A QM/MM Approach Using the AMOEBA Polarizable Embedding: From Ground State Energies to Electronic Excitations

Modeling infrared and vibrational circular dichroism spectra of complex systems: the DFTB/fluctuating charges route

Simulating vibrational spectra of large biomolecular systems in aqueous environments remains a challenge in computational chemistry due to the complex interactions between solutes and solvents. In this study, we employ the density functional tight-binding (DFTB) method, coupled with the fluctuating charges (FQ) force field, to simulate infrared (IR) and vibrational circular dichroism (VCD) spectra of solvated large biomolecules. We focus on three representative systems: the doxorubicin/DNA intercalation complex, ubiquitin, and hen egg white lysozyme. By using molecular dynamics (MD) trajectories to sample the conformational space, we compute spectra for multiple snapshots, employing different DFTB Hamiltonians, including SCC-DFTB, DFTB3, and GFN1-xTB. Our results demonstrate the accuracy and computational efficiency of the DFTB/FQ method in reproducing experimental spectral features, particularly for large, solvated systems which cannot be afforded by other ab initio methodologies. The results of this work highlight the potential of DFTB/FQ as a scalable method for simulating vibrational properties in complex molecular systems.

Lambda-ABF-OPES: Faster Convergence with High Accuracy in Alchemical Free Energy Calculations

Predicting the binding affinity between small molecules and target macromolecules while combining both speed and accuracy is a cornerstone of modern computational drug discovery, which is critical for accelerating therapeutic development. Despite recent progress in molecular dynamics (MD) simulations, such as advanced polarizable force fields and enhanced sampling techniques, estimating absolute binding free energies (ABFEs) remains computationally challenging. To overcome these difficulties, we introduce a highly efficient hybrid methodology that couples the Lambda-adaptive biasing force (Lambda-ABF) scheme with on-the-fly probability enhanced sampling (OPES). This approach achieves up to a 9-fold improvement in sampling efficiency and computational speed compared to the original Lambda-ABF when used in conjunction with the AMOEBA polarizable force field, yielding converged results at a fraction of the cost of standard techniques.

Modeling Raman Spectra in Complex Environments: From Solutions to Surface-Enhanced Raman Scattering

This perspective highlights the essential physicochemical factors required for accurate computational modeling of Raman and Resonance Raman signals in complex environments. It highlights the theoretical challenges for obtaining a balanced quantum mechanical description of the molecular target, integration of target-environment interactions into the Hamiltonian, and explicit treatment of strong interactions such as hydrogen bonding. The dynamical sampling of solute-solvent phase space and the incorporation of plasmonic effects for Surface-Enhanced Raman Scattering (SERS) are also addressed. Through selected applications, we illustrate how these factors influence Raman signals and propose a framework to tackle these challenges effectively, advancing the reliability of theoretical Raman spectroscopy in real-world scenarios.

Fluorescent Rhodopsins: A Challenging Test for Cost-Effective QM/MM Approaches

In this study, we evaluate the performance of two cost-effective models, namely, TD-DFT and ΔSCF methods, combined with different molecular mechanics models, to predict the photophysical and photochemical properties of a set of fluorescent mutants of the microbial rhodopsin Archaerhodopsin-3. We investigate absorption energies and excited-state isomerization barriers of the embedded retinal protonated Schiff-base chromophore by comparing different DFT functionals as well as different approximations of the embedding model. For absorption energies, CAM-B3LYP demonstrates the most consistent alignment with experiments among the functionals tested, whereas the embedding potentials exhibit similar accuracy. However, incorporating linear response corrections within the polarizable TD-DFT/MM framework enhances accuracy. The photoisomerization barriers, instead, exhibit a pronounced sensitivity to the choice of embedding model, underscoring the complex role that environmental factors play in modulating predictions of excited-state processes. For the two properties here investigated, ΔSCF/MM presents qualitatively similar behavior with respect to TD-DFT for all the tested embedding models.

Self-consistent Quantum Linear Response with a Polarizable Embedding Environment

Quantum computing presents a promising avenue for solving complex problems, particularly in quantum chemistry, where it could accelerate the computation of molecular properties and excited states. This work focuses on computing excitation energies with hybrid quantum-classical algorithms for near-term quantum devices, combining the quantum linear response (qLR) method with a polarizable embedding (PE) environment. We employ the self-consistent operator manifold of quantum linear response (q-sc-LR) on top of a unitary coupled cluster (UCC) wave function in combination with a Davidson solver. The latter removes the need to construct the entire electronic Hessian, improving computational efficiency when going toward larger molecules. We introduce a new superposition-state-based technique to compute Hessian-vector products and show that this approach is more resilient toward noise than our earlier gradient-based approach. We demonstrate the performance of the PE-UCCSD model on systems such as butadiene and para-nitroaniline in water and find that PE-UCCSD delivers comparable accuracy to classical PE-CCSD methods on such simple closed-shell systems. We also explore the challenges posed by hardware noise and propose simple error mitigation techniques to maintain accurate results on noisy quantum computers.

