Mass-weighted cartesian coordinates potential energy

The matrix L 2) connects the 3n-6 n is the number of atoms in the [nonlinear] molecule) normal coordinates with the set of 3n mass-weighted Cartesian coordinates, (2, the vectors and correspond to the stationary points on the adiabatic potential surfaces of states 1 and 2, respectively. Then, the normal mode displacements Ag 2) are obtained by projecting the displacements = q,<°> - q2<° onto the normal-mode vectors. Finally, substituting the calculated quantities into Equations 1.1.10, 1.1.13, and 1.1.14 provides the total relaxation energy. 

Use of Mass-weighted Coordinates. In this treatment ordinary cartesian displacement coordinates have been used, whereas in the earlier part of the chapter mass-weighted cartesian coordinates were emplo3 ed. If ([i = (mi) i Axi and 52 = (m.2) Aa 2, the kinetic and potential energies become... 

Equation (3.16) presents expressions for the kinetic T and potential V energies in Cartesian mass weighted displacement coordinates x( and corresponding velocities x(... 

The elements of column k of the matrix L are the normalized mass-weighted Cartesian amplitudes of the atoms in the kth vibrational mode with frequency. This procedure is the same as that used in normal mode analysis (NMA) with the difference that in NMA the matrix of the second derivatives of the energy is written in internal coordinates. When the minimization includes lattice parameters, one obtains 3N - 3 real vibrational frequencies if a true minimum has been found, where N is the number of atoms in the simulation box. Finding an imaginary frequency indicates that the minimization ended on a saddle point on the potential energy surface instead of a minimum. 

The set of coordinates q can be any sort of internal or Cartesian coordinates. The positions of the stationary points are independent of the choice of coordinates. However, away from the stationary points the shape of the potential energy surface no longer is independent of the choice of coordinates it depends among other things on the choice of internal or Cartesian coordinates, on whether the coordinates are mass-weighted, and on whether angles are expressed in degrees or radians. 

We pointed out in the preceding section that a realistic potential energy function may not be easily expressible in Cartesian coordinates, but may be written more naturally in terms of 3JV generalized coordinates related to the mass-weighted coordinates by a linear transformation (6.40). In fact, only 3N — 6 such coordinates are required to fully specify the potential (3N — 5 in linear molecules) 2K is not sensitive to the center-of-mass position or molecular orientation in space, and a polyatomic molecule exhibits only 3N — 6 (3N — 5) independent bond lengths and bond angles. Such a truncated set of 3N — 6 (iN — 5) generalized coordinates is called an internal coordinate basis, and is commonly denoted S. To illustrate how an internal coordinate basis may be used to evaluate normal modes, we consider the bent H2O molecule in Fig. 6.3. The three internal coordinates are conveniently chosen to be... 

---