Rotational and Vibrational Constants. Internuclear Distance

Geometry optimization for determining rg and evaluating the rotational and vibrational constants are included in numerous quantum-chemical ab initio calculations on NH. These are essentially the same studies that are quoted in the next section on potential energy functions (see table on p. 55). Further references may be found in the bibliographies on quantum-chemical calculations given on p. 31. 

ND in the States X and a A. Rotational and Vibrational Constants from Purely Rotational and Rotational-Vibrational Spectra in the Far IR and IR. 

Rotational and vibrational constants in cm equilibrium internuclear distance r in A. 

ND in the States X A Tlj, a A, b c TI, d Rotational and Vibrational Constants from Rovibronic Spectra in the Near IR, Visible, and UV. 

Rydberg States. Vibrationally and rotationally resolved, (2 +1)-resonance-enhanced multiphoton ionization (REMPI) spectra and REMPI-PES of NH, ND(a A) and NH, ND(X S ) radicals in the UV (cf. pp. 83/7) revealed the existence of some singlet and triplet Rydberg states. Using literature data for the spectroscopic constants of the a and states, 

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PH, PD. Ground State X Rotational and Vibrational Constants, Internuclear Distance. B, D, tte, cOe, cOeXe in cm" re in A three standard deviation in parentheses. 

Rotational and Vibrational Constants. Internuclear Distance. Experimental results are available for the ground state X 211 and the excited state A 2A, which have been obtained from rotational analyses of the UV-visible A-X system. Rotational constants Be or Bq, ae, Ye, Dq, vibrational constants cOe, cOgXe, or AG1/2, and internuclear distances re or ro are given in Table 4. 

PH+. Theoretical Rotational and Vibrational Constants and Internuclear Distances. Be, Bq, a, cOe, cOeXe in crrr rg in A. 

As mentioned in the previous papers, spectroscopy (sp) and electron diffraction (ed) have their own merits and demerits and, in many cases, a combined use of them should result in the most accurate structure for free molecules. The connections among the structural parameters determined by the ed and sp methods are shown in Table 1. Information about geometry comes from the bonded and non-bonded internuclear distances determined by ED and also from the rotational constants determined by sp. One has to test in the first place whether these values are consistent with each other. One of the experimental examinations is shown in Figure 1.t For BF3, the discrepancies between the measured ed intensities (dots) and those calculated from the rotational constants B0 (measured by Ginn etaL by high-resolution infrared spectroscopy), with corrections for vibrational effects, are well within the estimated limits of experimental error. The rz (B-F) distances determined by ed and sp independently are 1311 0.0012 A and 13113 ... 

Here, Be is called the equilibrium rotation constant, as it corresponds to the rotation constant of a hypothetical molecule with a fixed (i.e. vibration-free) internuclear distance r, which is the equilibrium internuclear distance. The quantity a is a vibration-rotation constant. Thus, in general, the two different vibrational states involved in a transition have values of By that, for a fundamental transition, differ by a in Eq. 7.11 above. Suitable manipulation of measured transition energies (such as a combination of P- and R-branch line positions) allows us to derive Bq, Bi and their difference directly from the spectrum. 

From precise wavelength measurements of the fluorescence spectrum (which may be performed e. g. by interferometric methods accurate values for the molecular constants can be obtained since the wavelength differences of subsequent lines in the fluorescence progression yield the energy separation of adjacent vibrational and rotational levels as a function of v . From these spectroscopically deduced molecular constants, the internuclear distance can be calculated A special computer programm developed by Zare ) allows the potential curve to be constructed from the measured constants and, if the observed fluorescence progression... 

The fifth term in (4.67) represents an interaction between vibration and rotation, and ae is called a vibration-rotation coupling constant. [Do not confuse ae with a in (4.26).] As the vibrational quantum number increases, the average internuclear distance increases, because of the anharmonicity of the potential-energy curve (Fig. 4.4). This increases the effective moment of inertia, and therefore decreases the rotational energy. We can define a mean rotational constant Bv for states with vibrational quantum number v by... 

The values of p and T can now be used for the statistical mechanical calculations. In order to calculate the rotational characteristic temperature t with Eq. (20), use the literature value for the rotational constant Bo = 0.037315 cm [or calculate Bo from the internuclear distance in the molecule, rg = 0.2667 nm, with Eqs. (17) to (19)]. From the literature value of the molecular vibrational frequency in the gas phase, Tg = 213.3 cm , calculate the vibrational characteristic temperature vu, with Eq. (22). From the phonon dispersion data in Table 1, calculate the 12 vibrational characteristic temperatures , -. 

The equilibrium internuclear separation (r ) of HgO(g) is estimated from the corresponding quantity for PbO(g) and comparison of Hg-X and Pb-X bond distances for cases in which values of both distances are known. The rotational constant 8 is calculated from r. The fundamental vibrational frequency (d is estimated from Guggenhelmer s relation for multiple bonded molecules ( ). 

Vibrational-rotational molecular constants, and structural quantities (harmonic force constant, k, and equilibrium internuclear distance, Re) derived from these constants are listed in Table 1.1 for the electronic ground states of three... 

This table lists the leading spectroscopic constants and equilibrium internuclear distance in the ground electronic state for selected diatomic molecules. The constants are those describing the vibrational and rotational energy through the expressions ... 

Because molecules are not rigid, the rotational energy levels for diatomic molecules differ slightly from rigid-rotor levels. From (6.52) and (6.55), the two-particle rigid-rotor levels are = BhJ J -l-1). Because of the anharmonicity of molecular vibration (Fig. 4.6), the average internuclear distance increases with increasing vibrational quantum number v, so as v increases, the moment of inertia I increases and the rotational constant B decreases. To allow for the dependence of B on v, one replaces B in E by The mean rotational constant B for vibrational level v is - Ug v + 1/2),... 

Rotational constants, centrifugal distortion constants, rotation-vibration interaction constants, Dunham energy-parameters and potential-coefficients, parameters of the breakdown of the Bom-Oppenheimer approximation and of the nuclear field shift, and equilibrium internuclear distances. 

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