Nuclear Magnetic Resonance Shielding Research Articles

Accurate NMR Shieldings with σ-Functionals.

In recent years, density-functional methods relying on a new type of fifth-rung correlation functionals called σ-functionals have been introduced. σ-Functionals are technically closely related to the random phase approximation and require the same computational effort but yield distinctively higher accuracies for reaction and transition state energies of main group chemistry and even outperform double-hybrid functionals for these energies. In this work, we systematically investigate how accurate σ-functionals can describe nuclear magnetic resonance (NMR) shieldings. It turns out that σ-functionals yield very accurate NMR shieldings, even though in their optimization, exclusively, energies are employed as reference data and response properties such as NMR shieldings are not involved at all. This shows that σ-functionals combine universal applicability with accuracy. Indeed, the NMR shieldings from a σ-functional using input orbitals and eigenvalues from Kohn-Sham calculations with the exchange-correlation functional of Perdew, Burke and Ernzerhof (PBE) turned out to be the most accurate ones among the NMR shieldings calculated with various density-functional methods including methods using double-hybrid functionals. That σ-functionals can be used for calculating both reliable energies and response properties like NMR shieldings characterizes them as all-purpose functionals, which is appealing from an application point of view.

Molecular NMR shieldings, J-couplings, and magnetizabilities from numeric atom-centered orbital based density-functional calculations

This paper reports and benchmarks a new implementation of nuclear magnetic resonance shieldings, magnetizabilities, and J-couplings for molecules within semilocal density functional theory, based on numeric atom-centered orbital (NAO) basis sets. NAO basis sets are attractive for the calculation of these nuclear magnetic resonance (NMR) parameters because NAOs provide accurate atomic orbital representations especially near the nucleus, enabling high-quality results at modest computational cost. Moreover, NAOs are readily adaptable for linear scaling methods, enabling efficient calculations of large systems. The paper has five main parts: (1) It reviews the formalism of density functional calculations of NMR parameters in one comprehensive text to make the mathematical background available in a self-contained way. (2) The paper quantifies the attainable precision of NAO basis sets for shieldings in comparison to specialized Gaussian basis sets, showing similar performance for similar basis set size. (3) The paper quantifies the precision of calculated magnetizabilities, where the NAO basis sets appear to outperform several established Gaussian basis sets of similar size. (4) The paper quantifies the precision of computed J-couplings, for which a group of customized NAO basis sets achieves precision of ∼Hz for smaller basis set sizes than some established Gaussian basis sets. (5) The paper demonstrates that the implementation is applicable to systems beyond 1000 atoms in size.

Computational Spectroscopy of Aqueous Solutions: The Underlying Role of Conformational Sampling.

Fully atomistic multiscale polarizable quantum mechanics (QM)/molecular mechanics (MM) approaches, combined with techniques to sample the solute-solvent phase space, constitute the most accurate method to compute spectral signals in aqueous solution. Conventional sampling strategies, such as classical molecular dynamics (MD), may encounter drawbacks when the conformational space is particularly complex, and transition barriers between conformers are high. This can lead to inaccurate sampling, which can potentially impact the accuracy of spectral calculations. For this reason, in this work, we compare classical MD with enhanced sampling techniques, i.e., replica exchange MD and metadynamics. In particular, we show how the different sampling techniques affect computed UV, electronic circular dichroism, nuclear magnetic resonance shielding, and optical rotatory dispersion of N-acetylproline-amide in aqueous solution. Such a system is a model peptide characterized by complex conformational variability. Calculated values suggest that spectral properties are influenced by solute conformers, relative population, and solvent effects; therefore, particular care needs to be paid for when choosing the sampling technique.

On the utmost importance of the geometry factor of accuracy in the quantum chemical calculations of 31P NMR chemical shifts: New efficient pecG-n (n = 1, 2) basis sets for the geometry optimization procedure.

