Gaussian Multipole Model (GMM)

PCMRESP: A Method for Polarizable Force Field Parameter Development and Transferability of the Polarizable Gaussian Multipole Models Across Multiple Solvents

The transferability of force field parameters is a crucial aspect of high-quality force fields. Previous investigations have affirmed the transferability of electrostatic parameters derived from polarizable Gaussian multipole models (pGMs) when applied to water oligomer clusters, polypeptides across various conformations, and different sequences. In this study, we introduce PCMRESP, a novel method for electrostatic parametrization in solution, intended for the development of polarizable force fields. We utilized this method to assess the transferability of three models: a fixed charge model and two variants of pGM models. Our analysis involved testing these models on 377 small molecules and 100 tetra-peptides in five representative dielectric environments: gas, diethyl ether, dichloroethane, acetone, and water. Our findings reveal that the inclusion of atomic polarization significantly enhances transferability and the incorporation of permanent atomic dipoles, in the form of covalent bond dipoles, leads to further improvements. Moreover, our tests on dual-solvent strategies demonstrate consistent transferability for all three models, underscoring the robustness of the dual-solvent approach. In contrast, an evaluation of the traditional HF/6-31G* method indicates poor transferability for the pGM-ind and pGM-perm models, suggesting the limitations of this conventional approach.

Assessment of Amino Acid Electrostatic Parametrizations of the Polarizable Gaussian Multipole Model

Accurate parametrization of amino acids is pivotal for the development of reliable force fields for molecular modeling of biomolecules such as proteins. This study aims to assess amino acid electrostatic parametrizations with the polarizable Gaussian Multipole (pGM) model by evaluating the performance of the pGM-perm (with atomic permanent dipoles) and pGM-ind (without atomic permanent dipoles) variants compared to the traditional RESP model. The 100-conf-combterm fitting strategy on tetrapeptides was adopted, in which (1) all peptide bond atoms (-CO-NH-) share identical set of parameters and (2) the total charges of the two terminal N-acetyl (ACE) and N-methylamide (NME) groups were set to neutral. The accuracy and transferability of electrostatic parameters across peptides with varying lengths and real-world examples were examined. The results demonstrate the enhanced performance of the pGM-perm model in accurately representing the electrostatic properties of amino acids. This insight underscores the potential of the pGM-perm model and the 100-conf-combterm strategy for the future development of the pGM force field.

Streamlining and Optimizing Strategies of Electrostatic Parameterization

Accurate characterization of electrostatic interactions is crucial in molecular simulation. Various methods and programs have been developed to obtain electrostatic parameters for additive or polarizable models to replicate electrostatic properties obtained from experimental measurements or theoretical calculations. Electrostatic potentials (ESPs), a set of physically well-defined observables from quantum mechanical (QM) calculations, are well suited for optimization efforts due to the ease of collecting a large amount of conformation-dependent data. However, a reliable set of QM ESP computed at an appropriate level of theory and atomic basis set is necessary. In addition, despite the recent development of the PyRESP program for electrostatic parameterizations of induced dipole-polarizable models, the time-consuming and error-prone input file preparation process has limited the widespread use of these protocols. This work aims to comprehensively evaluate the quality of QM ESPs derived by eight methods, including wave function methods such as Hartree-Fock (HF), second-order Møller-Plesset (MP2), and coupled cluster-singles and doubles (CCSD), as well as five hybrid density functional theory (DFT) methods, used in conjunction with 13 different basis sets. The highest theory levels CCSD/aug-cc-pV5Z (a5z) and MP2/aug-cc-pV5Z (a5z) were selected as benchmark data over two homemade data sets. The results show that the hybrid DFT method, ωB97X-D, combined with the aug-cc-pVTZ (a3z) basis set, performs well in reproducing ESPs while taking both accuracy and efficiency into consideration. Moreover, a flexible and user-friendly program called PyRESP_GEN was developed to streamline input file preparation. The restraining strengths, along with strategies for polarizable Gaussian multipole (pGM) model parameterizations, were also optimized. These findings and the program presented in this work facilitate the development and application of induced dipole-polarizable models, such as pGM models, for molecular simulations of both chemical and biological significance.

Fitting Potential Energy Surfaces by Learning the Charge Density Matrix with Permutationally Invariant Polynomials

The electronic energy in the Hartree-Fock (HF) theory is the trace of the product of the charge density matrix (CDM) with the one-electron and two-electron matrices represented in an atomic orbital basis, where the two-electron matrix is also a function of the same CDM. In this work, we examine a formalism of analytic representation of a generic molecular potential energy surface (PES) as a sum of a linearly parameterized HF and a correction term, the latter formally representing the electron correlation energy, also linearly parameterized, by expressing the elements of CDM using permutationally invariant polynomials (PIPs). We show on a variety of numerical examples, ranging from exemplary two-electron systems HeH+ and H3+ to the more challenging cases of methanium (CH5+) fragmentation and high-energy tautomerization of formamide to formimidic acid that such a formulation requires significantly fewer, 10-20% of PIPs, to accomplish the same accuracy of the fit as the conventional representation at practically the same computational cost.

