Polarization propagators: A powerful theoretical tool for a deeper understanding of NMR spectroscopic parameters

Recent advances in liquid-phase NMR of the coal-derived products

Present review focuses on the most recent advances in a liquid-phase nuclear magnetic resonance (NMR) of the coal-derived products-coal tar pitches, asphaltenes, and humic and fulvic acids, covering exclusively the results in the liquid-phase NMR studies leaving apart an overwhelming amount of publications dealing with the solid-state NMR investigations in this field (which are comprehensively reviewed elsewhere). Owing to the complexity of the coal-derived products, their 1H and 13C NMR spectra consist of a number of overlapping signals belonging to different hydrocarbon types. Comprehensive studies of coal tar pitches, asphaltenes, and humic and fulvic acids by means of NMR over the past several decades revealed characteristic functional groups of those fractions together with spectral regions in which they resonate. Quantitative 1H and 13C NMR spectra characterize aromatic and saturated carbons spread over many structural moieties, which provides a solid guideline into molecular structure of the coal-derived products. Nowadays, quantitative 13C NMR measurements yield information about a variety of structural parameters such as functional group distribution, aromaticity, degree of condensation of aromatic rings, and medium chain lengths together with many other more specific parameters. The structural NMR studies of coal and coal-derived products are developing on a backdrop of a marked progress in computational NMR. At present, we are witnessing an unprecedentedly fast development of theoretical and computational methods in the field of NMR spectroscopy. Discussed in the present review are the most recent advances in the NMR studies of the processing products of peat, lignite or brown coal, anthracite or hard coal, and graphite in solution, like coal tar pitches, asphaltenes, and humic and fulvic acids.

Relativistic and quantum electrodynamics effects on NMR shielding tensors of TlX (X = H, F, Cl, Br, I, At) molecules

The results of relativistic calculations of nuclear magnetic resonance shielding tensors (σ) for the thallium monocation (Tl+), thallium hydride (TlH), and thallium halides (TlF, TlCl, TlBr, TlI, and TlAt) are presented as obtained within a four-component polarization propagator formalism and a two-component linear response approach within the zeroth-order regular approximation. In addition to a detailed analysis of relativistic effects performed in this work, some quantum electrodynamical (QED) effects on those nuclear magnetic resonance shieldings and other small contributions are estimated. A strong dependence of σ(Tl) on the bonding partner is found, together with a very weak dependence of QED effects with them. In order to explain the trends observed, the excitation patterns associated with relativistic ee (or paramagnetic-like) and pp (or diamagnetic-like) contributions to σ are analyzed. For this purpose, the electronic spin-free and spin-dependent contributions are separated within the two-component zeroth-order regular approximation, and the influence of spin-orbit coupling on involved molecular orbitals is studied, which allows for a thorough understanding of the underlying mechanisms.

Coordination of Mercury(II) in Water Promoted over Hydrolysis in Solvated Clusters [Hg(H2O)1-6](aq)2+: Insights from Relativistic Effects and Free Energy Analysis

Understanding the nature of the interaction between mercury(II) ions, Hg2+, and water molecules is crucial to describe the stability and chemical behavior of structures formed during solvation, as well as the conditions that favor the Hg2+ coordination or inducing water hydrolysis. In our study, we explored exhaustively the potential energy surface of Hg2+ with up to six water molecules. We analyzed electronic and Gibbs free energies, binding, and nuclear magnetic resonance parameters. We used the zeroth-order regular approximation Hamiltonian, including scalar and spin-orbit relativistic corrections for free energy calculations and geometry optimizations to explore the interplay between electron correlation and relativistic effects. We analyzed intermolecular interactions with energy decomposition analysis, quantum theory of atoms in molecules, and natural bond orbital. Additionally, we used the four-component Dirac Hamiltonian to compute solvent effect on the magnetic shielding and J-coupling constants. Our results revealed that the water hydrolysis by Hg2+ requires a minimum of three water molecules. We found that the interaction between Hg2+ and water molecules is an orbital interaction due to relativistic effects and the most stable structures are opened-shape clusters, reducing the number of oxygen-mercury contacts and maximizing the formation of hydrogen bonds among water molecules. In these types of clusters, Hg2+ promotes the water hydrolysis over coordination with oxygen atoms. However, when we considered the change associated with the transfer of a cluster from the ideal gas to a solvated system, our solvation free energy analysis revealed that closed-shape clusters are more favorable, maximizing the number of oxygen-mercury contacts and reducing the formation of hydrogen bonds among water molecules. This finding suggests that, under room conditions, the coordination of Hg2+ is more favorable than hydrolysis. Our results have significant implications for understanding Hg2+ behavior in water, helping to develop targeted strategies for mercury remediation and management, and contributing to advancements in the broader field of environmental chemistry.

