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

A polarizable valence electron density based force field for high-energy interactions between atoms and molecules

High-accuracy molecular force field models suited for hot gases and plasmas are not as abundant as those geared toward ambient pressure and temperature conditions. Here, we present an improved version of our previous electron-density based force field model that can now account for polarization effects by adjusting the atomic valence electron contributions to match ab initio calculated Mulliken partial charges. Using a slightly modified version of the Hohenberg-Kohn theorem, we also include an improved theoretical formulation of our model when applied to systems with degenerate ground states. We present two variants of our polarizable model, fitted from ab initio reference data calculated at CCSD(T)/cc-pVTZ and CCSD(T)/CEP-31G levels of theory, that both accurately model water dimer interaction energies. Further improvements include the additional interaction components with fictitious non-spherically symmetric, yet atom-centered, electron densities and fitting the exchange and correlation coefficients against analytical expressions. The latter removes all unphysical oscillations that are observed in the previous non-polarizable variant of our force field.

Seamless integration of GEM, a density based-force field, for QM/MM simulations via LICHEM, Psi4, and Tinker-HP

Hybrid quantum mechanics/molecular mechanics (QM/MM) simulations have become an essential tool in computational chemistry, particularly for analyzing complex biological and condensed phase systems. Building on this foundation, our work presents a novel implementation of the Gaussian Electrostatic Model (GEM), a polarizable density-based force field, within the QM/MM framework. This advancement provides seamless integration, enabling efficient and optimized QM/GEM calculations in a single step using the LICHEM Code. We have successfully applied our implementation to water dimers and hexamers, demonstrating the ability to handle water systems with varying numbers of water molecules. Moreover, we have extended the application to describe the double proton transfer of the aspartic acid dimer in a box of water, which highlights the method's proficiency in investigating heterogeneous systems. Our implementation offers the flexibility to perform on-the-fly density fitting or to utilize pre-fitted coefficients to estimate exchange and Coulomb contributions. This flexibility enhances efficiency and accuracy in modeling molecular interactions, especially in systems where polarization effects are significant.

Beyond isotropic repulsion: Classical anisotropic repulsion by inclusion of p orbitals

Accurate modeling of intermolecular repulsion is an integral component in force field development. Although repulsion can be explicitly calculated by applying the Pauli exclusion principle, this approach is computationally viable only for systems of limited sizes. Instead, it has previously been shown that repulsion can be reformulated in a "classical" picture: the Pauli exclusion principle prohibits electrons from occupying the same state, leading to a depletion of electronic charge between atoms, giving rise to an enhanced nuclear-nuclear electrostatic repulsion. This classical picture is called the isotropic S2/R approximation, where S is the overlap and R is the interatomic distance. This approximation accurately captures the repulsion of isotropic atoms such as noble gas dimers; however, a key deficiency is that it fails to capture the angular dependence of the repulsion of anisotropic molecules. To include directionality, the wave function must at least be a linear combination of s and p orbitals. We derive a new anisotropic S2/R repulsion model through the inclusion of the anisotropic p orbital term in the total wave function. Because repulsion is pairwise and decays rapidly, it can be truncated at a short range, making it amenable for efficient calculation of energy and forces in complex biomolecular systems. We present a parameterization of the S101 dimer database against the ab initio benchmark symmetry-adapted perturbation theory, which yields an rms error of only 0.9 kcal/mol. The importance of the anisotropic term is demonstrated through angular scans of water-water dimers and dimers involving halobenzene. Simulation of liquid water shows that the model can be computed efficiently for realistic system sizes.

A Simple Electron-Density Based Force Field Model for High-Energy Interactions between Atoms and Molecules

In high-energy molecular dynamics or Monte Carlo simulations, standard force fields optimized for simulations at ambient temperatures are inadequate. This is largely because their repulsive parts have been regarded as not very significant, even well below zero interaction energies. It is, therefore, not obvious which force fields to resort to for simulating hot gases or plasmas. A force field model that uses the electronic densities of noninteracting atoms or molecules within the pair approximation is introduced. We start by deriving a naïve model that neglects any exchange and correlation effects between the electronic clouds and then correct this model by adding a term calibrated from ab initio calculations using the CCSD(T)/cc-pVTZ level of theory. The resulting expression for this term can be regarded as a simple exchange-correlation function. We compare the results for the repulsive part of the potential energy hypersurfaces with the force fields commonly used on some dimers of small molecules.

