Temperature effects in first-principles solid state calculations of the chemical shielding tensor made simple

The EFG Rosetta Stone: translating between DFT calculations and solid state NMR experiments

We present a comprehensive study on the best practices for integrating first principles simulations in experimental quadrupolar solid-state nuclear magnetic resonance (SS-NMR), exploiting the synergies between theory and experiment for achieving the optimal interpretation of both. Most high performance materials (HPMs), such as battery electrodes, exhibit complex SS-NMR spectra due to dynamic effects or amorphous phases. NMR crystallography for such challenging materials requires reliable, accurate, efficient computational methods for calculating NMR observables from first principles for the transfer between theoretical material structure models and the interpretation of their experimental SS-NMR spectra. NMR-active nuclei within HPMs are routinely probed by their chemical shielding anisotropy (CSA). However, several nuclear isotopes of interest, e.g.7Li and 27Al, have a nuclear quadrupole and experience additional interactions with the surrounding electric field gradient (EFG). The quadrupolar interaction is a valuable source of information about atomistic structure, and in particular, local symmetry, complementing the CSA. As such, there is a range of different methods and codes to choose from for calculating EFGs, from all-electron to plane wave methods. We benchmark the accuracy of different simulation strategies for computing the EFG tensor of quadrupolar nuclei with plane wave density functional theory (DFT) and study the impact of the material structure as well as the details of the simulation strategy. Especially for small nuclei with few electrons, such as 7Li, we show that the choice of physical approximations and simulation parameters has a large effect on the transferability of the simulation results. To the best of our knowledge, we present the first comprehensive reference scale and literature survey for 7Li quadrupolar couplings. The results allow us to establish practical guidelines for developing the best simulation strategy for correlating DFT to experimental data extracting the maximum benefit and information from both, thereby advancing further research into HPMs.

Effect of Dynamical Motion in ab Initio Calculations of Solid-State Nuclear Magnetic and Nuclear Quadrupole Resonance Spectra

Solid-state nuclear magnetic resonance (SSNMR) and nuclear quadrupole resonance (NQR) spectra provide detailed information about the electronic and atomic structure of solids. Modern ab initio methods such as density functional theory (DFT) can be used to calculate NMR and NQR spectra from first-principles, providing a meaningful avenue to connect theory and experiment. Prediction of SSNMR and NQR spectra from DFT relies on accurate calculation of the electric field gradient (EFG) tensor associated with the potential of electrons at the nuclear centers. While static calculations of EFGs are commonly seen in the literature, the effects of dynamical motion of atoms in molecules and solids have been less explored. In this study, we develop a method to calculate EFGs of solids while taking into account the dynamics of atoms through DFT-based molecular dynamics simulations. The method we develop is general, in the sense that it can be applied to any material at any desired temperature and pressure. Here, we focus on application of the method to NaNO2 and study in detail the EFGs of 14N, 17O, and 23Na. We find in the cases of 14N and 17O that the dynamical motion of the atoms can be used to calculate mean EFGs that are in closer agreement with experiments than those of static calculations. For 23Na, we find a complex behavior of the EFGs when atomic motion is incorporated that is not at all captured in static calculations. In particular, we find a distribution of EFGs that is influenced strongly by the local (changing) bond environment, with a pattern that reflects the coordination structure of 23Na. We expect the methodology developed here to provide a path forward for understanding materials in which static EFG calculations do not align with experiments.

