Poles first order

The —(/i /2p)W (Rx) matrix does not have poles at conical intersection geometries [as opposed to W (R )] and furthermore it only appears as an additive term to the diabatic energy matrix (q ) and does not increase the computational effort for the solution of Eq. (55). Since the neglected gradient term is expected to be small, it can be reintroduced as a first-order perturbation afterward, if desired. 

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. 

It is cumbersome to write the partial fraction with complex numbers. With complex conjugate poles, we commonly combine the two first order terms into a second order term. With notations that we will introduce formally in Chapter 3, we can write the second order term as... 

The real part of a complex pole in (3-19) is -Zjx, meaning that the exponential function forcing the oscillation to decay to zero is e- x as in Eq. (3-23). If we draw an analogy to a first order transfer function, the time constant of an underdamped second order function is x/t,. Thus to settle within 5% of the final value, we can choose the settling time as 1... 

This is a form that serves many purposes. The term in the denominator introduces a negative pole in the left-hand plane, and thus probable dynamic effects to the characteristic polynomial of a problem. The numerator introduces a positive zero in the right-hand plane, which is needed to make a problem to become unstable. (This point will become clear when we cover Chapter 7.) Finally, the approximation is more accurate than a first order Taylor series expansion.1... 

Let say we have a high order transfer function that has been factored into partial fractions. If there is a large enough difference in the time constants of individual terms, we may try to throw away the small time scale terms and retain the ones with dominant poles (large time constants). This is our reduced-order model approximation. From Fig. E3.3, we also need to add a time delay in this approximation. The extreme of this idea is to use a first order with dead time function. It obviously cannot do an adequate job in many circumstances. Nevertheless, this simple... 

A second order function with dead time generally provides a better estimate, and this is how we may make a quick approximation. Suppose we have an -th order process which is broken down into n first-order processes in series with time constants Xi, t2,..., t . If we can identify, say, two dominant time constants (or poles) xi and t2, we can approximate the process as... 

The choice of the time constant and dead time is meant as an illustration. The fit will not be particularly good in this example because there is no one single dominant pole in the fifth order function with a pole repeated five times. A first order with dead time function will never provide a perfect fit. 

With only open-loop poles, examples (a) to (c) can only represent systems with a proportional controller. In case (a), the system contains a first orders process, and in (b) and (c) are overdamped and critically damped second order processes. 

Since it was desired not to lose the advantage already gained from using a first-order lag on Q, the scheme shown in Figure 6 was actually used for the pole placement tests. Figure 6 differs from Figure 5 only in that the Q lag of Figure 2 is included. 

Time-dependent response theory concerns the response of a system initially in a stationary state, generally taken to be the ground state, to a perturbation turned on slowly, beginning some time in the distant past. The assumption that the perturbation is turned on slowly, i.e. the adiabatic approximation, enables us to consider the perturbation to be of first order. In TD-DFT the density response dp, i.e. the density change which results from the perturbation dveff, enables direct determination of the excitation energies as the poles of the response function dP (the linear response of the KS density matrix in the basis of the unperturbed molecular orbitals) without formally having to calculate a(co). 

This particular type of transfer function is called a first-order lag. It tells us how the input affects the output C/, both dynamically and at steadystate. The form of the transfer function (polynomial of degree one in the denominator, i.e., one pole), and the numerical values of the parameters (steadystate gain and time constant) give a complete picture of the system in a very compact and usable form. The transfer function is property of the system only and is applicable for any input. 

Assume holdups and flow rates are constant. The reaction is an irreversible, first-order consumption of reactant A, The system is isothermal. Solve for the transfer function relating and C. What are the eros and poles of the transfer function What is the steadystate gain ... 

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. 

Exeitation energies are readily obtained as poles of a polarization propagator [37-40], whereas the transition moments are known as first-order non-adiabatie... 

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, First-order quasi-phase matched LiNbOs wave-guide periodically poled by applying an external-field for efficient blue second-harmonic generation. Applied Physics Letters 62(5), 435-436 (1993). 

The effect of substituents in the 5-, 6-, 7-, or 8-position of quinazo-line was summed up in the earlier review.38 In general, (—1) substituents promote hydration of the 3,4-bond by lowering the electron density on C-4. Later it was found that a (—1) substituent in the 2-position had the opposite effect. The addition of the negatively charged pole of a water molecule to C-4 is favored by the polarization of the 3,4-bond in this sense —C4 =N—4V But a (—1) group in the 2-position can oppose this polarization. In a study of twenty 2-substituted quinazolines,23 it was found that hydration was helped by (+1) substituents, not greatly affected by (+M), and much diminished by (—I) substituents. The pH rate profile (first-order kinetics) for the hydration of 2-aminoquinazoline, measured from pH 2 to 10, was parabolic,23 typical of molecules that undergo reverse covalent hydration.315... 

The closed contour C goes around the point a in the complex-number plane. The essential point is that the quantity (z - a) in the denominator creates a mathematical pole of the "first order" [first power in (z - a)] at position (z = a). In application of the theorem to the summation of free energies g(coj), the derivative d In[D(cn)]/dcn automatically creates first-order poles... 

We shall first briefly describe the phase-integral approximation referred to in item (i). Then we collect connection formulas pertaining to a single transition point [first-order zero or first-order pole of Q2(z) and to a real potential barrier, which can be derived by... 

When the first-order approximation is used, it is often convenient to choose the constant lower limit of integration in the definition (4.4) of w(z) to be a zero or a first-order pole of Q2(z). This is, however, in general not possible when a higher-order approximation is used, since the integral in (4.4) would then in general be divergent. If the lower limit of integration in (4.4) is an odd-order zero or an odd-order pole of Q2(z), it is possible and convenient to replace the... 

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