Transferability of the Electrostatic Parameters of the Polarizable Gaussian Multipole Model

Accuracy and transferability are the two highly desirable properties of molecular mechanical force fields. Compared with the extensively used point-charge additive force fields that apply fixed atom-centered point partial charges to model electrostatic interactions, polarizable force fields are thought to have the advantage of modeling the atomic polarization effects. Previous works have demonstrated the accuracy of the recently developed polarizable Gaussian multipole (pGM) models. In this work, we assessed the transferability of the electrostatic parameters of the pGM models with (pGM-perm) and without (pGM-ind) atomic permanent dipoles in terms of reproducing the electrostatic potentials surrounding molecules/oligomers absent from electrostatic parameterizations. Encouragingly, both the pGM-perm and pGM-ind models show significantly improved transferability than the additive model in the tests (1) from water monomer to water oligomer clusters; (2) across different conformations of amino acid dipeptides and tetrapeptides; (3) from amino acid tetrapeptides to longer polypeptides; and (4) from nucleobase monomers to Watson-Crick base pair dimers and tetramers. Furthermore, we demonstrated that the double-conformation fittings using amino acid tetrapeptides in the αR and β conformations can result in good transferability not only across different tetrapeptide conformations but also from tetrapeptides to polypeptides with lengths ranging from 1 to 20 repetitive residues for both the pGM-ind and pGM-perm models. In addition, the observation that the pGM-ind model has significantly better accuracy and transferability than the point-charge additive model, even though they have an identical number of parameters, strongly suggest the importance of intramolecular polarization effects. In summary, this and previous works together show that the pGM models possess both accuracy and transferability, which are expected to serve as foundations for the development of next-generation polarizable force fields for modeling various polarization-sensitive biological systems and processes.

Mechanistic study of the formation of arbutin polymorphs and solvates

Six solid forms of arbutin are successfully obtained and four single crystal structures are reported for the first time. Further, the formation mechanism of arbutin solvates is proposed and rationalized.

Accurate Reproduction of Quantum Mechanical Many-Body Interactions in Peptide Main-Chain Hydrogen-Bonding Oligomers by the Polarizable Gaussian Multipole Model

A key advantage of polarizable force fields is their ability to model the atomic polarization effects that play key roles in the atomic many-body interactions. In this work, we assessed the accuracy of the recently developed polarizable Gaussian Multipole (pGM) models in reproducing quantum mechanical (QM) interaction energies, many-body interaction energies, as well as the nonadditive and additive contributions to the many-body interactions for peptide main-chain hydrogen-bonding conformers, using glycine dipeptide oligomers as the model systems. Two types of pGM models were considered, including that with (pGM-perm) and without (pGM-ind) permanent atomic dipoles. The performances of the pGM models were compared with several widely used force fields, including two polarizable (Amoeba13 and ff12pol) and three additive (ff19SB, ff15ipq, and ff03) force fields. Encouragingly, the pGM models outperform all other force fields in terms of reproducing QM interaction energies, many-body interaction energies, as well as the nonadditive and additive contributions to the many-body interactions, as measured by the root-mean-square errors (RMSEs) and mean absolute errors (MAEs). Furthermore, we tested the robustness of the pGM models against polarizability parameterization errors by employing alternative polarizabilities that are either scaled or obtained from other force fields. The results show that the pGM models with alternative polarizabilities exhibit improved accuracy in reproducing QM many-body interaction energies as well as the nonadditive and additive contributions compared with other polarizable force fields, suggesting that the pGM models are robust against the errors in polarizability parameterizations. This work shows that the pGM models are capable of accurately modeling polarization effects and have the potential to serve as templates for developing next-generation polarizable force fields for modeling various biological systems.

Learning Atomic Multipoles: Prediction of the Electrostatic Potential with Equivariant Graph Neural Networks

The accurate description of electrostatic interactions remains a challenging problem for classical potential-energy functions. The commonly used fixed partial-charge approximation fails to reproduce the electrostatic potential at short range due to its insensitivity to conformational changes and anisotropic effects. At the same time, possibly more accurate machine-learned (ML) potentials struggle with the long-range behavior due to their inherent locality ansatz. Employing a multipole expansion offers in principle an exact treatment of the electrostatic potential such that the long-range and short-range electrostatic interactions can be treated simultaneously with high accuracy. However, such an expansion requires the calculation of the electron density using computationally expensive quantum-mechanical (QM) methods. Here, we introduce an equivariant graph neural network (GNN) to address this issue. The proposed model predicts atomic multipoles up to the quadrupole, circumventing the need for expensive QM computations. By using an equivariant architecture, the model enforces the correct symmetry by design without relying on local reference frames. The GNN reproduces the electrostatic potential of various systems with high fidelity. Possible uses for such an approach include the separate treatment of long-range interactions in ML potentials, the analysis of electrostatic potential surfaces, and static multipoles in polarizable force fields.

