Frequency external perturbation

Applications The potential use of 2D correlation spectroscopy is very wide [1007], Most multidimensional techniques arise from the correlation of frequency domains in the presence of external perturbations, as in NMR. For applications of multidimensional NMR spectroscopy and NMR diffusion measurements, see Sections 5.4.1 and 5.4.1.1. 

There are several possible ways of deriving the equations for TDDFT. The most natural way departs from density-functional perturbation theory as outlined above. Initially it is assumed that an external perturbation is applied, which oscillates at a frequency co. The linear response of the system is then computed, which will be oscillating with the same imposed frequency co. In contrast with the standard static formulation of DFPT, there will be special frequencies cov for which the solutions of the perturbation theory equations will persist even when the external field vanishes. These particular solutions for orbitals and frequencies describe excited electronic states and energies with very good accuracy. 

This equation describes the electronic reaction to the oscillating external perturbation. In principle, it has a solution for any frequency co. One special class of solutions is, however, of particular interest. If at a frequency co, there is a solution i//(l that satisfies the above equation also at zero perturbation strength (hp — 0), then the unperturbed system will also be stable in the state described by this particular solution + hf/ K In such circumstances the term X is no longer bound to the perturbation strength. Instead, it can take any value, as long as it is sufficiently small to remain in the linear response region of the system. Such a new state, however, is nothing but an excited state. 

Thus, the goal is to find frequencies co and corresponding orbitals y/(m) for which the DFPT equation without the external perturbation Hamiltonian ... 

Here, we consider the response theory which has been successful for many investigations. For simplicity, we consider the A-electron system to be initially in the ground state, which is subjected to an external TD electric field. The density change Sp(r, t) induced by an external perturbation 6vext(r, t) can be written in the response theory framework, in terms of quantities in frequency domain as... 

Equation (4.65) provides the maximal change of R achievable by an external perturbation, since it does not involve any averaging (smoothing) of G(m) incurred by the width of Ffa>) the modified/ can even vanish, if the shifted frequency is beyond the cutoff frequency of the coupling, where G(m) = 0 (Figure 4.6d). 

In this paper I shall discuss, in a general way, some basic dynamical properties of biochemical reaction schemes that are subjected to an external perturbation that oscillates in time. The first part of this paper will deal with the ways in which a weak external oscillating stimulus is able to alter the concentrations of metabolites in a biochemical system. This part will, in particular, consider what happens when the reaction system already oscillates in a limit cycle mode due to non-linearities in its reaction kinetics. It will be shown that such an autonomous biochemical oscillator may exhibit an enhanced sensitivity to a narrow range of externally applied frequencies. 

For nonlinear reaction schemes, maintained far from chemical equilibrium, a variety of more interesting interactions are possible (2) These include threshold phenomena in which a small transitory external perturbation may induce a permanent change in the steady state concentrations of metabolites. In such a case the magnitude of the change may be independent of that of the stimulus beyond a certain threshold value. Nonlinear reactions may also display a form of resonance when the perturbation oscillates in time. This can be inferred by examining the stability properties of linearized forms of nonlinear reaction schemes (2, 3) A complete description of this form of interaction, however, usually requires numerical computations ( ). I shall now describe the results of some computations in which a nonlinear reaction scheme that is capable of autonomous oscillations was perturbed by an oscillating stimulus applied over a range of frequencies ( ) ... 

Numerical integration of equations (2) and (3) with initial values for X,Y on the limit cycle and with one of the rate constants oscillating as in equation (4) or (5) may result in a transition of the X,Y trajectory across the separatrix towards the stationary state. The occurrence of a transition is dependent on the parameters g, u) and 0. For extremely small amplitude perturbations (g - -0), the trajectory continues to oscillate close to the limit cycle. As g is increased, however, transitions may occur. The time taken for a transition is then primarily a function of the frequency of the perturbation. The time from the onset of the oscillating perturbation to the time at which the trajectory attains the lower steady state (At) is plotted in Figure 3 as a function of with all other parameters held constant. The arrow marks the minimum value for At which occurs when the frequency of the external perturbation exactly equals that of the unperturbed limit cycle itself. The second minimum occurs at the first harmonic of the limit cycle. Qualitatively similar results are obtained when numerical integration is carried out with differing values for g and 0. 

Using the ability to displace the trajectory of a limit cycle across a fixed boundary (the separatrix) as a measure of sensitivity to an external perturbation, it can therefore be seen that nonlinear oscillating reaction systems are able to respond most sensitively to a range of externally applied frequencies close to their endogenous frequency. 

