Equilibrium rotational constants

The equilibrium rotational constants and the theoretical harmonic frequencies calculated by Sherrill and Schaefer424 were in good agreement with the experimental ground-state rotational constants obtained by Leclercq and Dubois420 and in reasonable accord with the experimental fundamental vibrational frequencies obtained by Bengali and Leopold425. 

The conclusion is that if the spectrum can be analysed in terms of equations (3)—(7), then the force constants can be determined. The bond length re can be determined from the equilibrium rotational constant Bc then the quadratic force constant /3 can be determined either from the harmonic wavenumber centrifugal distortion constant De then the cubic force constant /3 can be determined from aB and finally the quartic force constant /4 can be determined from x. It is necessary to determine the force constants in this order since in each case we depend upon already knowing the preceding constants of lower order. The values of re,f2,f3, and /4 calculated in this way for a number of diatomic molecules are shown in Table 2. 

The vibration-rotation spectra of the v, and v bands of CO C1 j have been measured (using a tunable semiconductor-diode laser), and assigned with the aid of Stark modulation spectra [2224]. The precise values of these bands were determined to be 1828.2012 and 851.0105 cm", respectively, and the equilibrium rotational constants for CO Clj were calculated as = 7950.35, Bg = 3490.22, and Cg = 2425.44 MHz cf. 

The details of spectral analysis need not concern us here standard sources should be consulted [16,18]. All we need to know for our purposes is that analysis of rotational spectra or vibration-rotation bands leads to what are called the effective rotational constants as symbolized in Eqs. (1). For simplicity we shall often write simply Ay, By and Cy, but it is imperative to remember that the simple subscript signifies all the vibrational quantum numbers. The effective rotational constants are related to the equilibrium rotational constants by... 

It is worth noting the experimental task involved in evaluating equilibrium rotational constants. For the diatomic molecule, it is necessary to measure B in a minimum of two states, e.g., v = 0 and v = 1. Then Eq. (15) yields... 

Le Guennec et al. determined equilibrium rotational constants and moments of inertia for FCIO3 [51] and 74gcH3F [52] and calculated the vibration-rotation contributions s for each principal axis. They used them to estimate s for other isotopomers of FCIO3 and 74gcH3F using the relation... 

CINO, experimental a constants were available for only two isotopomers to determine the equilibrium rotational constants. The discrepancy in the case of COCI2 is suspected by the authors to arise om inconsistencies in the rotational constants because of the quadrupole coupling. For these twelve molecules not containing hydrogen, the standard deviations of the fits are significantly smaller than the standard deviations of /-q fits, with ratios o(ro)/a(r, ,(2)) ranging fi-om as low as 20 to several hundreds. The ratios with the standard deviations from fits, o r a rjP) ), range fi-om 2 to 30. structures were determined for the linear hydrides... 

If the Hamiltonian in Eq. (1) is adequate to interpret the rotational spectrum of the molecule, the experimental frequencies may be used to obtain values of the rotational constants. In most cases rotational constants for the ground vibrational state (i.e., A0, B0y C0) will be available. To obtain equilibrium rotational constants, rotational spectra in the first excited state of every vibrational mode must be analyzed. 

In modem papers ground state constants are frequently reported with cited uncertainties lxl0" cm" (3 kHz) from infrared work and 1 x lO cm" (0.3 MHz) from Raman studies. In band spectra, two sets of rotational constants are obtained, those of the upper and lower states involved in the transition, and a statistical treatment allows the differences between the constants to be determined to precisions approaching or eqnal to microwave uncertainties (1 kHz or less). Thus equilibrium rotational constants of polar molecules can be quite precisely calculated by using microwave-determined Bq constants and infrared-determined a constants. When the values of some of these a constants are missing, they can be substituted by reliable ab initio values. Despite the recent instmmental improvements, the resolution available from both infrared and Raman studies is still much lower than that from microwave spectroscopy, and therefore, studies are limited to fairly small and simple molecules. However, these techniques are not restricted to polar molecules as is the case for microwave spectroscopy, and thus... 

The vibration-rotation spectra and/or the rotational spectra in excited vibrational states provide the af constants and, when all the a/ constants are determined, the equilibrium rotational constants can be obtained by extrapolation. This method has often been hampered by anharmonic or harmonic resonance interactions in excited vibrational states, such as Fermi resonances arising from cubic and higher anharmonic force constants in the vibrational potential, or by Coriolis resonances. Equihbrium rotational constants have so far been determined only for a limited number of simple molecules. To be even more precise, one has further to consider the contributions of electrons to the moments of inertia, and to correct for the small effects of centrifugal distortion which arise from transformation of the original Hamiltonian to eliminate indeterminacy terms [11]. Higher-order time-independent effects such as the breakdown of the Bom-Oppenheimer separation between the electronic and nuclear motions have been discussed so far only for diatomic molecules [12]. 

The equilibrium rotational constants were calculated using experimental a, constants for three isotopomers completed by values deduced from an empirical anharmonic force field based on an ab initio surface. The structure was obtained by a fit of all equilibrium rotational constants. Other calculations and experimental data give credit to the structure. 

The re structure is derived from a set of equilibrium rotational constants, obtained from experimental ground state constants and ab initio constants. 

The equilibrium rotational constants were calculated using experimental Ot constants. The stmcture was deduced from the values of and Ce only. 

The equilibrium rotational constant was obtained from Bq by using two experimental and two ab initio Oi values. The structure was deduced from Bg. 

Subscripts on B refer to vibrational states, e.g. B (equilibrium rotational constant) B, Bi, etc. (rotational constant in vibrational state labelled 0, 1, V, etc.). The symbol used for a general moment of inertia is I. I is related to B by... 

Summary.— The observed rotational constants for the ground vibrational state, or indeed any other vibrational State, are functions of the interatomic distances which are averaged in a complex and subtle way over the molecular vibrations. If the rotational constants can be extrapolated empirically to equilibrium rotational constants, good re structures can be calculated from them but the labour of doing this is considerable, and it has so far proved possible only for simple polyatomic molecules. 

If, because equilibrium rotational constants are not available, structural parameters are calculated from rotational constants for the ground vibrational state, then the ts structure is the most satisfactory. The ra structure is self-consistent and is easy to calculate, but it does not have a clear physical significance and rg parameters cannot easily be compared with rg parameters. The i-g structure has a clear physical significance only for diatomic molecules, where rg = and it should be avoided otherwise. 

Equilibrium rotational constants [5,14] Rotation-vibration interaction constants [5, 14] ... 

This means that estimates of the effective potential minimum and the corresponding equilibrium rotational constant are available in the form... 

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