Efficient Polarizable QM/MM Using the Direct Reaction Field Hamiltonian with Electrostatic Potential Fitted Multipole Operators

Electronic polarization and dispersion are decisive actors in determining interaction energies between molecules. These interactions have a particularly profound effect on excitation energies of molecules in complex environments, especially when the excitation involves a significant degree of charge reorganization. The direct reaction field (DRF) approach, which has seen a recent revival of interest, provides a powerful framework for describing these interactions in quantum mechanics/molecular mechanics (QM/MM) models of systems, where a small subsystem of interest is described using quantum chemical methods and the remainder is treated with a simple MM force field. In this paper we show how the DRF approach can be combined with the electrostatic potential fitted (ESPF) multipole operator description of the QM region charge density, which significantly improves the efficiency of the method, particularly for large MM systems, and for typical calculations effectively eliminates the dependence on MM system size. We also show how the DRF approach can be combined with fluctuating charge descriptions of the polarizable environment, as well as previously used atom-centered dipole-polarizability based models. We further show that the ESPF-DRF method provides an accurate description of molecular interactions in both ground and excited electronic states of the QM system and apply it to predict the gas to aqueous solution solvatochromic shifts in the UV/visible absorption spectrum of acrolein.

A Polarizable CASSCF/MM Approach Using the Interface Between OpenMMPol Library and Cfour

We present a polarizable embedding quantum mechanics/molecular mechanics (QM/MM) framework for ground- and excited-state Complete Active Space Self-Consistent Field (CASSCF) calculations on molecules within complex environments, such as biological systems. These environments are modeled using the AMOEBA polarizable force field. This approach is implemented by integrating the OpenMMPol library with the CFour quantum chemistry software suite. The implementation supports both single-point energy evaluations and geometry optimizations, facilitated by the availability of analytical gradients. We demonstrate the methodology by applying it to two distinct photoreceptors, exploring the impact of the protein environment on the structural and photophysical properties of their embedded chromophores.

Improving the Efficiency of Electrostatic Embedding Using the Fast Multipole Method

This paper reports the improvement in the efficiency of embedded-cluster model (ECM) calculations in ORCA thanks to the implementation of the fast multipole method. Our implementation is based on state-of-the-art algorithms and revisits certain aspects, such as efficiently and accurately handling the extent of atomic orbital shell pairs. This enables us to decompose near-field and far-field terms in what we believe is a simple and effective manner. The main result of this work is an acceleration of the evaluation of electrostatic potential integrals by at least one order of magnitude, and up to two orders of magnitude, while maintaining excellent accuracy (always better than the chemical accuracy of 1 kcal/mol). Moreover, the implementation is versatile enough to be used with molecular systems through QM/MM approaches. The code has been fully parallelized and is available in ORCA 6.0.

Development of artificial photosystems based on designed proteins for mechanistic insights into photosynthesis

This review aims to provide an overview of the progress in protein-based artificial photosystem design and their potential to uncover the underlying principles governing light-harvesting in photosynthesis. While significant advances have been made in this area, a gap persists in reviewing these advances. This review provides a perspective of the field, pinpointing knowledge gaps and unresolved challenges that warrant further inquiry. In particular, it delves into the key considerations when designing photosystems based on the chromophore and protein scaffold characteristics, presents the established strategies for artificial photosystems engineering with their advantages and disadvantages, and underscores the recent breakthroughs in understanding the molecular mechanisms governing light-harvesting, charge separation, and the role of the protein motions in the chromophore's excited state relaxation. By disseminating this knowledge, this article provides a foundational resource for defining the field of bio-hybrid photosystems and aims to inspire the continued exploration of artificial photosystems using protein design.

Structural and Electronic Properties of Novel Azothiophene Dyes: A Multilevel Study Incorporating Explicit Solvation Effects

Among azobenzene derivatives, azothiophenes represent a relatively recent family of compounds that exhibit similar characteristics as dyes and photoreactive systems. Their technological applications are extensive thanks to the additional design flexibility conferred by the heteroaromatic ring. In this study, we present a comprehensive investigation of the structural and electronic properties of novel dyes derived from 3-thiophenamine, utilizing a multilevel approach. We thoroughly examined the potential energy surfaces of the E and Z isomers for three molecules, each bearing different substituents on the phenyl ring at the para position relative to the diazo group. This exploration was conducted through quantum chemistry calculations at various levels of theory, employing a continuum solvent model. Subsequently, we incorporated an explicit solvent (a dimethyl sulfoxide-water mixture) to simulate the most stable isomers using classical molecular dynamics, delivering a clear picture of the local solvation structure and intermolecular interactions. Finally, a hybrid quantum mechanics/molecular mechanics (QM/MM) approach was employed to accurately describe the evolution of the solutes' properties within their environment, accounting for finite temperature effects.

Importance of Polarizable Embedding for Computing Optical Rotation: The Case of Camphor in Ethanol

Solvation effects on optical rotation are notoriously challenging to model for computational chemistry, as the specific rotatory power of a molecule can vary wildly going from apolar to polar or even protic solvents. To address such a problem, we present a polarizable embedding implementation of an electric and magnetic response property based on density functional theory and the AMOEBA polarizable force field, and apply such an implementation to the study of the optical rotation of camphor in ethanol. By comparing a continuum model, and electrostatic and polarizable embedding QM/MM models, we observe that accounting for the environment's polarization gives rise to not only a different quantitative prediction, in very good agreement with experiments for the QM/AMOEBA model, but also to a very different qualitative picture, with the values of the optical rotation computed along a classical molecular dynamics trajectory with electrostatic embedding being statistically uncorrelated to the ones obtained with the polarizable description.