31P nuclear magnetic resonance (NMR) chemical shifts were shown to be very sensitive to the basis set used at the geometry optimization stage. Commonly used energy-optimized basis sets for a phosphorus atom containing only one polarization d-function were shown to be unable to provide correct equilibrium geometries for the calculations of phosphorus chemical shifts. The use of basis sets with at least two polarization d-functions on a phosphorus atom is strongly recommended. In this paper, an idea of creating the basis sets purposed for the geometry optimization that provide the least possible error coming from the geometry factor of accuracy in the resultant NMR shielding constants is proposed. The property-energy consisted algorithm with the target function in the form of the molecular energy gradient relative to P-P bond lengths was applied to create new geometry-oriented pecG-n (n = 1, 2) basis sets for a phosphorus atom. New basis sets have demonstrated by far superior performance as compared to the other commonly used energy-optimized basis sets in massive calculations of 31P NMR chemical shifts carried out at the gauge-including atomic orbital-coupled cluster singles and doubles/pecS-2 level of the theory by taking into account solvent, vibrational, and relativistic corrections.

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

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

Analytical Second-Order Properties for the Random Phase Approximation: Nuclear Magnetic Resonance Shieldings.

A method for the analytical computation of nuclear magnetic resonance (NMR) shieldings within the direct random phase approximation (RPA) is presented. As a starting point, we use the RPA ground-state energy expression within the resolution-of-the-identity approximation in the atomic-orbital formalism. As has been shown in a recent benchmark study using numerical second derivatives [Glasbrenner, M. J. Chem. Theory Comput. 2022, 18, 192], RPA based on a Hartree-Fock reference shows accuracies comparable to coupled cluster singles and doubles (CCSD) for NMR chemical shieldings. Together with the much lower computational cost of RPA, it has emerged as an accurate method for the computation of NMR shieldings. Therefore, we aim to extend the applicability of RPA NMR to larger systems by introducing analytical second-order derivatives, making it a viable method for the accurate and efficient computation of NMR chemical shieldings.

An in-silico NMR laboratory for nuclear magnetic shieldings computed via finite fields: Exploring nucleus-specific renormalizations of MP2 and MP3.

We developed and implemented a method-independent, fully numerical, finite difference approach to calculating nuclear magnetic resonance shieldings, using gauge-including atomic orbitals. The resulting capability can be used to explore non-standard methods, given only the energy as a function of finite-applied magnetic fields and nuclear spins. For example, standard second-order Møller-Plesset theory (MP2) has well-known efficacy for 1H and 13C shieldings and known limitations for other nuclei such as 15N and 17O. It is, therefore, interesting to seek methods that offer good accuracy for 15N and 17O shieldings without greatly increased compute costs, as well as exploring whether such methods can further improve 1H and 13C shieldings. Using a small molecule test set of 28 species, we assessed two alternatives: κ regularized MP2 (κ-MP2), which provides energy-dependent damping of large amplitudes, and MP2.X, which includes a variable fraction, X, of third-order correlation (MP3). The aug-cc-pVTZ basis was used, and coupled cluster with singles and doubles and perturbative triples [CCSD(T)] results were taken as reference values. Our κ-MP2 results reveal significant improvements over MP2 for 13C and 15N, with the optimal κ value being element-specific. κ-MP2 with κ = 2 offers a 30% rms error reduction over MP2. For 15N, κ-MP2 with κ = 1.1 provides a 90% error reduction vs MP2 and a 60% error reduction vs CCSD. On the other hand, MP2.X with a scaling factor of 0.6 outperformed CCSD for all heavy nuclei. These results can be understood as providing renormalization of doubles amplitudes to partially account for neglected triple and higher substitutions and offer promising opportunities for future applications.

Characterization of the 1-(5-(4,5-Dimethyl-1,3,2-dioxoborolan-2-yl)thiophen-2-yl)ethanone Using NMR 13C, 1H and 11B through the Density Functional Theory.

The use of computational methods that allow us to perform characterization on new compounds is not a novelty; nevertheless, the degree of complexity of the structures makes their study more challenging since new techniques and methods are required to adjust to the new structural model. The case of nuclear magnetic resonance characterization of boronate esters is fascinating because of its widespread use in materials science. In this paper, we use density functional theory to characterize the structure of the compound 1-[5-(4,5-Dimethyl-1,3,2-dioxaborolan-2-yl)thiophen-2-yl]ethanonea by means of nuclear magnetic resonance. We studied the compound in its solid form with the PBE-GGA and PBEsol-GGA functionals, with a set of plane wave functions and an augmented wave projector, which included gauge in CASTEP and its molecular structure with the B3LYP functional using the package Gaussian 09. In addition, we performed the optimization and calculation of the chemical shifts and isotropic nuclear magnetic resonance shielding of 1H, 13C, and 11B. Finally, we analyzed and compared the theoretical results with experimental diffractometric data observing a good approximation.

