Expansion of the propagator

Kosloff [1] showed how we can propagate the wavefunction forward in time in a more efficient way than in eq. (1.3) above. Instead of expanding the exponential operator in a Taylor series he proposed that it be expanded in a Chebyshev series. This series has the form 

The Pn are Chebyshev polynomials of complex argument. They obey the recursion formula 

Hnorm IS a normalised hamiltonian. It is normalised in such a way as to limit its spectrum to lie between -1 and +1. The spectrum of the hamiltonian is the range of possible eigenvalues it can have. This normalisation is performed by finding the range of the hamiltonian operator, 

The Jn (in eq. 2.1) are Bessel functions. These play a very important role in the convergence of the expansion. For n values greater than the argument, At/2H these Bessel functions decrease exponentially in value. We can therefore predict that the number of terms needed in the expansion is approximately 

This is in fact an important conclusion. The number of terms required to expand the time evolution operator is proportional to the range of the hamiltonian operator. Or equivalently, this is the number of operations of 

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Using this transformation, it has been shown in Refs. [54,72] that the effective-mode Hamiltonian Heg by itself reproduces the short-time dynamics of the overall system exactly. This is reflected by an expansion of the propagator, for which it can be shown that the first few terms of the expansion - relating to the first three moments of the overall Hamiltonian - are exactly reproduced by the reduced-dimensional Hamiltonian Heg. 

Successive orders H(n ) can be shown to correspond to successive orders in a moment (or cumulant) expansion of the propagator, which takes one to increasing times. Truncation of the chain at a given order n (i.e., 3 + 3n modes) leads to an approximate, lower-dimensional representation of the dynamical process, which reproduces the true dynamics up to a certain time. In Ref. [51], we have demonstrated explicitly that the nth-order (3n+3 mode) truncated HEP Hamiltonian exactly reproduces the first (2n + 3)rd order moments (cumulants) of the total Hamiltonian. A related analysis is given in Ref. [73],... 

The correlated collisions may be between a solute A (or B) molecule and a solvent molecule, while A and B are statically correlated. The scattering processes that contribute to the vertices are shotvn schematically in Fig. 7.4. Once again, expansion of the propagator in (7.29) in powers of yields a repeated ring series, but now one with a richer structure due to the variety of collisional processes that enter into each vertex. We explore the structure of this operator in more detail in connection with the rate coefficient. 

Expansion of the propagator about Gaas leads to a variety of collision events. The series of terms with the form... 

Chebyshev expansion of the propagator. First consider the function of energy, f(E) = exp(-/ Ar), in which E is an energy belonging to the interval [7 , Emax, and where At represents a small time interval. In order to perform a Chebyshev expansion, we need to define a function restricted to the interval [-1, 1]. We will define the new variable e ... 

The higher moments in the moment expansion of the propagator or the propagator matrix can become quite comphcated and approximations are necessary. The simplest approximation that yields useful results proceeds by approximating higher moments as powers of the first moment F = (X HX). Denote S = (X X) and obtain the approximation... 

The Hamiltonian and the coordinates are discretized by means of the generalized pseudospectral (GPS) method in prolate spheroidal coordinates [44-47], allowing optimal and nonuniform spatial grid distribution and accurate solution of the wave functions. The time-dependent Kohn-Sham Equation 3.5 can be solved accurately and efficiently by means of the split-operator method in the energy representation with spectral expansion of the propagator matrices [44-46,48]. We employ the following split operator, second-order short-time propagation formula [40] ... 

As the quinone stabilizer is consumed, the peroxy radicals initiate the addition chain propagation reactions through the formation of styryl radicals. In dilute solutions, the reaction between styrene and fumarate ester foUows an alternating sequence. However, in concentrated resin solutions, the alternating addition reaction is impeded at the onset of the physical gel. The Hquid resin forms an intractable gel when only 2% of the fumarate unsaturation is cross-linked with styrene. The gel is initiated through small micelles (12) that form the nuclei for the expansion of the cross-linked network. 

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. 

The RWP method also has features in common with several other accurate, iterative approaches to quantum dynamics, most notably Mandelshtam and Taylor s damped Chebyshev expansion of the time-independent Green s operator [4], Kouri and co-workers time-independent wave packet method [5], and Chen and Guo s Chebyshev propagator [6]. Kroes and Neuhauser also implemented damped Chebyshev iterations in the time-independent wave packet context for a challenging surface scattering calculation [7]. The main strength of the RWP method is that it is derived explicitly within the framework of time-dependent quantum mechanics and allows one to make connections or interpretations that might not be as evident with the other approaches. For example, as will be shown in Section IIB, it is possible to relate the basic iteration step to an actual physical time step. 

Since most types of experimental inadequacy or incompetence produce more ethanol in the hydrolysate than could be derived from fert.-oxonium ions, one must conclude from this evidence that the propagating species is a secondary oxonium ion. It follows necessarily that the ring-expansion mechanism is the best representation of the propagation reaction, that tert.-oxonium ions do not play an essential role in the polymerisation of DCA by perchloric acid, and that therefore this part of the old controversy appears now to be settled. 

It is interesting to note that stratified combustible gas mixtures can exist in tunnel-like conditions. The condition in a coal mine tunnel is an excellent example. The marsh gas (methane) is lighter than air and accumulates at the ceiling. Thus a stratified air-methane mixture exists. Experiments have shown that under the conditions described the flame propagation rate is very much faster than the stoichiometric laminar flame speed. In laboratory experiments simulating the mine-like conditions the actual rates were found to be affected by the laboratory simulated tunnel length and depth. In effect, the expansion of the reaction products of these type laboratory experiments drives the flame front developed. The overall effect is similar in context to the soap bubble type flame experiments discussed in Section C5c. In the soap bubble flame experiment measurements, the ambient condition is about 300 K and the stoichiometric flame temperature of the flame products for most hydrocarbon fuels... 

Our main concern in this section is with the actual propagation forward in time of the wavepacket. The standard ways of solving the time-dependent Schrodinger equation are the Chebyshev expansion method proposed and popularised by Kossloff [16,18,20,37 0] and the split-operator method of Feit and Fleck [19,163,164]. I will not discuss these methods here as they have been amply reviewed in the references just quoted. Comparative studies [17-19] show conclusively that the Chebyshev expansion method is the most accurate and stable but the split-operator method allows for explicit time dependence in the Hamiltonian operator and is often faster when ultimate accuracy is not required. All methods for solving the time propagation of the wavepacket require the repeated operation of the Hamiltonian operator on the wavepacket. It is this aspect of the propagation that I will discuss in this section. 

A different way, developed extensively by Schwartz and his coworkeis, - is to use approximate quantum propagators, based on expansions of the exponential operators. These approximations have been tested for a number of systems, including comparison with the numerically exact results of Ref 38 for the rate in a double well potential, with satisfying results. 

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