Symmetry coordinates, vibrational

Symmetry coordinates Vibrational quantum numbers FiJ V, k (varies) Fy = d V/dS,dSj... 

Symmetry coordinates can be generated from the internal coordinates by the use of the projection operator introduced in Chapter 4. Both the symmetry coordinates and the normal modes of vibration belong to an irreducible representation of the point group of the molecule. A symmetry coordinate is always associated with one or another type of internal coordinate—that is pure stretch, pure bend, etc.—whereas a normal mode can be a mixture of different internal coordinate changes of the same symmetry. In some cases, as in H20, the symmetry coordinates are good representations of the normal vibrations. In other cases they are not. An example for the latter is Au2C16 where the pure symmetry coordinate vibrations would be close in energy, so the real normal vibrations are mixtures of the different vibrations of the same symmetry type [7], The relationship between the symmetry coordinates and the normal vibrations can be... 

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. 

To see how use is made of symmetry coordinates as the bases of the vibrational problem, reconsider the kinetic and potential energies as given earlier, e.g. 

As both F and G are partitioned by the use of symmetry coordinates, the secular determinant is factored accordingly. The problem of calculating the vibrational frequencies is thus divided into two parts solution of a linear equation for the single frequency of species B2 and of a quadratic equation for the pair of frequencies of species Aj. 

It has been shown that the potential energy distribution provides an approximate method to evaluate the relative contribution of each symmetry coordinate to a given normal mode of vibration. From the definition of the symmetry coordinates, the relation... 

Pj and p2 represent the displacement vectors of the nuclei A and D (the corresponding polar coordinates are p1 cji, and p2, < )2, respectively) p, and pc are the displacement vectors and pT, r and pc, <[)f the corresponding polar coordinates of the terminal nuclei at the (collective) trans-bending and cis-bending vibrations, respectively. As a consequence of the use of these symmetry coordinates the nuclear kinetic energy operator for small-amplitude bending vibrations represents the kinetic energy of two uncoupled 2D harmonic oscillators ... 

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. 

Here the Fjt are again force constants but pertain to vibrations described by the symmetry coordinates Sr Sh and so on. From the standpoint of physical insight, it is the fik that have meaning for us, whereas mathematically the Ffi and the associated symmetry coordinates provide the easiest route to calculations because of symmetry factorization of the secular equation. Clearly, if we could express the Ffs in terms of the fik s we would have an optimum situation. The following considerations will show how to do this. 

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. 

An investigation of the vibrational spectrum of cyclopropylcarbonyl fluoride was carried out by Durig and coworkers using HF/3-21G theory115. The authors could assign all frequencies of cis and trans conformations and analyse normal modes in terms of potential energy contributions using appropriate symmetry coordinates. The calculated conformational stability and rotational barriers [HF/6-31G(d) and HF/3-21G] were compared with results obtained from the far-infrared spectrum. 

In order to apply equation (44) to a specific vibration in a given molecule it is necessary to know the relative amplitudes of motion of the atoms, i. e., it is necessary to know the normal coordinate. Since in most cases this is not known, it would seem difficult to make use of this isotope rule. In many cases, however, the normal coordinate is quite closely approximated by a suitably chosen symmetry coordinate. Since the latter is usually determined quite readily, and since the frequencies are stationary with respect to small changes in the normal coordinates [Bernstein 14)], it can be expected that equation (44) will apply to a good approximation. This of course depends on the chosen symmetry coordinate being reasonably close to the actual normal coordinate. 

Here q and Q symbolize the sets of all electronic and nuclear coordinates q (i = 1,. . ., 3 ) and Qt (i = 1,2,.. . , 3AT — 6), respectively. The derivatives are taken at the coordinate values Qf and the summation runs over all nuclear coordinates of independent vibrations. The expansion may be carried out with respect to different types of nuclear coordinates, i.e. symmetry coordinates and normal coordinates of the ground or the excited states. If the Q, s are normal coordinates and the Q s are taken at the potential minimum of an electronic state E the coordinate values are by definition Q = 0 for all i. In this case the matrix elements of the electron dependent part in the second term of Eq. (1) should vanish due to the minimal condition, i.e. 

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]. 

Based on the experimental frequencies and isotope shifts, a Quantum-Chemistry Assisted Normal Coordinate Analysis (QCA-NCA) has been performed. Details of the QCA-NCA procedure of I, including the f-matrix and the definition of the symmetry coordinates, have been described previously (12a). The NCA is based on model I (vide supra). Assignments of the experimentally observed vibrations and frequencies obtained with the QCA-NCA procedure are presented in Table II. The symbolic F-matrix for model I is shown in Scheme 3. Table III collects the force constants of the central N-N-M-N-N unit of I resulting from QCA-NCA. As evident from Table II, good agreement between measured and calculated frequencies is achieved, demonstrating the success of this method. 

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