Calculating solution redox free energies with ab initio quantum mechanical/molecular mechanical minimum free energy path method

Redox Properties of Flavin in BLUF and LOV Photoreceptor Proteins from Hybrid QM/MM Molecular Dynamics Simulation

Flavins play an important role in many oxidation and reduction processes in biological systems. For example, flavin adenine dinucleotide (FAD) and flavin mononucleotide (FMN) are common cofactors found in enzymatic proteins that use the special redox properties of these flavin molecules for their catalytic or photoactive functions. The redox potential of the flavin is strongly affected by its (protein) environment; however, the underlying molecular interactions of this effect are still unknown. Using hybrid quantum mechanics/molecular mechanics (QM/MM) simulation techniques, we have studied the redox properties of flavin in the gas phase, aqueous solution, and two different protein environments, in particular, a BLUF and a LOV photoreceptor domain. By mapping the changes in electrostatic potential and solvent structure, we gain insight into how specific polarization of the flavin by its environment tunes the reduction potential. We find also that accurate calculation of the reduction potentials of these systems by using the hybrid QM/MM approach is hampered by a too limited sampling of the counterion configurations and by artifacts at the QM/MM boundary. We make suggestions for how these issues can be overcome.

DFT-based QM/MM with Particle-Mesh Ewald for Direct, Long-Range Electrostatic Embedding

We present a DFT-based, QM/MM implementation with long-range electrostatic embedding achieved by direct real-space integration of the particle mesh Ewald (PME) computed electrostatic potential. The key transformation is the interpolation of the electrostatic potential from the PME grid to the DFT quadrature grid, from which integrals are easily evaluated utilizing standard DFT machinery. We provide benchmarks of the numerical accuracy with choice of grid size and real-space corrections, and demonstrate that good convergence is achieved while introducing nominal computational overhead. Furthermore, the approach requires only small modification to existing software packages, as is demonstrated with our implementation in the OpenMM and Psi4 software. After presenting convergence benchmarks, we evaluate the importance of long-range electrostatic embedding in three solute/solvent systems modeled with QM/MM. Water and BMIM/BF4 ionic liquid were considered as ``simple' and ``complex' solvents respectively, with water and p-phenylenediamine (PPD) solute molecules treated at QM level of theory. While electrostatic embedding with standard real-space truncation may introduce negligible error for simple systems such as water solute in water solvent, errors become more significant when QM/MM is applied to complex solvents such as ionic liquids. An extreme example is the electrostatic embedding energy for oxidized PPD in BMIM/BF4 for which real-space truncation produces severe error even at 2-3 nm cutoff distances. This latter example illustrates that utilization of QM/MM to compute redox potentials within concentrated electrolytes/ionic media requires carefully chosen long-range electrostatic embedding algorithms, with our presented algorithm providing a general and robust approach.

Artificial Photosynthesis: Is Computation Ready for the Challenge Ahead?

A tremendous effort is currently devoted to the generation of novel hybrid materials with enhanced electronic properties for the creation of artificial photosynthetic systems. This compelling and challenging problem is well-defined from an experimental point of view, as the design of such materials relies on combining organic materials or metals with biological systems like light harvesting and redox-active proteins. Such hybrid systems can be used, e.g., as bio-sensors, bio-fuel cells, biohybrid photoelectrochemical cells, and nanostructured photoelectronic devices. Despite these efforts, the main bottleneck is the formation of efficient interfaces between the biological and the organic/metal counterparts for efficient electron transfer (ET). It is within this aspect that computation can make the difference and improve the current understanding of the mechanisms underneath the interface formation and the charge transfer efficiency. Yet, the systems considered (i.e., light harvesting protein, self-assembly monolayer and surface assembly) are more and more complex, reaching (and often passing) the limit of current computation power. In this review, recent developments in computational methods for studying complex interfaces for artificial photosynthesis will be provided and selected cases discussed, to assess the inherent ability of computation to leave a mark in this field of research.

