Multipolar Ewald methods, 2: applications using a quantum mechanical force field

Software Infrastructure for Next-Generation QM/MM-ΔMLP Force Fields

We present software infrastructure for the design and testing of new quantum mechanical/molecular mechanical and machine-learning potential (QM/MM-ΔMLP) force fields for a wide range of applications. The software integrates Amber's molecular dynamics simulation capabilities with fast, approximate quantum models in the xtb package and machine-learning potential corrections in DeePMD-kit. The xtb package implements the recently developed density-functional tight-binding QM models with multipolar electrostatics and density-dependent dispersion (GFN2-xTB), and the interface with Amber enables their use in periodic boundary QM/MM simulations with linear-scaling QM/MM particle-mesh Ewald electrostatics. The accuracy of the semiempirical models is enhanced by including machine-learning correction potentials (ΔMLPs) enabled through an interface with the DeePMD-kit software. The goal of this paper is to present and validate the implementation of this software infrastructure in molecular dynamics and free energy simulations. The utility of the new infrastructure is demonstrated in proof-of-concept example applications. The software elements presented here are open source and freely available. Their interface provides a powerful enabling technology for the design of new QM/MM-ΔMLP models for studying a wide range of problems, including biomolecular reactivity and protein-ligand binding.

Amber free energy tools: Interoperable software for free energy simulations using generalized quantum mechanical/molecular mechanical and machine learning potentials

We report the development and testing of new integrated cyberinfrastructure for performing free energy simulations with generalized hybrid quantum mechanical/molecular mechanical (QM/MM) and machine learning potentials (MLPs) in Amber. The Sander molecular dynamics program has been extended to leverage fast, density-functional tight-binding models implemented in the DFTB+ and xTB packages, and an interface to the DeePMD-kit software enables the use of MLPs. The software is integrated through application program interfaces that circumvent the need to perform "system calls" and enable the incorporation of long-range Ewald electrostatics into the external software's self-consistent field procedure. The infrastructure provides access to QM/MM models that may serve as the foundation for QM/MM-ΔMLP potentials, which supplement the semiempirical QM/MM model with a MLP correction trained to reproduce ab initio QM/MM energies and forces. Efficient optimization of minimum free energy pathways is enabled through a new surface-accelerated finite-temperature string method implemented in the FE-ToolKit package. Furthermore, we interfaced Sander with the i-PI software by implementing the socket communication protocol used in the i-PI client-server model. The new interface with i-PI allows for the treatment of nuclear quantum effects with semiempirical QM/MM-ΔMLP models. The modular interoperable software is demonstrated on proton transfer reactions in guanine-thymine mispairs in a B-form deoxyribonucleic acid helix. The current work represents a considerable advance in the development of modular software for performing free energy simulations of chemical reactions that are important in a wide range of applications.

DeePMD-kit v2: A software package for deep potential models

DeePMD-kit is a powerful open-source software package that facilitates molecular dynamics simulations using machine learning potentials known as Deep Potential (DP) models. This package, which was released in 2017, has been widely used in the fields of physics, chemistry, biology, and material science for studying atomistic systems. The current version of DeePMD-kit offers numerous advanced features, such as DeepPot-SE, attention-based and hybrid descriptors, the ability to fit tensile properties, type embedding, model deviation, DP-range correction, DP long range, graphics processing unit support for customized operators, model compression, non-von Neumann molecular dynamics, and improved usability, including documentation, compiled binary packages, graphical user interfaces, and application programming interfaces. This article presents an overview of the current major version of the DeePMD-kit package, highlighting its features and technical details. Additionally, this article presents a comprehensive procedure for conducting molecular dynamics as a representative application, benchmarks the accuracy and efficiency of different models, and discusses ongoing developments.