A Machine Learning Force Field for Bio-Macromolecular Modeling Based on Quantum Chemistry-Calculated Interaction Energy Datasets

Accurate energy data from noncovalent interactions are essential for constructing force fields for molecular dynamics simulations of bio-macromolecular systems. There are two important practical issues in the construction of a reliable force field with the hope of balancing the desired chemical accuracy and working efficiency. One is to determine a suitable quantum chemistry level of theory for calculating interaction energies. The other is to use a suitable continuous energy function to model the quantum chemical energy data. For the first issue, we have recently calculated the intermolecular interaction energies using the SAPT0 level of theory, and we have systematically organized these energies into the ab initio SOFG-31 (homodimer) and SOFG-31-heterodimer datasets. In this work, we re-calculate these interaction energies by using the more advanced SAPT2 level of theory with a wider series of basis sets. Our purpose is to determine the SAPT level of theory proper for interaction energies with respect to the CCSD(T)/CBS benchmark chemical accuracy. Next, to utilize these energy datasets, we employ one of the well-developed machine learning techniques, called the CLIFF scheme, to construct a general-purpose force field for biomolecular dynamics simulations. Here we use the SOFG-31 dataset and the SOFG-31-heterodimer dataset as the training and test sets, respectively. Our results demonstrate that using the CLIFF scheme can reproduce a diverse range of dimeric interaction energy patterns with only a small training set. The overall errors for each SAPT energy component, as well as the SAPT total energy, are all well below the desired chemical accuracy of ~1 kcal/mol.

Intermolecular Non-Bonded Interactions from Machine Learning Datasets

Accurate determination of intermolecular non-covalent-bonded or non-bonded interactions is the key to potentially useful molecular dynamics simulations of polymer systems. However, it is challenging to balance both the accuracy and computational cost in force field modelling. One of the main difficulties is properly representing the calculated energy data as a continuous force function. In this paper, we employ well-developed machine learning techniques to construct a general purpose intermolecular non-bonded interaction force field for organic polymers. The original ab initio dataset SOFG-31 was calculated by us and has been well documented, and here we use it as our training set. The CLIFF kernel type machine learning scheme is used for predicting the interaction energies of heterodimers selected from the SOFG-31 dataset. Our test results show that the overall errors are well below the chemical accuracy of about 1 kcal/mol, thus demonstrating the promising feasibility of machine learning techniques in force field modelling.

Accuracy of One- and Two-Photon Intensities with the Extended Polarizable Density Embedding Model

The recently developed extended polarizable density embedding (PDE-X) model is evaluated for the spectroscopic properties of organic chromophores solvated in water, including both one- and two-photon absorption properties. The PDE-X embedding model systematically improves vertical excitation energies over the preceding polarizable density embedding model (PDE). PDE-X shows more modest improvements over existing embedding models for oscillator strengths and two-photon absorption cross-sections, which are more sensitive properties. We argue that the origin of these discrepancies is related to the description of polarization effects, suggesting directions for future development of the embedding model.

Improving the Selectivity of the C-C Coupled Product Electrosynthesis by Using Molecularly Imprinted Polymer─An Enhanced Route from Phenol to Biphenol

We developed a procedure for selective 2,4-dimethylphenol, DMPh, direct electro-oxidation to 3,3',5,5'-tetramethyl-2,2'-biphenol, TMBh, a C-C coupled product. For that, we used an electrode coated with a product-selective molecularly imprinted polymer (MIP). The procedure is reasonably selective toward TMBh without requiring harmful additives or elevated temperatures. The TMBh product itself was used as a template for imprinting. We followed the template interaction with various functional monomers (FMs) using density functional theory (DFT) simulations to select optimal FM. On this basis, we used a prepolymerization complex of TMBh with carboxyl-containing FM at a 1:2 TMBh-to-FM molar ratio for MIP fabrication. The template-FM interaction was also followed by using different spectroscopic techniques. Then, we prepared the MIP on the electrode surface in the form of a thin film by the potentiodynamic electropolymerization of the chosen complex and extracted the template. Afterward, we characterized the fabricated films by using electrochemistry, FTIR spectroscopy, and AFM, elucidating their composition and morphology. Ultimately, the DMPh electro-oxidation was performed on the MIP film-coated electrode to obtain the desired TMBh product. The electrosynthesis selectivity was much higher at the electrode coated with MIP film in comparison with the reference nonimprinted polymer (NIP) film-coated or bare electrodes, reaching 39% under optimized conditions. MIP film thickness and electrosynthesis parameters significantly affected the electrosynthesis yield and selectivity. At thicker films, the yield was higher at the expense of selectivity, while the electrosynthesis potential increase enhanced the TMBh product yield. Computer simulations of the imprinted cavity interaction with the substrate molecule demonstrated that the MIP cavity promoted direct coupling of the substrate to form the desired TMBh product.

