Simple harmonic expansions

In this Chapter, we present step-by-step derivations of the explicit expressions for matrix elements based on the spherical-harmonic expansion of the tip wavefunction in the gap region. The result — derivative rule is extremely simple and intuitively understandable. Two independent proofs are presented. The mathematical tool for the derivation is the spherical modified Bessel functions, which are probably the simplest of all Bessel functions. A concise summary about them is included in Appendix C. 

The simple harmonic motion of a diatomic molecule was treated in Chapter 1, and will not be repeated here. Instead, emphasis is placed on polyatomic molecules whose electronic energy s dependence on the 3N Cartesian coordinates of its N atoms can be written (approximately) in terms of a Taylor series expansion about a stable local minimum. We therefore assume that the molecule of interest exists in an electronic state for which the geometry being considered is stable (i.e., not subject to spontaneous geometrical distortion). 

In the following we show that a simple description of the (quantum or classical) dynamics can be obtained in a multidimensional system close to a stationary point. Thus, the system can be described by a set of uncoupled harmonic oscillators. The formalism is related to the generalization of the harmonic expansion in Eq. (1.7) to multidimensional systems. 

Let us now imagine that the function p(0,axial symmetry with respect to the z-axis, as, e.g., in Fig. 4.1(6), and that it is created by pulsed excitation at t = 0. In this case its expansion over multipoles includes transversal polarization moments pq (these are complex quantities) where Q 0. If the precession frequency uj< is much larger than the relaxation rate, then the components pQ will depend on time, in the time scale wj,1, following (4.9), according to the simple harmonic law ... 

Such a potential energy function gives rise to the famihar parabolic curve (Figure 22) where the curvature of the function is related to the force constant. The success of this simple harmonic model in treating surface atom vibrations lies in the relatively small displacement of surface atoms during a period of vibration. For some crystal properties, such as thermal expansion at elevated temperature, anharmoitic contributions to the potential must be included for an accurate description. 

The strategies employed by CHARMM and MM3 in the formulations of bond stretch terms are repeated in the formulation of angle bending terms. Thus, CHARMM uses simple harmonic terms, whereas MM3 uses an expansion up to sextic terms. 

In some respects, this approach is very attractive since, if the spherical harmonic expansions of the correlation functions are sufficiently rapidly convergent, the approximate solution of the Ornstein-Zernike equation for a molecular fluid can be placed upon essentially the same footing as that for a simple atomic fluid. The question of convergence of the spherical harmonic expansions turns out to be the key issue in determining the efficacy of the approach, so it is worthwhile to review briefly the available evidence on this question. Most of the work on this problem has concerned the spherical harmonic expansion of (1,2) for linear molecules. This work was pioneered by Streett and Tildesley, who showed how it was possible to write the spherical harmonic expansion coefficients as ensemble averages obtainable from a Monte Carlo or molecular dynamics simulation via... 

For class-1 states, a simple harmonic representation of U leads to a complete set of eigenfunctions ( ) this harmonic oscillator basis set is used to diagonalize equation (6). In this case, it is sufficient to construct U( 4>k) using a standard approach involving mass fluctuation (or nuclear ) coordinates and the corresponding electronic state dependent Hessian. The higher terms in the Taylor expansion define anharmonic contributions to the transition moments. These diabatic states are confining and only one stationary point in -space would be found for each... 

In HTST, a harmonic expansion of the PES is invoked both in the IS and in the saddle point separating the IS and the FS. The HTST is therefore appUcable under the same general assumptions as mentioned for TST but further demands that the PES is smooth enough for a local harmonic expansion of the PES to be reasonable. This means that it is necessary that the potential is reasonably well represented by its second-order Taylor expansion around these two expansion configurations. The general idea is that the partition functions in Equation (4.20) can be evaluated analytically for the harmonic expansion of the PES around the expansion points. This leads to very simple expressions for the rate constants and gives reasonable rate constants for... 

This effect, which is in a loose sense the nonlinear analog of linear optical rotation, is based on using linearly polarized fundamental light and measuring the direction of the major axis of the ellipse that describes the state of polarization of the second-harmonic light. For a simple description of the effect, we assume that the expansion coefficients are real, as would be the case for nonresonant excitation within the electric dipole approximation.22 In this case, the second-harmonic light will also be linearly polarized in a direction characterized by the angle... 

The heat capacity models described so far were all based on a harmonic oscillator approximation. This implies that the volume of the simple crystals considered does not vary with temperature and Cy m is derived as a function of temperature for a crystal having a fixed volume. Anharmonic lattice vibrations give rise to a finite isobaric thermal expansivity. These vibrations contribute both directly and indirectly to the total heat capacity directly since the anharmonic vibrations themselves contribute, and indirectly since the volume of a real crystal increases with increasing temperature, changing all frequencies. The constant volume heat capacity derived from experimental heat capacity data is different from that for a fixed volume. The difference in heat capacity at constant volume for a crystal that is allowed to relax at each temperature and the heat capacity at constant volume for a crystal where the volume is fixed to correspond to that at the Debye temperature represents a considerable part of Cp m - Cv m. This is shown for Mo and W [6] in Figure 8.15. 

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