A General Route to Include Pauli Repulsion and Quantum Dispersion Effects in QM/MM Approaches

A Vision for the Future of Multiscale Modeling

The rise of modern computer science enabled physical chemistry to make enormous progresses in understanding and harnessing natural and artificial phenomena. Nevertheless, despite the advances achieved over past decades, computational resources are still insufficient to thoroughly simulate extended systems from first principles. Indeed, countless biological, catalytic and photophysical processes require ab initio treatments to be properly described, but the breadth of length and time scales involved makes it practically unfeasible. A way to address these issues is to couple theories and algorithms working at different scales by dividing the system into domains treated at different levels of approximation, ranging from quantum mechanics to classical molecular dynamics, even including continuum electrodynamics. This approach is known as multiscale modeling and its use over the past 60 years has led to remarkable results. Considering the rapid advances in theory, algorithm design, and computing power, we believe multiscale modeling will massively grow into a dominant research methodology in the forthcoming years. Hereby we describe the main approaches developed within its realm, highlighting their achievements and current drawbacks, eventually proposing a plausible direction for future developments considering also the emergence of new computational techniques such as machine learning and quantum computing. We then discuss how advanced multiscale modeling methods could be exploited to address critical scientific challenges, focusing on the simulation of complex light-harvesting processes, such as natural photosynthesis. While doing so, we suggest a cutting-edge computational paradigm consisting in performing simultaneous multiscale calculations on a system allowing the various domains, treated with appropriate accuracy, to move and extend while they properly interact with each other. Although this vision is very ambitious, we believe the quick development of computer science will lead to both massive improvements and widespread use of these techniques, resulting in enormous progresses in physical chemistry and, eventually, in our society.

Time-Dependent Multilevel Density Functional Theory

We present a novel three-layer approach based on multilevel density functional theory (MLDFT) and polarizable molecular mechanics to simulate the electronic excitations of chemical systems embedded in an external environment within the time-dependent DFT formalism. In our method, the electronic structure of a target system, the chromophore, is determined in the field of an embedded inactive layer, which is treated as frozen. Long-range interactions are described by employing the polarizable fluctuating charge (FQ) force field. The resulting MLDFT/FQ thus accurately describes both electrostatics (and polarization) and non-electrostatic target-environment interactions. The robustness and reliability of the approach are demonstrated by comparing our results with experimental data reported for various organic molecules in solution.

Close-Up Look at Electronic Spectroscopic Signatures of Common Pharmaceuticals in Solution

Simulating electronic properties and spectral signals requires robust computational approaches that need tuning with the system's peculiarities. In this paper, we test implicit and fully atomistic solvation models for the calculation of UV-vis and electronic circular dichroism (ECD) spectra of two pharmaceutically relevant molecules, namely, (2S)-captopril and (S)-naproxen, dissolved in aqueous solution. Room temperature molecular dynamics simulations reveal that these two drugs establish strong contacts with the surrounding solvent molecules via hydrogen bonds. Such specific interactions, which play a major role in the spectral response and are neglected in implicit approaches, are further characterized and quantified with natural bond orbital methods. Our calculations show that simulated spectra, and especially ECD, are in good agreement with experiments solely when conformational and configurational dynamics, mutual polarization, and solute-solvent repulsion effects are considered.

Effective fully polarizable QM/MM approaches to compute Raman and Raman Optical Activity spectra in aqueous solution

Raman and Raman Optical Activity (ROA) signals are amply affected by solvent effects, especially in the presence of strongly solute-solvent interactions such as Hydrogen Bonding (HB). In this work, we extend the fully atomistic polarizable Quantum Mechanics/Molecular Mechanics approach, based on the Fluctuating Charges and Fluctuating Dipoles force field to the calculation of Raman and ROA spectra. Such an approach is able to accurately describe specific HB interactions, by also accounting for anisotropic contributions due to the inclusion of fluctuating dipoles. To highlight the potentiality of the novel approach, Raman and ROA spectra of L-Serine and L-Cysteine dissolved in aqueous solution are computed and compared both with alternative theoretical approaches and experimental measurements.

