Term values rotational

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 ... 

Figure 5.2 shows a set of rotational energy levels, or, strictly, term values, for the CO molecule. 

Figure 5.2 Rotational term values F J) (horizontal lines) relative populations Nj/Nq (calculated from Equation 5.15) and transition wavenumbers v (for the transitions indicated by the vertical arrows) for CO...

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... 

Although these molecules form much the largest group we shall take up the smallest space in considering their rotational spectra. The reason for this is that there are no closed formulae for their rotational term values. Instead, these term values can be determined accurately only by a matrix diagonalization for each value of J, which remains a good quantum number. The selection mle A/ = 0, 1 applies and the molecule must have a permanent dipole moment. 

At a simple level, the rotational transitions of near-symmetric rotors (see Equations 5.8 and 5.9) are easier to understand. For a prolate or oblate near-symmetric rotor the rotational term values are given, approximately, by... 

Examples of prolate near-symmetric rotors are the s-trans and s-cis isomers of crotonic acid, shown in Figure 5.8, the a axis straddling a chain of the heavier atoms in both species. The rotational term values for both isomers are given approximately by Equation (5.37) but, because A and B are different for each of them, their rotational transitions are not quite coincident. Figure 5.9 shows a part of a low-resolution microwave spectmm of crotonic acid in which the weaker series of lines is due to the less abundant s-cis isomer and the stronger series is due to the more abundant s-trans isomer. 

Starting from an expression for rotational term values show that... 

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),... 

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. 

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

Although, as in linear and symmetric rotor molecules, the term values are slightly modified by Coriolis forces in a degenerate (T2) state, the rotational selection rules... 

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,... 

In Figure 7.25 are shown stacks of rotational levels associated with two electronic states between which a transition is allowed by the -F -F and, if it is a homonuclear diatomic, g u selection rules of Equations (7.70) and (7.71). The sets of levels would be similar if both were states or if the upper state were g and the lower state u The rotational term values for any X state are given by the expression encountered first in Equation (5.23), namely... 

For an oblate symmetric rotor, such as NH3, the rotational term values are given by... 

Any effects of centrifugal distortion will show up as slight curvature of the A2F(J) versus (J + f) graphs. If the term —DJ2(J + l)2 is included, as in Equation (6.24), in the rotational term value expression we get... 

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