Independent symmetry coordinates

One has to point out that, for the sake of simplicity and brevity, the basis of the molecular vibration theory is directly presented in a given system of independent symmetry coordinates, and one assumes the following assumptions ... 

X column matrix of Cartesian coordinates column matrix of independent symmetry coordinates Q column matrix of vibrational normal coordinates... 

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. 

It is often convenient to use the symmetry coordinates that form the irreducible basis of the molecular symmetry group. This is because the potential-energy surface, being a consequence of the Born-Oppenheimer approximation and as such independent of the atomic masses, must be invariant with respect to the interchange of equivalent atoms inside the molecule. For example, the application of the projection operators for the irreducible representations of the symmetry point group D3h (whose subgroup... 

A diatomic molecule has only the internuclear distance Q as an internal coordinate (F = I). Unless //, vanishes because d>, and are of different symmetry, it is in general impossible to find a value of Q that would satisfy simultaneously both conditions. The energies , and E, therefore are different, and in one dimension, two states of the same electronic symmetry cannot cross. In a system with two independent internal coordinates Q, and Q2 (F = 2), for... 

The internal coordinates shown in Figure 2.1 do not transform according to the symmetry operations of the point group. They are neither left unchanged, nor are they simply reversed in sign instead, they interchange their values. However, it is not difficult to construct linear combinations of these coordinates that do have the desired property [5]. From the 10 internal coordinates, we obtain 10 linearly independent combinations called symmetry coordinates (Thble 2.2), each having the property we seek. 

One use of the symmetry coordinate classification is that it can tell us which types of distortion are expected to be coupled to other types. Each representative point can be associated with a value of the molecular potential energy, a structural invariant B that must be independent of such matters as the choice of coordinate system or the way the labels of the various atoms or bonds have been chosen. Consider, for the spiro-ketal example, the general quadratic energy expression for the S2 and Sg coordinates, which both transform as B2, symmetric with respect to the (yz) plane, antisymmetric with respect to the (xz) plane ... 

In Table 4.8 we present the experimental results [4] for U Fe, as compared with the Hessian eigenvalues, based on extensive relativistic calculations [5]. The eigenfunctions of the Hessian matrix are the corresponding normal modes. The Hessian matrix will be block-diagonal over the irreps of the group and, within each irrep, over the individual components of the irrep. Moreover, the blocks are independent of the components. All this illustrates the power of symmetry, and the reasons for it will be explained in detail in the next chapter. As an immediate consequence, symmetry coordinates, which belong to irreps that occur only once, are exact normal modes of the Hessian. Five irreps fulfil this criterion the T g mode, which corresponds to the overall rotations, and the vibrational modes, A g + Eg + l2g + 72 . Only the T u irrep gives rise to a triple multiplicity. In this case, the actual normal modes will depend on the matrix elements in the Hessian. Let us study this in detail... 

In the present case (for q = 0) we use twelve displacement coordinates Aj(0A ) of which eight are independent, as already discussed, and the four relations of (2.53) represent the redundancy conditions R of (2.56). In this case the 4>j, for the four redundant coordinates will not be zero in the potential expansion (2.54). In fact, Walmsley and Pople (1964) construct symmetry coordinates... 

We also have made use of the fact that the transformation T is independent of unit cell for q = 0. An equation can be written paralleling (4.26) for librational symmetry coordinates but has been found not to contribute in the cases investigated (Ron and Schnepp, 1967). 

Returning now to the form of (4.5) referred to above, after substitution in it of (4.6) we note that the sum of the intensity contributions of all normal coordinates belonging to the same irreducible representation is independent of the force constant problem. This is so because the transformation is always orthogonal. Moreover, this sum of intensities is equal to the sum of intensity contributions by the symmetry coordinates belonging to this irreducible representation. For this reason, such sums should be compared to experimental measurements, if the intensity theory is to be tested independently of the force constant problem. 

It is probably easier, how ever, to determine the number of independent force constants from a knowledge of the size of the factors of the secular determinant, made possible by the introduction of symmetry coordinates. ... 

The number of independent constants does not depend on the set of coordinates used to describe V, since a transformation of coordinates changes the original set of force constants Fw into a new set, whose members Fkk- are simply linear combinations of the original ones. Thus if there are symmetry coordinates of species there will be... 

Symmetrically Complete Sets. It is easiest to construct each internal symmetry coordinate out of equivalent internal coordinates, i.e., internal coordinates which arc exchanged by the symmetry operations of the molecules, such as the four CII bond extensions in CH4. Then the construction of the symmetry coordinates for a given molecule breaks down into separate problems for the different equivalent sets. Further, it is very desirable to utilize symmetrically complete sets i.e., sets containing all the coordinates resulting from the application of the symmetry operations of the molecule to an arbitrarily chosen coordinate. Thus the six IICH bond angles in CH4 form such a set and should all bo used, even though only five are independent. This use of redundant coordinates will introduce zero roots into the secular equation, but they are most... 