Multiscale Frozen Density Embedding/Molecular Mechanics Approach for Simulating Magnetic Response Properties of Solvated Systems

We present a three-layer hybrid quantum mechanical/quantum embedding/molecular mechanics approach for calculating nuclear magnetic resonance (NMR) shieldings and J-couplings of molecular systems in solution. The model is based on the frozen density embedding (FDE) and polarizable fluctuating charges (FQ) and fluctuating dipoles (FQFμ) force fields and permits the accurate ab initio description of short-range nonelectrostatic interactions by means of the FDE shell and cost-effective treatment of long-range electrostatic interactions through the polarizable force field FQ(Fμ). Our approach's accuracy and potential are demonstrated by studying NMR spectra of Brooker's merocyanine in aqueous and nonaqueous solutions.

Free Energy Profile Decomposition Analysis for QM/MM Simulations of Enzymatic Reactions

In enzyme mechanistic studies and mutant design, it is highly desirable to know the individual residue contributions to the reaction free energy and barrier. In this work, we show that such free energy contributions from each residue can be readily obtained by postprocessing ab initio quantum mechanical molecular mechanical (ai-QM/MM) free energy simulation trajectories. Specifically, through a mean force integration along the minimum free energy pathway, one can obtain the electrostatic, polarization, and van der Waals contributions from each residue to the free energy barrier. Separately, a similar analysis procedure allows us to assess the contribution from different collective variables along the reaction coordinate. The chorismate mutase reaction is used to demonstrate the utilization of these two trajectory analysis tools.

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.

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.

Integrated Multiscale Multilevel Approach to Open Shell Molecular Systems

We present a novel multiscale approach to study the electronic structure of open shell molecular systems embedded in an external environment. The method is based on the coupling of multilevel Hartree-Fock (MLHF) and Density Functional Theory (MLDFT), suitably extended to the unrestricted formalism, to Molecular Mechanics (MM) force fields (FF). Within the ML region, the system is divided into active and inactive parts, thus describing the most relevant interactions (electrostatic, polarization, and Pauli repulsion) at the quantum level. The surrounding MM part, which is formulated in terms of nonpolarizable or polarizable FFs, permits a physically consistent treatment of long-range electrostatics and polarization effects. The approach is extended to the calculation of hyperfine coupling constants and applied to selected nitroxyl radicals in an aqueous solution.

Assessing the quality of QM/MM approaches to describe vacuo-to-water solvatochromic shifts

The performance of different quantum mechanics/molecular mechanics embedding models to compute vacuo-to-water solvatochromic shifts is investigated. In particular, both nonpolarizable and polarizable approaches are analyzed and computed results are compared to reference experimental data. We show that none of the approaches outperform the others and that errors strongly depend on the nature of the molecular transition to be described. Thus, we prove that the best choice of embedding model highly depends on the molecular system and that the use of a specific approach as a black box can lead to significant errors and, sometimes, totally wrong predictions.

Multiple Facets of Modeling Electronic Absorption Spectra of Systems in Solution

In this Perspective, we outline the essential physicochemical aspects that need to be considered when building a reliable approach to describe absorption properties of solvated systems. In particular, we focus on how to properly model the complexity of the solvation phenomenon, arising from dynamical aspects and specific, strong solute-solvent interactions. To this end, conformational and configurational sampling techniques, such as Molecular Dynamics, have to be coupled to accurate fully atomistic Quantum Mechanical/Molecular Mechanics (QM/MM) methodologies. By exploiting different illustrative applications, we show that an effective reproduction of experimental spectral signals can be achieved by delicately balancing exhaustive sampling, hydrogen bonding, mutual polarization, and nonelectrostatic effects.

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.