Toward a numerically efficient description of bulk-solvated anionic states

We investigate the vertical electron attachment energy (VAE) of 1-methyl-4-nitroimidazole, a model radiosensitizer, employing quantum mechanics/molecular mechanics (QM/MM) and QM/polarized continuum (QM/PCM) solvation models. We considered the solvent-excluded surface (QM/PCM-SES) and Van der Waals (QM/PCM-VDW) cavities within the PCM framework, the electrostatic embedding QM/MM (EE-QM/MM) model, and the self-consistent sequential QM/MM polarizable electrostatic embedding (scPEE-S-QM/MM) model. Due to slow VAE convergence concerning the number of QM solvent molecules, full QM calculations prove inefficient. Ensemble averages in these calculations do not align with VAEs computed for the representative solute-solvent configuration. QM/MM and QM/PCM calculations show agreement with each other for sufficiently large QM regions, although the QM/PCM-VDW model exhibits artifacts linked to the cavity. QM/MM models demonstrate good agreement between ensemble averages and VAEs calculated with the representative configuration. Notably, the VAE computed with the scPEE-S-QM/MM model achieves faster convergence concerning the number of QM water molecules compared to the EE-QM/MM model, attributed to enhanced efficiency from MM charge polarization in the scPEE-S-QM/MM approach. This emphasizes the importance of QM/classical models with accurate solute-solvent and solvent-solvent mutual polarization for obtaining converged VAEs at a reasonable computational cost. The full-QM approach is very inefficient, while the microsolvation model is inaccurate. Computational savings in QM/MM models result from electrostatic embedding and the representative configuration, with the scPEE-S-QM/MM approach emerging as an efficient tool for describing bulk-solvated anions within the QM/MM framework. Its potential extends to improving transient anion state descriptions in biomolecules and radiosensitizers, especially given the frequent employment of microsolvation models.

Importance of Polarizable Embedding for Absorption Spectrum Calculations of Arabidopsis thaliana Cryptochrome 1

Cryptochromes are essential flavoproteins for circadian rhythms and avian magnetoreception. Flavin adenine dinucleotide (FAD), a chromophore within cryptochromes, absorbs blue light, initiating electron transfer processes that lead to a biological signaling cascade. A key step in this cascade is the formation of the FAD semiquinone radical (FADH•), characterized through a specific red-light absorption. The absorption spectra of FADH• in cryptochromes are, however, significantly different from those recorded for the cofactor in solution, primarily due to protein-induced shifts in the absorption peaks. This study employs a multiscale approach, combining molecular dynamics (MD) simulations with quantum mechanical/molecular mechanical (QM/MM) methodologies, to investigate the influence of protein dynamics on embedded FADH• absorption. We emphasize the role of the protein's polarizable environment in the shaping of the absorption spectrum, crucial for accurate spectral predictions in cryptochromes. Our findings provide valuable insights into the absorption process, advancing our understanding of cryptochrome functioning.

Building polarization into protein-inhibitor binding dynamics in rational drug design for rheumatoid arthritis

Standard force field-based simulations to accomplish structure-based evaluations of lead molecules is a powerful tool. Combining protein fragmentation into tractable sub-systems with continuum solvation method is envisaged to enable quantum mechanics-based electronic structure calculations of macromolecules in their realistic environment. This along with incorporation of many-body polarization effect in molecular dynamics simulations may augment an accurate description of electrostatics of protein-inhibitor systems for effective drug design. Rheumatoid arthritis (RA) is a complex autoimmune disorder plagued by the ceiling effect of current targeted therapies, encouraging identification of new druggable targets and corresponding drug design to tackle the refractory form of disease. In this study, polarization-inclusive force field approach has been used to model protein solvation and ligand binding for 'Mitogen-activated protein kinase' (MAP3K8), a regulatory node of notable pharmacological relevance in RA synovial biology. For MAP3K8 inhibitors belonging to different scaffold series, the calculations illustrated differential electrostatic contribution to their relative binding affinities and successfully explained examples from available structure-activity relationship studies. Results from this study exemplified i) the advantage of this approach in reliably ranking inhibitors having close nanomolar range activities for the same target; and ii) its prospective application in lead molecule identification aiding drug discovery efforts in RA.Communicated by Ramaswamy H. Sarma.

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.

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.

The OpenMMPol library for polarizable QM/MM calculations of properties and dynamics

We present a new library designed to provide a simple and straightforward way to implement QM/AMOEBA (Atomic Multipole Optimized Energetics for Biomolecular Applications) and other polarizable QM/MM (Molecular Mechanics) methods based on induced point dipoles. The library, herein referred to as OpenMMPol, is free and open-sourced and is engineered to address the increasing demand for accurate and efficient QM/MM simulations. OpenMMPol is specifically designed to allow polarizable QM/MM calculations of ground state energies and gradients and excitation properties. Key features of OpenMMPol include a modular architecture facilitating extensibility, parallel computing capabilities for enhanced performance on modern cluster architectures, a user-friendly interface for intuitive implementation, and a simple and flexible structure for providing input data. To show the capabilities offered by the library, we present an interface with PySCF to perform QM/AMOEBA molecular dynamics, geometry optimization, and excited-state calculation based on (time-dependent) density functional theory.

