Potential energy well, harmonic

A parabolic potential energy well (harmonic oscillator) reduces this tendency, and the energy levels are equidistant. The distance decreases if the parabola gets wider (less restrictive). 

A lot of theoretical work on displacive phase transitions has focussed on a simple model in which atoms are connected by harmonic forces to their nearest neighbors, and each neighbor also sees the effect of the rest of the crystal by vibrating independently in a local potential energy well (Bruce and Cowley 1980). For a phase transition to occur, this double well must have two minima, and can be described by the following function ... 

This value of electron affinity, calculated from the bottom of the potential energy well, must be corrected for the zero-point energy (ZPE) effects. It may be expected that the normal modes of the anion will be softer than in SF6 and the ZPE correction will increase the computed value of electron affinity. In order to establish the magnitude of ZPE and determine the character of the stationary point for the anion, harmonic vibrational analysis was performed. Due to limited computer resources, only the DF1 calculations were done using the extended, (3dl/, 3dlf) polarized basis set. Zero-point energies were found to be 0.51 eV and 0.32 eV for SF6 and SFg, respectively the ZPE correction to the computed value of electron affinity is thus about 0.2 eV, and the ZPE-corrected value of electron affinity is 1.6 eV. 

The classical harmonic approximation is adequate at low enough temperatures, where most of the contribution to S. comes from the bottom part of the potential energy well (except near absolute zero, where quantum effects become important " ). This approximation is expected to be less adequate at higher temperatures, where the contribution of the anharmonic wings of a localized microstate become significant. Also, the contribution of the higher frequencies should be calculated quantum mechanically. 

Fig. 6.11. The bound, continuum, and resonance (metastable) states of an anbarmonic oscillator. Two discrete bound slates are shown (energy levels and wave functions) in the lower part of the image. The continuum (shaded area) extends above the dissociation limit i.e.. the system may have any of the energies above the limit. Tliere is me resonance state in the continuum, whidi corresponds to the third level in the potential energy well of the oscillator. Within the well, the wave function is veiy similar to the third slate of the harmonic oscillator, but there are differences. One is that the function has some low-amplitude oscillatims rai the right side. They indicate that the function is non-normalizable and that the system will dissociate sooner or later.

Fig. 12.7. Sadlej relation. The electric field mainly causes a shift of the electronic charge distribution toward the anode (a). A GTO represents the eigenfunction of a harmonic oscillator. Suppose that an electron oscillates in a parabolic potential energy well (with the force constant I ). In this situation, a homogeneous electric field corresponds to the perturbation x, that conserres the hannonicity with unchanged force constant k (b).

Fig. 7.7. A ball oscillating in a potential energy well (scheme), (a) and (b) show the normal vibrations (normal modes) about a point /fo = being a minimum of the potential energy function V(/ o + ) of two variables = (xj, X2). This function is first approximated by a quadratic function i.e., a paraboloid V X, X2)- Computing the normal modes is equivalent to such a rotation of the Cartesian coordinate system (a), that the new axes (b) xj and x become the principal axes of any section of V by a plane V = const (i.e., ellipses). Then, we have V(xi,X2) = V Rq = 0) + j/ti (xj) + k2 The problem then becomes equivalent to the two-dimensional harmonic oscillator (cf.,...

Table 16.2 Equilibrium distance (Re) in pm, harmonic vibrational frequency (u)e) in cm and potential energy well depth (De) in eV of the ground state of TIH obtained from various relativistic approaches. The abbreviation var." indicates a variational and pert. perturbational treatment of spin-orbit coupling (SOC). Corn elec. denotes the number of correlated electrons. Note that all data from Ref. [282] was obtained for a limited orbital space of 94 active orbitals.

In order to evaluate the contribution from the nuclear component it is necessary to know the vibrational wavefunctions. For a simple harmonic oscillator, the potential energy well for intemuclear distances is given by a parabolic curve where the vibrational energy levels are quantized (Fig. 1.4). The curves for v = 0,1,2, and 3 represent the wavefunctions for each vibrational level. When a transition occurs from the ground to an excited state, it is necessary to consider the potential energy wells of both states. Since the electronic structure of the excited state is different from that of the ground state, the vibrational energy profiles will be different. In... 

In the previous examples we only considered electronic energy changes and approximated the entropy as all or nothing. In essence, we assumed that gas-phase species have 100% of their standard state entropy and surface species possess no entropy at all. These assumptions can certainly be improved and in order to construct thermodynamically consistent microkinetic models this is not just optional, but absolutely necessary. Entropy and enthalpy corrections for surface species can be calculated using statistical thermodynamics from knowledge of the vibrational frequencies, and the translational and rotational degrees of freedom (DOF). In contrast to gas-phase molecules, adsorbates cannot freely rotate and move across the surface, but the translational and rotational DOF are frustrated within the potential energy well imposed by the surface. In the harmonic limit the frustrated translational and rotational DOF can conveniently be described as vibrational modes, which in turn means that any surface adsorbate will have iN vibrational DOFs that are all treated equally. 

Figure 3.3 A h5 othetical free energy relationship for the oxidation of a snrface H species. Both potential energy wells are represented as harmonic, and the product well is drawn at two electrode potentials to illustrate how the reaction barrier change with potential is some fraction of the reaction energy change.

---