Method to Compute the Solute-Solvent Dispersion Contribution to the Electronic Excitation Energy in Solution

A method formulated within the polarizable continuum model of the solvent and a quantum Monte Carlo treatment of the electronic states of the solute molecule is presented for the calculation of the solute–solvent dispersion contribution to the electronic excitation energy in solution. Variational quantum Monte Carlo is exploited to measure the fluctuations of the electronic electric field of the solute molecule to compute the London’s dispersion forces with the solvent. The method previously applied to the ground state of the solute is here extended to excited states. To perform the Casimir–Polder integration, we introduce a positive parameter Ω whose value is properly chosen for this purpose. We derive a general expression that for Ω = 0 reduces to that already proposed for the ground state. For an excited state, Ω must be less than the first transition electronic energy of the solvent molecule but greater than the transition energy from the ground to excited electronic state of the solute molecule. Benchmark calculations were performed on the n → π* transition for formaldehyde, acrolein, and acetone in six solvents, including water, ethanol, cyclohexane, chloroform, carbon tetrachloride, and toluene, and the π → π* transition of acrolein in cyclohexane. Solvents are characterized by their ionization potential and the refractive index at frequency Ω. In all cases, we found that the dispersion solute–solvent interaction stabilizes the excited state of the solutes leading to red (negative) solvatochromic shifts.

Amide Spectral Fingerprints are Hydrogen Bonding-Mediated

The origin of the peculiar amide spectral features of proteins in aqueous solution is investigated, by exploiting a combined theoretical and experimental approach to study UV Resonance Raman (RR) spectra of peptide molecular models, namely N-acetylglycine-N-methylamide (NAGMA) and N-acetylalanine-N-methylamide (NALMA). UVRR spectra are recorded by tuning Synchrotron Radiation at several excitation wavelengths and modeled by using a recently developed multiscale protocol based on a polarizable QM/MM approach. Thanks to the unparalleled agreement between theory and experiment, we demonstrate that specific hydrogen bond interactions, which dominate hydration dynamics around these solutes, play a crucial role in the selective enhancement of amide signals. These results further argue the capability of vibrational spectroscopy methods as valuable tools for refined structural analysis of peptides and proteins in aqueous solution.

Evaluation of molecular photophysical and photochemical properties using linear response time-dependent density functional theory with classical embedding: Successes and challenges

Time-dependent density functional theory (TDDFT) based approaches have been developed in recent years to model the excited-state properties and transition processes of the molecules in the gas-phase and in a condensed medium, such as in a solution and protein microenvironment or near semiconductor and metal surfaces. In the latter case, usually, classical embedding models have been adopted to account for the molecular environmental effects, leading to the multi-scale approaches of TDDFT/polarizable continuum model (PCM) and TDDFT/molecular mechanics (MM), where a molecular system of interest is designated as the quantum mechanical region and treated with TDDFT, while the environment is usually described using either a PCM or (non-polarizable or polarizable) MM force fields. In this Perspective, we briefly review these TDDFT-related multi-scale models with a specific emphasis on the implementation of analytical energy derivatives, such as the energy gradient and Hessian, the nonadiabatic coupling, the spin–orbit coupling, and the transition dipole moment as well as their nuclear derivatives for various radiative and radiativeless transition processes among electronic states. Three variations of the TDDFT method, the Tamm–Dancoff approximation to TDDFT, spin–flip DFT, and spin-adiabatic TDDFT, are discussed. Moreover, using a model system (pyridine–Ag20 complex), we emphasize that caution is needed to properly account for system–environment interactions within the TDDFT/MM models. Specifically, one should appropriately damp the electrostatic embedding potential from MM atoms and carefully tune the van der Waals interaction potential between the system and the environment. We also highlight the lack of proper treatment of charge transfer between the quantum mechanics and MM regions as well as the need for accelerated TDDFT modelings and interpretability, which calls for new method developments.

