Direct Validation of the Single Step Classical to Quantum Free Energy Perturbation

Abstract

The use of the Zwanzig equation in the calculation of single-step perturbations to provide first principles (ab initio) Quantum Mechanics (QM) correction terms to Molecular Mechanics (MM) free energy cycles is well established. A rigorous test of the ability to converge such calculations would be very useful in this context. In this work, we perform a direct assessment of the convergence of the MM to QM perturbation, by attempting the reverse QM to MM perturbation. This required the generation of extensive QM molecular dynamics trajectories, using Density Functional Theory (DFT), within the representative biological system of a DNA Adenosine-Thymidine dimer.Over 100 ps of dynamics with the PBE functional and 6.25 ps with the LDA functional were generated. We demonstrate that calculations with total potential energies are very poorly convergent due to a lack of overlap of phase space distributions between ensembles. Whilst not theoretically rigorous, the use of interaction energies provides far superior convergence, despite the presence of non-classical charge transfer effects within the DFT trajectories. The source of poor phase space overlap for total energies is diagnosed, the approximate quantification of overlaps suggesting that even for the comparatively simple system considered here convergence of total energy calculations within a reasonable simulation time is unfeasible.

Absolute proton hydration free energy, surface potential of water, and redox potential of the hydrogen electrode from first principles: QM/MM MD free-energy simulations of sodium and potassium hydration

“…It has attracted renewed interest in recent years, 84,86 in the form of sampling approaches involving either the reweighting or perturbation of MM trajectories to the QM/MM level [185][186][187][188][189][190][191][192][193][194] (including the so-called paradynamics [195][196][197][198] or the continuous switching between MM and QM/MM interactions. 199 The advantage of adjusting the parameters of the MM model to improve the convergence has also been noted in this context.…”

Capturing the Role of Temperature and Entropy in Crystal Polymorphs Through Molecular Simulations

“…The simplest method to obtain free energy differences via Hamiltonian reweighting involves reweighting from one sampled to one unsampled state, generally referred to as exponential reweighting or the Zwanzig equation [191,238,240,[242][243][244][247][248][249][250]252], which is simply eq 6.1. This method produces asymptotically unbiased free energy differences in the limit of large sampling.…”

“…Warshel et al [238] introduced a method based on the pioneering work of Kirkwood [239] and Zwanzig [191] to calculate free energies in complex Hamiltonians indirectly by simulating in a cheap potential and calculating ensemble averages in the expensive potential using Boltzmann reweighting, thus avoiding direct simulation in the expensive potential [240][241][242][243][244][245][246][247][248][249][250][251][252][253][254][255][256][257][258]. The formula for estimating the ensemble average in a target state given samples in another state is [191]:…”

“…In previous studies employing Boltzmann reweighting, poor overlap was attributed to differences in the intramolecular interactions between the cheap and expensive potential [247,253]. This poor overlap has led researchers to using alternate reweighting strategies such as replacing the total energy difference in equation 4.1 with just the difference in intermolecular interactions, while holding the intramolecular interactions constant [247][248][249][250]. The target potential with this approach has intermolecular interactions that correspond with the expensive potential and intramolecular interactions at the cheaper level of theory.…”

“…The additional free energy cost to convert the intramolecular interactions from the cheap to expensive Hamiltonian is assumed to be negligible. The bias introduced by this approximation is in many cases small, [247,249,250] but there is presently no rigorous bounds or systematic prediction of the error introduced by this approximation. Therefore we will not investigate this approach here.…”