Quantum numbers centrifugal distortions

Since even very low levels of theoiy can give fairly accurate geometries, rotational spectra are quite simple to address computationally, at least over low rotational quantum numbers. For higher-energy rotational levels, molecular centrifugal distortion becomes an issue, and more sophisticated solutions of Eq. (9.37) are required. 

Vibration-rotation interaction causes the rotational constants to vary with the vibrational quantum numbers [Eq. (5.72)]. Correction terms for centrifugal distortion can also be added to the energy expressions for asymmetric tops. 

Up to this point, the molecule has been considered to be a rigid rotor, but the work in Chapter 4 on diatomics shows that we must add corrections for rotation-vibration interaction and centrifugal distortion. For a polyatomic molecule, there are several normal modes of vibration, each with its own vibrational quantum number (see Chapter 6). By analogy to (4.75), we write for polyatomic molecules... 

It should be noted that the rotational spectroscopy of CO confined to a single vibrational level, usually the ground v = 0 level, provides only a limited amount of information about molecular structure. In the field of vibration-rotation spectroscopy, however, CO has been studied extensively and particular attention paid to the variation of the rotational and centrifugal distortion constants with vibrational quantum number. Vibrational transitions involving v up to 37 have been studied with high accuracy [78, 79, 80], and the measurements extended to other isotopic species [81] to test the conventional isotopic relationships. CO is, however, an extremely important and widespread molecule in the interstellar medium. CO distribution maps are now commonplace and with the advent of far-inffared telescopes, it is also an important... 

Spectroscopists often write Dj 3 Dg, risking confusion with the potential-well depth Dg in Fig. 14.1.) Centrifugal distortion is usually not very significant until rotational quantum numbers around J = 30 are reached. A more important effect is the dependence of the rotational constant on the vibrational quantum number. This can be parametrized as... 

Here u>k and xkk> k = 1, 2,.. . m, are constants. and T pg-g- correspond to the rotational and the centrifugal distortion constants, since Jg = Pg/h has been introduced for the angular and large amplitude momenta in units of ft (g = x, y, z, 1,2,...n, m + n = 3 N-6). It must be emphasized, however, that they are not constants, since they depend on the large amplitude coordinates, pf / = 1.2.. .. . Furthermore the matrix of fi -elements is in general non-diagonal. Despite this we shall call these quantities by there usual names, but use citation marks around the word constant to prevent confusion. In the present approach it is assumed that and the effective potential depend linearly on the vibrational quantum numbers, vk. 

Here v is the transition frequency, B is the rotational constant, D is the centrifugal distortion parameter, and J is the rotational quantum number. Students were first asked to calculate the frequencies of HCCCN based on these constants and equation 1. Three rotational transitions were chosen by the students to be measured at the telescope in the range of 72 to 163 GHz prior to the observations. This task allowed the students to actively participate in planning the observations. These frequencies were then used at the telescope to detect HCCCN towards a dense cloud, W51M, located approximately 23,000... 

The most important use of rotational spectroscopy is the determination of rotation constants. A problem besetting rotational spectroscopy is centrifugal distortion however, it is possible to take account of this if precise data over a range of quantum numbers is available. 

The molecule PH3 (C3V symmetry) is an oblate symmetric top (Crotational constants C (refers to rotation around the C3 axis) and B (perpendicular to C3). Since the permanent electric dipole moment is pointed parallel to the C3 axis, only pure rotational transitions with the selection rule AK=0 are allowed (K is the quantum number of the component about the C3 axis of the total angular momentum J). Their analysis leads to the parameters B, Dj, Djk, and Hjk. From the perturbation-allowed transitions AK= 3n (n=1,2,...), which become weakly allowed by centrifugal distortion effects (inducing a small dipole moment of about 8x10 D perpendicular to the C3 axis [1, 2, 3]), the K-related constants (C, Dk, Hk) were obtained see, e.g. [1, 3, 4]. 

Centrifugal distortion constants are not of prime importance in the determination of molecular stmcture, but it may be necessary to determine them in order to interpret an observed speetmm. To determine a quartic distortion constant it is generally necessary to have very precise data, preferably over a good range of the quantum numbers (7, and also K for a symmetric top). Otherwise, there is no partieular difficulty in obtaining reliable values for Dj and Djx for a symmetrie top, or D for a linear molecule. ) has very little influence on the form of the speetmm for a symmetric top, and is correspondingly difficult to determine. 

FIGURE 14.16 (a) Normal bond length in adiatomic molecule, (b) As the rotational quantum number increases, a centrifugal distortion causes the bond to stretch. This adds some nonideal component to the predicted rotational spectrum. 

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