Gaussian wavepacket calculations calculation

In this seiniclassical calculation, we use only one wavepacket (the classical path limit), that is, a Gaussian wavepacket, rather than the general expansion of the total wave function. Equation (39) then takes the following form ... 

Quantum chemical methods, exemplified by CASSCF and other MCSCF methods, have now evolved to an extent where it is possible to routinely treat accurately the excited electronic states of molecules containing a number of atoms. Mixed nuclear dynamics, such as swarm of trajectory based surface hopping or Ehrenfest dynamics, or the Gaussian wavepacket based multiple spawning method, use an approximate representation of the nuclear wavepacket based on classical trajectories. They are thus able to use the infoiination from quantum chemistry calculations required for the propagation of the nuclei in the form of forces. These methods seem able to reproduce, at least qualitatively, the dynamics of non-adiabatic systems. Test calculations have now been run using duect dynamics, and these show that even a small number of trajectories is able to produce useful mechanistic infomiation about the photochemistry of a system. In some cases it is even possible to extract some quantitative information. 

Fig. 7.12. Comparison of the measured and the calculated absorption spectra for the So — Si transition in CH3ONO. The quantum number n denotes vibrational excitation of the NO moiety in the complex. The theoretical spectrum is obtained in a three-dimensional wavepacket calculation including the ONO bending angle in addition to the two N-0 stretching coordinates. The spectrum is convoluted with a Gaussian function with width AEres = 0.02 eV in order roughly to mimic thermal broadening and is artificially shifted along the energy axis. Reproduced from Untch, Weide, and Schinke (1991a).

In the following calculations, the laser field is assumed to be linearly polarized parallel to the transition dipole moment /x, which is almost perpendicular to the molecular plane. The initial wavepacket is a two-dimensional symmetrical Gaussian wavepacket of form... 

Figure 13. Result of the two-dimensional calculation for Ca-HBr. The calculation and the spectrum of the Pb(A A")transition are compared on the left. The two-dimensional surface built on the reaction coordinate Rh-x and the Ca-HX bending angle is displayed in the top-right panel. The propagation was performed over 1.25 ps. Each contour corresponds to 500 cm. The contour of the Gaussian wavepacket issued from the ground state by vertical excitation is also shown in this surface. The bottom panel on the right shows the resulting modulus of the autocorrelation function with clear recurrences. Adapted from Ref. [243].

From a practical point of view the classical method using directly the probability density function is not convenient, and it is computationally preferable to use an approach that involves trajectory calculations. A derivation of such formulation can be made by starting from the quantum-mechanical TDSCF, and using semiclassical (gaussian) wavepackets. Here we merely quote the final result. In analogy to (62), the single-mode classical SCF potentials are given by... 

As long as the photodissociation reaction is fairly direct, the time-dependent formulation is fruitful and provides insight into both the process itself and the relationship of the final-state distributions to the absorption spectrum features. Moreover, solution of the time-dependent Schrodinger equation is feasible for these short-time evolutions, and total and partial cross sections may be calculated numerically.5 Finally, in those cases where the wavepacket remains well localized during the entire photodissociation process, a semi-classical gaussian wavepacket propagation will yield accurate results for the various physical quantities of interest.6... 

All the initial electronic state populations are set to be localized at the energetically highest adiabatic state to be compatible with the calculation for PSANB and the semiclassical Ehrenfest theory. The momentum of the wavepacket is varied by changing the initial wave number as k =16.05, 32.11, 48.16, 64.22, 80.28, 96.33, 112.39, 128.44, 144.50 and 160.56 a.u., which specifies the initial Gaussian wavepacket of minimum uncertainty... 

Other quantum mechanical approaches based on Gaussian wavepackets or coherent-state basis sets are those by Methiu and co-workers [46] and Martinazzo and co-workers [47] as well as the multiple spawning method developed by Martinez et al. [48] by which the moving wavepacket is expanded on a variable number of frozen Gaussians. Elsewhere [49] such an approach, especially conceived to be run on the fly, has been utilized for computing the ethylene spectrum by directly coupling it with electronic structure calculations. 

This resolution can be carried out at a number of energies in an interval around the energy where the Gaussian wavepacket is centered, i.e., around Eq = Tp klllm. When the flux ratio is calculated at a given energy using V (jc, y, z) and eq. (5.10), it is simply divided by the amount Ck of that energy component present in the initial wavepacket [156]. 

In this expression, Erf denotes the error function, while M2 is the full width at half maximum (FWHM) of the Gaussian probe pulse. The calculation of the total ionization probability S(t) therefore only requires the knowledge of the excited state wavepacket x e(r,R, f) at time t = T. Note that the origin of time (t = 0) is chosen here as the peak intensity of the pump pulse, and consequently the quantity Tin Eq. (1) corresponds to the pump-probe delay. 

Fig. 1. The autocorrelation function C(t) = (U (O)l I (i) is shown for a wavepacket initially prepared on the upper diabatic surface [7]. Panels (a) and (b) C(t) for the four core modes calculated by the standard MCTDH method for the model Hamiltonian Hy of Eq. (9), shown on different scales in the two panels. Panel (c) G-MCTDH calculation (bold line) as compared with standard MCTDH calculation (dotted line) for the composite system with four core modes (combined into two 2-dimensional particles

In practical calculations it is always possible to take the wavepacket y(0)) as a Gaussian function... 

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