 1 NMR – a short overview
 2 Theoretical nonrelativistic (NR) and relativistic calculations of nuclear shielding
 3 Prediction of nuclear shielding and chemical shift in the gas phase and solution in chemistry and biology
 3.1 Chemical applications
 3.2 Biological applications
 4 Modeling nuclear magnetic shieldings for the solid state
 5 Calculation of nuclear shieldings in the presence of “exotic” halogen, chalcogen, pnicogen, and tetrel interactions
 Abbreviations
Theory and computation of nuclear shielding

Published:23 Nov 2020

Special Collection: 2020 ebook collection
T. Kupka, in Nuclear Magnetic Resonance
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A review of the literature published from January to December 2019 on theoretical aspects of nuclear magnetic shielding is presented. It covers both nonrelativistic and relativistic prediction of nuclear shielding at both DFT and ab initio levels of theory. Benchmark studies on small molecular systems, corrections due to solvent effect and rovibrational averaging, as well as experimental studies on absolute shielding scale determination are covered.
1 NMR – a short overview
As result of eight decades of tremendous hardware and software development, nuclear magnetic resonance (NMR) technique become a well established tool, indispensable in many fields of basic and applied science, and technology. On the other hand, understanding the complexity of molecular systems and phenomena studied using NMR has been constantly supported by theoretical modeling of observed magnetic parameters, including positions of spectral peaks and their fine structure. Thus, prediction of nuclear magnetic shieldings (σ) and indirect spin–spin coupling constants (SSCC) has become an important activity of theoretical chemists and physicists working close with experimentalists for the last five decades. This was nicely illustrated in reviews on computational prediction of ^{1}H and ^{13}C chemical shifts published in 2012 by Tantillo et al.,^{1 } and later by Ananikov^{2 } and Tormena.^{3 } From the theoretical side, modeling of small molecules with advanced method like CCSD(T)^{4,5 } and large basis set in the gas phase is significantly easier and more accurate than in the condensed phases, i.e. in the presence of solvent or of a crystal lattice.^{6,7 } However, the majority of NMR experiments are performed in solution. This short review attempts to cover the computational NMR studies in the gas phase, solution and solid state, reported in 2019.
2 Theoretical nonrelativistic (NR) and relativistic calculations of nuclear shielding
In the reporting period nonrelativistic nuclear magnetic shielding at Hartree–Fock, MP2 and DFT levels was mainly calculated using recent versions of the versatile and userfriendly Gaussian program.^{8–11 } More sophisticated coupled cluster calculations of nonrelativistic nuclear shieldings were performed with the CFOUR software.^{12 } Some calculations with Keal–Tozer KTn density functionals^{13–15 } were conducted in DALTON^{16 } and ADF.^{17 } The latter software also allowed the twocomponent relativistic (2c, scalar, SC and spin–orbit, SO) ZORA^{18,19 } calculations of nuclear shielding tensors. Both nonrelativistic and full four component (4c) relativistic HF and DFT calculations of nuclear shielding tensor were reported using DIRAC^{20 } and ReSpect^{21 } program packages.
Aucar et al.^{22 } continued research within quantum field theory^{23–25 } and relativistic effects. His numerous earlier works were widely reviewed, for example in some editions of Nuclear Magnetic Resonance.^{26,27 } Their recent work^{22 } concentrates on relativistic and quantum electrodynamic (QED) effects on nuclear shieldings of atoms, their ions and diatomics. They calculated relativistic effect on nuclear magnetic shieldings for atoms and their ions with atomic numbers, Z, between 10 and 86 using the fourcomponent approach within the polarization propagator formalism at its random phase level of approach, as well as reporting on the first estimation of QED effects and Breit interactions in those atomic systems using two theoretical successful models. They also analyzed two types of diatomics: X_{2} and HX, where X=Br, I and At. They noticed a faster increase of the QED effect (as Z^{5}) on nuclear shielding than the relativistic effect with growing Z. The authors reviewed in detail the theory of nuclear shielding and the computational issues and used the dyall.acv4z basis set^{28 } for Ne, Ar, Br, Kr, I, Xe, At, and Rn, and dyall.cv4z for Zn Cd and Hg in Dirac 2017.^{20 } They noticed that the QED effects contribution to relativistic effects varied from −0.10% for Ne to −1.13% for Rn.
In a review of polarization propagators and their physical (and mathematical) basis Aucar et al.^{29 } showed the history of their conceptual development, starting in Denmark with Jens Oddershede in early 1970s and 1980s,^{30–32 } and later on partly transferred to Argentina. The South American story of polarization propagators started in a group of late Ruben Contreras.^{33 } The development of initial theory provided new tools for both nuclear shieldings and SSCC calculation at NR and relativistic level of theory. In this reporting period Aucar and Aucar^{34 } reviewed developments in absolute shielding scales for NMR spectroscopy. At the beginning of NMR the basic parameter, the nuclear magnetic shielding tensor, σ, was derived from a combined experimental (microwave spectroscopy) and theoretical approach. Thus, the accuracy of this semiempirically obtained parameter was not so high and soon benchmark theoretical estimations allowed its prediction in the gas phase with higher accuracy.^{6,35 } At the beginning of this century problems with obtaining nuclear shieldings for heavier atoms (Xe or some metals) forced a search to find a way of including relativistic effects. Recently, experiments in the gas phase with varied density allowed its extrapolation to infinite zerodensity limit.^{36,37 } As result of breakthroughs in this field, there are currently three modern fourcomponent relativistic approaches: wave function, Density Functional Theory (DFT) and polarization propagator methods, as well as twocomponent and scalar methods.
