Simple harmonic approximations

These features are illustrated for H2O in Figure 2.5, where the exact form is taken firom a parametric fit to a large number of spectroscopic data. The simple harmonic approximation (P2) is seen to be accurate to about 20° from the equilibrium geometry and the cubic approximation (P3) up to 40°. Enforcing the cubic polynomial to have a zero derivative at 180° (P3 ) gives a qualitative correct behaviour, but reduces the overall fit, although it still is better than a simple harmonic approximation. 

In a simple harmonic approximation the first term gives the 0-0 band if the transition is allowed for non-distorted symmetrical structure. But in the case of benzene phosphorescence the 3Blu transition is forbidden in Deh symmetry even if the SOC... 

Depending on the character of the molecular motions, one can distinguish several physical situations. In most cases, the molecules are trapped in relatively deep potential wells. Then, they perform small translational and orientational oscillations around well-defined equilibrium positions and orientations. Such motions are reasonably well described by the harmonic approximation. The collective vibrational excitations of the crystal, which are considered as a set of harmonic oscillators, are called phonons. Those phonons that represent pure angular oscillations, or libra-tions, are called librons. For some properties it turns out to be necessary to look at the effects of anharmonicities. Anharmonic corrections to the harmonic model can be made by perturbation theory or by the self-consistent phonon method. These methods, which are summarized in Section III under the name quasi-harmonic theories, can be considered to be the standard tools in lattice dynamics calculations, in addition to the harmonic model. They are only applicable in the case of fairly small amplitude motions. Only the simple harmonic approximation is widely used the calculation of anharmonic corrections is often hard in practice. For detailed descriptions of these methods, we refer the reader to the books and reviews by Maradudin et al. (1968, 1971, 1974), Cochran and Cowley (1967), Barron and Klein (1974), Birman (1974), Wallace (1972), and Cali-fano et al. (1981). 

The results of Figure 5 from n = 10 are now unstable to single atom attachment as shown by the calculations of Figure 4, thus confirming the suggestion that, once nuclear delocalisation is somewhat accounted for by even a simple harmonic approximation, the growth process presumably occurs by multiple-atom attachments that stabilize the larger clusters. 

Figure 2.2 compares the performance of various functional forms for the stretch energy in CHi. The exact form is taken from electronic structure calculations ([8,8]-CASSCF/aug-cc-pVTZ). The simple harmonic approximation (P2) is seen to be accurate to about 0.1 A from the equilibrium geometry and the quartic approximation (P4) up to 0.3 A. The Morse potential reproduces the real curve quite accurately up to an elongation of 0.8 A, and becomes exact again in the dissociation limit. 

Here, the bond stretch and angle bend terms are represented by simple harmonic approximations each with their respective force constant k and an energy contribution that is proportional to the square of the deviation of the bond length or angle from the respective expectation values. The torsional term has a well-defined well depth V and discrete maxima and minima governed by the geometry of the torsion. 

In Co,-based CG representations of backbone chains, the two-dimensional Ramachandran space is reduced to one bending parameter (angle y in Fig. 5). The most common secondary structure elements a-helices and p-sheets correspond to two well-separated values of y ( 90° for the a-helix and between 120° and 140° for the p-sheet). This simple example is sufficient to explain how a CG potential that aims at exploring thermodynamically accessible protein conformations and/or folding events cannot use a simple harmonic approximation for the angular terms, but it requires more complicated functional forms allowing multiple minima. 

---