Response - first order

If, as is eommon, the atomie orbital bases used to earry out the MCSCF energy optimization are not explieitly dependent on the external field, the third term also vanishes beeause (9xv/3)i)o = 0. Thus for the MCSCF ease, the first-order response is given as the average value of the perturbation over the wavefunetion with X=0 ... 

It ean be seen in Figure 5.17 that the pole at the origin and the zero at. v = —1 dominate the response. With the eomplex loei, ( = 0.7 gives K a value of 15. ITowever, this value of K oeeurs at —0.74 on the dominant real loeus. The time response shown in Figure 5.20 shows the dominant first-order response with the oseillatory seeond-order response superimposed. The settling time is 3.9 seeonds, whieh is outside of the speeifieation. 

Although a calculation of the wave function response can be avoided for the first derivative, it is necessary for second (and higher) derivatives. Eq. (10.29) gives directly an equation for determining the (first-order) response, which is structurally the same as eq. (10.36). For an HF wave function, an equation of the change in the MO coefficients may also be formulated from the Hartree-Fock equation, eq. (3.50). 

The linear polarizability, a, describes the first-order response of the dipole moment with respect to external electric fields. The polarizability of a solute can be related to the dielectric constant of the solution through Debye s equation and molar refractivity through the Clausius-Mosotti equation [1], Together with the dipole moment, a dominates the intermolecular forces such as the van der Waals interactions, while its variations upon vibration determine the Raman activities. Although a corresponds to the linear response of the dipole moment, it is the first quantity of interest in nonlinear optics (NLO) and particularly for the deduction of stracture-property relationships and for the design of new... 

As a consequence, field methods, which consist of computing the energy or dipole moment of the system for external electric field of different amplitudes and then evaluating their first, second derivatives with respect to the field amplitude numerically, cannot be applied. Similarly, procedures such as the coupled-perturbed Hartree-Fock (CPHF) or time-dependent Hartree-Fock (TDHF) approaches which determine the first-order response of the density matrix with respect to the perturbation cannot be applied due to the breakdown of periodicity. 

To find the power series expansion of Eq. (30) in ub, ojc, u>d we can thus replace the first-order responses of the cluster amplitudes and Lagrangian multipliers and the second-order responses of the cluster amplitudes by the expansions in Eqs. (37), (39) and (44) and express OJA as —ojb ojc — ojd- However, doing so starting from Eq. (30) leads to expressions which involve an unneccessary large number of second-order Cauchy vectors C m,n). To keep the number of second-order... 

The polarizability expresses the capacity of a system to be deformed under the action of electric field it is the first-order response. The hyperpolarizabilities govern the non linear processes which appear with the strong fields. These properties of materials perturb the propagation of the light crossing them thus some new phenomenons (like second harmonic and sum frequency generation) appear, which present a growing interest in instrumentation with the lasers development. The necessity of prediction of these observables requires our attention. 

First-Order Response to an Input Step-Change Disturbance... 

However, this simplified reasoning does not meet the real situation of the analysis as a whole, where both the preparation of the sample and the renewal of the titration medium cause a certain and often considerable loss of time therefore, the following set of more or less well defined symbols are used (see Fig. 5.2. a and b for a signal S of first-order response) ... 

Figure 2.10. Illustration of a first order response (2-51) normalized by MKp. The curve is plotted with...

Most galvanostatic transients followed a first order response with reasonable accuracy, in agreement with (25) ... 

Because of the separation into a time-independent unperturbed wavefunction and a time-dependent perturbation correction, the time derivative on the right-hand side of the time-dependent Kohn-Sham equation will act only on the response orbitals. From this perturbed wavefunction the first-order response density follows as ... 

Figure 4.11 A solid-state electrode showing a first-order response. An electrode designed to measure the activity of silver ions uses a crystalline membrane of silver sulphide. An equilibrium between the mobile silver ions of the membrane and the silver ions in the solutions results in the development of a potential difference across the membrane.

The simplest solid-state membranes are designed to measure test ions, which are also the mobile ions of the crystal (first-order response) and are usually single-substance crystals (Figure 4.11). Alternatively, the test substance may be involved in one or two chemical reactions on the surface of the electrode which alter the activity of the mobile ion in the membrane (Figures 4.12 and 4.13). Such membranes, which are often mixtures of substances, are said to show second- and third-order responses. While only a limited number of ions can gain access to a particular membrane, a greater number of substances will be able to react at the surface of the membrane. As a result, the selectivity of electrodes showing second- and third-order responses is reduced. 

The calculation of frequency-dependent linear-response properties may be an expensive task, since first-order response equations have to be solved for each considered frequency [1]. The cost may be reduced by introducing the Cauchy expansion in even powers of the frequency for the linear-response function [2], The expansion coefficients, or Cauchy moments [3], are frequency independent and need to be calculated only once for a given property. The Cauchy expansion is valid only for the frequencies below the first pole of the linear-response function. 

In the second step, the solution vector Xb(w), corresponding to the first-order response of the wave function, is contracted with property gradient to form the... 

ALTERNATING CURRENT PERTURBATION. FIRST-ORDER RESPONSE 2.3.1 The impedance concept... 

Section 2.3 was exclusively dedicated to the first-order response to a sinusoidal perturbation that is filtered from the total response by tuning the detection device to the fundamental frequency, oj. Due to the non-... 

Analytical gradients and Hessians are available for CASSCF, and it is expected that this technology will be extended to the MR-CI and MP2 methods soon. Further, by virtue of the multireference approach, a balanced description of ground and excited states is achieved. Unfortunately, unlike black boxes such as first-order response methods (e.g., time-dependent DFT), CAS-based methods require considerable skill and experience to use effectively. In the last section of this chapter, we will present some case studies that serve to illustrate the main conceptual issues related to computation of excited state potential surfaces. The reader who is contemplating performing computations is urged to study some of the cited papers to appreciate the practical issues. 

The non-stiff, low-dimensional model (6.63) is ideally suited for control purposes. We used it to design an input-output linearizing controller with integral action that manipulates QH and enforces a first-order response in the Tr dynamics, namely... 

The term Cb + 71,1 dCB/dt is statically equivalent to Cb, and corresponds to requesting a first-order response in Cb when using a standard input-output linearizing controller. However, such a controller would lead to closed-loop instability, and the output requires a statically equivalent addition that would allow one to overcome this limitation. 

With the outputs y and M, using the reduced-order model (7.38), a multi-variable input-output linearizing controller with integral action (Daoutidis and Kravaris 1994) was designed for the product purity and reactor holdup, requesting a decoupled first-order response ... 

A straight line on such a plot indicates a first-order response. This is a particularly sensitive way to plot the results, since differences of comparable quantities are being taken. The relative error Is largest for turndown to 80%. Within the accuracy used to satisfy the boundary condition at z - L, the three curves cannot be distinguished from the result for turndown to 50%. The apparent first-order time... 

In the identification of W 3) we have used the first-order response equations [Eq. (64)] to eliminate A(2). From Eqs. (68) and (69) we see that the first-order correction to the wave function determines the energy through third order. In general the nth-order response of the wave function determines the energy through order 2n + 1. 

The first set of equations determines the first-order response of the cluster amplitudes. Once this has been determined we may construct the effective operator H l and determine the second-order amplitude response using the same set of equations (with HU) replaced by These equations may be solved using an iterative technique. The key steps would be the calculation of... 

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