Redox Potentials of Disulfide Bonds in LOXL2 Studied by Nonequilibrium Alchemical Simulation

Lysyl oxidase-like 2 (LOXL2) is a metalloenzyme that catalyzes the oxidative deamination ε-amino group of lysine. It is found that LOXL2 is a promotor for the metastasis and invasion of cancer cells. Disulfide bonds are important components in LOXL2, and they play a stabilizing role for protein structure or a functional role for regulating protein bioactivity. The redox potential of disulfide bond is one important property to determine the functional role of disulfide bond. In this study, we have calculated the reduction potential of all the disulfide bonds in LOXL2 by non-equilibrium alchemical simulations. Our results show that seven of seventeen disulfide bonds have high redox potentials between -182 and -298 mV and could have a functional role, viz., Cys573-Cys625, Cys579-Cys695, Cys657-Cys673, and Cys663-Cys685 in the catalytic domain, Cys351-Cys414, Cys464-Cys530, and Cys477-Cys543 in the scavenger receptor cysteine-rich (SRCR) domains. The disulfide bond of Cys351-Cys414 is predicted to play an allosteric function role, which could affect the metastasis and invasion of cancer cells. Other functional bonds have a catalytic role related to enzyme activity. The rest of disulfide bonds are predicted to play a structural role. Our study provides an important insight for the classification of disulfide bonds in LOXL2 and can be utilized for the drug design that targets the cysteine residues in LOXL2.

Microscopic Picture of the Solvent Reorganization During Electron Transfer to Flavin in Water

The redox potential of molecular species is largely modulated by its molecular environment so that a change of the environment will lead to a different redox potential. However, a detailed molecular picture of reorganization of the environment upon reduction is still unclear. To unravel the details of the solvent reorganization during electron transfer, we have performed density functional theory-based molecular dynamics (DFT-MD) and hybrid quantum mechanics/molecular mechanics (QM/MM) simulations of the reduction of lumiflavin. Previously, we have calculated the reduction free energy curves of the redox half reactions of lumiflavin in water as a function of the instantaneous gap energy (ΔE) ( J. Chem. Theory Comput. 2013 , 9 , 3889 - 3899 ). In this work, we focus on finding the changes in the solvent environment that correlate with this ΔE reaction coordinate. Comparing the QM/MM simulations, in which the solvent is modeled with an empirical force field, with the (full) DFT-MD simulations, we find that the response through electronic polarization plays a significant role in the latter case. Also a small charge transfer between flavin and solvent is observed in the full DFT treatment. As a result, we find only in the case of the QM/MM model a strong correlation between ΔE and the (pairwise computed) electrostatic potential (ESP) at the flavin due to the solvent. By analyzing the contribution of the ESP at the flavin per solvent molecule, we cannot only distinguish between the different modes of hydration by solvent molecules that coordinate at the hydrophilic and hydrophobic sides of the flavin molecule but also quantify their contribution to the reorganization free energy by measuring the ESP fluctuations per solvent molecule.

Accurate Quantum Mechanical/Molecular Mechanical Calculations of Reduction Potentials in Azurin Variants

Understanding the regulation mechanism and molecular determinants of the reduction potential of metalloprotein is a major challenge. An ab initio quantum mechanical/molecular mechanical (QM/MM) method combining the minimum free energy path (MFEP) and fractional number of electron (FNE) approaches has been developed in our group to simulate the redox processes of large systems. The FNE scheme provides an efficient unique description for the redox process, while the MFEP method provides improved conformational sampling on complex environments such as protein in the QM/MM calculations. The reduction potentials of wild-type and seven mutants of azurin, a type 1 copper metalloprotein, were simulated with the QM/MM-MFEP+FNE approach in this paper. A range of 350 mV for the variations of the reduction potentials of these azurin proteins was reproduced faithfully with relative errors around 20 mV. The correlation between structural interactions and reduction potentials observed in simulations provides in-depth insight into the regulation of reduction potentials, which potentially can also be very useful to the engineering of metalloprotein-based electrocatalysts in artificial photosynthesis. The excellent accuracy and efficiency of the QM/MM-MFEP+FNE approach demonstrate the potential for simulations of many electron transfer processes in condensed phases and biochemical systems.

