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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.

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.

Iterative energy minimization with the aim of achieving self-consistency is a common feature of Born-Oppenheimer molecular dynamics (BOMD) and classical molecular dynamics with polarizable force fields. In the former, the electronic degrees of freedom are optimized, while the latter often involves an iterative determination of induced point dipoles. The computational effort of the self-consistency procedure can be reduced by re-using converged solutions from previous time steps. However, this must be done carefully, as not to break time-reversal symmetry, which negatively impacts energy conservation. Self-consistent schemes based on the extended Lagrangian formalism, where the initial guesses for the optimized quantities are treated as auxiliary degrees of freedom, constitute one elegant solution. We report on the performance of two integration schemes with the same underlying extended Lagrangian structure, which we both employ in two radically distinct regimes-in classical molecular dynamics simulations with the AMOEBA polarizable force field and in BOMD simulations with the Onetep linear-scaling density functional theory (LS-DFT) approach. Both integration schemes are found to offer significant improvements over the standard (unpropagated) molecular dynamics formulation in both the classical and LS-DFT regimes.

Iterative energy minimization with the aim of achieving self-consistency is a common feature of Born-Oppenheimer molecular dynamics (BOMD) and classical molecular dynamics with polarizable force fields. In the former, the electronic degrees of freedom are optimized, while the latter often involves an iterative determination of induced point dipoles. The computational effort of the self-consistency procedure can be reduced by re-using converged solutions from previous time steps. However, this must be done carefully, as not to break time-reversal symmetry, which negatively impacts energy conservation. Self-consistent schemes based on the extended Lagrangian formalism, where the initial guesses for the optimized quantities are treated as auxiliary degrees of freedom, constitute one elegant solution. We report on the performance of two integration schemes with the same underlying extended Lagrangian structure, which we both employ in two radically distinct regimes-in classical molecular dynamics simulations with the AMOEBA polarizable force field and in BOMD simulations with the Onetep linear-scaling density functional theory (LS-DFT) approach. Both integration schemes are found to offer significant improvements over the standard (unpropagated) molecular dynamics formulation in both the classical and LS-DFT regimes.

The VV10 and rVV10 nonlocal correlation functionals are consistently implemented and assessed, with the goal of determining if the rVV10 nonlocal correlation functional can replace the VV10 nonlocal correlation functional in the recently developed B97M-V density functional, to give the B97M-rV density functional. Along the way, four density functionals are simultaneously tested: VV10, rVV10, B97M-V, and B97M-rV. An initial assessment is carried out across the S22 data set, and the short-range damping variable, b, is varied for all four density functionals in order to determine the sensitivity of the functionals to the empirical parameter. The results of this test indicate that a value of b = 6 (fortuitously the same as that in B97M-V) is suitable for B97M-rV. The functionals are then compared across an extensive database of interaction energies, and it is demonstrated that B97M-rV either matches or outperforms B97M-V for all of the tests considered. Finally, the optimization of b across the S22 data set is extended to two range-separated hybrid density functionals, ωB97X-V and ωB97M-V, and a value of b = 6.2 is recommended for both ωB97X-rV and ωB97M-rV.

The VV10 and rVV10 nonlocal correlation functionals are consistently implemented and assessed, with the goal of determining if the rVV10 nonlocal correlation functional can replace the VV10 nonlocal correlation functional in the recently developed B97M-V density functional, to give the B97M-rV density functional. Along the way, four density functionals are simultaneously tested: VV10, rVV10, B97M-V, and B97M-rV. An initial assessment is carried out across the S22 data set, and the short-range damping variable, b, is varied for all four density functionals in order to determine the sensitivity of the functionals to the empirical parameter. The results of this test indicate that a value of b = 6 (fortuitously the same as that in B97M-V) is suitable for B97M-rV. The functionals are then compared across an extensive database of interaction energies, and it is demonstrated that B97M-rV either matches or outperforms B97M-V for all of the tests considered. Finally, the optimization of b across the S22 data set is extended to two range-separated hybrid density functionals, ωB97X-V and ωB97M-V, and a value of b = 6.2 is recommended for both ωB97X-rV and ωB97M-rV.

