Modeling Solvatochromic Shifts Using the Orbital-Free Embedding Potential at Statistically Mechanically Averaged Solvent Density

Embedding Nonrigid Solutes in an Averaged Environment: A Case Study on Rhodopsins

Many simulation methods concerning solvated molecules are based on the assumption that the solvated species and the solvent can be characterized by some representative structures of the solute and some embedding potential corresponding to this structure. While the averaging of the solvent configurations to obtain an embedding potential has been studied in great detail, this hinges on a single solute structure representation. This assumption is re-examined and generalized for conformationally flexible solutes and tested on 4 nonrigid systems. In this generalized approach, the solute is characterized by a set of representative structures and the corresponding embedding potentials. The representative structures are identified by means of subdividing the statistical ensemble, which in this work is generated by a constant-temperature molecular dynamics simulation. The embedding potential defined in the Frozen-Density Embedding Theory is used to characterize the average effect of the solvent in each subensemble. The numerical examples concern the vertical excitation energies of protonated retinal Schiff bases in protein environments. It is comprehensively shown that subensemble averaging leads to huge computational savings compared with explicit averaging of the excitation energies in the whole ensemble while introducing only minor errors in the case of the systems examined.

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.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

Adaptive Subsystem Density Functional Theory

Subsystem density functional theory (DFT) is emerging as a powerful electronic structure method for large-scale simulations of molecular condensed phases and interfaces. Key to its computational efficiency is the use of approximate nonadditive noninteracting kinetic energy functionals. Unfortunately, currently available nonadditive functionals lead to inaccurate results when the subsystems interact strongly such as when they engage in chemical reactions. This work disrupts the status quo by devising a workflow that extends subsystem DFT's applicability also to strongly interacting subsystems. This is achieved by implementing a fully automated adaptive definition of subsystems which is realized during geometry optimizations or ab initio molecular dynamics simulations. The new method prescribes subsystem merging and splitting events redistributing the resources (both for work and data) in an efficient way making use of modern parallelization strategies and object-oriented programming. We showcase the method with examples probing from moderate-to-strong inter-subsystem interactions, opening the door to using subsystem DFT for modeling chemical reactions in molecular condensed phases with a black box computational tool.

Role of Dielectric Screening in Calculating Excited States of Solvated Azobenzene: A Benchmark Study Comparing Quantum Embedding and Polarizable Continuum Model for Representing the Solvent

The low energy excited states of the conformational isomers of solvated azobenzene are calculated with several DFT methods accounting for the solute-solvent interaction implicitly with the polarizable continuum model or explicitly with subsystem DFT. For the latter, embedding potentials are calculated for 21 sampled snapshots of the solvent molecules. First, we find that accounting for the solvent implicitly or explicitly has little effect on the predicted cis-trans S₁ excitation energy gap. Second, we find that azobenzene's S₁ cis and trans energies are accurate as long as a screened range-separated hybrid exchange-correlation functional is employed. Finally, we also tested a simplified workflow whereby a single, averaged, embedding potential is used. Unfortunately, we find larger deviations against the experiment for the simplified workflow. This highlights a basic flaw in the approach, where the time scale of solvent averaging is much longer than that of the solute's electronic polarization.

Accelerating the 3D reference interaction site model theory of molecular solvation with treecode summation and cut-offs

The 3D reference interaction site model (3D-RISM) of molecular solvation is a powerful tool for computing the equilibrium thermodynamics and density distributions of solvents, such as water and co-ions, around solute molecules. However, 3D-RISM solutions can be expensive to calculate, especially for proteins and other large molecules where calculating the potential energy between solute and solvent requires more than half the computation time. To address this problem, we have developed and implemented treecode summation for long-range interactions and analytically corrected cut-offs for short-range interactions to accelerate the potential energy and long-range asymptotics calculations in non-periodic 3D-RISM in the AmberTools molecular modeling suite. For the largest single protein considered in this work, tubulin, the total computation time was reduced by a factor of 4. In addition, parallel calculations with these new methods scale almost linearly and the iterative solver remains the largest impediment to parallel scaling. To demonstrate the utility of our approach for large systems, we used 3D-RISM to calculate the solvation thermodynamics and density distribution of 7-ring microtubule, consisting of 910 tubulin dimers, over 1.2 million atoms.

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.