Aucar et al.^{38 } also published a study on application of structural modeling, hydrogen bond interaction assessment and calculated NMR parameters (nuclear magnetic shieldings and SSCC) in studies of malonaldehyde and similar molecular systems with intramolecular hydrogen bonds.
The presence of heavy atoms can significantly deteriorate the accuracy of calculated neighboring light atom nuclear shieldings.^{39–41 } This socalled heavy atom on light atom relativistic effect is called the HALA^{41 } effect. Rusakova and Rusakov^{42 } reported on detailed theoretical analysis of the HALA effect in selected phosphine tellurides. They performed nonrelativistic geometry optimization at MP2/ATZP level of theory. Gaugeincluding atomic orbital^{43,44 } (GIAO) nuclear magnetic shielding constants were calculated with a fourcomponent relativistic Hamiltonian at the DFT KT1/dyall.av3z level of theory. The overall HALA effect results from socalled spinfree HALA effects due to scalar relativistic effects (massvelocity and Darwincorrection) and spinorbit Fermi contact interactions (SO/FC). The second effect, spinorbit (SO) dominate over scalar (SC) in the calculated relativistic corrections. The authors concluded that the onebond HALA effect depends on the spatial deformation of the heavy chalcogen atom lone electron pair. The calculated shielding constant is controlled by the spatial arrangement of light substituents on phosphorus, resulting in the deformation of the lone electron pairs of tellurium.
Catalysts often contain complexes of heavy metal ions with organic ligands, and due to their open shell electronic structure such molecular systems are very challenging for theoretical characterization. In addition, corrections due to relativistic effects originating from the presence of heavier nuclei in such systems are important. Squareplanar iridium catalysts were selected as model system, and carefully analyzed by Marek et al.^{45 } They demonstrate that NMR (and EPR) parameters are very sensitive magnetic probes for the electronic spin density and they could be used to test the space near the heavymetal atom ions, as well as the metal–ligand bonding. The authors used the recent implementation of the fully relativistic Dirac–Kohn–Sham (DKS) method with the hybrid PBE0 functional and an implicit solvent model for prediction of hyperfine NMR shifts. The change of metal–ligand bond character is reflected in the “longrange” throughbond Fermicontact (FC) contributions to the ligand ^{1}H and ^{13}C hyperfine couplings. In case of complexes formed by paramagnetic ions and organic ligands the corresponding ^{13}C and ^{1}H signals are significantly shifted with respect to free ligands due to nucleus–electron hyperfine interaction. Another observed effect of this coupling is broadening of ligand resonances due to fast nuclearspin relaxation. The authors used fourcomponent relativistic theory to predict the hyperfine coupling tensor, A, calculated as a sum of four contributions: the Fermicontact (FC), the spindipole (SD), the paramagnetic nuclearspin–electron–orbit (PSO), and the pure relativistic (REL) terms.
The structures were optimized at the DFT (UPBE0/def2ECP/def2TZVPP and the COnductorlike Screening MOdel,^{46 } COSMO, for the toluene solvent) level of theory and compared with crystal data. Next, the hyperfine coupling constants were calculated using a PBE0/2ζ in vacuum approach.
Mercury is a very toxic element and so its solvation in water is important from a practical point of view. Since this element is very heavy, both scalar and spin–orbit relativistic effects should be included in theoretical modeling of structural and spectroscopic properties of mercury compounds. ^{199}Hg is a spin ½ and fairly abundant (about 17%) nucleus, and its NMR spectrum covers a very wide window of chemical shifts (several thousands of ppm). As result, ^{199}Hg NMR spectroscopy is a sensitive experimental technique which can easily follow intermolecular interactions. Jakubowska and Pecul^{47 } studied the Hg⋯H_{2}O dimer using the coupled cluster method with singles, doubles, and perturbationally included triple excitations,^{4,48 } CCSD(T), and also using DFT methods. Initially they optimized the distance between oxygen and mercury using the coupled cluster method with single and double excitations,^{4,48 } CCSD and CCSD(T) methods, and a four component Hamiltonian.^{49 } The Dyall double zeta basis set^{28 } was applied for mercury and augccpVDZ for the remaining atoms. B3LYP, BLYP, B3P86, PBE0 and PBE38 density functionals with the same basis sets were used for fourcomponent relativistic calculations. Nonrelativistic geometry optimization was performed using DFT (B3LYP, PBE0 and CAMB3LYP) and CCSD methods with augccpVDZ basis set for water and MDF60^{50 } effective core potential for mercury. NMR parameters were calculated using fourcomponent Dirac–Kohn–Sham Hamiltonian and B3LYP hybrid density functional with upcJ3^{51 } basis set on H and O atoms and Dyall triple zeta basis set for Hg. It was observed that relativistic effects change the nuclear shieldings of water by 10% or less, but contribute about 40% to the total shielding of mercury. In addition, ^{1}J(OH) decreases with Hg moving further from water molecule.
Sixmembered phosphorus compounds are popular in organic chemistry and in practical applications. For example, cyclophosphazenes are used as flame retardants.^{52 } Alkorta et al.^{53 } optimized the geometry of 13 λ5phosphinines compounds at the B3LYP/6311++G** level of theory. Subsequent nonrelativistic GIAO calculations were performed at the same level of theory. Linear correlations between experimental and calculated ^{1}H, ^{13}C, ^{15}N and ^{19}F were observed and a new regression formula for ^{31}P nuclear shieldings was proposed. In case of heavier substituents (chlorine and bromine) the deviations of ^{13}C chemical shifts from experiment were about −7 and −30 ppm. The largest difference was observed for two bromine atoms (about −330 ppm originating from PBr_{2} structural fragment). Additional calculations with the twocomponent zeroth order regular approximation (ZORA)^{18,19 } Hamiltonian removed this discrepancy between theory and experiment. The HALA effect was only observed for Pipso atoms; the effect was negligible for the ortho ring position (for nitrogen atom).
