Molecular vibrations symmetry coordinate

Determining the number and symmetries of molecular vibrations Normal coordinate analysis (symmetry coordinates)... 

Coordinates such as these, which have the symmetry properties of the point group are known as symmetry coordinates. As they transform in the same manner as the IRs when used as basis coordinates, they factor the secular determinant into block-diagonal form. Thus, while normal coordinates most be found to diagonalize the secular determinant, the factorization resulting horn the use of symmetry coordinates often provides considerable simplification of the vibrational problem. Furthermore, symmetry coordinates can be chosen a priori by a simple analysis of the molecular structure. 

Qualitative ways of analyzing a problem in molecular vibrations, that is, methods for determining the number of normal modes of each symmetry type which will arise in the molecule as a whole and in each set of equivalent internal coordinates, have been developed. There is also the quantitative problem of how the frequencies of these vibrations, which can be obtained by experiment, are related to the masses of the atoms, the bond angles and bond lengths, and most particularly the force constants of the individual bonds and interbond angles. In this section we shall show how to set up the equations which express these relationships, making maximum use of symmetry to simplify the task at every stage. 

Because chemists seem to have become increasingly interested in employing vibration spectra quantitatively—or at least semiquantitatively—to obtain information on bond strengths, it seemed mandatory to augment the previous treatment of molecular vibrations with a description of the efficient F and G matrix method for conducting vibrational analyses. The fact that the convenient projection operator method for setting up symmetry coordinates has also been introduced makes inclusion of this material particularly feasible and desirable. 

In addition, also the equilibrium NCoN angle is changed due to the contribution of these symmetry coordinates in the Q(at) coordinate given above. The deuterated complex treated in similar way has almost equal molecular constants although, as we have seen in the corresponding complex with NH3 ligands, their vibrational constants are quite different [97]. 

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... 

Before we apply the formalism developed in Section 3 to the vibration—inversion-rotation spectra of ammonia, we shall discuss in this section certain group theoretical problems concerning the classification of the states of ammonia, the construction of the symmetry coordinates, the symmetry properties of the molecular parameters, and the GF matrix problem for the ammonia molecule. 

For a molecule that has little or no symmetry, it is usually correct to assume that all its vibrational modes are both IR and Raman active. However, when the molecule has considerable symmetry, it is not always easy to picture whether the molecular dipole moment and polarizability will change during the vibration, especially for large and complex molecules. Fortunately, we can easily solve this problem by resorting to simple symmetry selection mles. The molecular vibration is active in IR absorption if it belongs to the same representation as at least one of the dipole moment components fjix, iiy, jj z) or, since the dipole moment is a vector, as one of the Cartesian coordinates (x, y, z). In contrast, the molecular vibration is active in Raman scattering if it belongs to the same representation as at least one of the polarizability components, etc.) or, since the polarizability is a tensor, as... 

We shall not describe the internal coordinate system nor the symmetry coordinates, the symmetry force constants and the kinetic energy matrix, as they are detailed in McCullough s paper (/). Wilson s book, "Molecular vibrations , gives further explanations of the methods used (2). 

Molecular vibrations can often be conveniently treated in terms of symmetry coordinates, which leads to factorization of the secular determinant to the maximum extent possible by symmetry. For N and NJ the symmetry coordinates are identical to the normal coordinates. The vibrations will, therefore, be described under the following symmetry coordinates [5] ... 

In this case, analytic formulas can also be obtained for the second derivatives that govern the harmonic vibrations [10]. Table 2 lists the vibrational force constants and normal mode frequencies determined [10, 12] using Wilson s FG matrix treatment, the procedure standard for molecular vibrations [14]. The radial displacements from the minimum are combined in the usual way to form symmetry coordinates,... 

Thus, reaction rate coefficients can be estimated from the thermochemistry of the transition states, whose molecular properties can be calculated with quantum chemical programs. In calculating reaction rate coefficients, the only negative second derivative of energy with respect to atomic coordinates (called imaginary vibrational frequency ) from the transition state is ignored, so that there are only 37/-7 molecular vibrations in the transition structure (37/ — 6 if linear) and all internal and external symmetry numbers have to be included in the rotational partition functions (then any reaction path degeneracy is usually included automatically). 

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