The Seamless Connection of Local and Collective Excited States in Subsystem Time-Dependent Density Functional Theory

The theoretical understanding of photoinduced processes in multichromophoric systems requires, as an essential ingredient, the possibility of accurately describing their electronically excited states. However, the size of these systems often prohibits the usage of conventional electronic-structure methods, so that often multiscale approaches based on phenomenologically motivated models are employed. In contrast, subsystem time-dependent density functional theory (sTDDFT) allows for a subsystem-based ab initio description of multichromophoric systems and therefore allows for, in principle, an exact description of photoinduced processes. This Perspective aims to outline the theoretical foundations and commonly used practical realizations as well as to illustrate benefits of recent developments and open issues in the field of sTDDFT. Prospective, potential future applications and possible methodological developments are discussed.

Quantifying Fluctuations of Average Solvent Environments for Embedding Calculations

The viability and effectiveness of replacing an ensemble of embedded solute calculations by a single calculation using an average description of the solvent environment are evaluated. This work explores the fluctuations of the average description of the system obtained in two ways: from calculations on an ensemble of geometries and from an average environment constructed from the same ensemble. To this end, classical molecular dynamics simulations of a rigid acetone solute in SPCE water are performed in order to generate an ensemble of solvent environments. From this ensemble of solvent configurations, a number of different approaches for constructing an average solvent environment are employed. We perform a thorough numerical analysis of the fluctuations of the electrostatic potential experienced by the solute, as well as the resulting fluctuations of the solute's electronic density, measured through its dipole moment and fitted atomic point charges. At the same time, we inspect the accuracy of the methods used to construct average environments. Finally, the proposed method for generating the embedding potential from an average environment density is applied to estimate the solvatochromic shift of the first excitation of acetone. In order to account for quantum confinement effects, which may be important in certain cases, the fluctuations in the shift due to the interaction with the solvent are evaluated using frozen-density-embedding theory. Our results demonstrate that, for normally distributed environments, the constructed average environment is a reasonably good representation of a fluctuating molecular solvent environment. We then provide guidance for future comparisons between these theoretical treatments of solute/solvent systems to experimental measurements.

Solvation Free Energies in Subsystem Density Functional Theory

For many chemical processes the accurate description of solvent effects are vitally important. Here, we describe a hybrid ansatz for the explicit quantum mechanical description of solute-solvent and solvent-solvent interactions based on subsystem density functional theory and continuum solvation schemes. Since explicit solvent molecules may compromise the scalability of the model and transferability of the predicted solvent effect, we aim to retain both, for different solutes as well as for different solvents. The key for the transferability is the consistent subsystem decomposition of solute and solvent. The key for the scalability is the performance of subsystem DFT for increasing numbers of subsystems. We investigate molecular dynamics and stationary point sampling of solvent configurations and compare the resulting (Gibbs) free energies to experiment and theoretical methods. We can show that with our hybrid model reaction barriers and reaction energies are accurately reproduced compared to experimental data.

Benchmarking Free Energy Calculations in Liquid Aliphatic Ketone Solvents Using the 3D-RISM-KH Molecular Solvation Theory

The three-dimensional reference interaction site model of the molecular solvation theory with the Kovalenko–Hirata closure is used to calculate the free energy of solvation of organic solutes in liquid aliphatic ketones. The ketone solvent sites were modeled using a modified united-atom force field. The successful application of these solvation models in calculating ketone–water partition coefficients of a large number of solutes supports the validation and benchmarking reported here.

Frontiers in Multiscale Modeling of Photoreceptor Proteins

This perspective article highlights the challenges in the theoretical description of photoreceptor proteins using multiscale modeling, as discussed at the CECAM workshop in Tel Aviv, Israel. The participants have identified grand challenges and discussed the development of new tools to address them. Recent progress in understanding representative proteins such as green fluorescent protein, photoactive yellow protein, phytochrome, and rhodopsin is presented, along with methodological developments.

Tackling Solvent Effects by Coupling Electronic and Molecular Density Functional Theory

Solvation effects can have a tremendous influence on chemical reactions. However, precise quantum chemistry calculations are most often done either in vacuum neglecting the role of the solvent or using continuum solvent model ignoring its molecular nature. We propose a new method coupling a quantum description of the solute using electronic density functional theory with a classical grand-canonical treatment of the solvent using molecular density functional theory. Unlike a previous work, both densities are minimized self-consistently, accounting for mutual polarization of the molecular solvent and the solute. The electrostatic interaction is accounted using the full electron density of the solute rather than fitted point charges. The introduced methodology represents a good compromise between the two main strategies to tackle solvation effects in quantum calculation. It is computationally more effective than a direct quantum mechanics/molecular mechanics coupling, requiring the exploration of many solvent configurations. Compared to continuum methods, it retains the full molecular-level description of the solvent. We validate this new framework onto two usual benchmark systems: a water solvated in water and the symmetrical nucleophilic substitution between chloromethane and chloride in water. The prediction for the free energy profiles are not yet fully quantitative compared to experimental data, but the most important features are qualitatively recovered. The method provides a detailed molecular picture of the evolution of the solvent structure along the reaction pathway.