Embedding Beyond Electrostatics: The Extended Polarizable Density Embedding Model

The polarizable density embedding (PDE) model is a focused QM/QM fragment-based embedding model designed to model solvation effects on molecular properties. We extend the PDE model to include exchange and nonadditive exchange-correlation (for DFT) in the embedding potential in addition to the existing electrostatic, polarization, and nonelectrostatic effects already present. The resulting model, termed PDE-X, yields localized electronic excitation energies that accurately capture the range dependence of the solvent interaction and gives close agreement with full quantum mechanical (QM) results, even when using minimal QM regions. We show that the PDE-X embedding description consistently improves the accuracy of excitation energies for a diverse set of organic chromophores. The improved embedding description leads to systematic solvent effects that do not average out when applying configurational sampling.

Conical Intersections in Solution with Polarizable Embedding: Integral-Exact Direct Reaction Field

A common strategy to exploring the properties and reactivity of complex systems is to use quantum mechanics/molecular mechanics (QM/MM) embedding, wherein a QM region is defined and treated with electronic structure theory, and the remainder of the system is treated with a force field. Important to the description of electronic excited states, especially those of charge-transfer character, is the treatment of the coupling between the QM and MM subsystems. The state of the art is to use a polarizable force field for the MM region and mutually couple the QM wavefunction and MM induced dipoles, in addition to the usual electrostatic embedding, yielding a polarizable embedding (QM/MM-Pol) approach. However, we showed previously that current popular QM/MM-Pol approaches exhibit issues of root flipping and/or incorrect descriptions of electronic crossings in multistate calculations [J. Chem. Theory Comput. 14, 2137 (2018)]. Here, we demonstrate a solution to these problems with an integral-exact reformulation of the direct reaction field approach of Thole and Van Duijnen (QM/MM-IEDRF). The resulting embedding potential includes one- and two-electron operators and many-body dipole-induced dipole interactions and thus includes a natural description of the screening of electron-electron interactions by the MM induced dipoles. Pauli repulsion from the environment is mimicked by effective core potentials on the MM atoms. Inherent to the DRF approach is the assumption that MM dipoles respond instantaneously to the positions of the QM electrons; therefore, dispersion interactions are captured approximately. All electronic states are eigenfunctions of the same Hamiltonian, while the polarization induced in the environment and the associated energetic stabilization are unique to each state. This allows for a consistent definition of transition properties and state crossings. We demonstrate QM/MM-IEDRF by exploring the influence of a (polarizable) inert xenon matrix environment on the conical intersection underlying the photoisomerization of ethylene.

A Minimum Quantum Chemistry CCSD(T)/CBS Data Set of Dimeric Interaction Energies for Small Organic Functional Groups: Heterodimers

We extend our previous quantum chemistry calculations of interaction energies for 31 homodimers of small organic functional groups (the SOFG-31 data set) by including 239 heterodimers with monomers selected within the SOFG-31 data set, thus resulting in the SOFG-31+239 data set. The minimum-level theoretical scheme contains (1) the basis set superposition error corrected supermolecule (BSSE-SM) approach for intermolecular interactions; (2) the second-order Møller-Plesset perturbation theory (MP2) with the Dunning's aug-cc-pVXZ (X = D, T, Q) basis sets for the geometry optimization and correlation energy calculations; and (3) the single-point energy calculations with the coupled cluster with single, double, and perturbative triple excitations method at the complete basis set limit [CCSD(T)/CBS] using the well-tested extrapolation methods for the MP2 energy calibrations. In addition, we have performed a parallel series of energy decomposition calculations based on the symmetry adapted perturbation theory (SAPT) in order to gain chemical insights. That the above procedure cannot be further reduced has been proven to be very crucial for constructing reliable data sets of interaction energies. The calculated CCSD(T)/CBS interaction energy data can serve as a benchmark for testing or training less accurate but more efficient calculation methods, such as the electronic density functional theory. As an application, we employ a segmental SAPT model previously developed for the SOFG-31 data set to predict binding energies of large heterodimer complexes. These model energy "quanta" can be used in coarse-grained molecular dynamics simulations by avoiding large-scale calculations.

Improvement of the Gaussian Electrostatic Model by separate fitting of Coulomb and exchange-repulsion densities and implementation of a new dispersion term

The description of each separable contribution of the intermolecular interaction is a useful approach to develop polarizable force fields (polFFs). The Gaussian Electrostatic Model (GEM) is based on this approach, coupled with the use of density fitting techniques. In this work, we present the implementation and testing of two improvements of GEM: the Coulomb and exchange-repulsion energies are now computed with separate frozen molecular densities and a new dispersion formulation inspired by the Sum of Interactions Between Fragments Ab initio Computed polFF, which has been implemented to describe the dispersion and charge-transfer interactions. Thanks to the combination of GEM characteristics and these new features, we demonstrate a better agreement of the computed structural and condensed properties for water with experimental results, as well as binding energies in the gas phase with the ab initio reference compared with the previous GEM* potential. This work provides further improvements to GEM and the items that remain to be improved and the importance of the accurate reproduction for each separate contribution.