Lattice Dynamics in the NASICON NaZr2(PO4)3 Solid Electrolyte from Temperature-Dependent Neutron Diffraction, NMR, and Ab Initio Computational Studies

Natrium super ionic conductor (NASICON) compounds form a rich and highly chemically tunable family of crystalline materials that are of widespread interest because they include exemplars with high ionic conductivity, low thermal expansion, and redox tunability. This makes them suitable candidates for applications ranging from solid-state batteries to nuclear waste storage materials. The key to an understanding of these properties, including the origins of effective cation transport and low, anisotropic (and sometimes negative) thermal expansion, lies in the lattice dynamics associated with specific details of the crystal structure. Here we closely examine the prototypical NASICON compound, NaZr₂(PO₄)₃, and obtain detailed insights into such behavior via variable-temperature neutron diffraction and ²³Na and ³¹P solid-state NMR studies, coupled with comprehensive density functional theory-based calculations of NMR parameters. Temperature-dependent NMR studies yield some surprising trends in the chemical shifts and the quadrupolar coupling constants that are not captured by computation unless the underlying vibrational modes of the crystal are explicitly taken into account. Furthermore, the trajectories of the sodium, zirconium, and oxygen atoms in our dynamical simulations show good qualitative agreement with the anisotropic thermal parameters obtained at higher temperatures by neutron diffraction. The work presented here widens the utility of NMR crystallography to include thermal effects as a unique probe of interesting lattice dynamics in functional materials.

Importance of Nuclear Quantum Effects for NMR Crystallography

The resolving power of solid-state nuclear magnetic resonance (NMR) crystallography depends heavily on the accuracy of computational predictions of NMR chemical shieldings of candidate structures, which are usually taken to be local minima in the potential energy. To test the limits of this approximation, we systematically study the importance of finite-temperature and quantum nuclear fluctuations for ¹H, ¹³C, and ¹⁵N shieldings in polymorphs of three paradigmatic molecular crystals: benzene, glycine, and succinic acid. The effect of quantum fluctuations is comparable to the typical errors of shielding predictions for static nuclei with respect to experiments, and their inclusion improves the agreement with measurements, translating to more reliable assignment of the NMR spectra to the correct candidate structure. The use of integrated machine-learning models, trained on first-principles energies and shieldings, renders rigorous sampling of nuclear fluctuations affordable, setting a new standard for the calculations underlying NMR structure determinations.

NMR crystallography of molecular organics

Developments of NMR methodology to characterise the structures of molecular organic structures are reviewed, concentrating on the previous decade of research in which density functional theory-based calculations of NMR parameters in periodic solids have become widespread. With a focus on demonstrating the new structural insights provided, it is shown how "NMR crystallography" has been used in a spectrum of applications from resolving ambiguities in diffraction-derived structures (such as hydrogen atom positioning) to deriving complete structures in the absence of diffraction data. As well as comprehensively reviewing applications, the different aspects of the experimental and computational techniques used in NMR crystallography are surveyed. NMR crystallography is seen to be a rapidly maturing subject area that is increasingly appreciated by the wider crystallographic community.

A Bayesian approach to NMR crystal structure determination

Nuclear Magnetic Resonance (NMR) spectroscopy is particularly well suited to determine the structure of molecules and materials in powdered form. Structure determination usually proceeds by finding the best match between experimentally observed NMR chemical shifts and those of candidate structures. Chemical shifts for the candidate configurations have traditionally been computed by electronic-structure methods, and more recently predicted by machine learning. However, the reliability of the determination depends on the errors in the predicted shifts. Here we propose a Bayesian framework for determining the confidence in the identification of the experimental crystal structure, based on knowledge of the typical errors in the electronic structure methods. We demonstrate the approach on the determination of the structures of six organic molecular crystals. We critically assess the reliability of the structure determinations, facilitated by the introduction of a visualization of the similarity between candidate configurations in terms of their chemical shifts and their structures. We also show that the commonly used values for the errors in calculated ¹³C shifts are underestimated, and that more accurate, self-consistently determined uncertainties make it possible to use ¹³C shifts to improve the accuracy of structure determinations. Finally, we extend the recently-developed ShiftML model to render it more efficient, accurate, and, most importantly, to evaluate the uncertainties in its predictions. By quantifying the confidence in structure determinations based on ShiftML predictions we further substantiate that it provides a valid replacement for first-principles calculations in NMR crystallography.