Evaluation of an Efficient 3D-RISM-SCF Implementation as a Tool for Computational Spectroscopy in Solution

The 3D-RISM-SCF solvent-model implementation of Gusarov et al. [ J. Phys. Chem. A 2006, 110, 6083-6090] in the Amsterdam density functional program has been improved and extended. In particular, an accurate yet efficient representation of the solute electrostatic potential is provided. The Coulomb-potential fitting of many DFT codes can be used advantageously in this context. The extra effort compared to a point-charge representation is small for a given SCF cycle and compensated by faster SCF convergence. This allows applications to large solutes, as demonstrated by evaluation of the solvatochromism of Reichardt's dye. In general, TDDFT applications to excitation energies in solution stand out and are highlighted. Applications to the ¹⁷O NMR chemical shifts of N-methylformamide in different solvents also demonstrate the distinct advantages of 3D-RISM over continuum solvents. Limitations are observed in this case for water solvent, where the solvent shielding is overestimated. This shortcoming applies also to the ¹⁷O gas-to-liquid shift of water, where we used localized molecular orbital analyses for a deeper understanding. For such cases of extremely strong solute-solvent interactions, couplings between solute and solvent orbitals induced by the magnetic perturbation are relevant. These clearly require a quantum-mechanical treatment of the most closely bound solvent molecules. Except for such extreme cases, 3D-RISM-SCF is very well suited to treat solvent effects on NMR parameters. More serious limitations pertain to the treatment of vibrational spectra, where the absence of the coupling between solute and solvent vibrational modes limits the accuracy of applications of 3D-RISM-SCF. The reported extended, efficient, and numerically accurate 3D-RISM-SCF implementation should provide a useful tool to study chemical and spectroscopic properties of molecules of appreciable size in a realistic solvent environment.

Embedding-theory-based simulations using experimental electron densities for the environment

For the first time, the use of experimentally derived molecular electron densities as ρ_B(r) in calculations based on frozen-density embedding theory (FDET) of environment-induced shifts of electronic excitations for chromophores in clusters is demonstrated. ρ_B(r) was derived from X-ray restrained molecular wavefunctions of glycylglycine to obtain environment densities for simulating electronic excitations in clusters.

The basic idea of frozen-density embedding theory (FDET) is the constrained minimization of the Hohenberg–Kohn density functional E^HK[ρ] performed using the auxiliary functional , where Ψ_A is the embedded N_A-electron wavefunction and ρ_B(r) is a non-negative function in real space integrating to a given number of electrons N_B. This choice of independent variables in the total energy functional makes it possible to treat the corresponding two components of the total density using different methods in multi-level simulations. The application of FDET using ρ_B(r) reconstructed from X-ray diffraction data for a molecular crystal is demonstrated for the first time. For eight hydrogen-bonded clusters involving a chromophore (represented as Ψ_A) and the glycylglycine molecule [represented as ρ_B(r)], FDET is used to derive excitation energies. It is shown that experimental densities are suitable for use as ρ_B(r) in FDET-based simulations.

The electrochemical interface in first-principles calculations

First-principles predictions play an important role in understanding chemistry at the electrochemical interface. Electronic structure calculations are straightforward for vacuum interfaces, but do not easily account for the interfacial fields and solvation that fundamentally change the nature of electrochemical reactions. Prevalent techniques for first-principles prediction of electrochemical processes range from expensive explicit solvation using ab initio molecular dynamics, through a hierarchy of continuum solvation techniques, to neglecting solvation and interfacial field effects entirely. Currently, no single approach reliably captures all relevant effects of the electrochemical double layer in first-principles calculations. This review systematically lays out the relation between all major approaches to first-principles electrochemistry, including the key approximations and their consequences for accuracy and computational cost. Focusing on ab initio methods for thermodynamic properties of aqueous interfaces, we first outline general considerations for modeling electrochemical interfaces, including solvent and electrolyte dynamics and electrification. We then present the specifics of various explicit and implicit models of the solvent and electrolyte. Finally, we discuss the compromise between computational efficiency and accuracy, and identify key outstanding challenges and future opportunities in the wide range of techniques for first-principles electrochemistry.