Fast Evaluation of Two-Center Integrals over Gaussian Charge Distributions and Gaussian Orbitals with General Interaction Kernels

We present efficient algorithms for computing two-center integrals and integral derivatives, with general interaction kernels K(r₁₂), over Gaussian charge distributions of general angular momenta l. While formulated in terms of traditional ab initio integration techniques, full derivations and required secondary information, as well as a reference implementation, are provided to make the content accessible to other fields. Concretely, the presented algorithms are based on an adaption of the McMurchie-Davidson Recurrence Relation (MDRR) combined with analytical properties of the solid harmonic transformation; this obviates all intermediate recurrences except the adapted MDRR itself, and allows it to be applied to fully contracted auxiliary kernel integrals. The technique is particularly well-suited for semiempirical molecular orbital methods, where it can serve as a more general and efficient replacement of Slater-Koster tables, and for first-principles quantum chemistry methods employing density fitting. But the formalism's high efficiency and ability of handling general interaction kernels K(r₁₂) and multipolar Gaussian charge distributions may also be of interest for modeling electrostatic interactions and short-range exchange and charge penetration effects in classical force fields and model potentials. With the presented technique, a 4894 × 4894 univ-JKFIT Coulomb matrix J_AB = (A|1/r₁₂|B) (183 MiB) can be computed in 50 ms on a Q2'2018 notebook CPU, without any screening or approximations.

Molecular Dynamics Simulations of Ionic Liquids and Electrolytes Using Polarizable Force Fields

Many applications in chemistry, biology, and energy storage/conversion research rely on molecular simulations to provide fundamental insight into structural and transport properties of materials with high ionic concentrations. Whether the system is comprised entirely of ions, like ionic liquids, or is a mixture of a polar solvent with a salt, e.g., liquid electrolytes for battery applications, the presence of ions in these materials results in strong local electric fields polarizing solvent molecules and large ions. To predict properties of such systems from molecular simulations often requires either explicit or mean-field inclusion of the influence of polarization on electrostatic interactions. In this manuscript, we review the pros and cons of different treatments of polarization ranging from the mean-field approaches to the most popular explicit polarization models in molecular dynamics simulations of ionic materials. For each method, we discuss their advantages and disadvantages and emphasize key assumptions as well as their adjustable parameters. Strategies for the development of polarizable models are presented with a specific focus on extracting atomic polarizabilities. Finally, we compare simulations using polarizable and nonpolarizable models for several classes of ionic systems, discussing the underlying physics that each approach includes or ignores, implications for implementation and computational efficiency, and the accuracy of properties predicted by these methods compared to experiments.

Development of Polarizable Gaussian Model for Molecular Mechanical Calculations I: Atomic Polarizability Parameterization To Reproduce ab Initio Anisotropy

A set of atomic polarizability parameters for a new polarizable Gaussian model (pGM) has been developed with the goal to accurately reproduce the polarizability anisotropy, taking advantage of its ability to attenuate all short-range electrostatic interactions, by fitting the ab initio molecular polarizability tensors ( A _pq) calculated at the B3LYP/aug-cc-pVTZ level. For comparison, we also rederived the parameters for three Thole models in which the 1-2 (bonded), 1-3 (separated by two bonds), and 1-4 (separated by three bonds) interactions are fully included. The average percent errors (APEs) of molecular polarizability tensors for 4842 molecules or dimers are 2.98, 3.76, 3.28, and 3.82% for the pGM, Thole linear, Thole exponential, and Thole Amoeba models, respectively, with atom-type independent, universal screening factors (USF). The APEs are reduced further to 2.30, 2.69, 2.25, and 2.48% for the four corresponding polarizable models with atom-type dependent, variable screening factors (VSF). It is encouraging that the pGM with variable screening factors achieved APEs of 1.83 for 1155 amino acid analogs, dipeptides, and tetrapeptides, 1.39 for 28 nucleic acid bases, 0.708 for 1464 water clusters, and 1.99 for 85 dimers of water and biological building blocks. Compared to the new set of models, the APEs of the old Thole models that were fitted to isotropic molecular polarizabilities are 8.7% for set A (without the 1-2 and 1-3 interactions) and 6.3% for set D (with the 1-2 and 1-3 interactions) models, respectively. MPAD, a metric of molecular polarization anisotropy difference based on the diagonal terms of molecular polarizability tensors was defined and applied to assess the polarizable models in reproducing the ab initio molecular polarization anisotropy. The MPADs are 3.71, 4.70, 4.11, and 4.77% for the pGM, Thole linear, Thole exponential, and Thole Amoeba USF models, respectively. The APEs are reduced further to 2.85, 3.58, 2.90, and 3.15% for the four corresponding VSF models. Thus, the new pGM and Thole models notably improve molecular polarization anisotropy. Since pGM attenuates all short-range electrostatic interactions, its application is expected to improve stability in charge fitting, energy, and force calculations and the accuracy of multibody polarization.

Polarizable Drude Model with s-Type Gaussian or Slater Charge Density for General Molecular Mechanics Force Fields

Gas-phase electric properties of molecules can be computed routinely using wave function methods or density functional theory (DFT). However, these methods remain computationally expensive for high-throughput screening of the vast chemical space of virtual compounds. Therefore, empirical force fields are a more practical choice in many cases, particularly since force field methods allow one to routinely predict the physicochemical properties in the condensed phases. This work presents Drude polarizable models, to increase the physical realism in empirical force fields, where the core particle is treated as a point charge and the Drude particle is treated either as a 1 s-Gaussian or a ns-Slater ( n = 1, 2, 3) charge density. Systematic parametrization to large high-quality quantum chemistry data obtained from the open access Alexandria Library ( https://doi.org/10.5281/zenodo.1004711 ) ensures the transferability of these parameters. The dipole moments and isotropic polarizabilities of the isolated molecules predicted by the proposed Drude models are in agreement with experiment with accuracy similar to DFT calculations at the B3LYP/aug-cc-pVTZ level of theory. The results show that the inclusion of explicit polarization into the models reduces the root-mean-square deviation with respect to DFT calculations of the predicted dipole moments of 152 dimers and clusters by more than 50%. Finally, we show that the accuracy of the electrostatic interaction energy of the water dimers can be improved systematically by the introduction of polarizable smeared charges as a model for charge penetration.

