Internal coordinates, vibration-rotation

Electronic spectra are almost always treated within the framework of the Bom-Oppenlieimer approxunation [8] which states that the total wavefiinction of a molecule can be expressed as a product of electronic, vibrational, and rotational wavefiinctions (plus, of course, the translation of the centre of mass which can always be treated separately from the internal coordinates). The physical reason for the separation is that the nuclei are much heavier than the electrons and move much more slowly, so the electron cloud nonnally follows the instantaneous position of the nuclei quite well. The integral of equation (BE 1.1) is over all internal coordinates, both electronic and nuclear. Integration over the rotational wavefiinctions gives rotational selection rules which detemiine the fine structure and band shapes of electronic transitions in gaseous molecules. Rotational selection rules will be discussed below. For molecules in condensed phases the rotational motion is suppressed and replaced by oscillatory and diflfiisional motions. 

The assignment of (hr) - 5) vibrational modes for a linear molecule and (hr) - 6) vibrational modes for a nonlinear molecule comes from a consideration of the number of degrees of freedom in the molecule. It requires hr) coordinates to completely specify the position of all t) atoms in the molecule, and each coordinate results in a degree of freedom. Three coordinates (x, y, and z) specify the movement of the center of mass of the molecule in space. They set the translational degrees of freedom, since translational motion is associated with movement of the molecule as a whole. Two internal coordinates (angles) are required to specify the orientation of the axis of a linear molecule during rotation, while three angles are required for a nonlinear... 

The QCRNA database is viewable and searchable with a web browser on the internet and it is also contained as a MySQL database that is easily incorporated with parameter optimization software to allow for the rapid development of specific reaction parameters. Molecular structures can be viewed with the JMOL [47, 48] or MOLDEN [49, 50] programs as viewers for chemical MIME types. If the web browser is JAVA-enabled, then the JMOL software will automatically load as a web applet. Both programs allow the structure to be manipulated, i.e., rotated, scaled, and translated, and allow for measurement of internal coordinates, e.g., bond lengths, angles, and dihedral angles. Similarly, animations of the vibrational frequencies are available and can be viewed with either program. 

We describe as rigid-body rotation any molecular motion that leaves the centre of mass at rest, leaves the internal coordinates unaltered, but otherwise changes the positions of the atomic nuclei with respect to a reference frame. Whereas in a simple molecule, such as carbon monoxide, it is easy to visualize the two atoms vibrating about a mean position, i.e. with the bond length changing periodically, we may sometimes find it easier to see the vibration in our mind s eye if we think of one atom being stationary while the other atom moves relative to it. 

Clodius, W. B., and Quade, C. R. (1985), Internal Coordinate Formulation for the Vibration-Rotation Energies of Polyatomic Molecules. III. Tetrahedral and Octahedral Spherical Top Molecules, /. Chem. Phys. 82, 2365. 

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 described treatment has the disadvantage of being based on Cartesian coordinates, which depend on the system of axes used to localize the molecule. As an example, a methyl group can have different coordinates (CH3 in toluene or ethane), while the chemical and spectro.scopic properties of both are very similar. In order to take advantage of this chemical information, internal coordinates were introduced, which refer to chemically relevant quantities. A molecule with n atoms has 3n degrees of freedom, six of which correspond to the overall translations and rotations of the molecule. Only 3n 6 coordinates are necessary to describe the vibrational motions of the system. Five types of coordinates can be defined ... 

For the calculation of the normal mode spectra external and internal coordinates were assumed to be dynamically decoupled. Translational and rotational coordinates were extracted from the trajectories while all vibrational coordinates were set to zero. Dynamical matrices were set up for 50 configurations generated by molecular dynamics simulation. Long-range Coulombic interactions were treated using the Ewald summation technique. In Figure 2 the instantaneous normal mode spectra are depicted while in Table 3 some of their integral properties are compiled. 

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 vibrational structure may be explained as follows For each state of a molecule there is a wave function that depends on time, as well as on the internal space and spin coordinates of all electrons and all nuclei, assuming that the overall translational and rotational motions of the molecule have been separated from internal motion. A set of stationary states exists whose observable properties, such as energy, charge density, etc., do not change in time. These states may be described by the time-independent part of their wave functions alone. Their wave functions are the solutions of the time-independent Schrddinger equation and depend only on the internal coordinates q = 9, Qz,. . . of all electrons and the internal coordinates Q = Q, Qz, of all nuclei. 

T. J. Lukka, A simple method for the derivation of exact quantum-mechanical vibration-rotation Hamiltonians in terms of internal coordinates. J. Chem. Phys. 102, 3945—3955 (1995). 

S. M. Colwell and N. C. Handy, The derivation of vibration-rotation kinetic energy operators in internal coordinates II. Mol. Phys. 92, 317—330 (1997). 

The dependence of rag on the internal coordinates is not restricted by requirements other than the center of mass conditions (2.4) and that Eq. (2.6) is invertible. In expressing the rag functions we may therefore also consider how the final Hamiltonian is influenced, so that we obtain an operator of optimum suitability characterized by e.g. rapid convergence of the perturbing terms. In this respect there are two particular concerns, the vibration-rotation interaction and the potential energy expansion. 

The vibration-rotation interaction is the effect arising from coupling terms between angular and vibrational momenta as well as from the dependence of the rotational G-matrix elements (the /u-tensor) on the internal coordinates. The importance of this effect may to some extent be reduced provided an appropriate axis convention is used. The axis convention is the set of rules defining the orientation of the molecular axes, eg, g = x,y, z, relative to an arbitrary configuration as given by the position vectors, Ra, a. = 1, 2,... N. These rules can be expressed in three relations between the rag components, similar to the center of mass conditions(2.4). We shall refer to these relations as the axial constraints . Usually Eckart-condi-tions39 are imposed, but other possibilities may be considered. 

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