Chebyshev expansion method

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. 

The key to performing a wavepacket calculation is the propagation of the wavepacket forward in time so as to solve the time-dependent Schrodinger equation. In 1983, Kosloff proposed the Chebyshev expansion technique [5, 6, 7, 8] for evaluating the action of the time evolution operator on a wavepacket. This led to a huge advance in time-dependent wavepacket dynamics [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Several studies have compared different propagation methods [30, 31, 32] and these show that the Chebyshev expansion method is the most accurate. 

Subroutine Mixing is called by subroutine Pitzer, and calculates the variables dij and given the ionic strength. These variables were defined by Pitzer (1975, 1983), and account for the interaction of ions of like but different charge, that is, where / and j are both cations or both anions (such as Na+ and Ca " ", or Cl and in this case). The term was derived by Pitzer from the statistical mechanics theory of Friedman (1962). Its evaluation involves integrals [functions Jq, 7, in the notation of Harvie and Weare (1980)] which have no analytical solution, and Pitzer (1975, 1995) gives approximate methods. Harvie (1981) and Harvie and Weare (1980) used a more accurate scaled Chebyshev expansion method, also described in Pitzer (1987, 1991), which is used in this subroutine. 

The above time propagation scheme can be carried out exactly within a given numerical accuracy. An advantage of the Chebyshev expansion method is its rapid convergence. In particular, it shows one of the best performances among others when the time step At is taken to be long within a given tolerance. 

The Chebyshev expansion method [430] is applied to the propagation of the nuclear wavepacket subject to the above Hamiltonian for both the two-and three-state models. Nuclear wavefimctions, the potential functions, the matrix to represent the first and second nuclear derivatives arising from T of Eq. (6.70) are all expressed in the sine discrete variable representation (sinc-DVR) [92]. The time length for one-step integration of nuclear wavepacket is set to 0.02 fs. The 1200 DVR grid points are employed within a range from —3 to 14 Bohrs. For a practical reason, the potential function is cut off in the range shorter than 1.2 Bohrs. 

The electron dynamics has been integrated with the Chebyshev method. [430] The time propagation of multi-state coupled nuclear wavefunction has been performed with use of the Chebyshev expansion method. [430] Since the ground electronic state jl) involves molecular dissociation, we... 

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. 

We will here no more dwell on the characteristic polynomial except for mentioning additional publications that interested readers can explore and follow. There have been several publications on Le Verrier-Fadeev s method [60,61], which appears to be the most general technique for obtaiuiug the characteristic polynomial. Related to the above is the method of Frame [62], which received some attention in chemical literature [63]. Also techniques for obtaining the characteristic polynomial large graphs have received some attention in chanical literature [45,64-67]. The Chebyshev expansion of characteristic polynomials in terms of L polynomials on paths has received... 

As a useful polynomial expansion method we show the Chebyshev expansion scheme below. We consider the general case such as a dissociation process, in which a wavepacket may proceed out of the grid region set in advance. The time propagation operator of the wavepacket x) is expanded in terms of the modified Chebyshev polynomial [255],... 

We may be able to make some progress in solving the ABC system by appealing to the Newton polynomial expansion, which is a useful method for approximating scalar functions. A special case is the Chebyshev polynomial expansion, which is widely used in commercial algorithms for evaluating special functions [37, 38]. Based on the Chebyshev expansion for u x) = for x real, Tal-Ezer and Kosloff [5]... 

In order to compute the evolving state i(/)), Tal-Ezer and Kosloff (77) were the first to propose an expansion of the evolution operator in terms of Chebyshev polynomials. They initially developed this method for wavepacket calculations on spatial grids. More recently, this procedure has been adapted and applied to bound systems (20). It involves breaking the total integration time (for instance 2 ps) into smaller time steps At (each about 25 fs), and using a polynomial expansion of the evolution operator (/(Af) over each small time step. This efficient method provides all the transition probabilities Plf(t) from initial state i) in one calculation because it directly provides the evolving state /(r)). 

Usually, the propagator (7(r, to) is approximated by various schemes [55,60,137], and there are plenty of wonderful articles that have explained each in detail, such as the split operator method and higher order split operator methods [11, 36, 130], Chebyshev polynomial expansion [131], Faber polynomial expansion [51, 146], short iterative Lanczos propagation method [95], Crank-Nicholson second-order differencing [10,56,57], symplectic method [14,45], recently proposed real Chebyshev method [24,44,125], and Multi-configuration Time-Dependent Hartree (MCTDH) Method [ 12,73,81-83]. For details, one may refer to the corresponding references. 

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