The indirect spin–spin coupling (Jcoupling) mechanism is transmitted via electrons of chemical bonds within a molecule and depends on the lowest lying electrons, as well as electron density at the nuclei, although through space Jcoupling is also known. An interesting question was addressed by Saielli et al.^{54 } Inspired by the existence of through space spin–spin coupling they decided to look for a through space HALA effect. In other words, is the HALA effect present in the absence of a direct covalent bond between the heavy and light atoms? The authors selected several ionic liquids as models with chlorine, bromine and iodine anions and their geometry optimization was performed with the VoskoWilkNusair (VWN) local density functional^{55 } in conjunction with the Becke Exchange^{56 } and the Perdew correlation functionals^{57 } (BP86) with Grimme's^{58 } additive D3(BJ) term for dispersion and Becke–Johnson damping,^{59 } combined with a Slatertype TZ2P basis set.^{60–62 } The presence of solvent was modeled via the COSMO approach. Relativistic effects, modeling the impact of heavy atom on light atom (HALA), were included via SC and SOZORA calculations in ADF. For 1methylpyridinium chlorine, bromine and iodine the optimized HALA distances were 2.268 to 2.547 Å to 2.854 Å. Boltzmannaveraged contributions to ^{13}C and ^{1}H NMR chemical shifts were small but not negligible. In case of I_{3}, the carbon shifts changed by about 1 ppm.
Martin^{63 } introduced a series of three important review papers on theoretical calculations of ^{1}H nuclear magnetic shieldings by Leonid Krivdin.^{64 } In the first review article by Krivdin^{64,65 } the importance of theoretical prediction of ^{1}H nuclear magnetic shieldings was demonstrated. At the beginning, a selection of methods and basis sets for efficient calculation of ^{1}H nuclear shieldings was given. Both ab initio and DFT methods for GIAO calculation of nuclear shieldings were briefly introduced. Recent computational works on accurate calculations of ^{1}H nuclear magnetic parameters were reviewed. These included both gas phase and solution studies, as well as complete basis set limit (CBS) estimations using the locally dense basis set^{66,67 } (LDBS) approach for accurate prediction of NMR parameters in larger molecules using the socalled “raisins in a dough model”. In the latter approach the selected atoms are calculated with large basis sets and the rest of molecule is treated with relatively small and inexpensive ones. This model resembles Morokuma's ONIOM method.^{68 } In addition, Krivdin discussed solvent effects, zeropoint vibration corrections (ZPVCs) and relativistic effects. The latter effects were demonstrated on model compounds containing selenium or tellurium.
Phosphorus mononitride is a small diatomic molecule. Interestingly, it was recently discovered to be the first molecule containing phosphorus in dense clouds of interstellar space, as well as in the atmospheres of Jupiter and Saturn.^{69 } Kaminský et al.^{70 } analyzed the convergence of ^{15}N and ^{31}P nuclear magnetic shieldings in phosphorus mononitride, calculated with the simplest theory without electron correlation, HF, as well as the advanced CCSD(T) method with several series of very good quality basis sets, including correlationconsistent (Dunningtype) augccpVXZ, augccpV(X+d)Z and augccpCVXZ ones. As expected, a fairly regular and smooth convergence of ^{15}N nuclear shieldings toward the complete basis set limit (CBS) was observed for both HF and CCSD(T) methods combined with all basis set families. In contrast, ^{31}P shieldings calculated with augccpVXZ and augccpV(X+d)Z families of basis sets showed a very irregular behavior with scatter of data of 80 ppm! This erratic behavior as result of changing X was markedly improved by applying basis sets recovering corevalence effects (augccpCVXZ). This study shows the importance of proper selection of basis set for accurate calculation of phosphorus nuclear shieldings.
Routine NMR studies for chemical applications rely on chemical shifts (δ), mainly of proton and carbon measured with respect to tetramethylsilane (TMS) as an internal reference in nonaqueous solvents. Thus, relative chemical shifts are the basic parameters reported by chemists as the results of NMR experiments. These parameters can be converted to nuclear magnetic shieldings (σ) using the simple formula: δ_{i}=σ_{ref}−σ_{i}. However, each measured nucleus needs different reference compounds. Jackowski et al.^{71,72 } strongly encouraged the use of ^{3}He as an universal reference of nuclear magnetic shielding in NMR. This could be achieved by using deuterium signal of deuterated solvent as secondary reference. Recently, Garbacz and Jackowski^{73 } reported on direct measurements of ^{1}H, ^{2}H and ^{13}C NMR shieldings performed for 13 deuterated liquid solvents using ^{3}He as the primary reference. They reported ^{1}H resonance frequencies, chemical shifts and magnetic shielding values for residual hydrogen atoms in 27 liquid deuterated solvents, as well as ^{13}C resonance frequencies, carbon chemical shifts and magnetic shielding of ^{13}C nuclei in 31 liquid deuterated solvents. They also measured ^{2}H and ^{3}He resonance frequencies, chemical shifts, the reference standard values and primary isotope effects on shielding for 29 liquid deuterated solvents. To check the proposed approach, they measured proton shieldings of 2butanone and 2butanol using the deuterium lock signal originating from perdeuterated benzene, chloroform, dioxane and tetrachloroethane as secondary reference. Thus, it should be stressed that their experimental approach allows for a direct comparison of theoretically predicted nuclear magnetic shieldings with experimental parameters directly measured in solution.