Theoretical estimation of redox potential of biological quinone cofactors

Redox potentials are essential to understand biological cofactor reactivity and to predict their behavior in biological media. Experimental determination of redox potential in biological system is often difficult due to complexity of biological media but computational approaches can be used to estimate them. Nevertheless, the quality of the computational methodology remains a key issue to validate the results. Instead of looking to the best absolute results, we present here the calibration of theoretical redox potential for quinone derivatives in water coupling QM + MM or QM/MM scheme. Our approach allows using low computational cost theoretical level, ideal for long simulations in biological systems, and determination of the uncertainties linked to the calculations. © 2017 Wiley Periodicals, Inc.

Methods for Efficiently and Accurately Computing Quantum Mechanical Free Energies for Enzyme Catalysis

Enzyme activity is inherently linked to free energies of transition states, ligand binding, protonation/deprotonation, etc.; these free energies, and thus enzyme function, can be affected by residue mutations, allosterically induced conformational changes, and much more. Therefore, being able to predict free energies associated with enzymatic processes is critical to understanding and predicting their function. Free energy simulation (FES) has historically been a computational challenge as it requires both the accurate description of inter- and intramolecular interactions and adequate sampling of all relevant conformational degrees of freedom. The hybrid quantum mechanical molecular mechanical (QM/MM) framework is the current tool of choice when accurate computations of macromolecular systems are essential. Unfortunately, robust and efficient approaches that employ the high levels of computational theory needed to accurately describe many reactive processes (ie, ab initio, DFT), while also including explicit solvation effects and accounting for extensive conformational sampling are essentially nonexistent. In this chapter, we will give a brief overview of two recently developed methods that mitigate several major challenges associated with QM/MM FES: the QM non-Boltzmann Bennett's acceptance ratio method and the QM nonequilibrium work method. We will also describe usage of these methods to calculate free energies associated with (1) relative properties and (2) along reaction paths, using simple test cases with relevance to enzymes examples.

Quantum Mechanics/Molecular Mechanics Method Combined with Hybrid All-Atom and Coarse-Grained Model: Theory and Application on Redox Potential Calculations

We developed a new multiresolution method that spans three levels of resolution with quantum mechanical, atomistic molecular mechanical, and coarse-grained models. The resolution-adapted all-atom and coarse-grained water model, in which an all-atom structural description of the entire system is maintained during the simulations, is combined with the ab initio quantum mechanics and molecular mechanics method. We apply this model to calculate the redox potentials of the aqueous ruthenium and iron complexes by using the fractional number of electrons approach and thermodynamic integration simulations. The redox potentials are recovered in excellent accordance with the experimental data. The speed-up of the hybrid all-atom and coarse-grained water model renders it computationally more attractive. The accuracy depends on the hybrid all-atom and coarse-grained water model used in the combined quantum mechanical and molecular mechanical method. We have used another multiresolution model, in which an atomic-level layer of water molecules around redox center is solvated in supramolecular coarse-grained waters for the redox potential calculations. Compared with the experimental data, this alternative multilayer model leads to less accurate results when used with the coarse-grained polarizable MARTINI water or big multipole water model for the coarse-grained layer.

Practically Efficient QM/MM Alchemical Free Energy Simulations: The Orthogonal Space Random Walk Strategy

The difference between free energy changes occurring at two chemical states can be rigorously estimated via alchemical free energy (AFE) simulations. Traditionally, most AFE simulations are carried out under the classical energy potential treatment; then, accuracy and applicability of AFE simulations are limited. In the present work, we integrate a recent second-order generalized ensemble strategy, the orthogonal space random walk (OSRW) method, into the combined quantum mechanical/molecular mechanical (QM/MM) potential based AFE simulation scheme. Thereby, within a commonly affordable simulation length, accurate QM/MM alchemical free energy simulations can be achieved. As revealed by the model study on the equilibrium of a tautomerization process of hydrated 3-hydroxypyrazole and by the model calculations of the redox potentials of two flavin derivatives, lumichrome (LC) and riboflavin (RF) in aqueous solution, the present OSRW-based scheme could be a viable path toward the realization of practically efficient QM/MM AFE simulations.

