Wave-packet algorithm

Section II discusses the real wave packet propagation method we have found useful for the description of several three- and four-atom problems. As with many other wave packet or time-dependent quantum mechanical methods, as well as iterative diagonalization procedures for time-independent problems, repeated actions of a Hamiltonian matrix on a vector represent the major computational bottleneck of the method. Section III discusses relevant issues concerning the efficient numerical representation of the wave packet and the action of the Hamiltonian matrix on a vector in four-atom dynamics problems. Similar considerations apply to problems with fewer or more atoms. Problems involving four or more atoms can be computationally very taxing. Modern (parallel) computer architectures can be exploited to reduce the physical time to solution and Section IV discusses some parallel algorithms we have developed. Section V presents our concluding remarks. 

In general, the topology of interprocessor communication reflects both the structure of the mathematical algorithms being employed and the way that the wave packet is distributed. For example, our very first implementation of parallel algorithms in a study of planar OH - - CO [47] used fast Fourier transforms (FFTs) to compute the action of 7, which also required all-to-all communication but in a topology that is very different from the simple ring-like structure shown in Fig. 5. 

There have been several successful approaches [41, 42] to improving the scaling of parallel algorithms that employ DVRs or FFTs. These rely on more sophisticated methods of distributing the wave packet so as to reduce the amount of communication required. One should consult Refs. [41] and [42] for details. 

It can be seen from the algorithm model stated above that in the Krotov method the electric field obtained in the feth iteration is used immediately to propagate /(f), which has a direct contribution to the new electric field in the next time step. In one iteration, the Krotov method involves three wave packet propagations, that is, the forward propagations of and x (t) in Steps 8.3... 

Avoiding Long Propagation Times in Wave Packet Calculations on Scattering with Resonances A New Algorithm Involving Filter Diagonalization. 

Fig. 6.4. Final wavepackets driven by the optimal field calculated quantum mechanically and semiclassically after ten iterations of the optimization algorithm, a Target state wave packet, b Quantum result. Semiclassical results c correlation functions are obtained with (6.34) d correlation functions are obtained with the simple formula (6.35)...

Further, regarding ijrn), it is also possible to have the explicit Lanczos algorithm by deriving the expression that holds the whole result with no recourse to recurrence relations. For example, applying the explicit Lanczos polynomial operator Q (U) from Eq. (177 to tq) = 0) will generate the wave packet IV n) according to ijrn) = Q (U) V o) as in Eq. (91). Therefore, the final result is the following expression for the explicit Lanczos states Vr ) ... 

To calculate numerically the quantum dynamics of the various cations in time-dependent domain, we shall use the multiconfiguration time-dependent Hartree method (MCTDH) [79-82, 113, 114]. This method for propagating multidimensional wave packets is one of the most powerful techniques currently available. For an overview of the capabilities and applications of the MCTDH method we refer to a recent book [114]. Additional insight into the vibronic dynamics can be achieved by performing time-independent calculations. To this end Lanczos algorithm [115,116] is a very suitable algorithm for our purposes because of the structural sparsity of the Hamiltonian secular matrix and the matrix-vector multiplication routine is very efficient to implement [6]. 

It is important to emphasize that, in the above examples, knowledge of the PES was not required for the optimization process. The adaptive-control learning algorithm explores the available phase space and optimizes the evolution of the wave packet on the excited state PES without any prior knowledge of the surface. Thus, the intrinsic information about the excited-state dynamics of these polyatomic systems remains concealed in the detailed shape and phase of the optimized pulse. Inevitably, however, scientific curiosity, together with a desire to imder-stand how chemical reactions can be controlled, has led to pioneering studies that aim to identify the underlying rules and rationale that lead to a particular pulse shape or phase relationship that produces the optimum yield. 

---