Model harmonic

A different approach comes from the idea, first suggested by Flelgaker et al. [77], of approximating the PES at each point by a harmonic model. Integration within an area where this model is appropriate, termed the trust radius, is then trivial. Normal coordinates, Q, are defined by diagonalization of the mass-weighted Flessian (second-derivative) matrix, so if... 

Harmonic analysis (normal modes) at given temperature and curvature gives complete time behavior of the system in the harmonic limit [1, 2, 3]. Although the harmonic model may be incomplete because of the contribution of anharmonic terms to the potential energy, it is nevertheless of considerable importance because it serves as a first approximation for which the theory is highly developed. This model is also useful in SISM which uses harmonic analysis. 

The harmonic model thus predicts that the "fundamental" (v=0 v = 1) and "hot band" (v=l V = 2) transition should occur at the same energy, and the overtone (v=0 v=2) transitions should occur at exactly twice this energy. 

The simple harmonic model gives a force constant of and since the... 

Vibrational levels are often analyzed in terms of an harmonic model, but in practice this is not a good approximation for real chemical bonds. An empirical expression that provides a more realistic description of a stretched bond, is the Morse function... 

As pointed out in Ref. [4], no entropy variation appears in the description given by the harmonic model, apart from the weak contribution arising from the frequency shifts of the oscillators. The applications of this model are then a priori restricted to redox reactions in which entropic contributions can be neglected. We shall see in Sect. 3 that the current interpretations of most electron transfer processes which take place in bacterial reaction centers are based on this assumption. 

One may wonder whether a purely harmonic model is always realistic in biological systems, since strongly unharmonic motions are expected at room temperature in proteins [30,31,32] and in the solvent. Marcus has demonstrated that it is possible to go beyond the harmonic approximation for the nuclear motions if the temperature is high enough so that they can be treated classically. More specifically, he has examined the situation in which the motions coupled to the electron transfer process include quantum modes, as well as classical modes which describe the reorientations of the medium dipoles. Marcus has shown that the rate expression is then identical to that obtained when these reorientations are represented by harmonic oscillators in the high temperature limit, provided that AU° is replaced by the free energy variation AG [33]. In practice, tractable expressions can be derived only in special cases, and we will summarize below the formulae that are more commonly used in the applications. 

Fig. 6.31 Normalised intermediate scattering function from C-phycocyanin (CPC) obtained by spin-echo [335] compared to a full MD simulation (solid line) exhibiting a good quantitative matching. In contrast the MD results from simplified treatments as from protein without solvent (long dash-short dash /me), with point-like residues (Cpt-atoms) (dashed line) or coarse grained harmonic model (dash-dotted line) show similar slopes but deviate in particular in terms of the amplitude of initial decay. The latter deviation are (partly) explained by the employed technique of Fourier transformation. (Reprinted with permission from [348]. Copyright 2002 Elsevier)...

In order to give a more quantitative view of the relative perturbation of each vibrator (C=C stretching and CH2 scissoring), the authors performed calculations for this complex based on simplified harmonic models involving mechanical couplings between the C=C... 

TABLE 4. Comparison between observed and calculated wavenumbers (cm ) for Li(C2H4)(N2) using a 6 X 6 harmonic model including the C=C stretch and two CH2 bond angle deformations of the ethylene subunits ... 

The simple harmonic model gives a force constant of 4tt2CqO)Iia and since the reduced mass pL = 1.139 x 10-27kg, ks = 1901.5 Nm 1. The potential energy is therefore... 

Suppose for example that we want to test the validity of our data against the harmonic model... 

At room temperature the value of Boltzmann factor (kT) is about 200 cm-1, so the vast majority of molecules are in the their vibrational ground state (n = 0). Thus in the case of the simple harmonic model described above, only transitions to n= 1) are allowed, and NIR spectroscopy would not exist. 

---