Mass weighted coordinates dynamics

The choice of reaction path definition used as the reference for such a constrained dynamics is arbitrary any path may be used in practice. However, a natural choice in order to ensure that the simulation moves along the bottom of the potential energy valley connecting reactants/products with TS is the intrinsic reaction path (IRP) of Fukui.46,47 IRP by definition goes along the bottom of such a valley. IRP simply corresponds to a steepest descent path in a mass-weighted coordinates ... 

The most common assumption is one of a reaction path in hyperspace (Miller et al. 1980). A saddle point on the PES is found and the steepest descent path (in mass-weighted coordinates) from this saddle point to reactants and products is defined as the reaction path. The information needed, except for the path and the energies along it, is the local quadratic PES for motion perpendicular to the path. The reaction-path Hamiltonian is only a weakly local method since it can be viewed as an approximation to the full PES and since it is possible to use any of the previously defined global-dynamical methods with this potential. However, it is local because the approximate PES restricts motion to lie around the reaction path. The utility of a reaction-path formalism involves convenient approximations to the dynamics which can be made with the formalism as a starting point. 

Here is the generalized Laplace operator, defined by equation (37), while R is the hyperradius (equation (5)). In a later chapter of this book. Professor Fano will discuss the application of the hyperspherical method to nonseparable dynamical problems. Here we shall only note that if mass-weighted coordinates axe used, the Schrodinger equation for any system interacting through Coulomb forces can be written in the form ... 

Investigation of the total RP by using appropriate coordinates The RP is explored by proceeding on it in a stepwise manner using appropriate mathematical techniques and an appropriate mass-weighted coordinate system. Simplified, but useful, descriptions of the reaction dynamics become possible. Depending on the number of points calculated along the RP and the ab initio method used, computational costs are still tolerable. 

Raman intensities were calculated by differentiation of the molecular polarizability with respect to nuclear coordinates. It is often sufficient or desirable to calculate only Raman intensities for selected modes instead of for all 3N - 6 vibrational modes of a large molecule, which can be achieved if the normal modes of the molecule are already known. Therefore, a frequency analysis was performed using numerical differentiation of analytical gradients with respect to Cartesian nuclear coordinates in the first step. This yields vibrational frequencies and normal modes. Then, we used displacements along selected mass-weighted normal coordinates Q/t, for which (static and/or dynamic) polarizabilities are calculated. With a step size SQk,... 

Suppose we have n data points associated with k variables—in our case, n instantaneous structures of 3N Cartesian coordinates of N particles along a molecular dynamics (MD) trajectory. Let D be a 3N x n matrix, whose elements Dim are defined as the deviation of the (mass-weighted) zth Cartesian coordinate qi(tm) at a time tm(l time average (qi), that is,... 

The actual path mapped out by the MEP on the PES is dependent on coordinate system. However, changes in coordinate system do not alter the nature of the stationary points on the PES (i.e. minima, TSs, etc.). One coordinate system, mass-weighted Cartesian coordinates (see Section 10.2.3), is especially significant for reaction dynamics, and the MEP in this coordinate system is known as the intrinsic reaction coordinate (IRC) [162]. In this section, we use the terms MEP, IRC, steepest descent path, and reaction path synonymously. 

Such a discrete frequency spectrum can be obtained from the diagonalization of the dynamic matrix. We shall not go into the details here except to mention that the dynamic matrix is a 3V x 3N matrix formed by the mass weighted position fluctuations along the three spatial coordinates of each individual atom. However, if the spectrum is continuous, then the addition can be replaced by integration and a function describing the probability distribution of the frequencies of the spectmm appears. This new function is called the density of state (DOS) and the integration of this function over the whole frequency range is the total number of modes of vibration, 3N 6. 

All the analyses of van der Waals molecule predissociation begin with the same system Hamiltonian. We will denote the mass-weighted van der Waals coordinates by r. Then, neglecting the influence of rotation on the system dynamics, the Hamiltonian is... 

In addition to the reaction coordinate mapped out by the DRC, the minimum energy path from transition state to reactants or products is of interest. As with the DRC originating at the transition state, this path is coordinate-system independent. The calculation to determine the minimum energy path starts in a similar manner to the DRC calculation, only after the initial displacement, all velocities are annulled at every step. This results in the system moving perpendicular to the energy contours in mass-weighted coor nate space. Sutime independent, and are called intrinsic reaction coordinates (IRC). For a review of potential energy surfaces for polyatomic reaction dynamics, see ref. 59. 

The mass-weighted path of steepest descent in Cartesian coordinates has been termed the "intrinsic reaction coordinate" [13]. In the past, it was used by a number of scientists [14] to describe some kind of reaction dynamics. Since then, the path has been analyzed, particularly by Fukui et al. [13,15,16]. At present, it is well-accepted, and its calculation is, for instance, implemented in the GAUSSIAN program package [17], using a procedure established by Schlegel [18]. In the coordinate space, diis path... 

We present the results of the calculation of the cumulative reaction probability for coUineaj H-f-H2 over the total energy range of 0.37 to 1.27 eV, using the method described above. The availability of accurate PES s and dynamics calculations makes it a good benchmark system to use to study a new method. We use the Liu-Siegbahn-Truhlar-Horowitz [100, 101] (LSTH) PES for the calculations. The coordinates used for the calculations were the mass-weighted rectilinear normal modes [56, 102] referenced to the transition state on the LSTH PES. We denote the two dimensional coordinates by q = (x y) where x is the reaction coordinate and y is the perpendicular vibrational coordinate, i.e. the anti-symmetric and symmetric stretch, respectively. 

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