Nuclear Magnetic Shielding Constants from Quantum Mechanical/Molecular Mechanical Calculations Using Polarizable Embedding: Role of the Embedding Potential

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.

Modeling solvent effects and convergence of 31P-NMR shielding calculations with COBRAMM

Solvent effects on 31P-NMR parameters for triphenylphosphine oxide and triphenylphosphine in chloroform have been extensively investigated by testing different solvation models. The solvent is described implicitly, mixed implicitly/explicitly, and using full explicit models. Polarizable continuum model (PCM), molecular dynamic (MD) simulations, and hybrid quantum mechanics/molecular mechanics (QM/MM) calculations are used to disclose the effects of solute/solvent interactions and, more generally, the role of the embedding in NMR simulations. The results show the beneficial effect of carrying out QM/MM optimizations on top of geometries directly extracted from classical MD simulations, used to ensure representative conformational sampling. The nuclear shielding convergence has been tested against a different number of snapshots and with the inclusion of solvent shells into the QM region. An automated MD//QM/MM//GIAO protocol, implemented in the COBRAMM package, is here proposed and tested on trimethyl phosphite showing that our approach boosts the convergence of nuclear shielding satisfactorily. The present work aims to be a stepping-stone to assess proper QM/MM computational strategies in simulating chemical shifts in non-homogeneous systems like supramolecular and biological systems.

Polarizable Embedding Potentials through Molecular Fractionation with Conjugate Caps Including Hydrogen Bonds

Polarizable embedding (PE) refers to classical embedding approaches, such as those used in quantum mechanics/molecular mechanics (QM/MM), that allow mutual polarization between the quantum and classical regions. The quality of the embedding potential is critical to provide accurate results, e.g., for spectroscopic properties and dynamical processes. High-quality embedding-potential parameters can be obtained by dividing the classical region into smaller fragments and deriving the parameters from ab initio calculations on the fragments. For solvents and other systems composed of small molecules, the fragments can be individual molecules, while a more complicated fragmentation procedure is needed for larger molecules, such as proteins and nucleic acids. One such fragmentation strategy is the molecular fractionation with conjugate caps (MFCC) approach. As is widely known, hydrogen bonds play a key role in many biomolecular systems, e.g., in proteins, where they are responsible for the secondary structure. In this work, we assess the effects of including hydrogen-bond fragmentation in the MFCC procedure [MFCC(HB)] for deriving the embedding-potential parameters. The MFCC(HB) extension is evaluated on several molecular systems, ranging from small model systems to proteins, directly in terms of molecular electrostatic potentials and embedding potentials and indirectly in terms of selected properties of chromophores embedded in water and complex protein environments.

Polarizable Embedding without Artificial Boundary Polarization

We present a fully self-consistent polarizable embedding (PE) model that does not suffer from unphysical boundary polarization. This is achieved through the use of the minimum-image convention (MIC) in the induced electrostatics. It is a simple yet effective approach that includes a more physically accurate description of the polarization throughout the molecular system. Using PE with MIC (PE-MIC), we shed new light on the limitations of commonly employed cutoff models, such as the droplet model, when used in PE calculations. Specifically, we investigate the effects of the unphysical polarization at the outer boundary by comparing induced dipoles and the associated electrostatic potentials, as well as some optical properties of solute-solvent and biomolecular systems. We show that the magnitude of the inaccuracies caused by the unphysical polarization depends on multiple parameters: the nature of the quantum subsystem and of the environment, the cutoff model and distance, and the calculated property.

Nuclear Magnetic Shielding Constants with the Polarizable Density Embedding Model

We extend the polarizable density embedding (PDE) model to support the calculation of nuclear magnetic resonance (NMR) shielding constants using gauge-including atomic orbitals (GIAOs) within a density functional theory (DFT) framework. The PDE model divides the total system into fragments, describing some by quantum mechanics (QM) and the others through an embedding model. The PDE model uses anisotropic polarizabilities, inter-fragment two-electron Coulomb integrals, and a non-local repulsion operator to emulate the QM effects. The terms involving Coulomb integrals are straightforwardly extended with GIAOs. In contrast, we consider two approaches to handle the gauge dependency of the non-local operator, employing either simple symmetrization or a gauge transformation. We find the latter approach to be most stable with respect to increasing the basis set size of the QM region. We examine the accuracy of the PDE model for calculating NMR shielding constants on several solutes in a water solution. The performance is compared with the classical polarizable embedding (PE) model in addition to supermolecular reference calculations. Based on these systems, we address the basis set convergence characteristics and the QM region size requirements. Furthermore, we investigate the performance of the PDE model for a system with significant electron spill-out. In many cases, we find that the PDE model outperforms the PE model, especially regarding the accuracy of nuclear shielding constants when using small QM region sizes and in systems with significant electron spill-out.

