Coordinates, mass-weighted

Tran sl orm in g to mass-weighted coordinates, equation (210) can be rewritten into a set oI 3N sim u Itan eon s lin ear differen tial equation s... 

The potential energy of vibration is a function of the coordinates, xj,. .., z hence it is a function of the mass-weighted coordinates, qj,. .., q3N. For a molecule, the vibrational potential energy, U, is given by the sum of the electronic energy and the nuclear repulsion energy ... 

Using second order perturbation theory [3], the mean and mean square values of the mass weighted coordinate x in the vibrational state Ij) with quantum number j are explicitely given by ... 

In Eq. (13), the vector q denotes a set of mass-weighted coordinates in a configuration space of arbitrary dimension N, U(q) is the potential of mean force governing the reaction, T is a symmetric positive-definite friction matrix, and , (/) is a stochastic force that is assumed to represent white noise that is Gaussian distributed with zero mean. The subscript a in Eq. (13) is used to label a particular noise sequence For any given a, there are infinitely many... 

A quantitative physical picture of the vibrations of the system illustrated in Fig. 2 can be obtained with the use of normal coordinates. However, it is first customary to introduce the so-called mass-weighted coordinates by Si = y/mixh where of course mt =m2 =m in this example. The normal coordinates Q can then be defined by the relation... 

It is convenient to now convert to mass-weighted coordinates x = p1/2x and s (and then drop the overbars hereafter), so that the Hamiltonian is... 

For convenience we introduce mass weighted coordinates (designated by a prime)... 

A complementary approach to the parabolic barrier problem is obtained by considering the Hamiltonian equivalent representation of the GLE. If the potential is parabolic, then the Hamiltonian may be diagonalized" using a normal mode transformation. One rewrites the Hamiltonian using mass weighted coordinates q Vmd. An orthogonal transformation matrix... 

INMs are obtained by diagonalizing the Hessian matrix generated by expanding the system potential energy to quadratic order in mass-weighted coordinates. 

The geometrical properties of the PES in the vicinity of a transition state mean that the steepest descent path down from the transition state (also generally calculated in mass-weighted coordinates, and called the intrinsic reaction coordinate or IRC) will usually lead only to a single reactant in one direction and a single product (or intermediate) in the other. However, a transition state can sometimes be shared by more than one reactant and/or product. One of these cases arises when the PES possesses a so-called valley-ridge inflection point (VRI). °... 

In mass-weighted coordinates, the hessian matrix becomes the harmonic force constant matrix, from which a normal coordinate analysis may be carried out to yield harmonic frequencies and normal modes, essentially a prediction of the fundamental IR transition... 

We see that Hk in (6.46) is a harmonic-oscillator Hamiltonian with Qk corresponding to x, with k corresponding to the force constant k, and with m = 1. (The normal coordinate Qk is a combination of mass-weighted coordinates, which is why m is absent from Hk.) Although the boundary conditions for (6.50) differ from harmonic-oscillator boundary conditions, the difference is unimportant (Section 4.1). From (1.133)—(1.135) and (1.140), we have... 

A normal-mode representation of the Hamiltonian for the reduced system involves the diagonalization of the projected force constant matrix, which in turn generates a reduced-dimension potential-energy surface in terms of the mass-weighted coordinates of the reaction path [64] ... 

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