Polarizable Water Potential Derived from a Model Electron Density

A new empirical potential for efficient, large scale molecular dynamics simulation of water is presented. The HIPPO (Hydrogen-like Intermolecular Polarizable POtential) force field is based upon the model electron density of a hydrogen-like atom. This framework is used to derive and parametrize individual terms describing charge penetration damped permanent electrostatics, damped polarization, charge transfer, anisotropic Pauli repulsion, and damped dispersion interactions. Initial parameter values were fit to Symmetry Adapted Perturbation Theory (SAPT) energy components for ten water dimer configurations, as well as the radial and angular dependence of the canonical dimer. The SAPT-based parameters were then systematically refined to extend the treatment to water bulk phases. The final HIPPO water model provides a balanced representation of a wide variety of properties of gas phase clusters, liquid water, and ice polymorphs, across a range of temperatures and pressures. This water potential yields a rationalization of water structure, dynamics, and thermodynamics explicitly correlated with an ab initio energy decomposition, while providing a level of accuracy comparable or superior to previous polarizable atomic multipole force fields. The HIPPO water model serves as a cornerstone around which similarly detailed physics-based models can be developed for additional molecular species.

Atomistic Polarizable Embeddings: Energy, Dynamics, Spectroscopy, and Reactivity

The computational modeling of realistic extended systems, relevant in, e.g., Chemistry and Biophysics, is a fundamental problem of paramount importance in contemporary research. Enzymatic catalysis and photoinduced processes in pigment-protein complexes are typical problems targeted by computer-aided approaches, to complement experiments as interpretative tools at a molecular scale. The daunting complexity of this task lies in between the opposite stringent requirements of results' reliability for structural/dynamical properties and related intermolecular interactions, and a mandatory principle of realism in the modeling strategy. Therefore, in practice, a truly realistic computational model of a biologically relevant system can easily fail to meet the accuracy requirement, in order to balance the excessive computational cost necessary to reach the desired precision.To address such an "accuracy vs reality" dualistic requirement, mixed quantum mechanics/classical mechanics approaches within Atomistic (i.e., preserving the discrete particle configuration) Polarizable Embeddings (QM/APEs) methods have been proposed over the years. In this Account, we review recent developments in the design and application of general QM/APE methods, targeting situations where a local intrinsically quantum behavior is coupled to a large molecular system (i.e., an environment), often involving processes with different dynamical time scales, in order to avoid brute-force, unpractical quantum chemistry calculations on the complete system.In the first place, our interest is devoted to the available APEs models presently implemented in computational software, highlighting the quantum chemistry methods that can be used to treat the QM subsystem. We review the coupling strategy between the QM subsystem and the APE, which requires to examine the way the QM/MM mutual interactions are accounted for and how the polarization of the classical environment is considered with respect to (wrt) the quantum variables. Because of the need of reliable molecular and macromolecular structures, a pivotal aspect to address here is the handling of the system dynamics (i.e., gradients wrt nuclear positions are required), especially for large molecular assemblies composed by an overwhelming number of atoms, exploring many conformations on a complex energy landscape.Alongside, we highlight our views on the necessary steps to take toward more accurate general-purposes and transferable explicit embeddings. The main objective to achieve here is to design a more physically grounded multiscale approach. To do so, one should apply advanced new generation classical models to account for refined induction effects that are able to (i) improve the quality of QM/MM interaction energies; (ii) enhance transferability by avoiding the compulsory partial (or total) reparameterization of the classical model. Moreover, the extension of recent developments originating from the field of advanced classical molecular dynamics (MD) to the realm of QM/APE methods is a key direction to improve both speed and efficiency for the phase space exploration of systems of growing size and complexity.Lastly, we point out specific research topics where an advanced QM/APE dynamics can certainly shed some light. For example, we discuss chemical reactions in "harsh" environments and the case of spectroscopic theoretical modeling where the inclusion of refined environment effects is often mandatory.

Development and application of quantum mechanics/molecular mechanics methods with advanced polarizable potentials

Quantum mechanics/molecular mechanics (QM/MM) simulations are a popular approach to study various features of large systems. A common application of QM/MM calculations is in the investigation of reaction mechanisms in condensed-phase and biological systems. The combination of QM and MM methods to represent a system gives rise to several challenges that need to be addressed. The increase in computational speed has allowed the expanded use of more complicated and accurate methods for both QM and MM simulations. Here, we review some approaches that address several common challenges encountered in QM/MM simulations with advanced polarizable potentials, from methods to account for boundary across covalent bonds and long-range effects, to polarization and advanced embedding potentials.