QDπ: A Quantum Deep Potential Interaction Model for Drug Discovery

We report QDπ-v1.0 for modeling the internal energy of drug molecules containing H, C, N, and O atoms. The QDπ model is in the form of a quantum mechanical/machine learning potential correction (QM/Δ-MLP) that uses a fast third-order self-consistent density-functional tight-binding (DFTB3/3OB) model that is corrected to a quantitatively high-level of accuracy through a deep-learning potential (DeepPot-SE). The model has the advantage that it is able to properly treat electrostatic interactions and handle changes in charge/protonation states. The model is trained against reference data computed at the ωB97X/6-31G* level (as in the ANI-1x data set) and compared to several other approximate semiempirical and machine learning potentials (ANI-1x, ANI-2x, DFTB3, MNDO/d, AM1, PM6, GFN1-xTB, and GFN2-xTB). The QDπ model is demonstrated to be accurate for a wide range of intra- and intermolecular interactions (despite its intended use as an internal energy model) and has shown to perform exceptionally well for relative protonation/deprotonation energies and tautomers. An example application to model reactions involved in RNA strand cleavage catalyzed by protein and nucleic acid enzymes illustrates QDπ has average errors less than 0.5 kcal/mol, whereas the other models compared have errors over an order of magnitude greater. Taken together, this makes QDπ highly attractive as a potential force field model for drug discovery.

Development of Range-Corrected Deep Learning Potentials for Fast, Accurate Quantum Mechanical/Molecular Mechanical Simulations of Chemical Reactions in Solution

We develop a new deep potential─range correction (DPRc) machine learning potential for combined quantum mechanical/molecular mechanical (QM/MM) simulations of chemical reactions in the condensed phase. The new range correction enables short-ranged QM/MM interactions to be tuned for higher accuracy, and the correction smoothly vanishes within a specified cutoff. We further develop an active learning procedure for robust neural network training. We test the DPRc model and training procedure against a series of six nonenzymatic phosphoryl transfer reactions in solution that are important in mechanistic studies of RNA-cleaving enzymes. Specifically, we apply DPRc corrections to a base QM model and test its ability to reproduce free-energy profiles generated from a target QM model. We perform these comparisons using the MNDO/d and DFTB2 semiempirical models because they differ in the way they treat orbital orthogonalization and electrostatics and produce free-energy profiles which differ significantly from each other, thereby providing us a rigorous stress test for the DPRc model and training procedure. The comparisons show that accurate reproduction of the free-energy profiles requires correction of the QM/MM interactions out to 6 Å. We further find that the model's initial training benefits from generating data from temperature replica exchange simulations and including high-temperature configurations into the fitting procedure, so the resulting models are trained to properly avoid high-energy regions. A single DPRc model was trained to reproduce four different reactions and yielded good agreement with the free-energy profiles made from the target QM/MM simulations. The DPRc model was further demonstrated to be transferable to 2D free-energy surfaces and 1D free-energy profiles that were not explicitly considered in the training. Examination of the computational performance of the DPRc model showed that it was fairly slow when run on CPUs but was sped up almost 100-fold when using NVIDIA V100 GPUs, resulting in almost negligible overhead. The new DPRc model and training procedure provide a potentially powerful new tool for the creation of next-generation QM/MM potentials for a wide spectrum of free-energy applications ranging from drug discovery to enzyme design.

Automated fragmentation quantum mechanical calculation of 13C and 1H chemical shifts in molecular crystals