Measurement of frequency response is important for several reasons. The response of an acoustic-wave device to an external perturbation, for example in a chemical sensing application, can be better understood if the device s frequency response is known in advance. Measurement of the frequency response is also important if the most stable and accurate measurement system is to be designed for a particular device. Finally, the change in the frequency response of a device that results from some significant modification of its surface environment, such as the deposition of a polymer layer or immersion of the surface in a liquid, can... 

Let us return to the problem of solving the response of the quantum mechanical system to an external electric field. The zeroth-order wave function of the quantum mechanical system is obtained by use of any of the standard approximate methods in quantum chemistry and the coupling to the field is described by the electtic dipole operator. There exist a number of ways to determine the response functions, some of which differ in formulation only, whereas others will be inherently different. We will give a short review of the characteristics of tire most common formulations used for the calculation of molecular polarizabilities and hyperpolarizabilities. The survey begins with the assumption that the external perturbing fields arc non-oscillatory, in which case we may determine molecular properties at zero frequencies, and then continues with the general situation of time-dependent fields and dynamic properties. 

Frequency-dependent higher-order properties can now be obtained as derivatives of the real part of the time-average of the quasi-energy W j- with respect to the field strengths of the external perturbations. To derive computational efficient expressions for the derivatives of the coupled cluster quasi-energy, which obey the 2n-(-1- and 2n-(-2-rules of variational perturbation theory [44, 45, 93], the (quasi-) energy is combined with the cluster equations to a Lagrangian ... 

Analytic response theory, which represents a particular formulation of time-dependent perturbation theory, has constituted a core technology in much of the this development. Response functions provide a universal representation of the response of a system to perturbations, and are applicable to all computational models, density-functional as well as wave-function models, and to all kinds of perturbations, dynamic as well as static, internal as well as external perturbations. The analytical character of the theory with properties evaluated from analytically derived expressions at finite frequencies, makes it applicable for a large range of experimental conditions. The theory is also model transferable in that, once the computational model has been defined, all properties are obtained on an equal footing, without further approximations. 

As an example of sensor applications, for an optical fiber with a core diameter 20 pm, ciad = 1.45, and core = 1.46, the field of the modal power forLP/ modes is shown in Fig. 28a,b for the modal orders I = 6 and /= 13, at an optical wavelength X = 0.75 pm. In the presence of external perturbations applied to the fiber, resulted in changing the cladding index to ciad = 1.455, the modal power redistribution of Z = 6 and / = 13 modes is shown in Fig. 28c, d. This theoretical analysis is based on the use of a single frequency (laser) light source. 

A noise that has a clearly distinct origin from noise discussed in previous sections is the electric noise that originates in modulation of ion transport by fluctuations in system conductance. These temporal fluctuations can be measured, at least in principle, even in systems at equilibrium. Such a measurement was conducted by Voss and Clark in continuous metal films (44). The idea of the Voss and Clark experiment was to measure low-frequency fluctuations of the mean-square Johnson noise of the object. In accordance with the Nyquist formula, fluctuations in the system conductance result in fluctuations in the spectral density of its equilibrium noise. Measurement of these fluctuations (that is, measurement of the noise of noise) yields information on conductance fluctuations of the system without the application of any external perturbations. The samples used in these experiments require rather large amplitude conductance fluctuations to be distinguished from Johnson noise fluctuations because of the intrinsic limitation of statistics. Voss and... 

As shown in previous sections of this chapter, when an external perturbation is applied to the polymer film (such as irradiation), the ATR guided modes shift their angular positions and the reflectivity is modulated (Fig. 31b). These angular shifts are very small in the case of electrooptic experiments they correspond to refractive index variations of the order of 10 . One has then to modulate the measuring electric field at a low frequency Q( = cos fit) and to detect the modulated signal with lock-in amplifiers. The lock-in signals detected at the modulation frequency and its second harmonic give, respectively, the linear (or Pockels) and the quadratic (or Kerr) electrooptic effects. The amplitude of the modulation of the thickness and the refractive indices is evaluated by a computer fit, and allows the determination of Pockels (r) and Kerr (s) coefficients (Eqs. 28) ... 

The time of plasma reaction to external pertnrbation (l/wp) corresponds to a time required by a thermal electron (velocity. JTTjm) to travel the distance r, required to shield the external perturbation. The plasma frequency depends only on plasma density and numerically can be calculated as... 

Experimentation with step-scan interferometry in electrochemistry began in the early 1990s (cf Ref. [23]), and interest has grown steadily ]24, 29-31, 51-54]. Step-scan FTIR spectroscopy provides a means to investigate time- and frequency-dependent processes. Measurements are hmited to reversible systems. However, a great deal of insight can be gained into the molecular transformations that accompany the external perturbation [181-183]. 

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