Assessing the Performance of the Diffusion Monte Carlo Method As Applied to the Water Monomer, Dimer, and Hexamer

Abstract

The diffusion Monte Carlo (DMC) method is applied to the water monomer, dimer, and hexamer using q-TIP4P/F, one of the most simple empirical water models with flexible monomers. The bias in the time step (Δτ) and population size (Nw) is investigated. For the binding energies, the bias in Δτ cancels nearly completely, whereas a noticeable bias in Nw remains. However, for the isotope shift (e.g, in the dimer binding energies between (H2O)2 and (D2O)2), the systematic errors in Nw do cancel. Consequently, very accurate results for the latter (within ∼0.01 kcal/mol) are obtained with moderate numerical effort (Nw ∼ 10(3)). For the water hexamer and its (D2O)6 isotopomer, the DMC results as a function of Nw are examined for the cage and prism isomers. For a given isomer, the issue of the walker population leaking out of the corresponding basin of attraction is addressed by using appropriate geometric constraints. The population size bias for the hexamer is more severe, and to maintain accuracy similar to that of the dimer, Nw must be increased by ∼2 orders of magnitude. Fortunately, when the energy difference between the cage and prism is taken, the biases cancel, thereby reducing the systematic errors to within ∼0.01 kcal/mol when using a population of Nw = 4.8 × 10(5) walkers. Consequently, a very accurate result for the isotope shift is also obtained. Notably, both the quantum and isotope effects for the prism-cage energy difference are small.

Binding energies from diffusion Monte Carlo for the MB-pol H2O and D2O dimer: A comparison to experimental values

“…The curves follow a pattern similar to that reported in Ref. 28 for the q-TIP4P/F dimer and monomer. A clear bias does exist for the ground state energies (E 0 ) in both ∆τ and N W , but for the reported range of time steps and walker numbers, the energy change is in the second or third positions beyond the decimal.…”

“…Here, we also comment that our walker number studies employ larger values of N W than might seem to be needed, as including more walkers in the population simultaneously reduces the statistical uncertainty as well as the systematic error 28 . Fig.…”

“…Additionally, in accordance with Ref. 28, the ground state energies for the isomers of the MB-pol hexamer will be computed.…”

“…In the limit of τ → ∞, ∆τ → 0, and N W → ∞, this distribution converges to the ground state wavefunction with E ref = E 0 , the ground state energy. In a recent paper 28 , we have undertaken a thorough analysis of the behavior and extent of the bias (systematic error) arising from the finite time step ∆τ , as well as the bias caused by a finite random walker population N W for the q-TIP4P/F 7 water monomer, dimer and hexamer. The time step bias in the estimate of E 0 vanishes slowly, and for the water monomer at ∆τ = 10.0 au is 0.015 kcal/mol.…”

“…In this work, we compute the energies for the MB-pol water monomer and dimer adhering to the DMC protocol of Ref. 28 and performing studies for the time step with ∆τ = 5, 10, and 15 au and for the random walker population with N W = 1.6×10 3 , 3.6×10 3 , and 1.96×10 4 . All of the DMC simulations were run to a maximum projection time of 2.0 × 10 6 au.…”