Cartesian displacements

Thus the transfonnation matrix for the gradient is the inverse transpose of that for the coordinates. In the case of transfonnation from Cartesian displacement coordmates (Ax) to internal coordinates (Aq), the transfonnation is singular becanse the internal coordinates do not specify the six translational and rotational degrees of freedom. One conld angment the internal coordinate set by the latter bnt a simpler approach is to rise the generalized inverse [58]... 

Here, we discuss the motion of a system of three identical nuclei in the vicinity of the D3/, configuration. The conventional coordinates for the in-plane motion are employed, as shown in Figure 5. The noraial coordinates Qx, Qy, Qz), the plane polar coordinates (p,(p,z), and the Cartesian displacement coordinates (xi,yhZi of the three nuclei (t = 1,2,3) are related by [20,94]... 

Wfi cri you perform a single poin t sem i-cmpirical or ah initio ealeu-laliori, you obtain th c en ei gy and tli e first dci ivalives of the eu ei gy with respect to Cartesian displacement of the atoms. Since the wave function for the molecule is computed in the process, there are a n urn ber of oth er molecti lar properties th at could be available to you. Molecularproperties arc basically an average over th e wave fun ction of certain operatorsdescribin g the property. For exam pie, the electron ic dipole operator is basical ly ju st the operator for the position of an electron and the electron ic con tribution to the dipole iTi otTi en t is... 

Defining mass-weighted Cartesian displacement coordinates r/ . 

More generally, it is possible to combine sets of Cartesian displacement coordinates qk into so-called symmetry adapted coordinates Qrj, where the index F labels the irreducible representation and j labels the particular combination of that symmetry. These symmetry adapted coordinates can be formed by applying the point group projection operators to the individual Cartesian displacement coordinates. 

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. 

With the use of the Cartesian displacement coordinates defined in Fig. 3 a basis vector is of the form... 

To illustrate the application of Eq. (37), consider the ammonia molecule with the system of 12 Cartesian displacement coordinates given by Eq. (19) as the basis. The reducible representation for the identity operation then corresponds to the unit matrix of order 12, whose character is obviously equal to 12. The symmetry operation A = Cj of Eq. (18) is represented by the matrix of Eq. (20) whore character is equal to zero. Hie same result is of course obtained for die operation , as it belongs to the same class. For the class 3av the character is equal to two, as exemplified by the matrices given by Eqs. (21) and (22) for the operations C and Z), respectively. The representation of the operation F is analogous to D (problem 12). 

Table 9 The symmetry group and the reduction of the representations of the Cartesian displacements for...

The effect of the symmetry operations on the Cartesian displacement coordinates of the two hydrogen atoms in die water molecule. The sharp ( ) indicates the inversion of a coordinate axis, resulting in a change in handedness of the Cartesian coordinate system. 

Fig, 3 Cartesian displacement coordinates for the ammonia molecule, The Z(CO a is perpendicular to the plane of the paper (which is not a plane of symmetry)... 

The characters Xj for the examples in the previous section were calculated following the method described in Section 8.9, that is, on the basis of Cartesian displacement coordinates. Alternatively, it is often desirable to employ a set of internal coordinates as the basis. However, they must be well chosen so that they are sufficient to describe the vibrational degrees of freedom of the molecule and that they are linearly independent The latter condition is necessary to avoid the problem of redundancy. Even when properly chosen, the internal coordinates still do not usually transform following the symmetry of the molecule. Once again, the water molecule provides a very simple example of this problem. 

The matrix of Cartesian displacements could in principle be found by multiplying Eq. (80) by the inverse of B. However, B is not, in general square and nonsingular, and hence cannot be inverted. Nevertheless, as G sb it... 

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

Take an N-atomic molecule with the nuclei each at their equilibrium internuclear position. Establish a Cartesian x, y, z coordinate system for each of the nuclei such that, for Xj with i = 1,.., 3N, xi is the Cartesian x displacement of nucleus 1, x2 is the Cartesian y displacement coordinate for nucleus 1, X3 is the Cartesian z displacement coordinate for nucleus 1,..., x3N is the Cartesian z displacement for nucleus N. Use of one or another quantum chemistry program yields a set of force constants I ij in Cartesian displacement coordinates... 

In fact, the result of Equation 3.43 not only applies to internal displacement coordinates but also to Cartesian displacements. The kinetic energy in terms of Cartesian coordinates (Equation 3.11) can easily be transformed into an expression in terms of Cartesian momenta (Equation 3.28)... 

The matrix to be diagonalized for finding the vibrational frequencies is the matrix product of the above G matrix for Cartesian coordinates and the corresponding F matrix for Cartesian displacement coordinates. It is noted in passing that the GF matrix is generally not symmetric, i.e. 

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)... 

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) ... 

An approximate form of the NMO model, the atomic polar tensor (APT) model, has also proved effective (42). Rather than considering local contributions, this model considers contributions from the derivative of the total molecular dipole moment with respect to Cartesian displacement of a given nucleus, (3p/9R )o. The latter ate the elements of the atomic polar tensor for atom n. 

So, of the 9 cartesian displacements, 3 are of aj symmetry, 3 of b2,2 of bi, and 1 of a2- Of these, there are three translations (aj, b2, and b j) and three rotations (b2, b j, and a2). This leaves two vibrations of aj and one of bj symmetry. For the H2O example treated here, the three non zero eigenvalues of the mass-weighted Hessian are therefore of aj b2, and aj symmetry. They describe the symmetric and asymmetric stretch vibrations and the bending mode, respectively as illustrated below. 

Stable adsorption complexes are characterized by local minima on the potential energy hypersurface. The reaction pathway between two stable minima is determined by computation of a transition state structure, a saddle point on the potential energy hypersurface, characterized by a single imaginary vibrational mode. The Cartesian displacements of atoms that participate in this vibration characterize movements of these atoms along the reaction coordinate between sorption complexes. 

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

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