Unraveling Water Solvation Effects with Quantum Mechanics/Molecular Mechanics Semiclassical Vibrational Spectroscopy: The Case of Thymidine

We introduce a quantum mechanics/molecular mechanics semiclassical method for studying the solvation process of molecules in water at the nuclear quantum mechanical level with atomistic detail. We employ it in vibrational spectroscopy calculations because this is a tool that is very sensitive to the molecular environment. Specifically, we look at the vibrational spectroscopy of thymidine in liquid water. We find that the C═O frequency red shift and the C═C frequency blue shift, experienced by thymidyne upon solvation, are mainly due to reciprocal polarization effects, that the molecule and the water solvent exert on each other, and nuclear zero-point energy effects. In general, this work provides an accurate and practical tool to study quantum vibrational spectroscopy in solution and condensed phase, incorporating high-level and computationally affordable descriptions of both electronic and nuclear problems.

Multistate, Polarizable QM/MM Embedding Scheme Based on the Direct Reaction Field Method: Solvatochromic Shifts, Analytical Gradients and Optimizations of Conical Intersections in Solution

We recently introduced a polarizable embedding scheme based on an integral-exact reformulation of the direct reaction field method (IEDRF) that accounts for the differential solvation of ground and excited states in QM/MM simulations. The polarization and dispersion interactions between the quantum-mechanical (QM) and molecular-mechanical (MM) regions are described by the DRF Hamiltonian, while the Pauli repulsion between explicitly treated QM electrons and the implicit electron density around MM atoms is modeled with effective core potentials. A single Hamiltonian is used for all electronic states so that Born-Oppenheimer states belonging to the same geometry are orthogonal and state crossings are well-defined. In this work, we describe the implementation of the method using graphical processing unit acceleration in TeraChem, where it is combined with multiple electronic structure methods, including Hartree-Fock, time-dependent density functional theory, and complete active space self-consistent field. In contrast with older implementations of the DRF method, integrals of the polarization operators are evaluated exactly. Expressions for ingredients needed to construct analytical gradients and nonadiabatic coupling vectors are derived and tested by optimizing a conical intersection between two excited states in the presence of a polarizable solvent shell. The method is applied to estimate the solvent shifts of absorption energies of a series of donor-acceptor dyes having low-lying charge-transfer states. Even for a nonpolar solvent such as n-hexane, the inclusion of its static polarizability leads to non-negligible shifts that improve the agreement to essentially quantitative levels (0.03 eV) with full-system calculations. Good agreement with the positions of the experimental absorption maxima measured in solution is also observed.

Deciphering Photoreceptors Through Atomistic Modeling from Light Absorption to Conformational Response

In this review, we discuss the successes and challenges of the atomistic modeling of photoreceptors. Throughout our presentation, we integrate explanations of the primary methodological approaches, ranging from quantum mechanical descriptions to classical enhanced sampling methods, all while providing illustrative examples of their practical application to specific systems. To enhance the effectiveness of our analysis, our primary focus has been directed towards the examination of applications across three distinct photoreceptors. These include an example of Blue Light-Using Flavin (BLUF) domains, a bacteriophytochrome, and the orange carotenoid protein (OCP) employed by cyanobacteria for photoprotection. Particular emphasis will be placed on the pivotal role played by the protein matrix in fine-tuning the initial photochemical event within the embedded chromophore. Furthermore, we will investigate how this localized perturbation initiates a cascade of events propagating from the binding pocket throughout the entire protein structure, thanks to the intricate network of interactions between the chromophore and the protein.

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.

Theoretical and Numerical Comparison of Quantum- and Classical Embedding Models for Optical Spectra

Quantum-mechanical (QM) and classical embedding models approximate a supermolecular quantum-chemical calculation. This is particularly useful when the supermolecular calculation has a size that is out of reach for present QM models. Although QM and classical embedding methods share the same goal, they approach this goal from different starting points. In this study, we compare the polarizable embedding (PE) and frozen-density embedding (FDE) models. The former is a classical embedding model, whereas the latter is a density-based QM embedding model. Our comparison focuses on solvent effects on optical spectra of solutes. This is a typical scenario where super-system calculations including the solvent environment become prohibitively large. We formulate a common theoretical framework for PE and FDE models and systematically investigate how PE and FDE approximate solvent effects. Generally, differences are found to be small, except in cases where electron spill-out becomes problematic in the classical frameworks. In these cases, however, atomic pseudopotentials can reduce the electron-spill-out issue.