Analytic Excited State Gradients for the QM/MM Polarizable Embedded Second-Order Algebraic Diagrammatic Construction for the Polarization Propagator PE-ADC(2)

An implementation of a QM/MM embedding in a polarizable environment is presented for second-order Møller-Plesset perturbation theory, MP2, for ground state energies and molecular gradients and for the second-order Algebraic Diagrammatic Construction, ADC(2), for excitation energies and excited state molecular gradients. In this implementation of PE-MP2 and PE-ADC(2), the polarizable embedded Hartree-Fock wave function is used as uncorrelated reference state. The polarization-correlation cross terms for the ground and excited states are included in this model via an approximate coupling density. A Lagrangian formulation is used to derive the relaxed electron densities and molecular gradients. The resolution-of-the-identity approximation speeds up the calculation of four-index electron repulsion integrals in the molecular orbital basis. As a first application, the method is used to study the photophysical properties of host-guest complexes where the accuracy and weaknesses of the model are also critically examined. It is demonstrated that the ground state geometries of the full quantum mechanical calculation for the supermolecule can be well reproduced. For excited state geometries, the deviations from the supermolecular calculation are slightly larger, but still the environment effects are captured qualitatively correctly, and energy gaps between the ground and excited states are obtained with sufficient accuracy.

QM/MM Simulations with the Gaussian Electrostatic Model: A Density-based Polarizable Potential

The use of advanced polarizable potentials in quantum mechanical/molecular mechanical (QM/MM) simulations has been shown to improve the overall accuracy of the calculation. We have developed a density-based potential called the Gaussian electrostatic model (GEM), which has been shown to provide very accurate environments for QM wave functions in QM/MM. In this contribution we present a new implementation of QM/GEM that extends our implementation to include all components (Coulomb, exchange-repulsion, polarization, and dispersion) for the total intermolecular interaction energy in QM/MM calculations, except for the charge-transfer term. The accuracy of the method is tested using a subset of water dimers from the water dimer potential energy surface reported by Babin et al. ( J. Chem. Theory Comput. 2013 9, 5395-5403). Additionally, results of the new implementation are contrasted with results obtained with the classical AMOEBA potential. Our results indicate that GEM provides an accurate MM environment with average root-mean-square error <0.15 kcal/mol for every intermolecular interaction energy component compared with SAPT2+3/aug-cc-pVTZ reference calculations.

A GPU-Accelerated Parameter Interpolation Thermodynamic Integration Free Energy Method

There has been a resurgence of interest in free energy methods motivated by the performance enhancements offered by molecular dynamics (MD) software written for specialized hardware, such as graphics processing units (GPUs). In this work, we exploit the properties of a parameter-interpolated thermodynamic integration (PI-TI) method to connect states by their molecular mechanical (MM) parameter values. This pathway is shown to be better behaved for Mg2+ → Ca2+ transformations than traditional linear alchemical pathways (with and without soft-core potentials). The PI-TI method has the practical advantage that no modification of the MD code is required to propagate the dynamics, and unlike with linear alchemical mixing, only one electrostatic evaluation is needed (e.g., single call to particle-mesh Ewald) leading to better performance. In the case of AMBER, this enables all the performance benefits of GPU-acceleration to be realized, in addition to unlocking the full spectrum of features available within the MD software, such as Hamiltonian replica exchange (HREM). The TI derivative evaluation can be accomplished efficiently in a post-processing step by reanalyzing the statistically independent trajectory frames in parallel for high throughput. We also show how one can evaluate the particle mesh Ewald contribution to the TI derivative evaluation without needing to perform two reciprocal space calculations. We apply the PI-TI method with HREM on GPUs in AMBER to predict p Ka values in double stranded RNA molecules and make comparison with experiments. Convergence to under 0.25 units for these systems required 100 ns or more of sampling per window and coupling of windows with HREM. We find that MM charges derived from ab initio QM/MM fragment calculations improve the agreement between calculation and experimental results.