Removing artifacts in polarizable embedding calculations of one- and two-photon absorption spectra of fluorescent proteins

The multiscale calculations involving excited states may suffer from the electron spill-out (ESO) problem. This seems to be especially the case when the environment of the core region, described with the electronic structure method, is approximated by a polarizable force field. The ESO effect often leads to incorrect physical character of electronic excitations, spreading outside the quantum region, which, in turn, results in erroneous absorption spectra. In this work, we investigate means to remove the artifacts in one-photon absorption (OPA) and two-photon absorption (TPA) spectra of green and yellow fluorescent protein representatives. This includes (i) using different basis sets, (ii) extending the core subsystem beyond the chromophore, (iii) modification of polarization interaction between the core region and its environment, and (iv) including the Pauli repulsion through effective core potentials (ECPs). Our results clearly show that ESO is observed when diffuse functions are used to assemble the multielectron wave function regardless of the exchange-correlation functional used. Furthermore, extending the core region, thus accounting for exchange interactions between the chromophore and its environment, leads to even more spurious excited states. Also, damping the interactions between the core subsystem and the polarizable force field is hardly helpful. In contrast, placing ECPs in the position of sites creating the embedding potential leads to the removal of artificious excited states that presumably should not be observed in the OPA and TPA spectra. We prove that it is a reliable and cost-effective approach for systems where the covalent bond(s) between the core region and its environment must be cut.

A polarizable three-layer frozen density embedding/molecular mechanics approach

We present a novel multilayer polarizable embedding approach in which the system is divided into three portions, two of which are treated using density functional theory and their interaction is based on frozen density embedding (FDE) theory, and both also mutually interact with a polarizable classical layer described using an atomistic model based on fluctuating charges (FQ). The efficacy of the model is demonstrated by extending the formalism to linear response properties and applying it to the simulation of the excitation energies of organic molecules in aqueous solution, where the solute and the first solvation shell are treated using FDE, while the rest of the solvent is modeled using FQ charges.

A simplified charge projection scheme for long-range electrostatics in ab initio QM/MM calculations

In a previous work [Pan et al., Molecules 23, 2500 (2018)], a charge projection scheme was reported, where outer molecular mechanical (MM) charges [>10 Å from the quantum mechanical (QM) region] were projected onto the electrostatic potential (ESP) grid of the QM region to accurately and efficiently capture long-range electrostatics in ab initio QM/MM calculations. Here, a further simplification to the model is proposed, where the outer MM charges are projected onto inner MM atom positions (instead of ESP grid positions). This enables a representation of the long-range MM electrostatic potential via augmentary charges (AC) on inner MM atoms. Combined with the long-range electrostatic correction function from Cisneros et al. [J. Chem. Phys. 143, 044103 (2015)] to smoothly switch between inner and outer MM regions, this new QM/MM-AC electrostatic model yields accurate and continuous ab initio QM/MM electrostatic energies with a 10 Å cutoff between inner and outer MM regions. This model enables efficient QM/MM cluster calculations with a large number of MM atoms as well as QM/MM calculations with periodic boundary conditions.

Combining multilevel Hartree–Fock and multilevel coupled cluster approaches with molecular mechanics: a study of electronic excitations in solutions

We present the coupling of different quantum-embedding approaches with a third molecular-mechanics layer, which can be either polarizable or non-polarizable.

A Many-Body, Fully Polarizable Approach to QM/MM Simulations

We present a new development in quantum mechanics/molecular mechanics (QM/MM) methods by replacing conventional MM models with data-driven many-body (MB) representations rigorously derived from high-level QM calculations. The new QM/MM approach builds on top of mutually polarizable QM/MM schemes developed for polarizable force fields with inducible dipoles and uses permutationally invariant polynomials to effectively account for quantum-mechanical contributions (e.g., exchange-repulsion and charge transfer and penetration) that are difficult to describe by classical expressions adopted by conventional MM models. Using the many-body MB-pol and MB-DFT potential energy functions for water, which include explicit two-body and three-body terms fitted to reproduce the corresponding CCSD(T) and PBE0 two-body and three-body energies for water, we demonstrate a smooth energetic transition as molecules are transferred between QM and MM regions, without the need of a transition layer. By effectively elevating the accuracy of both the MM region and the QM/MM interface to that of the QM region, the new QM/MB-MM approach achieves an accuracy comparable to that obtained with a fully QM treatment of the entire system.

