Rotator, diatomic molecule

A rotating diatomic molecule consists of masses mi and m2 circling the centre of mass at distances 7 1 and r2 respectively. The moment of inertia is / = mir + m2r. By definition, the centre of mass is located such that m r = m2r2, and hence... 

In three dimensions the rotating diatomic molecule is equivalent to a particle moving on the surface of a sphere. Since V — 0 the Schrodinger equation is... 

Only those problems that can be reduced to one-dimensional one-particle problems can be solved in closed form by the methods of wave mechanics, which excludes all systems of chemical interest. As shown before, several chemical systems can be approximated by one-dimensional model systems, such as a rotating diatomic molecule modelled in terms of a rotating particle in a fixed orbit. The trick is to find a one-dimensional potential function, V that provides an approximate model of the interaction of interest, in the Schrodinger formulation... 

Mizushima, M. (1975), The Theory of Rotating Diatomic Molecules, Wiley, New York. 

For a real rotating diatomic molecule, known as a nonrigid rotor, Eq. (D) becomes... 

Table IX-2.—Characteristic Temperature for Rotation, Diatomic Molecules...

It is instructive to start with the simplest possible model of a rotating diatomic molecule, the so-called dumbbell model, as illustrated in figure 6.19. The two atoms, of masses ni and m2, are regarded as point-like, and are fastened a distance R apart... 

The simplest model of a rotating diatomic molecule is a rigid rotor or dumbbell model in which the two atoms of mass and m2 are considered to be joined by a rigid, weightless rod. The allowed energy levels for a rigid rotor may be shown by quantum mechanics to be... 

Let us start with the following picture of laser light interaeting with diatomic molecule. We have a rotating diatomic molecule with a large total angular rnoinen-... 

Mizushima M (1975) The theory of rotating diatomic molecules, Wiley, Chichester... 

In dealing simultaneously with space- and molecule-fixed coordinate systems, one must explicitly define the transformations between coordinate systems and specify the relative phases of all basis functions. Larsson (1981) has reviewed the numerous phase conventions currently in use for rotating diatomic molecules. (See also Wolf, 1969.)... 

Many detailed derivations of the RKR equations have been published (Zare, 1964 Miller, 1971 Elander, et al, 1979). The unknown potential energy for a rotating diatomic molecule,... 

The "statistical formulation (67.Ill) cannot be applied to unimolecular reactions for which the classical activation energy and the reaction heat are equal (E = Q) without introducing some additio-nal assumptions which are necessary for the definition of the transition state. One usually considers the "activated complex (AB) as a rotating "diatomic molecule in which the centrifugal force is balanced by an attractive dipole-induced dipole or dispersion force /HO/. This "diatomic model implies that the angular momentum... 

Consider a rotating diatomic molecule, as shown in Fig. 3.5, with the atomic masses m and mg at distances r and rg from the centre of gravity. The moment of inertia with respect to the rotational axis is I. We have... 

We have seen that the two-particle system of an electron and a nucleus rotating about a center of mass (COM) can be transformed to the one-particle system of a reduced mass rotating about a fixed point. However, this transformation can be made for any two-mass system, and so it applies also to the case of the nuclei of a rotating diatomic molecule. As we now show, the mathematical outcome for the rotating diatomic molecule is strikingly similar to that for the hydrogenlike ion. 

Fig. 1.17. The radiation s electric field interacting with the rotating dipole moment of a rotating diatomic molecule.

In a nonrigid rotating diatomic molecule, centrifugal distortion will elongate the bond as the rotational frequency increases. To accurately fit the energy levels, a correction term must be added to Eq. (1.55)... 

The selection rule of Eq. (23.3-2b) can be understood classically. In order for a molecule to interact with the electric field of the radiation, it must exhibit a periodically varying electric dipole moment of the correct frequency. A rotating diatomic molecule with a permanent dipole moment does present a periodically varying dipole to the radiation, so it should absorb or emit radiation if it is rotating with the correct frequency. If a molecule has no permanent dipole moment it does not exhibit any periodic variation in the dipole moment. 

Consider a system of four distinguishable rigid rotating diatomic molecules with a total energy of 20hB, where B is the rotational constant. 

Consider a two-particle system with a potential energy that depends only on the distance between the particles. This case applies to the hydrogen atom and to the nuclei of a rotating diatomic molecule in the Bom-Oppenheimer approximation. The equation of motion of such a system can be separated into two separate equations. We first treat the case in which there is motion only in the x direction and apply the Lagrangian method. The Lagrangian of the system is... 

The Schrodinger equahon for a vibrating and rotating diatomic molecule, or generally for a two-body problem, with p as the reduced mass, is... 

The states of a vibrating-rotating diatomic molecule are distinguished by three quantum numbers, n,, and M. Under the assumption of rigid rotation and a harmonic stretching potential, the selection rules derived for a harmonic oscillator and a rigid rotator apply to a diatomic molecule. An, the change in n for an allowed transition, is +1 (absorption) or -1 (emission). A/ is +1 or -1, and AM = 0. To see this, we need to make explicit use of the wavefunctions, which are products of radial and spherical harmonic functions ... 

The Schrodinger equation for this two-body problem starts out the same as the general two-body Schrodinger equation (Equation 9.18) however, the potential function, V(r), is different from that of the vibrating-rotating diatomic molecule. It is an electrostatic attraction of two point charges, and its form is... 

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