Approximation third-order

The above theory is usually called the generalized linear response theory because the linear optical absorption initiates from the nonstationary states prepared by the pumping process [85-87]. This method is valid when pumping pulse and probing pulse do not overlap. When they overlap, third-order or X 3 (co) should be used. In other words, Eq. (6.4) should be solved perturbatively to the third-order approximation. From Eqs. (6.19)-(6.22) we can see that in the time-resolved spectra described by x"( ), the dynamics information of the system is contained in p(Af), which can be obtained by solving the reduced Liouville equations. Application of Eq. (6.19) to stimulated emission monitoring vibrational relaxation is given in Appendix III. 

J. Schirmer, A.B. Trofimov, G. Stelter, A non-Dyson third-order approximation scheme for the electron propagator, J. Chem. Phys. 109 (1998) 4734. 

As one of the alternative methods for treating ultrafast pump-probe experiments [50], the third-order susceptibility method is often used. A main feature of this method is that it can treat the case of the overlapping region between the pump and the probe lasers, but one has to carry out the perturbation calculations to the third-order approximations. 

A third-order approximation to CCSDT (in which I, is counted as a zeroth-order quantity)... 

In this chapter we collect and present, without derivation, in explicit, Anal form the relevant phase-integral quantities and their partial derivatives with respect to E and Z expressed in terms of complete elliptic integrals for the first, third and fifth order of the phase-integral approximation. For the first- and third-order approximations some of the formulas were first derived by means of analytical calculations, and then all formulas were obtained by means of a computer program. In practical calculations it is most convenient to work with real quantities. For the phase-integral quantities associated with the r -equation we therefore give different formulas for the sub-barrier and the super-barrier cases. As in Chapter 6 we use instead of L2 , L2n, K2n the notations LAn+1 >, L( 2n+1 KAn+l). 

For the third-order approximation we introduce the universal functions... 

Our starting point is the microscopic Hamiltonian for the chemical reacting system in solution. After the transformation of the coordinates, it can be expressed within the third order approximation in the neighborhood of any equilibrium geometry as follows 3,14... 

The functions G u), S u), and C u) have been tabulated 1, 8) for u up to 25.00. Use of these tables provides an easy method for evaluating In s/sj and leaves little to be desired with respect to the rate of convergence. For the pair of isotopic molecules HD/Ho, for example, the first, second, and third order approximations of Equation 3 compute In s/sj to —0.15, 0.088, and —0.010%, respectively, at room temperature. Vojta (25) later extended this expansion to an infinite series... 

This expression indicates that the inversion of (0 can be carried out by a differentiation process of infinite order, an approach that is not feasible. However, an approximation of order k can be obtained. The first-, second-and third-order approximations are given by (5)... 

Derive an expression for the first-, second-, and third-order approximation of the relaxation spectrum for a Maxwell element in shear. 

Higher-order integration schemes in time and space can be used to improve on the accuracy of the calculations, for example, using the higher order convection schemes presented above. For reactor simulations, any appropriate second- and third-order approximations are recommended. 

The series is a linear model with no accounting for curvature. A plot of this surface is shown in Fig. A.2. Notice that the series is a quite good approximation to the true response surface near the evaluation point and up to around x, y = 0.8, 0.8. At this point, the function begins to fail because the true function begins to grow exponentially whereas the linear approximation remains linear in its growth. Figure A.2 presents a third-order approximation... 

Equations 8.57 and 8.58 in Appendix 8.2 use a third-order approximation to estimate a backward first derivative. Use the methodology of Example 8.6 to estimate the order of convergence for this derivative. 

A more accurate estimate of the first derivative is obtained from a third-order approximation. Fit a cubic through the points T(R), T(R — Ar), T R —2 Ar), and T(R — 3 Ar) and differentiate to estimate the slope at point R. The derivative approximation is... 

In Fig. 1 the results of Markovian approximations are compared with the results of exact andytical calculation. One can see that the use of Markovian approximations can be rather efficient in the case of a retarding nei boring-groups effect and in the case of small accelerations. The most optimal is evidently a second-order Markovian approximation, as the accuracy of a first-order approximation is not sufficient, while the accuracy of a third-order approximation is the same as the second order, but twice as many equations are required. The Markovian approximatirms are not very useful for greater accelerations, as they permit only the calMation of the probabilities of very diort sequences with sufficient accuracy. 

The perturbation series can be truncated to various orders and one indicates the accuracy of MP methods applied within the Restricted Hartee-Fock (RHF) scheme by referring to the highest-orderterm allowed in the energy expansion. Thus a truncation to second-order corresponds to an MP2 approach, to third-order to an MP3 approach and so forth [27]. MP theory may also be used in the spin-Unrestricted Hartree-Fock (UHF) model. In this case, second- and third-order approximations of MP theory are indicated as UMP2andUMP3. 

Third-order approximations for the derivative of first order ... 

Refs. [44, 46, 49, 51], the second- and third-order approximate coupled cluster (CC2 and CC3) calculations of Ref. [50], the equation of motion coupled cluster with single, double and perturbative triple excitations calculations of Ref. [45], the symmetry adapted cluster configuration interaction calculations of Ref. [48] and the time-dependent density functional theory calculations using the B3LYP functional of Ref. [39]. 

An example of the third-order approximations of Tschoegl, employing the third derivative of G(t), is... 

An example of a third-order approximation from the method of Tschoegl is, for m positive,... 

Finally, Roesler25 and Roesler and Twyman2 have outlined more complicated numerical methods, and Hlavdcek and Kotrba have presented an iterative procedure starting from a third-order approximation of Schwarzl and Staverman. ... 

These results may be written in terms of the ratios of other effective cross sections as has usually been done for monatomic species (Maitland et al. 1987) and has recently been performed for polyatomic molecules (Heck et al. 1994). Generally and depart from unity by no more than 2%. In the particular case of monatomic gases explicit results for the transport coefficients up to the third-order approximation are available (Maitland et al. 1987). 

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