Reducing the Cost of Energy Differences in Variational Monte Carlo with Spotlight Sampling

Abstract

We investigate an approximate sampling scheme that can significantly reduce the cost scaling of variational Monte Carlo when it is employed to predict the energy differences associated with local chemical changes. Inspired by side-chaining and embedding methods, this spotlight sampling approach adopts an approximate fragmented Hamiltonian and correlated sampling to reduce cost scaling to the point that it is essentially linear with system size, with the potential to go sub-linear if certain conditions are met. In tests on bond stretching energies in alcohols, hydrogen dimer chains, and molecules with various degrees of $π$-system delocalization, we observe the anticipated linear scaling as well as an explicit cost crossover with standard variational Monte Carlo.

Figures (10)

• Figure 1: An illustration of spotlight sampling showing three examples of how centering the spotlight on one fragment (region A) creates partially lit (region B), dimly lit (region C), and unlit (region D) portions of the overall system. The system's total energy is the sum of fragment energies, each of which is evaluated in its own Markov chain in which the spotlight centers on that fragment and in which region D electrons are frozen. See text for details.
• Figure 2: Predictions for the energy required to stretch an O-H bond by 0.2 Bohr from standard VMC and from spotlight sampling with different buffer region choices.
• Figure 3: Overall spotlight sampling cost for $\mathrm{(H_2)_n}$ when employing a fixed number of samples per fragment. Each calculation used 1 core on an Intel Xeon Gold 6230 processor.
• Figure 4: Each fragment's average uncertainty contribution to the O-H bond stretching energy in 1-octanol. Each fragment contains one C atom and any H or O-H bonded to it.
• Figure 5: Each fragment's average uncertainty contribution to the O-H bond stretching energy in methanol-$\mathrm{(H_2)_{16}}$. Each molecule is one fragment.