Coordinate transformations vibrational normal coordinates

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. 

The lattice vibrations for the simple tetrahedral lattice wore studied in Section 9-A. The state of the distortion of the lattice was specified by giving the displacement (5r, of each atom. We then made a transformation to normal coordinates u., each corresponding to a normal mode frequency w(k), and these were plotted as a function of k in Fig. 9-2. There were three curves for each atom in the primitive cell. We see immediately that there will be difficulties in complex structures in quartz there arc 27 sets of modes, and oven in the simple molecular lattice there are 9. This complexity suggests that one should proceed by computer. One such approach was taken by Bell, Bird, and Dean (1968). They took a large cluster... 

Since this transformation to normal coordinates is invertible, one can readily determine the functional dependencies of the terms in Eq. (1) using either the normal or internal coordinates. Interestingly, in our study of vibrational states of the well-known local mode molecule H20 and its deuterated analogs we found only minor differences between the results of CVPT in the internal and normal mode representations (46). The normal mode calculations, however, required significantly less computer time to run, since many terms in the Hamiltonian are constrained to zero by symmetry. For this reason we chose to use the normal mode coordinates for all subsequent studies. 

Many problems in vibration analysis are difficult to solve using a Cartesian co-ordinate system. These coordinates change in a complex manner with the translation, vibration and rotation of molecules or other bodies because of interaction terms. When Cartesian coordinates are transformed into normal coordinates, as shown below, the interaction terms disappear and a motion can be expressed in terms of a single normal coordinate. 

Here, the vibrational normal coordinate Qi (Q2) transforms according to the Bi B2) irreducible representation, with the corresponding coupling constant denoted k ( 2), and Hq collects the pertinent harmonic oscillator Hamiltonians. The adiabatic potential energy surfaces V read as follows... 

In order to give the concept of an irreducible representation a more concrete reality for the reader, the transformations of normal coordinates have been used throughout as examples, but the concept itself is quite independent of the idea of normal coordinates or the problem of molecular vibrations. It arises whenever a set of linear transformations has the properties of a group, no matter what the meaning of the transformation variables. Later, it will be necessary to regard the transformations of a set of wave functions as forming a group, and the present theory will be applied. 

The mathematics behind the transformation to normal coordinates, which are represented by the arrow diagrams, provides certain rules that can be used to guess the qualitative form of the normal modes of molecules. First, a normal mode is a vibrational motion, and so any motion along the normal coordinate (following the arrows in the diagrams) must not lead to a rotation or translation of the molecule. For carbon dioxide, the following two diagrams are examples of pure translation and pure rotation ... 

The symmetry properties of a fundamental vibrational wave function are the same as those of the corresponding normal coordinate Q. For example, when the C3 operation is carried out on Qi, the normal coordinate for Vj, it is transformed into Q[, where... 

The first Raman and infrared studies on orthorhombic sulfur date back to the 1930s. The older literature has been reviewed before [78, 92-94]. Only after the normal coordinate treatment of the Sg molecule by Scott et al. [78] was it possible to improve the earlier assignments, especially of the lattice vibrations and crystal components of the intramolecular vibrations. In addition, two technical achievements stimulated the efforts in vibrational spectroscopy since late 1960s the invention of the laser as an intense monochromatic light source for Raman spectroscopy and the development of Fourier transform interferometry in infrared spectroscopy. Both techniques allowed to record vibrational spectra of higher resolution and to detect bands of lower intensity. 

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. 

It is now fundamental to define the normal coordinates of this vihrational system - that is to say, the nuclear displacements in a polyatomic molecule. Again in the limit of small amplitudes of vibration, the normal coordinates in the form of the vector Q, are related to the internal coordinates by a linear transformation, viz. 

Just as group theory enables one to find symmetry-adapted orbitals, which simplify the solution of the MO secular equation, group theory enables one to find symmetry-adapted displacement coordinates, which simplify the solution of the vibrational secular equation. We first show that the matrices describing the transformation properties of any set of degenerate normal coordinates form an irreducible representation of the molecular point group. The proof is based on the potential-energy expression for vibration, (6.23) and (6.33) ... 

The transformed normal coordinate 0RQjVk can be expressed as some linear combination of the 3N mass-weighted displacement coordinates using (6.21), we can express 0RQjVk as a linear combination of all the normal coordinates of the molecule. If in this linear combination, the coefficient of a normal coordinate whose vibrational frequency differed... 

We shall now briefly review the Kubo-Oxtoby theory of vibrational line-shape. The starting point for most theories of vibrational dephasing is the stochastic theory of lineshape first developed by Kubo [131]. This theory gives a simple expression for the broadened isotropic Raman line shape (/(< )) in terms of the Fourier transform of the normal coordinate time correlation function by... 

In the theory of small vibrations it is shown that, by a linear transformation of the displacements xj yj Zj to a set of normal coordinates Qk, the kinetic energy and the potential energy may be transformed simultaneously into diagonal form so that... 

The transformation from L, to Ap simply consists in dividing each component of L by the square root of the mass of its corresponding atom. Defining A as the second derivative of the curve of the energy versus these normal coordinates permits to obtain the vibrational frequency by... 

Within the HA, the prediction of a vibrational absorption spectrum amounts to the calculation of the harmonic normal mode frequencies, vi7 and dipole strengths, Dt. The frequencies are obtained from the harmonic force field (HFF). With respect to Cartesian displacement coordinates, this is the Hessian (d2WG/dXXadXx,a,)0. Diagonalization (after mass-weighting) yields the force constants ky the frequencies, vy and the normal coordinates, <2 , i.e. the transformation matrices, SXa4. The dipole strengths depend in addition on the APTs these require calculation of (dtpG/dXXa)0. 

In vibration-rotation theory, the /., / and contributions to the contact-transformed Hamiltonian are commonly evaluated directly from the relationships (7.59), (7.63), (7.65) and (7.66). This is because the particularly simple commutation relationships which exist between the normal coordinate operator Q, its conjugate... 

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