Internal displacement coordinate kinetic energy

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 ... 

In fact, the result of Equation 3.43 not only applies to internal displacement coordinates but also to Cartesian displacements. The kinetic energy in terms of Cartesian coordinates (Equation 3.11) can easily be transformed into an expression in terms of Cartesian momenta (Equation 3.28)... 

The classical kinetic energy of the system has now been separated into the effect of displacement of the center of mass of the system, with momentum P and that of the relative movement of the two particles, with momentum p. In the absence of external forces, the interaction of the two (spherical) particles is only a function of (heir separation, r. That is, the potential function appearing in Eq. (37) depends only on the internal coordinates x, y, z. 

Equation (2.24) shows that, because of the kinetic energy coupling, in internal coordinates the energy is not separable at small displacements. Thus, it is necessary to search for 3N new coordinates (2 called normal mode coordinates, which define a new column vector Q. To begin it will be assumed that the mass-weighted Cartesian coordinates and normal mode coordinates are related by a linear transformation of the form... 

The potential energy is written in terms of internal coordinates or here explicitly atom-atom distances, and it takes the form of a sum of terms as in (2.2). All these distances (including those within the molecule and those between different molecules) form the components of a column vector R. The harmonic force constant matrix in terms of the corresponding displacements r is As already noted, however, it is advantageous to use cartesian displacement coordinates since the kinetic energy matrix G is then diagonal. It is therefore necessary to express also the potential 4> in terms of the cartesian displacements (vector x) and to write the force constant matrix accordingly ... 

Changes in interatomic distances or in the angles between chemical bonds, or both, can be used to provide a set of 3A — 6 (or 37V — 5 for linear molecules) internal coordinates (Sec. 2-6), i.e., coordinates which are unaffected by translations or rotations of the molecule as a whole. These are particularly important because they provide the most physically significant set for use in describing the potential emwgy of the molecule. The kinetic energy, on the other hand, is more easily set up in terms of cartesian displacement coordinates of the atoms (Sec. 2-6). A relation between the two types is therefore needed. 

If the displacement vector of the nth cell in Cartesian and internal coordinates is denoted by x and r , respectively, we can express the kinetic-energy term as a function of the vectors x in the form... 

The unidimensional tunnelhng calculations described above presume that movanent along the reaction coordinate is electronically and vibrationally adiabatic. In real systems there are two sources of non-adiabatic effects on the reaction coordinate (i) changes in the frequencies of the internal displacements (perpendicular to the reaction coordinate) (ii) curvilinear effects on the reaction coordinate. Such effects are more pronounced in exothermic reactions, which may lead to vibrationally excited products, and in systems where a hght atom is transferred between two heavy atoms (e.g. I-H-I), because the diag-onalisation of the kinetic energy in such systems leads to a substantial curvature of the PES. 

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