Rotational Constants. Potential Energy Functions

4 Vibrational and Rotational Constants. Potential Energy Functions 

Experimental Results. The experimentalspectroscopic constants for NF(X32 ,a A, b 2 , c n) are given in Table 16, p. 279, along with some explanatory comments regarding the methods of derivation. 

Theoretical Results. Generally, the ab initio cOe values for the states X, a, and b are larger 

Calculated potential energy curves were depicted for the three lowest states [10,11, 13, 14,16]. 

Three Cl calculations dealt with the potential curves for a number of unknown excited states (cf. pp. 274/5) [10, 14, 16]. 

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Diatomic molecules provide a simple introduction to the relation between force constants in the potential energy function, and the observed vibration-rotation spectrum. The essential theory was worked out by Dunham20 as long ago as 1932 however, Dunham used a different notation to that presented here, which is chosen to parallel the notation for polyatomic molecules used in later sections. He also developed the theory to a higher order than that presented here. For a diatomic molecule the energy levels are observed empirically to be well represented by a convergent power-series expansion in the vibrational quantum number v and the rotational quantum number J, the term... 

When, the geometry of the molecule is fully optimized at each conformation, the geometry of the whole molecule changes during the rotation. As a result, the rotational constants change with the rotation angles, and have to be fitted in a Fourier series. Since the deformation does not influence the dynamical symmetry properties of the molecule, the rotational B constants may be fitted to a symmetry adapted functional form identical to that of the potential energy function of (113). 

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. 

The choice of Internal coordinates as an object for optimisation Is obvious use of rotational constants maybe less so. They certainly do not give very detailed Information about the conformation of a molecule, but they are the primary structural Information derived from rotational and ro-vlb spectroscopy on small molecules. The Inclusion of dipole moments Is a must when Coulomb terms are present In the potential energy function. Charges are Included, although they are not experimentally observable quantities, because It may be desirable to lock a parameter set to data derived from photoelectron spectroscopy or from ab Initio calculations with a large basis set. Quite naturally we want to optimise on vibrational spectra, and we shall see below that It Is a bit more cumbersome In the consistent force field context than In traditional normal coordinate analysis. 

PF and A for the pure solvent) and will be cancelled out when computing the binding constants or the correlation function. The quantity Eq( ) is essentially the rotational potential energy of the empty molecule, i.e., the doubly ionized acid, as given in Eqs. (4.8.26) V ,(< >) in Eq. (4.8.26) is the rotational potential energy of ethane (Eliel and Wilen, 1994) and is given by... 

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... 

Y, and Z are connected by bonds of fixed length joined at fixed valence angles, that atoms W, X, and Y are confined to fixed positions in the plane of the paper, and that torsional rotation 0 occurs about the X-Y bond which allows Z to move on the circular path depicted. If the rotation 0 is "free such that the potential energy is constant for all values of 0, then all points on the circular locus are equally probable, and the mean position of Z, i.e., the terminus of , lies at point z. The mean vector would terminate at z for any potential function symmetric in 0 for any potential function at all, except one that allows absolutely no rotational motion, the vector will terminate at a point that is not on the circle. Thus, the mean position of Z as seen from W is not any one of the positions that Z can actually adopt, and, while the magnitude ll may correspond to some separation that W and Z can in fact achieve, it is incorrect to attribute the separation to any real conformation of the entity W-X-Y-Z. Mean conformations tiiat would place Z at a position z relative to the fixed positions of W, X, and Y have been called "virtual" conformations.i9,20it is clear that such conformations can never be identified with any conformation that the molecule can actually adopt... 

The first term on the right is the translational kinetic energy of the molecule as a whole this simply adds a constant to the total energy, and we shall omit this term. The second and third terms are ihe rotational and vibrational kinetic energies of the molecule. The final term is the energy of interaction between rotation and vibration. To get the classical-mechanical Hamiltonian function, we add the potential energy V to (5.2), where U is a function of the relative positions of the nuclei. 

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