Linear harmonic approximation

Molecular structure enters into the rotational entropy component, and vibrational frequencies into the vibrational entropy component. The translational entropy component cancels in a (mass) balanced reaction, and the electronic component is most commonly zero. Note that the vibrational contribution to the entropy goes to oo as v goes to 0. This is a consequence of the linear harmonic oscillator approximation used to derive equation 7, and is inappropriate. Vibrational entropy contributions from frequencies below 300 cm should be treated with caution. 

J.O. Nordling, J.S. Faulkner, Approximate Linear Dependence, Scaling and Operator Convergence in the Harmonic Oscillator Problem, J. Mol. Spectros. 12 (1964) 171. 

The statistical mechanical form of transition theory makes considerable use of the concepts of translational, rotational and vibrational degrees of freedom. For a system made up of N atoms, 3N coordinates are required to completely specify the positions of all the atoms. Three coordinates are required to specify translational motion in space. For a linear molecule, two coordinates are required to specify rotation of the molecule as a whole, while for a non-linear molecule three coordinates are required. These correspond to two and three rotational modes respectively. This leaves 3N — 5 (linear) or 3N — 6 (non-linear) coordinates to specify vibrations in the molecule. If the vibrations can be approximated to harmonic oscillators, these will correspond to normal modes of vibration. 

When the molecule is rigid for the applied bias, approximations of harmonic motions for nuclei and linear couplings with electrons will be sufficient. Then we can adopt the (nonlocal) Holstein model [2], and the total Hamiltonian is expressed by... 

The equation for P(p) may be treated by the same general method as was employed for the equation of the linear harmonic oscillator in Section llo. The first step is to obtain an asymptotic solution for large values of p, in which region Equation 17-13 becomes approximately... 

The potential around its minimum is approximately a harmonic potential with a linear restoring force. The atoms perform a radial harmonic oscillation. 

Fig. 12a for the homopolymer and the elastomer at low c, and in Fig. 12b for higher crosslink concentrations. The FWHM of the quasi-Bragg peaks is not resolution limited, and the central part can be well described by a Gaussian. For the first-order peak of the homopolymer this indicates smectic domains with a finite size along the layer normal, L 0.6-0.7 pm. Away from the center of the peak, algebraic decay is observed with an exponent rjjn = rj = 0.15 0.02, similar to that reported for other smectic polymers [106]. For the elastomer with c = 10%, three harmonics are displayed in Fig. 9. Interestingly, the peak width Aq increases approximately linearly with the harmonic number n. In the tails of the peak, algebraic decay is nicely preserved and no evidence of true long-range order is found.

The vibrational potential energy, V, can be written as a function of any of the coordinate sets, V = V(qc) or V = V(qi) or V = V(( n), and coordinate sets transform one into another using linear combinations that involve geometrical relationships. The potential energy can be written as a Taylor series the first derivatives are null at equilibrium, and if the vibrations are as a first approximation considered harmonic, for any set of coordinates [g] the second derivatives are the force constants,/(i,j), and higher-order derivatives are neglected ... 

To use Eq. [55], one must estimate the nuclear speed R along the reaction coordinate in question. A harmonic approximation for the reaction coordinate (e.g., the S-S stretching coordinate for the example in Figure 2) affords a simple means to do this, " and one that is consistent with the semiclassical nature of Landau-Zener theory. Having computed the (linear) harmonic frequency v for the mode in question, using some flavor of quantum chemistry, one may compute the classical turning points of the harmonic potential ... 

The harmonic approximation restricts the dipole moment expansion to the constant and linear terms. Thus, the selection rule associated with the approximation of electrical hannonicity, states that transitions involving a change by 1 of just one of the vibrational quantum numbers of the linear harmonic oscillator functions defining the vibrational states of molecules are only allowed. Aside from this, for a transition to take place at least one of tiie Cartesian components of the (dp/dQk)o derivatives should differ from... 

A physical system, making oscillations, is referred to as an oscillator. If it complies with eq. (2.4.6), it is referred to as a one-dimensional harmonic oscillator. In the first approximation any molecule can be considered as a classical one-dimensional harmonic oscillator this is the simplest physical model explaining some (but by no means aU) particularities of atom vibrations in the molecule. In Chapter 7 it will be shown that in qnantum mechanics a much better approximation is given by a model of a quantum linear harmonic oscillator. However, the next best approximation is a nonharmonic (nonlinear) model (this model is more compUcated it describes atomic vibrations in more detail and introduces new phenomena). 

In his classical paper, Renner [7] first explained the physical background of the vibronic coupling in triatomic molecules. He concluded that the splitting of the bending potential curves at small distortions of linearity has to depend on p, being thus mostly pronounced in H electronic state. Renner developed the system of two coupled Schrbdinger equations and solved it for H states in the harmonic approximation by means of the perturbation theory. 

The role of two-phonon processes in the relaxation of tunneling systems has been analyzed by Silbey and Trommsdorf [1990]. Unlike the model of TLS coupled linearly to a harmonic bath (2.39), bilinear coupling to phonons of the form Cijqiqja was considered. In the deformation potential approximation the coupling constant Cij is proportional to (y.cUj. There are two leading two-phonon processes with different dependence of the relaxation rate on temperature and energy gap, A = (A Two-phonon emission prevails at low temperatures, and it is... 

It is important to note that in all these methods, the first term in the series solution constitutes the so-called approximation of zero order. This is generally the solution of a simple linear problem e.g., the harmonic oscillator the second term appears as the first approximation, and so on. The amount of labor increases very rapidly with the order of approximation, but the additional information obtained from approximations of higher orders (beginning with the second) does not increase our knowledge from the qualitative point of view. It merely adds small quantitative corrections to the first approximation, and in most applied problems, these corrections are scarcely worth the considerable complication in calculations. For that reason the first approximation is generally sufficient in exploring a new problem, or in investigating the qualitative aspect of a phenomenon. 

Because the degrees of freedom decouple in the linear approximation, it is easy to describe the dynamics in detail. There is the motion across a harmonic barrier in one degree of freedom and N — 1 harmonic oscillators. Phase-space plots of the dynamics are shown in Fig. 1. The transition from the reactant region at q <0 to the product region at q >0 is determined solely by the dynamics in (pi,qi), which in the traditional language of reaction dynamics is called the reactive mode. 

So far, the discussion of the dynamics and the associated phase-space geometry has been restricted to the linearized Hamiltonian in eq. (5). However, in practice the linearization will rarely be sufficiently accurate to describe the reaction dynamics. We must then generalize the discussion to arbitrary nonlinear Hamiltonians in the vicinity of the saddle point. Fortunately, general theorems of invariant manifold theory [88] ensure that the qualitative features of the dynamics are the same as in the linear approximation for every energy not too high above the energy of the saddle point, there will be a NHIM with its associated stable and unstable manifolds that act as separatrices between reactive and nonreactive trajectories in precisely the manner that was described for the harmonic approximation. 

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