A relativistic relationship between parity-violating nuclear spin-rotation tensors and parity-violating NMR shielding tensors.

The nuclear-spin-dependent parity-violation contributions to the nuclear magnetic resonance shielding and nuclear spin-rotation tensors (σPV and MPV, respectively) are known to be formally related to one another in the non-relativistic regime. In this work, the polarization propagator formalism and the linear response within the elimination of small components model are used to show a new and more general relationship between them, which is valid within the relativistic framework. The full set of the zeroth- and first-order relativistic contributions to σPV and MPV are also given here for the first time, and these results are compared with previous findings. According to four-component relativistic calculations, the electronic spin-orbit effects are the most significant ones for the isotropic values of σPV and MPV in the H2X2 series of molecules (with X = O, S, Se, Te, and Po). When only scalar relativistic effects are taken into account, the non-relativistic relationship between σPV and MPV does hold. However, when the spin-orbit effects are taken into consideration, this old non-relativistic relationship breaks down, and therefore, the new one must be considered.

Current density functional framework for spin–orbit coupling

Relativistic two-component density functional calculations are carried out in a non-collinear formalism to describe spin-orbit interactions, where the exchange-correlation functional is constructed as a generalization of the non-relativistic density functional approximation. Contrary to non-relativistic density functional theory (DFT), spin-orbit coupling, however, leads to a non-vanishing paramagnetic current density. Density functionals depending on the kinetic energy density, such as meta-generalized gradient approximations, should therefore be constructed in the framework of current DFT (CDFT). The latter has previously exclusively been used in the regime of strong magnetic fields. Herein, we present a consistent CDFT approach for relativistic DFT, including spin-orbit coupling. Furthermore, we assess the importance of the current density terms for ground-state energies, excitation energies, nuclear magnetic resonance shielding, and spin-spin coupling constants, as well as hyperfine coupling constants, Δg-shifts, and the nuclear quadrupole interaction tensor in electron paramagnetic resonance (EPR) spectroscopy. The most notable changes are found for EPR properties. The impact of the current-dependent terms rises with the number of unpaired electrons, and consequently, the EPR properties are more sensitive toward CDFT. Considerable changes are observed for the strongly constrained and appropriately normed functionals, as well as the B97M family and TASK. The current density terms are less important when exact exchange is incorporated. At the same time, the current-dependent kernel ensures the stability of response calculations in all cases. We, therefore, strongly recommend to use the framework of CDFT for self-consistent spin-orbit calculations.

Reference interaction site model self-consistent field with constrained spatial electron density approach for nuclear magnetic shielding in solution.

We propose a new hybrid approach combining quantum chemistry and statistical mechanics of liquids for calculating the nuclear magnetic resonance (NMR) chemical shifts of solvated molecules. Based on the reference interaction site model self-consistent field with constrained spatial electron density distribution (RISM-SCF-cSED) method, the electronic structure of molecules in solution is obtained, and the expression for the nuclear magnetic shielding tensor is derived as the second-order derivative of the Helmholtz energy of the solution system. We implemented a method for calculating chemical shifts and applied it to an adenine molecule in water, where hydrogen bonding plays a crucial role in electronic and solvation structures. We also performed the calculations of 17O chemical shifts, which showed remarkable solvent dependence. While converged results could not be sometimes obtained using the conventional method, in the present framework with RISM-SCF-cSED, an adequate representation of electron density is guaranteed, making it possible to obtain an NMR shielding constant stably. This introduction of cSED is key to extending the method's applicability to obtain the chemical shift of various chemical species. The present demonstration illustrates our approach's superiority in terms of numerical robustness and accuracy.

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.

Ammonia: The molecule for establishing 14N and 15N absolute shielding scales and a source of information on nuclear magnetic moments.