A minimum quantum chemistry CCSD(T)/CBS dataset of dimeric interaction energies for small organic functional groups

We have performed a quantum chemistry study on the bonding patterns and interaction energies for 31 dimers of small organic functional groups (dubbed the SOFG-31 dataset), including the alkane-alkene-alkyne (6 + 4 + 4 = 14, AAA) groups, alcohol-aldehyde-ketone (4 + 4 + 3 = 11, AAK) groups, and carboxylic acid-amide (3 + 3 = 6, CAA) groups. The basis set superposition error corrected super-molecule approach using the second order Møller-Plesset perturbation theory (MP2) with the Dunning's aug-cc-pVXZ (X = D, T, Q) basis sets has been employed in the geometry optimization and energy calculations. To calibrate the MP2 calculated interaction energies for these dimeric complexes, we perform single-point calculations with the coupled cluster with single, double, and perturbative triple excitations method at the complete basis set limit [CCSD(T)/CBS] using the well-tested extrapolation methods. In order to gain more physical insights, we also perform a parallel series of energy decomposition calculations based on the symmetry adapted perturbation theory (SAPT). The collection of these CCSD(T)/CBS interaction energy values can serve as a minimum quantum chemistry dataset for testing or training less accurate but more efficient calculation methods. As an application, we further propose a segmental SAPT model based on chemically recognizable segments in a specific functional group. These model interactions can be used to construct coarse-grained force fields for larger molecular systems.

Exchange Repulsion in Quantum Mechanical/Effective Fragment Potential Excitation Energies: Beyond Polarizable Embedding

Hybrid quantum mechanical and molecular mechanical (QM/MM) approaches facilitate computational modeling of large biological and materials systems. Typically, in QM/MM, a small region of the system is modeled with an accurate quantum mechanical method and its surroundings with a more efficient alternative, such as a classical force field or the effective fragment potential (EFP). The reliability of QM/MM calculations depends largely on the treatment of interactions between the two subregions, also known as embedding. The polarizable embedding, which allows mutual polarization between solvent and solute, is considered to be essential for describing electronic excitations in polar solvents. In this work, we employ the QM/EFP model and extend the polarizable embedding by incorporating two short-range terms-a charge penetration correction to the electrostatic term and the exchange-repulsion term-both of which are modeled with one-electron contributions to the quantum Hamiltonian. We evaluate the accuracy of these terms by computing excitation energies across 37 molecular clusters consisting of biologically relevant chromophores surrounded by polar solvent molecules. QM/EFP excitation energies are compared to the fully quantum mechanical calculations with the configuration interaction singles (CIS) method. We find that the charge penetration correction diminishes the accuracy of the QM/EFP calculations. On the other hand, while the effect of exchange-repulsion is negligible for most ππ* transitions, the exchange-repulsion significantly improves description of nπ* transitions with blue solvatochromic shifts. As a result, addition of the exchange-repulsion term improves the overall accuracy of QM/EFP. Performances of QM/EFP models remain similar when excitation energies are modeled with cc-pVDZ and aug-cc-pVDZ basis sets.

Molecular spectroscopy of aqueous solutions: a theoretical perspective

Computational spectroscopy is an invaluable tool to both accurately reproduce the spectra of molecular systems and provide a rationalization for the underlying physics. However, the inherent difficulty to accurately model systems in aqueous solutions, owing to water's high polarity and ability to form hydrogen bonds, has severely hampered the development of the field. In this tutorial review we present a technique developed and tested in recent years based on a fully atomistic and polarizable classical modeling of water coupled with a quantum mechanical description of the solute. Thanks to its unparalleled accuracy and versatility, this method can change the perspective of computational and experimental chemists alike.

Fragment Exchange Potential for Realizing Pauli Deformation of Interfragment Interactions

In fragment-based methods, the lack of explicit short-range exchange interactions between monomers can result in unphysical deformation in charge density. In this study, we describe a fragment exchange potential (XFP) to explicitly account for interfragmental Pauli deformation. In our implementation, a Kohn-Sham exchange potential is adopted along with the Yukawa potential. The method has been validated by comparison of the computed exchange energies using the XFP potential with results obtained from antisymmetrized fragmental orbitals on the S66×8 data set containing 528 bimolecular interactions of equilibrium and arbitrary geometries. It was also found that it is only necessary to deploy numerical grids on atoms within their van der Waals contacts, significantly reducing the small, albeit extra, computational cost. We anticipate that the XFP presented here may be applied to molecular dynamics simulations of macromolecules using a fragment-based quantum mechanical potential with improved SCF convergence and computational accuracy.

Absorption spectra of xanthines in aqueous solution: a computational study

We present a detailed computational analysis of the UV/Vis spectra of caffeine, paraxanthine and theophylline in aqueous solution. A hierarchy of solvation approaches for modeling the aqueous environment have been tested, ranging from the continuum model to the non-polarizable and polarizable quantum mechanical (QM)/molecular mechanics (MM) models, with and without the explicit inclusion of water molecules in the QM portion. The computed results are directly compared with the experimental data, thus highlighting the role of electrostatic, polarization and hydrogen boding solute-solvent interactions.

