Harmonic potential, definition

Various other ways to incorporate the out-of-plane bending contribution are possible. For e3plane bend involves a cakulation of the angle between a bond from the central atom and the plane defined by I he central atom and the other two atoms (Figure 4.10). A value of 0° corresponds to all four atoms being coplanar. A third approach is to calculate the height of the central atom above a plane defined by the other three atoms (Figure 4.10). With these two definitions the deviation of the out-of-plane coordinate (be it an angle or a distance) can be modelled Lt ing a harmonic potential of the form... 

According to cla.ssical mechanics, a harmonic oscillator may vibrate with any amplitude, which means that it can possess any amount of energy, large or small. Quantum mechanics, however, shows that molecules can only exist in definite energy states. In the case of harmonic potentials, these states are equidistant. 

In spectroscopic analysis the first term, Vo, is usually made zero by definition. This is done by assuming that each internal coordinate, let it be a given bond distance or bond angle, is strainless at its equilibriiun value. For example, the standard sp3-sp3 carbon-carbon bond length zq is 1.53 A. The harmonic stretching potential then is represented by ... 

This review shows how the photochemistry of ketones can be rationalized through a single model, the Tunnel Effect Theory (TET), which treats reactions of ketones as radiationless transitions from reactant to product potential energy curves (PEC). Two critical approximations are involved in the development of this theory (i) the representation of reactants and products as diatomic harmonic oscillators of appropriate reduced masses and force constants (ii) the definition of a unidimensional reaction coordinate (RC) as the sum of the reactant and product bond distensions to the transition state. Within these approximations, TET is used to calculate the reactivity parameters of the most important photoreactions of ketones, using only a partially adjustable parameter, whose physical meaning is well understood and which admits only predictable variations. 

The expansion of the electrostatic potential into spherical harmonics is at the basis of the first quantum-continuum solvation methods (Rinaldi and Rivail, 1973 Tapia and Goschinski, 1975 Hylton McCreery et al., 1976). The starting points are the seminal Kirkwood s and Onsager s papers (Kirkwood 1934 Onsager 1936) the first one introducing the concept of cavity in the dielectric, and of the multipole expansion of the electrostatic potential in that spherical cavity, the second one the definition of the solvent reaction field and of its effect on a point dipole in a spherical cavity. The choice of this specific geometrical shape is not accidental, since multipole expansions work at their best for spherical cavities (and, with a little additional effort, for other regular shapes, such as ellipsoids or cylinders). 

A transition from a definite to a non-definite matrix occurs even for the simple case of the harmonic oscillator as a function of the total time. In general, the shorter the time, the smaller the term with the potential derivatives and the matrix as a whole is more positive. At sufficiently long time we expect some of the eigenvalues to reverse their sign and become negative, making the matrix indefinite. 

At moderate deviations from the equilibrium, we can chose the adiabatic potential of the movable quantum dot as detailed above. The first part is the potential energy Upot of the dispersion interaction in the harmonic approximation. The second part accounts for the electrostatic energy of charging. By definition, it is the minimum energy required for adding charge Q to the quantum dot ... 

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