Typically, NMR studies are performed for molecular systems in their ground state and theoretical calculations are used to predict NMR parameters in such molecular systems. An interesting theoretical study was reported by Atsumi^{74 } on excited state ^{13}C nuclear magnetic shielding constants for singlet states in CH_{2}CCH_{2}, CH_{2}O, CH_{3}CHO, CH_{3}NH_{2}, and CO. The ground state geometries were optimized at the coupled cluster level with single, double, and perturbative triple excitations^{75 } (CCSD(T)) and the 6311++G(2d,2p) basis set. The NMR spectroscopic parameters of the ground state and the singlet excited states were calculated at the statespecific complete active space selfconsistent field (SSCASSCF) level of theory.^{76 } To account for static electron correlation in excited states, Atsumi applied the multiconfigurational selfconsistent field (MCSCF) method. According to Ramsey,^{77 } the nuclear shielding can be decomposed into dia and para magnetic terms (σ_{d} and σ_{p}). With one exception (360 ppm for the aldehyde carbon in CH_{3}CHO), the ^{13}C diamagnetic term for all compounds was very similar (300±20 ppm) in both the excited and ground state, while the paramagnetic term was different. For CH_{2}O, CH_{3}CHO, CH_{3}NH_{2} and CO the n→π* transitions were important and had an impact on the paramagnetic term. The calculated ^{13}C shielding constants in their excited states were higher than in their ground states. The author observed an approximately linear relationship between the inverse gap to the ground state (ΔE=E_{1}−E_{o}) and σ_{p}.
3 Prediction of nuclear shielding and chemical shift in the gas phase and solution in chemistry and biology
3.1 Chemical applications
Molecular modeling of nuclear shieldings and chemical shifts has been used by numerous research groups. However, there is some danger in using the software and methodology as a black box without understanding some essentials of the approach and meaning of the calculated numbers. On the other hand, there is a wellknown tradeoff between the accuracy of predicted nuclear magnetic shieldings^{35,78 } and the calculation time required. Thus, one needs to balance the required accuracy and available computer speed and storage space. Hartree–Fock (HF) calculations, which do not include electron correlation, scale with the number of basis functions N as N^{4}, MP2 (the simplest ab initio method including electron correlation) as N^{5} and CCSD(T), considered the gold standard method of theoretical calculations, as N^{7}. Density functional theory includes electron correlation but is somewhat semiempirical in its nature. Calculation with density functionals are faster than MP2 and the produced results are of similar quality. Thus, an endless “available set” of density functionals offers a cheaper way of predicting various molecular and spectroscopic parameters but their performance is hard to control and they work in a somewhat unpredictable way. Including electron correlation via MP2 in doublehybrid density functionals^{79 } (DHDF) leads to more accurate prediction at additional cost. Resolution of identity^{80 } (RI) for MP2 and CCSD(T) methods allow significant shortening of CPU time without loss of performance. Taking into account the importance of speeding up very accurate prediction of nuclear magnetic shielding, the work by Neese et al.^{81 } published in 2018, is also included in this review. The authors implemented analytic calculation of GIAO shielding tensors for DHDF with inclusion of the MP2RI approximation. The proposed approach was tested on 15 molecules containing ^{1}H, ^{13}C, ^{15}N, ^{17}O, ^{19}F, and ^{31}P nuclei. The performance of the modified density functionals was compared with HF, MP2, pure and hybrid functionals, as well as the CCSD(T) method. The best results were observed for the DSDPBEP86 doublehybrid functional (1.9% mean absolute error from benchmark data). This accuracy is better than for MP2 (4.1%) or M062X (5.4%). DHDF can be routinely applied to molecules containing 100–400 electrons, and are 1–2 orders of magnitude slower than standard density functionals.
Fullerenes^{82 } have been very popular molecular systems for both basic and applied research for the last three and half decades. ^{13}C, as well as ^{3}He and ^{129}Xe NMR studies successfully helped with determining the size and symmetry of fullerenes.^{83–87 } However, differentiation of various disubstituted fullerenes is experimentally very challenging.^{88 } Tulyabaev and Khalilov^{89 } successfully applied DFT calculations at relatively low level of theory, X3LYP/631G, for both geometry optimization and GIAO calculations to distinguish fullerene C_{60} biscycloadducts. They analyzed the results and reported underestimated theoretical ^{13}C nuclear shieldings in comparison to experimental values for sp^{2} hybridized carbon atoms. Halogenated fullerenes are more reactive than their parent carbon structures and therefore they could be used as chemical subtracts for further derivatization. Lee et al.^{90 } reported on theoretical prediction of structural and NMR nuclear shieldings for fullerene and bromofullerene. They modeled fullerene C_{60} and its bromo derivative C_{60}Br_{6} using B3LYP/6311G** methodology. GIAO NMR calculations were conducted at lower level of theory (B3LYP/631G) in ethanol solution, modeled by polarized conductorlike model of solvent^{91–93 } (the PCM approach) with TMS as reference. A single resonance peak was predicted for initial fullerene at about 150 ppm and its bromoderivative was characterized by two groups of resonances, in the range of 41 to 44 ppm and from 132 to 188 ppm.
