Mass-weighted Cartesian displacement coordinates

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... 

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 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... 

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. 

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