Progress and challenges in simulating and understanding electron transfer in proteins

This Review presents an overview of the most common numerical simulation approaches for the investigation of electron transfer (ET) in proteins. We try to highlight the merits of the different approaches but also the current limitations and challenges. The article is organized into three sections. Section 2 deals with direct simulation algorithms of charge migration in proteins. Section 3 summarizes the methods for testing the applicability of the Marcus theory for ET in proteins and for evaluating key thermodynamic quantities entering the reaction rates (reorganization energies and driving force). Recent studies interrogating the validity of the theory due to the presence of non-ergodic effects or of non-linear responses are also described. Section 4 focuses on the tunneling aspects of electron transfer. How can the electronic coupling between charge transfer states be evaluated by quantum chemistry approaches and rationalized? What interesting physics regarding the impact of protein dynamics on tunneling can be addressed? We will illustrate the different sections with examples taken from the literature to show what types of system are currently manageable with current methodologies. We also take care to recall what has been learned on the biophysics of ET within proteins thanks to the advent of atomistic simulations.

Complete mineralization of perfluorooctanoic acid (PFOA) by γ-irradiation in aqueous solution

Decomposition of perfluorooctanoic acid (C₇F₁₅COOH, PFOA) has been gaining increasing interests because it is a ubiquitous environmental contaminant and resistant to the most conventional treatment processes. In this work, the rapid and complete mineralization of PFOA and simultaneous defluorination were achieved by γ-ray irradiation with a ⁶⁰Co source. The degradation rate of PFOA by γ-ray irradiation would be high, and a pseudo-first-order kinetic rate constant of 0.67 h⁻¹ could be achieved in the N₂ satured condition at pH 13.0. The experimental results and quantum chemical calculation confirmed that two radicals, i.e., hydroxyl radical (·OH) and aqueous electrons (e_aq⁻), were responsible for the degradation of PFOA, while only either e_aq⁻ or ·OH might not be able to accomplish complete mineralization of PFOA. The synergistic effects of ·OH and e_aq⁻ involved in the cleavage of C-C and C-F bonds, and therefore complete mineralization of PFOA were achieved. The intermediate products were identified and the degradation pathway was also proposed. The results of this study may offer a useful, high-efficient approach for complete mineralizing fluorochemicals and other persistent pollutants.

Computational electrochemistry: prediction of liquid-phase reduction potentials

This article reviews recent developments and applications in the area of computational electrochemistry. Our focus is on predicting the reduction potentials of electron transfer and other electrochemical reactions and half-reactions in both aqueous and nonaqueous solutions. Topics covered include various computational protocols that combine quantum mechanical electronic structure methods (such as density functional theory) with implicit-solvent models, explicit-solvent protocols that employ Monte Carlo or molecular dynamics simulations (for example, Car-Parrinello molecular dynamics using the grand canonical ensemble formalism), and the Marcus theory of electronic charge transfer. We also review computational approaches based on empirical relationships between molecular and electronic structure and electron transfer reactivity. The scope of the implicit-solvent protocols is emphasized, and the present status of the theory and future directions are outlined.