Enhanced solvation force extrapolation for speeding up molecular dynamics simulations of complex biochemical liquids

We propose an enhanced approach to the extrapolation of mean potential forces acting on atoms of solute macromolecules due to their interactions with solvent atoms in complex biochemical liquids. It improves and extends our previous extrapolation schemes by additionally including new techniques such as an exponential scaling transformation of coordinate space with weights complemented by an automatically adjusted balancing between the least square minimization of force deviations and the norm of expansion coefficients in the approximation. The expensive mean potential forces are treated in terms of the three-dimensional reference interaction site model with Kovalenko-Hirata closure molecular theory of solvation. During the dynamics, they are calculated only after every long (outer) time interval, i.e., quite rarely to reduce the computational costs. At much shorter (inner) time steps, these forces are extrapolated on the basis of their outer values. The equations of motion are then solved using a multiple time step integration within an optimized isokinetic Nosé-Hoover chain thermostat. The new approach is applied to molecular dynamics simulations of various systems consisting of solvated organic and biomolecules of different complexity. For example, we consider hydrated alanine dipeptide, asphaltene in toluene solvent, miniprotein 1L2Y, and protein G in aqueous solution. It is shown that in all these cases, the enhanced extrapolation provides much better accuracy of the solvation force approximation than the existing approaches. As a result, it can be used with much larger outer time steps, leading to a significant speedup of the simulations.

Solvent Composition Effects on the Structural Properties of the Aβ42 Monomer from the 3D-RISM-KH Molecular Theory of Solvation

Structural characterization of amyloid (A)β peptides implicated in Alzheimer's disease is a challenging problem due to their intrinsically disordered nature and their high propensity for aggregation. Only limited information is currently available from experiments on conformational properties and aggregation pathways of the peptides in cellular environments. In silico modeling complements experimental information, providing atomistic insight into structure and dynamics of different Aβ species. All-atom explicit solvent molecular dynamics (MD) simulations with a properly selected force field can deliver reliable structural and dynamic information. In the case of intrinsically disordered Aβ peptides, enhanced sampling simulations beyond the nanosecond time scale are required to obtain statistically meaningful results even for simple solvent conditions. To overcome the challenges of conformational sampling in crowded cellular environments, alternative approaches have to be used, including postprocessing of MD data. In this study, we employ the statistical-mechanical, three-dimensional reference interaction site model with the Kovalenko-Hirata closure integral equation molecular theory of solvation to describe solvent composition effects on the conformational equilibrium in a structural ensemble of the Aβ42 (covering residues 1-42) monomer based on a statistical reweighting technique. The methodology enables a computationally efficient prediction on how different factors in the cellular environment, such as solvent composition, nonpolar solvation, and macromolecular crowding, affect the structural properties of the monomer. Similarities have been identified between changes in the structural ensemble caused by nonpolar solvation and crowded environments modeled by ionic solution with large negative ions. In particular, both solvent conditions reduce the random coil content and enhance the helical structure content of the monomer. In contrast to the previous studies, which reported increased α-helical content of peptides in crowded environments, this work attributes these structural features to the difference in solvent exposure of hydrophilic residues of the monomer for different secondary structure elements, rather than to (entropic) excluded volume effects.

Multiscale methods framework: self-consistent coupling of molecular theory of solvation with quantum chemistry, molecular simulations, and dissipative particle dynamics

In this work, we will address different aspects of self-consistent field coupling of computational chemistry methods at different time and length scales in modern materials and biomolecular science.

Quantum Embedding Theories

In complex systems, it is often the case that the region of interest forms only one part of a much larger system. The idea of joining two different quantum simulations-a high level calculation on the active region of interest, and a low level calculation on its environment-formally defines a quantum embedding. While any combination of techniques constitutes an embedding, several rigorous formalisms have emerged that provide for exact feedback between the embedded system and its environment. These three formulations: density functional embedding, Green's function embedding, and density matrix embedding, respectively, use the single-particle density, single-particle Green's function, and single-particle density matrix as the quantum variables of interest. Many excellent reviews exist covering these methods individually. However, a unified presentation of the different formalisms is so far lacking. Indeed, the various languages commonly used, functional equations for density functional embedding, diagrammatics for Green's function embedding, and entanglement arguments for density matrix embedding, make the three formulations appear vastly different. In this Account, we introduce the basic equations of all three formulations in such a way as to highlight their many common intellectual strands. While we focus primarily on a straightforward theoretical perspective, we also give a brief overview of recent applications and possible future developments. The first section starts with density functional embedding, where we introduce the key embedding potential via the Euler equation. We then discuss recent work concerning the treatment of the nonadditive kinetic potential, before describing mean-field density functional embedding and wave function in density functional embedding. We finish the section with extensions to time-dependence and excited states. The second section is devoted to Green's function embedding. Here, we use the Dyson equation to obtain equations that parallel as closely as possible the density functional embedding equations, with the hybridization playing the role of the embedding potential. Embedding a high-level self-energy within a low-level self-energy is treated analogously to wave function in density functional embedding. The numerical computation of the high-level self-energy allows us to briefly introduce the bath representation in the quantum impurity problem. We then consider translationally invariant systems to bring in the important dynamical mean-field theory. Recent developments to incorporate screening and long-range interactions are discussed. The third section concerns density matrix embedding. Here, we first highlight some mathematical complications associated with a simple Euler equation derivation, arising from the open nature of fragments. This motivates the density matrix embedding theory, where we use the Schmidt decomposition to represent the entanglement through bath orbitals. The resulting impurity plus bath formulation resembles that of dynamical mean-field theory. We discuss the numerical self-consistency associated with using a high-level correlated wave function with a mean-field low-level treatment, and connect the resulting numerical inversion to that used in density functional embedding. We finish with perspectives on the future of all three methods.