Computational 19 F NMR of trifluoromethyl derivatives of alkenes, pyrimidines, and indenes

The ¹⁹ F NMR chemical shifts of 13 trifluoromethyl derivatives of alkenes, pyrimidines, and indenes were calculated at the DFT level using the BhandHLYP, BHandH, PBE, PBE0, O3LYP, B3LYP, KT2, and KT3 functionals in combination with the pcS-2 basis set. Best result was documented for the BHandHLYP functional: The mean absolute error (MAE) of 0.66 ppm for the scaled values was achieved for the range of about 20 ppm. Solvent, vibrational, and relativistic corrections were found to be rather small, especially when taken in combination, generally demonstrating a slight decrease in the difference between calculated and experimental fluorine chemical shifts. As a measure of the practical importance of these compounds, one should recall that the growing number of life science products that contain trifluoromethyl groups provides a continuing driving force for the development of an effective methodology that enables both regio- and stereoselective introduction of trifluoromethyl groups into both aliphatic and aromatic systems.

On the Utmost Importance of the Basis Set Choice for the Calculations of the Relativistic Corrections to NMR Shielding Constants

The investigation of the sensitivity of the relativistic corrections to the NMR shielding constants (σ) to the configuration of angular spaces of the basis sets used on the atoms of interest was carried out within the four-component density functional theory (DFT). Both types of relativistic effects were considered, namely the so-called heavy atom on light atom and heavy atom on heavy atom effects, though the main attention was paid to the former. As a main result, it was found that the dependence of the relativistic corrections to σ of light nuclei (exemplified here by 1H and 13C) located in close vicinity to a heavy atom (exemplified here by In, Sn, Sb, Te, and I) on the basis set used on the light spectator atom was very much in common with that of the Fermi-contact contribution to the corresponding nonrelativistic spin-spin coupling constant (J). In general, it has been shown that the nonrelativistic J-oriented and σ-oriented basis sets, artificially saturated in the tight s-region, provided much better accuracy than the standard nonrelativistic σ-oriented basis sets when calculating the relativistic corrections to the NMR shielding constants of light nuclei at the relativistic four-component level of the DFT theory.

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 (σ^PV and M^PV, 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 σ^PV and M^PV 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 σ^PV and M^PV 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 σ^PV and M^PV 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.

Recent advances in the liquid-phase 6,7 Li nuclear magnetic resonance

The present review is focused on experimental methods and structural applications, including computational aspects, of classical lithium liquid-phase nuclear magnetic resonance (NMR). It consists of four parts covering accordingly a brief overview, early experimental reports (papers of up to about 2015) and more recent (papers appearing in the interim of about 2015 until 2022) results, together with very few but highly prospective computational results.

The LRESC-Loc Model to Analyze Magnetic Shieldings with Localized Molecular Orbitals

The leading electronic mechanisms of relativistic effects in the NMR magnetic shieldings of heavy-atom (HA) containing molecules are well described by the linear response with elimination of small components model (LRESC). We show here first results from a new version of the LRESC model written in terms of localized molecular orbitals (LMOs) which is coined as LRESC-Loc. Those LMOs resemble "chemist's orbitals", representing lone-pairs, atomic cores, and bonds. The whole set of relativistic effects are expressed in terms of non-ligand-dependent and ligand-dependent contributions. We show the electronic origin of trends and behavior of different mechanisms in molecular systems which contain heavy elements that belong to any of the IB to VIIA groups of the periodic table. The SO mechanism has a well-defined dependence with the LPs (LPσ and LPπ) when the HAs have them, but the non-SO mechanisms mostly depend on other LMOs. In addition we propose here that the SO mechanism can be used to characterize interactions involving LPs and the non-SO mechanisms to characterize covalent and close-shell interactions. All our main results are in accord with previous findings, though we are now able to analyze them in a different manner.

