Internal symmetry coordinate

A consequence of the symmetry of the molecule is that states must transform according to representations of the appropriate symmetry group. In terms of coordinates, this implies that one must form internal symmetry coordinates. These are linear combinations of the internal coordinates. For example, denoting in Fig. 6.1 by sx, s2, s3,, v4, j5, s6 the stretching coordinates of the six C-H bonds, the internal symmetry coordinates are linear combinations... 

The kinetic energy is not easily expressed in terms of internal (symmetry) coordinates. Wilson (63) has shown that... 

In the case of H2O, Bis a 3 x 9matrix, whereas U,r, and Fr are 3 x 3 matrices. To take advantage of symmetry properties, internal coordinates are further transformed to internal symmetry coordinates (R ) by the relation... 

A Simple Rule for Nondegenerate Coordinates. Nondegenerate internal symmetry coordinates are very easily written down by the use of a rule involving the characters Xk for the given species What is desired... 

If the species is degenerate, the problem of orientation can be solved by the same method as used in Sec. 6-4 for internal symmetry coordinates that is, select to be a coordinate, or simple combination of coordinates, symmetrical (or antisymmetrical) to some chosen set of group operations constituting a subgroup. 

There arc several useful types of symmetry coordinates one of the simplest consists of linear combinations of internal coordinates, i.e., changes in interatomic distances and angles within the molecule. These combinations are chosen so as to conform to the requirements of Sec. 6-3 and therefore serve to factor the secular eejuation to the maximum extent possible from symmetry. The coordinates used in the water example of Sec. 6-3 are internal symmetry coordinates. 

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... 

In Sec. 6-3 the potential and kinetic enei gy expressions for the w ater molecule were obtained in terms of internal symmetry coordinates, and it rvas found that no cross terms occurred betiveen symmetry coordinates of different species. The method employed there ivas the obvious one of soh ing for the internal coordinates in terms of the symmetry coordinates and then substituting in the original expressions for T and V. This can ahvays be done, even in (implicated cases, although short cuts are available. 

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. 

Fig. 3. The aj, and e g carbon-carbon internal symmetry displacements of benzene. A counterclockwise rotation of the molecule by 60° about the z-axis replaces by —(1I2)S — (V3j2)S, and Si.j by [V3j2)S g — (l/2)S i , (k = 6, 8). (Please note the error in reference 12c the counterclockwise rotation of the molecule there used was 60° not 120° as stated. Also, in Fig. 1 of I2c the signs in both 0, and 0,.0 should be reversed.) The relationship connecting the Cartesian symmetry coordinates of Fig. 2 to the internal symmetry coordinates above is readily established by means of the vector addition of the appropriate displacement diagrams. This procedure yields 3i = Sj, 2( )= (1/8)V2/3 X (3S,(j)-b V3S,( )), and = (1I8)V2I3 x (S,(J) - - 3 /3Ss( ).

The L matrix given in reference 13b was determined using an orthonormal set of internal symmetry coordinates, rather than the cartesian coordinates listed. This use gives rise to the y 2 factor in Eq. (15) of reference 13b. 

Applications of the matrix method to the polyethylene crystal were described by Kitagawa (1968), Shiro and Miyazawa (1971). For orthorhombic polyethylene with the space group Pnam-Dl , the elastic constants Cn,C22, C33,C23,C3i and are for the Ag species, and C44, C55 and Qe are for the Big,B2g and B3, spedes, respectively. Accordingly, the treatment of elastic constants is significantly simplified by the use of internal symmetry-coordinate vector, internal symmetry-strain vector and external strain vector for each synunetry spedes. 

The group-theoretical approach is based on the fact that, if equation (5) contains products of internal (symmetry) coordinate displacements Sa(oi = /, y, k,...), then the product SaSfi - Sco and that obtained by any permutation of the indices are indistinguishable. Thus, the n-member products transform according to the permutation (symmetric) group Sn or in another, perhaps more appropriate, notation where... 

The solution of the inverse intensity problem, which in this formulation implies die evaluation of bond polar parameters from experimental dipole moment derivatives with respect to normal coordinates, can only be performed fm a molecule possessing higfrer symmetry. The situation in this respect is the same as in the alternative theoretical formulations. The requirement is that the direction of the vibrational transition dipole is fixed by symmetry. In odier words, there should be only one non>zero element in each colunm of the Pq matrix [Eq. (3.1)]. Again, all calculations are considerably simplified if the dp/dQi derivatives are first transformed into dipole moment derivatives with respect to internal symmetry coordinates. The determination of the elements of P(, can then be realized using die following general expression, in matrix notation... 

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