Weighted Cartesian Coordinates

To illustrate the effect of mass-weighting or mass-scaling the coordinates, consider two noninteracting hydrogen molecules. The potential energy is a function of the two internuclear separations, r and r2- For small displacements from the equilibrium positions, a contour plot of the potential energy surface looks like a series of concentric circles. If the hydrogen molecules are each 

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The MEP is defined as the path of steepest descent in mass-weighted Cartesian coordinates. This is also called intrinsic reaction coordinate (IRC). In reality, we know that many other paths close to the IRC path would also lead to a reaction and the percentage of the time each path is taken could be described by the Boltzmann distribution. 

The relationship between the force constant matrix in Cartesian displacement coordinates Fy, and the force constant matrix for mass weighted Cartesian coordinates F can be written as follows (only the first three rows and columns of the matrices are explicitly shown) ... 

One can simplify Equation 4.95 and obtain a very interesting result. We previously obtained the normal mode vibrational frequencies v by diagonalization of the matrix of the harmonic force constants in mass weighted Cartesian coordinates (Chapter 3). These force constants Fy were obtained from the force constants in Cartesian coordinates fq by using... 

These classical equations can more compactly be expressed in terms of the time evolution of a set of so-called mass weighted Cartesian coordinates defined as ... 

Let us summarize the results. Using the principal-axis system, we defined the nuclear mass-weighted Cartesian coordinates qt by (6.1) and (6.3) ... 

RDGmcpd has split into two new ridges (comprising the MCPD valley ) around 1.0 amu1/2bohr from the decay point (see Figure 17). Distances are measured in mass-weighted Cartesian coordinates as defined in Ref. 81. 

When the minimum energy path is expressed in the mass-weighted cartesian coordinates YJ. Eq. (33) reduces to a simpler expression... 

Using the mass-weighted Cartesian coordinates cr/ir defined as... 

The minimum energy path (MEP) in mass weighted Cartesian coordinates is the path through configuration space traced out by a hypothetical trajectory initiated at the saddle point with all inertia effects removed. It is the path that molasses would follow flowing downhill. The MEP satisfies the differential equation. 

The calculation of force constants at points along the MEP is often done separately from the determination of the path by numerical integration of Eq (2), but these two problems can profitably be combined. Methods recently have been proposed [6,7] that efficiently use the available force constants to better follow the path. To understand these methods and the relationship between them, consider two different Taylor series expansions about a point on the MEP. The first is the familiar expansion of the energy in the mass-weighted Cartesian coordinates,... 

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. 

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. 

If the calculation is carried out in mass-weighted Cartesian coordinates then overall translation and rotation are also projected out of the Hessian. 

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