Hybrid Quantum Mechanical/Molecular Mechanical Methods For Studying Energy Transduction in Biomolecular Machines

Hybrid quantum mechanical/molecular mechanical (QM/MM) methods have become indispensable tools for the study of biomolecules. In this article, we briefly review the basic methodological details of QM/MM approaches and discuss their applications to various energy transduction problems in biomolecular machines, such as long-range proton transports, fast electron transfers, and mechanochemical coupling. We highlight the particular importance for these applications of balancing computational efficiency and accuracy. Using several recent examples, we illustrate the value and limitations of QM/MM methodologies for both ground and excited states, as well as strategies for calibrating them in specific applications. We conclude with brief comments on several areas that can benefit from further efforts to make QM/MM analyses more quantitative and applicable to increasingly complex biological problems.

Evaluating the Transition State Stabilization/Destabilization Effects of the Electric Fields from Scaffold Residues by a QM/MM Approach

The protein scaffolds of enzymes not only provide structural support for the catalytic center but also exert preorganized electric fields for electrostatic catalysis. In recent years, uniform oriented external electric fields (OEEFs) have been widely applied to enzymatic reactions to mimic the electrostatic effects of the environment. However, the electric fields exerted by individual residues in proteins may be quite heterogeneous across the active site, with varying directions and strengths at different positions of the active site. Here, we propose a QM/MM-based approach to evaluate the effects of the electric fields exerted by individual residues in the protein scaffold. In particular, the heterogeneity of the residue electric fields and the effect of the native protein environment can be properly accounted for by this QM/MM approach. A case study of the O-O heterolysis reaction in the catalytic cycle of TyrH shows that (1) for scaffold residues that are relatively far from the active site, the heterogeneity of the residue electric field in the active site is not very significant and the electrostatic stabilization/destabilization due to each residue can be well approximated with the interaction energy between a uniform electric field and the QM region dipole; (2) for scaffold residues near the active site, the residue electric fields can be highly heterogeneous along the breaking O-O bond. In such a case, approximating the residue electric fields as uniform fields may misrepresent the overall electrostatic effect of the residue. The present QM/MM approach can be applied to evaluate the residues' electrostatic impact on enzymatic reactions, which also can be useful in computational optimization of electric fields to boost the enzyme catalysis.

Quantum chemical elucidation of a sevenfold symmetric bacterial antenna complex

The light-harvesting complex 2 (LH2) of purple bacteria is one of the most studied photosynthetic antenna complexes. Its symmetric structure and ring-like bacteriochlorophyll arrangement make it an ideal system for theoreticians and spectroscopists. LH2 complexes from most bacterial species are thought to have eightfold or ninefold symmetry, but recently a sevenfold symmetric LH2 structure from the bacterium Mch. purpuratum was solved by Cryo-Electron microscopy. This LH2 also possesses unique near-infrared absorption and circular dichroism (CD) spectral properties. Here we use an atomistic strategy to elucidate the spectral properties of Mch. purpuratum LH2 and understand the differences with the most commonly studied LH2 from Rbl. acidophilus. Our strategy exploits a combination of molecular dynamics simulations, multiscale polarizable quantum mechanics/molecular mechanics calculations, and lineshape simulations. Our calculations reveal that the spectral properties of LH2 complexes are tuned by site energies and exciton couplings, which in turn depend on the structural fluctuations of the bacteriochlorophylls. Our strategy proves effective in reproducing the absorption and CD spectra of the two LH2 complexes, and in uncovering the origin of their differences. This work proves that it is possible to obtain insight into the spectral tuning strategies of purple bacteria by quantitatively simulating the spectral properties of their antenna complexes.

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.

Fast Method for Excited-State Dynamics in Complex Systems and Its Application to the Photoactivation of a Blue Light Using Flavin Photoreceptor

The excited-state dynamics of molecules embedded in complex (bio)matrices is still a challenging goal for quantum chemical models. Hybrid QM/MM models have proven to be an effective strategy, but an optimal combination of accuracy and computational cost still has to be found. Here, we present a method which combines the accuracy of a polarizable embedding QM/MM approach with the computational efficiency of an excited-state self-consistent field method. The newly implemented method is applied to the photoactivation of the blue-light-using flavin (BLUF) domain of the AppA protein. We show that the proton-coupled electron transfer (PCET) process suggested for other BLUF proteins is still valid also for AppA.

Excitation Energies of Embedded Chromophores from Frozen-Density Embedding Theory Using State-Specific Electron Densities of the Environment

Starting from the Perdew-Levy theorem on extrema of the Hohenberg-Kohn functional, the expression for the vertical excitation energy is derived within the formal framework of Frozen-Density Embedding Theory (FDET) that makes it possible to use state-specific electron densities of the environment (ρ_B) of an embedded species. The derived general expression involves the embedded wave functions for ground and excited states that are orthogonal and is exact up to quadratic terms in the appropriate density expansion. It can be applied in practice using various methods differing in the treatment of the electron-electron correlation for embedded electrons, the method to evaluate different contributions to the excitation energy, the method to generate state-specific ρ_B, and the approximation used for the non-electrostatic component of the FDET embedding potential. The derived expression is applied for 47 local excitations in 10 embedded organic chromophores. The explicit treatment of the differential polarization of ρ_B improves indeed the accuracy of the excitation energy as compared to the implicit treatment in which the same ρ_B is used for all states of embedded chromophore. For 47 local excitations in 10 embedded organic chromophores, the average absolute errors in excitation energies drop from 0.04 to 0.03 eV and their standard deviations from 0.032 to 0.025 eV, respectively. The maximal errors show similar trends.