Performance of the AMOEBA Water Model in the Vicinity of QM Solutes: A Diagnosis Using Energy Decomposition Analysis

The importance of incorporating solvent polarization effects into the modeling of solvation processes has been well-recognized, and therefore a new generation of hybrid quantum mechanics/molecular mechanics (QM/MM) approaches that accounts for this effect is desirable. We present a fully self-consistent, mutually polarizable QM/MM scheme using the AMOEBA force field, in which the total energy of the system is variationally minimized with respect to both the QM electronic density and the MM induced dipoles. This QM/AMOEBA model is implemented through the Q-Chem/LibEFP code interface and then applied to the evaluation of solute-solvent interaction energies for various systems ranging from the water dimer to neutral and ionic solutes (NH₃, NH₄⁺, CN^-) surrounded by increasing numbers of water molecules (up to 100). In order to analyze the resulting interaction energies, we also utilize an energy decomposition analysis (EDA) scheme which identifies contributions from permanent electrostatics, polarization, and van der Waals (vdW) interaction for the interaction between the QM solute and the solvent molecules described by AMOEBA. This facilitates a component-wise comparison against full QM calculations where the corresponding energy components are obtained via a modified version of the absolutely localized molecular orbitals (ALMO)-EDA. The results show that the present QM/AMOEBA model can yield reasonable solute-solvent interaction energies for neutral and cationic species, while further scrutiny reveals that this accuracy highly relies on the delicate balance between insufficiently favorable permanent electrostatics and softened vdW interaction. For anionic solutes where the charge penetration effect becomes more pronounced, the QM/MM interface turns out to be unbalanced. These results are consistent with and further elucidate our findings in a previous study using a slightly different QM/AMOEBA model ( Dziedzic et al. J. Chem. Phys. 2016 , 145 , 124106 ). The implications of these results for further refinement of this model are also discussed.

Channeling through Two Stacked Guanine Quartets of One and Two Alkali Cations in the Li+, Na+, K+, and Rb+ Series. Assessment of the Accuracy of the SIBFA Anisotropic Polarizable Molecular Mechanics Potential

Stacking of guanine quartets (GQs) can trigger the formation of DNA or RNA quadruple helices, which play numerous biochemical roles. The GQs are stabilized by alkali cations, mainly K⁺ and Na⁺, which can reside in, or channel through, the central axis of the GQ stems. Further, ion conduction through GQ wires can be leveraged for nanochemistry applications. G-quadruplex systems have been extensively studied by classical molecular dynamics (MD) simulations using pair-additive force fields or by quantum-chemical (QC) calculations. However, the non-polarizable force fields are very approximate, while QC calculations lack the necessary sampling. Thus, ultimate description of GQ systems would require long-enough simulations using advanced polarizable molecular mechanics (MM). However, to perform such calculations, it is first mandatory to evaluate the method's accuracy using benchmark QC. We report such an evaluation for SIBFA polarizable MM, bearing on the channeling (movement) of an alkali cation (Li⁺, Na⁺, K⁺, or Rb⁺) along the axis of two stacked G quartets interacting with either one or two ions. The QC energy profiles display markedly different features depending upon the cation but can be retrieved in the majority of cases by the SIBFA profiles. An appropriate balance of first-order (electrostatic and short-range repulsion) and second-order (polarization, charge-transfer, and dispersion) contributions within ΔE is mandatory. With two cations in the channel, the relative weights of the second-order contributions increase steadily upon increasing the ion size. In the G8 complexes with two K⁺ or two Rb⁺ cations, the sum of polarization and charge-transfer exceeds the first order terms for all ion positions.

Toward polarizable AMOEBA thermodynamics at fixed charge efficiency using a dual force field approach: application to organic crystals

First principles prediction of the structure, thermodynamics and solubility of organic molecular crystals, which play a central role in chemical, material, pharmaceutical and engineering sciences, challenges both potential energy functions and sampling methodologies. Here we calculate absolute crystal deposition thermodynamics using a novel dual force field approach whose goal is to maintain the accuracy of advanced multipole force fields (e.g. the polarizable AMOEBA model) while performing more than 95% of the sampling in an inexpensive fixed charge (FC) force field (e.g. OPLS-AA). Absolute crystal sublimation/deposition phase transition free energies were determined using an alchemical path that grows the crystalline state from a vapor reference state based on sampling with the OPLS-AA force field, followed by dual force field thermodynamic corrections to change between FC and AMOEBA resolutions at both end states (we denote the three step path as AMOEBA/FC). Importantly, whereas the phase transition requires on the order of 200 ns of sampling per compound, only 5 ns of sampling was needed for the dual force field thermodynamic corrections to reach a mean statistical uncertainty of 0.05 kcal mol-1. For five organic compounds, the mean unsigned error between direct use of AMOEBA and the AMOEBA/FC dual force field path was only 0.2 kcal mol-1 and not statistically significant. Compared to experimental deposition thermodynamics, the mean unsigned error for AMOEBA/FC (1.4 kcal mol-1) was more than a factor of two smaller than uncorrected OPLS-AA (3.2 kcal mol-1). Overall, the dual force field thermodynamic corrections reduced condensed phase sampling in the expensive force field by a factor of 40, and may prove useful for protein stability or binding thermodynamics in the future.

Torque and atomic forces for Cartesian tensor atomic multipoles with an application to crystal unit cell optimization

New equations for torque and atomic force are derived for use in flexible molecule force fields with atomic multipoles. The expressions are based on Cartesian tensors with arbitrary multipole rank. The standard method for rotating Cartesian tensor multipoles and calculating torque is to first represent the tensor with n indexes and 3(n) redundant components. In this work, new expressions for directly rotating the unique (n + 1)(n + 2)/2 Cartesian tensor multipole components Θpqr are given by introducing Cartesian tensor rotation matrix elements X(R). A polynomial expression and a recursion relation for X(R) are derived. For comparison, the analogous rotation matrix for spherical tensor multipoles are the Wigner functions D(R). The expressions for X(R) are used to derive simple equations for torque and atomic force. The torque and atomic force equations are applied to the geometry optimization of small molecule crystal unit cells. In addition, a discussion of computational efficiency as a function of increasing multipole rank is given for Cartesian tensors. © 2016 Wiley Periodicals, Inc.

