Normal internal displacement 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. 

A nonlinear molecule of N atoms has 32V — 6 internal vibrational degrees of freedom, and therefore 3A — 6 normal modes of vibration (the three translational and three rotational degrees of freedom are not of vibrational spectroscopic relevance). Thus, there are 32V — 6 independent internal coordinates, each of which can be expressed in terms of Cartesian coordinates. To first order, we can write any internal displacement coordinate ry in the form... 

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... 

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, ... 

The determination of these normal frequencies, and the forms of the normal vibrations, thus becomes the primary problem in correlating the structure and internal forces of the molecule with the observed vibrational spectrum. It is the complexity of this problem for large molecules which has hindered the kind of detailed solution that can be achieved with small molecules. In the general case, a solution of the equations of motion in normal coordinates is required. Let the Cartesian displacement coordinates of the N nuclei of a molecule be designated by qlt q2,... qsN. The potential energy of the oscillating system is not accurately known in the absence of a solution to the quantum mechanical problem of the electronic energies, but for small displacements it can be quite well approximated by a power series expansion in the displacements ... 

Six of these normal coordinates (five for a linear molecule) have a frequency eigenvalue identically equal to zero. These motions are translations and rotations of the molecule. Although the approach through Cartesian displacement coordinates is theoretically elegant, it is generally more practical to express the vibrational motions in terms of internal coordinates, such as bond stretches and distortions of bond angles. The method is discussed in detail in Chapter 4 of Wilson, Decius and Cross [57]. Since the distortions of the molecule can be described in terms of 3A — 6 of these internal coordinates there are no redundant dimensions to be removed when the analysis is complete. 

The internal coordinates s are also expressed as a linear combination of the displacement coordinates. Both the normal coordinates Q and the internal coordinates s span the same space of 3A — 6 (or 3A — 5) dimensions and so each of the internal coordinates may be written as a linear combination of the normal coordinates ... 

As pointed out in Section IIB, it is possible to approach the lattice dynamics problem of a molecular crystal by choosing the cartesian displacement coordinates of the atoms as dynamical variables (Pawley, 1967). In this case, all vibrational degrees of freedom of the system are included, i.e., translational and librational lattice modes (external modes) as well as intramolecular vibrations perturbed by the solid (internal modes). It is then obviously necessary to include all intermolecular and intramolecular interactions in the potential function O. For the intramolecular part a force field derived from a molecular normal coordinate analysis is used. The force constants in such a case are calculated from the measured vibrational frequencies, The intermolecular part of O is usually expressed as a sum of terms, each representing the interaction between a pair of atoms on different molecules, as discussed in Section IIA. 

The eigenfrequencies are the (internal as well as external) vibrational excitation frequencies of the crystal in the harmonic approximation. The eigenvectors which express the crystal normal modes in terms of the displacement coordinates Ga( ) usually called the polarization vectors. 

Normal modes of vibration may be described using either cartesian displacement coordinates or internal coordinates. For every normal mode of vibration all the coordinates vary periodically with the same frequency and go through equilibrium simultaneously. The form of the normal mode of vibration is defined by specifying the relative amplitudes (which may be positive or negative) of the various coordinates in the set being used. For example, in Fig. 1.3, S1IS2 equals minus one and one for the two vibrations, respectively. 

For the diatomic molecule the same results apply except that the X displacement coordinate is replaced by the internal coordinate r — rj. For polyatomic molecules, each nonlinear molecule acts as though it consisted of 3N - 6 separate harmonic oscillators in each of which the X displacement coordinate is replaced by the appropriate normal coordinate Q. 

It can be shown by the methods of classical mechanics that the 3n — 6 (or 3n — 5) internal degrees of freedom of motion correspond to 3n — 6 (or 3n —5) different normal modes of vibration. In a normal mode of vibration the Cartesian displacement coordinates of every atom change periodically, each oscillating with the same frequency and passing through the equilibrium configuration at the same time. The molecule does not translate its center of mass or rotate. 

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.). ... 

This table gives the displacements for the normal mode corresponding to the imaginary frequency in terms of redundant internal coordinates (several zero-valued coordinates have been eliminated). The most significant values in this list are for the dihedral angles D1 through D6. When we examine the standard orientation, we realize that such motion corresponds to a rotation of the methyl group. 

Fig. 5.15 Schematic representation of the normal modes of the Fe(ni)-azide complex with the largest iron composition factors. The individual displacements of the Fe nucleus are depicted by a blue arrow. All vibrations except for V4 are characterized by a significant involvement of bond stretching and bending coordinates (red arrows and archlines), hi such a case, the length of the arrows and archlines roughly indicate the relative amplitude of bond stretching and bending, respectively. Internal coordinates vibrating in antiphase are denoted by inward and outward arrows respectively (taken from [63])...

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