Mass-weighted Cartesian displacement

Defining mass-weighted Cartesian displacement coordinates r/ . 

To illustrate, again consider the H2O molecule in the coordinate system described above. The 3N = 9 mass weighted Cartesian displacement coordinates (Xl, Yl, Zl, Xq, Yq, Zq, Xr, Yr, Zr) can be symmetry adapted by applying the following four projection operators ... 

It is convenient to define a set of 3N mass-weighted Cartesian displacement coordinates q, q2,..., q N such that the first three q s are the components of Qi, the fourth, fifth and sixth q s are the components of Q2, and so on. The kinetic energy T can therefore be written... 

This is clearly a matrix eigenvalue problem the eigenvalues determine tJie vibrational frequencies and the eigenvectors are the normal modes of vibration. Typical output is shown in Figure 14.10, with the mass-weighted normal coordinates expressed as Unear combinations of mass-weighted Cartesian displacements making up the bottom six Unes. 

We have introduced here mass-weighted Cartesian displacements which are thus defined as... 

The second rule for isotopomer harmonic frequencies is the so-called Sum Rule which follows from Equation 3.A1.8. Equation 3.A1.8 relates the sum of the squares of all the frequencies to the sum of the diagonal matrix elements of the (FG) matrix diagonalized to obtain the frequencies. When mass weighted Cartesian displacement coordinates are used to calculate the vibration frequencies, this means that the sum of the A s (A = 4n2v12)can be found as follows (Equation3.51)... 

In classical terms, if we use the mass-weighted Cartesian displacement coordinates, the kinetic energy of the moving nuclei isf... 

To simplify this equation, we define the mass-weighted Cartesian displacement coordinates qv...,q3N ... 

Thus if there are no degenerate vibrations, each normal coordinate is either unchanged or multiplied by — 1 upon application of a symmetry operation. Each Qk is a linear combination of the mass-weighted Cartesian displacement coordinates of the nuclei. If Qk is multiplied by — 1, each Cartesian displacement coordinate is multiplied by - 1, which reverses the directions of all the displacement vectors. If Qk is unchanged by a symmetry operation, then the symmetry operation sends the displacement vectors to a configuration indistinguishable from the original one. (The displacement vectors are defined relative to molecule-fixed axes, which in turn are defined relative to the nuclear positions. The effect of a symmetry... 

The calculations of the second derivatives of the energy with respect to the mass-weighted cartesian displacements, evaluated at the equilibrium nuclear configuration,... 

The expressions for T, Eq. (2.28), and V, Eq. (2.31), can be written in a simpler form using matrix notation. Using the column vector q, whose components are the 3N mass-weighted Cartesian displacement coordinates. 

Solving Eq. (2.45) is a standard problem in linear algebra [an example solution is outlined in Steinfeld et al. (1989)]. The solution gives A, which is a diagonal matrix of the 3N eigenvalues and the eigenvector matrix L with components which define the transformation between normal mode coordinates and the mass-weighted Cartesian displacement coordinates that is. 

Computational and interpretational aspects of vibrational spectroscopy are greatly simplified by the introduction of normal coordinates . First, mass-weighted Cartesian displacement coordinates for an N-atom molecule q, qi,. ..,qm are defined according to... 

Let S bo the column matrix of the internal coordinates (S may include redundant coordinates) and let the transformation from the mass-weighted cartesian displacement to internal coordinates bo given by... 

Figure 2 Contour line diagrams of the potential V for the system of two noninteracting molecules (a) V(X, X2) for W2 W2 (b) V(Xi,X2) for W2 O2 (c) V X[,X2) for Ht,--- D2. X and x denote normal and mass-weighted Cartesian displacement coordinates, respectively. The components of the acceleration vector are given at point (Xi, X2)...

ATh and Ay , are stretching and angular internal coordinates. X( ) and q( ) are combinations of the three ordinary and mass-weighted Cartesian displacement coordinates of atom a. 

Cartesians, mass-weighted Cartesian displacements, mass-scaled Cartesians, and mass-scaled Jacobis. In mass-weighted coordinates, mass is unity and unitless, and the coordinates have units of length times square root of mass in mass-scaled coordinates, the reduced mass for all coordinates is a constant p (with units of mass), and the coordinates have units of length. We almost always use mass-scaled coordinates the main exception is in the subsection on curvihnear internal coordinates, where much of the analysis involving internal coordinates is done in terms of unsealed coordinates. 

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