A closure relation to molecular theory of solvation for macromolecules

We propose a closure to the integral equations of molecular theory of solvation, particularly suitable for polar and charged macromolecules in electrolyte solution. This includes such systems as oligomeric polyelectrolytes at a finite concentration in aqueous and various non-aqueous solutions, as well as drug-like compounds in solution. The new closure by Kobryn, Gusarov, and Kovalenko (KGK closure) imposes the mean spherical approximation (MSA) almost everywhere in the solvation shell but levels out the density distribution function to zero (with the continuity at joint boundaries) inside the repulsive core and in the spatial regions of strong density depletion emerging due to molecular associative interactions. Similarly to MSA, the KGK closure reduces the problem to a linear equation for the direct correlation function which is predefined analytically on most of the solvation shells and has to be determined numerically on a relatively small (three-dimensional) domain of strong depletion, typically within the repulsive core. The KGK closure leads to the solvation free energy in the form of the Gaussian fluctuation (GF) functional. We first test the performance of the KGK closure coupled to the reference interaction site model (RISM) integral equations on the examples of Lennard-Jones liquids, polar and nonpolar molecular solvents, including water, and aqueous solutions of simple ions. The solvation structure, solvation chemical potential, and compressibility obtained from RISM with the KGK closure favorably compare to the results of the hypernetted chain (HNC) and Kovalenko-Hirata (KH) closures, including their combination with the GF solvation free energy. We then use the KGK closure coupled to RISM to obtain the solvation structure and thermodynamics of oligomeric polyelectrolytes and drug-like compounds at a finite concentration in electrolyte solution, for which no convergence is obtained with other closures. For comparison, we calculate their solvation structure from molecular dynamics (MD) simulations. We further couple the 3D-RISM integral equation with the 3D-version of the KGK closure, and solve it for molecular mixtures as well as oligomeric polyelectrolytes and drug-like molecules in electrolyte solutions.

MTS-MD of Biomolecules Steered with 3D-RISM-KH Mean Solvation Forces Accelerated with Generalized Solvation Force Extrapolation

We developed a generalized solvation force extrapolation (GSFE) approach to speed up multiple time step molecular dynamics (MTS-MD) of biomolecules steered with mean solvation forces obtained from the 3D-RISM-KH molecular theory of solvation (three-dimensional reference interaction site model with the Kovalenko-Hirata closure). GSFE is based on a set of techniques including the non-Eckart-like transformation of coordinate space separately for each solute atom, extension of the force-coordinate pair basis set followed by selection of the best subset, balancing the normal equations by modified least-squares minimization of deviations, and incremental increase of outer time step in motion integration. Mean solvation forces acting on the biomolecule atoms in conformations at successive inner time steps are extrapolated using a relatively small number of best (closest) solute atomic coordinates and corresponding mean solvation forces obtained at previous outer time steps by converging the 3D-RISM-KH integral equations. The MTS-MD evolution steered with GSFE of 3D-RISM-KH mean solvation forces is efficiently stabilized with our optimized isokinetic Nosé-Hoover chain (OIN) thermostat. We validated the hybrid MTS-MD/OIN/GSFE/3D-RISM-KH integrator on solvated organic and biomolecules of different stiffness and complexity: asphaltene dimer in toluene solvent, hydrated alanine dipeptide, miniprotein 1L2Y, and protein G. The GSFE accuracy and the OIN efficiency allowed us to enlarge outer time steps up to huge values of 1-4 ps while accurately reproducing conformational properties. Quasidynamics steered with 3D-RISM-KH mean solvation forces achieves time scale compression of conformational changes coupled with solvent exchange, resulting in further significant acceleration of protein conformational sampling with respect to real time dynamics. Overall, this provided a 50- to 1000-fold effective speedup of conformational sampling for these systems, compared to conventional MD with explicit solvent. We have been able to fold the miniprotein from a fully denatured, extended state in about 60 ns of quasidynamics steered with 3D-RISM-KH mean solvation forces, compared to the average physical folding time of 4-9 μs observed in experiment.

