Fighting the curse of dimensionality in first-principles semiclassical calculations: Non-local reference states for large number of dimensions

Abstract

Semiclassical methods face numerical challenges as the dimensionality of the system increases. In the general context of the theory of differential equations, this is known as the “curse of dimensionality.” In the present manuscript, we apply the recently-introduced multi-coherent states semiclassical initial value representation (MC-SC-IVR) approach to extend the applicability of first-principles semiclassical calculations. The proposed strategy involves the use of non-local coherent states with the goal of increasing accuracy in the Fourier transforms, and on the other hand, allows for the selection of peaks of different frequencies. The ability to filter desired peaks is important for analyzing the power spectra of complex systems. The MC-SC-IVR approach allows us to solve a 19-dimensional test system and to resolve on-the-fly the power spectra of the formaldehyde molecule with very few classical trajectories.

A novel Gaussian Binning (1GB) analysis of vibrational state distributions in highly excited ${\rm H}_\text{2}$H2O from reactive quenching of OH* by ${\rm H}_\text{2}$H2

“…In the case of FFT, momenta and positions are expressed in their Fourier series representations, while AS is based on a time-dependent perturbation of the initial normal mode Hamiltonian, which is smoothly increasing in time until the true Hamiltonian is obtained. AS has been successfully applied even to the reactive abstraction of hydrogen in the Cl+CH 4 reaction 11 and to H 2 CO. 12 Also, impressive calculations of quantum vibrational energies using the Semiclassical Initial Value Representation of Miller were recently reported for H 2 CO. 13,14 Nevertheless, the application of these methods to highly vibrationally excited states remains practically an unsolved problem.…”

Reproducing Deep Tunneling Splittings, Resonances, and Quantum Frequencies in Vibrational Spectra From a Handful of Direct Ab Initio Semiclassical Trajectories

“…In few words, the integration of eq (3) is reduced to a sum of trajectories starting from the convenient set of coherent state centers pictorially reported in Figure Figure 1. Single-well simulations 8,24,25 showed that the coherent state momenta do not need to be placed at an energy very close to the eigenvalues, because the Gaussian spreading of each coherent state is wide enough to include the peak energy shell 8,24 . The necessary quantum mechanical delocalization is provided by the presence of several coherent states on each well with energy both below and above the barrier.…”

“…To better identify each spectral peak, we enforce the A 1 symmetry into our coherent states combination of eq (5) by doubling the number of coherent states. 8,23 The A 1 vibrational levels are highlighted in Figure Figure 2 allowing us to prove the presence of tunneling splittings of the order at least of a few wavenumbers. Considering that the barrier height for the are examples of the possibility to correctly detect a deep tunneling effect.…”

“…The approximation of eq (4) (known as the separable approximation 23 ) has been proved to be much less computationally demanding than eq (3). 2,7,8,[23][24][25] The number of trajectories required for Monte Carlo convergence in eq (3) and eq (4) is of the order of thousands per each degree of freedom. 2 Unfortunately, this amount of computational demand is out of reach for carrying out direct ab initio molecular dynamics.…”

“…photodissociation or scattering processes. 6 In this manuscript, we report progress towards addressing these issues by developing the multiple coherent states time-averaging semiclassical initial value representation (MC-TA-SC-IVR) method [7][8][9][10] for multidimensional multi-well systems. In MC-TA-SC-IVR, the spectrum is computed from the autocorrelation function of a wavepacket evolved "on-the-fly".…”