Harmonic cosine function

A number of functions with multiple minima have been proposed to model the valence angles around metals. The molecular mechanics program DREIDING, based on a generic force field, uses a harmonic cosine function (Eq. 2.16)1321. 

The force constant kg of the harmonic cosine function is related to the force constant kg of the harmonic oscillator (Eq. 2.7) via Eq. 2.17. 

The TRIPOS, Chem-X, CHARMm, COSMIC, and CVFF force fields all use only the harmonic approximation, whereas in the DREIDING force field, angle bending is described by a harmonic cosine function where 0 and 0o in Eq. [6] are replaced by cos 0 and cos 0q, respectively, to avoid a zero slope as 0 approaches 180°. In the UFF force field, angle bending is described with a small Fourier expansion in 0 (Eq. [7]). This functional form was selected to better describe large-amplitude motions. 

The first solution has been implemented in the DREIDING program which employs a harmonic cosine function and in the SHAPES FF which uses a Fourier term. The VALBOND program implements a multiple minima expression based on Pauling s strength functions which describe the ideal angles between hybrid orbitals. 

Most types of motion due to vibration occur in periodic motion. Periodic motion repeats itself at equal time intervals. A typical periodic motion is shown in Figure 5-3. The simplest form of periodic motion is harmonic motion, which can be represented by sine or cosine functions. It is important to remember that harmonic motion is always periodic however, periodic motion is not always harmonic. Harmonic motion of a system can be represented by the following relationship ... 

The representation of tp(x, t) by the sine function is completely equivalent to the cosine-function representation the only difference is a shift by A/4 in the value of X when t = 0. Moreover, any linear combination of sine and cosine representations is also an equivalent description of the simple harmonic wave. The most general representation of the harmonic wave is the complex function... 

Fig. 2.8. Left oscillatory part of the reflectivity change of Bi (0001) surface at 8K (open circles). Fit to the double damped harmonic function (solid curve) shows that the Aig and Eg components (broken and dotted curves) are a sine and a cosine functions of time, respectively. Right pump polarization dependence of the amplitudes of coherent Aig and Eg phonons of Bi (0001). Adapted from [25]...

By Fourier transformation, a signal is decomposed into its sine and cosine components [Angl]. In this way, it is analysed in terms of the amplitude and the phase of harmonic waves. Sine and cosine functions are conveniently combined to form a complex exponential, coscot 4- i sinwt = exp icomplex amplitudes of these exponentials constitute the spectrum F((o) of the signal f(t), where co = In IT is the frequency in units of 2n of an oscillation with time period T. The Fourier transformation and its inverse are defined as... 

The term is a constant offset, or the average of the waveform. The b and c coefficients are the weights of the wth harmonic cosine and sine terms. If the function is purely even about t = 0 (this is a boundary condition like that discussed in Chapter 4), only cosines are required to represent it, and only the b terms would be nonzero. Similarly, if the function is odd, only the terms would be required. A general function Fper(0 will require sinusoidal harmonics of arbitrary amplitudes and phases. The magnitude and phase of the mth harmonic in the Fourier series can be found by ... 

According to (5.27) the amplitude of scattering F(q) for our one-dimensional crystal is given by Fourier transform of density function p(x). Since we have only the sum of cosine functions there are only discrete harmonics at wavevectors q = mqo = Inmla. The structure factor (5.25) is proportional to scattered light intensity F(q)F (q) and also consists of harmonics represented by 8-functions situated at the same wavevector values q = Inmla and having amplitude... 

Any well-behaved periodic function (such as a spectrum) can be represented by a Fourier series of sine and cosine functions of varying amplitudes and harmonically related frequencies. A typical NIR spectrum may be defined mathematically by a series of sines and cosines in the following equation ... 

Other modulation techniques are oscillation (tilting) of an interference filter [3] and modulation of the electron beam scan pattern in a vidicon or image-disk-sector photomultiplier spectrophotometer [34]. This was the first nonmechanical wavelength modulation. Wavelength modulation induces a synchronous modulation of the amplitude. If these intensities are expanded, for instance, in the form of a Taylor series in Aq, and the powers of the sine functions are expressed as sine and cosine functions of the corresponding multiple angles, then the derivatives can be obtained from the Fourier coefficients (see Sec. 2.1.3.3) of these series. The second derivative is obtained from the second harmonic of the induced intensity. 

