Basis sets spawning

Certain additional numerical considerations should be satisfied before a spawning attempt is successful. First, in order to avoid unnecessary basis set expansion, we require that the parent of a spawned basis function have a population greater than or equal to Fmln, where the population of the ktU basis function on electronic state / is defined as... 

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... 

When the FMS method was first introduced, a series of test calculations were performed using analytical PESs. These calculations tested the numerical convergence with respect to the parameters that define the nuclear basis set (number of basis functions and their width) and the spawning algorithm (e.g., Xo and MULTISPAWN). These studies were used to validate the method, and therefore we refrained from making any approximations beyond the use of a... 

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.

Similarly, improvement in the accuracy of the nuclear dynamics would be fruitful. While in this review we have shown that, in the absence of any approximations beyond the use of a finite basis set, the multiple spawning treatment of the nuclear dynamics can border on numerically exact for model systems with up to 24 degrees of freedom, we certainly do not claim this for the ab initio applications presented here. In principle, we can carry out sequences of calculations with larger and larger nuclear basis sets in order to demonstrate that experimentally observable quantities have converged. In the context of AIMS, the cost of the electronic structure calculations precludes systematic studies of this convergence behavior for molecules with more than a few atoms. A similar situation obtains in time-independent quantum chemistry—the only reliable way to determine the accuracy of a particular calculation is to perform a sequence of... 

Using a GVB-OA-CAS(2/2) S wave function and a double-f quahty basis set, we simulated ethylene photochemistiy following n- n excitation. The AIMS simulations treat the excitation as instantaneous and centered at the absorption maximum. Hence, the initial-state nuclear basis functions are sampled from the groimd-state Wigner distribution in the harmonic approximation. Ten basis functions are used to describe the initial state. Overall, approximately 100 basis functions are spawned diuing the dynamics, and we follow the dynamics up to 0.5 ps (picoseconds) (using a time-step of 0.25 fs [femtoseconds]). 

The fifil multiple spawning (FMS) method [12, 13, 54-58] is an adaptive-basis-set approach that uses classically driven Gaussian functions. Simulations start with a relatively small basis set on the initial electronic state. The spawning procedure... 

Yang S, Coe JD, Kaduk B, Martinez TJ (2009) An optimal spawning algorithm for adaptive basis set expansion in nonadiabatic dynamics. J Chem Phys 130 134113... 

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. 

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