State averaged CASSCF in AMOEBA polarizable water model for simulating nonadiabatic molecular dynamics with nonequilibrium solvation effects

This paper presents a state-averaged complete active space self-consistent field (SA-CASSCF) in the atomic multipole optimized energetics for biomolecular application (AMOEBA) polarizable water model, which enables rigorous simulation of non-adiabatic molecular dynamics with nonequilibrium solvation effects. The molecular orbital and configuration interaction coefficients of the solute wavefunction, and the induced dipoles on solvent atoms, are solved by minimizing the state averaged energy variationally. In particular, by formulating AMOEBA water models and the polarizable continuum model (PCM) in a unified way, the algorithms developed for computing SA-CASSCF/PCM energies, analytical gradients, and non-adiabatic couplings in our previous work can be generalized to SA-CASSCF/AMOEBA by properly substituting a specific list of variables. Implementation of this method will be discussed with the emphasis on how the calculations of different terms are partitioned between the quantum chemistry and molecular mechanics codes. We will present and discuss results that demonstrate the accuracy and performance of the implementation. Next, we will discuss results that compare three solvent models that work with SA-CASSCF, i.e., PCM, fixed-charge force fields, and the newly implemented AMOEBA. Finally, the new SA-CASSCF/AMOEBA method has been interfaced with the ab initio multiple spawning method to carry out non-adiabatic molecular dynamics simulations. This method is demonstrated by simulating the photodynamics of the model retinal protonated Schiff base molecule in water.

Understanding Charge Dynamics in Dense Electronic Manifolds in Complex Environments

Photoinduced charge transfer (CT) excited states and their relaxation mechanisms can be highly interdependent on the environment effects and the consequent changes in the electronic density. Providing a molecular interpretation of the ultrafast (subpicosecond) interplay between initial photoexcited states in such dense electronic manifolds in condensed phase is crucial for improving and understanding such phenomena. Real-time time-dependent density functional theory is here the method of choice to observe the charge density, explicitly propagated in an ultrafast time domain, along with all time-dependent properties that can be easily extracted from it. A designed protocol of analysis for real-time electronic dynamics to be applied to time evolving electronic density related properties to characterize both in time and in space CT dynamics in complex systems is here introduced and validated, proposing easy to be read cross-correlation maps. As case studies to test such tools, we present the photoinduced charge-transfer electronic dynamics of 5-benzyluracil, a mimic of nucleic acid/protein interactions, and the metal-to-ligand charge-transfer electronic dynamics in water solution of [Ru(dcbpy)2(NCS)2]4-, dcbpy = (4,4'-dicarboxy-2,2'-bipyridine), or "N34-", a dye sensitizer for solar cells. Electrostatic and explicit ab initio treatment of solvent molecules have been compared in the latter case, revealing the importance of the accurate modeling of mutual solute-solvent polarization on CT kinetics. We observed that explicit quantum mechanical treatment of solvent slowed down the charge carriers mobilities with respect to the gas-phase. When all water molecules were modeled instead as simpler embedded point charges, the electronic dynamics appeared enhanced, with a reduced hole-electron distance and higher mean velocities due to the close fixed charges and an artificially increased polarization effect. Such analysis tools and the presented case studies can help to unveil the influence of the electronic manifold, as well as of the finite temperature-induced structural distortions and the environment on the ultrafast charge motions.

Transferable Potential Function for Flexible H2O Molecules Based on the Single-Center Multipole Expansion

A potential function is presented for describing a system of flexible H₂O molecules based on the single-center multipole expansion (SCME) of the electrostatic interaction. The model, referred to as SCME/f, includes the variation of the molecular quadrupole moment as well as the dipole moment with changes in bond length and angle so as to reproduce results of high-level electronic structure calculations. The multipole expansion also includes fixed octupole and hexadecapole moments, as well as anisotropic dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole polarizability tensors. The model contains five adjustable parameters related to the repulsive interaction and damping functions in the electrostatic and dispersion interactions. Their values are adjusted to reproduce the lowest energy isomers of small clusters, (H₂O)_n with n = 2-6, as well as measured properties of the ice Ih crystal. Subsequent calculations of the energy difference between the various isomer configurations of the clusters show that SCME/f gives good agreement with results of electronic structure calculations and represents a significant improvement over the previously presented rigid SCME potential function. Analysis of the vibrational frequencies of the clusters and structural properties of ice Ih crystal show the importance of accurately describing the variation of the quadrupole moment with molecular structures.

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.

Trajectory Surface Hopping for a Polarizable Embedding QM/MM Formulation

We present the implementation of trajectory surface-hopping nonadiabatic dynamics for a polarizable embedding QM/MM formulation. Time-dependent density functional theory was used at the quantum mechanical level of theory, whereas the molecular mechanics description involved the polarizable AMOEBA force field. This implementation has been obtained by integrating the surface-hopping program Newton-X NS with an interface between the Gaussian 16 and the Tinker suites of codes to calculate QM/AMOEBA energies and forces. The implementation has been tested on a photoinduced electron-driven proton-transfer reaction involving pyrimidine and a hydrogen-bonded water surrounded by a small cluster of water molecules and within a large water droplet.

