Wavepacket method

A different approach is to represent the wavepacket by one or more Gaussian functions. When using a local harmonic approximation to the trae PES, that is, expanding the PES to second-order around the center of the function, the parameters for the Gaussians are found to evolve using classical equations of motion [22-26], Detailed reviews of Gaussian wavepacket methods are found in [27-29]. 

To add non-adiabatic effects to semiclassical methods, it is necessary to allow the trajectories to sample the different surfaces in a way that simulates the population transfer between electronic states. This sampling is most commonly done by using surface hopping techniques or Ehrenfest dynamics. Recent reviews of these methods are found in [30-32]. Gaussian wavepacket methods have also been extended to include non-adiabatic effects [33,34]. Of particular interest here is the spawning method of Martinez, Ben-Nun, and Levine [35,36], which has been used already in a number of direct dynamics studies. 

Finally, Gaussian wavepacket methods are described in which the nuclear wavepacket is described by one or more Gaussian functions. Again the equations of motion to be solved have the fomi of classical trajectories in phase space. Now, however, each trajectory has a quantum character due to its spread in coordinate space. 

The big advantage of the Gaussian wavepacket method over the swarm of trajectory approach is that a wave function is being used, which can be easily manipulated to obtain quantum mechanical information such as the spechum, or reaction cross-sections. The initial Gaussian wave packet is chosen so that it... 

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... 

S. C. Althorpe,/. Chem. Phys., 114,1601 (2001). Quantum Wavepacket Method for State-to-... 

The Grid and the Absorbing Potential The Real Wavepacket Method... 

The initial wavepacket, described in Section III.B is intrinsically complex (in the mathematical sense). Furthermore, the solution of the time-dependent Schrodinger equation [Eq. (4.23)] also involves an intrinsically complex time evolution operator, exp(—/Ht/ ). It therefore seems reasonable to assume that aU the numerical operations involved with generating and analyzing the time-dependent wavefunction will involve complex arithmetic. It therefore comes as a surprise to realize that this is in fact not the case and that nearly all aspects of the calculation can be performed using entirely real wavefunctions and real arithmetic. The theory of the real wavepacket method described in this section has been developed by S. K. Gray and the author [133]. 

Techniques similar to the those described in this section and in Ref. 133, but used within a time-independent framework, have been developed by Kouri and coworkers [188,189] and by Mandelshtam and Taylor [62,63]. Kroes and Neuhauser [65-68] have used the methods developed in these papers to perform time-independent wavepacket calculations using only real arithmetic. The iterative equation that lies at the heart of the real wavepacket method, Eq. (4.68), is in fact simply the Chebyshev recursion relationship [187]. This was realized by Guo, who developed similar techniques based on Chebyshev iterations [50,51]. 

Key topics covered in the review are the analysis of the wavepacket in the exit channel to yield product quantum state distributions, photofragmentation T matrix elements, state-to-state S matrices, and the real wavepacket method, which we have applied only to reactive scattering calculations. 

Figures 7.5 and 7.6 show the experimental absorption spectra of HCl and DCl, respectively, together with the spectra calculated using wavepacket propagation and the reflection principle. It can be seen that the wavepacket method is in good agreement with the experimental results. For both HCl and DCl the wavepacket propagation method yields the correct frequency for the absorption peak. The wavepacket propagation method is exact and the deviation from the experimental spectrum must be attributed to the use of only two electronic states and/or inaccurate transition dipole moments and/or potential energy curves and/or not treating the rotations of the molecule. The experimental results are quite accurate.

Althorpe, S.C. (2001) Quantum wavepacket method for state-to-state reactive cross sections,, 7. Chem. Phys. 114. 1601-1616. 

Hankel, M. (2001) Time-Dependent Wavepacket Methods for the Calculation of Statc-to-Statc Molecular Reactive Cross Sections, Ph.D. thesis, University of Bristol, Bristol. 

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