Exploring Accuracy Limits of Predictions of the 1H NMR Chemical Shielding Anisotropy in the Solid State

The 1H chemical shielding anisotropy (CSA) is an NMR parameter that is exquisitely sensitive to the local environment of protons in crystalline systems, but it is difficult to obtain it experimentally due to the need to concomitantly suppress other anisotropic interactions in the solid-state NMR (SSNMR) pulse sequences. The SSNMR measurements of the 1H CSA are particularly challenging if the fast magic-angle-spinning (MAS) is applied. It is thus important to confront the results of both the single-crystal (SC) and fast-MAS experiments with their theoretical counterparts. Here the plane-waves (PW) DFT calculations have been carried out using two functionals in order to precisely characterize the structures and the 1H NMR chemical shielding tensors (CSTs) of the solid forms of maleic, malonic, and citric acids, and of L-histidine hydrochloride monohydrate. The level of agreement between the PW DFT and either SC or fast-MAS SSNMR 1H CSA data has been critically compared. It has been found that for the eigenvalues of the 1H CSTs provided by the fast-MAS measurements, an accuracy limit of current PW DFT predictions is about two ppm in terms of the standard deviation of the linear regression model, and sources of this error have been thoroughly discussed.

Reducing the computational cost of NMR crystallography of organic powders at natural isotopic abundance with the help of 13 C-13 C dipolar couplings

Structure determination of functional organic compounds remains a formidable challenge when the sample exists as a powder. Nuclear magnetic resonance crystallography approaches based on the comparison of experimental and Density Functional Theory (DFT)-computed ¹ H chemical shifts have already demonstrated great potential for structure determination of organic powders, but limitations still persist. In this study, we discuss the possibility of using ¹³ C-¹³ C dipolar couplings quantified on powdered theophylline at natural isotopic abundance with the help of dynamic nuclear polarization, to realize a DFT-free, rapid screening of a pool of structures predicted by ab initio random structure search. We show that although ¹³ C-¹³ C dipolar couplings can identify structures possessing long range structural motifs and unit cell parameters close to those of the true structure, it must be complemented with other data to recover information about the presence and the chemical nature of the supramolecular interactions.

The use of variable temperature 13 C solid-state MAS NMR and GIPAW DFT calculations to explore the dynamics of diethylcarbamazine citrate

Experimental ¹³ C solid-state magic-angle spinning (MAS) Nuclear Magnetic Resonance (NMR) as well as Density-Functional Theory (DFT) gauge-including projector augmented wave (GIPAW) calculations were used to probe disorder and local mobility in diethylcarbamazine citrate, (DEC)⁺ (citrate)^- . This compound has been used as the first option drug for the treatment of filariasis, a disease endemic in tropical countries and caused by adult worms of Wuchereria bancrofti, which is transmitted by mosquitoes. We firstly present 2D ¹³ C─¹ H dipolar-coupling-mediated heteronuclear correlation spectra recorded at moderate spinning frequency, to explore the intermolecular interaction between DEC and citrate molecules. Secondly, we investigate the dynamic behavior of (DEC)⁺ (citrate)^- by varying the temperature and correlating the experimental MAS NMR results with DFT GIPAW calculations that consider two (DEC)⁺ conformers (in a 70:30 ratio) for crystal structures determined at 293 and 235 K. Solid-state NMR provides insights on slow exchange dynamics revealing conformational changes involving particularly the DEC ethyl groups.

Nuclear Magnetic Resonance Spectroscopy as a Dynamical Structural Probe of Hydrogen under High Pressure

An unambiguous crystallographic structure solution for the observed phases II-VI of high pressure hydrogen does not exist due to the failure of standard structural probes at extreme pressure. In this work we propose that nuclear magnetic resonance spectroscopy provides a complementary structural probe for high pressure hydrogen. We show that the best structural models available for phases II, III, and IV of high pressure hydrogen exhibit markedly distinct nuclear magnetic resonance spectra which could therefore be used to discriminate amongst them. As an example, we demonstrate how nuclear magnetic resonance spectroscopy could be used to establish whether phase III exhibits polymorphism. Our calculations also reveal a strong renormalization of the nuclear magnetic resonance response in hydrogen arising from quantum fluctuations, as well as a strong isotope effect. As the experimental techniques develop, nuclear magnetic resonance spectroscopy can be expected to become a useful complementary structural probe in high pressure experiments.

