External symmetry coordinates

Another type of symmetry coordinate that is often useful consists of linear combinations of cartesian displacement coordinates chosen to satisfy the requirements laid down for all symmetry coordinates in Sec. 6-3. These will be called external symmetry coordinates, and have the same advantages and disadvantages as cartesian coordinates compared with internal coordinates, Sec. 4-1. 

The Product Ride. The vibration frequencies of isotopic molecules are related by a rule which is a generalization of the methane example given above. Suppose that the kinetic and potential energies are expressed in terms of external symmetry coordinates (Chap. 6)... 

Here the kinetic energy expressed in terms of the momenta, Pt, conjugate to the coordinates Sa, contains no cross terms because the transformation from cartesian to external symmetry coordinates is orthogonal. Moreover, since each external symmetry coordinate is a linear combination of the cartesian displacement coordinates of a single equivalent set of atoms, the coefficient Gaa = ma- will be the reciprocal mass of an atom of the set from which Sa is constructed. Since the product of the characteristic values... 

The other quadratic factors can also be ivritten down at once and will not be given here. It should be noted, however, that a check on the G elements arises from the fact that the last term wiU always involve /iaMc and never or This can be seen if the factored secular equation in terms of external symmetry coordinates (combinations of cartesians) is visualized. In terms of these the kinetic energy is diagonal, with the appropriate atomic masses as diagonal coefficients. The Aig factor, according to Table 10-1, would involve one H mass and one C mass = 1, = 1, = n > — 0) so that the product of the roots... 

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). 

The approach is based on the premise that a thermally excited molecule will decompose preferentially along a pathway that incorporates the symmetry coordinate which mixes a low-lying excited state into the ground-state most effectively. In outline, his recipe is similar to our Equation 3.12, except that the configurations are not mixed by an external field, but by a vibrational distortion of the nuclear frame. The requirement can be stated " ... 

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 second term in Eqs. (9.75) and (9.76), die rotational atomic polarizability tensor reflects the contribution of molecular translation and rigid-body rotation to ax- The inclusion of the six external molecular coordinates in those equations - the diree translations Xy and X2, and the three rotations p, Py and P2, completes die set of molecular coordinates up to 3N. In diis vray polarizability dmivatives are transformed into quantities corresponding to a space-fixed Cartesian coordinate system. As already pointed out in section 4.1, the great advantage of such a step is that the imensity parameters defined in terms of a space-fixed coordinate system are independent on isotopic substitutions provided the symmetry of the molecule is preserved. This will be illustrated with an example in the succeeding section. By analogy with Eq. (9.77), die rotational polarizability tensor can be represented as... 

Summarizing, in the crystal there are 36 Raman active internal modes (symmetry species Ug, hig, 2g> and 26 infrared active internal modes (biw b2w hsu) as well as 12 Raman active and 7 infrared active external vibrations (librations and translations). Vibrations of the type are inactive because there appears no dipole moment along the normal coordinates in these vibrations of the crystal. 

Sources of external potential can be produced in a number of ways in which there is no need for special massive nuclei. As they identify external sources they are classical variables, namely, position coordinates for the sources. There is no quantum dynamics related to them yet. Symmetry constraints can be naturally defined. We formally write He(p a) to distinguish this situation from the standard approach. Since the primacy is given to the electronic wave function, and no Schrodinger equation is available at this point, its existence has to be taken as workinghypotheses. 

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