In this work, the automated fragmentation quantum mechanics/molecular mechanics (AF-QM/MM) approach was applied to calculate the ¹³C and ¹H nuclear magnetic resonance (NMR) chemical shifts in molecular crystals. Two benchmark sets of molecular crystals were selected to calculate the NMR chemical shifts. Systematic investigation was conducted to examine the convergence of AF-QM/MM calculations and the impact of various density functionals with different basis sets on the NMR chemical shift prediction. The result demonstrates that the calculated NMR chemical shifts are close to convergence when the distance threshold for the QM region is larger than 3.5 Å. For ¹³C chemical shift calculations, the mPW1PW91 functional is the best density functional among the functionals chosen in this study (namely, B3LYP, B3PW91, M06-2X, M06-L, mPW1PW91, OB98, and OPBE), while the OB98 functional is more suitable for the ¹H NMR chemical shift prediction of molecular crystals. Moreover, with the B3LYP functional, at least a triple-ζ basis set should be utilized to accurately reproduce the experimental ¹³C and ¹H chemical shifts. The employment of diffuse basis functions will further improve the accuracy for ¹³C chemical shift calculations, but not for the ¹H chemical shift prediction. We further proposed a fragmentation scheme of dividing the central molecule into smaller fragments. By comparing with the results of the fragmentation scheme using the entire central molecule as the core region, the AF-QM/MM calculations with the fragmented central molecule can not only achieve accurate results but also reduce the computational cost. Therefore, the AF-QM/MM approach is capable of predicting the ¹³C and ¹H NMR chemical shifts for molecular crystals accurately and effectively, and could be utilized for dealing with more complex periodic systems such as macromolecular polymers and biomacromolecules. The AF-QM/MM program for molecular crystals is available at https://github.com/shiman1995/NMR.

Prediction of Excited-State Properties of Oligoacene Crystals Using Fragment-Based Quantum Mechanical Method

A fundamental understanding of the excited-state properties of molecular crystals is of central importance for their optoelectronics applications. In this study, we developed the electrostatically embedded generalized molecular fractionation (EE-GMF) method for the quantitative characterization of the excited-state properties of locally excited molecular clusters. The accuracy of the EE-GMF method is systematically assessed for oligoacene crystals. Our result demonstrates that the EE-GMF method is capable of providing the lowest vertical singlet (S₁) and triplet excitation energies (T₁), in excellent agreement with the full-system quantum mechanical calculations. Using this method, we also investigated the performance of different density functionals in predicting the excited-state properties of the oligoacene crystals. Among the 13 tested functionals, B3LYP and MN15 give the two lowest overall mean unsigned errors with reference to the experimental S₁ and T₁ excitation energies. The EE-GMF approach can be readily utilized for studying the excited-state properties of large-scale organic solids at diverse ab initio levels.

Quantum mechanical force fields for condensed phase molecular simulations

Molecular simulations are powerful tools for providing atomic-level details into complex chemical and physical processes that occur in the condensed phase. For strongly interacting systems where quantum many-body effects are known to play an important role, density-functional methods are often used to provide the model with the potential energy used to drive dynamics. These methods, however, suffer from two major drawbacks. First, they are often too computationally intensive to practically apply to large systems over long time scales, limiting their scope of application. Second, there remain challenges for these models to obtain the necessary level of accuracy for weak non-bonded interactions to obtain quantitative accuracy for a wide range of condensed phase properties. Quantum mechanical force fields (QMFFs) provide a potential solution to both of these limitations. In this review, we address recent advances in the development of QMFFs for condensed phase simulations. In particular, we examine the development of QMFF models using both approximate and ab initio density-functional models, the treatment of short-ranged non-bonded and long-ranged electrostatic interactions, and stability issues in molecular dynamics calculations. Example calculations are provided for crystalline systems, liquid water, and ionic liquids. We conclude with a perspective for emerging challenges and future research directions.

Charge-dependent many-body exchange and dispersion interactions in combined QM/MM simulations