Computational 199 Hg NMR

Theoretical background and fundamental results dealing with the computation of mercury chemical shifts and spin-spin coupling constants are reviewed with a special emphasis on their stereochemical behavior and applications.

Computational 1 H and 13 C NMR in structural and stereochemical studies

Present review outlines the advances and perspectives of computational ¹ H and ¹³ C NMR applied to the stereochemical studies of inorganic, organic, and bioorganic compounds, involving in particular natural products, carbohydrates, and carbonium ions. The first part of the review briefly outlines theoretical background of the modern computational methods applied to the calculation of chemical shifts and spin-spin coupling constants at the DFT and the non-empirical levels. The second part of the review deals with the achievements of the computational ¹ H and ¹³ C NMR in the stereochemical investigation of a variety of inorganic, organic, and bioorganic compounds, providing in an abridged form the material partly discussed by the author in a series of parent reviews. Major attention is focused herewith on the publications of the recent years, which were not reviewed elsewhere.

Relativistic and QED corrections to one-bond indirect nuclear spin-spin couplings in X$_2^{2+}$ and X$_3^{2+}$ ions (X = Zn, Cd, Hg)

The indirect spin-spin coupling tensor, $\bm J$, between mercury nuclei in systems containing this element can be of the order of few kHz and one of the largest measured. We analysed the physics behind the electronic mechanisms that contribute to the one- and two-bond couplings $^n \bm{J}_{\mathrm{Hg}-\mathrm{Hg}}$ ($n=1, 2$). For doing so, we performed calculations for $J$-couplings in the ionized $X_2^{2+}$ and $X_3^{2+}$ linear molecules ($X$ = Zn, Cd, Hg) within polarization propagator theory, using the random phase approximation and the pure zeroth--order approximation with Dirac--Hartree--Fock and Dirac--Kohn--Sham orbitals, both at four-component and ZORA levels. We show that the ``paramagnetic-like' mechanism contribute with more than 99.98\% to the total isotropic value of the coupling tensor. By analyzing the molecular and atomic orbitals involved in the total value of the response function, we find that the $s$-type valence atomic orbitals have a predominant role in the description of the coupling. This fact allows us to develop an effective model from which QED effects on $J$-couplings in the aforementioned ions can be estimated. Those effects were found to be within the interval $(0.7;~1.7)$\% of the total relativistic effect on isotropic one-bond $^1\bm{J}$ coupling, though ranging those corrections between the interval $(-0.4;~-0.2)$\% in Zn-containing ions, to $(-1.2;~-0.8)$\% in Hg-containing ions, of the total isotropic coupling constant in the studied systems. The estimated QED corrections show a visible dependence on the nuclear charge $Z$ of each atom $X$ in the form of a power-law proportional to $Z_X^5$.

Quantum Chemical Approaches to the Calculation of NMR Parameters: From Fundamentals to Recent Advances

Quantum chemical methods for the calculation of indirect NMR spin–spin coupling constants and chemical shifts are always in progress. They never stay the same due to permanently developing computational facilities, which open new perspectives and create new challenges every now and then. This review starts from the fundamentals of the nonrelativistic and relativistic theory of nuclear magnetic resonance parameters, and gradually moves towards the discussion of the most popular common and newly developed methodologies for quantum chemical modeling of NMR spectra.

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%).

Computational NMR of charged systems

This review covers NMR computational aspects of charged systems-carbocations, heterocations, and heteroanions, which were extensively studied in a number of laboratories worldwide, first of all, at the Loker Hydrocarbon Research Institute in California directed for several decades by a distinguished scientist, the Nobel laureate George Andrew Olah. The first part of the review briefly outlines computational background of the modern theoretical methods applied to the calculation of chemical shifts and spin-spin coupling constants at the DFT and the non-empirical levels. The second part of the review deals with the historical results, advances, and perspectives of the computational NMR of classical carbocations like methyl cation, CH₃⁺ , and protonated methane, CH₅⁺ , together with their numerous homologs and derivatives. The third and the forth parts of this survey are focused on the NMR computational aspects of accordingly, heterocations and heteroanions, the organic and inorganic ions with a charge localized mainly on heteroatoms like boron, oxygen, nitrogen, and heavier elements.