Aromaticity with its multidimensional description is a very popular basic term in both organic and inorganic chemistry. One of widely used aromaticity indexes is a magnetic one – the nucleus independent chemical shift (NICS), introduced by Schleyer.^{94,95 } Giving a familiar picture of benzene shielding and deshielding cone shaped space around benzene molecule, it helps to explain position of the ^{1}H and ^{13}C NMR signals.^{96,97 } Interestingly, the largest deshielding of ^{3}He NMR signal is observed in C$60\u2212$ cage: the corresponding ^{3}He NMR signal appears at −48.7 ppm. In neutral fullerene this signal is significantly less shielded and is located at −6.3 ppm.^{83,84,98,99 } MuñozCastro and Gonzales^{100 } optimized C_{60} and C_{70} fullerenes with the BP86 density functional combined with the Slatertype TZ2P basis set^{101 } (using the ADF program) and calculated nuclear magnetic shielding constants of ^{3}He inside the cages using the GIAO OPBE/TZ2P approach, referenced with respect to free helium atom at 59.94 ppm. Their calculated chemical shifts reproduced well experimental values and earlier theoretical results.^{102 }
In contrast to planar aromatic hydrocarbons (PAHs), nonplanar (curved) molecules, such as carbon nanobelts (CNBs), are still very challenging molecules for chemical synthesis. Such systems show very interesting physical and electronic properties, related to their size and shape. Wu et al.^{103 } reported on the synthesis, physical characterization, and supramolecular properties of two fully conjugated bowlshaped carbon nanobelts. DFT calculations, in agreement with single crystal Xray data, predicted a nonplanar structure, with inner diameter of about 10 and 14 Å and a closedshell nature. The recorded ^{1}H NMR spectra showed sharp resonances in deuterated THF and toluene. An openshell singlet ground state of the smaller belt was deduced from broadened ^{1}H NMR signals at room temperature. This agreed with the results of unrestricted DFT CAMB3LYP/631G** calculations. NICS calculations were performed to understand global (anti)aromaticity and anisotropy of the induced current density. In case of the dication CNB1^{2+} a negative NICS(0) value of −10.3 ppm for the macrocycle center was predicted. The tetracation of the second nanobelt with 124 πconjugated electrons showed a global antiaromatic character (NICS(0)=+16.97 ppm). The authors demonstrated experimentally from ^{1}H NMR signal the changes of their binding properties with respect to C_{70} fullerene.
A novel approach to presenting the current density induced by magnetic field, perpendicular to the plane of planar ring molecules, was demonstrated for benzene and cyclopropane, chosen as archetypes of π and σaromatic molecules.^{104 }
Porphyrinoids are polycyclic structures, related to porphyrins, and of practical interest in organic electronic devices as well as in photodynamic therapy (PDT). Elucidation of their structure in solution heavily relays on ^{1}H NMR spectroscopy. In addition, the electronic structure of porphyrinoids makes them good candidates for molecular aromatic/antiaromatic switches. Theoretical prediction of ^{1}H shieldings could assist assignment of their experimental spectra and verify their (non/anti)aromatic character. Structures of selected new porphyrinoids were optimized by Agren et al.^{105 } at the B3LYP/631G* level of theory and next GIAO and CSGT methods of calculating nuclear magnetic shieldings were applied using B3LYP/6311++G** approach, modelling the presence of chloroform via PCM. Magnetically induced current densities and current strengths were calculated at the B3LYP/def2TZVP level of theory. In order to improve agreement between predicted and experimental proton chemical shifts, the authors divided signals into CH and NH groups and used separate linear regression analysis. Very little change upon including the solvent effect via the PCM model was observed. The authors also observed a better performance of B3LYP in comparison to M062X for predicting the ^{1}H NMR chemical shifts.
Reliable prediction of ^{1}H and ^{13}C NMR parameters in solution requires including of solvent effects and electron correlation. Sauer et al.^{106 } reported on challenges during theoretical modeling of NMR spectral parameters for protonated alkylpyrroles. The authors used B3LYP/ccpVDZ and MP2/ccpVDZ theory level for geometry optimization with solvent effects included via PCM, as well as the average solvent electrostatic configuration^{107 } (ASEC) model. GIAO nuclear shielding constants were calculated with 6311++G(2d,p) and Jensen's (aug)pcSsegn (n=1–3) basis sets. In addition, Monte Carlo (MC) simulations of solute, counterion and 500 molecules of solvent were also performed. Good linear correlations between observed chemical shifts and calculated nuclear shieldings were observed (R^{2}>0.99). Only a small improvement in accuracy was observed upon introducing solvent effects via the PCM or ASEC approaches. Moreover, they noticed a lack of improvement upon averaging solvent over many configurations produced by Monte Carlo simulation. B3LYP shieldings obtained with polarizationconsistent basis sets still markedly differed from experiment. Finally, a comparison of HF, B3LYP and MP2predicted nuclear shieldings proved the importance of direct inclusion of electron correlation of the most expensive method on the quality of the predicted nuclear shieldings.
A very interesting combined experimental and theoretical study was published by Straka et al.^{108 } about a hydrogen bonding to gold. The PBE0 functional with def2TZVPP basis set for light atoms and relativistic effective core potential (ECP) for the Au atoms and dispersion correction (D3) by Grimme were used for structure optimization. Solvent effects were included via the COSMO approach. Twocomponent SOZORA relativistic calculations of nuclear magnetic shieldings were calculated with PBE0 density functional in the ADF program. This was the first proof of such an exotic noncovalent bond. It should be stressed that relativistic ^{1}H NMR shielding calculations were necessary to prove the presence of this controversial phenomenon.
Determination of solution structure of natural products is closely related to ^{1}H and ^{13}C NMR spectroscopy; their spectral assignments benefit from accurate theoretical predictions of their chemical shifts and coupling constants. Hehre et al.^{109 } reported on attempts to find a reliable methodology for accurate calculation of ^{13}C chemical shifts of conformationally flexible natural products. They started with structures optimized using fast molecular mechanics calculations with inclusion of a conformational search. Subsequently they obtained DFTrefined geometries using the ωB97XD/631G* level of theory and removed conformers higher in energy by 40 kJ mol^{−1} than the global minimum. Next the authors selected the ωB97XD functional in combination with several small basis sets (631G*, 631G**, 6311G*, 6311G** and 6311G(2d,2p)) and calculated GIAO nuclear shieldings of strychnine. From a set of 24 molecules, different linear correlations for aliphatic and aromatic ^{13}C chemical shifts were observed. An empirical correction to chemical shifts of carbon signals, varying by hybridization, was applied and markedly improved the initial picture. Energy calculations at the ωB97XV/6311+G(2df,2p)//6311G* level of theory allowed conformers for NMR calculations to be selected, using a Boltzmann weighting of their populations. The methodology was verified on over 900 structures with available NMR chemical shifts (RMS of 4 ppm or less) and an automated procedure designed.
