Cosine propagator

The propagator nature of the Chebyshev operator is not merely a formality it has several important numerical implications.136 Because of the similarities between the exponential and cosine propagators, any formulation based on time propagation can be readily transplanted to one that is based on the Chebyshev propagation. In addition, the Chebyshev propagation can be implemented easily and exactly with no interpolation errors using Eq. [56], whereas in contrast the time propagator has to be approximated. 

As pointed out in the previous section, the Chebyshev operator can be viewed as a cosine propagator. By analogy, both the energy wave function and the spectrum can also be obtained using a spectral method. More specifically, the spectral density operator can be defined in terms of the conjugate Chebyshev order (k) and Chebyshev angle (0) 128 132... 

The cosine iterative equation [8] is the ancestor of the RWP method. Consider propagating a wave packet at time t forward in time to I + r. 

Equation (4) is a three-term recursion for propagating a wave packet, and, assuming one starts out with some 4>(0) and (r) consistent with Eq. (1), then the iterations of Eq. (4) will generate the correct wave packet. The difficulty, of course, is that the action of the cosine operator in Eq. (4) is of the same difficulty as evaluating the action of the exponential operator in Eq. (1), requiring many evaluations of H on the current wave packet. Gray [8], for example, employed a short iterative Lanczos method [9] to evaluate the cosine operator. However, there is a numerical simplification if the representation of H is real. In this case, if we decompose the wave packet into real and imaginary parts. 

What is the equivalent of Eq. (7) in the cosine iterative scheme Simply applying A to the result, Eq. (5), of a cosine iteration proved to be less stable than a different absorption algorithm, which can be derived on the basis of time-reversal ideas as follows [8]. Consistent with Eq. (7), the backward propagated result should be... 

The cosine form of the Chebyshev propagator also affords symmetry in the effective time domain, which allows for doubling of the autocorrelation function. In particular, 2K values of autocorrelation function can be obtained from a E-step propagation 147... 

We can now use the results that we have obtained as a guide to the general problem of waves propagated in an arbitrary direction. To describe the direction of propagation, imagine a unit vector along the wave normal. Let the x, y, and z components of this unit vector be Z, m, n. These quantities are often called direction cosines, for it is obvious that they are equal respectively to the cosines of the angles between the direction of the wave normal and the x, y, z axes. Then in... 

Abstract. The Chebyshev operator is a discrete cosine-type propagator that bears many formal similarities with the time propagator. It has some unique and desirable numerical properties that distinguish it as an optimal propagator for a wide variety of quantum mechanical stndies of molecnlar systems. In this contribution, we discuss some recent applications of the Chebyshev propagator to scattering problems, including the calcnlation of resonances, cumulative reaction probabilities, S-matrix elements, cross-sections, and reaction rates. 

A closer look at the Chebyshev operator ((H)) reveals that it possesses many similar properties with the better known time propagator. Because of the cosine mapping in Eq. (2), one can equate the Chebyshev operator to the real part of an effective time propagator [15,16]... 

Where A, A, v, t, and 0are the amplitude, wavelength, velocity, time, and phase, respectively. This equation describes the propagation of a cosine curve [A cos(2jt/A)x] along the x-axis. Introducing the frequency v = v/A, the general formula of wave can be written as ... 

Level set and VOF methods are often applied to a number of test problems such as propagation of a curve (e.g., a cosine function) in the normal... 

It leads to the solution of the wave propagation Equation 9.187 in terms of the harmonic function cosine when the initial value is not zero (or sine when this is not the case) ... 

The theoretical model for photodetachment is similar to that used to describe photodissociation outlined in the last section. As illustrated in Fig. 3.7, the initial wave packet on the neutral PES was chosen as the ground vibrational state of cis-HOCO, which has a lower energy than its tram counterpart. The anion vibrational eigenfunction was determined on a newly developed anion PES at the same CCSD(T)-F12/AVTZ level [130], as used to construct the neutral PES [100, 101]. The neutral wave packet was propagated to yield probabilities to both the HO-I-CO and H-I-CO2 asymptotes with a flux method [108] and the cosine Fourier transform of the Chebyshev autocorrelation function yielded the energy spectrum [44]. The discretization of the Hamiltonian and wavepacket, and the propagation were essentially the same as in our recent reaction dynamics study [107]. 

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