Quantum mechanics spawning

Most of the AIMD simulations described in the literature have assumed that Newtonian dynamics was sufficient for the nuclei. While this is often justified, there are important cases where the quantum mechanical nature of the nuclei is crucial for even a qualitative understanding. For example, tunneling is intrinsically quantum mechanical and can be important in chemistry involving proton transfer. A second area where nuclei must be described quantum mechanically is when the BOA breaks down, as is always the case when multiple coupled electronic states participate in chemistry. In particular, photochemical processes are often dominated by conical intersections [14,15], where two electronic states are exactly degenerate and the BOA fails. In this chapter, we discuss our recent development of the ab initio multiple spawning (AIMS) method which solves the elecronic and nuclear Schrodinger equations simultaneously this makes AIMD approaches applicable for problems where quantum mechanical effects of both electrons and nuclei are important. We present an overview of what has been achieved, and make a special effort to point out areas where further improvements can be made. Theoretical aspects of the AIMS method are... 

One can also ask about the relationship of the FMS method, as opposed to AIMS, with other wavepacket and semiclassical nonadiabatic dynamics methods. We first compare FMS to previous methods in cases where there is no spawning, and then proceed to compare with previous methods for nonadiabatic dynamics. We stress that we have always allowed for spawning in our applications of the method, and indeed the whole point of the FMS method is to address problems where localized nuclear quantum mechanical effects are important. Nevertheless, it is useful to place the method in context by asking how it relates to previous methods in the absence of its adaptive basis set character. There have been many attempts to use Gaussian basis functions in wavepacket dynamics, and we cannot mention all of these. Instead, we limit ourselves to those methods that we feel are most closely related to FMS, with apologies to those that are not included. A nice review that covers some of the... 

Figure 20. The (So —> S2) absorption spectrum of pyrazine for reduced three- and four-dimensional models (left and middle panels) and for a complete 24-vibrational model (right panel). For the three- and four-dimensional models, the exact quantum mechanical results (full line) are obtained using the Fourier method [43,45]. For the 24-dimensional model (nearly converged), quantum mechanical results are obtained using version 8 of the MCTDH program [210]. For all three models, the calculations are done in the diabatic representation. In the multiple spawning calculations (dashed lines) the spawning threshold 0,o) is set to 0.05, the initial size of the basis set for the three-, four-, and 24-dimensional models is 20, 40, and 60, and the total number of basis functions is limited to 900 (i.e., regardless of the magnitude of the effective nonadiabatic coupling, we do not spawn new basis functions once the total number of basis functions reaches 900).

Figure 21. The (So — S2) absorption spectrum of pyrazine for the reduced three-dimensional model using different spawning thresholds. Full line Exact quantum mechanical results. Dashed line Multiple spawning results for — 2.5, 5.0, 10, and 20. (All other computational details are as in Fig. 20.) As the spawning threshold is increased, the number of spawned basis functions decreases, the numerical effort decreases, and the accuracy of the result deteriorates (slowly). In this case, the final size of the basis set (at t — 0.5 ps) varies from 860 for 0 = 2.5 to 285 for 0 = 20.

M. D. Hack, A. M. Wensmann, D. G. Truhlar, M. Ben-Nun, and T. J. Martinez (2001) Comparison of full multiple spawning, trajectory surface hopping and converged quantum mechanics for electronically nonadiabatic dynamics. J. Chem. Phys. 115, p. 1172... 

Virshup AM, Punwong C, Pogorelov TV, Lindquist BA, Ko C, Martinez TJ (2009) Photodynamics in complex environments ab initio multiple spawning quantum mechanical/molecular mechanical dynamics. J Phys Chem B 113 3280-3291... 

The full multiple spawning (FMS) method has been developed as a genuine quantum mechanical method based on semiclassical considerations. The FMS method can be seen as an extension of semiclassical methods that brings back quantum character to the nuclear motion. Indeed, the nuclear wave function is not reduced to a product of delta functions centered on the nuclear positions but retains a minimum uncertainty relationship. The nuclear wave function is expressed as a sum of Born-Oppenheimer states ... 

The self-consistent field method was introduced in many-electron quantum mechanics by D. R. Hartree in Proc. Camb. Phil Soc. 24, 105 (1928), and his ideas spawned an intensive development, particularly by J. C. Slater and V. Fock. 

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. 

Figure 19. Diabatic reaction probability as a function of time (in femtoseconds) for the collinear A - - BC AB + C model of Fig. 1, at an excess energy of 0.109 eV. In both panels, the full line designates the exact quantum mechanical results and the dashed lines are multiple spawning results with different initial conditions. Right panel TDB using an initial basis set with 30 basis functions and 10 seeds. Left panel Regular basis set using an initial basis set with 10 basis functions. (Figure adapted from Ref. 41.)...

The conceptual framework for the - semiclassical simulation of ultrafast spectroscopic observables is provided by the Wigner representation of quantum mechanics [2, 3]. Specifically, for the ultrafast pump-probe spectroscopy using classical trajectories, methods based on the semiclassical limit of the Liouville-von Neumann equation for the time evolution of the vibronic density matrix have been developed [4-8]. Our approach [4,6-8] is related to the Liouville space theory of nonlinear spectroscopy developed by Mukamel et al. [9]. It is characterized by the ability to approximately describe quantum phenomena such as optical transitions by averaging over the ensemble of classical trajectories. Moreover, quantum corrections for the nuclear dynamics can be introduced in a systematic manner, e.g. in the framework of the entangled trajectory method [10,11]. Alternatively, these effects can be also accounted for in the framework of the multiple spawning method [12]. In general, trajectory-based methods require drastically less computational effort than full quantum mechanical calculations and provide physical insight in ultrafast processes. Additionally, they can be combined directly with quantum chemistry methods for the electronic structure calculations. 

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