SAMPL5: 3D-RISM partition coefficient calculations with partial molar volume corrections and solute conformational sampling

Implicit solvent methods for classical molecular modeling are frequently used to provide fast, physics-based hydration free energies of macromolecules. Less commonly considered is the transferability of these methods to other solvents. The Statistical Assessment of Modeling of Proteins and Ligands 5 (SAMPL5) distribution coefficient dataset and the accompanying explicit solvent partition coefficient reference calculations provide a direct test of solvent model transferability. Here we use the 3D reference interaction site model (3D-RISM) statistical-mechanical solvation theory, with a well tested water model and a new united atom cyclohexane model, to calculate partition coefficients for the SAMPL5 dataset. The cyclohexane model performed well in training and testing ([Formula: see text] for amino acid neutral side chain analogues) but only if a parameterized solvation free energy correction was used. In contrast, the same protocol, using single solute conformations, performed poorly on the SAMPL5 dataset, obtaining [Formula: see text] compared to the reference partition coefficients, likely due to the much larger solute sizes. Including solute conformational sampling through molecular dynamics coupled with 3D-RISM (MD/3D-RISM) improved agreement with the reference calculation to [Formula: see text]. Since our initial calculations only considered partition coefficients and not distribution coefficients, solute sampling provided little benefit comparing against experiment, where ionized and tautomer states are more important. Applying a simple [Formula: see text] correction improved agreement with experiment from [Formula: see text] to [Formula: see text], despite a small number of outliers. Better agreement is possible by accounting for tautomers and improving the ionization correction.

Revealing electronic open quantum systems with subsystem TDDFT

Open quantum systems (OQSs) are perhaps the most realistic systems one can approach through simulations. In recent years, describing OQSs with Density Functional Theory (DFT) has been a prominent avenue of research with most approaches based on a density matrix partitioning in conjunction with an ad-hoc description of system-bath interactions. We propose a different theoretical approach to OQSs based on partitioning of the electron density. Employing the machinery of subsystem DFT (and its time-dependent extension), we provide a novel way of isolating and analyzing the various terms contributing to the coupling between the system and the surrounding bath. To illustrate the theory, we provide numerical simulations on a toy system (a molecular dimer) and on a condensed phase system (solvated excimer). The simulations show that non-Markovian dynamics in the electronic system-bath interactions are important in chemical applications. For instance, we show that the superexchange mechanism of transport in donor-bridge-acceptor systems is a non-Markovian interaction between the donor-acceptor (OQS) with the bridge (bath) which is fully characterized by real-time subsystem time-dependent DFT.

3D-RISM-MP2 Approach to Hydration Structure of Pt(II) and Pd(II) Complexes: Unusual H-Ahead Mode vs Usual O-Ahead One

Solvation of transition metal complexes with water has been one of the fundamental topics in physical and coordination chemistry. In particular, Pt(II) complexes have recently attracted considerable interest for their relation to anticancer activity in cisplatin and its analogues, yet the interaction of the water molecule and the metal center has been obscured. The challenge from a theoretical perspective remains that both the microscopic solvation effect and the dynamical electron correlation (DEC) effect have to be treated simultaneously in a reasonable manner. In this work we derive the analytical gradient for the three-dimensional reference interaction site model Møller-Plesset second order (3D-RISM-MP2) free energy. On the basis of the three-regions 3D-RISM self-consistent field (SCF) method recently proposed by us, we apply a new layer of the Z-vector method to the CP-RISM equation as well as point-charge approximation to the derivatives with respect to the density matrix elements in the RISM-CPHF equation to remarkably reduce the computational cost. This method is applied to study the interaction of H2O with the d(8) square planar transition metal complexes in aqueous solution, trans-[Pt(II)Cl2(NH3)(glycine)] (1a), [Pt(II)(NH3)4](2+) (1b), [Pt(II)(CN)4](2-) (1c), and their Pd(II) analogues 2a, 2b, and 2c, respectively, to elucidate whether the usual H2O interaction through O atom (O-ahead mode) or unusual one through H atom (H-ahead mode) is stable in these complexes. We find that the interaction energy of the coordinating water and the transition metal complex changes little when switching from gas to aqueous phase, but the solvation free energy differs remarkably between the two interaction modes, thereby affecting the relative stability of the H-ahead and O-ahead modes. Particularly, in contrast to the expectation that the O-ahead mode is preferred due to the presence of positive charges in 1b, the H-ahead mode is also found to be more stable. The O-ahead mode is found to be more stable than the H-ahead one only in 2b. The energy decomposition analysis (EDA) at the 3D-RISM-MP2 level revealed that the O-ahead mode is stabilized by the electrostatic (ES) interaction, whereas the H-ahead one is mainly stabilized by the DEC effect. The ES interaction is also responsible for the difference between the Pd(II) and Pt(II) complexes; because the electrostatic potential is more negative along the z-axis in the Pt(II) complex than in the Pd(II) one, the O-ahead mode prefers the Pd(II) complexes, whereas the H-ahead becomes predominant in the Pt(II) complexes.