Relativistic study of parity-violating nuclear spin-rotation tensors

We present a four-component relativistic approach to describe the effects of the nuclear spin-dependent parity-violating (PV) weak nuclear forces on nuclear spin-rotation (NSR) tensors. The formalism is derived within the four-component polarization propagator theory based on the Dirac-Coulomb Hamiltonian. Such calculations are important for planning and interpretation of possible future experiments aimed at stringent tests of the standard model through the observation of PV effects in NSR spectroscopy. An exploratory application of this theory to the chiral molecules H₂X₂ (X = ¹⁷O, ³³S, ⁷⁷Se, ¹²⁵Te, and ²⁰⁹Po) illustrates the dramatic effect of relativity on these contributions. In particular, spin-free and spin-orbit effects are even of opposite signs for some dihedral angles, and the latter fully dominate for the heavier nuclei. Relativistic four-component calculations of isotropic nuclear spin-rotation constants, including parity-violating electroweak interactions, give frequency differences of up to 4.2 mHz between the H₂Po₂ enantiomers; on the nonrelativistic level of theory, this energy difference is 0.1 mHz only.

Computational NMR of Carbohydrates: Theoretical Background, Applications, and Perspectives

This review is written amid a marked progress in the calculation of NMR parameters of carbohydrates substantiated by a vast amount of experimental data coming from several laboratories worldwide. By no means are we trying to cover in the present compilation a huge amount of all available data. The main idea of the present review was only to outline general trends and perspectives in this dynamically developing area on the background of a marked progress in theoretical and computational NMR. Presented material is arranged in three basic sections: (1)-a brief theoretical introduction; (2)-applications and perspectives in computational NMR of monosaccharides; and (3)-calculation of NMR chemical shifts and spin-spin coupling constants of di- and polysaccharides.

Quantum chemical calculations of 77 Se and 125 Te nuclear magnetic resonance spectral parameters and their structural applications

An accurate quantum chemical (QC) modeling of ⁷⁷ Se and ¹²⁵ Te nuclear magnetic resonance (NMR) spectra is deeply involved in the NMR structural assignment for selenium and tellurium compounds that are of utmost importance both in organic and inorganic chemistry nowadays due to their huge application potential in many fields, like biology, medicine, and metallurgy. The main interest of this review is focused on the progress in QC computations of ⁷⁷ Se and ¹²⁵ Te NMR chemical shifts and indirect spin-spin coupling constants involving these nuclei. Different computational methodologies that have been used to simulate the NMR spectra of selenium and tellurium compounds since the middle of the 1990s are discussed with a strong emphasis on their accuracy. A special accent is placed on the calculations resorting to the relativistic methodologies, because taking into account the relativistic effects appreciably influences the precision of NMR calculations of selenium and, especially, tellurium compounds. Stereochemical applications of quantum chemical calculations of ⁷⁷ Se and ¹²⁵ Te NMR parameters are discussed so as to exemplify the importance of integrated approach of experimental and computational NMR techniques.

Recent advances in computational liquid-phase 77Se NMR

This review aims to highlight significant progress in the calculation of 77Se NMR chemical shifts and spin – spin coupling constants involving selenium substantiated with a vast amount of experimental data. The material is arranged in two basic sections: the first one dealing with the calculation of 77Se NMR chemical shifts and the second one dealing with the computation of spin – spin coupling constants involving 77Se nucleus, namely 77Se–1H, 77Se–13C and 77Se–77Se together with some more exotic types of couplings, 77Se – 15N, 77Se–19F, 77Se–29Si and 77Se–31P. A special attention is focused on the stereoelectronic effects involving selenium atom and their manifestation in the 77Se NMR spectra of organoselenium compounds studied with the aid of the modern calculation of 77Se NMR parametres in combination with experimental results.

The bibliography includes 114 references.