Covalent Inhibition of the Human 20S Proteasome with Homobelactosin C Inquired by QM/MM Studies

20S proteasome is a main player in the protein degradation pathway in the cytosol, thus intervening in multiple pivotal cellular processes. Over the years the proteasome has emerged as a crucial target for the treatment of many diseases such as neurodegenerative diseases, cancer, autoimmune diseases, developmental disorders, cystic fibrosis, diabetes, cardiac diseases, atherosclerosis, and aging. In this work, the mechanism of proteasome covalent inhibition with bisbenzyl-protected homobelactosin C (hBelC) was explored using quantum mechanics/molecular mechanics (QM/MM) methods. Molecular dynamic simulations were used to describe key interactions established between the hBelC and its unique binding mode in the primed site of the β5 subunit. The free energy surfaces were computed to characterize the kinetics and thermodynamics of the inhibition process. This study revealed that although the final inhibition product for hBelC is formed according to the same molecular mechanism as one described for hSalA, the free energy profile of the reaction pathway differs significantly from the one previously reported for γ-lactam-β-lactone containing inhibitors in terms of the height of the activation barrier as well as the stabilization of the final product. Moreover, it was proved that high stabilization of the covalent adduct formed between β5-subunit and hBelC, together with the presence of aminocarbonyl side chain in the structure of the inhibitor which prevents the hydrolysis of the ester bond from taking place, determines its irreversible character.

Absorption Properties of Large Complex Molecular Systems: The DFTB/Fluctuating Charge Approach

We report on the first formulation of a novel polarizable QM/MM approach, where the density functional tight binding (DFTB) is coupled to the fluctuating charge (FQ) force field. The resulting method (DFTB/FQ) is then extended to the linear response within the TD-DFTB framework and challenged to study absorption spectra of large condensed-phase systems.

Quantum Mechanics/Fluctuating Charge Protocol to Compute Solvatochromic Shifts

Despite the potentialities of the quantum mechanics (QM)/fluctuating charge (FQ) approach to model the spectral properties of solvated systems, its extensive use has been hampered by the lack of reliable parametrizations of solvents other than water. In this paper, we substantially extend the applicability of QM/FQ to solvating environments of different polarities and hydrogen-bonding capabilities. The reliability and robustness of the approach are demonstrated by challenging the model to simulate solvatochromic shifts of four organic chromophores, which display large shifts when dissolved in apolar, aprotic or polar, protic solvents.

A Joint Venture of Ab Initio Molecular Dynamics, Coupled Cluster Electronic Structure Methods, and Liquid-State Theory to Compute Accurate Isotropic Hyperfine Constants of Nitroxide Probes in Water

The isotropic hyperfine coupling constant (HFCC, Aiso) of a pH-sensitive spin probe in a solution, HMI (2,2,3,4,5,5-hexamethylimidazolidin-1-oxyl, C9H19N2O) in water, is computed using an ensemble of state-of-the-art computational techniques and is gauged against X-band continuous wave electron paramagnetic resonance (EPR) measurement spectra at room temperature. Fundamentally, the investigation aims to delineate the cutting edge of current first-principles-based calculations of EPR parameters in aqueous solutions based on using rigorous statistical mechanics combined with correlated electronic structure techniques. In particular, the impact of solvation is described by exploiting fully atomistic, RISM integral equation, and implicit solvation approaches as offered by ab initio molecular dynamics (AIMD) of the periodic bulk solution (using the spin-polarized revPBE0-D3 hybrid functional), embedded cluster reference interaction site model integral equation theory (EC-RISM), and polarizable continuum embedding (using CPCM) of microsolvated complexes, respectively. HFCCs are obtained from efficient coupled cluster calculations (using open-shell DLPNO-CCSD theory) as well as from hybrid density functional theory (using revPBE0-D3). Re-solvation of "vertically desolvated" spin probe configuration snapshots by EC-RISM embedding is shown to provide significantly improved results compared to CPCM since only the former captures the inherent structural heterogeneity of the solvent close to the spin probe. The average values of the Aiso parameter obtained based on configurational statistics using explicit water within AIMD and from EC-RISM solvation are found to be satisfactorily close. Using either such explicit or RISM solvation in conjunction with DLPNO-CCSD calculations of the HFCCs provides an average Aiso parameter for HMI in aqueous solution at 300 K and 1 bar that is in good agreement with the experimentally determined one. The developed computational strategy is general in the sense that it can be readily applied to other spin probes of similar molecular complexity, to aqueous solutions beyond ambient conditions, as well as to other solvents in the longer run.