Importance of polarization effect in the study of metalloproteins: application of polarized protein specific charge scheme in predicting the reduction potential of azurin

Molecular dynamics (MD) simulation is commonly used in the study of protein dynamics, and in recent years, the extension of MD simulation to the study of metalloproteins is gaining much interest. Choice of force field is crucial in MD studies, and the inclusion of metal centers complicates the process of accurately describing the electrostatic environment that surrounds the redox centre. Herein, we would like to explore the importance of including electrostatic contribution from both protein and solvent in the study of metalloproteins. MD simulations with the implementation of thermodynamic integration will be conducted to model the reduction process of azurin from Pseudomonas aeruginosa. Three charge schemes will be used to derive the partial charges of azurin. These charge schemes differ in terms of the amount of immediate environment, respective to copper, considered during charge fitting, which ranges from the inclusion of copper and residues in the first coordination sphere during density functional theory charge fitting to the comprehensive inclusion of protein and solvent effect surrounding the metal centre using polarized protein-specific charge scheme. From the simulations conducted, the relative reduction potential of the mutated azurins respective to that of wild-type azurin (ΔEcal) were calculated and compared with experimental values. The ΔEcal approached experimental value with increasing consideration of environmental effect hence substantiating the importance of polarization effect in the study of metalloproteins. This study also attests the practicality of polarized protein-specific charge as a computational tool capable of incorporating both protein environment and solvent effect into MD simulations.

What can density functional theory tell us about artificial catalytic water splitting?

Water splitting by artificial catalysts is a critical process in the production of hydrogen gas as an alternative fuel. In this paper, we examine the essential role of theoretical calculations, with particular focus on density functional theory (DFT), in understanding the water-splitting reaction on these catalysts. First, we present an overview of DFT thermochemical calculations on water-splitting catalysts, addressing how these calculations are adapted to condensed phases and room temperature. We show how DFT-derived chemical descriptors of reactivity can be surprisingly good estimators for reactive trends in water-splitting catalysts. Using this concept, we recover trends for bulk catalysts using simple model complexes for at least the first-row transition-metal oxides. Then, using the CoPi cobalt oxide catalyst as a case study, we examine the usefulness of simulation for predicting the kinetics of water splitting. We demonstrate that the appropriate treatment of solvent effects is critical for computing accurate redox potentials with DFT, which, in turn, determine the rate-limiting steps and electrochemical overpotentials. Finally, we examine the ability of DFT to predict mechanism, using ruthenium complexes as a focal point for discussion. Our discussion is intended to provide an overview of the current strengths and weaknesses of the state-of-the-art DFT methodologies for condensed-phase molecular simulation involving transition metals and also to guide future experiments and computations toward the understanding and development of novel water-splitting catalysts.

QM/MM investigations of organic chemistry oriented questions

About 35 years after its first suggestion, QM/MM became the standard theoretical approach to investigate enzymatic structures and processes. The success is due to the ability of QM/MM to provide an accurate atomistic picture of enzymes and related processes. This picture can even be turned into a movie if nuclei-dynamics is taken into account to describe enzymatic processes. In the field of organic chemistry, QM/MM methods are used to a much lesser extent although almost all relevant processes happen in condensed matter or are influenced by complicated interactions between substrate and catalyst. There is less importance for theoretical organic chemistry since the influence of nonpolar solvents is rather weak and the effect of polar solvents can often be accurately described by continuum approaches. Catalytic processes (homogeneous and heterogeneous) can often be reduced to truncated model systems, which are so small that pure quantum-mechanical approaches can be employed. However, since QM/MM becomes more and more efficient due to the success in software and hardware developments, it is more and more used in theoretical organic chemistry to study effects which result from the molecular nature of the environment. It is shown by many examples discussed in this review that the influence can be tremendous, even for nonpolar reactions. The importance of environmental effects in theoretical spectroscopy was already known. Due to its benefits, QM/MM can be expected to experience ongoing growth for the next decade.In the present chapter we give an overview of QM/MM developments and their importance in theoretical organic chemistry, and review applications which give impressions of the possibilities and the importance of the relevant effects. Since there is already a bunch of excellent reviews dealing with QM/MM, we will discuss fundamental ingredients and developments of QM/MM very briefly with a focus on very recent progress. For the applications we follow a similar strategy.