Computational study on the binding of Mango-II RNA aptamer and fluorogen using the polarizable force field AMOEBA

Fluorescent light-up aptamers (FLAPs) are well-performed biosensors for cellular imaging and the detection of different targets of interest, including RNA, non-nucleic acid molecules, metal ions, and so on. They could be easily designed and emit a strong fluorescence signal once bound to specified fluorogens. Recently, one unique aptamer called Mango-II has been discovered to possess a strong affinity and excellent fluorescent properties with fluorogens TO1-Biotin and TO3-Biotin. To explore the binding mechanisms, computational simulations have been performed to obtain structural and thermodynamic information about FLAPs at atomic resolution. AMOEBA polarizable force field, with the capability of handling the highly charged and flexible RNA system, was utilized for the simulation of Mango-II with TO1-Biotin and TO3-Biotin in this work. The calculated binding free energy using published crystal structures is in excellent agreement with the experimental values. Given the challenges in modeling complex RNA dynamics, our work demonstrates that MD simulation with a polarizable force field is valuable for understanding aptamer-fluorogen binding and potentially designing new aptamers or fluorogens with better performance.

INAQS, a Generic Interface for Nonadiabatic QM/MM Dynamics: Design, Implementation, and Validation for GROMACS/Q-CHEM simulations

The accurate description of large molecular systems in complex environments remains an ongoing challenge for the field of computational chemistry. This problem is even more pronounced for photoinduced processes, as multiple excited electronic states and their corresponding nonadiabatic couplings must be taken into account. Multiscale approaches such as hybrid quantum mechanics/molecular mechanics (QM/MM) offer a balanced compromise between accuracy and computational burden. Here, we introduce an open-source software package (INAQS) for nonadiabatic QM/MM simulations that bridges the sampling capabilities of the GROMACS MD package and the excited-state infrastructure of the Q-CHEM electronic structure software. The interface is simple and can be adapted easily to other MD codes. The code supports a variety of different trajectory-based molecular dynamics, ranging from Born-Oppenheimer to surface hopping dynamics. To illustrate the power of this combination, we simulate electronic absorption spectra, free-energy surfaces along a reaction coordinate, and the excited-state dynamics of 1,3-cyclohexadiene in 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.

Recent Developments in Polarizable Molecular Dynamics Simulations of Electrolyte Solutions

Polarizable molecular dynamics simulations are a fast progressing field in the scientific research of ionic liquids. The fundamentals of polarizable simulations, as well as their application to ionic liquids, were summarized in a review [Bedrov, D.; Piquemal, J.-P.; Borodin, O.; MacKerell, Jr., A. D.; Roux, B.; Schröder, C. Molecular Dynamics Simulations of Ionic Liquids and Electrolytes Using Polarizable Force Fields. Chem. Rev. 2019, 119, 7940–7995] in 2019. Since then, new methods to treat intermolecular interaction of induced dipoles in these highly charged systems were developed. This concerns the damping of these interactions and additional charge transfer as well as the prediction of ionic materials with ultrahigh refractive indices. In addition to the progress of the polarizable force fields, also thermostats and barostats for polarizable simulations evolved recently.

Polarizable Embedding Complex Polarization Propagator in Four- and Two-Component Frameworks

Explicit embedding methods combined with the complex polarization propagator (CPP) enable the modeling of spectroscopy for increasingly complex systems with a high density of states. We present the first derivation and implementation of the CPP in four- and exact-two-component (X2C) polarizable embedding (PE) frameworks. We denote the developed methods PE-4c-CPP and PE-X2C-CPP, respectively. We illustrate the methods by estimating the solvent effect on ultraviolet-visible (UV-vis) and X-ray atomic absorption (XAS) spectra of [Rh(H₂O)₆]³⁺ and [Ir(H₂O)₆]³⁺ immersed in aqueous solution. We moreover estimate solvent effects on UV-vis spectra of a platinum complex that can be photochemically activated (in water) to kill cancer cells. Our results clearly show that the inclusion of the environment is required: UV-vis and (to a lesser degree) XAS spectra can become qualitatively different from vacuum calculations. Comparison of PE-4c-CPP and PE-X2C-CPP methods shows that X2C essentially reproduces the solvent effect obtained with the 4c methods.

Multiscale molecular modelling: from electronic structure to dynamics of nanosystems and beyond