On the quantum origin of few response properties

In modern physics, the entanglement between quantum states is a well-established phenomenon. Going one step forward, one can conjecture the likely existence of an entanglement between excitations of one-particle quantum states. Working with a density matrix that is well defined within the polarization propagator formalism, together with information theory, we found that the quantum origin of, at least, few molecular response properties can be described by the entanglement between two pairs of virtual excitations of molecular orbitals (MOs). With our model, we are able to bring new insights into the electronic mechanisms that are behind the transmission, and communication, of the effects of a given perturbation to the whole electronic system described by the Hamiltonian of an unperturbed quantum system. With our entanglement model, we analyzed the electronic origin of the Karplus rule of nuclear magnetic resonance spectroscopy, a well-known empirical phenomenon, and found that this rule is straightforwardly related to the behavior of entangled MO excitations. The model compound used to show it is the H₂O₂ molecule.

Recent advances in computational 31 P NMR: Part 1. Chemical shifts

This is the first part of two closely related reviews dealing with the computation of phosphorus-31 nuclear magnetic resonance chemical shifts in a wide series of organophosphorus compounds including complexes, clusters, and bioorganic phosphorus compounds. In particular, the analysis of the accuracy factors, such as substitution effects, solvent effects, vibrational corrections, and relativistic effects, is presented. This review is dedicated to the Full Member of the Russian Academy of Sciences Professor Boris A. Trofimov in view of his invaluable contribution to the field of synthesis, nuclear magnetic resonance, and computation studies of organophosphorus compounds.

Recent advances in computational 31 P NMR: Part 2. Spin-spin coupling constants

This is the second part of two closely related reviews dealing with the computation of ³¹ P nuclear magnetic resonance (NMR) parameters in a wide range of phosphorous containing compounds. The first part of this review concentrated primarily on the computation of ³¹ P NMR chemical shifts, whereas the second part concerns the calculation of spin-spin coupling constants involving phosphorus nucleus, focusing primarily on their stereochemical dependencies and stereodynamic behavior in particular classes of organophosphorus compounds. This review is dedicated to the Full Member of the Russian Academy of Sciences Professor Boris A. Trofimov in view of his invaluable contribution to the field of synthesis, NMR, and computation studies of organophosphorus compounds.

Performance of the LRESC Model on top of DFT Functionals for Relativistic NMR Shielding Calculations

The linear response within the elimination of the small component model (LRESC) is an insightful and computationally efficient method for including relativistic effects on molecular properties like the nuclear magnetic shielding constants, spin-rotation constant, g-tensor, and electric field gradient of heavy atom containing molecules with atoms belonging up to the sixth row of the periodic table. One of its main advantages is its capacity to analyze the electronic origin of the different relativistic correcting terms. Until now, it was always applied on top of Hartree-Fock ground-state wave functions (LRESC/HF) to calculate and analyze NMR shieldings. In this work, we show the performance of the LRESC formalism on top of some density functional theory (DFT) functionals to compute tin shielding constants in SnX₄ (X = H, F, Cl, Br, I) molecular systems. We analyze the performance of each LRESC/DFT scheme on reproducing the electronic mechanisms of the shieldings, taking as a benchmark the results of relativistic calculations at the RPA level of approach (4c/RPA). As in previous works, we divide the LRESC relativistic correcting terms into two groups: core-dependent and ligand-dependent contributions. It is shown here that core-dependent corrections are well-reproduced for the selected DFT functionals, but some differences arise in the ligand-dependent ones. We focus on the performance of different functionals, including the same electron correlation part but containing different amounts of HF exchange. The best results are obtained for the BHandHLYP functional (50% of HF exchange) and the worst for BLYP (0%). When the percentage of HF exchange increases, ligand-dependent contributions are better described, and the final LRESC/DFT results are closer to those obtained with LRESC/HF and 4c/RPA methods. The spin-orbit correction to the shielding constant is one of the main ligand-dependent contributions (there are two more) with total value depending on the amount of HF exchange included in the functional. When the amount of HF exchange decreases, the spin-orbit contribution becomes larger, overestimating the shielding constant even when nonrelativisitc DFT values are much smaller than the nonrelativistic HF ones, as it happens for the heaviest molecular system studied here (SnI₄).