Beyond Born-Mayer: Improved Models for Short-Range Repulsion in ab Initio Force Fields

Short-range repulsion within intermolecular force fields is conventionally described by either Lennard-Jones (A/r(12)) or Born-Mayer (A exp(-Br)) forms. Despite their widespread use, these simple functional forms are often unable to describe the interaction energy accurately over a broad range of intermolecular distances, thus creating challenges in the development of ab initio force fields and potentially leading to decreased accuracy and transferability. Herein, we derive a novel short-range functional form based on a simple Slater-like model of overlapping atomic densities and an iterated stockholder atom (ISA) partitioning of the molecular electron density. We demonstrate that this Slater-ISA methodology yields a more accurate, transferable, and robust description of the short-range interactions at minimal additional computational cost compared to standard Lennard-Jones or Born-Mayer approaches. Finally, we show how this methodology can be adapted to yield the standard Born-Mayer functional form while still retaining many of the advantages of the Slater-ISA approach.

Generalized and efficient algorithm for computing multipole energies and gradients based on Cartesian tensors

Accurate representation of intermolecular forces has been the central task of classical atomic simulations, known as molecular mechanics. Recent advancements in molecular mechanics models have put forward the explicit representation of permanent and/or induced electric multipole (EMP) moments. The formulas developed so far to calculate EMP interactions tend to have complicated expressions, especially in Cartesian coordinates, which can only be applied to a specific kernel potential function. For example, one needs to develop a new formula each time a new kernel function is encountered. The complication of these formalisms arises from an intriguing and yet obscured mathematical relation between the kernel functions and the gradient operators. Here, I uncover this relation via rigorous derivation and find that the formula to calculate EMP interactions is basically invariant to the potential kernel functions as long as they are of the form f(r), i.e., any Green's function that depends on inter-particle distance. I provide an algorithm for efficient evaluation of EMP interaction energies, forces, and torques for any kernel f(r) up to any arbitrary rank of EMP moments in Cartesian coordinates. The working equations of this algorithm are essentially the same for any kernel f(r). Recently, a few recursive algorithms were proposed to calculate EMP interactions. Depending on the kernel functions, the algorithm here is about 4-16 times faster than these algorithms in terms of the required number of floating point operations and is much more memory efficient. I show that it is even faster than a theoretically ideal recursion scheme, i.e., one that requires 1 floating point multiplication and 1 addition per recursion step. This algorithm has a compact vector-based expression that is optimal for computer programming. The Cartesian nature of this algorithm makes it fit easily into modern molecular simulation packages as compared with spherical coordinate-based algorithms. A software library based on this algorithm has been implemented in C++11 and has been released.

A Universal and Straightforward Approach to Include Penetration Effects in Electrostatic Interaction Energy Estimation

To compensate for the lack of the explicit treatment of charge penetration in classical force fields, we propose a new charge-distribution model based on a promolecule augmented with point charges (aug-PROmol). It relies on a superposition of spherical atomic electron densities obtained for each chemical element from SCF energy optimized atomic orbitals. Atomic densities are further rescaled by partial point charges computed from fits to the molecular electrostatic potential. Aug-PROmol was tested on the S66 benchmark dataset extended to nonequilibrium geometries (J. Chem. Theory Comput., 2011, 7, 3466). The model does not need any additional parametrization other than point charges. Despite its simplicity, aug-PROmol approximates the electrostatic energy with good agreement (RMSE=0.76 kcal mol(-1) to DFT-SAPT with B3LYP/aug-cc-pVTZ).

Scalable improvement of SPME multipolar electrostatics in anisotropic polarizable molecular mechanics using a general short-range penetration correction up to quadrupoles

We propose a general coupling of the Smooth Particle Mesh Ewald SPME approach for distributed multipoles to a short-range charge penetration correction modifying the charge-charge, charge-dipole and charge-quadrupole energies. Such an approach significantly improves electrostatics when compared to ab initio values and has been calibrated on Symmetry-Adapted Perturbation Theory reference data. Various neutral molecular dimers have been tested and results on the complexes of mono- and divalent cations with a water ligand are also provided. Transferability of the correction is adressed in the context of the implementation of the AMOEBA and SIBFA polarizable force fields in the TINKER-HP software. As the choices of the multipolar distribution are discussed, conclusions are drawn for the future penetration-corrected polarizable force fields highlighting the mandatory need of non-spurious procedures for the obtention of well balanced and physically meaningful distributed moments. Finally, scalability and parallelism of the short-range corrected SPME approach are addressed, demonstrating that the damping function is computationally affordable and accurate for molecular dynamics simulations of complex bio- or bioinorganic systems in periodic boundary conditions.

Introducing DDEC6 atomic population analysis: part 1. Charge partitioning theory and methodology

We introduce a new atomic population analysis method that performs exceptionally well across an extremely broad range of periodic and non-periodic material types.