Chalcones (unsaturated α,βketones) are important natural products. They are biologically active, colored compounds which contain the ketoethylenic group CO–CHCH. B3LYP/6311++G** modeling of (E)–3(4hydroxy3methoxyphenyl)–1(4hydroxyphenyl)prop2en1one structure and GIAOcalculated ^{1}H and ^{13}C shieldings of selected chalcone molecules in DMSO solution (included via the PCM model) were reported.^{110 } TMS was used as a reference molecule. The obtained proton and carbon chemical shifts were typical for similar aromatic compounds. Among numerous biologically active compounds possessing enzyme inhibitory properties are various derivatives of propanamide and pyridine. 2,2dimethylN(2pyridinyl)propanamide^{111 } was characterized experimentally and theoretically. The corresponding IR, Raman and ^{1}H, as well ^{13}C NMR studies in DMSO were reported. Four different conformers of the selected amide were characterized theoretically at B3LYP/6311++G** level of theory. The predicted vibrational and NMR spectral data well reproduced the experimental results.
Numerous experimental and theoretical studies on amides have been performed in order to determine, via conformational analysis, the preferred structure in the gas phase and solution. Theoretically predicted lowest energy minima agree well with available experimental Xray structures. Such studies are important in the search for small dipeptides which mimic the biological action of natural compounds in the human body. Of particular interest are modified dipeptides with a residue able to force their structural rigidity, for example, using the CC fragment. Broda et al.^{112 } reported on the synthesis and conformational studies of Ntbutoxycarbonylglycine(E/Z)dehydrophenylalanine N′methylamides (BocGly(E/Z)ΔPheNHMe) in chloroform using efficient NMR and IR spectroscopic techniques and lowtemperature singlecrystal diffraction studies. In addition, the structure in the gas phase and in the presence of chloroform (via PCM) were fully optimized using DFT (B3LYP and M062X) and MP2 methods combined with 6311++G** Popletype basis sets, containing both diffuse and polarization functions. To better describe dispersion interactions, singlepoint calculations in the gas phase and solution on previously determined minima were performed using MP2/6311++G** and M062X/6311++G** calculations. The GIAO method at the B3LYP/augccpVTZJ level of theory in chloroform allowed prediction of fairly accurate magnetic shieldings, which were converted to proton and carbon chemical shifts using TMS and benzene as references. It is worth mentioning that the augccpVTZJ basis set^{113,114 } was originally developed for efficient calculation of indirect spin–spin coupling constants. However, its good performance in predicting nuclear magnetic shieldings was also reported.^{115,116 } In addition, another compact basis set, STO3G_{mag}, designed by Leszczyński et al.^{117 } for calculating nuclear shieldings was also used. Interestingly, theoretical proton and carbon chemical shifts obtained with both kinds of basis sets correlated very well with experiment in chloroform (linear correlation with R^{2}>0.99). The good performance of the relatively small STO3G_{mag} basis set, in comparison to the larger augccpVTZJ one, was stressed (CPU time of 17 h vs. 34 days) with comparable RMS deviations from experiment (2.74 vs. 2.76 ppm for isomer E). A newer version of the Slatertype basis set,^{101 } STO(1M)3G, was also tested on prediction of nuclear shieldings and chemical shifts in 5fluorocytosine.^{118 } Structural optimization was carried out in the gas phase and DMSO (modeled by PCM) using several basis sets: STO3G, STO(1M)3G, as well as medium and large Popletype ones, 6311++G** and 6311++G(3df,2pd). GIAOB3LYP calculations of shieldings, converted to chemical shifts, were carried out using the same level of theory as for geometry optimization. In addition, Jensentype basis sets, augpcS2, augpcS3, augpcJ2 and augpcJ3, designed for nuclear shielding and SSCC calculations, were also applied. Surprisingly, both structural parameters and harmonic frequencies obtained at B3LYP/STO(1M)3G level of theory were better than those obtained with larger and more expensive basis sets. The performance of B3LYP/STO(1M)3G calculated chemical shifts in the presence of DMSO solvent is significantly better than for the original STO3G basis set and comparable to those obtained with more expensive schemes. Similar performance was observed for cytosine in the gas phase and DMSO. Abraham and Cooper^{119 } performed combined MM/QM calculations of ^{13}C and ^{15}N chemical shifts in the conformational analysis of alkyl substituted anilines. Initial structures were obtained using MMFF94 force field^{120 } and subsequently refined at the B3LYP/631+G* level of theory. The authors observed good agreement between ^{13}C NMR chemical shifts calculated with the GIAOB3LYP/631+G* approach and experimental data. However, worse performance was observed for ^{15}N chemical shifts.
