Mass-weighted internal coordinates

The IRC in mass-weighted internal coordinates (amu1/2 bohr) is determined by the method of C. Gonzalez and H. B. Schlegel, J. Phys. Chem. 94 (1990), 5523. 

Gonzalez C, Schegel HBJ, Reaction path following in mass-weighted internal coordinates, J Phys Chem, 94, 5523-5527 (1990)... 

Vibrational analyses were performed in order to characterize the optimized structures as local minima or transition states. All the calculated frequencies, the zero point, and the thermal energies correspond to harmonic oscillators. The calculations of the intrinsic reaction coordinates (IRCs) for each transition state with mass-weighted internal coordinates were performed in order to check whether the transition states under consideration connect the expected reactants and products [ 14]. 

Most computational investigations of the IRC path use mass-weighted internal coordinates rather than Cartesian coordinates. This is done by directly solving the equations of motion in internal coordinates r ) using the Wilson G... 

In this way, the IRC path in terms of mass-weighted internal coordinates r is obtained ... 

X is a matrix whose elements Xu give the mass-weighted internal displacements of each atomic coordinate i from its average position at a given time step t. N is the total number of integration steps. 

The mass-weighted Cartesian coordinates are denoted Using Eq. (3), the numerical value of the Blk matrix elements are determined for any value of the internal coordinates via a finite difference scheme. Once the B-matrix is constructed, expansions for Gtj and V, about the equilibrium configuration of the molecule, can be evaluated using higherordering differencing techniques. 

Equation (2.24) shows that, because of the kinetic energy coupling, in internal coordinates the energy is not separable at small displacements. Thus, it is necessary to search for 3N new coordinates (2 called normal mode coordinates, which define a new column vector Q. To begin it will be assumed that the mass-weighted Cartesian coordinates and normal mode coordinates are related by a linear transformation of the form... 

Equation (3.7) is an equation for the path of steepest descent in the 3N mass-weighted Cartesian coordinates. In fact, we can write similar equations for the path of steepest descent in any set of coordinates, say some complete set of (3A - 6)-internal coordinates, such as interatomic distances, R ... 

For linearly driven harmonic oscillators, we make as before a transformation from the atomic displacements u = b = 1,..., N to the internal bond variables s = 5, / = 1,..., /y, and from the latter to the mass-weighted normal coordinates Q = Qj =, fy with frequencies Wj. The... 

To verify that the transition state identified truly cormects the designated reactant and product, IRC calculations should be carried out. The IRC calculation starts from the transition state and then steps towards the reactant and then towards the product. IRC uses mass-weighted Cartesian coordinates even if the geometry is provided in terms of internal coordinates. It explicitly computes the Hessian along the reaction path. The IRC method is the most advanced method to trace the path of a chemical reaction. 

Fig. 4.2. Contour lines (0.35-3.0 eV) for the H+H2 Porter-Karplus potential (a) internal coordinates (RaBi Rbc) (t>) mass-weighted Jacobi coordinates Ra-BCi BC)-...

By convention the internal coordinates are expressed as lengths, and so (because of the mass weighting), the coefficients 0C,j have the dimensions M V2. The motion of a particular internal coordinate is therefore a superposition of several harmonic motions with different amplitudes and frequencies. This motion is illustrated for the stretching of the N-Cl bond in the molecule NO2CI in Fig. 6 using the parameters determined by Slater and tabulated section 3.8 of his book [56],... 

The elements of column k of the matrix L are the normalized mass-weighted Cartesian amplitudes of the atoms in the kth vibrational mode with frequency. This procedure is the same as that used in normal mode analysis (NMA) with the difference that in NMA the matrix of the second derivatives of the energy is written in internal coordinates. When the minimization includes lattice parameters, one obtains 3N - 3 real vibrational frequencies if a true minimum has been found, where N is the number of atoms in the simulation box. Finding an imaginary frequency indicates that the minimization ended on a saddle point on the potential energy surface instead of a minimum. 

An undesirable property of Cartesian coordinates (with or without mass-weighting) is that they generate PESs with strongly coupled coordinates. An alternative to the Cartesian coordinate systems is internal coordinates, sometimes referred to as Z-matrix coordinates. This set of coordinates defines the molecular structure in terms of bond lengths, angles, and dihedral angles. For a non-linear molecule, a unique structure is defined by — 6 internal coordinates for a linear molecule — 5 internal... 

Qi is the mass-weighted coordinate of the ith normal mode 0), is the frequency of the associated normal mode N is the number of internal degrees of freedom in the system... 

The set of coordinates q can be any sort of internal or Cartesian coordinates. The positions of the stationary points are independent of the choice of coordinates. However, away from the stationary points the shape of the potential energy surface no longer is independent of the choice of coordinates it depends among other things on the choice of internal or Cartesian coordinates, on whether the coordinates are mass-weighted, and on whether angles are expressed in degrees or radians. 

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