Screened Electrostatic Interactions in Molecular Mechanics

In a typical application of molecular mechanics (MM), the electrostatic interactions are calculated from parametrized partial atomic charges treated as point charges interacting by radial Coulomb potentials. This does not usually yield accurate electrostatic interactions at van der Waals distances, but this is compensated by additional parametrized terms, for example Lennard-Jones potentials. In the present work, we present a scheme involving radial screened Coulomb potentials that reproduces the accurate electrostatics much more accurately. The screening accounts for charge penetration of one subsystem's charge cloud into that of another subsystem, and it is incorporated into the interaction potential in a way similar to what we proposed in a previous article (J. Chem. Theory Comput. 2010, 6, 3330) for combined quantum mechanical and molecular mechanical (QM/MM) simulations, but the screening parameters are reoptimized for MM. The optimization is carried out with electrostatic-potential-fitted partial atomic charges, but the optimized parameters should be useful with any realistic charge model. In the model we employ, the charge density of an atom is approximated as the sum of a point charge representing the nucleus and inner electrons and a smeared charge representing the outermost electrons; in particular, for all atoms except hydrogens, the smeared charge represents the two outermost electrons in the present model. We find that the charge penetration effect can cause very significant deviations from the popular point-charge model, and by comparison to electrostatic interactions calculated by symmetry-adapted perturbation theory, we find that the present results are considerably more accurate than point-charge electrostatic interactions. The mean unsigned error in electrostatics for a large and diverse data set (192 interaction energies) decreases from 9.2 to 3.3 kcal/mol, and the error in the electrostatics for 10 water dimers decreases from 1.7 to 0.5 kcal/mol. We could have decreased the average errors further, but at the cost of sometimes significantly overestimating the screening; instead we chose a more conservative (safer) parametrization that systematically underestimates the screening (which by definition means it improves over point charges) and only occasionally overestimates it. Despite this conservative choice, we find that the screened MM method is even more accurate for the electrostatics than unscreened QM/MM calculations. This new method is easy to implement in any MM program, and it can be used to develop more physical force fields for molecular simulations.

An efficient algorithm for multipole energies and derivatives based on spherical harmonics and extensions to particle mesh Ewald

Next-generation molecular force fields deliver accurate descriptions of non-covalent interactions by employing more elaborate functional forms than their predecessors. Much work has been dedicated to improving the description of the electrostatic potential (ESP) generated by these force fields. A common approach to improving the ESP is by augmenting the point charges on each center with higher-order multipole moments. The resulting anisotropy greatly improves the directionality of the non-covalent bonding, with a concomitant increase in computational cost. In this work, we develop an efficient strategy for enumerating multipole interactions, by casting an efficient spherical harmonic based approach within a particle mesh Ewald (PME) framework. Although the derivation involves lengthy algebra, the final expressions are relatively compact, yielding an approach that can efficiently handle both finite and periodic systems without imposing any approximations beyond PME. Forces and torques are readily obtained, making our method well suited to modern molecular dynamics simulations.

On combining Thole's induced point dipole model with fixed charge distributions in molecular mechanics force fields

The Thole induced point dipole model is combined with three different point charge fitting methods, Merz-Kollman (MK), charges from electrostatic potentials using a grid (CHELPG), and restrained electrostatic potential (RESP), and two multipole algorithms, distributed multipole analysis (DMA) and Gaussian multipole model (GMM), which can be used to describe the electrostatic potential (ESP) around molecules in molecular mechanics force fields. This is done to study how the different methods perform when intramolecular polarizability contributions are self-consistently removed from the fitting done in the force field parametrization. It is demonstrated that the polarizable versions of the partial charge models provide a good compromise between accuracy and computational efficiency in describing the ESP of small organic molecules undergoing conformational changes. For the point charge models, the inclusion of polarizability reduced the the average root mean square error of ESP over the test set by 4-10%.

Multipolar Ewald methods, 1: theory, accuracy, and performance

The Ewald, Particle Mesh Ewald (PME), and Fast Fourier–Poisson (FFP) methods are developed for systems composed of spherical multipole moment expansions. A unified set of equations is derived that takes advantage of a spherical tensor gradient operator formalism in both real space and reciprocal space to allow extension to arbitrary multipole order. The implementation of these methods into a novel linear-scaling modified “divide-and-conquer” (mDC) quantum mechanical force field is discussed. The evaluation times and relative force errors are compared between the three methods, as a function of multipole expansion order. Timings and errors are also compared within the context of the quantum mechanical force field, which encounters primary errors related to the quality of reproducing electrostatic forces for a given density matrix and secondary errors resulting from the propagation of the approximate electrostatics into the self-consistent field procedure, which yields a converged, variational, but nonetheless approximate density matrix. Condensed-phase simulations of an mDC water model are performed with the multipolar PME method and compared to an electrostatic cutoff method, which is shown to artificially increase the density of water and heat of vaporization relative to full electrostatic treatment.

Analysis of the conformational profiles of fenamates shows route towards novel, higher accuracy, force-fields for pharmaceuticals

The conformational barriers of the fenamates which lead to conformational polymorphism can be represented by a novel, physically motivated, model intramolecular potential suitable for extension to other pharmaceuticals.

