Computational protocol: Evaluation of Representations and Response Models for Polarizable Force Fields

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Protocol publication

[…] Atom-centered point charges were first fitted to the baseline, unpolarized ESP using the standard RESP, then atom-centered point dipoles, superimposed on the point charges, were optimized separately for each polarized QM ESP. This model reports on the maximal accuracy attainable by any polarization model that uses the atom-centered point-dipole representation of polarization along with RESP baseline point charges. However, like the optimal-point-charge model above, it cannot be employed in simulations because it requires a new quantum calculation for each molecular configuration. The procedure for computing optimal point dipoles is described below.The ESP at rm due to the baseline RESP partial charges, qi, and the atom-centered point dipoles, μi, is given by6where well-known physical constants are omitted for simplicity and rij is as given in and7For each external charge position rk, where index k refers to the inducing charge, the error metric χk2 () may be written as8where ϕmk′ contains all quantities independent of the dipoles.Optimal dipoles are obtained by setting for all μi. This yields the following system of linear equations9Solving this matrix equation yields the desired atom-centered point dipoles optimized for the external charge position rk. The NumPy linalg.norm() function was used to find the solution to the matrix equation. [...] All models require global parameter optimization. For model 0 and models 3–6, the parameters were optimized to minimize R2, the mean of the squared potential deviations across all inducing charge sites k, for each molecule of interest (). For models 1 and 2, the parameters were optimized to minimize Rk2, the mean-squared potential deviations for each separate inducing charge position (). All optimizations were performed with a SciPy implementation of L-BFGS-B,, a gradient-based constrained minimization method. Charges are left unconstrained, whereas polarizabilities are restricted to positive values. For the inducible dipole models with fixed RESP charges (models 3 and 5), only the atom-typed polarizabilities, αti, require adjustment; for those with co-optimized point charges, the atom-typed charges, qti, are adjusted along with the polarizabilities.For each molecule, multiple optimizations were run with initial parameter values drawn from a uniform distribution using numpy.random.rand().() For model 0, five optimizations were run using initial charges randomly drawn from the range −1.0e to 1.0e. For models 3–6, 50 optimizations were run using initial polarizabilities randomly drawn from 0 to 10 bohr3 (0–1.482 Å3). The parameter set with the lowest value of R2 was selected as the optimum. When charges were co-optimized (models 4 and 6), the baseline RESP charges were used as their starting values. Only 10 optimizations were run for tyrosine as the calculations became time-consuming for this relatively large molecule. […]

Pipeline specifications

Software tools Numpy, SciPy
Applications Miscellaneous, WGS analysis