Diatomic molecule translational motion

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... 

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... 

The application of the Bom-Oppenheimer and the adiabatic approximations to separate nuclear and electronic motions is best illustrated by treating the simplest example, a diatomic molecule in its electronic ground state. The diatomic molecule is sufficiently simple that we can also introduce center-of-mass coordinates and show explicitly how the translational motion of the molecule as a whole is separated from the internal motion of the nuclei and electrons. 

The total number of spatial coordinates for a molecule with Q nuclei and N electrons is 3(Q + N), because each particle requires three cartesian coordinates to specify its location. However, if the motion of each particle is referred to the center of mass of the molecule rather than to the external spaced-fixed coordinate axes, then the three translational coordinates that specify the location of the center of mass relative to the external axes may be separated out and eliminated from consideration. For a diatomic molecule (Q = 2) we are left with only three relative nuclear coordinates and with 3N relative electronic coordinates. For mathematical convenience, we select the center of mass of the nuclei as the reference point rather than the center of mass of the nuclei and electrons together. The difference is negligibly small. We designate the two nuclei as A and B, and introduce a new set of nuclear coordinates defined by... 

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.

Problem 7-8. Consider the case of a heteronuclear diatomic molecule constrained to move in one dimension. Let the masses of the nuclei be denoted by m and M, and the force constant by k. Set up and solve the secular equation determine that the allowed modes of motion are the overall translation and vibration. Determine the vibrational frequency in terms of m, M and k. 

Our treatment of the nuclear Schrodinger equation for diatomic molecules has shown that the wave function for nuclear motion can be separated into rotational, vibrational, and translational wave functions ... 

Thus, in this review we present the desorption phenomena focused on the rotational and translational motions of desorbed molecules. That is, we describe the DIET process stimulated by ultraviolet (UV) and visible nanosecond pulsed lasers for adsorbed diatomic molecules of NO and CO from surfaces. Non-thermal laser-induced desorption of NO and CO from metal surfaces occurs via two schemes of DIET and DIMET (desorption induced by multiple electronic transitions). DIET is induced by nanosecond-pulsed lasers and has been observed in the following systems NO from Pt(0 0 1) [4, 5],... 

In the absence of external fields, we may take axes that move laterally with the molecule, thus eliminating translational motion. In these co-ordinates a diatomic molecule becomes equivalent to a single particle with mass ti=MaMjs/(Ma+Mb), moving in a spherically symmetrical potential U(R), where R is the intranuclear separation. The Schrodinger equation is therefore... 

In order to discuss the spectroscopic properties of diatomic molecules it is useful to transform the kinetic energy operators (2.5) or (2.6) so that the translational, rotational, vibrational, and electronic motions are separated, or at least partly separated. In this section we shall discuss transformations of the origin of the space-fixed axis system the following choices of origin have been discussed by various authors (see figure 2.1) ... 

Now we want to determine the relation between temperature and the energy involved in other kinds of molecular motions that depend on molecular structure, not just the translation of the molecule. This relation is provided by the Boltzmann energy distribution, which relies on the quantum description of molecular motions. This section defines the Boltzmann distribution and uses it to describe the vibrational energy of diatomic molecules in a gas at temperature T. 

CO its rotation, must occur such that the angular momentum vector of CO is always perpendicular to the OCS plane, while the recoil direction is always in plane. Therefore, v is perpendicular to j this correlation in OCS photodissociation has in fact been seen [31], The departing CO resembles a tumbling cartwheel. Such vj correlations are very commonly observed. They are expected from the simple impulsive model or indeed from any model where the forces all lie within a plane. Any impulsive force imparted to a diatomic fragment that is directed at a position displaced from the center of mass of the diatomic molecule will lead to both translational and rotational energy and a cartwheeling motion will result. 

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