Vibrational kinetic energy

The vibrational kinetic energy can also be expressed in terms of the velocities in internal coordinates by taking the partial derivatives of Eq. (49). Thus, S = GP and, as G is square and nonsingular, P G lS and its transpose... 

The actual calculation consists of minimizing the intramolecular potential energy, or steric energy, as a function of the nuclear coordinates. The potential-energy expressions derive from the force-field concept that features in vibrational spectroscopic analysis according to the G-F-matrix formalism [111]. The G-matrix contains as elements atomic masses suitably reduced to match the internal displacement coordinates (matrix D) in defining the vibrational kinetic energy T of a molecule ... 

So by measuring the second-order Doppler shift of the Mossbauer nuclei in a material it is possible to determine their average velocity and thus their average vibrational kinetic energy, /2, where the mass of the Mossbauer nucleus. The... 

The classical roto-vibrational kinetic energy can be defined in internal coordinates as a function of the angular momentum [2-3] ... 

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. 

This is a very important result because the first term describes the vibrational kinetic energy of the nuclei, whilst the second and third terms represent the rotational kinetic energy. The transformation is straightforward provided one takes proper note of the non-commutation of the operator products which arise. 

The rotational and vibrational kinetic energies of the nuclei are represented by the term —(7/2/2/x)V ( in equation (2.37) we now seek its explicit form and the relation between the momentum operators PR and Pa in equation (2.6). If we take components of Pr in a space-fixed frame, we have the straightforward relationship ... 

The first term consists of both the kinetic energy of the electrons and the complete Coulomb energy, the second is the vibrational kinetic energy, the third is the rotational kinetic energy, the fourth is the mass polarisation energy involving the total linear momentum of the electrons P = Pi > the fifth and sixth are the parts of the electronic... 

In all of these computations, there is a dense manifold of excited states present [83], Thus the computations are sensitive to dynamic electron correlation and the details of the reaction coordinates involved. In the cytosine-guanine base pair simulations, trajectory calculations proved to be necessary to determine the extent of the conical intersection that is actually accessible. Subsequent improvements in the level of theory used for the static calculation of single molecules will be possible, but these should be balanced against a more realistic treatment of vibrational kinetic energy and environmental effects (solvent/protein). 

A. A Familiar Expression for the Rotation-Vibration Kinetic Energy ... 

J. H. Frederick and C. Woywood, General formulation of the vibrational kinetic energy operator in internal bond-angle coordinates. J. Chem. Phys. Ill, 7255-7271 (1999). 

For all but the smallest molecules, the procedure outlined in the previous paragraph is impractical. If we can locate or approximate the vibrational band origins from the experimental data, simplifications result. We then treat the J = 0 states in which case all terms in Eq. (3.1) involving rotational angular momenta vanish. Rewriting the pure vibrational term we obtain the vibrational kinetic energy Tv ... 

The definitions of the terms can be found in [30], but it is sufficient here to note that Ka represents the vibrational kinetic energy operator, Ep the electronic energy, Vn the nuclear repulsion operator, and the terms b and bo are elements of a matrix closely related to the inverse of the instantaneous inertia operator matrix. It should also be noted that the y terms arise from the interaction of the rotational with the electronic motion and tend to couple electronic states, even those diagonal in k. 

Figure 3 Calculated primary ct probability PpTim (upper panel) for all simulated eventB (+) and orientation-averaged (thick line), mean tkel AE and mean vibrational kinetic energy E,b (middle panel), as well as upper and lower limit for the final, i. e. exclusive ct probability obtained along with the limiting fragmentation probabilities (lower panel) as functions of the impact parameter b. The length of the error bars is given by two times the standard deviation of the orientation average.

Except for the neglect of the center of mass motions and the additional term accounting for the vibrational kinetic energy, Eq. (3.b), this Hamiltonian closely corresponds to the Hamiltonian given in Eq. (IV.48). This Hamiltonian is written in a form appropriate for direct translation into quantum mechanics by replacing the conjugate momenta by the corresponding differential operators ... 

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