Normal coordinate transformation

The coupling of electronic and vibrational motions is studied by two canonical transformations, namely, normal coordinate transformation and momentum transformation on molecular Hamiltonian. It is shown that by these transformations we can pass from crude approximation to adiabatic approximation and then to non-adiahatic (diabatic) Hamiltonian. This leads to renormalized fermions and renotmahzed diabatic phonons. Simple calculations on H2, HD, and D2 systems are performed and compared with previous approaches. Finally, the problem of reducing diabatic Hamiltonian to adiabatic and crude adiabatic is discussed in the broader context of electronic quasi-degeneracy. 

In order to show that this procedure leads to acceptable results, reference is briefly made to the normal coordinate transformation mentioned at the end of Section 2.2. By this transformation the set of coordinates of junction points is transformed into a set of normal coordinates. These coordinates describe the normal modes of motion of the model chain. It can be proved that the lowest modes, in which large parts of the chain move simultaneously, are virtually uninfluenced by the chosen length of the subchains. This statement remains valid even when the subchains are chosen so short that their end-to-end distances no longer display a Gaussian distribution in a stationary system [cf. a proof given in the appendix of a paper by Ham (75)]. As a consequence, the first (longest or terminal) relaxation time and some of the following relaxation times will be quite insensitive for the details of the chain... 

The form of eq. (5.25) is particularly suitable, as it lends itself to Zimm s normal coordinate transformation. In contrast, however, to Zimm s original equations, an extra unknown function is retained, which couples the x- and they-direction. This function is the angular velocity Q, which yields... 

Cerf omitted to prove that a normal coordinate transformation will, in general, be possible for a subchain model when internal friction is introduced10. 

There is some similarity between Ferry s treatment of concentrated systems (14), (123) [eq. (4.4)] and Cerf s just mentioned approach. In both cases the normal coordinate transformation is assumed to be possible along the lines given for infinitely dilute solutions of kinetically perfectly flexible chains (Rouse, Zimm). Only afterwards, different external (Ferry) or internal (Cerf) friction factors are ascribed to the various normal modes. 

The internal frictional force on the -th junction, e.g. in thex-direction, is obtained by differentiating R with respect to xit which is the -component of the velocity of the i-th bead. In this way, the equation which is analogous to eqs. (5.25) and (5.28), becomes clearly non-linear. As a consequence, a normal coordinate transformation becomes impossible. The authors give an approximate solution of the problem by having resource to a perturbation method ... 

There are several possible approaches to simplifying the vibrational Hamiltonian given by Eq. (3.13). Some of these will be outlined here. The first approach consists of removing the harmonic potential energy cross terms and the kinetic energy terms by the equivalent of a normal coordinate transformation. We define a new set of coordinates Q... 

Use is made here [38] of the dynamical quantities presented in Chapter 2. Let Go, Fo and Lo be the kinetic, potential, and normal coordinate transformation matrices of one molecule of the series taken as reference [39]. In going from one molecule to a chemically similar one within the same class, one may expect small changes in the geometry (AG) or in the force constants (AF), or in both, which may be the cause of the observed changes of v,. The corresponding matrices of the molecule so modified can be written ... 

The 3 m equations have to be uncoupled using a normal coordinate transformation, to obtain eigenfunctions that are linear combinations of the positions of the submolecules. Each eigenfunction then describes a single viscoelastic element with characteristic time-dependent properties. 

The identification of the IRs according to which the normal coordinates transform can greatly reduce the computational labor associated with implementing the FG matrix method. It is frequently easy to set up symmetry coordinates, as linear combinations of internal coordinates, which transform according to IRs of the point group G. For CIF3, one choice of symmetry coordinates would be... 

In principle a similarity transformation can be found which will change the cartesian coordinate transformation matrices [Eq. (3.10)] into the completely reduced normal coordinate transformation matrices which include... 

The normal coordinate transformation requires diagonahzation of the matrix A, appearing in Eq. (4.46), in such a way that the transformed matrix will have only the diagonal components. The method of obtaining such a matrix is well documented in many standard textbooks. An excellent exposition of this method can be found in the textbook by Hildebrand (1952). 

Ls is a normal coordinate transformation matrix of rank 3N-3 which includes the three rotations. It defines the relation between the bond displacement coordinates Xg and an... 

The evaluation of the first and second derivatives of molecular energy with respect to an tqtpropriate set of coordinates defining die positions of nuclei is reqtiired for calculations of frequencies and intensities in vibrational spectra. The second derivatives of energy, the force constants, are used in determining the frequencies and tiie normal coordinates. The normal coordinate transformation matrix is applied together with theoretical estimates of the dipole moment derivatives in evaluating vibrational absorption intensities. 

The transfrumation of vibrational intensities in Raman spectra into molecular parameto s involves sevoal calculation stages. An essential initial step is the reduction of intensity data to polarizability derivatives with respect to symmetiy vibrational coordinates. As pointed out in previous ciutyters, the inverse electro-optical problem of vibrational intensities can be performed with success only for molecules possessing sufficient symmetry. The transformation between da/dQ and do/dSj derivatives is carried out widi die aid of the normal coordinate transformation matrix Lg according to the expression ... 

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