Taylor series oscillation

Aris and Amundson (1958) solved the coupled, time-dependent material and energy balances, linearizing the equations about the operating point by a Taylor series expansion. This made the solution possible by the method of characteristic equations. The solution yielded two equations, one the slope condition and the other recognized by Gilles and Hofmann (1961) as the condition that sets the limits to avoid rate oscillation. This is called the... 

In the lowest approximation the molecular vibrations may be described as those of a harmonic oscillator. These can be derived by expanding the energy as a function of the nuclear coordinates in a Taylor series around the equilibrium geometry. For a diatomic molecule this is the intemuclear distance R. 

Hyper)polarizabilities are defined as the coefficients in the Taylor series expansion of the dipole moment - or the energy - in the presence of static and/or oscillating electric fields ... 

We have not failed to recognize that appropriately designed (6,0) carbon and C/B/N nanotubes may display considerably enhanced nonlinear optical activity. This term refers to the response of the dipole moment of a molecule (or the polarization of bulk material) to the oscillating electric field of electromagnetic radiation.82 85 The component of the dipole moment along an axis i in the presence of an electric field e can be represented by a Taylor series ... 

In section 6.8.2 we described and solved the Schrodinger equation for a harmonic oscillator, equation (6.178). The potential energy was expressed in terms of a vibrational coordinate q which was equal to R - Re, Re being the equilibrium bond length. The dependence of the electric dipole moment on the internuclear distance may be expressed as a Taylor series,... 

When the region of expansion is properly chosen, the error e x) oscillates on both sides of the abscissa. Thus, choosing an orthogonal polynomial P x) is equivalent to demanding that the error of the approximation be zero at a finite set of points. This is in contrast to the Taylor series for which the error is zero only at one point. In this sense the orthogonal expansion is an interpolating approximation. 

Molecular electric properties give the response of a molecule to the presence of an applied field E. Dynamic properties are defined for time-oscillating fields, whereas static properties are obtained if the electric field is time-independent. The electronic contribution to the response properties can be calculated using finite field calculations , which are based upon the expansion of the energy in a Taylor series in powers of the field strength. If the molecular properties are defined from Taylor series of the dipole moment /x, the linear response is given by the polarizability a, and the nonlinear terms of the series are given by the nth-order hyperpolarizabilities ()6 and y). 

Paralleling the harmonic oscillator expansion of the potential function of a mechanical system, we next approximate the equilibrium entropy function 5(Tp A) by its quadratic order Taylor series expansion about A (the point at which 5 has its maximum). That is, we assume... 

To obtain the perturbative solutions for the (J = 0) Hamiltonian we take as our starting point the vibrational Hamiltonian of Eq. (2), where the coordinate-dependent terms G, V, and V are expanded in a Taylor series in the dimensionless normal coordinates. The coordinates Q, and corresponding momenta P, are reexpressed in terms of harmonic oscillator raising and lowering operators aj and ah where... 

This is the heat capacity of a one-dimensional oscillator according to Einstein. The heat capacity deviates at low temperatures. It is not possible to expand into a Taylor series around T 0. In other words, the function has a pole at zero, which emerges as an essential singular point. A more accurate formula is due to Debye, n... 

Show that if one expands U R) in Fig. 4.5 in a Taylor series about R = and neglects terms containing (R - Ref and hi er powers (these terms are small for R near / <,) then one obtains a harmonic-oscillator potential with k = d U/dR n=R. ... 

Question. The next term in the Taylor series for the potential energy is — —0 . Treat this as a perturbation to the harmonic oscillator wavefunction and compute the first-order cmrection to the energy. 

To express the vibronic coupling strength quantitatively, we require an analytical expression for the coupling operator. It is standard practice to assume this operator to be a linear function of the vibrational coordinate. In other words, the analytical form of this operator is approximated by the first nonvanishing term in a Taylor series expansion. It may appear that this approximation is consistent with the harmonic oscillator approximation, which also amounts to a Taylor series expansion truncated after the first nonvanishing Q-dependent term. However, this analogy does not hold, as follows directly from the form of the (two-state) vibronic Hamiltonian matrix, namely,... 

For small-amplitude oscillations about the synchronous molecule, the motion is harmonic. Expanding the right-hand side of Equation 15.29 in a Taylor series about z = 0, gives d z/dt - (W (zon) - Wizos))z/(MD) = 0 where W ia) = dW/dz evaluated at z = a. The angular oscillation frequency for small-amplitude axial oscillations is therefore... 

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. 

There is a fair match between the harmonic oscillator potential and the real molecular potential energy curve we saw in Chapter 5 at low potential energy (this is our requirement that R should be near R in the Taylor series). The match becomes worse as we go up in energy and the potential becomes anharmonic. At low vibrational energy, the harmonic oscillator is a good approximation to most molecular vibrations. 

From the dimensional analysis, the vibration frequency is proportional to the square root of the bond stiffness. Equaling the vibration energy of an ideal harmonic oscillator to the corresponding term in the Taylor series of the interatomic potential around its equilibrium, one can find. 

At the molecular level, the application of an external electric field F of high intensity induces changes in the dipole moment of the individual molecules, which can be expressed by the Taylor series in the electric field amplitudes. Thus, in the presence of electric fields containing static and dynamic (oscillating) contributions, the z-component of the dipole moment reads ... 

Tacoma Narrows bridge % tangent 16 Taylor s series 32-34 tests of series convergence 35-36 thermodynamics applications 56-57, 81 first law 38-39 Jacobian notation 160-161 systems of constant composition 38 three-dimensional harmonic oscillator 125-128... 

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