Noncovalent Interaction Analysis in Fluctuating Environments

Noncovalent interactions play a central role in many chemical and biological systems. In a previous study, Johnson et al developed a NonCovalent Interaction (NCI) index to characterize and visualize different types of weak interactions. To apply the NCI analysis to fluctuating environments as in solution phase, we here develop a new Averaged NonCovalent Interaction (i.e., aNCI) index along with a fluctuation index to characterize magnitude of interactions and fluctuations. We applied aNCI for various systems including solute-solvent and ligand-protein noncovalent interactions. For water and benzene molecules in aqueous solution, solvation structures and the specific hydrogen bond patterns were visualized clearly. For the Cl^-+CH₃Cl S_N2 reaction in aqueous solution, charge reorganization influences over solvation structure along S_N2 reaction were revealed. For ligand-protein systems, aNCI can recover several key fluctuating hydrogen bond patterns that have potential applications for drug design. Therefore, aNCI, as a complementary approach to the original NCI method, can extract and visualize noncovalent interactions from thermal noise in fluctuating environments.

Fragment-based Quantum Mechanical/Molecular Mechanical Simulations of Thermodynamic and Kinetic Process of the Ru2+-Ru3+ Self-Exchange Electron Transfer

A fragment-based fractional number of electron (FNE) approach, is developed to study entire electron transfer (ET) processes from the electron donor region to the acceptor region in condensed phase. Both regions are described by the density-fragment interaction (DFI) method while FNE as an efficient ET order parameter is applied to simulate the electron transfer process. In association with the QM/MM energy expression, the DFI-FNE method is demonstrated to describe ET processes robustly with the Ru2+-Ru3+ self-exchange ET as a proof-of-concept example. This method allows for systematic calculations of redox free energies, reorganization energies, and electronic couplings, and the absolute ET rate constants within the Marcus regime.

Theoretical study of catalytic mechanism for single-site water oxidation process

Water oxidation is a linchpin in solar fuels formation, and catalysis by single-site ruthenium complexes has generated significant interest in this area. Combining several theoretical tools, we have studied the entire catalytic cycle of water oxidation for a single-site catalyst starting with [Ru(II)(tpy)(bpm)(OH(2))](2+) (i.e., [Ru(II)-OH(2)](2+); tpy is 2,2':6',2''-terpyridine and bpm is 2,2'-bypyrimidine) as a representative example of a new class of single-site catalysts. The redox potentials and pK(a) calculations for the first two proton-coupled electron transfers (PCETs) from [Ru(II)-OH(2)](2+) to [Ru(IV) = O](2+) and the following electron-transfer process to [Ru(V) = O](3+) suggest that these processes can proceed readily in acidic or weakly basic conditions. The subsequent water splitting process involves two water molecules, [Ru(V) = O](3+) to generate [Ru(III)-OOH](2+), and H(3)O(+) with a low activation barrier (~10 kcal/mol). After the key O-O bond forming step in the single-site Ru catalysis, another PECT process oxidizes [Ru(III)-OOH](2+) to [Ru(IV)-OO](2+) when the pH is lower than 3.7. Two possible forms of [Ru(IV)-OO](2+), open and closed, can exist and interconvert with a low activation barrier (< 7 kcal/mol) due to strong spin-orbital coupling effects. In Pathway 1 at pH = 1.0, oxygen release is rate-limiting with an activation barrier ~12 kcal/mol while the electron-transfer step from [Ru(IV)-OO](2+) to [Ru(V)-OO](3+) becomes rate-determining at pH = 0 (Pathway 2) with Ce(IV) as oxidant. The results of these theoretical studies with atomistic details have revealed subtle details of reaction mechanisms at several stages during the catalytic cycle. This understanding is helpful in the design of new catalysts for water oxidation.

