Internal displacement coordinate normal 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... 

It is now fundamental to define the normal coordinates of this vihrational system - that is to say, the nuclear displacements in a polyatomic molecule. Again in the limit of small amplitudes of vibration, the normal coordinates in the form of the vector Q, are related to the internal coordinates by a linear transformation, viz. 

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 ... 

In equation 27 is the distance vector of the equilibrium internuclear distance between the atoms k and i. S and S are the vector components associated with the matrix that transforms normal coordinate displacements to internal coordinate displacements (along and perpendicular to bond directions). The sum is over the atoms and transformation vectors associated with the yth vibrational normal coordinate. 

Any procedure to define an amplitude / must guarantee that normal and internal vibrational modes are related in a physically reasonable way [20]. The internal mode vector Vn describes how the molecule vibrates when internal coordinate qn that initiates ("leads") the internal motion is slightly distorted from its equilibrium value. From the NMA, one obtains normal mode vectors each of which shows how the atoms of a molecule move when the normal coordinate Q is changed. By comparing the normal mode with the internal mode Vp the amplitude /ln i is obtained that describes in terms of the vibration of the smaller structural unit 0n represented by displacement vector Vp. Clearly, amplitude / p has to be defined as a function of and Vp ... 

The Bom-Oppenheimer separation of the electronic and nuclear motions is a cornerstone in computational chemistry. Once the electronic Schrodinger equation has been solved for a large number of nuclear geometries (and possibly also for several electronic states), the potential energy surface (PES) is known. The motion of the nuclei on the PES can then be solved either classically (Newton) or by quantum (Schrodinger) methods. If there are N nuclei, the dimensionality of the PES is 3N, i.e. there are 3N nuclear coordinates that define the geometry. Of these coordinates, three describe the overall translation of the molecule, and three describe the overall rotation of the molecule with respect to three axes. For a linear molecule, only two coordinates are necessary for describing the rotation. This leaves 3N - 6(5) coordinates to describe the internal movement of the nuclei, which for small displacements may be chosen as vibrational normal coordinates . 

The decomposition of coupled harmonic oscillators into a collection of independent oscillators is known as a normal mode expansion and the independent oscillators are called normal modes. Normal modes are defined as modes of vibration where the respective atomic motions of the atoms are in harmony , i.e., they all reach their maximum and minimum displacements at the same time. These normal modes can be expressed in terms of bond stretches and angle deformation (termed internal coordinates) and can be calculated by using a procedure called normal coordinate analysis. 

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. 

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