How does the NMR thermometer work? Application of combined quantum molecular dynamics and GIPAW calculations into the study of lead nitrate

Lead nitrate is an inorganic salt, commonly used for the accurate temperature determination in the solid state NMR spectroscopy, due to the strong temperature dependence of the ²⁰⁷ Pb chemical shift. As the reason for this phenomenon remained unknown, the main purpose of this study was to explain this temperature dependence at the molecular level. To achieve this, combined CASTEP geometry optimization, quantum molecular dynamics at chosen temperatures and GIPAW NMR computations were performed. Due to the previous literature reports on inaccuracy in the calculation of ²⁰⁷ Pb NMR parameters using GIPAW, a large emphasis was put on the optimization of computational method. The application of quantum molecular dynamics provided the simulation of the temperature-dependent vibrational motions and enabled to accurately compute the changes in the value of Pb δ_iso resulting from them. © 2018 Wiley Periodicals, Inc.

Electron-phonon coupling from finite differences

The interaction between electrons and phonons underlies multiple phenomena in physics, chemistry, and materials science. Examples include superconductivity, electronic transport, and the temperature dependence of optical spectra. A first-principles description of electron-phonon coupling enables the study of the above phenomena with accuracy and material specificity, which can be used to understand experiments and to predict novel effects and functionality. In this topical review, we describe the first-principles calculation of electron-phonon coupling from finite differences. The finite differences approach provides several advantages compared to alternative methods, in particular (i) any underlying electronic structure method can be used, and (ii) terms beyond the lowest order in the electron-phonon interaction can be readily incorporated. But these advantages are associated with a large computational cost that has until recently prevented the widespread adoption of this method. We describe some recent advances, including nondiagonal supercells and thermal lines, that resolve these difficulties, and make the calculation of electron-phonon coupling from finite differences a powerful tool. We review multiple applications of the calculation of electron-phonon coupling from finite differences, including the temperature dependence of optical spectra, superconductivity, charge transport, and the role of defects in semiconductors. These examples illustrate the advantages of finite differences, with cases where semilocal density functional theory is not appropriate for the calculation of electron-phonon coupling and many-body methods such as the GW approximation are required, as well as examples in which higher-order terms in the electron-phonon interaction are essential for an accurate description of the relevant phenomena. We expect that the finite difference approach will play a central role in future studies of the electron-phonon interaction.

How Strong Is the Hydrogen Bond in Hybrid Perovskites?

Hybrid organic-inorganic perovskites represent a special class of metal-organic framework where a molecular cation is encased in an anionic cage. The molecule-cage interaction influences phase stability, phase transformations, and the molecular dynamics. We examine the hydrogen bonding in four AmBX₃ formate perovskites: [Am]Zn(HCOO)₃, with Am⁺ = hydrazinium (NH₂NH₃⁺), guanidinium (C(NH₂)₃⁺), dimethylammonium (CH₃)₂NH₂⁺, and azetidinium (CH₂)₃NH₂⁺. We develop a scheme to quantify the strength of hydrogen bonding in these systems from first-principles, which separates the electrostatic interactions between the amine (Am⁺) and the BX₃^- cage. The hydrogen-bonding strengths of formate perovskites range from 0.36 to 1.40 eV/cation (8-32 kcalmol^-1). Complementary solid-state nuclear magnetic resonance spectroscopy confirms that strong hydrogen bonding hinders cation mobility. Application of the procedure to hybrid lead halide perovskites (X = Cl, Br, I, Am⁺ = CH₃NH₃⁺, CH(NH₂)₂⁺) shows that these compounds have significantly weaker hydrogen-bonding energies of 0.09 to 0.27 eV/cation (2-6 kcalmol^-1), correlating with lower order-disorder transition temperatures.