Accurate and computationally efficient exchange-correlation functionals are critical to the successful application of linear-scaling density functional theory (DFT). Local and semi-local functionals of the density are naturally compatible with linear-scaling approaches, having a general form which assumes the locality of electronic interactions and which can be efficiently evaluated by numerical quadrature. Presently, the most sophisticated and flexible semi-local functionals are members of the meta-generalized-gradient approximation (meta-GGA) family, and depend upon the kinetic energy density, τ, in addition to the charge density and its gradient. In order to extend the theoretical and computational advantages of τ-dependent meta-GGA functionals to large-scale DFT calculations on thousands of atoms, we have implemented support for τ-dependent meta-GGA functionals in the ONETEP program. In this paper we lay out the theoretical innovations necessary to implement τ-dependent meta-GGA functionals within ONETEP's linear-scaling formalism. We present expressions for the gradient of the τ-dependent exchange-correlation energy, necessary for direct energy minimization. We also derive the forms of the τ-dependent exchange-correlation potential and kinetic energy density in terms of the strictly localized, self-consistently optimized orbitals used by ONETEP. To validate the numerical accuracy of our self-consistent meta-GGA implementation, we performed calculations using the B97M-V and PKZB meta-GGAs on a variety of small molecules. Using only a minimal basis set of self-consistently optimized local orbitals, we obtain energies in excellent agreement with large basis set calculations performed using other codes. Finally, to establish the linear-scaling computational cost and applicability of our approach to large-scale calculations, we present the outcome of self-consistent meta-GGA calculations on amyloid fibrils of increasing size, up to tens of thousands of atoms.

Accurate and computationally efficient exchange-correlation functionals are critical to the successful application of linear-scaling density functional theory (DFT). Local and semi-local functionals of the density are naturally compatible with linear-scaling approaches, having a general form which assumes the locality of electronic interactions and which can be efficiently evaluated by numerical quadrature. Presently, the most sophisticated and flexible semi-local functionals are members of the meta-generalized-gradient approximation (meta-GGA) family, and depend upon the kinetic energy density, τ, in addition to the charge density and its gradient. In order to extend the theoretical and computational advantages of τ-dependent meta-GGA functionals to large-scale DFT calculations on thousands of atoms, we have implemented support for τ-dependent meta-GGA functionals in the ONETEP program. In this paper we lay out the theoretical innovations necessary to implement τ-dependent meta-GGA functionals within ONETEP's linear-scaling formalism. We present expressions for the gradient of the τ-dependent exchange-correlation energy, necessary for direct energy minimization. We also derive the forms of the τ-dependent exchange-correlation potential and kinetic energy density in terms of the strictly localized, self-consistently optimized orbitals used by ONETEP. To validate the numerical accuracy of our self-consistent meta-GGA implementation, we performed calculations using the B97M-V and PKZB meta-GGAs on a variety of small molecules. Using only a minimal basis set of self-consistently optimized local orbitals, we obtain energies in excellent agreement with large basis set calculations performed using other codes. Finally, to establish the linear-scaling computational cost and applicability of our approach to large-scale calculations, we present the outcome of self-consistent meta-GGA calculations on amyloid fibrils of increasing size, up to tens of thousands of atoms.

We present a novel quantum mechanical/molecular mechanics (QM/MM) approach in which a quantum subsystem is coupled to a classical subsystem described by the AMOEBA polarizable force field. Our approach permits mutual polarization between the QM and MM subsystems, effected through multipolar electrostatics. Self-consistency is achieved for both the QM and MM subsystems through a total energy minimization scheme. We provide an expression for the Hamiltonian of the coupled QM/MM system, which we minimize using gradient methods. The QM subsystem is described by the onetep linear-scaling DFT approach, which makes use of strictly localized orbitals expressed in a set of periodic sinc basis functions equivalent to plane waves. The MM subsystem is described by the multipolar, polarizable force field AMOEBA, as implemented in tinker. Distributed multipole analysis is used to obtain, on the fly, a classical representation of the QM subsystem in terms of atom-centered multipoles. This auxiliary representation is used for all polarization interactions between QM and MM, allowing us to treat them on the same footing as in AMOEBA. We validate our method in tests of solute-solvent interaction energies, for neutral and charged molecules, demonstrating the simultaneous optimization of the quantum and classical degrees of freedom. Encouragingly, we find that the inclusion of explicit polarization in the MM part of QM/MM improves the agreement with fully QM calculations.