Cisplatinum complexes have been used for a long time in cancer therapy. Palladium is a closely related element to platinum. In a quest for new compounds with interesting biological activity, 2hydroxyacetophenoneprophanesulfonylhydrazone, 5Cl2hydroxyacetophenoneprophane sulfonylhydrazone, 3,5ditertbutyl2hydroxybenzaldehydeprophanesulfonyl hydrazone and their Pd(ii) complexes were synthesized.^{121 } Both the free organic ligands and their complexes were fully characterized using IR, NMR and mass spectrometry. In some cases singlecrystal Xray studies were also conducted. ^{1}H and ^{13}C nuclear magnetic shielding tensors were calculated in the presence of DMSO (modeled by the PCM method) at the B3LYP/6311++G** level of theory with TMS as reference. In case of the metal complexes B3LYP calculations using LANL2DZ efficient pseudopotential^{122 } were performed. Linear correlations between predicted and experimental proton and carbon chemical shifts, with a very small scatter of points, were observed. Toušek et al.^{123 } reported on a quest for the most suitable method for calculating ^{31}P NMR chemical shifts of platinum(ii) complexes. The authors first examined the impact of allelectron and ECP basis sets, DFT functionals, and solvent effects on the optimized geometry. Among numerous density functionals used were BP86, B3LYP, PBE0, TPSSh, CAMB3LYP and ωB97XD, combined with allelectron basis sets 631G, 631G*, 631G**, 6311G** and the TZVP basis set, as well as with SDD, LanL2DZ and CEP31G ECP pseudopotential basis sets. Finally, BP86, PBE0 and B3LYP functionals with the Slatertype TZ2P basis set^{101 } were used in GIAO calculations of ^{31}P nuclear shieldings. Both scalar and spin–orbit contributions to nuclear shieldings were calculated using the ZORA Hamiltonian with the ADF^{17 } program. The COSMO approximation was used for including solvent effects. The author stressed the importance of dfunctions during geometry optimization (6311G** produced the best results). The best performance for the prediction of ^{31}P nuclear shieldings was observed for PBE0, TPSSh and CAMB3LYP density functionals, together with inclusion of solvent effects. The spin–orbit term in these molecular systems was fairly large (6–40 ppm) and its inclusion in the calculations was important.
Correction for relativistic effects enables more accurate prediction of multinuclear magnetic shieldings for ligands directly bonded to platinum. Swart et al.^{124 } highlighted the importance of solvent and dynamic effects for accurate prediction of fourcomponent ^{31}P nuclear magnetic shieldings for transplatinum(ii) complexes. Their experimental ^{1}H and ^{13}C NMR studies in solution were of limited value for interpretation due to severe overlap with solvents in case of trans.^{125} The ^{31}P spectra measured were easier to interpret. However, due to the HALA effect, the predicted phosphorus shifts did not reproduce experimental chemical shifts. The effect of platinum scalar relativistic (SR) and spin–orbit (SO) relativistic effects need to be included. ZORA calculations in the ADF program were performed within twocomponent approximation using PBE and KT2 density functionals in combination with eventempered Slatertype (STOtype ETpVQZ) and the special STOtype DZP, TZ2P and QZ4P basis sets used in ADF. The SOZORA results differed markedly more than the SCZORA ones, but the basis set size had a relatively small effect. Fourcomponent relativistic corrections were obtained with the ReSpect program^{21 } (Relativistic Spectroscopy) using the fourcomponent matrix Dirac–Kohn–Sham method (mDKS) and the Dirac–Coulomb Hamiltonian. These calculations were done with mixed Dyall basis sets (cvtz+vdz). Introduction of solvent effects via the COSMO model only slightly improved the results. Explicit introduction of solvent molecules also changed the ^{31}P chemical shifts only a little. The authors stressed that dynamic averaging, combined with addition of 3 or 5 molecules of solvent, significantly improved the accuracy of theoretical prediction (the deviation of about 30 ppm with two component calculations decreased to less than 10 ppm for four component predictions).
Combined spectroscopic and DFT studies on 5methyl4(2thiazolylazo)resorcinol were reported.^{126 } The structures of eight conformers were optimized at the B3LYP/6311G** level of theory. GIAOB3LYP calculations predicted reasonably well the experimental proton and carbon chemical shifts (obtained from calculated nuclear magnetic shieldings in DMSO, modeled by PCM). In a similar study, the structure and spectroscopic properties of 3trifluoromethylphenylchloroformate were investigated.^{127 } DFT modeling (B3LYP with 6311++G** and ccpVTZ basis sets) of structure, subsequent vibration analysis and GIAO B3LYP/ccpVTZ calculations of nuclear magnetic shieldings were completed. Theoretical chemical shifts were obtained from the calculated nuclear magnetic shieldings using TMS as reference. Good agreement between predicted proton and carbon NMR chemical shifts and experimental values, measured in deuterochloroform, was observed.
Nuclear shielding of proton signals in organic molecules depends on hydrogen atom localization in the molecule and the possibility to interchange electron density with neighboring molecular fragments. Thus, the chemical shift of ketone α–protons is related to its pK_{a} value. In other words, the acidity of this hydrogen atom is an important indicator of ketone reactivity. Obviously, this is also related to ^{1}H nuclear shielding in the corresponding hydroxyl group of the tautomeric enol form. Zhan et al.^{128 } optimized structures of 30 ketones at the B3LYP/631+G* level of theory in solution using both the universal solvation model based on solute electron density^{129 } (SMD) and PCM models of solvent. In addition, they optimized 1 : 1 clusters formed by solute and DMSO. Next, GIAO calculations were conducted at the B3LYP/631+G* level of theory in the presence of solvent, included via the SMD model. They observed a linear correlation between pK_{a} values of these ketones and the calculated nuclear shielding values of the acidic protons. However, the correlations were not impressive (R^{2} from 0.2 to 0.8).