Catalytic mechanism of 4-oxalocrotonate tautomerase: significances of protein-protein interactions on proton transfer pathways

4-Oxalocrotonate tautomerase (4-OT), a member of tautomerase superfamily, is an essential enzyme in the degradative metabolism pathway occurring in the Krebs cycle. The proton transfer process catalyzed by 4-OT has been explored previously using both experimental and theoretical methods; however, the elaborate catalytic mechanism of 4-OT still remains unsettled. By combining classical molecular mechanics with quantum mechanics, our results demonstrate that the native hexametric 4-OT enzyme, including six protein monomers, must be employed to simulate the proton transfer process in 4-OT due to protein-protein steric and electrostatic interactions. As a consequence, only three out of the six active sites in the 4-OT hexamer are observed to be occupied by three 2-oxo-4-hexenedioates (2o4hex), i.e., half-of-the-sites occupation. This agrees with experimental observations on negative cooperative effect between two adjacent substrates. Two sequential proton transfers occur: one proton from the C3 position of 2o4hex is initially transferred to the nitrogen atom of the general base, Pro1. Subsequently, the same proton is shuttled back to the position C5 of 2o4hex to complete the proton transfer process in 4-OT. During the catalytic reaction, conformational changes (i.e., 1-carboxyl group rotation) of 2o4hex may occur in the 4-OT dimer model but cannot proceed in the hexametric structure. We further explained that the docking process of 2o4hex can influence the specific reactant conformations and an alternative substrate (2-hydroxymuconate) may serve as reactant under a different reaction mechanism than 2o4hex.

Conical intersections in solution: formulation, algorithm, and implementation with combined quantum mechanics/molecular mechanics method

The significance of conical intersections in photophysics, photochemistry, and photodissociation of polyatomic molecules in gas phase has been demonstrated by numerous experimental and theoretical studies. Optimization of conical intersections of small- and medium-size molecules in gas phase has currently become a routine optimization process, as it has been implemented in many electronic structure packages. However, optimization of conical intersections of small- and medium-size molecules in solution or macromolecules remains inefficient, even poorly defined, due to large number of degrees of freedom and costly evaluations of gradient difference and nonadiabatic coupling vectors. In this work, based on the sequential quantum mechanics and molecular mechanics (QM/MM) and QM/MM-minimum free energy path methods, we have designed two conical intersection optimization methods for small- and medium-size molecules in solution or macromolecules. The first one is sequential QM conical intersection optimization and MM minimization for potential energy surfaces; the second one is sequential QM conical intersection optimization and MM sampling for potential of mean force surfaces, i.e., free energy surfaces. In such methods, the region where electronic structures change remarkably is placed into the QM subsystem, while the rest of the system is placed into the MM subsystem; thus, dimensionalities of gradient difference and nonadiabatic coupling vectors are decreased due to the relatively small QM subsystem. Furthermore, in comparison with the concurrent optimization scheme, sequential QM conical intersection optimization and MM minimization or sampling reduce the number of evaluations of gradient difference and nonadiabatic coupling vectors because these vectors need to be calculated only when the QM subsystem moves, independent of the MM minimization or sampling. Taken together, costly evaluations of gradient difference and nonadiabatic coupling vectors in solution or macromolecules can be reduced significantly. Test optimizations of conical intersections of cyclopropanone and acetaldehyde in aqueous solution have been carried out successfully.

Redox potentials and pKa for benzoquinone from density functional theory based molecular dynamics

The density functional theory based molecular dynamics (DFTMD) method for the computation of redox free energies presented in previous publications and the more recent modification for computation of acidity constants are reviewed. The method uses a half reaction scheme based on reversible insertion/removal of electrons and protons. The proton insertion is assisted by restraining potentials acting as chaperones. The procedure for relating the calculated deprotonation free energies to Brønsted acidities (pK(a)) and the oxidation free energies to electrode potentials with respect to the normal hydrogen electrode is discussed in some detail. The method is validated in an application to the reduction of aqueous 1,4-benzoquinone. The conversion of hydroquinone to quinone can take place via a number of alternative pathways consisting of combinations of acid dissociations, oxidations, or dehydrogenations. The free energy changes of all elementary steps (ten in total) are computed. The accuracy of the calculations is assessed by comparing the energies of different pathways for the same reaction (Hess's law) and by comparison to experiment. This two-sided test enables us to separate the errors related with the restrictions on length and time scales accessible to DFTMD from the errors introduced by the DFT approximation. It is found that the DFT approximation is the main source of error for oxidation free energies.