Split operator propagation

An alternative relax-and-drive procedure can be based on a strictly unitary treatment where the advance from Iq to t is done with a norm-conserving propagation such as provided by the split-operator propagation technique.(49, 50) This however is more laborious, and although it conserves the norm of the density matrix, it is not necessarily more accurate because of possible inaccuracies in the individual (complex) density matrix elements. It can however be used to advantage when the dimension of the density matrix is small and exponentiation of matrices can be easily done.(51, 52)... 

The idea behind split operator propagation [91,94,95] is to split the action of the time evolution operator such that the kinetic energy operator T and the potential energy operator V are separated into different exponentials, causing a small error since T and V do not commute. In second order potential referenced split operator propagation the evolution operator can be approximated as... 

For multidimensional systems it is necessary to split the kinetic energy operator into several parts, in order to decouple the degrees of freedom in the kinetic energy operator. The split operator propagation has been implemented in e.g. spherical [91,94] and hyper spherical coordinates [95]. 

The wavefunction is propagated using the split-operator propagator. 

The split-operator propagator [6] is used to carry out the time propagation of the wavepacket,... 

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... 

The initial wavefunction is then expanded as in Eq. (2), and the wavepacket is propagated using the split operator method42 ... 

PES evaluations. In either case (coupled or split-operator with frozen Gaussian propagator)... 

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. 

Z.G. Sun, S.Y. Lee, H. Guo, D.H. Zhang, Comparison of second-order split operator and Chebyshev propagator in wave packet based state-to-state reactive scattering calculations, J. Chem. Phys. 130 (2009) 174102. 

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]. 

The quantum propagation is performed using a symmetric split operator methodology where the free propagation is achieved using the DAF propagator [147-149,163,164] ... 

The split-operator (SP) method is extremely popular and has been widely used in many practical applications. It approximates the short-time propagator by the equation... 

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... 

The damping factor e y acts on a wave function, multiplying it by the function e Mr). The role of e 7recursion relations clearly is also analogous to that of the exponential damping factor used in conjunction with the split-operator time-propagation scheme (4). 

Time-propagation with split operator formalism... 

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. 

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