Multinuclear Nuclear Magnetic Resonance (NMR) studies of the gaseous mixtures 3He/14NH3 and 3He/15NH3 are reported. Precise analysis of the 3He, 14N, 15N, and 1H resonance frequencies show a linear dependence on the gas density. Extrapolation of these results to the zero-pressure limit gives ν0(1H), ν0(14N), and ν0(15N) resonance frequencies of the isolated ammonia molecule at 300K. The analogous value for 3He atoms in gaseous mixtures ν0(3He) was measured as well. The application of a new scheme to introduce the most important electronic effects on NMR shieldings, together with highly accurate quantum chemical shielding calculations, allows the 14/15N and 1H shielding of the isolated ammonia molecule to be obtained with the greatest accuracy and precision. For the first time, these studies were carried out on ammonia within the so-called four-component relativistic framework. The NMR frequency comparison method provides an approach for determining the 14N and 15N nuclear magnetic moments. The new shielding parameters in ammonia were used for re-evaluation of the entire nitrogen absolute shielding scale. Additionally, the absolute shielding values of several gaseous compounds and secondary reference substances in liquids were presented. It was established that 14N and 15N absolute shielding constants in 14NH3 and 15NH3 are very similar and only differ by less than 0.01 ppm, which is not usually measurable in NMR experiments. Precise calculations of 14N and 15N dipole moments were also made from these accurate shielding values.

A local hybrid exchange functional approximation from first principles.

Local hybrid functionals are a more flexible class of density functional approximations, allowing for a position-dependent admixture of exact exchange. This additional flexibility, however, comes with a more involved mathematical form and a more complicated design. A common denominator for previously constructed local hybrid functionals is the usage of thermochemical benchmark data to construct these functionals. Herein, we design a local hybrid functional without relying on benchmark data. Instead, we construct it in a more ab initio manner, following the principles of modern meta-generalized gradient approximations and considering theoretical constraints. To achieve this, we make use of the density matrix expansion and a local mixing function based on an approximate correlation length. The accuracy of the developed density functional approximation is assessed for thermochemistry, excitation energies, polarizabilities, magnetizabilities, nuclear magnetic resonance (NMR) spin-spin coupling constants, NMR shieldings, and shifts, as well as EPR g-tensors and hyperfine coupling constants. Here, the new exchange functional shows a robust performance and is especially well suited for atomization energies, barrier heights, excitation energies, NMR coupling constants, and EPR properties, whereas it loses some ground for the NMR shifts. Therefore, the designed functional is a major step forward for functionals that have been designed from first principles.

Impact of the current density on paramagnetic NMR properties.

Meta-generalized gradient approximations (meta-GGAs) and local hybrid functionals generally depend on the kinetic energy density τ. For magnetic properties, this necessitates generalizations to ensure gauge invariance. In most implementations, τ is generalized by incorporating the external magnetic field. However, this introduces artifacts in the response of the density matrix and does not satisfy the iso-orbital constraint. Here, we extend previous approaches based on the current density to paramagnetic nuclear magnetic resonance (NMR) shieldings and electron paramagnetic resonance (EPR) g-tensors. The impact is assessed for main-group compounds and transition-metal complexes considering 25 density functional approximations. It is shown that the current density leads to substantial improvements-especially for the popular Minnesota and strongly constrained and appropriately normed (SCAN) functional families. Thus, we strongly recommend to use the current density generalized τ in paramagnetic NMR and EPR calculations with meta-GGAs.

Statistically optimal analysis of the extended-system adaptive biasing force (eABF) method

The extended-system adaptive biasing force (eABF) method and its newer variants offer rapid exploration of the configuration space of chemical systems. Instead of directly applying the ABF bias to collective variables, they are harmonically coupled to fictitious particles, which separates the problem of enhanced sampling from that of free energy estimation. The prevalent analysis method to obtain the potential of mean force (PMF) from eABF is thermodynamic integration. However, besides the PMF, most information is lost as the unbiased probability of visited configurations is never recovered. In this contribution, we show how statistical weights of individual frames can be computed using the Multistate Bennett's Acceptance Ratio (MBAR), putting the post-processing of eABF on one level with other frequently used sampling methods. In addition, we apply this formalism to the prediction of nuclear magnetic resonance shieldings, which are very sensitive to molecular geometries and often require extensive sampling. The results show that the combination of enhanced sampling by means of extended-system dynamics with the MBAR estimator is a highly useful tool for the calculation of ensemble properties. Furthermore, the extension of the presented scheme to the recently published Gaussian-accelerated molecular dynamics eABF hybrid is straightforward and approximation free.