Important contemporary biological and materials problems often depend on interactions that span orders of magnitude differences in spatial and temporal dimensions. This Tutorial Review attempts to provide an introduction to such fascinating problems through a series of case studies, aimed at beginning researchers, graduate students, postdocs and more senior colleagues who are changing direction to focus on multiscale aspects of their research. The choice of specific examples is highly personal, with examples either chosen from our own work or outstanding multiscale efforts from the literature. I start with various embedding schemes, as exemplified by polarizable continuum models, 3-D RISM, molecular DFT and frozen-density embedding. Next, QM/MM (quantum mechanical/molecular mechanical) techniques are the workhorse of pm-to-nm/ps-to-ns simulations; examples are drawn from enzymes and from nanocatalysis for oil-sands upgrading. Using polarizable force-fields in the QM/MM framework represents a burgeoning subfield; with examples from ion channels and electron dynamics in molecules subject to strong external fields, probing the atto-second dynamics of the electrons with RT-TDDFT (real-time - time-dependent density functional theory) eventually coupled with nuclear motion through the Ehrenfest approximation. This is followed by a section on coarse graining, bridging dimensions from atoms to cells. The penultimate chapter gives a quick overview of multiscale approaches that extend into the meso- and macro-scales, building on atomistic and coarse-grained techniques to enter the world of materials engineering, on the one hand, and cell biology, on the other. A final chapter gives just a glimpse of the burgeoning impact of machine learning on the structure-dynamics front. I aim to capture the excitement of contemporary leading-edge breakthroughs in the description of physico-chemical systems and processes in complex environments, with only enough historical content to provide context and aid the next generation of methodological development. While I aim also for a clear description of the essence of methodological breakthroughs, equations are kept to a minimum and detailed formalism and implementation details are left to the references. My approach is very selective (case studies) rather than exhaustive. I think that these case studies should provide fodder to build as complete a reference tree on multiscale modelling as the reader may wish, through forward and backward citation analysis. I hope that my choices of cases will excite interest in newcomers and help to fuel the growth of multiscale modelling in general.

Probing aqueous ions with non-local Auger relaxation

Non-local analogues of Auger decay are increasingly recognized as important relaxation processes in the condensed phase. Here, we explore non-local autoionization, specifically Intermolecular Coulombic Decay (ICD), of a series of aqueous-phase isoelectronic cations following 1s core-level ionization. In particular, we focus on Na+, Mg2+, and Al3+ ions. We unambiguously identify the ICD contribution to the K-edge Auger spectrum. The different strength of the ion-water interactions is manifested by varying intensities of the respective signals: the ICD signal intensity is greatest for the Al3+ case, weaker for Mg2+, and absent for weakly-solvent-bound Na+. With the assistance of ab initio calculations and molecular dynamics simulations, we provide a microscopic understanding of the non-local decay processes. We assign the ICD signals to decay processes ending in two-hole states, delocalized between the central ion and neighbouring water. Importantly, these processes are shown to be highly selective with respect to the promoted water solvent ionization channels. Furthermore, using a core-hole-clock analysis, the associated ICD timescales are estimated to be around 76 fs for Mg2+ and 34 fs for Al3+. Building on these results, we argue that Auger and ICD spectroscopy represents a unique tool for the exploration of intra- and inter-molecular structure in the liquid phase, simultaneously providing both structural and electronic information.

The atomistic modeling of light-harvesting complexes from the physical models to the computational protocol

The function of light-harvesting complexes is determined by a complex network of dynamic interactions among all the different components: the aggregate of pigments, the protein, and the surrounding environment. Complete and reliable predictions on these types of composite systems can be only achieved with an atomistic description. In the last few decades, there have been important advances in the atomistic modeling of light-harvesting complexes. These advances have involved both the completeness of the physical models and the accuracy and effectiveness of the computational protocols. In this Perspective, we present an overview of the main theoretical and computational breakthroughs attained so far in the field, with particular focus on the important role played by the protein and its dynamics. We then discuss the open problems in their accurate modeling that still need to be addressed. To illustrate an effective computational workflow for the modeling of light harvesting complexes, we take as an example the plant antenna complex CP29 and its H111N mutant.

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.

Ultrafast Transient Infrared Spectroscopy of Photoreceptors with Polarizable QM/MM Dynamics

Ultrafast transient infrared (TRIR) spectroscopy is widely used to measure the excitation-induced structural changes of protein-bound chromophores. Here, we design a novel and general strategy to compute TRIR spectra of photoreceptors by combining μs-long MM molecular dynamics with ps-long QM/AMOEBA Born-Oppenheimer molecular dynamics (BOMD) trajectories for both ground and excited electronic states. As a proof of concept, the strategy is here applied to AppA, a blue-light-utilizing flavin (BLUF) protein, found in bacteria. We first analyzed the short-time evolution of the embedded flavin upon excitation revealing that its dynamic Stokes shift is ultrafast and mainly driven by the internal reorganization of the chromophore. A different normal-mode representation was needed to describe ground- and excited-state IR spectra. In this way, we could assign all of the bands observed in the measured transient spectrum. In particular, we could characterize the flavin isoalloxazine-ring region of the spectrum, for which a full and clear description was missing.

Energy, Structures, and Response Properties with a Fully Coupled QM/AMOEBA/ddCOSMO Implementation

We present the implementation of a fully coupled polarizable QM/MM/continuum model based on the AMOEBA polarizable force field and the domain decomposition implementation of the conductor-like screening model. Energies, response properties, and analytical gradients with respect to both QM and MM nuclear positions are available, and a generic, atomistic cavity can be employed. The model is linear scaling in memory requirements and computational cost with respect to the number of classical atoms and is therefore suited to model large, complex systems. Using three variants of the green-fluorescent protein, we investigate the overall computational cost of such calculations and the effect of the continuum model on the convergence of the computed properties with respect to the size of the embedding. We also demonstrate the fundamental role of polarization effects by comparing polarizable and nonpolarizable embeddings to fully QM ones.