Computational 1 H NMR: Part 3. Biochemical studies

This is the third and the last part of three closely interrelated reviews dealing with computation of ¹ H nuclear magnetic resonance chemical shifts and ¹ H-¹ H spin-spin coupling constants. Present review deals with the computation of these parameters in biologically active natural products, carbohydrates, and other molecules of biological origin focusing on stereochemical applications of computational ¹ H nuclear magnetic resonance to these objects.

Computational 1 H NMR: Part 1. Theoretical background

This is the first one of the three closely interrelated reviews to be published in Magnetic Resonance in Chemistry dealing with accordingly theoretical background, chemical applications, and biochemical studies of and by means of computational ¹ H NMR. Presented in the first part of the review is a general outline of the modern theoretical methods and accuracy factors of computational ¹ H NMR involving locally dense basis set schemes, solvent effects, vibrational corrections, and relativistic effects performed at the density functional theory and/or nonempirical levels. This review is dedicated to Prof. Stephan Sauer in view of his invaluable contribution to the field of computational nuclear magnetic resonance.

Computational protocols for calculating 13C NMR chemical shifts

The most recent results dealing with the computation of ¹³C NMR chemical shifts in chemistry (small molecules, saturated, unsaturated and aromatic compounds, heterocycles, functional derivatives, coordination complexes, carbocations, and natural products) are reviewed, paying special attention to theoretical background and accuracy, the latter involving solvent effects, vibrational corrections, and relativistic effects.

Relativistic and QED effects on NMR magnetic shielding constant of neutral and ionized atoms and diatomic molecules

We show here results of four-component calculations of nuclear magnetic resonance σ for atoms with 10 ≤ Z ≤ 86 and their ions, within the polarization propagator formalism at its random phase level of approach, and the first estimation of quantum electrodynamic (QED) effects and Breit interactions of those atomic systems by using two theoretical effective models. We also show QED corrections to σ(X) in simple diatomic HX and X₂ (X = Br, I, At) molecules. We found that the Z dependence of QED corrections in bound-state many-electron systems is proportional to Z⁵, which is higher than its dependence in H-like systems. The analysis of relativistic ee (or paramagneticlike) and pp (or diamagneticlike) terms of σ exposes two different patterns: the pp contribution arises from virtual electron-positron pair creation/annihilation and the ee contribution is mainly given by 1s → ns and 2s → ns excitations. The QED effects on shieldings have a negative sign, and their magnitude is larger than 1% of the relativistic effects for high-Z atoms such as Hg and Rn, and up to 0.6% of its total four-component value for neutral Rn. Furthermore, percentual contributions of QED effects to the total shielding are larger for ionized than for neutral atoms. In a molecule, the contribution of QED effects to σ(X) is determined by its highest-Z atoms, being up to -0.6% of its total σ value for astatine compounds. It is found that QED effects grow faster than relativistic effects with Z.

Polarization propagator theory and the entanglement between MO excitations

Entanglement is at the core of quantum physics and so, one may conjecture that it should have some influence on atomic and molecular response properties. The usual way of treating entanglement is by applying information theory via the von Newman entropy. Given that the principal propagator is the operator that contains the physical information that arises due to the transmission of the effects of two external perturbations through the electronic framework of a quantum system, it should have in it the information necessary to quantify the likely entanglement among molecular orbital excitations. In this article we first propose a proper density matrix and from it, the way to quantify entangled excitations by using information theory. The NMR J-couplings are among the best candidates to learn about the potentialities of this formalism. We applied this new tool to analyze the famous Karplus rule and found a relationship between the dihedral angular dependence and the entanglement. We also found that the entangled excitations are related to electron correlation. The new formalism can be applied to all other response properties.

Theoretical calculations of carbon-hydrogen spin-spin coupling constants

Structural applications of theoretical calculations of carbon-hydrogen spin-spin coupling constants are reviewed covering papers published mainly during the last 10-15 years with a special emphasis on the most notable studies of hybridization, substitution and stereoelectronic effects together with the investigation of hydrogen bonding and intermolecular interactions. The wide scope of different applications of calculated carbon-hydrogen couplings in the structural elucidation of particular classes of organic and bioorganic molecules is reviewed, concentrating mainly on saturated, unsaturated, aromatic and heteroaromatic compounds and their functional derivatives, as well as on natural compounds and carbohydrates. The review is dedicated to Professor Emeritus Michael Barfield in view of his invaluable pioneering contribution to this field.