Improved parameterization of interatomic potentials for rare gas dimers with density-based energy decomposition analysis

We examine interatomic interactions for rare gas dimers using the density-based energy decomposition analysis (DEDA) in conjunction with computational results from CCSD(T) at the complete basis set (CBS) limit. The unique DEDA capability of separating frozen density interactions from density relaxation contributions is employed to yield clean interaction components, and the results are found to be consistent with the typical physical picture that density relaxations play a very minimal role in rare gas interactions. Equipped with each interaction component as reference, we develop a new three-term molecular mechanical force field to describe rare gas dimers: a smeared charge multipole model for electrostatics with charge penetration effects, a B3LYP-D3 dispersion term for asymptotically correct long-range attractions that is screened at short-range, and a Born-Mayer exponential function for the repulsion. The resulted force field not only reproduces rare gas interaction energies calculated at the CCSD(T)/CBS level, but also yields each interaction component (electrostatic or van der Waals) which agrees very well with its corresponding reference value.

A Comparative Study of Transferable Aspherical Pseudoatom Databank and Classical Force Fields for Predicting Electrostatic Interactions in Molecular Dimers

Accurate and fast evaluation of electrostatic interactions in molecular systems is one of the most challenging tasks in the rapidly advancing field of macromolecular chemistry and drug design. Electrostatic interactions are of crucial importance in biological systems. They are well represented by quantum mechanical methods; however, such calculations are computationally expensive. In this study, we have evaluated the University of Buffalo Pseudoatom Databank (UBDB)1,2 approach for approximation of electrostatic properties of macromolecules and their complexes. We selected the S663 and JSCH-20054 data sets (208 molecular complexes in total) for this study. These complexes represent a wide range of chemical and biological systems for which hydrogen bonding, electrostatic, and van der Waals interactions play important roles. Reference electrostatic energies were obtained directly from wave functions at the B3LYP/aug-cc-pVTZ level of theory using the SAPT (Symmetry-Adapted Perturbation Theory) scheme for calculation of electrostatic contributions to total intermolecular interaction energies. Electrostatic energies calculated on the basis of the UBDB were compared with corresponding reference results. Results were also compared with energies computed using a point charge model from popular force fields (AM1-BCC and RESP used in AMBER and CGenFF from CHARMM family). The energy trends are quite consistent (R2 ≈ 0.98) for the UBDB method as compared to the AMBER5 and CHARMM force field methods6(R2 ≈ 0.93 on average). The RSMEs do not exceed 3.2 kcal mol-1 for the UBDB and are in the range of 3.7-7.6 kcal mol-1 for the point charge models. We also investigated the discrepancies in electrostatic potentials and magnitudes of dipole moments among the tested methods. This study shows that estimation of electrostatic interaction energies using the UBDB databank is accurate and reasonably fast when compared to other known methods, which opens potential new applications to macromolecules.

Parametrization of an Orbital-Based Linear-Scaling Quantum Force Field for Noncovalent Interactions

We parametrize a linear-scaling quantum mechanical force field called mDC for the accurate reproduction of nonbonded interactions. We provide a new benchmark database of accurate ab initio interactions between sulfur-containing molecules. A variety of nonbond databases are used to compare the new mDC method with other semiempirical, molecular mechanical, ab initio, and combined semiempirical quantum mechanical/molecular mechanical methods. It is shown that the molecular mechanical force field significantly and consistently reproduces the benchmark results with greater accuracy than the semiempirical models and our mDC model produces errors twice as small as the molecular mechanical force field. The comparisons between the methods are extended to the docking of drug candidates to the Cyclin-Dependent Kinase 2 protein receptor. We correlate the protein-ligand binding energies to their experimental inhibition constants and find that the mDC produces the best correlation. Condensed phase simulation of mDC water is performed and shown to produce O-O radial distribution functions similar to TIP4P-EW.

QM/MM investigations of organic chemistry oriented questions

About 35 years after its first suggestion, QM/MM became the standard theoretical approach to investigate enzymatic structures and processes. The success is due to the ability of QM/MM to provide an accurate atomistic picture of enzymes and related processes. This picture can even be turned into a movie if nuclei-dynamics is taken into account to describe enzymatic processes. In the field of organic chemistry, QM/MM methods are used to a much lesser extent although almost all relevant processes happen in condensed matter or are influenced by complicated interactions between substrate and catalyst. There is less importance for theoretical organic chemistry since the influence of nonpolar solvents is rather weak and the effect of polar solvents can often be accurately described by continuum approaches. Catalytic processes (homogeneous and heterogeneous) can often be reduced to truncated model systems, which are so small that pure quantum-mechanical approaches can be employed. However, since QM/MM becomes more and more efficient due to the success in software and hardware developments, it is more and more used in theoretical organic chemistry to study effects which result from the molecular nature of the environment. It is shown by many examples discussed in this review that the influence can be tremendous, even for nonpolar reactions. The importance of environmental effects in theoretical spectroscopy was already known. Due to its benefits, QM/MM can be expected to experience ongoing growth for the next decade.In the present chapter we give an overview of QM/MM developments and their importance in theoretical organic chemistry, and review applications which give impressions of the possibilities and the importance of the relevant effects. Since there is already a bunch of excellent reviews dealing with QM/MM, we will discuss fundamental ingredients and developments of QM/MM very briefly with a focus on very recent progress. For the applications we follow a similar strategy.