Internal coordinates, vibration-rotation derivatives

T. J. Lukka, A simple method for the derivation of exact quantum-mechanical vibration-rotation Hamiltonians in terms of internal coordinates. J. Chem. Phys. 102, 3945—3955 (1995). 

S. M. Colwell and N. C. Handy, The derivation of vibration-rotation kinetic energy operators in internal coordinates II. Mol. Phys. 92, 317—330 (1997). 

This complicated paper describes the separation of electronic and nuclear coordinates in a molecular quantum-mechanical problem. The internal coordinates of the molecule are indicated by the translation and rotation coordinates by The equation for the minimum of the electronic energy is derived [Eq. (40)], and the problem of molecular vibration [Eq. (46)] as well as molecular rotation [Eq. (69)] is discussed. 

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. 

The result of all of the vibrational modes contributions to la (3 J-/3Ra) is a vector p-trans that is termed the vibrational "transition dipole" moment. This is a vector with components along, in principle, all three of the internal axes of the molecule. For each particular vibrational transition (i.e., each particular X and Xf) its orientation in space depends only on the orientation of the molecule it is thus said to be locked to the molecule s coordinate frame. As such, its orientation relative to the lab-fixed coordinates (which is needed to effect a derivation of rotational selection rules as was done earlier in this Chapter) can be described much as was done above for the vibrationally averaged dipole moment that arises in purely rotational transitions. There are, however, important differences in detail. In particular. 

There is also an immediate interest in coordinate derivatives of chemical shifts in situations where temperature effects, isotope chemical shifts, vibrational or rotational corrections are investigated. For these purposes chemical shielding hyper-surfaces have been calculated using ab initio methods (for a discussion see Jameson, Bennett and Raynes, Raynes and Bennett, Chesnut and Wright, de Dios et Sundholm et Sundholm and Gauss,and Auer et and the slope with respect to any internal... 

Thus, reaction rate coefficients can be estimated from the thermochemistry of the transition states, whose molecular properties can be calculated with quantum chemical programs. In calculating reaction rate coefficients, the only negative second derivative of energy with respect to atomic coordinates (called imaginary vibrational frequency ) from the transition state is ignored, so that there are only 37/-7 molecular vibrations in the transition structure (37/ — 6 if linear) and all internal and external symmetry numbers have to be included in the rotational partition functions (then any reaction path degeneracy is usually included automatically). 

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