Accurate modeling of the molecular environment is critical in condensed phase simulations of chemical reactions. Conventional quantum mechanical/molecular mechanical (QM/MM) simulations traditionally model non-electrostatic non-bonded interactions through an empirical Lennard-Jones (LJ) potential which, in violation of intuitive chemical principles, is bereft of any explicit coupling to an atom's local electronic structure. This oversight results in a model whereby short-ranged exchange-repulsion and long-ranged dispersion interactions are invariant to changes in the local atomic charge, leading to accuracy limitations for chemical reactions where significant atomic charge transfer can occur along the reaction coordinate. The present work presents a variational, charge-dependent exchange-repulsion and dispersion model, referred to as the charge-dependent exchange and dispersion (QXD) model, for hybrid QM/MM simulations. Analytic expressions for the energy and gradients are provided, as well as a description of the integration of the model into existing QM/MM frameworks, allowing QXD to replace traditional LJ interactions in simulations of reactive condensed phase systems. After initial validation against QM data, the method is demonstrated by capturing the solvation free energies of a series of small, chlorine-containing compounds that have varying charge on the chlorine atom. The model is further tested on the SN2 attack of a chloride anion on methylchloride. Results suggest that the QXD model, unlike the traditional LJ model, is able to simultaneously obtain accurate solvation free energies for a range of compounds while at the same time closely reproducing the experimental reaction free energy barrier. The QXD interaction model allows explicit coupling of atomic charge with many-body exchange and dispersion interactions that are related to atomic size and provides a more accurate and robust representation of non-electrostatic non-bonded QM/MM interactions.

Ambient-Potential Composite Ewald Method for ab Initio Quantum Mechanical/Molecular Mechanical Molecular Dynamics Simulation

A new approach for performing Particle Mesh Ewald in ab initio quantum mechanical/molecular mechanical (QM/MM) simulations with extended atomic orbital basis sets is presented. The new approach, the Ambient-Potential Composite Ewald (CEw) method, does not perform the QM/MM interaction with Mulliken charges nor electrostatically fit charges. Instead the nuclei and electron density interact directly with the MM environment, but in a manner that avoids the use of dense Fourier transform grids. By performing the electrostatics with the underlying QM density, the CEw method avoids self-consistent field instabilities that have been encountered with simple charge mapping procedures. Potential of mean force (PMF) profiles of the p-nitrophenyl phosphate dissociation reaction in explicit solvent are computed from PBE0/6-31G* QM/MM molecular dynamics simulations with various electrostatic protocols. The CEw profiles are shown to be stable with respect to real-space Ewald cutoff, whereas the PMFs computed from truncated and switched electrostatics produce artifacts. PBE0/6-311G**, AM1/d-PhoT, and DFTB2 QM/MM simulations are performed to generate two-dimensional PMF profiles of the phosphoryl transesterification reactions with ethoxide and phenoxide leaving groups. The semiempirical models incorrectly produce a concerted ethoxide mechanism, whereas PBE0 correctly produces a stepwise mechanism. The ab initio reaction barriers agree more closely to experiment than the semiempirical models. The failure of Mulliken-charge QM/MM-Ewald is analyzed.

Semiempirical Quantum Mechanical Methods for Noncovalent Interactions for Chemical and Biochemical Applications

Semiempirical (SE) methods can be derived from either Hartree-Fock or density functional theory by applying systematic approximations, leading to efficient computational schemes that are several orders of magnitude faster than ab initio calculations. Such numerical efficiency, in combination with modern computational facilities and linear scaling algorithms, allows application of SE methods to very large molecular systems with extensive conformational sampling. To reliably model the structure, dynamics, and reactivity of biological and other soft matter systems, however, good accuracy for the description of noncovalent interactions is required. In this review, we analyze popular SE approaches in terms of their ability to model noncovalent interactions, especially in the context of describing biomolecules, water solution, and organic materials. We discuss the most significant errors and proposed correction schemes, and we review their performance using standard test sets of molecular systems for quantum chemical methods and several recent applications. The general goal is to highlight both the value and limitations of SE methods and stimulate further developments that allow them to effectively complement ab initio methods in the analysis of complex molecular systems.

