in subvolmne II/29E. For a more systematic presentation of their physical properties we chose to order the paramagnetic species in a way which deviates from Hill s rules. 

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