Chebyshev expansion

Now we replace tlie first exponential in the right-hand side of (A3.11.129) by a Chebyshev expansion as follows ... 

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. 

Interestingly, the spectral transform Lanczos algorithm can be made more efficient if the filtering is not executed to the fullest extent. This can be achieved by truncating the Chebyshev expansion of the filter,76,81 or by terminating the recursive linear equation solver prematurely.82 In doing so, the number of vector-matrix multiplications can be reduced substantially. 

Figure 5 Gaussian filter approximated by Chebyshev expansion with various numbers of terms. Adapted with permission from Ref. 148. 

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. 

Np, total number of processors Nq, number of terms in Chebyshev expansion. 

L = 5 are shown in Tables VI and VII, respectively. For each value of L, the best polynomial naturally gives better RMSE than Tn x) does, for every range and order. Comparison of Tables VI and VIII shows, however, that for smaller ranges and higher orders, T (x) with L = 0 yields lower values of the RMSE than T,7 x) with L = 5. On the other hand the use of sub-divided ranges of the expansion variable improves the Chebyshev expansion when u covers a wide range and the order of the expansion is less than 2. 

Comparison of the accuracies of the Chebyshev and Bernoulli approximations in Table XI shows the Chebyshev (L = 0) to be better than the Bernoulli by a factor of 5 to 10 at n = 1 the improvement increases to about 300 at n = 4. This was generally observed for all other isotopic substitutions tested the rate of convergence of the Chebyshev expansion is better than the Bernoulli expansion at any order, at any temperature. The Chebyshev expansion exists at any temperature, while the Bernoulli series diverges for most of the molecules at room temperature. Table XI also shows that the Bigeleisen-Mayer approximation. 

Table XI reveals many interesting features of the Chebyshev expansion common to all deuterium-for-protium substitutions in all the hydrocarbons tested at room temperature where Umax is about 15 or higher. In...

With Eqs. (32)-(35), one can construct the basis functions (Ej), filtered out of the initial state near the energies Ei. Before applying the modified Chebyshev expansion, one has to rescale the original Hamiltonian H, because Chebyshev polynomials converge only for la < 1 [236]. In fact, H in the equations above has to be substituted with the operator. ffnorm H H))/AH, whose eigenvalues lie between —1 and 1. Then, a basis function can be written as [208]... 

TABLE 1. Comparison of the order of the Chebyshev expansion for the exponential... 

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. 

Chebyshev expansion of a real function /(x). We will first define the Chebyshev polynomial of degree n (n = 0, 1, 2,. ..) denoted T (x),... 

Since the Tn(x) s are all bounded between —1 and 1, the difference between the estimated function and the function f(x) can be no larger than the sum of the absolute values of neglected coefficients, a (k > N). In fact, these coefficients usually decrease rapidly. As a result, the error is dominated by the first omitted term aNJtlTN+l(x), which is a small oscillatory function with N + 2 extrema (equal in absolute value) distributed uniformly over the interval [—1, +1]. Because of the uniform character of the Chebyshev expansion, the error usually decreases exponentially once N is large enough. 

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 initial condition for the dynamics at / = 0 was again chosen to be a v = 3 overtone excited CH oscillator. The subsequent dynamics was then generated by propagating the time-dependent wave function in the active space. If we let C(/) denote the column vector of the expansion coefficients of the various basis functions at time t, then this state vector evolves according to the equation C(t) = U(r)C(r), where U(r) = exp[—iH/] is the time propagator associated with the vibrational Hamiltonian matrix H. In order to evaluate the propagator, we used the Chebyshev expansion (Sec. II.C.3). 

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 Coefficients of Characteristic Polynomials in Chebyshev Expansion for Isomers of Hexene... 

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],... 

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... 

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