In panel a, p(x)p(0) is a constant plus a cosine function with a period of a. This correlation function is observed when p(x) changes sinusoidally. The Fourier transform converts the constant into d(k) and cos 2irx/a) into Sik-lv/a). In part b, (p(jc)p(0)) has a harmonic sAk = Air/a. The density correlation is slightly distorted from the cosine function. 

Because every periodic function can be represented as a Fourier series, that is, as a sum of sinus (or cosine) functions (see textbooks of mathematics), such a function now contains the basic frequency as well as higher harmonics. Consequently, the measured heat flow rate function also contains the higher harmonics. With proper mathematical procedures, the heat capacities of the different components can be individually determined and show possible frequency dependence. Pak and Wunderlich (2001) introduced the sawtooth modulation to investigate the heat capacity of polymers at different frequencies. Other types of periodic functions were successfully used to determine the heat capacity of polymers at different frequencies (Kamasa et al, 2002). 

The question arises whether these series of Q branches could be interpreted on a slightly anharmonic potential function rather than on a cosine function type of pseudo-rotation. For a low barrier ( 7.5 kJ mol ) the contribution to the entropy at 298 K of a hindered pseudo-rotator differs from that of a harmonic oscillator of the same frequency by 6 J K mol , which is certainly measurable. But for a high barrier ( 14.6 kJ mol ) the difference between the entropy contributions is 0.6 J K mol at 298 K and only 1.2 JK mol at 398 K. Therefore only for a low barrier can the two possibilities be distinguished by thermodynamic measurements. For Y-butyrolactone and ethylene carbonate the barriers are calculated to be 46 kJ mol and 75 kJ mol , respectively, which would negate the effect of pseudo-rotation at reasonable temperatures. Thus, in molecules with a sufficiently high barrier the motion can be treated as an ordinary vibration in which the puckering oscillates about a stable configuration. 

If the barrier is very high, 3 00, the internal motion of the methyl group corresponds to simple harmonic torsional oscillation in each well. The cosine function in Eq. (79) may be expanded, giving (a) = (9 3/4)a, and the form of Eq. (80) is like that for a simple harmonic oscillator. Solution gives for the energy... 

The problem is heated in elementary physical chemishy books (e.g., Atkins, 1998) and leads to a set of eigenvalues (energies) and eigenfunctions (wave functions) as depicted in Fig. 6-1. It is solved by much the same methods as the hamionic oscillator in Chapter 4, and the solutions are sine, cosine, and exponential solutions just as those of the harmonic oscillator are. This gives the wave function in Fig. 6-1 its sinusoidal fonn. 

Periodicity in space means that it repeats at regular intervals, known as the wavelength, A. Periodicity in time means that it moves past a fixed point at a steady rate characterised by the period r, which counts the crests passing per unit time. By definition, the velocity v = A/r. It is custom to use the reciprocals of wavelength 1/X — (k/2-ir) or 9, known as the wavenumber (k = wave vector) and 1/t — v, the frequency, or angular frequency u = 2itv. Since a sine or cosine (harmonic) wave repeats at intervals of 2n, it can be described in terms of the function... 

Instead of using harmonic functions the torsional potential is more often described with a small cosine expansion in ... 

The factor r1 enters because the Cartesian spherical harmonics clmp are defined in terms of the direction cosines in a Cartesian coordinate system. The expressions for clmp are listed in appendix D. As an example, the c2mp functions have the form 3z2 — 1, xz, yz, (x2 — y2)/2 and xy, where x, y and z are the direction cosines of the radial vector from the origin to a point in space. 

D.1 Real Spherical Harmonic Functions and Associated Normalization Constants (x, y, and z are Direction Cosines)... 

Working independently, A.Abakonovicz in 1878 and C.V. Boys in 1882 devised the integraph, an instrument that drew the integral of an arbitrary function when the latter was plotted on a suitable scale on paper. A device for finding trigonometric functions (sines and cosines), known as harmonic analyzer was devised in 1876 by Lord Kelvin. 

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