Diatomic molecules nuclear motion

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.

Condon E U 1928 Nuclear motion associated with electron transitions in diatomic molecules Phys. Rev. 32 858-72... 

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... 

The potentiul energy for the nuclear motion of a diatomic molecule is closely approximated hy the Morse function... 

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... 

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... 

Unlike the case of simple diatomic molecules, the reaction coordinate in polyatomic molecules does not simply correspond to the change of a particular chemical bond. Therefore, it is not yet clear for polyatomic molecules how the observed wavepacket motion is related to the reaction coordinate. Study of such a coherent vibration in ultrafast reacting system is expected to give us a clue to reveal its significance in chemical reactions. In this study, we employed two-color pump-probe spectroscopy with ultrashort pulses in the 10-fs regime, and investigated the coherent nuclear motion of solution-phase molecules that undergo photodissociation and intramolecular proton transfer in the excited state. 

Combining (4.6) and (4.24), we have as the approximate wave function for the internal nuclear motion of a diatomic molecule... 

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 ... 

Whereas for diatomic molecules the vibration-rotation interaction added only a small correction to the energy, for a number of polyatomic molecules the vibration-rotation interaction leads to relatively large corrections. Similarly, although the Born-Oppenheimer separation of electronic and nuclear motions holds extremely well for diatomic molecules, it occasionally breaks down for polyatomic molecules, leading to substantial interactions between electronic and nuclear motions. 

If the rotational quantum number J is zero, the molecule possesses no angular momentum arising from the motion of the nuclei nuclear motion is purely vibrational. The vibrational energy levels depend on the shape of the potential function 1/(7 ), most often of the well-known diatomic form. Near the minimum the potential function approximates to a parabola. The eigenvalues and functions are thus approximately those appropriate for a harmonic oscillator,... 

We require only three nuclear coordinates to define the nuclear motion and we choose these to be R, the internuclear distance, and 0 the third Euler angle x is a redundant coordinate. In fact, because there are no nuclei lying off-axis in a diatomic molecule, X is undefineable it is, however, expedient to retain it because of simplification in the final form of the rotational Hamiltonian. We shall examine this point in more detail in... 

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