Displacement coordinates, vibration-rotation

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. 

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... 

There are two difficulties with the use of cartesian displacement coordinates. The first is that the 3AT solutions for A include six nongenuine vibrations with zero frequency. These are the translations and rotations. This... 

Further improvements can be made in the vibrational-rotational energy level expression for a diatomic by carrying through the next (third) term in the truncation of Equation 9.29 via perturbation theory. This term is quadratic in the displacement coordinate, s. With second order as a chosen point of truncation of the perturbative corrections to the energy, the energy level expression is... 

Each vibration a molecule exhibits corresponds to two quadratic terms in the Hamiltonian as in Equation 7.35. There is a quadratic momentum term and a quadratic term in the associated displacement coordinate. Thus, each vibrational mode contributes 2 X NkT/2 to U. We can collect this information concisely by saying that for a molecular gas, U is 3NkT/2 (translation), plus either NkT/2 for linear molecules or 3NkT/2 for nonlinear molecules (rotation), plus NkT times the number of modes of vibration. The gas of an ideal diatomic molecule has U = 7NkT/2 as in Equation 11.59. A linear triatomic molecule has U = 13NkT/2 because there are three vibrational modes, one of which is twofold degenerate. 

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. 

The Eckart Sayvetz conditions imply that, if during the vibration a small translation of the center of masses is invoked, the origin of the Cartesian reference system is displaced so that no linear momentum is produced. The second Sayvetz condition, expressed in the last diree equations of (2.8), imposes the constraint that, during vibrational displacements, no angular momentum is produced. Eq. (2.8) implies that the reference Cartesian system translates and rotates with the molecule in such a way that the displacement coordinates Ax, Ay and Az reflect pure vibrational distortions. It is evident that through Eq. (2.8) certain mass-dependency is imposed on the atomic Cartesian displacement coordinates. 

The Eckart-Sayvetz conditions can easily be expressed in tmms of die coordinates q g (g = X, y, z). Summarizing, the vibrational motion of an N-atomic molecule widi 3N-6 vibrational degree of freedom can be described by 3N nuclear Cartesian displacement coordinates forming a column matrix X. Six degrees of fireedmn are related widi translational and rotational motions of the molecule. These motions can be described by the external corndinates p (diree translations and duee rotations). In a transposed form die different types of vibrational coordinates may be presented as follows... 

The 3N atomic Cartesian displacement coordinates describe not only vibrational motion but die translation and rotation of die molecule in space as well. Therefore, Qt in expression (4.5) include also the six rototranslational normal coordinates. Thus, (dp/dQth divided into two parts. The derivatives of p with respect to normal... 

The entire representation of (dp/dQt)o in Eq. (4.5) in terms of Cartesian displacement coordinates is simply the sum of vibrational and rotational polar tensors... 

Vibrations may be decomposed into three orthogonal components Ta (a = x, y, z) in three directions. These displacements have the same symmetry properties as cartesian coordinates. Likewise, any rotation may be decomposed into components Ra. The i.r. spanned by translations and rotations must clearly follow the appropriate symmetry type of the point-group character table. In quantum formalism, a transition will be allowed only if the symmetry product of the initial and final-state wave functions contains the symmetry species of the operator appropriate to the transition process. Definition of the symmetry product will be explained in terms of a simple example. 

Of the 3n coordinates needed to describe an n-atom molecule, three are used for center of mass motion, three describe angular displacement (rotation, hindered rotation, or libration) (two if the molecule is linear, 0 if monatomic), the remaining 3n—6 (3n—5, if linear, 3n—3 = 0, if monatomic) describe atom-atom displacements (vibrations). In some cases it may not be possible to separate translation cleanly from rotation and vibration, but when the separation can be made it is a convenience. Elementary treatments assume... 

There are two totally symmetric ( ,) normal modes and one b2 normal mode. (The convention is to use lowercase letters for the symmetry species of the normal modes.) The symmetry species of the normal modes have been found without solving the vibrational secular equation. Moreover, since there is only one b2 normal mode, the form of this vibration must be determined from symmetry considerations together with the requirement that the vibration have no translational or rotational energy associated with it. Thus (Fig 6.1), any bent XYX molecule has a b2 normal mode with the X atoms vibrating along the X—Y bonds and the Y atom vibrating in the plane of the molecule and perpendicular to the symmetry axis. On the other hand, there are two ax symmetry coordinates and the two ax normal vibrations are linear combinations of the ax symmetry coordinates, where the coefficients are dependent on the nuclear masses and the force constants. Thus the angles between the displacement vectors of the X atoms and the X—Y bonds for the ax modes of a bent XYX molecule vary from molecule to molecule. 

In VFF the molecular vibrations are considered in terms of internal coordinates qs (s = 1..3N — 6, where N is the number of atoms), which describe the deformation of the molecule with respect to its equilibrium geometry. The advantage of using internal coordinates instead of Cartesian displacements is that the translational and rotational motions of the molecule are excluded explicitly from the very beginning of the vibrational analysis. The set of internal coordinates q = qs is related to the set of Cartesian atomic displacements x = Wi by means of the Wilson s B-matrix [1] q = Bx. In the harmonic approximation the B-matrix depends only on the equilibrium geometry of the molecule. 

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