An efficient and robust procedure to calculate absorption spectra of aqueous charged species applied to NO2. Phys Chem Chem Phys 2021;23:14857-14872. [PMID: 34223573 DOI: 10.1039/d1cp00652e]

Accurate calculation of absorption spectra of aqueous NO2- requires rigorously sampling the quantum potential energy surfaces for microsolvation of NO2- with at least five explicit water molecules and embedding the resulting clusters in a continuum solvent accounting for the statistical weighted contributions of individual isomers. This method, which we address as ASCEC + PCM, introduces several desired features when compared against MD simulations derived QM/MM spectra: comparatively fewer explicit solvent molecules to be treated with expensive QM methods, the identification of equilibrium structures in the quantum PES to be used in further vibrational spectroscopy, and the unequivocal identification of cluster orbitals undergoing electronic transitions and charge transfer that originate the spectral bands.

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.

Dispersion without Many-Body Density Distortion: Assessment on Atoms and Small Molecules

The "fixed diagonal matrices" (FDM) dispersion formalism [Kooi, D. P.; et al. J. Phys. Chem. Lett. 2019, 10, 1537] is based on a supramolecular wave function constrained to leave the diagonal of the many-body density matrix of each monomer unchanged, reducing dispersion to a balance between kinetic energy and monomer-monomer interaction. The corresponding variational optimization leads to expressions for the dispersion energy in terms of the ground-state pair densities of the isolated monomers only, providing a framework to build new approximations without the need for polarizabilities or virtual orbitals. Despite the underlying microscopic real space mechanism being incorrect, as in the exact case there is density relaxation, the formalism has been shown to give extremely accurate (or even exact) dispersion coefficients for H and He. The question we answer in this work is how accurate the FDM expressions can be for isotropic and anisotropic C6 dispersion coefficients when monomer pair densities are used from different levels of theory, namely Hartree-Fock, MP2, and CCSD. For closed-shell systems, FDM with CCSD monomer pair densities yield a mean average percent error for isotropic C6 dispersion coefficients of about 7% and a maximum absolute error within 18%, with a similar accuracy for anisotropies. The performance for open-shell systems is less satisfactory, with CCSD pair densities performing sometimes worse than Hartree-Fock or MP2. In the present implementation, the computational cost on top of the monomer's ground-state calculations is O(N4). The results show little sensitivity to the basis set used in the monomer's calculations.

Ab initio effective one-electron potential operators: Applications for charge-transfer energy in effective fragment potentials

The concept of effective one-electron potentials (EOPs) has proven to be extremely useful in efficient description of electronic structure of chemical systems, especially extended molecular aggregates such as interacting molecules in condensed phases. Here, a general method for EOP-based elimination of electron repulsion integrals is presented, that is tuned toward the fragment-based calculation methodologies such as the second generation of the effective fragment potentials (EFP2) method. Two general types of the EOP operator matrix elements are distinguished and treated either via the distributed multipole expansion or the extended density fitting (DF) schemes developed in this work. The EOP technique is then applied to reduce the high computational costs of the effective fragment charge-transfer (CT) terms being the bottleneck of EFP2 potentials. The alternative EOP-based CT energy model is proposed, derived within the framework of intermolecular perturbation theory with Hartree-Fock noninteracting reference wavefunctions, compatible with the original EFP2 formulation. It is found that the computational cost of the EFP2 total interaction energy calculation can be reduced by up to 38 times when using the EOP-based formulation of CT energy, as compared to the original EFP2 scheme, without compromising the accuracy for a wide range of weakly interacting neutral and ionic molecular fragments. The proposed model can thus be used routinely within the EFP2 framework.

