Internal displacement coordinate symmetry coordinates

As a second set of internal displacement coordinates we may choose the increments or decrements to the three OCO angles. Before using this set to form a representation which will tell us the normal coordinates involving inplane OCO bends, we must be careful to note that all of the coordinates in the set are not independent. If all three of the angles were to increase by the same amount at the same time, the motion would have A symmetry. It is obviously impossible, however, for all three angles simultaneously to expand within the plane. Thus the A representation which we shall find when we have reduced the representation is to be discarded as spurious. 

Consider an ideally infinite helical chain whose crystallographic repeat contains N chemical repeat units, each with P atoms. A screw symmetry operation transforms one chemical unit into the next, with a being the rotation about the helix axis and d the translation along the axis. Let r" denote the ith internal displacement coordinate associated with the nth chemical repeat unit. The potential energy, by analogy with Eq. (3), is given by... 

Linear transformations, including symmetry transformations in configuration space are analogous to those in Cartesian space (see Sections 1.2.2 and 1.2.3). For symmetrical reference structures, it is usually better to use not the internal coordinates themselves but to choose a new coordinate system in which the basis vectors are symmetry adapted linear combinations of the internal displacement coordinates with the special property that they transform according to the irreducible representations of the point group of the idealized, reference molecule (symmetry coordinates, see Chapter 2). 

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. 

For the water molecule, a reasonable set of internal coordinates would be the lengths of the 0-H bonds - let us call them r and V2 - and the H-O-H angle, 9. Displacements of these coordinates form a basis for a reducible representation of C2v that is composed of symmetry species of vibrational motions only. 

Symmetry Types of the Normal Modes. For this nonlinear four-atomic molecule there are 3(4) -6 = 6 genuine internal vibrations. Using a set of three Cartesian displacement coordinates on each atom, we obtain the following representation of the group C3l, ... 

When one bond is stretched, no moments are produced in other bonds. In other words, the total moment resulting from the simultaneous displacement of several bonds is assumed to be the vector sum of the moments produced by each individual bond. Since the molecular symmetry coordinates are linear combinations of internal bond coordinates R 135),... 

Another type of symmetry coordinate that is often useful consists of linear combinations of cartesian displacement coordinates chosen to satisfy the requirements laid down for all symmetry coordinates in Sec. 6-3. These will be called external symmetry coordinates, and have the same advantages and disadvantages as cartesian coordinates compared with internal coordinates, Sec. 4-1. 

In the second-order methods we have described, the choice of coordinate system was not made explicit. Prom a quantum-chemical perspective, analytical derivatives are most conveniently computed in Cartesian (or symmetry-adapted Cartesian) coordinates. Indeed, second-order methods are not particularly sensitive to the choice of coordinate system and second-order implementations based on Cartesian coordinates usually perform quite well. As we discussed above, however, if the Hessian is to be estimated empirically, a representation in which the Hessian is diagonal, or close to diagonal, is highly desirable. This is certainly not true for Cartesian coordinates some set of internal coordinates that better resemble normal coordinates would be required. Two related choices are popular. The first choice is the internal coordinates suggested by Wilson, Decius and Cross [25], which comprise bond stretches, bond angle bends, motion of a bond relative to a plane defined by several atoms, and torsional (dihedral) motion of two planes, each defined by a triplet of atoms. Commonly, the molecular geometry is specified in Cartesian coordinates, and a linear transformation between Cartesian displacement coordinates and internal displacement coordinates is either supplied by the user or generated automatically. Less often, the (curvilinear) transformation from Cartesian coordinates to internals may be computed. The second choice is Z-matrix coordinates, popularized by a number of semiempirical... 

Fig. 3. The aj, and e g carbon-carbon internal symmetry displacements of benzene. A counterclockwise rotation of the molecule by 60° about the z-axis replaces by —(1I2)S — (V3j2)S, and Si.j by [V3j2)S g — (l/2)S i , (k = 6, 8). (Please note the error in reference 12c the counterclockwise rotation of the molecule there used was 60° not 120° as stated. Also, in Fig. 1 of I2c the signs in both 0, and 0,.0 should be reversed.) The relationship connecting the Cartesian symmetry coordinates of Fig. 2 to the internal symmetry coordinates above is readily established by means of the vector addition of the appropriate displacement diagrams. This procedure yields 3i = Sj, 2( )= (1/8)V2/3 X (3S,(j)-b V3S,( )), and = (1I8)V2I3 x (S,(J) - - 3 /3Ss( ).

The group-theoretical approach is based on the fact that, if equation (5) contains products of internal (symmetry) coordinate displacements Sa(oi = /, y, k,...), then the product SaSfi - Sco and that obtained by any permutation of the indices are indistinguishable. Thus, the n-member products transform according to the permutation (symmetric) group Sn or in another, perhaps more appropriate, notation where... 

Experimental intensity data, reference Cartesian system, internal coordinates, force fields and L matrices for methyl chloride are the same as given in section m.C. Bond displacement coordinates are defined in Fig. 4.6. Rotational corrections to the dipole moment derivatives with respect to symmetry coordinates are evaluated using the heavy isotope method [34]. The rotational correction terms are given in Table 4.8. To illustrate the calculations in more detail the entire V matrix of methyl chloride is presented in Table 4.9. To remove die rotational terms fi om the sets of linear equations for symmetiy... 

Build geometry, reference coordinate system atom 4 in origo, atom 7 on x-axis, atom 1 in xy--plane, do the transformation after minimisation, charge zero, symmetry number six, fifty steepest descents, no davidons, up to twenty newtons, do this, print normal coordinates in sorted internal displacements, with IR intensity (only when you have charge parameters), do thermodynamics for gas at BOOK, put in five torsions to start with,..., statistical weight one (trivial here, inportant when more than one conformer or isomer are present.). ... 

The first step in the symmetry determination of the dynamic properties is the selection of the appropriate basis. Appropriate here means the correct representation of the changes in the properties examined. In the investigation of molecular vibrations (Chapter 5), either Cartesian displacement vectors or internal coordinate vectors are used. In the description of the molecular electronic structure (Chapter 6), the angular components of the atomic orbitals are frequently used... 

Examine now the symmetries of these different types of vibration. For this purpose a new type of basis set is used. Since we are interested in the changes of the geometrical parameters, these changes are an obvious choice for basis set. The geometrical parameters are also called internal coordinates, and the basis is the displacement of these internal coordinates. 

To start with, let us determine the stress and the deformation of a hollow sphere (outer radius J 2, inner radius R ) under a sudden increase in internal pressure if the material is elastic in compression but a standard solid (spring in series with a Kelvin-Voigt element) in shear (Fig. 16.1). As a consequence of the radial symmetry of the problem, spherical coordinates with the origin in the center of the sphere will be used. The displacement, obviously radial, is a function of r alone as a consequence of the fact that the components of the strain and stress tensors are also dependent only on r. As a consequence, the Navier equations, Eq. (4.108), predict that rot u = 0. Hence, grad div u = 0. This implies that... 

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