Avoiding Electron Spill-Out in QM/MM Calculations on Excited States with Simple Pseudopotentials

QM/MM calculations of electronic excitations with diffuse basis sets often have large errors due to spill-out of electrons from the quantum subsystem. The Pauli repulsion of the electrons by the environment has to be included to avoid this. We propose transferable atomic all-electron pseudopotentials that can readily be combined with most MM force fields to avoid electron spill-out. QM/MM excitation energies computed with time-dependent Hartree-Fock and the algebraic diagrammatic construction through second-order are benchmarked against supermolecular calculations to validate these new pseudopotentials. The QM/MM calculations with pseudopotentials give accurate results that are stable with augmentation of the basis set with diffuse functions. We show that the largest contribution to residual deviations from full QM calculations is caused by the missing London dispersion interaction.

Embedding Methods for Quantum Chemistry: Applications from Materials to Life Sciences

Quantum mechanical embedding methods hold the promise to transform not just the way calculations are performed, but to significantly reduce computational costs and improve scaling for macro-molecular systems containing hundreds if not thousands of atoms. The field of embedding has grown increasingly broad with many approaches of different intersecting flavors. In this perspective, we lay out the methods into two streams: QM:MM and QM:QM, showcasing the advantages and disadvantages of both. We provide a review of the literature, the underpinning theories including our contributions, and we highlight current applications with select examples spanning both materials and life sciences. We conclude with prospects and future outlook on embedding, and our view on the use of universal test case scenarios for cross-comparisons of the many available (and future) embedding theories.

Quantum Confinement Effects on Solvatochromic Shifts of Molecular Solutes

We demonstrate the pivotal role of quantum mechanics density confinement effects on solvatochromic shifts. In particular, by resorting to a quantum mechanics/molecular mechanics (QM/MM) approach capable of accounting for confinement effects we successfully reproduce vacuo-to-water solvatochromic shifts for dark n → π* and bright π → π* transitions of acrolein and dark n → π* transitions of pyridine and pyrimidine without the need of including explicit water molecules in the QM portion. Remarkably, our approach is also able to dissect the effects of the single forces acting on the solute-solvent couple and allows for a rationalization of the experimental findings in terms of physicochemical quantities.

Ewald-based methods for Gaussian integral evaluation: application to a new parameterization of GEM

The development of accurate potentials for computational simulations has been an active area of research. Our group has been involved in the development of the Gaussian electrostatic model (GEM), a force field based on molecular densities. The philosophy of GEM is based on the pioneering work of N. Gresh and co-workers of the reproduction of individual inter-molecular interaction components obtained from quantum mechanical (QM) energy decomposition analysis (EDA). The molecular densities used in GEM are represented by fitting accurate QM molecular densities using auxiliary basis sets (comprised of Hermite Gaussians). The use of these molecular densities results in the need to evaluate a large number of Gaussian integrals. We have previously shown that the particle-mesh Ewald (PME), and fast Fourier Poisson (FFP) methods can be used for efficiently evaluating these types of integrals. Here, we present the latest parameterization of GEM* and its application for an extensive study of PME and FFP for molecular dynamics (MD) simulations using a hybrid version of our potential, GEM*. The temperature dependence of various bulk properties is presented and discussed, as well as the effect of various parameters affecting the performance/accuracy of both methods.

Describing Molecular Polarizability by a Bond Capacity Model

We propose a bond capacity model for describing molecular polarization in force field energy functions at the charge-only level. Atomic charges are calculated by allowing charge to flow between atom pairs according to a bond capacity and a difference in electrostatic potential. The bond capacity is closely related to the bond order and decays to zero as the bond distance is increased. The electrostatic potential is composed of an intrinsic potential, identified as the electronegativity, and a screened Coulomb potential from all other charges. The bond capacity model leads to integer fragment charges upon bond dissociation and displays linear scaling of the polarizability with system size. Bond capacity parameters can be derived from reference molecular polarizabilities, while electronegativity parameters can be derived from reference atomic charges or a reference molecular electrostatic potential. Out-of-plane polarization for planar systems is modeled by off-nuclei charge sites. The model is shown to be able to reproduce anisotropic reference molecular polarizabilities with an accuracy of ∼10% using a limited set of bond capacity parameters and can describe both inter- and intramolecular polarization.

LICHEM 1.1: Recent Improvements and New Capabilities

The QM/MM method has become a useful tool to investigate various properties of complex systems. We previously introduced the layered interacting chemical models (LICHEM) package to enable QM/MM simulations with advanced potentials by combining various (unmodified) QM and MM codes ( J. Comp. Chem., 2016, 37, 1019). LICHEM provides several capabilities such as the ability to use polarizable force fields, such as AMOEBA, for the MM environment. Here, we describe an updated version of LICHEM (v1.1), which includes several new functionalities including a new method to account for long-range electrostatic effects in QM/MM (QM/MM-LREC), a new implementation for QM/MM with the Gaussian electrostatic model (GEM), and new capabilities for path optimizations using the quadratic string model (QSM) coupled with restrained MM environment optimization.

