Page 270 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

VIBRATIONAL MODULATION EFFECTS ON EPR SPECTRA 253
In the above equation is the array of conserved quantum numbers for
the SA modes, and (neglected in this study) accounts for anharmonic effects and non
orthogonality between the path tangent and the energy gradient [16,23]. In fact, the so
called intrinsic reaction path (IRP) is always parallel to the gradient, so that the last
contribution vanishes [23]. For intramolecular dynamics, however, the distinguished
coordinate (DC) approach has the advantage of being isotope independent and well defined
also beyond energy minima, while still retaining an almost negligible coupling between the
gradient and the path tangent. This model corresponds to the construcion of the one-
dimensional path through the optimization of all the other geometrical parameters at selected
values of a specific internal coordinate. In the present context, the distinguished coordinate
is the out-of-plane angle defined in Figure 1. Furthermore, the distance s along the path is
set to zero at a suitable reference configuration (in the present case the planar structure
where

When the IRP is traced, successive points are obtained following the energy gradient.
Because there is no external force or torque, the path is irrotational and leaves the center of
mass fixed. Sets of points coming from separate geometry optimizations (as in the case of
the DC model) introduce the additional problem of their relative orientation. In fact, the
distance in MW coordinates between adjacent points is altered by the rotation or translation
of their respective reference axes. The problem of translation has the trivial solution of
centering the reference axes at the center of mass of the system. On the other hand, for non
planar systems, the problem of rotations does not have an analytical solution and must be
solved by numerical minimization of the distance between successive points as a function of
the Euler angles of the system [16,24].
In the scenario just sketched, the large amplitude vibration along s is governed by the
following equation:

where is the kinetic energy operator and is a generic vibrational eigenstate with
energy The so-called vibrationally adiabatic zero curvature (VAZC) approximation is
obtained neglecting the small couplings between the path tangent and the local vibrational
coordinates appearing in the kinetic energy operator [23,25]. Since the arc length is
measured in the space of mass weighted cartesian coordinates, we obtain a Schrödinger