Chebyshev operator

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... 

Apart from the delta filter discussed here, one can define other filters using the same Chebyshev operators. In fact, any analytic function of the Hamiltonian can be expressed as an expansion in terms of the Chebyshev operator.148 For instance, the Green filter can be expressed as follows 126,127 149... 

Chen, R. and Guo, H. (1996) Evolution of quantum system in order domain of Chebyshev operator. Chem. Phys. 105. 3569-3578. 

Abstract. The Chebyshev operator is a diserete eosine-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 studies of molecular systems. In this contribution, we discuss some recent applications of the Chebyshev propagator to scattering problems, including the calculation of resonances, cumulative reaction probabilities, S-matrix elements, cross-sections, and reaction rates. 

A eloser look at the Chebyshev operator reveals that it possesses... 

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]... 

Pig. 4. Photo dissociation of ArHCl. Left hand side the number of force field evaluations per unit time. Right hand side the number of Fast-Fourier-transforms per unit time. Dotted line adaptive Verlet with the Chebyshev approximation for the quantum propagation. Dash-dotted line with the Lanczos iteration. Solid line stepsize controlling scheme based on PICKABACK. If the FFTs are the most expensive operations, PiCKABACK-like schemes are competitive, and the Lanczos iteration is significantly cheaper than the Chebyshev approximation. 

Antilinear operator, antiunitary, 688 Antiunitary operators, 727 A-operation, 524 upon Dirac equation, 524 Approximation, 87 methods, successive minimax (Chebyshev), 96 problem of, 52 Arc, 258... 

When solving the model problem concerned, the transition from the A th iteration to the (k + l)th iteration is performed either in 9 steps or in 26 steps 5 operations of addition and 4 operations of multiplication during the course of the explicit Chebyshev s method and 12 operations of addition and 14 operations of multiplication in the case of ADM in connection with the double elimination (first, along the rows and then along the columns). This provides reason enough to conclude that in the case of noncommutative operators the first method is rather economical than the second one. Both 1 1... 

The intention is to use such a factorized operator in the explicit method with optimal set of Chebyshev s parameters ... 

It is worth noting here that the same estimate for no( ) was established before for ATM with optimal set of Chebyshev s parameters, but other formulas were used to specify r] in terms of and A - If R = —A, where A is the difference Laplace operator, and the Dirichlet problem is posed on a square grid in a unit square, then... 

For Chebyshev s scheme with such an operator we obtain... 

The real wave packet (RWP) method, developed by Gray and Bahnt-Kuiti [ 1], is an approach for obtaining accurate quantum dynamics information. Unlike most wave packet methods [2] it utilizes only the real part of the generally complex-valued, time-evolving wave packet, and the effective Hamiltonian operator generating the dynamics is a certain function of the actual Hamiltonian operator of interest. Time steps in the RWP method are accomphshed by a simple three-term Chebyshev... 

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. 

The usefulness of the Chebyshev polynomials being both efficient and accurate building blocks in numerically approximating operator functions was realized first by Tal-Ezer and Kosloff,79,123 and later by Kouri, Hoffman, and coworkers.124-129 Aspects of their pioneering work will be discussed later in this review. 

How does one extract eigenpairs from Chebyshev vectors One possibility is to use the spectral method. The commonly used version of the spectral method is based on the time-energy conjugacy and extracts energy domain properties from those in the time domain.145,146 In particular, the energy wave function, obtained by applying the spectral density, or Dirac delta filter operator (8(E — H)), onto an arbitrary initial wave function ( (f)(0)))1 ... 

The most successful strategy for approximating the Liouville-von Neumann propagator is to interpolate the operator with polynomial operators. To this end, Newton and Faber polynomials have been suggested to globally approximate the propagator,126,127,225,232-234 as in Eq. [95]. For short-time propagation, short-iterative Arnoldi,235 dual Lanczos,236 and Chebyshev... 

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. 

In concluding this discussion it is worth noting that the type of the original equation Au = f and the operator B have no influence on a universal method of numbering the parameters r1 ..., rn that can be obtained through the use of the ordered set M n of zeroes of Chebyshev s polynomial of degree n, whose description and composition were made in Section 2 of the present chapter. 

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