Harmonic curve

The French physicist and mathematician Jean Fourier determined that non-harmonic data functions such as the time-domain vibration profile are the mathematical sum of simple harmonic functions. The dashed-line curves in Figure 43.4 represent discrete harmonic components of the total, or summed, non-harmonic curve represented by the solid line. 

For a cubic site, relations between the cumulants and the coefficients of the OPP model have been derived by Kontio and Stevens (1982), and applied to the Al(4) atom in the alloy VA110 4.2 The coordination of Al(4) is illustrated in Fig. 2.4(a), while the potential along [111], derived from the thermal parameter refinement, is shown in Fig. 2.4(b). It is clear from these figures that higher than third-order terms contribute to the potential, because the deviation from the harmonic curve is not exactly antisymmetric with respect to the equilibrium configuration. The potential appears steeper at the higher temperature, which is opposite to what is expected on the basis of the thermal expansion of the solid. 

Harmonic curves, 1487. 1495 solvent ion bonds, 1504 Harmonic electron transfer, 1504 Heat of adsorption. 940... 

Figure 6. The guided mode dispersion curves for a birefringent film and an optically isotropic substrate. Both the fundamental and harmonic curves are shown. The TE mode utilizes the ordinary refractive index and TM primarily the extraordinary index. Note the change in horizontal axis needed to plot both the fundamental and harmonic dispersion curves. Phase-matching of the TEq(co) to the TMo(2o>) is obtained at the intersection of the appropriate fundamental and harmonic curves.

The definition of a free form curve entity applies control points (Figure 7-4a), interpolation points (Figure 7-4b), or a hand sketch (Figure 7-4c) as input information. Unevenly spaced points are fitted accurately by correct modeling procedures. A hand sketch is processed into a harmonic curve of the similar shape. 

Figure 7 Potential energy curve along the symmetric 0=Re=0 coordinate of trans-Re02(ethylenediamine)2 from DFT calculations, shown as dots and interpolated solid line. The dotted line shows a harmonic curve calculated from the observed vibrational frequency of 880cm ...

Figure 8 Experimental luminescence spectrum of crystalline fraw5-Re02(ethylenediamine)2Cl at room temperature. The calculated spectra were obtained from the one-dimensional potential energy curve calculated by DFT and from the harmonic curve, both shown in Figure 7.

The r.h.s. corresponds to the radius of the curve, and its center is M=(-G /A, -G2/A22) We note that, away from the stationary points, the higher order terms are more important and the harmonic curve (6) can give only a crude imagination of the valley structure. Nevertheless, we can treat some special cases (cf.Fig.3). The point (Y-itY-y) (0 0) is the initial point of the curve other interesting points are y at y = i/ ii... 

In his classical paper, Renner [7] first explained the physical background of the vibronic coupling in triatomic molecules. He concluded that the splitting of the bending potential curves at small distortions of linearity has to depend on p, being thus mostly pronounced in H electronic state. Renner developed the system of two coupled Schrbdinger equations and solved it for H states in the harmonic approximation by means of the perturbation theory. 

Comparison of the simple harmonic potential (Hooke s law) with the Morse curve. 

The reason that does not change with isotopic substitution is that it refers to the bond length at the minimum of the potential energy curve (see Figure 1.13), and this curve, whether it refers to the harmonic oscillator approximation (Section 1.3.6) or an anharmonic oscillator (to be discussed in Section 6.1.3.2), does not change with isotopic substitution. Flowever, the vibrational energy levels within the potential energy curve, and therefore tq, are affected by isotopic substitution this is illustrated by the mass-dependence of the vibration frequency demonstrated by Equation (1.68). 

Figure 6.4 Potential energy curve and energy levels for a diatomic molecule behaving as an anharmonic oscillator compared with those for a harmonic oscillator (dashed curve)...

Owing to the effects of mechanical anharmonicity - to which we shall refer in future simply as anharmonicity since we encounter electrical anharmonicity much less frequently -the vibrational wave functions are also modified compared wifh fhose of a harmonic oscillator. Figure 6.6 shows some wave functions and probabilify densify functions (iA A ) for an anharmonic oscillator. The asymmefry in and (iA A ) 5 compared wifh fhe harmonic oscillator wave functions in Figure f.i3, increases fheir magnitude on the shallow side of the potential curve compared with the steep side. 

The BQ term alone, wifh B positive, would give a pofenfial resembling fhe harmonic oscillator pofenfial in Figure 6.4 (dashed curve) buf wifh steeper sides. The inclusion of fhe AQ term, wifh A negative, adds an upside-down parabola af 0 = 0 and fhe resulf is a W-shaped pofenfial. The barrier heighf b is given by... 

Figure 43.4 Discrete (harmonic) and total (non-harmonic) time-domain vibration curves...

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