A new way of analyzing vibrational spectra. I. Derivation of adiabatic internal modes

Abstract

A new way of analyzing measured or calculated vibrational spectra in terms of internal vibrational modes associated with the internal parameters used to describe geometry and conformation of a molecule is described. The internal modes are determined by solving the Euler᎐Lagrange equations for molecular fragments n described by internal parameters . An internal mode is localized in a molecular n fragment by describing the rest of the molecule as a collection of massless points that just define molecular geometry. Alternatively, one can consider the new fragment motions as motions that are obtained after relaxing all parts of the vibrating molecule but the fragment under consideration. Because of this property, the internal modes are called adiabatic internal modes, and the associated force constants k , adiabatic force constants. a Minimization of the kinetic energy of the vibrating fragment yields the adiabatic mass n Ž . m corresponding to 1rG of Wilson's G matrix and, by this, adiabatic frequencies . a n n aAdiabatic modes are perfectly suited to analyze and understand the vibrational spectra of a molecule in terms of internal parameter modes in the same way as one understands molecular geometry in terms of internal coordinates.

Generalization of the Badger Rule Based on the Use of Adiabatic Vibrational Modes

“…Because of these reasons, AICoMs [79] were introduced to obtain local modes that are associated with a specific structural unit of a molecule without being contaminated by coupling with other vibrational modes.…”

“…The construction of an AICoM is based on how an internal coordinate mode v n would vibrate if the associated internal coordinate were to be displaced by an amount q à n in such a way that the increase in the potential energy becomes minimal. To accomplish this objective, mode v n , led by q à n (leading parameter principle [79]), must be constrained to the molecular fragment associated with q n , that is, the rest of the molecule is allowed to relax upon applying a perturbation q à n . This is equivalent to minimizing the potential energy given in normal coordinates Q under the constraint that the internal coordinate displacement q n is kept constant (Equation 4.33a):…”

“…While the potential energy has already been used to derive the AICoM vectors, so far nothing has been said with regard to the kinetic energy T. It has been shown that upon perturbation of the equilibrium geometry caused by a change in the leading parameter, q à n , the atoms of the molecule move in such a way that the kinetic energy adopts a minimum and the generalized velocity _ q n becomes identical to _ q à n . Again, this leads to a constrained minimization problem, the solution of which is found with the help of another Lagrange multiplier [79]. The results of the derivation are Tð_ q à n Þ ¼ 1 2 m a n ð_ q à n Þ 2 ð4:48Þ and m a n ¼ ðb n a n Þ 2 b n M À1 b n ð4:49Þ…”

“…The leading parameter principle [79] implies a new set of Euler-Lagrange equations because the generalized momenta for all other internal coordinates q m ðm 6 ¼ nÞ become zero [79], which can be pictured in the way that all atomic masses outside the molecular fragment are considered as massless points. Hence, the derivation of the AICoMs follows the same procedure as the derivation of normal modes and it has been shown that the solutions of the Euler-Lagrange equations for the AICoMs are obtained by requiring that the potential energy V is minimized for a geometric perturbation under the constraint that the perturbation is defined by q à n [79]. The second advantage of the AICoMs is that their properties, namely, adiabatic force constant, adiabatic mass, and adiabatic frequency, are clearly defined and easy to calculate.…”

“…This has to do with the fact that the derivation of v BG n does not explicitly revert to a dynamic principle. AICoM frequencies correspond to AICoM vectors, which in turn are based on the leading parameter principle (the dynamic principle [79]) and the modified Euler-Lagrange equations for the vibrational problem expressed in terms of local modes.…”