Potential energy curves rotational motion

The potential-energy curves for IBr are displayed in Fig. 1. The calculation reported by Brumer and Shapiro includes the effects arising from rotational motion of the molecule. Figure 2 shows the results obtained when the projec-... 

This may be written F(J) = B J(J+l)- DVJ2(J+1)2. If rotational levels are required to a greater degree of accuracy, higher terms may be included, but this is rarely justified by the experimental data. Thus, in principle, the energy levels for nuclear motion may be calculated exactly from a potential-energy curve, either by numerical solution of equation (1) for different values of J, or by numerical solution of the simplified rotationless equation,... 

Indeed, Dunham s energy-level formula [Eq. (2.1.1)] is based both on the concept of a potential energy curve, which rests on the separability of electronic and nuclear motions, and on the neglect of certain couplings between the angular momenta associated with nuclear rotation, electron spin, and electron orbital motion. The utility of the potential curve concept is related to the validity of the Born-Oppenheimer approximation, which is discussed in Section 3.1. 

Figure 30. Potential energy curve for the rotational motion of a molecular motor (upper panel). A hindered rotation takes place in a triple potential well. Starting from the bottom of the lowest potential well, the absorption of energy from a control field leads to free rotational motion. The case where a clockwise directional motion is obtained is indicated as (+). For a weaker field, the motion is forced to take place in the counterclockwise direction (—). The lower panel shows the dipole moment entering into the construction of the control field.

Measurement of the rotational absorption frequencies allows B to be found. From B, we get the molecule s moment of inertia I, and from I we get the bond distance d. The value of d found is an average over the w = 0 vibrational motion. Because of the asymmetry of the potential-energy curve in Figs. 4.5 and 13.1, d is very slightly longer than the equilibrium bond length in Fig. 13.1. 

As shown in Fig. 6.8 rotational excitations may lead to a qualitative change of the potential energy curve for the motion of the nuclei. Rotational excitations lower the dissociation energy of the molecule. They may also create metastable vibrational states (vibrational resonances). 

Figure 2.11 Vibrational and rotational motion of the I2 molecule. Left potential energy curves of the ground-state X Eg, the excited intermediate Ballou state reached by the pump laser and the final exdted (ion-pair) state fOg of I2 populated by the probe laser. The vibrational (and rotational) motion of the superposition of levels / of I2 in the Ballou state is monitored by the probe laser-induced fluorescence. Right laser-induced fluorescence intensity as a function of the delay time between probe pulse and pump pulse, showing the oscillation of the wave packet for the vibrational (top panel) and rotational (bottom panel) motion. For further details see text. Data adapted from Gruebele et al, Chem. Phys. Lett, 1990, 166 459, with permission of Elsevier...

This chapter is the first of three that dissect the molecular Hamiltonian. By solving just the electronic part of the Schrodinger equation, we construct the potential energy curves that dictate the motions of the atomic nuclei. From here, therefore, we can explain much about the vibrations of molecules (Chapter 8). And we draw on both of these terms in the Hamiltonian to arrive at an average overall geometry of the molecule, which then controls the rotational energies (Chapter 9). That will complete our picture of molecular structure, which provides the basis for our understanding of molecular interactions. 

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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