The application of tailor-made force fields and molecular dynamics for NMR crystallography: a case study of free base cocaine

Motional averaging has been proven to be significant in predicting the chemical shifts in ab initio solid-state NMR calculations, and the applicability of motional averaging with molecular dynamics has been shown to depend on the accuracy of the molecular mechanical force field. The performance of a fully automatically generated tailor-made force field (TMFF) for the dynamic aspects of NMR crystallography is evaluated and compared with existing benchmarks, including static dispersion-corrected density functional theory calculations and the COMPASS force field. The crystal structure of free base cocaine is used as an example. The results reveal that, even though the TMFF outperforms the COMPASS force field for representing the energies and conformations of predicted structures, it does not give significant improvement in the accuracy of NMR calculations. Further studies should direct more attention to anisotropic chemical shifts and development of the method of solid-state NMR calculations.

Crystallographic and Dynamic Aspects of Solid-State NMR Calibration Compounds: Towards ab Initio NMR Crystallography

The excellent results of dispersion-corrected density functional theory (DFT-D) calculations for static systems have been well established over the past decade. The introduction of dynamics into DFT-D calculations is a target, especially for the field of molecular NMR crystallography. Four (13) C ss-NMR calibration compounds are investigated by single-crystal X-ray diffraction, molecular dynamics and DFT-D calculations. The crystal structure of 3-methylglutaric acid is reported. The rotator phases of adamantane and hexamethylbenzene at room temperature are successfully reproduced in the molecular dynamics simulations. The calculated (13) C chemical shifts of these compounds are in excellent agreement with experiment, with a root-mean-square deviation of 2.0 ppm. It is confirmed that a combination of classical molecular dynamics and DFT-D chemical shift calculation improves the accuracy of calculated chemical shifts.

Combining solid-state NMR spectroscopy with first-principles calculations - a guide to NMR crystallography

Recent advances in the application of first-principles calculations of NMR parameters to periodic systems have resulted in widespread interest in their use to support experimental measurement. Such calculations often play an important role in the emerging field of "NMR crystallography", where NMR spectroscopy is combined with techniques such as diffraction, to aid structure determination. Here, we discuss the current state-of-the-art for combining experiment and calculation in NMR spectroscopy, considering the basic theory behind the computational approaches and their practical application. We consider the issues associated with geometry optimisation and how the effects of temperature may be included in the calculation. The automated prediction of structural candidates and the treatment of disordered and dynamic solids are discussed. Finally, we consider the areas where further development is needed in this field and its potential future impact.

Temperature Dependence of NMR Parameters Calculated from Path Integral Molecular Dynamics Simulations

The influence of temperature on NMR chemical shifts and quadrupolar couplings in model molecular organic solids is explored using path integral molecular dynamics (PIMD) and density functional theory (DFT) calculations of shielding and electric field gradient (EFG) tensors. An approach based on convoluting calculated shielding or EFG tensor components with probability distributions of selected bond distances and valence angles obtained from DFT-PIMD simulations at several temperatures is used to calculate the temperature effects. The probability distributions obtained from the quantum PIMD simulations, which includes nuclear quantum effects, are significantly broader and less temperature dependent than those obtained with conventional DFT molecular dynamics or with 1D scans through the potential energy surface. Predicted NMR observables for the model systems were in excellent agreement with experimental data.

K5Mo4O14F: A Novel Fluorinated Polyoxomolybdate and Its Structural Stability

We successfully synthesized a novel fluorinated polyoxomolybdate, K5Mo4O14F, in which the unusual polyanion [Mo4O14F](5-) consists of face-sharing [Mo2O8F] bioctahedra linked with [MoO4] tetrahedra. This unique structural feature provides the very rare case that simultaneously violates Pauling's electrostatic valence (II) and atomic coordination (IV) rules, as well as the stable tendency governed by the polyhedral sharing (III) rule. The structural stability of K5Mo4O14F was confirmed from thermal experiments over a wide temperature range and further elucidated by first-principles calculations.