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.

Multilevel Density Functional Theory

Following recent developments in multilevel embedding methods, we introduce a novel density matrix-based multilevel approach within the framework of density functional theory (DFT). In this multilevel DFT, the system is partitioned in an active and an inactive fragment, and all interactions are retained between the two parts. The decomposition of the total system is performed upon the density matrix. The orthogonality between the two parts is maintained by solving the Kohn–Sham equations in the MO basis for the active part only, while keeping the inactive density matrix frozen. This results in the reduction of computational cost. We outline the theory and implementation and discuss the differences and similarities with state-of-the-art DFT embedding methods. We present applications to aqueous solutions of methyloxirane and glycidol.

Energy-Based Molecular Orbital Localization in a Specific Spatial Region

We present a novel energy-based localization procedure able to localize molecular orbitals into predefined spatial regions. The method is defined in a multiscale framework based on the multilevel Hartree-Fock approach. In particular, the system is partitioned into active and inactive fragments. The localized molecular orbitals are obtained maximizing the repulsion between the two fragments. The method is applied to several cases including both conjugated and non-conjugated systems. Our multiscale approach is compared with reference values for both ground-state properties, such as dipole moments, and local excitation energies. The proposed approach is useful to extend the application range of high-level electron correlation methods. In fact, the reduced number of molecular orbitals can lead to a large reduction in the computational cost of correlated calculations.

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.

Simulating Absorption Spectra of Flavonoids in Aqueous Solution: A Polarizable QM/MM Study

We present a detailed computational study of the UV/Vis spectra of four relevant flavonoids in aqueous solution, namely luteolin, kaempferol, quercetin, and myricetin. The absorption spectra are simulated by exploiting a fully polarizable quantum mechanical (QM)/molecular mechanics (MM) model, based on the fluctuating charge (FQ) force field. Such a model is coupled with configurational sampling obtained by performing classical molecular dynamics (MD) simulations. The calculated QM/FQ spectra are compared with the experiments. We show that an accurate reproduction of the UV/Vis spectra of the selected flavonoids can be obtained by appropriately taking into account the role of configurational sampling, polarization, and hydrogen bonding interactions.

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.

Calculation of Linear and Non-linear Electric Response Properties of Systems in Aqueous Solution: A Polarizable Quantum/Classical Approach with Quantum Repulsion Effects

We present a computational study of polarizabilities and hyperpolarizabilities of organic molecules in aqueous solutions, focusing on solute-water interactions and the way they affect a molecule's linear and non-linear electric response properties. We employ a polarizable quantum mechanics/molecular mechanics (QM/MM) computational model that treats the solute at the QM level while the solvent is treated classically using a force field that includes polarizable charges and dipoles, which dynamically respond to the solute's quantum-mechanical electron density. Quantum confinement effects are also treated by means of a recently implemented method that endows solvent molecules with a parametric electron density, which exerts Pauli repulsion forces upon the solute. By applying the method to a set of aromatic molecules in solution we show that, for both polarizabilities and first hyperpolarizabilities, observed solution values are the result of a delicate balance between electrostatics, hydrogen-bonding, and non-electrostatic solute solvent interactions.

Theory and algorithms for chiroptical properties and spectroscopies of aqueous systems

Chiroptical properties and spectroscopies are valuable tools to study chiral molecules and assign absolute configurations. The spectra that result from chiroptical measurements may be very rich and complex, and hide much of their information content. For this reason, the interplay between experiments and calculations is especially useful, provided that all relevant physico-chemical interactions that are present in the experimental sample are accurately modelled. The inherent difficulty associated to the calculation of chiral signals of systems in aqueous solutions requires the development of specific tools, able to account for the peculiarities of water-solute interactions, and especially its ability to form hydrogen bonds. In this perspective we discuss a multiscale approach, which we have developed and challenged to model the most used chiroptical techniques.

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.

