Diatomic molecule rotational motion

Simultaneously with the oscillations with respect to the internuclear axis, a diatomic molecule rotates as a whole with a frequency ft (not to be confused with the quantum number of the projection of the total momentum, as in Section 1.2) around an axis which is perpendicular to the direction of the internuclear axis, exhibiting a total angular momentum J see Fig. 1.3. Hence in the laboratory coordinate system, with respect to which the light wave possesses a certain polarization, electrons participate simultaneously in two types of motion dipole oscillations with respect to the internuclear axis, and rotation of the molecule as a whole. 

For diatomic and linear polyatomic molecules, rotational motion is restricted to an axis perpendicular to the molecular longitudinal axis. A corresponding diagram is shown in... 

Figure Al.2.2. Internal nuclear motions of a diatomic molecule. Top the molecule in its equilibrium configuration. Middle vibration of the molecule. Bottom rotation of the molecule.

Consider a gas of N non-interacting diatomic molecules moving in a tln-ee-dimensional system of volume V. Classically, the motion of a diatomic molecule has six degrees of freedom—tln-ee translational degrees corresponding to the centre of mass motion, two more for the rotational motion about the centre of mass and one additional degree for the vibrational motion about the centre of mass. The equipartition law gives (... 

These results do not agree with experimental results. At room temperature, while the translational motion of diatomic molecules may be treated classically, the rotation and vibration have quantum attributes. In addition, quantum mechanically one should also consider the electronic degrees of freedom. However, typical electronic excitation energies are very large compared to k T (they are of the order of a few electronvolts, and 1 eV corresponds to 10 000 K). Such internal degrees of freedom are considered frozen, and an electronic cloud in a diatomic molecule is assumed to be in its ground state f with degeneracy g. The two nuclei A and... 

Suppose that W(r,Q) describes the radial (r) and angular (0) motion of a diatomic molecule constrained to move on a planar surface. If an experiment were performed to measure the component of the rotational angular momentum of the diatomic molecule perpendicular to the surface (Lz= -ih d/dQ), only values equal to mh (m=0,1,-1,2,-2,3,-3,...) could be observed, because these are the eigenvalues of ... 

The rotational motion of a linear polyatomic molecule can be treated as an extension of the diatomic molecule case. One obtains the Yj m (0,(1)) as rotational wavefunctions and, within the approximation in which the centrifugal potential is approximated at the equilibrium geometry of the molecule (Re), the energy levels are ... 

If the rotational motion of the molecules is assumed to be entirely unhindered (e.g., by any environment or by collisions with other molecules), it is appropriate to express the time dependence of each of the dipole time correlation functions listed above in terms of a "free rotation" model. For example, when dealing with diatomic molecules, the electronic-vibrational-rotational C(t) appropriate to a specific electronic-vibrational transition becomes ... 

A linear molecule, such as any diatomic molecule, carbon dioxide, and ethyne (acetylene, HC=CH), can rotate about two axes perpendicular to the line of atoms, and so it has two rotational modes of motion. Its average rotational energy is therefore 2 X jkT = kT, and the contribution to the molar internal energy is NA times this value ... 

Let us consider a diatomic molecule in such a crystal. As a first approximation we may neglect the translational oscillations of the molecule under consideration and both the translational and rotational motion of the other molecules in the crystal. The wave equation then may be written... 

A single particle of (reduced) mass p in an orbit of radius r = rq + r2 (= interatomic distance) therefore has the same moment of inertia as the diatomic molecule. The classical energy for such a particle is E = p2/2m and the angular momentum L = pr. In terms of the moment of inertia I = mr2, it follows that L2 = 2mEr2 = 2EI. The length of arc that corresponds to particle motion is s = rep, where ip is the angle of rotation. The Schrodinger equation is1... 

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.

The electronic contributions to the g factors arise in second-order perturbation theory from the perturbation of the electronic motion by the vibrational or rotational motion of the nuclei [19,26]. This non-adiabatic coupling of nuclear and electronic motion, which exemplifies a breakdown of the Born-Oppenheimer approximation, leads to a mixing of the electronic ground state with excited electronic states of appropriate symmetry. The electronic contribution to the vibrational g factor of a diatomic molecule is then given as a sum-over-excited-states expression... 

The radial motion of a diatomic molecule in its lowest (J=0) rotational level can be described by the following Schrodinger equation ... 

Thus far, exactly soluble model problems that represent one or more aspects of an atom or molecule s quantum-state structure have been introduced and solved. For example, electronic motion in polyenes was modeled by a particle-in-a-box. The harmonic oscillator and rigid rotor were introduced to model vibrational and rotational motion of a diatomic molecule. 

Treatment of the rotational motion at the zeroth-order level described above introduces the so-called rigid rotor energy levels and wavefunctions Ej = h2 J(J+l)/(2 4,Re2) and Yjm (0,( )) these same quantities arise when the diatomic molecule is treated as a rigid rod of length Re. The spacings between successive rotational levels within this approximation are... 

In this section, we describe the motion of vibronic WPs created in a diatomic molecule that has only one vibrational mode. The influence of other degrees of motions such as rotation and nuclear spins are omitted for simplicity. Since our studies deal with the quantum property of the system in which relaxations can be neglected, the decoherence process is not taken into account in the following formulations. Assuming that the molecule occupies a single vibrational level v = 0 as an initial state, the WP generated by the absorption of a pump laser pulse is given as... 

The Vibration and Rotation of Molecules.—The nature of the vibrational motion and the values of the vibrational energy levels of a molecule are determined by the electronic energy function, such as that shown in Figure VII-1. The simplest discussion of the vibrational motion of a diatomic molecule is based upon the approximation of the energy curve in the neighborhood of its minimum by a parabola that is, it is assumed that the force between the atoms of the molecule is proportional to the displacement of the internuclear distance from its equilibrium value r.. This corresponds to the approximate potential function... 

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