Energy levels of a molecule

The rotational eigenfunctions and energy levels of a molecule for which all three principal moments of inertia are distinct (a so-called asymmetric top) can not easily be expressed in terms of the angular momentum eigenstates and the J, M, and K quantum numbers. However, given the three principal moments of inertia la, Ib, and Ic, a matrix representation of each of the three contributions to the rotational Hamiltonian... 

Fig. 1. Schematic of the vibrational energy levels of a molecule where (-) indicate the changes effected by the excitation photon, and (—) those of...

As already introduced in section I of this chapter, in a CARS process (Figures 7.9a-c see also Figure 7.1c), a Raman transition between two vibrational energy levels of a molecule is coherently driven by two optical laser fields (frequencies co and co) and subsequently probed by interaction with a third field at frequency co, . This generates the anti-Stokes signal at the blue-shifted frequency cars = p- The... 

On the other hand, there are many instances when the rotating wave approximation cannot be used. For example, in order to find the energy levels of a molecule placed in a strong microwave field, it is necessary to diagonalize a large piece of the full Floquet matrix involving multiple n-states and multiple eigenstates of Hq, as discussed in Section 8.3.4. 

The energy levels of a molecule placed in an off-resonant microwave field can be calculafed by diagonalizing fhe mafrix of fhe Floquef Hamiltonian in the basis of direct products y) ), where y) represents in the eigenstates of the molecule in the absence of the field and ) - fhe Fourier componenfs in Eq. (8.21). The states k) are equivalent to photon number states in the alternative formalism using the quantum representation of the field [11, 15, 26], The eigensfales of the Floquet Hamiltonian are the coherent superpositions... 

IR spectroscopy involves the study of transitions between the vibrational energy levels of a molecule and the interaction of the oscillating electric vector of the IR light with the oscillating dipole moment of the molecule. 

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... 

Fig. 6.4 Vibrational energy levels of a molecule with >, 3800 cm-1, v2wk 1600 cm-1, and p3 3900 cm-1. (These are roughly the values for H20.) The energy is calculated from (6.54), and does not include anharmonicity corrections.

The development of a new form of spectroscopy based on the exploitation of the time evolution of the coherence associated with the rotational motion of an excited molecule. Conventional spectroscopies depend on the measurement of differences between the energy levels of a molecule, which become more and more difficult to measure and to interpret as the size of the molecule increases. In contrast, the intervals between recurrences in the coherent rotational motions of large molecules are directly related to the moments of inertia of the molecules and can be used to determine their structures. 

The number of vibrational modes of a molecule composed of N atoms is 3N — 6 (or 3N — 5 if linear). We may find which of these are infrared and Raman active by the application of a few simple symmetry arguments. First, infrared energy is absorbed for certain changes in the vibrational energy levels of a molecule. For a vibration to be infrared active, there must be a change in the dipole moment vector... 

Molecular absorption-A higher energy-level change involving a change in the energy levels of a molecule and characterized by isotropic absorption in the visible or ultraviolet range. 

Estimates of the triplet energy level of a molecule may be made by observing whether it can accept or transfer energy to several other molecules. Hammond and co-workers42 have shown that ds-trans interconversions of piperylene, 2-pentene, and 1 2 dichloroethylene may be affected by many photosensitizers. The stationary states of the sensitized ds-trans ratios of piperylene with various donors were found to form a coherent pattern if triplet energy transfer was assumed as the key step in the photochemical reaction. From these results they were able to infer the presence and the energies of the triplet states of acetone, phenanthrenequinone, and fluorenone, for which phosphorescence data are not available. The triplet levels were estimated as >70, 65, and 62 kcal./mole, respectively. 

Photoelectron spectroscopy is a very valuable tool in measuring the different energy levels of a molecule. This is however based on the assumption that the ionization potentials which are determined experimentally are directly correlated with the MO s of a certain molecule, i.e. Koopman s theorem must be valid. The few azo compounds studied so far already allow a clear assignment of the orbitals concerned and a deeper insight into the ordering of the n and -states in azo compounds. 

As is obvious from the preceding discussion, calculation of the electronic energy levels for a periodic sohd is no more complex than calculation of the electronic energy levels of a molecule. The only difference is that in the case of a periodic solid the calculations have to be carried out for a large number of k values, sampling the first BZ of the solid, that is, from —Ttja to r/a for a ID system. However, in this case, because ofthe general relationship e, (/c) = e, (—fc), calculations can be restricted to the k values Q irreducible Brillouin zone (IBZ) of the system. 

FIGURE 1. Jablonski diagram for the relative positions of the electronic energy levels of a molecule, where F = fluorescence rate, P = phosphorescence rate and VR = vihrational relaxation... 

Fig. 2-1. A diagram of the lower energy levels of a molecule which contains an even number of electrons without orbital degeneracy and the primary processes induced upon photon absorption by this molecule.

Band spectra are often produced in spectral sources because of the presence of gaseous radicals or small molecules. For example, in Figure 24-19, bands for OH, MgOH, and MgO are labeled and consist of a series of closely spaced lines that are not fully resolved by the instrument used to obtain the spectrum. Bands arise from the numerous quantized vibrational levels that are superimposed on the ground-state electronic energy level of a molecule. For further discussion of band spectra, see Section 28B-3. 

The energy levels of a molecule confined to a cubic box of side l are given, as indicated in Section 9.1, by... 

The energy levels of a molecule vibrating with simple harmonic motion are given by sv-lb — hv(v + ), where v is the vibrational frequency and v the vibrational quantum number. To obtain the vibrational partition function we sum the energy levels measuring the energy from the lowest available level, e0 %hv (the zero-point energy), to obtain... 

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