Short time propagator

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 short-time propagator U ht) = becomes progressively more valid... 

The application of the kinetic energy part of the short-time propagator proceeds as follows ... 

The Lanczos algorithm can also be used to approximate a short-time propagator. The so-called short-iterative Lanczos (SIL) method of Park and Light constructs a small set of Lanczos vectors,226 which can be summarized by Eq. [96] ... 

The most successful strategy for approximating the Liouville-von Neumann propagator is to interpolate the operator with polynomial operators. To this end, Newton and Faber polynomials have been suggested to globally approximate the propagator,126,127,225,232-234 as in Eq. [95]. For short-time propagation, short-iterative Arnoldi,235 dual Lanczos,236 and Chebyshev... 

For short time propagators entering into this equation one usually... 

The short-time propagators can be solved through application of the Dyson identity truncated to first order. The subsequent dynamics of the quantity of interest are obtained by propagating the classical variables along a surface that corresponds to the quantum state (aa1) followed by Monte Carlo sampling of the nonadiabatic transition events ... 

We remark that the simulation scheme for master equation dynamics has a number of attractive features when compared to quantum-classical Liouville dynamics. The solution of the master equation consists of two numerically simple parts. The first is the computation of the memory function which involves adiabatic evolution along mean surfaces. Once the transition rates are known as a function of the subsystem coordinates, the sequential short-time propagation algorithm may be used to evolve the observable or density. Since the dynamics is restricted to single adiabatic surfaces, no phase factors... 

D. Mac Kernan, R. Kapral, and G. Ciccotti. Sequential short-time propagation of quantum-classical dynamics sequential short-time propagation of quantum-classical dynamics. J. Phys. Condensed Matter, 14 9069, 2002. 

Since the Liouville operator is time independent and commutes with itself we may write the propagator exactly as the product of N short time propagators as... 

From left to right, the short-time propagator describes classical propagation on the Sj i surface through a time interval 6/2, a transition Sj 1 —> Sj determined by the elements of A4 and classical propagation on the Sj surface for a time interval 6/2. 

D. MacKernan, G. Ciccotti, and R. Kapral (2002) Sequential Short-Time Propagation of Quantum-Classical Dynamics. J. Phys. Condens. Matt. 14, pp. 9069-9076... 

We can now factorize the short time propagator in a different way (confr. 

In this new form, the widely oscillating potential energy is replaced by the local energy which is much smoother for an accurate We need to find the short time propagator of the importance sampling hamiltonian T-i which is nothing but the solution of the corresponding Bloch equation [19,21]... 

For N large enough, each short-time propagator, < j exp( - iHSt/h) qf j >,... 

In many physical applications the Hamiltonian is explicitly time dependent. The common solution for propagation in these explicitly time-dependent problems is to use very small grid spacing in time, such that within each time step the Hamiltonian H(r) is almost stationary. Under these semistationary conditions a short-time propagation method in the time-energy phase space is employed. The drawback of this solution is that it is based on extrapolation therefore the errors accumulate. Moreover, time ordering errors add with the usual numerical dispersion errors (108). 

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 Chebychev method converges exponentially with the number of expansion terms n for a given step size A and is particularly advantageous and efficient when A is large. However, unlike short-time propagators such as SOD or SP, the Chebychev method is not directly applicable to time-dependent or non-Hermitian Hamiltonians. 

To disentangle the short-time propagator, we use a (symmetrized) Trotter formula,... 

---