Efficiently Computing NMR 1H and 13C Chemical Shifts of Saccharides in Aqueous Environment

Determining the structure of saccharides in their native environment is crucial to understanding their function and more accurately targeting their utilization. Nuclear magnetic resonance observables such as the nuclear Overhauser effect or spin-spin coupling constants are routinely utilized to study saccharides in their native water environment. However, while highly sensitive to the local environment, chemical shifts are mostly overlooked, despite being commonly measured for compounds identification. Although chemical shifts carry considerable structural information, their direct association with structure is notoriously difficult. This is mostly due to the similarity in the chemical nature of most saccharides causing similar physicochemical environments close to sugar C and H atoms, resulting in comparable chemical shifts. The rise of computational power allows one to compute reliable chemical shifts and use them to determine atomistic details of these sugars in solution. However, any prediction is severely limited by the computational protocol used and its accuracy. In this work, we studied a set of 31 saccharides on which we evaluated various computational protocols to calculate the total number of 375 ¹H and 327 ¹³C chemical shifts of sugars in an aqueous environment. Our study proposes two cost-effective protocols for simulating ¹H and ¹³C chemical shifts that we recommend for further use. These protocols can help with the interpretation of experimental spectra, but we also show that they are also capable of structure prediction independently. This is possible because of the low mean absolute deviations of calculated shifts from the experiment (0.06 ppm for ¹H and 1.09 ppm for ¹³C). We explore different solvation methods, basis sets, and optimization schemes to reach such accuracy. A correct sampling of the conformation phase space of flexible sugar molecules is also key to obtaining accurately converged theoretical chemical shifts. The linear regression method was applied to convert the calculated isotropic nuclear magnetic shielding constants to simulated chemical shifts comparable with the experiment. The achieved level of accuracy can help in utilizing chemical shifts for elucidating the 3D atomistic structure of saccharides in aqueous solutions. All linear regression parameters obtained on our extensive set of sugars for all the tested protocols can be reutilized in future works.

Polarizable continuum models provide an effective electrostatic embedding model for fragment-based chemical shift prediction in challenging systems

Ab initio nuclear magnetic resonance chemical shift prediction provides an important tool for interpreting and assigning experimental spectra, but it becomes computationally prohibitive in large systems. The computational costs can be reduced considerably by fragmentation of the large system into a series of contributions from many smaller subsystems. However, the presence of charged functional groups and the need to partition the system across covalent bonds create complications in biomolecules that typically require the use of large fragments and careful descriptions of the electrostatic environment. The present work shows how a model that combines chemical shielding contributions from non-overlapping monomer and dimer fragments embedded in a polarizable continuum model provides a simple, easy-to-implement, and computationally inexpensive approach for predicting chemical shifts in complex systems. The model's performance proves rather insensitive to the continuum dielectric constant, making the selection of the optimal embedding dielectric less critical. The PCM-embedded fragment model is demonstrated to perform well across systems ranging from molecular crystals to proteins.

Predicting 19 F NMR Chemical Shifts: A Combined Computational and Experimental Study of a Trypanosomal Oxidoreductase-Inhibitor Complex

The absence of fluorine from most biomolecules renders it an excellent probe for NMR spectroscopy to monitor inhibitor-protein interactions. However, predicting the binding mode of a fluorinated ligand from a chemical shift (or vice versa) has been challenging due to the high electron density of the fluorine atom. Nonetheless, reliable 19 F chemical-shift predictions to deduce ligand-binding modes hold great potential for in silico drug design. Herein, we present a systematic QM/MM study to predict the 19 F NMR chemical shifts of a covalently bound fluorinated inhibitor to the essential oxidoreductase tryparedoxin (Tpx) from African trypanosomes, the causative agent of African sleeping sickness. We include many protein-inhibitor conformations as well as monomeric and dimeric inhibitor-protein complexes, thus rendering it the largest computational study on chemical shifts of 19 F nuclei in a biological context to date. Our predicted shifts agree well with those obtained experimentally and pave the way for future work in this area.