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.

Polarizable embedding QM/MM: the future gold standard for complex (bio)systems?

We provide a perspective of the induced dipole formulation of polarizable QM/MM, showing how efficient implementations will enable their application to the modeling of dynamics, spectroscopy, and reactivity in complex biosystems.

Polarizable Embedding with a Transferable H2O Potential Function II: Application to (H2O)n Clusters and Liquid Water

The incorporation of polarization in multiscale quantum-mechanics/molecular-mechanics (QM/MM) simulations is important for a variety of applications, for example, charge-transfer reactions. A recently developed formalism based on a density functional theory description of the QM region and a potential energy function for H₂O molecules that includes quadrupole as well as dipole polarizability of the MM region is used to simulate liquid water and water clusters. Analysis of the energy, atomic forces, MM polarization, and structure is presented. A quantitative assessment of the QM/MM-MM/MM interaction energy differences of all possible QM/MM configurations of (H₂O)_n clusters shows that the interquartile range of the distributions of the QM/MM binding energies is never more than 20 meV/molecule higher or lower than the binding energies produced with either of the single-model results. Comparing these interaction energy differences with the QM/MM induction differences show that they are not systematically caused by the induced MM moments of our polarizable embedding scheme. Optimized hexamer geometries as well as the liquid water structure are shown to be improved in comparison with results obtained using point-charge based embedding models neglecting polarization.

Electronic transitions for a fully polarizable QM/MM approach based on fluctuating charges and fluctuating dipoles: Linear and corrected linear response regimes

The fully polarizable Quantum Mechanics/Molecular Mechanics (QM/MM) approach based on fluctuating charges and fluctuating dipoles, named QM/FQFμ [T. Giovannini et al., J. Chem. Theory Comput. 15, 2233 (2019)], is extended to the calculation of vertical excitation energies of solvated molecular systems. Excitation energies are defined within two different solvation regimes, i.e., linear response (LR), where the response of the MM portion is adjusted to the QM transition density, and corrected-Linear Response (cLR) in which the MM response is adjusted to the relaxed QM density, thus being able to account for charge equilibration in the excited state. The model, which is specified in terms of three physical parameters (electronegativity, chemical hardness, and polarizability) is applied to vacuo-to-water solvatochromic shifts of aqueous solutions of para-nitroaniline, pyridine, and pyrimidine. The results show a good agreement with their experimental counterparts, thus highlighting the potentialities of this approach.

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.

A combined experimental and theoretical study of optical rotatory dispersion for (R)-glycidyl methyl ether in aqueous solution

The dispersive optical activity for aqueous solutions of non-rigid (R)-glycidyl methyl ether (R-GME) has been explored synergistically from experimental and theoretical perspectives. Density functional theory analyses performed with the polarizable continuum model for implicit solvation identified nine low-lying stable conformers that are interconverted by rotation about two large-amplitude torsional coordinates. The antagonistic chiroptical signatures predicted for these structural isomers were averaged under a Boltzmann-weighting ansatz to estimate the behavior expected for a thermally equilibrated ensemble. This led to optical rotatory dispersion profiles that reproduced the overall shape of observations but failed to achieve uniform agreement with measured specific-rotation values even when anharmonic vibrational corrections were applied. A mixed QM/FQ paradigm, whereby quantum-mechanical (QM) calculations of optical activity were combined with classical molecular dynamics simulations of explicit solvation that included mutual-polarization effects by means of fluctuating charges (FQ), was enlisted to elucidate the microsolvation environment and gauge its impact upon conformer distributions and response properties. Although quantitative accord with experiments remained elusive, this approach revealed strong variations in the magnitude and sign of rotatory powers for R-GME as the configuration of surrounding water molecules evolved, thereby highlighting the inherently dynamical nature of the solvated chiroptical response, calling into question the validity of "static" descriptions based on the presumption of distinct energy minima, and giving insight into the inherent complexity posed by the modeling of such properties for solvated systems.

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.

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

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