Split-operator method

A more powerfiil method for evaluating the time derivative of the wavefiinction is the split-operator method [39]. Flere we start by fomially solving ihd ild. = /7 / with the solution V fD = e Note that //is... 

Once the grid (or two grids) are prepared, there are two similar types of approaches to propagate the initial wavefiinction forward with time. One approach is split-operator methods, [59] where the short-time propagator is divided into a kinetic and potential parts so that... 

An alternative to split operator methods is to use iterative approaches. In these metiiods, one notes that the wavefiinction is fomially "tt(0) = exp(-i/7oi " ), and the action of the exponential operator is obtained by repetitive application of //on a function (i.e. on the computer, by repetitive applications of the sparse matrix... 

The following two sections present a brief overview of the split-operator method, as used in several recent applications [41, 42, 61, 62], and of the basis set expansion approach. 

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 the present work, we monitor the laser-driven dynamics designed by the present formulation by numerically solving the time-dependent Schrodinger (5.2). It is solved by the split operator method [52] with the fast Fourier transform technique [53]. In order to prevent artificial reflections of the wavepacket at the edges, a negative imaginary absorption potential is placed at the ends of the grid [54]. The envelope of the pulses employed is taken as... 

The time evolution operator exp(—/HAf/ft) acting on ( ) propagates the wave function forward in time. A number of propagation methods have been developed and we will briefly describe the following the split operator method [91,94,95], the Lanzcos method [96] and the polynomial methods such as Chebychev [93,97], Newtonian [98], Faber [99] and Hermite [100,101]. A classical comparison between the three first mentioned methods was done by Leforestier et al. [102]. 

G. Barinovs, N. Markovic, G. Nyman, Split operator method in hyperspherical coordinates Application to CH2I2 and OCIO, /. Chem. Phys. Ill (15) (1999) 6705-6711. 

A more powerful method for evaluating the time derivative of the wavefunction is the split-operator method... 

The split-operator method of (9) was used to carry out the wave packet propagation where the reference Hamiltonian H0 and the potential operator are defined as... 

A much more detailed account for the split operator method and the accompanying asymptotic analysis is given in Ref. [4]. 

On the other hand, approximations to Eq. (8) and time-integration techniques, suitable especially for time-independent Hamiltonians, under the requirement of only a few degrees of freedom and short-time evolution, have been developed and applied extensively in connection with grid-type techniques (see Section 2), by focusing on appropriate algebraic expansions of fhe exponenfial form. For example, such a approach is effected by the split-operator method [4] and references there in. 

As an example of the split-operator method, we here consider the time propagation of the vibrational wavefunction of a triatomic molecule, such as NO2 (Sec. 5.4). The two bond lengths of the molecule are ri and r2, and the bond angle is / . The total kinetic energy operator TV is represented in a convenient form by taking the Jacobi coordinates (r, R, 6) as depicted in Fig. 3.4, in which the length R is the distance between the upper O atom and the centroid of the N and the lower O atoms, and r is the bond length of one NO moiety. It is... 

We apply the split-operator method Eq. (3.7) to separately handle the potential and kinetic energy terms of the Hamiltonian. We further apply the split-operator scheme to separate the kinetic energy term into the two exponentially noncommutative parts Tr + Tr and Tg to obtain a numerical short-time propagation method. 

We may apply the split-operator method (Sec. 3.2.1) to the three matrices lijV) V/), and Vo for a short-time numerical propagation scheme. Ionization is now described by population of the neutral state Xn R,t) transferring to the ionized state partial-wave components Xc,kjix R,t) over time through the interaction represented by the matrix Vo(i ,f)-... 

The initial conditions for the quantum dynamics corresponding to those in the preceding section are as follows. Suppose that a total wavefunction totai(r, i , t) is expanded as in Eq. (6.104) and we are interested only in the nuclear wavepackets xi R,t) and x2 R,t). We propagate them with the extended split operator method. [77] The initial nuclear wavepackets xi R,t) are chosen to be a coherent-type Gaussian function only on the adiabatic ground state as... 

The Hamiltonian and the coordinates are discretized by means of the generalized pseudospectral (GPS) method in prolate spheroidal coordinates [44-47], allowing optimal and nonuniform spatial grid distribution and accurate solution of the wave functions. The time-dependent Kohn-Sham Equation 3.5 can be solved accurately and efficiently by means of the split-operator method in the energy representation with spectral expansion of the propagator matrices [44-46,48]. We employ the following split operator, second-order short-time propagation formula [40] ... 

Several different propagation methods exist in the market. We will briefly mention two of these a modified Crank-Nicholson scheme, and the split-operator method. 

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