Vibrational term values for

The vibrational term values for a polyatomic anharmonic oscillator with only nondegenerate vibrations are modified from the harmonic oscillator values of Equation (6.41) to... 

The vibrational term values for any electronic state, ground or excited, can be expressed, as in Equation (6.16), by... 

Figure 1.13 shows the potential function, vibrational wave functions and energy levels for a harmonic oscillator. Just as for rotation it is convenient to use term values instead of energy levels. Vibrational term values G(v) invariably have dimensions of wavenumber, so we have, from Equation (1.69),... 

In an approximation which is analogous to that which we have used for a diatomic molecule, each of the vibrations of a polyatomic molecule can be regarded as harmonic. Quantum mechanical treatment in the harmonic oscillator approximation shows that the vibrational term values G(v ) associated with each normal vibration i, all taken to be nondegenerate, are given by... 

As for diatomic molecules, there are stacks of rotational energy levels associated with all vibrational levels of a polyatomic molecule. The resulting term values S are given by the sum of the rotational and vibrational term values... 

Neglecting centrifugal distortion, the rotational term values for a spherical rotor in an A j vibrational state are... 

From the table a series of averaged values of vibrational term value differences, G ( v + 1) — (Hr), can be obtained for both electronic states and, from Equation (6.18), values of cue, cuexe, etc., for both states. For example, in the All and X1D+ states of CO, coe is 1518.2 cm-1 and 2169.8 cm-1, respectively. The large decrease in coe in the A state is a consequence of promoting an electron from a bonding to an antibonding orbital, greatly reducing the force constant. 

Relationships connecting ae, //., ye with coe, coexe and Be have been given by Pekeris [67] and Dunham [65], In a very detailed study of vibration-rotation interactions, Dunham [65] has shown that the term values for a vibrating rotor should actually be expressed as a double power series, given by... 

FT i.r, spectroscopy has been used to determine the solvent-induced frequency shifts for the C—H stretching bands of n-octane. Measurement of the FT i.r. spectra for HCl and DCl over the range 2840—8450cm and for HI and DI over the range 3000—10 380 cm" allowed accurate prediction of the TCI and TI spectra. For D2 0, the high resolution (5 x 10 cm ) available using FT i.r. spectroscopy enabled an extended and more precise set of rotational levels to be derived for the vibrational states (000), (020), (100), and (001). Rotational constants and vibrational term values were evaluated for from FT i.r. 

Turning our attention to the demanding triply bonded case of Ni, we find again a dramatic improvement in the agreement between the experimentally determined and computed vibrational term values as we proceed from standard CCSD to 4R-RMR, and to 8R-RMR CCSD, as illustrated for a few levels (for v=0, 5, 10, 15, 20 and 25) in Table II [actually, the highest experimentally observed level corresponds to v=15 and the values up to v=25 were obtained via the RKR procedure (43), For example, for the highest available v=25 level, the... 

Table II. Vibrational term values G(v) for selected levels v of the N2 molecule, and the deviations from the experimental values, G(v)-G< V), as obtained with the CCSD, 4R- and 8R-RMR CCSD...

For non-linear molecules, when treated as rigid (i.e., having fixed bond lengths, usually taken to be the equilibrium values or some vibrationally averaged values), the rotational Hamiltonian can be written in terms of rotation about three axes. If these axes (X,Y,Z) are located at the center of mass of the molecule but fixed in space such that they do not move with the molecule, then the rotational Hamiltonian can be expressed as ... 

There is a stack of rotational levels, with term values such as those given by Equation (5.19), associated with not only the zero-point vibrational level but also all the other vibrational levels shown, for example, in Figure 1.13. However, the Boltzmann equation (Equation 2.11), together with the vibrational energy level expression (Equation 1.69), gives the ratio of the population of the wth vibrational level to Nq, that of the zero-point level, as... 

The reason for the subscript 2 in the A FiJ) symbol is that these are the differences between rotational term values, in a particular vibrational state, with J differing by 2. 

For an anharmonic oscillator with degenerate vibrations the term values are modified from those of Equation (6.88) to... 

Just as in the ground electronic state a molecule may vibrate and rotate in excited electronic states. The total term value S for a molecule with an electronic term value T,... 

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