Internal displacement coordinate molecular vibrations

The procedure Split selects the internal displacement coordinates, q, and momenta, tt, (describing vibrations), the coordinates, r, and velocities, v, of the centers of molecular masses, angular velocities, a>, and directional unit vectors, e, of the molecules from the initial Cartesian coordinates, q, and from momenta, p. Thus, the staring values for algorithm loop are prepared. Step 1 Vibration... 

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. 

The derivatives of the Cartesian PAS coordinates h of the parent with respect to the internal coordinates 5(, required for Eq. 68, are well known from molecular vibrational theory as the derivatives of the respective displacement coordinates. They are usually arranged as the elements of a 3Na x Ns dimensional matrix, which has been called A [59] and gives the linear relation between the increments of the internal and Cartesian coordinates ... 

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... 

The expressions for the s. are available from molecular vibration theory, for the common internal coordinates of bond-stretch, angle-bend, and torsion. As described earlier, s. points in the direction in which a displacement of particle i produces the greatest change in Equation [77] provides a physical picture of the SHAKE displacements for internal coordinate constraints. In the language of Wilson vectors, the singularity condition for internal coordinate constraints takes the form... 

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 intermolecular potential as it is given in Eq. (3) for example, does not depend explicitly on the (external) molecular displacements or on the (internal) normal coordinates as required by Eq. (10). The atom-atom potential in Eq. (5) does not even depend explicitly on the molecular orientations Q. All these dependencies have to be brought out, by expansion and transformation of the potentials in Eq. (3) and Eq. (5), before these can actually be used in lattice dynamics calculations. The way this is performed depends on the lattice dynamics method chosen (see below). If one is not interested in the internal molecular vibrations, the free-molecule Hamiltonians may be omitted from Eq. (10) and the potential may be averaged over the molecular vibrational states. The effective potential thus obtained no longer depends on the coordinates and Q. ... 

These coordinates represent changes of (n — 1) internal coordinates Rp the changes which minimize the potential energy of the system for a given displacement of Ri (compare with the method of reaction coordinate). The method uses the force constants of molecular vibrations (interaction compliance) calculated somewhat differently than in Eq. (1.4), namely as the forces that need to be applied to achieve measurable distortions in with the potential energy minimized with respect to other coordinates ... 

It can be shown (Chapter 14) that the 3N — 6 internal degrees of freedom of motion of a nonlinear molecule correspond to 37V — 6 independent normal modes of vibration. In each normal mode of vibration all the atoms in the molecule vibrate with the same frequency and all atoms pass through their equilibrium positions simultaneously. The relative vibrational amplitudes of the individual atoms may be different in magnitude and direction but the center of gravity does not move and the molecule does not rotate. If the forces holding the molecule together are linear functions of the displacement of the nuclei from their equilibrium configurations, then the molecular vibrations will be harmonic. In this case each cartesian coordinate of each atom plotted as a function of time will be a sine or cosine wave when the molecule performs one normal mode of vibration (see Fig. 1.1). 

A much more detailed discussion of the choice of basis for a quantitative description of molecular vibrations is given in the text by Bright Wilson et al. referenced in this chapter s Further Reading section. This covers the use of mass-weighted coordinates and systems of internal coordinates based on bond vectors, bond angles and dihedral angles. Here, we are interested in the application of symmetry to vibrational spectroscopy to understand selection rules, and usually the much simpler basis of a few carefully chosen atom or bond displacements will suffice. 

For molecular vibrations there is a well-established technique called the 6F technique, which was developed by WILSON et al. [4.19]. An internal coordinate r can always be expressed in terms of the components of the Cartesian displacement coordinates u of the molecule (k = 1,2,...N a = x,y,z) ... 

Instead of the usual Cartesians, we can also apply a complete and non-redundant set of the so-called internal coordinates. In order to understand their application in vibrational calculations, let us consider a molecular system consisting of N nuclei let the Cartesian displacement vectors of the nuclei bedi,d2.---./ A around their equilibrium geometry in the usual three-dimensional Euclidean space E3. (The expression for the n-th Cartesian displacement vector is dn = Pn — P% where p is the instantaneous position vector of the n-th nucleus, and /0° is the position vector of the same nucleus at equilibrium. Hereafter, these position vectors correspond to an arbitrary origin. Note that for simplicity, we omit the explicit use of the atomic masses, i.e., do not use mass-weighted Cartesians.) A single-point 5 of a hypothetical 3iV-dimensional space (5 Iftsw), defined as... 

Thus, a rule of thumb says that valid normal coordinates must be either entirely symmetric or entirely antisymmetric over the most common symmetry operations if m and n label any two internal coordinates related by symmetry, then either Ukm = cikn or akm = —akn- This rule of thumb holds also in the construction of the coefficients of atomic orbitals in molecular orbitals (see Section 3.5). The bending mode of water straddles the mirror plane and is already symmetry-adapted if the displacements of the two hydrogen atoms are equal and in opposite directions (if the motions were in the same direction, that would be a molecular rotation, not a vibration). 

---