Accurate Prediction of Nuclear Magnetic Resonance Parameters via the XYG3 Type of Doubly Hybrid Density Functionals.

Nuclear magnetic resonance (NMR) spectroscopy is one of the most powerful and versatile tools in elucidating molecular structures. To eliminate ambiguities of experimental assignments, accurate calculations of NMR spectra are of great importance. Here, a method for theoretical evaluation of the NMR shielding constants by analytic derivatives using gauge including atomic orbitals (GIAO) has been implemented for the XYG3 type of doubly hybrid density functionals (xDH), namely, the GIAO-xDH method. Benchmark calculations on shielding constants and chemical shifts demonstrate the remarkable accuracy of the GIAO-xDH method, compared to the accurate CCSD(T) references. It is shown here that the XYGJ-OS functional is able to give a mean absolute deviation (MAD) of ∼3.0 ppm in the calculated shielding constants for 13C, 15N, 17O, 19F, while both XYGJ-OS and xDH-PBE0 functionals are able to provide a satisfactory estimation of chemical shifts with MADs of ∼0.03 and 1.0 ppm for 1H and 13C, respectively. The basis set influence upon the method has been examined and a computational scheme considering both accuracy and efficiency has been proposed and tested to predict the experimental 13C chemical shifts of five medium-sized natural product molecules, yielding a MAD of ∼1.0 ppm, which demonstrates the practical feasibility of the GIAO-xDH method.

Four-component relativistic calculations of NMR shielding constants of the transition metal complexes. Part 1: Pentaammines of cobalt, rhodium, and iridium.

The nonrelativistic and four-component fully relativistic calculations of 1 H, 15 N, 59 Co, 103 Rh, and 193 Ir shielding constants of pentaammineaquacomplexes of cobalt(III), rhodium(III), and iridium(III) were carried out at the density functional theory (DFT) level of theory. The noticeable deshielding relativistic corrections were observed for nitrogen shielding constants (chemical shifts), whereas those corrections were found to be negligible for protons. For the transition metals cobalt, rhodium, and iridium, relativistic corrections to their nuclear magnetic resonance (NMR) shielding constants were found to be rather small for cobalt and rhodium (some 5-10%), whereas they are essentially larger for iridium (up to 70%).

Efficient low-scaling computation of NMR shieldings at the second-order Møller-Plesset perturbation theory level with Cholesky-decomposed densities and an attenuated Coulomb metric.

A method for the computation of nuclear magnetic resonance (NMR) shieldings with second-order Møller-Plesset perturbation theory (MP2) is presented which allows to efficiently compute the entire set of shieldings for a given molecular structure. The equations are derived using Laplace-transformed atomic orbital second-order Møller-Plesset perturbation theory as a starting point. The Z-vector approach is employed for minimizing the number of coupled-perturbed self-consistent-field equations that need to be solved. In addition, the method uses the resolution-of-the-identity approximation with an attenuated Coulomb metric and Cholesky decomposition of pseudo-density matrices. The sparsity in the three-center integrals is exploited with sparse linear algebra approaches, leading to reduced computational cost and memory demands. Test calculations show that the deviations from NMR shifts obtained with canonical MP2 are small if appropriate thresholds are used. The performance of the method is illustrated in calculations on DNA strands and on glycine chains with up to 283 atoms and 2864 basis functions.

NMR chemical shift computations at second-order Møller-Plesset perturbation theory using gauge-including atomic orbitals and Cholesky-decomposed two-electron integrals.

We report on a formulation and implementation of a scheme to compute nuclear magnetic resonance (NMR) shieldings at second-order Møller-Plesset (MP2) perturbation theory using gauge-including atomic orbitals (GIAOs) to ensure gauge-origin independence and Cholesky decomposition (CD) to handle unperturbed and perturbed two-electron integrals. We investigate the accuracy of the CD for the derivatives of the two-electron integrals with respect to an external magnetic field and for the computed NMR shieldings, before we illustrate the applicability of our CD-based GIAO-MP2 scheme in calculations involving up to about 100 atoms and more than 1000 basis functions.