Propagator, second-order

In words, equation (Al.6.89) is saying that the second-order wavefunction is obtained by propagating the initial wavefunction on the ground-state surface until time t", at which time it is excited up to the excited state, upon which it evolves until it is returned to the ground state at time t, where it propagates until time t. NRT stands for non-resonant tenn it is obtained by and cOj -f-> -cOg, and its physical interpretation is... 

In this chapter the regimes of mechanical response nonlinear elastic compression stress tensors the Hugoniot elastic limit elastic-plastic deformation hydrodynamic flow phase transformation release waves other mechanical aspects of shock propagation first-order and second-order behaviors. 

The poles con espond to excitation energies, and the residues (numerator at the poles) to transition moments between the reference and excited states. In the limit where cj —> 0 (i.e. where the perturbation is time independent), the propagator is identical to the second-order perturbation formula for a constant electric field (eq. (10.57)), i.e. the ((r r))Q propagator determines the static polarizability. 

This expression suggests a rate-controlling step in which RM reacts with an intermediate. If so, [Int] °c [RM] /2. To be consistent with this, the initiation step should be first-order in [RM] and the termination step second-order in [Int]. Since O2 is not involved in the one propagation step deduced, it must appear in the other, because it is consumed in the overall stoichiometry. On the other hand, given that one RM is consumed by reaction with the intermediate, another cannot be introduced in the second propagation step, since the stoichiometry [Eq. (8-3)] would disallow that. Further, we know that the initiation and propagation steps are not the reverse of one another, since the system is not well-behaved. From this logic we write this skeleton ... 

The contributions of the second order terms in for the splitting in ESR is usually neglected since they are very small, and in feet they correspond to the NMR lines detected in some ESR experiments (5). However, the analysis of the second order expressions is important since it allows for the calculation of the indirect nuclear spin-spin couplings in NMR spectroscoi. These spin-spin couplings are usually calcdated via a closed shell polarization propagator (138-140), so that, the approach described here would allow for the same calculations to be performed within the electron Hopagator theory for open shell systems. 

The second procedure, several aspects of which are reviewed in this paper, consists of directly computing the asymptotic value by employing newly-developed polymeric techniques which take advantage of the one-dimensional periodicity of these systems. Since the polarizability is either the linear response of the dipole moment to the field or the negative of the second-order term in the perturbation expansion of the energy as a power series in the field, several schemes can be proposed for its evaluation. Section 3 points out that several of these schemes are inconsistent with band theory summarized in Section 2. In Section 4, we present the main points of the polymeric polarization propagator approaches we have developed, and in Section 5, we describe some of their characteristics in applications to prototype systems. 

Linear response function approaches were introduced into the chemistry literature about thirty years ago Ref. [1,2]. At that time they were referred to as Green functions or propagator approaches. Soon after the introduction it became apparent that they offered a viable and attractive alternative to the state specific approaches for obtaining molecular properties as excitation energies, transition moments and second order molecular properties. 

The propagation rate is assumed to be second order with respect to the end-group concentration,. p = ka. The boundary conditions are a specified inlet concentration, zero flux at the wall, and symmetry at the centerline. 

Eqs. (26) and (27) apply irrespective of the nature of the initiation process it is required merely that the propagation and termination processes be of the second order. They emphasize the very general inverse dependence of the kinetic chain length on the radical concentration and therefore on the rate of polymerization. The kinetic chain length may be calculated from the ratios k /kt as given in Table XI and the rate of polymerization. Thus, for pure styrene at 60°C... 

Hirose et al. [26] proposed a homodyne scheme to achieve the background-free detection of the fourth-order field. With pump irradiation in a transient grating configuration, the fourth-order field propagates in a direction different from that of the second-order field because of different phase match conditions. The fourth-order field is homodyned to make ffourth(td. 2 D) and spatially filtered from the second-order response hecond td, 2 D). 

In order for the overall rate expression to be 3/2 order in reactant for a first-order initiation process, the chain terminating step must involve a second-order reaction between two of the radicals responsible for the second-order propagation reactions. In terms of our generalized Rice-Herzfeld mechanistic equations, this means that reaction (4a) is the dominant chain breaking process. One may proceed as above to show that the mechanism leads to a 3/2 order rate expression. 

The first and third order terms in odd powers of the applied electric field are present for all materials. In the second order term, a polarization is induced proportional to the square of the applied electric field, and the. nonlinear second order optical susceptibility must, therefore, vanish in crystals that possess a center of symmetry. In addition to the noncentrosymmetric structure, efficient second harmonic generation requires crystals to possess propagation directions where the crystal birefringence cancels the natural dispersion leading to phase matching. 

This equation is second order in time, and therefore remains invariant under time reversal, that is, the transformation t - — t. A movie of a wave propagating to the left, run backwards therefore pictures a wave propagating to the right. In diffusion or heat conduction, the field equation (for concentration or temperature field) is only first order in time. The equation is not invariant under time reversal supporting the observation that diffusion and heat-flow are irreversible processes. 

This is indeed a system of three second-order differential equations. The tensor elements Cyki may be complex-valued in case of viscoelasticity. Analysis shows that the propagation can be split into three orthogonally polarized planar waves propagating along a wave vector k. Those three waves may have different propagating celerities. Phase celerity and polarization ilj are connected through Christoffel equation ... 

When developing this equation in terms of the rate expressions for the individual reactions it has been the usual practice to assume that there is only one propagating species in the system - an assumption which will be discussed in Section 6, and that the propagation reaction is of second order ... 

N.B. The important point to note is that, in favourable cases, the manner in which the DP depends on the monomer concentration can show unambiguously whether the propagation is a first- or a second-order reaction. This is very useful because often the evidence from rate measurements, especially the analysis of individual rate curves, may be uncertain and difficult to interpret. 

Case 2. If the dependence of DP on [P3] shows that the propagation is a second order reaction (equation (12)), and if the whole reaction curves are of first order, it follows that [Pn+] must be constant throughout the reaction, i.e., there is a stationary state. This can be of the First Kind, V = Vt 0, or of the Second Kind, V,- = Vt = 0. The diagnosis... 

Case 3. If the propagation is a second-order reaction and the whole reaction curve is of second order, we have again necessarily a stationary state and this can only be of the First Kind. Thus, if we are dealing with formation of high polymers, such that rate = VPt... 

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