Influence of the nuclear charge distribution and electron correlation effects on magnetic shieldings and spin-rotation tensors of linear molecules

The nuclear charge distribution effects (NChDE) on two response properties, the NMR magnetic shielding ( σ ) and the nuclear spin-rotation (SR) constants ( M ), are analyzed. We do it employing point-like and Gaussian-like models for describing the nuclear charge density of three linear molecules: HBr, HI and HAt. According to our results, both properties are sensitive to the NChDE. We show that the NChDE are almost completely relativistic, i.e., they nearly vanish in the non-relativistic limit of both properties. We calculated the NChDE on σ and M , and analyzed the differences between them in terms of a relativistic relation between these two properties. Using that relation we found that the electronic core mechanisms are the main ones for the NChDE on the shielding of nuclei of both, molecules and free atoms. The NChDE are smaller on SR constants than on shieldings. Nevertheless, within the relativistic polarization propagator formalism at the RPA level of approach they are very important for SR constants of nuclei in heavy-atom-containing compounds. Astatine in HAt has the largest influence: M At = -9.95 kHz for a point-like model and -50.10 kHz for a Gaussian-like model. Correlation effects must be included and we do it using different DFT schemes. The PBE0 functional gives results that are closest to experiments for Br and I, though the LDA gives the closest for hydrogen. The value of the SR constant of At is reduced among 350 kHz and 500 kHz from its RPA value, when different and usual functionals are applied. Given that the NChDE on M and σ are mostly relativistic in their origin, these effects are also dependent on electron correlation. They have also a nonvanishing dependence with the Gaunt electron-electron interactions.

Carbon-carbon spin-spin coupling constants: Practical applications of theoretical calculations

Practical applications of theoretical calculations of carbon-carbon spin-spin coupling constants in particular classes of organic and bioorganic molecules are reviewed, concentrating mainly on saturated, unsaturated, aromatic and heteroaromatic compounds and their functional derivatives as well as on carbohydrates and natural compounds.

Role of Spin-Dependent Terms in the Relationship among Nuclear Spin-Rotation and NMR Magnetic Shielding Tensors

The broadly accepted procedure to obtain the experimental absolute scale of NMR magnetic shieldings, σ, is well-known for nonheavy atom-containing molecules. It was uncovered more than 40 years ago by the works of Ramsey and Flygare. They found a quite accurate relationship among σ and the nuclear spin-rotation constants. Its relativistic extension was very recently proposed, although it has an intrinsic weakness because a new SO-S two-component term needs to be considered. We show how to overcome this problem. We found that (ν_Y^S - ν_Y^atom,S) generalizes the SO-S term, where ν_Y^S = ⟨⟨[((r - r_Y) × α)/(|r - r_Y|³)]; S⁽⁴⁾⟩⟩, r - r_Y is the electron position with respect to the position of nucleus Y, and S⁽⁴⁾ is the four-component total electron spin. When including this new term, one finds that the best of our relativistic Flygare-like models fits quite well with the results of the most accurate method available at the moment. We also show that the difference among the parallel component of σ(Xe) in XeF₂ and σ(Xe) of the free atom is almost completely described by that new term.

Microsolvation of methylmercury: structures, energies, bonding and NMR constants (199Hg,13C and17O)

Hartree–Fock (HF) and second order perturbation theory (MP2) calculations within the scalar and full relativistic frames were carried out in order to determine the equilibrium geometries and interaction energies between cationic methylmercury (CH₃Hg⁺) and up to three water molecules.

Toward an absolute NMR shielding scale using the spin-rotation tensor within a relativistic framework

How can one extend Flygare's rule to the relativistic framework? Three models are proposed here. The best of them shows that σ is related with the spin-rotation tensor, the atomic shielding and a new term coined as σ^SO-S.