Frozen-density embedding theory with average solvent charge densities from explicit atomistic simulations

Besides molecular electron densities obtained within the Born–Oppenheimer approximation (ρ_B(r)) to represent the environment, the ensemble averaged density (〈ρ_B〉(r)) is also admissible in frozen-density embedding theory (FDET) [Wesolowski, Phys. Rev. A, 2008, 77, 11444].

3D RISM theory with fast reciprocal-space electrostatics

The calculation of electrostatic solute-solvent interactions in 3D RISM ("three-dimensional reference interaction site model") integral equation theory is recast in a form that allows for a computational treatment analogous to the "particle-mesh Ewald" formalism as used for molecular simulations. In addition, relations that connect 3D RISM correlation functions and interaction potentials with thermodynamic quantities such as the chemical potential and average solute-solvent interaction energy are reformulated in a way that calculations of expensive real-space electrostatic terms on the 3D grid are completely avoided. These methodical enhancements allow for both, a significant speedup particularly for large solute systems and a smoother convergence of predicted thermodynamic quantities with respect to box size, as illustrated for several benchmark systems.

Octanol-Water Partition Coefficient from 3D-RISM-KH Molecular Theory of Solvation with Partial Molar Volume Correction

The octanol-water partition coefficient is an important physical-chemical characteristic widely used to describe hydrophobic/hydrophilic properties of chemical compounds. The partition coefficient is related to the transfer free energy of a compound from water to octanol. Here, we introduce a new protocol for prediction of the partition coefficient based on the statistical-mechanical, 3D-RISM-KH molecular theory of solvation. It was shown recently that with the compound-solvent correlation functions obtained from the 3D-RISM-KH molecular theory of solvation, the free energy functional supplemented with the correction linearly related to the partial molar volume obtained from the Kirkwood-Buff/3D-RISM theory, also called the "universal correction" (UC), provides accurate prediction of the hydration free energy of small compounds, compared to explicit solvent molecular dynamics [ Palmer , D. S. ; J. Phys.: Condens. Matter 2010 , 22 , 492101 ]. Here we report that with the UC reparametrized accordingly this theory also provides an excellent agreement with the experimental data for the solvation free energy in nonpolar solvent (1-octanol) and so accurately predicts the octanol-water partition coefficient. The performance of the Kovalenko-Hirata (KH) and Gaussian fluctuation (GF) functionals of the solvation free energy, with and without UC, is tested on a large library of small compounds with diverse functional groups. The best agreement with the experimental data for octanol-water partition coefficients is obtained with the KH-UC solvation free energy functional.

Role of water in ligand binding to maltose-binding protein: insight from a new docking protocol based on the 3D-RISM-KH molecular theory of solvation

Maltose-binding protein is a periplasmic binding protein responsible for transport of maltooligosaccarides through the periplasmic space of Gram-negative bacteria, as a part of the ABC transport system. The molecular mechanisms of the initial ligand binding and induced large scale motion of the protein's domains still remain elusive. In this study, we use a new docking protocol that combines a recently proposed explicit water placement algorithm based on the 3D-RISM-KH molecular theory of solvation and conventional docking software (AutoDock Vina) to explain the mechanisms of maltotriose binding to the apo-open state of a maltose-binding protein. We confirm the predictions of previous NMR spectroscopic experiments on binding modes of the ligand. We provide the molecular details on the binding mode that was not previously observed in the X-ray experiments. We show that this mode, which is defined by the fine balance between the protein-ligand direct interactions and solvation effects, can trigger the protein's domain motion resulting in the holo-closed structure of the maltose-binding protein with the maltotriose ligand in excellent agreement with the experimental data. We also discuss the role of water in blocking unfavorable binding sites and water-mediated interactions contributing to the stability of observable binding modes of maltotriose.

Frozen density embedding with non-integer subsystems' particle numbers

We extend the frozen density embedding theory to non-integer subsystems' particles numbers. Different features of this formulation are discussed, with special concern for approximate embedding calculations. In particular, we highlight the relation between the non-integer particle-number partition scheme and the resulting embedding errors. Finally, we provide a discussion of the implications of the present theory for the derivative discontinuity issue and the calculation of chemical reactivity descriptors.