Dalton Project: A Python platform for molecular- and electronic-structure simulations of complex systems

The Dalton Project provides a uniform platform access to the underlying full-fledged quantum chemistry codes Dalton and LSDalton as well as the PyFraME package for automatized fragmentation and parameterization of complex molecular environments. The platform is written in Python and defines a means for library communication and interaction. Intermediate data such as integrals are exposed to the platform and made accessible to the user in the form of NumPy arrays, and the resulting data are extracted, analyzed, and visualized. Complex computational protocols that may, for instance, arise due to a need for environment fragmentation and configuration-space sampling of biochemical systems are readily assisted by the platform. The platform is designed to host additional software libraries and will serve as a hub for future modular software development efforts in the distributed Dalton community.

Magnetic properties and core electron binding energies of liquid water

The magnetic properties and the core and inner valence electron binding energies of liquid water are investigated. The adopted methodology relies on the combination of molecular dynamics and electronic structure calculations. Born-Oppenheimer molecular dynamics with the Becke and Lee-Yang-Parr functionals for exchange and correlation, respectively, and includes an empirical correction (BLYP-D3) functional and classical molecular dynamics with the TIP4P/2005-F model were carried out. The Keal-Tozer functional was applied for predicting magnetic shielding and spin-spin coupling constants. Core and inner valence electron binding energies in liquid water were calculated with symmetry adapted cluster-configuration interaction. The relationship between the magnetic shielding constant σ(¹⁷O), the role played by the oxygen atom as a proton acceptor and donor, and the tetrahedral organisation of liquid water are investigated. The results indicate that the deshielding of the oxygen atom in water is very dependent on the order parameter (q) describing the tetrahedral organisation of the hydrogen bond network. The strong sensitivity of magnetic properties on changes of the electronic density in the nuclei environment is illustrated by a correlation between σ(¹⁷O) and the energy gap between the 1a₁[O1s] (core) and the 2a₁ (inner valence) orbitals of water. Although several studies discussed the eventual connection between magnetic properties and core electron binding energies, such a correlation could not be clearly established. Here, we demonstrate that for liquid water this correlation exists although involving the gap between electron binding energies of core and inner valence orbitals.

Automated Fragmentation Polarizable Embedding Density Functional Theory (PE-DFT) Calculations of Nuclear Magnetic Resonance (NMR) Shielding Constants of Proteins with Application to Chemical Shift Predictions

Full-protein nuclear magnetic resonance (NMR) shielding constants based on ab initio calculations are desirable, because they can assist in elucidating protein structures from NMR experiments. In this work, we present NMR shielding constants computed using a new automated fragmentation (J. Phys. Chem. B 2009, 113, 10380-10388) approach in the framework of polarizable embedding density functional theory. We extend our previous work to give both basis set recommendations and comment on how large the quantum mechanical region should be to successfully compute ¹³C NMR shielding constants that are comparable with experiment. The introduction of a probabilistic linear regression model allows us to substantially reduce the number of snapshots that are needed to make comparisons with experiment. This approach is further improved by augmenting snapshot selection with chemical shift predictions by which we can obtain a representative subset of snapshots that gives the smallest predicted error, compared to experiment. Finally, we use this subset of snapshots to calculate the NMR shielding constants at the PE-KT3/pcSseg-2 level of theory for all atoms in the protein GB3.

A Projector-Embedding Approach for Multiscale Coupled-Cluster Calculations Applied to Citrate Synthase

Projector-based embedding has recently emerged as a robust multiscale method for the calculation of various electronic molecular properties. We present the coupling of projector embedding with quantum mechanics/molecular mechanics modeling and apply it for the first time to an enzyme-catalyzed reaction. Using projector-based embedding, we combine coupled-cluster theory, density-functional theory (DFT), and molecular mechanics to compute energies for the proton abstraction from acetyl-coenzyme A by citrate synthase. By embedding correlated ab initio methods in DFT we eliminate functional sensitivity and obtain high-accuracy profiles in a procedure that is straightforward to apply.

The Effect of Molecular Conformation on the Accuracy of Theoretical (1)H and (13)C Chemical Shifts Calculated by Ab Initio Methods for Metabolic Mixture Analysis

