Small-amplitude vibration

As foe molecule executes small-amplitude vibrations with respect to the equilibrium intenmclear distance, it is appropriate to develop the potential function in a Taylor series about that position. Thus,... 

In the electron-diffraction jargon it is often referred to the framework vibration in contrast to the large amplitude motion. The idea is to try to separate the large amplitude motion, as for example a torsional motion, from the small amplitude vibration also taking place in rigid molecules. This practical approach does not lead to semantic difficulties, but the approach, of course, meets with the well known difficulty in any theoretical treatment of this kind, namely the problem of separability of the energy and consequently of the Hamiltonian operator. 

Small-amplitude vibrations, normal-mode coordinates... 

For small-amplitude vibrations in which the transition moments are relatively smooth, the expansion can be terminated at the linear second term. A vibronic transition between a set of e v e"v" electronic/vibrational levels is defined by the transition moment... 

We shall discuss briefly the physical meaning of various terms which occur in the expansion of our Hamiltonian and the corresponding approximations. The sim-pliest approximation to is obtained when all the small-amplitude vibrational coordinates (k = 1, 3a, 3b, 4a, Ab) are put equal to zero. We shall call this Hamiltonian the zeroth-order inversion—rotation Hamiltonian, or the rigid bender Hamiltonian. It follows from Eqs. (3.23) and (3.33)—(3.38) that it can be written in the following form ... 

Once Eq. (3.15) has been reached, it is generally possible to separate the small-amplitude (harmonic) modes from the large-amplitude modes and treat any remaining anharmonic coupling terms by perturbation methods. The resulting small-amplitude vibrational Hamiltonian is given by ... 

These definitions are similar to Wilson s3, but more general. The usual s-vectors, here written S(°a, have special properties because they are formed from derivatives evaluated in the equilibrium configuration. The constant 5-elements in the treatment of small amplitude vibrations are their components in a molecular system fixed to this equilibrium configuration,... 

As an example consider a planar, near oblate symmetric top molecule. If small amplitude vibrations are assumed, we may expect that the ju-tensor elements are of the order of the reciprocal principal moments of inertia. However, for a planar configuration, withIc-Ia+Ib, it is easily seen that the generally smallest element, ncc, may reach extreme values when evaluated in a PAS using Eq. (2.63). A numerical example illustrates this ... 

The assertion that the PAS is convenient for separating rotations and vibrations can be rejected, therefore. We shall see below (Sect. 4) that the small amplitude vibrations are always treated most simply using Eckart conditions, whereas large amplitude motions must be specially taken care of. Principal inertial axes may only be relevant in relation to the reference structure of the Eckart conditions. 

The first step is to formulate the relationship between Cartesian displacement coordinates, dag, and internal displacement coordinates, qk k = 1, 2,... 3 N—6. For rigid molecules undergoing small amplitude vibrations we can assume that an expansion from the equilibrium configuration,... 

We shall first consider the /-matrix and in particular the elements introduced by including p-labelling. Comparing Eqs. (4.10) and (4.14) with the formulae for rigid molecules, Eqs. (3.14) and (3.51) in particular, we see that once more one may introduce a/ -matrix with elements linear in the normal coordinates of the small amplitude vibrations,... 

In relation to the Van Vleck transformation (Sect. 4.8) we recapitulate that most of the formulae applying to rigid molecules could be generalized with only small adjustments [Eqs. (4.32), (4.50), (4.51), 4.58)-(4.60), (4.63)-(4.65) and (4.69)]. This indicates that the treatment without particular complications may be extended to cover a case where the small amplitude vibrational level is degenerate. This, however, is an object for future developments. 

A novel data analysis procedure is described, based on a variational solution of the Schrddinger equation, that can be used to analyze gas electron diffraction (GED) data obtained from molecular ensembles in nonequilibrium (non-Boltzmann) vibrational distributions. The method replaces the conventional expression used in GED studies, which is restricted to molecules with small-amplitude vibrations in equilibrium distributions, and is important in time-resolved (stroboscopic) GED, a new tool developed to study the nuclear dynamics of laser-excited molecules. As an example, the new formalism has been used to investigate the structural and vibrational kinetics of C=S, using stroboscopic GED data recorded during the first 120 ns following the 193 nm photodissociation of CS2. Temporal changes of vibrational population are observed, which can... 

The atoms in a molecule undergo vibrations around their equilibrium configuration within the quantum mechanical picture, even at zero temperature. The application of elementary djmamical principles to these small amplitude vibrations leads to normal mode analysis. Crystalline solids can naively be thought of as big molecules but solving the equations becomes impossible imless the periodicity of the unit cell is included whereupon major simplifications of the algebra are introduced. 

Here B"k is the B matrix of Eq. (3) evaluated at the equilibrium configuration. In the limit of small-amplitude vibrations the internal coordinates S are equivalent to the bond-angle extension coordinates R. This is demonstrated in Fig. 3, where the angle extension in HCN is compared to the corresponding extension of S,. The internal coordinates S are also related to the rectilinear normal coordinates via the linear transformation... 

For decades high-resolution rotational-vibrational spectroscopy treated nuclear motion in terms of near-rigid rotations and small-amplitude vibrations, relying heavily on perturbation theory (FT) [10-16]. While the formulas [11,14,15]... 

Under these conditions, the molecule is supposed to be in equilibrium with respect to the small amplitude vibration modes. As a result, the total energy of each conformation of the non-rigid coordinate space is determined by a reliable ab initio procedure with full optimization of the geometry with respect to the other coordinates. 

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