Multiple time step molecular dynamics in the optimized isokinetic ensemble steered with the molecular theory of solvation: accelerating with advanced extrapolation of effective solvation forces

We develop efficient handling of solvation forces in the multiscale method of multiple time step molecular dynamics (MTS-MD) of a biomolecule steered by the solvation free energy (effective solvation forces) obtained from the 3D-RISM-KH molecular theory of solvation (three-dimensional reference interaction site model complemented with the Kovalenko-Hirata closure approximation). To reduce the computational expenses, we calculate the effective solvation forces acting on the biomolecule by using advanced solvation force extrapolation (ASFE) at inner time steps while converging the 3D-RISM-KH integral equations only at large outer time steps. The idea of ASFE consists in developing a discrete non-Eckart rotational transformation of atomic coordinates that minimizes the distances between the atomic positions of the biomolecule at different time moments. The effective solvation forces for the biomolecule in a current conformation at an inner time step are then extrapolated in the transformed subspace of those at outer time steps by using a modified least square fit approach applied to a relatively small number of the best force-coordinate pairs. The latter are selected from an extended set collecting the effective solvation forces obtained from 3D-RISM-KH at outer time steps over a broad time interval. The MTS-MD integration with effective solvation forces obtained by converging 3D-RISM-KH at outer time steps and applying ASFE at inner time steps is stabilized by employing the optimized isokinetic Nosé-Hoover chain (OIN) ensemble. Compared to the previous extrapolation schemes used in combination with the Langevin thermostat, the ASFE approach substantially improves the accuracy of evaluation of effective solvation forces and in combination with the OIN thermostat enables a dramatic increase of outer time steps. We demonstrate on a fully flexible model of alanine dipeptide in aqueous solution that the MTS-MD/OIN/ASFE/3D-RISM-KH multiscale method of molecular dynamics steered by effective solvation forces allows huge outer time steps up to tens of picoseconds without affecting the equilibrium and conformational properties, and thus provides a 100- to 500-fold effective speedup in comparison to conventional MD with explicit solvent. With the statistical-mechanical 3D-RISM-KH account for effective solvation forces, the method provides efficient sampling of biomolecular processes with slow and/or rare solvation events such as conformational transitions of hydrated alanine dipeptide with the mean life times ranging from 30 ps up to 10 ns for "flip-flop" conformations, and is particularly beneficial for biomolecular systems with exchange and localization of solvent and ions, ligand binding, and molecular recognition.

Excited-state free energy surfaces in solution: time-dependent density functional theory∕reference interaction site model self-consistent field method

Constructing free energy surfaces for electronically excited states is a first step toward the understanding of photochemical processes in solution. For that purpose, the analytic free energy gradient is derived and implemented for the linear-response time-dependent density functional theory combined with the reference interaction site model self-consistent field method. The proposed method is applied to study (1) the fluorescence spectra of aqueous acetone and (2) the excited-state intramolecular proton transfer reaction of ortho-hydroxybenzaldehyde in an acetonitrile solution.

Simple electrolyte solutions: comparison of DRISM and molecular dynamics results for alkali halide solutions

Using the dielectrically consistent reference interaction site model (DRISM) of molecular solvation, we have calculated structural and thermodynamic information of alkali-halide salts in aqueous solution, as a function of salt concentration. The impact of varying the closure relation used with DRISM is investigated using the partial series expansion of order-n (PSE-n) family of closures, which includes the commonly used hypernetted-chain equation (HNC) and Kovalenko-Hirata closures. Results are compared to explicit molecular dynamics (MD) simulations, using the same force fields, and to experiment. The mean activity coefficients of ions predicted by DRISM agree well with experimental values at concentrations below 0.5 m, especially when using the HNC closure. As individual ion activities (and the corresponding solvation free energies) are not known from experiment, only DRISM and MD results are directly compared and found to have reasonably good agreement. The activity of water directly estimated from DRISM is nearly consistent with values derived from the DRISM ion activities and the Gibbs-Duhem equation, but the changes in the computed pressure as a function of salt concentration dominate these comparisons. Good agreement with experiment is obtained if these pressure changes are ignored. Radial distribution functions of NaCl solution at three concentrations were compared between DRISM and MD simulations. DRISM shows comparable water distribution around the cation, but water structures around the anion deviate from the MD results; this may also be related to the high pressure of the system. Despite some problems, DRISM-PSE-n is an effective tool for investigating thermodynamic properties of simple electrolytes.