Vibrational term values

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 rotational constants B and D are both slightly vibrationally dependent so that the term values of Equation (5.19) should be written... 

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),... 

However, unlike electrical anharmonicity, mechanical anharmonicity modifies the vibrational term values and wave functions. The harmonic oscillator term values of Equation (6.3) are modified to a power series in (u + ) ... 

When a molecule has both vibrational and rotational energy the total term values S are given by the sum of the rotational term values F J), given in Equation (5.23), and the vibrational term values G v), given in Equation (6.16) ... 

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. 

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... 

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

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... 

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

The only types of anharmonic potential function we have encountered so far are the two illustrated in Figure 6.38, both of which show only a single minimum. There are, however, some vibrations whose potential functions do not resemble either of those but show more than one minimum and whose term values are neither harmonic, nor are they given by Equation (6.88) or Equation (6.89). Such vibrations can be separated into various types, which will now be discussed individually. 

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,... 

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

If a sufficient number of vibrational term values are known in any electronic state the dissociation energy Dq can be obtained from a Birge-Sponer extrapolation, as discussed in... 

Section 6.13.2 and illustrated in Figure 6.5. The possible inaccuracies of the method were made clear and it was stressed that these are reduced by obtaining term values near to the dissociation limit. Whether this can be done depends very much on the relative dispositions of the various potential curves in a particular molecule and whether electronic transitions between them are allowed. How many ground state vibrational term values can be obtained from an emission spectrum is determined by the Franck-Condon principle. If r c r" then progressions in emission are very short and few term values result but if r is very different from r", as in the A U — system of carbon monoxide discussed in Section 7.2.5.4, long progressions are observed in emission and a more accurate value of Dq can be obtained. 

It has been demonstrated in several benchmark calculations that the CR-CCSD(T) (completely renormalized CCSD(T)) and CR-CCSD(TQ) (completely renormalized CCSD(TQ)) methods provide an excellent description of entire PESs involving single and double bond dissociation (P, 13, 15, 17-19, 21, 111), highly-excited vibrational term values near dissociation 17, 18, 21, 111), and... 

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