Polarizable Force Fields for Biomolecular Simulations: Recent Advances and Applications

Realistic modeling of biomolecular systems requires an accurate treatment of electrostatics, including electronic polarization. Due to recent advances in physical models, simulation algorithms, and computing hardware, biomolecular simulations with advanced force fields at biologically relevant timescales are becoming increasingly promising. These advancements have not only led to new biophysical insights but also afforded opportunities to advance our understanding of fundamental intermolecular forces. This article describes the recent advances and applications, as well as future directions, of polarizable force fields in biomolecular simulations.

Classical Pauli repulsion: An anisotropic, atomic multipole model

Pauli repulsion is a key component of any theory of intermolecular interactions. Although Pauli or exchange repulsion has its origin in the quantum mechanical nature of electrons, it is possible to describe the resulting energetic effects via a classical model in terms of the overlap of electron densities. In fact, closed shell intermolecular repulsion can be explained as a diminution of election density in the internuclear region resulting in decreased screening of nuclear charges and increased nuclear-nuclear repulsion. We provide a concise anisotropic repulsion formulation using the atomic multipoles from the Atomic Multipole Optimized Energetics for Biomolecular Applications force field to describe the electron density at each atom in a larger system. Mathematically, the proposed model consists of damped pairwise exponential multipolar repulsion interactions truncated at short range, which are suitable for use in compute-intensive biomolecular force fields and molecular dynamics simulations. Parameters for 26 atom classes encompassing most organic molecules are derived from a fit to Symmetry Adapted Perturbation Theory exchange repulsion energies for the S101 dimer database. Several applications of the multipolar Pauli repulsion model are discussed, including noble gas interactions, analysis of stationary points on the water dimer potential surface, and the directionality of several halogen bonding interactions.

Semi-Empirical Born-Oppenheimer Molecular Dynamics (SEBOMD) within the Amber Biomolecular Package

Semi-empirical quantum methods from the neglect of differential diatomic overlap (NDDO) family such as MNDO, AM1, or PM3 are fast albeit approximate quantum methods. By combining them with linear scaling methods like the divide & conquer (D&C) method, it is possible to quickly evaluate the energy of systems containing hundreds to thousands of atoms. We here present our implementation in the Amber biomolecular package of a SEBOMD module that provides a way to run semi-empirical Born-Oppenheimer molecular dynamics. At each step of a SEBOMD, a fully converged self-consistent field (SCF) calculation is performed to obtain the semiempirical quantum potential energy of a molecular system encaged or not in periodic boundary conditions. We describe the implementation and the features of our SEBOMD implementation. We show the requirements to conserve the total energy in NVE simulations, and how to accelerate SCF convergence through density matrix extrapolation. Specific ways of handling periodic boundary conditions using mechanical embedding or electrostatic embedding through a tailored quantum Ewald summation is developed. The parallel performance of SEBOMD simulations using the D&C scheme are presented for liquid water systems of various sizes, and a comparison between the traditional full diagonalization scheme and the D&C approach for the reproduction of the structure of liquid water illustrates the potentiality of SEBOMD to simulate molecular systems containing several hundreds of atoms for hundreds of picoseconds with a quantum mechanical potential in a reasonable amount of CPU time.

Interplay between conformational and solvent effects in UV-visible absorption spectra: curcumin tautomers as a case study

We present a combined theoretical and experimental study on the UV-vis spectra of enol-keto (EK) and keto-keto (KK) tautomeric forms of curcumin dissolved in aqueous solution. Solvent effects have been investigated by resorting to the implicit polarizable continuum model (QM/PCM) and non-polarizable and fully polarizable QM/MM approaches, the latter based on the fluctuating charges (FQ) force-field. In particular, all methods are challenged to rationalize the contribution of conformational, electrostatic and polarization effects in the calculation of the vertical excitation spectra of curcumin tautomers. The obtained results highlight that for both tautomers specific solute-solvent hydrogen-bond interactions play a minor role with respect to conformational and electrostatic effects.

Higher Accuracy Achieved in the Simulations of Protein Structure Refinement, Protein Folding, and Intrinsically Disordered Proteins Using Polarizable Force Fields

The accuracy of molecular mechanics force fields is of vital importance in biomolecular simulations. However, the admittedly more accurate polarizable force fields were recently reported to be less able to reproduce the experimental properties in comparison to additive force fields in some cases. Here, we perform long-time-scale molecular dynamics simulations to systematically evaluate the effect of explicit electronic polarization in polarizable force fields. The results show that the inclusion of electrostatic polarization effect in polarizable force fields can improve their accuracies in protein structure refinement and generate conformational ensembles more approximate to experiments for intrinsically disordered proteins. In contrast, it is difficult for polarizable force fields to approach the native structure, let alone to predict the native state when it is unknown a priori in the real protein structure predictions. We speculate that these effects might be attributed to the preference of protein-water interactions in polarizable force fields.