NMR spectroscopy is a powerful method for analyzing metabolic mixtures. The information obtained from an NMR spectrum is in the form of physical parameters, such as chemical shifts, and construction of databases for many metabolites will be useful for data interpretation. To increase the accuracy of theoretical chemical shifts for development of a database for a variety of metabolites, the effects of sets of conformations (structural ensembles) and the levels of theory on computations of theoretical chemical shifts were systematically investigated for a set of 29 small molecules in the present study. For each of the 29 compounds, 101 structures were generated by classical molecular dynamics at 298.15 K, and then theoretical chemical shifts for 164 (1)H and 123 (13)C atoms were calculated by ab initio quantum chemical methods. Six levels of theory were used by pairing Hartree-Fock, B3LYP (density functional theory), or second order Møller-Plesset perturbation with 6-31G or aug-cc-pVDZ basis set. The six average fluctuations in the (1)H chemical shift were ±0.63, ± 0.59, ± 0.70, ± 0.62, ± 0.75, and ±0.66 ppm for the structural ensembles, and the six average errors were ±0.34, ± 0.27, ± 0.32, ± 0.25, ± 0.32, and ±0.25 ppm. The results showed that chemical shift fluctuations with changes in the conformation because of molecular motion were larger than the differences between computed and experimental chemical shifts for all six levels of theory. In conclusion, selection of an appropriate structural ensemble should be performed before theoretical chemical shift calculations for development of an accurate database for a variety of metabolites.

Excited states in large molecular systems through polarizable embedding

Using the polarizable embedding model enables rational design of light-sensitive functional biological materials.

Fragment-based (13)C nuclear magnetic resonance chemical shift predictions in molecular crystals: An alternative to planewave methods

We assess the quality of fragment-based ab initio isotropic (13)C chemical shift predictions for a collection of 25 molecular crystals with eight different density functionals. We explore the relative performance of cluster, two-body fragment, combined cluster/fragment, and the planewave gauge-including projector augmented wave (GIPAW) models relative to experiment. When electrostatic embedding is employed to capture many-body polarization effects, the simple and computationally inexpensive two-body fragment model predicts both isotropic (13)C chemical shifts and the chemical shielding tensors as well as both cluster models and the GIPAW approach. Unlike the GIPAW approach, hybrid density functionals can be used readily in a fragment model, and all four hybrid functionals tested here (PBE0, B3LYP, B3PW91, and B97-2) predict chemical shifts in noticeably better agreement with experiment than the four generalized gradient approximation (GGA) functionals considered (PBE, OPBE, BLYP, and BP86). A set of recommended linear regression parameters for mapping between calculated chemical shieldings and observed chemical shifts are provided based on these benchmark calculations. Statistical cross-validation procedures are used to demonstrate the robustness of these fits.

Molecular quantum mechanical gradients within the polarizable embedding approach--application to the internal vibrational Stark shift of acetophenone

We present an implementation of analytical quantum mechanical molecular gradients within the polarizable embedding (PE) model to allow for efficient geometry optimizations and vibrational analysis of molecules embedded in large, geometrically frozen environments. We consider a variational ansatz for the quantum region, covering (multiconfigurational) self-consistent-field and Kohn-Sham density functional theory. As the first application of the implementation, we consider the internal vibrational Stark effect of the C=O group of acetophenone in different solvents and derive its vibrational linear Stark tuning rate using harmonic frequencies calculated from analytical gradients and computed local electric fields. Comparisons to PE calculations employing an enlarged quantum region as well as to a non-polarizable embedding scheme show that the inclusion of mutual polarization between acetophenone and water is essential in order to capture the structural modifications and the associated frequency shifts observed in water. For more apolar solvents, a proper description of dispersion and exchange-repulsion becomes increasingly important, and the quality of the optimized structures relies to a larger extent on the quality of the Lennard-Jones parameters.

Converging nuclear magnetic shielding calculations with respect to basis and system size in protein systems

Ab initio chemical shielding calculations greatly facilitate the interpretation of nuclear magnetic resonance (NMR) chemical shifts in biological systems, but the large sizes of these systems requires approximations in the chemical models used to represent them. Achieving good convergence in the predicted chemical shieldings is necessary before one can unravel how other complex structural and dynamical factors affect the NMR measurements. Here, we investigate how to balance trade-offs between using a better basis set or a larger cluster model for predicting the chemical shieldings of the substrates in two representative examples of protein-substrate systems involving different domains in tryptophan synthase: the N-(4'-trifluoromethoxybenzoyl)-2-aminoethyl phosphate (F9) ligand which binds in the α active site, and the 2-aminophenol quinonoid intermediate formed in the β active site. We first demonstrate that a chemically intuitive three-layer, locally dense basis model that uses a large basis on the substrate, a medium triple-zeta basis to describe its hydrogen-bonding partners and/or surrounding van der Waals cavity, and a crude basis set for more distant atoms provides chemical shieldings in good agreement with much more expensive large basis calculations. Second, long-range quantum mechanical interactions are important, and one can accurately estimate them as a small-basis correction to larger-basis calculations on a smaller cluster. The combination of these approaches enables one to perform density functional theory NMR chemical shift calculations in protein systems that are well-converged with respect to both basis set and cluster size.

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.