VR-SCOSMO: A smooth conductor-like screening model with charge-dependent radii for modeling chemical reactions

To better represent the solvation effects observed along reaction pathways, and of ionic species in general, a charge-dependent variable-radii smooth conductor-like screening model (VR-SCOSMO) is developed. This model is implemented and parameterized with a third order density-functional tight binding quantum model, DFTB3/3OB-OPhyd, a quantum method which was developed for organic and biological compounds, utilizing a specific parameterization for phosphate hydrolysis reactions. Unlike most other applications with the DFTB3/3OB model, an auxiliary set of atomic multipoles is constructed from the underlying DFTB3 density matrix which is used to interact the solute with the solvent response surface. The resulting method is variational, produces smooth energies, and has analytic gradients. As a baseline, a conventional SCOSMO model with fixed radii is also parameterized. The SCOSMO and VR-SCOSMO models shown have comparable accuracy in reproducing neutral-molecule absolute solvation free energies; however, the VR-SCOSMO model is shown to reduce the mean unsigned errors (MUEs) of ionic compounds by half (about 2-3 kcal/mol). The VR-SCOSMO model presents similar accuracy as a charge-dependent Poisson-Boltzmann model introduced by Hou et al. [J. Chem. Theory Comput. 6, 2303 (2010)]. VR-SCOSMO is then used to examine the hydrolysis of trimethylphosphate and seven other phosphoryl transesterification reactions with different leaving groups. Two-dimensional energy landscapes are constructed for these reactions and calculated barriers are compared to those obtained from ab initio polarizable continuum calculations and experiment. Results of the VR-SCOSMO model are in good agreement in both cases, capturing the rate-limiting reaction barrier and the nature of the transition state.

Quantum Fragment Based ab Initio Molecular Dynamics for Proteins

Developing ab initio molecular dynamics (AIMD) methods for practical application in protein dynamics is of significant interest. Due to the large size of biomolecules, applying standard quantum chemical methods to compute energies for dynamic simulation is computationally prohibitive. In this work, a fragment based ab initio molecular dynamics approach is presented for practical application in protein dynamics study. In this approach, the energy and forces of the protein are calculated by a recently developed electrostatically embedded generalized molecular fractionation with conjugate caps (EE-GMFCC) method. For simulation in explicit solvent, mechanical embedding is introduced to treat protein interaction with explicit water molecules. This AIMD approach has been applied to MD simulations of a small benchmark protein Trpcage (with 20 residues and 304 atoms) in both the gas phase and in solution. Comparison to the simulation result using the AMBER force field shows that the AIMD gives a more stable protein structure in the simulation, indicating that quantum chemical energy is more reliable. Importantly, the present fragment-based AIMD simulation captures quantum effects including electrostatic polarization and charge transfer that are missing in standard classical MD simulations. The current approach is linear-scaling, trivially parallel, and applicable to performing the AIMD simulation of proteins with a large size.

Improving intermolecular interactions in DFTB3 using extended polarization from chemical-potential equalization

Semi-empirical quantum mechanical methods traditionally expand the electron density in a minimal, valence-only electron basis set. The minimal-basis approximation causes molecular polarization to be underestimated, and hence intermolecular interaction energies are also underestimated, especially for intermolecular interactions involving charged species. In this work, the third-order self-consistent charge density functional tight-binding method (DFTB3) is augmented with an auxiliary response density using the chemical-potential equalization (CPE) method and an empirical dispersion correction (D3). The parameters in the CPE and D3 models are fitted to high-level CCSD(T) reference interaction energies for a broad range of chemical species, as well as dipole moments calculated at the DFT level; the impact of including polarizabilities of molecules in the parameterization is also considered. Parameters for the elements H, C, N, O, and S are presented. The Root Mean Square Deviation (RMSD) interaction energy is improved from 6.07 kcal/mol to 1.49 kcal/mol for interactions with one charged species, whereas the RMSD is improved from 5.60 kcal/mol to 1.73 for a set of 9 salt bridges, compared to uncorrected DFTB3. For large water clusters and complexes that are dominated by dispersion interactions, the already satisfactory performance of the DFTB3-D3 model is retained; polarizabilities of neutral molecules are also notably improved. Overall, the CPE extension of DFTB3-D3 provides a more balanced description of different types of non-covalent interactions than Neglect of Diatomic Differential Overlap type of semi-empirical methods (e.g., PM6-D3H4) and PBE-D3 with modest basis sets.

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.