Advanced potential energy surfaces are defined as theoretical models that explicitly include many-body effects that transcend the standard fixed-charge, pairwise-additive paradigm typically used in molecular simulation. However, several factors relating to their software implementation have precluded their widespread use in condensed-phase simulations: the computational cost of the theoretical models, a paucity of approximate models and algorithmic improvements that can ameliorate their cost, underdeveloped interfaces and limited dissemination in computational code bases that are widely used in the computational chemistry community, and software implementations that have not kept pace with modern high-performance computing (HPC) architectures, such as multicore CPUs and modern graphics processing units (GPUs). In this Feature Article we review recent progress made in these areas, including well-defined polarization approximations and new multipole electrostatic formulations, novel methods for solving the mutual polarization equations and increasing the MD time step, combining linear-scaling electronic structure methods with new QM/MM methods that account for mutual polarization between the two regions, and the greatly improved software deployment of these models and methods onto GPU and CPU hardware platforms. We have now approached an era where multipole-based polarizable force fields can be routinely used to obtain computational results comparable to state-of-the-art density functional theory while reaching sampling statistics that are acceptable when compared to that obtained from simpler fixed partial charge force fields.

We present a novel quantum mechanical/molecular mechanics (QM/MM) approach in which a quantum subsystem is coupled to a classical subsystem described by the AMOEBA polarizable force field. Our approach permits mutual polarization between the QM and MM subsystems, effected through multipolar electrostatics. Self-consistency is achieved for both the QM and MM subsystems through a total energy minimization scheme. We provide an expression for the Hamiltonian of the coupled QM/MM system, which we minimize using gradient methods. The QM subsystem is described by the onetep linear-scaling DFT approach, which makes use of strictly localized orbitals expressed in a set of periodic sinc basis functions equivalent to plane waves. The MM subsystem is described by the multipolar, polarizable force field AMOEBA, as implemented in tinker. Distributed multipole analysis is used to obtain, on the fly, a classical representation of the QM subsystem in terms of atom-centered multipoles. This auxiliary representation is used for all polarization interactions between QM and MM, allowing us to treat them on the same footing as in AMOEBA. We validate our method in tests of solute-solvent interaction energies, for neutral and charged molecules, demonstrating the simultaneous optimization of the quantum and classical degrees of freedom. Encouragingly, we find that the inclusion of explicit polarization in the MM part of QM/MM improves the agreement with fully QM calculations.

Advanced potential energy surfaces are defined as theoretical models that explicitly include many-body effects that transcend the standard fixed-charge, pairwise-additive paradigm typically used in molecular simulation. However, several factors relating to their software implementation have precluded their widespread use in condensed-phase simulations: the computational cost of the theoretical models, a paucity of approximate models and algorithmic improvements that can ameliorate their cost, underdeveloped interfaces and limited dissemination in computational code bases that are widely used in the computational chemistry community, and software implementations that have not kept pace with modern high-performance computing (HPC) architectures, such as multicore CPUs and modern graphics processing units (GPUs). In this Feature Article we review recent progress made in these areas, including well-defined polarization approximations and new multipole electrostatic formulations, novel methods for solving the mutual polarization equations and increasing the MD time step, combining linear-scaling electronic structure methods with new QM/MM methods that account for mutual polarization between the two regions, and the greatly improved software deployment of these models and methods onto GPU and CPU hardware platforms. We have now approached an era where multipole-based polarizable force fields can be routinely used to obtain computational results comparable to state-of-the-art density functional theory while reaching sampling statistics that are acceptable when compared to that obtained from simpler fixed partial charge force fields.

Recently developed density functionals have good accuracy for both thermochemistry (TC) and non-covalent interactions (NC) if very large atomic orbital basis sets are used. To approach the basis set limit with potentially lower computational cost, a new self-consistent field (SCF) scheme is presented that employs minimal adaptive basis (MAB) functions. The MAB functions are optimized on each atomic site by minimizing a surrogate function. High accuracy is obtained by applying a perturbative correction (PC) to the MAB calculation, similar to dual basis approaches. Compared to exact SCF results, using this MAB-SCF (PC) approach with the same large target basis set produces

Recently developed density functionals have good accuracy for both thermochemistry (TC) and non-covalent interactions (NC) if very large atomic orbital basis sets are used. To approach the basis set limit with potentially lower computational cost, a new self-consistent field (SCF) scheme is presented that employs minimal adaptive basis (MAB) functions. The MAB functions are optimized on each atomic site by minimizing a surrogate function. High accuracy is obtained by applying a perturbative correction (PC) to the MAB calculation, similar to dual basis approaches. Compared to exact SCF results, using this MAB-SCF (PC) approach with the same large target basis set produces