Polynomial expansion method

We next describe another propagation method that is also frequently used. In this scheme, the better accuracy is attained by the higher expansion order of the propagator in terms of the energy scaled Hamiltonian in the spectral range between [—1,1] [480]. An advantage of this method is that it allows a relatively large time step At. This expansion is also applied to 

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], 

To numerically absorb the wavepacket components at the boundaries, which are set beforehand to prevent the packet from flowing out to the asymptotic ranges, a damping function e is introduced in the Chebyshev recursion scheme such that 

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. 

4 Multiconfigurationcd time-dependent Hartree (MCTDH) approach 

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There is a range of iterative diagonalization routines to choose between, including classical orthogonal polynomial expansion methods [48], Davidson iteration[58] and Krylov subspace iteration methods. Here the popular Lanezos method[59] will be discussed in the context of finding the eigenstates of the surface Hamiltonian appearing in the hyperspherical coordinate method. 

Another widely used propagation scheme is the Chebychev polynomial expansion method introduced by Kosloff and Kosloff (8). This is a global propagator in the sense that it expands the propagator e u h)H in the interval [0, /]. The method is based on the Chebychev expansion relation for the function exp(iRX) (X E [—1, 1]) (13),... 

Equation 2.28 can be solved analytically at the limit of long straight rods by a polynomial expansion method [3], yielding... 

The described direct derivation of shape functions by the formulation and solution of algebraic equations in terms of nodal coordinates and nodal degrees of freedom is tedious and becomes impractical for higher-order elements. Furthermore, the existence of a solution for these equations (i.e. existence of an inverse for the coefficients matrix in them) is only guaranteed if the elemental interpolations are based on complete polynomials. Important families of useful finite elements do not provide interpolation models that correspond to complete polynomial expansions. Therefore, in practice, indirect methods are employed to derive the shape functions associated with the elements that belong to these families. 

A computationally efficient method of function fitting using an orthogonal polynomial expansion is presented for approximating continuous wall temperature profiles. 

As already mentioned, one of the main weaknesses of the simple reflection method is the fact that the electronic transition dipole moment, (or the transition dipole moment surface, TDMS for polyatomic molecules in Section 4) is assumed to be constant. This weakness will remain in the Formulae (12), (27) and (29) derived below. The average value of the square of the TDM (or TDMS) is then included in amplitude A and A = A /V. In Formulae (3), (3 ) and (3") the mass (or isotopologue) dependent parameters are p and the ZPE. In contrast, W and V., which define the upper potential, are mass independent. This Formula (3) is already known even if different notations have been used by various authors. As an example, Schinke has derived the same formula in his book [6], pages 81, 102 and 111. Now, the model will be improved by including the contribution of the second derivative of the upper potential at Re- The polynomial expansion of the upper potential up to second order in R - Re) can be expressed as ... 

This expression is used as a trial-function expansion for T in much the same way as the Lagrange interpolation polynomial is in the polynomial collocation method of Villadsen and Stewart (10,1 1 There are four unknown constants associated with each node, giving a total of 4(n+l)(nH-1) unknowns in the expansion. 

The expansions in even powers of normal frequencies are of special interest, because they provide means for obtaining explicit relations between the equations of motion and the thermodynamic quantities, through the use of the method of moments The sum of over all the normal vibrations can be expressed as the trace, or the sum of all the diagonal elements, of a matrix H" obtained by multiplying the Hamiltonian matrix H of the system by itself (n — 1) times. Such expansions thus enable us to estimate the thermodynamic functions and their isotope effects from known force fields and structures without solving the secular equations, or alternatively, to estimate the force fields from experimental data on the thermodynamic quantities and their isotope effects. The expansions explicitly correlate the motions of particles with the thermodynamic quantities. They can also be used to evaluate analytically a characteristic temperature associated with the system, such as the cross-over temperature of an isotope exchange equilibrium. Such possible applications, however, are useful only if the expansion yields a sufficiently close approximation. The precision of results obtainable with orthogonal polynomial expansions will be explored later. 

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)). 

We now turn to the polynomial expansion, all-energy method of evaluating ijr/s from Eq. (9). 

Unlike the traditional Taylor s series expansion method, the Galerkin approach utilizes basis functions, such as linear piecewise polynomials, to approximate the true solution. For example, the Galerkin approximation to the sample problem Equation 23.1 would require evaluating Equation 23.13 for the specific grid formation and specific choice of basis function ... 

Based on the criterion used to obtain the formula Taylor series expansion or polynomial approximation methods. 

Experience indicates that approximations by linear combinations of functions yield results that are in good agreement with Laguerre or related polynomial expansions when both methods are applicable. By the comparison of the results obtained with different weight functions one is led to the estimate that the flux can be calculated with an accuracy of 1-5% at a distance of four mean free paths from the source, and with an accuracy of 15-25% at a distance of twenty mean free paths. Theoretical error estimates are lacking at present, so that the extension of the moment method based on approximations of the type (14)—although a valuable engineering tool—is in need of further mathematical development. 

Theoretical curve by moment method (polynomial expansion). Experimental values for iron and lead from [8], for water from [9]. 

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