Moreover, each independent set is distinguished by its species under 3C. In fact, the reorganized coordinates can be constructed as if they were nondegenerate symmetry coordinates under X, employing the usual formula for such cases,... 

If the molecule has any symmetry, these considerations can be applied separately to each species. The number of independent coordinates in each species can be obtained by reducing the representation formed by the cartesian coordinates and subtracting the translations and rotations appropriately. The number of internal symmetry coordinates is similarly obtained for each species, and any excess represents redundancy. Likewise the rank of G " should equal the number of independent coordinates of species y. 

In order to illustrate the vibrational motions of a molecule belonging to a non-commutative symmetry point group, we return to the considerations of Section 2.3.2 and once more use as our example the square-planar complex, NiFj. A non-linear penta-atomic molecule has nine independent vibrational coordinates, distributed among the symmetry species of T>4h. These can be fully specified by standard methods [7], but the following simple qualitative considerations allow us to conclude that there are seven in-plane and two out-of-plane vibrations. Fig. 4.10 depicts several of the in-plane modes the motion of the nickel atom to conserve the center of mass is implied. 

In the GVFF there is no inherent limit to the number of FyS that are included. There is only the restraint that the number of Fs should not exceed the number of observable frequencies in fact, it should generally be much smaller so as to overdetermine the force field. Since an independent set of coordinates can be chosen, e.g., local symmetry coordinates, and the redundancy conditions explicitly given, there is no need to include linear terms in F, and Eq. (5-1) is the most general representation of the force field. Our discussion will focus on applications of the GVFF. 

Finally, there are certain implications imposed by crystal symmetry. According to Neumaim s principle the number of independent tensor coordinates characterizing a certain physical property of a crystal is limited by the symmetry of the crystal in its reference state. However, subjected to an external action the crystal, in general, has lower symmetry. In this case the set of symmetry elements is the intersection of the symmetry elements of the reference state crystal and those of the external action before being applied to the crystal. 

The entries 2C4, and 2aj in the top row of the character table for the C4V point group mean that this point group has two independent symmetry operations in each of the C4, a and a classes. The C4 class includes both the C4 operation itself, and the inverse of this operation, C4 , which is the same as 4. The basis functions for the two-dimensional irreducible representation (E) in the last row are pairs of coordinate values (x, y) or pairs of products of these values. The character 2 here means that the identity symmehy preserves both values, as it should, and the character -2 indicates that the C2 operation changes the sign of both values. 

There are only three vibrations for CHCI3 and thus only three symmetry coordinates are necessary, indicating that one of the four is redundant and is either equal to zero or not independent of the others. This point will be discussed later in this section. The coordinates must first be shown to be orthogonal and transform properly. S i is orthogonal with 6 2 since... 

In the previous. section we discussed the reference force field of /-PA (see Table 6.2) derived from the force field of butadiene. In the Og symmetry block, the high frequency C—H stretch is decoupled from the other modes and thus from tt electrons. We are left with three relevant Ug modes their reference and experimental frequencies are reported in Table 6.3 and, as discussed in Section II, fix the matrix and the x cd) curves in Fig. 6.4. The A matrix is written on the basis of the reference normal coordinates Q . It consequently depends on both the G and F matrices and varies with molecular or polymeric structure. The e-ph coupling constants g, thus vary even with isotopic substitution. To define coupling constants independent of mass, we use the symmetry coordinates to solve the GF problem for the reference state. In fact, diagonalization of GF gives both eigenvector matrix L in the S basis. The L matrix is used to transform Jin Eq. (12) back to the S basis ... 

The total number of elements 1 of the atomic polar tensor of a molecule is, therefore, equal to the number of Cartesian symmetry coordinates in the infrared active species. The set of Cartesian symmetry coordinates describes, in the general case, vibrational distortions as well as translations and rotations belonging to the same symmetry species as the infrared active modes. The translational and rotational conditions can be explicitly written as shown in Table 4.3. The important conclusion is that the net number of independent atomic polar tensor elements is exactly equal to the number of infrared active modes. In the case of AB2 (C2v) molecule 1 = 3+5 = S. For such molecules, however, there are three translational and two rotational ctmditions relating the APT elements as shown in Table 4.3. Subtracting these from 1 yields exactly the number of infrared active vibrations of the molecule. 

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