Theoretical analysis of NMR shieldings of group-11 metal halides on MX (M = Cu, Ag, Au; X = H, F, Cl, Br, I) molecular systems, and the appearance of quasi instabilities on AuF

Accurate calculations of nuclear magnetic shieldings of group-11 metal halides, σ(M; MX) (M = Cu, Ag, Au; X = H, F, Cl, Br, I), were performed with relativistic and nonrelativistic theoretical schemes in order to learn more about the importance of the involved electronic mechanisms that underlie such shieldings. We applied state of the art schemes: polarization propagators at a random phase level of approach (PP-RPA); spin-free Hamiltonian (SF); linear response elimination of small component (LRESC) and density functional theory (DFT) with two different functionals: B3LYP and PBE0. The results from DFT calculations are not close to those from the relativistic polarization propagator calculations at the RPA level of approach (RelPP-RPA), in line with previous results. The spin-orbit (SO) contribution to a shielding constant is important only for MF molecules (M = Cu, Ag, Au). Different electronic mechanisms are considered within the LRESC method, bunched into two groups: core- and ligand-dependent. For the analysed shieldings the core-dependent electronic mechanisms are the most important ones; the ligand-dependent being only important for MF molecules. An out of range value for σ(Au) is found in AuF. It was previously reported in the literature, either originated in the large fluorine electronegativity together with large spin-orbit coupling contributions; or, due to Fermi-contact contributions. We argue here that such an unexpected large value is an artifact originated in the appearance of quasi instabilities, and show how to handle this apparent problem.

Relativistic and electron-correlation effects on the nuclear magnetic resonance shieldings of molecules containing tin and lead atoms

The reference values for NMR magnetic shieldings, σ(ref), are of the highest importance when theoretical analysis of chemical shifts are envisaged. The fact that the nonrelativistically valid relationship among spin-rotation constants and magnetic shieldings is not any longer valid for heavy atoms requires that the search for σ(ref) for such atoms needs new strategies to follow. We present here results of σ(ref) that were obtained by applying our own simple procedure which mixes accurate experimental chemical shifts (δ) and theoretical magnetic shieldings (σ). We calculated σ(Sn) and σ(Pb) in a family of heavy-halogen-containing molecules. We found out that σ(ref)[Sn;Sn(CH3)4] in gas phase should be close to 3864.11 ± 20.05 ppm (0.5%). For Pb atom, σ(ref)[Pb;Pb(CH3)4] should be close to 14475.1 ± 500.7 ppm. Such theoretical values correspond to calculations with the relativistic polarization propagator method, RelPPA, at the RPA level of approach. They are closer to experimental values as compared to those obtained applying few different functionals such as PBE0, B3LYP, BLYP, BP86, KT2, and KT3 of the density functional theory, DFT. We studied tin and lead shieldings of the XY(4-n)Z(n) (X = Sn, Pb; Y, Z = H, F, Cl, Br, I) and PbH(4-n)I(n) (n = 0, 1, 2, 3, 4) family of compounds with four-component functionals as implemented in the DIRAC code. For these systems results of calculations with RelPPA-RPA are more reliable than DFT ones. We argue about why those DFT functionals must be modified in order to obtain more accurate results of NMR magnetic shieldings within the relativistic regime: first, there is a dependence among both electron-correlation and relativistic effects that should be introduced in some way in the functionals; and second, the DIRAC code uses standard nonrelativistic functionals and the functionals B3LYP and PBE0 were parametrized only with data taken from light elements. It can explain why they are not able to properly introduce relativistic effects on nuclear magnetic shieldings. We finally show that in the analysis of magnetic shieldings for the family of compounds mentioned above, one must consider the newest and so-called heavy-atom effect on vicinal heavy atoms, HAVHA. Such effects are among the most important relativistic effects in these kind of compounds.

Relativistic calculations of magnetic resonance parameters: background and some recent developments

This article outlines some basic concepts of relativistic quantum chemistry and recent developments of relativistic methods for the calculation of the molecular properties that define the basic parameters of magnetic resonance spectroscopic techniques, i.e. nuclear magnetic resonance shielding, indirect nuclear spin-spin coupling and electric field gradients (nuclear quadrupole coupling), as well as with electron paramagnetic resonance g-factors and electron-nucleus hyperfine coupling. Density functional theory (DFT) has been very successful in molecular property calculations, despite a number of problems related to approximations in the functionals. In particular, for heavy-element systems, the large electron count and the need for a relativistic treatment often render the application of correlated wave function ab initio methods impracticable. Selected applications of DFT in relativistic calculation of magnetic resonance parameters are reviewed.