Boron is present in numerous compounds of practical interest and the structure and reactivity of these molecules is very often studied using ^{11}B NMR spectroscopy. Experimental ^{11}B NMR studies are often supported by theoretical prediction of boron nuclear magnetic shielding tensors. However, benchmark calculations and standardizations are needed for evaluating boron nuclear shieldings in a variety of chemical surroundings. To check the performance of selected density functionals and basis sets in vacuum and in solution for the accurate determination of ^{11}B NMR parameters using two simple implicit solvent models, SMD and CPCM, Yu et al.^{130 } selected a large set of boron compounds with available experimental spectra. These included 16 molecules containing only B and H atoms and 21 molecules with B–C, B–O, B–S, B–N, B–Cl or B–F bonds. Geometry was optimized with B3LYP and M062X density functionals, combined with selected Popletype basis sets (631+G** and 6311+G(2d,p)) and the smallest correlationconsistent one (ccpVDZ). GIAO NMR calculations of shielding constants were conducted in THF with mPW1PW91, PBE0 and B3LYP density functionals, combined with 6311+G(2d,p), ccpVDZ and ccpVTZ basis sets. Fitted empirical scaling factors from linear plots of theoretical vs. experimental boron chemical shifts were derived. No effect of solvent during geometry optimization was noticed. The accuracy of ^{11}B chemical shift prediction for both solvent models (SMD and CPCM) was comparable. The best performance was observed for geometry optimization in the gas phase at M062X/631+G(d,p) or B3LYP/ccpVDZ levels of theory and NMR GIAO calculation using mPW1PW91/6311+G(2d,p) or B3LYP/ccpVDZ schemes with an implicit solvent model.
Continuing earlier studies,^{131 } Abraham and Cooper^{132 } combined molecular modeling (Pcmod 9.1/MMFF94) and GIAO DFT studies (B3LYP/DFT/631+G*) to determine conformations of substituted benzenes. For better performance, the chemical shifts of sp^{3} and sp^{2} hybridized carbon atoms were referenced separately. The set of studied molecules included benzene, toluene, ethylbenzene, benzylfluoride, benzylchloride and benzyl alcohol. Next they applied the same calculation procedure to several more substituted benzenes, as well as compounds with three and four fused aromatic rings. Aboulmouhajir et al.^{133 } studied the six lowest energy conformers of isooctane applying B3LYP and MP2 methodology combined with a 6311++G** basis set. Proton and carbon nuclear magnetic shieldings of the six optimized conformers were calculated using the GIAO approach at the B3LYP/6311++G** level of theory in chloroform modeled by the PCM method. The higher energy conformers (i.e. less likely to be observed in experiment) were shown to give lower agreement between predicted and experimental chemical shifts. Elguero et al.^{134 } studied experimentally and theoretically selected Nbenzyl azoles and benzazoles, concentrating on ^{1}H, ^{13}C and ^{15}N nuclear magnetic shieldings. They synthesized most of the 18 structures and analyzed 313 NMR signals. The GIAO method was used for B3LYP/6311++G** calculation of nuclear shieldings, and chemical shifts were obtained using TMS and nitromethane as references. Excellent linear correlations between calculated and observed chemical shifts were observed.
Many reports deal with combined experimental and theoretical studies on the chemical and spectroscopic properties of potential drugs. Both diffraction, vibrational and NMR studies are often supported by initial theoretical structural modeling, including semiempirical methods subsequently refined with DFT approaches. For example, Eren et al.^{135 } used the MOPAC2002 program^{136 } with the PM5 semiempirical method^{137 } to optimize the initial structure of N(phenyl)dimethyldisulfonimide. In the second step they continued geometrical refinement at the B3LYP/6311++G** level of theory. The minimum PES structure was confirmed by the lack of imaginary vibrational frequencies. The theoretically obtained conformer 1 closely reproduced the experimental Xray structure of the title compound. The presence of DMSO solvent was included in the theoretical geometry optimization using the PCM methodology, and the GIAO method was used to calculate the ^{1}H and ^{13}C shieldings at the same level of theory. Chemical shifts were derived from a linear relationship reported by Alyar et al.^{138 } The calculated proton and carbon chemical shifts for conformer 1 closely resembled the experimental NMR spectra. However, the authors did not compare theoretical structural and spectroscopic parameters with experimental data using some statistical tools, for example R^{2} from linear correlations, and RMS values. Gholivand et al.^{139 } reported on the prediction of ^{1}H, ^{13}C and ^{31}P NMR chemical shifts of some phosphoryl benzamide derivatives. The methodology used followed some earlier works on systematic DFT prediction of NMR parameters.^{140,141 } The structures of the studied molecules were fully optimized using B3LYP, BP86, PBE1PBE, M062X, MPWB1K and MP2 methods with the 631+G* basis set. The best prediction of structural parameters was obtained by the BP86 density functional. Remarkably, the def2TZVP and augccpVDZ basis sets produced structural parameters inferior to obtained with relatively small 631+G* one. Theoretical GIAO DFT nuclear shieldings were obtained at the same level of theory in vacuum and compared with experimental data in solution. The best performance was observed for BP86/631+G* calculations (linear fits with R^{2}=0.9627). The same authors^{142 } used a similar approach to find a relationship between ^{1}H, ^{13}C, ^{31}P chemical shifts, ^{2}J(PH), and the bonding structure of different phosphoryl benzamides. The selected compounds were optimized using B3LYP, BP86, PBE1PBE, M062X and MPWB1K density functionals, as well as a more expensive ab initio method including electron correlation (MP2), combined with the 6–31+G* basis set. In addition, the performance of two more flexible basis sets, def2TZVP and augccpVDZ, was analyzed. Surprisingly, the smallest basis set outperformed the two larger ones. However, we should keep in mind that accidental cancellation of errors sometimes produces excellent (but fortuitous) agreement with experiment. BP86/631+G* calculated shieldings in the gas phase, converted to chemical shift with respect to TMS and phosphoric acid (H_{3}PO_{4}), also produced the best agreement with observed spectral parameters in solution, as evidenced by linear scaling. Indirect spin–spin coupling constants were calculated using Dalton 2.0 program package.^{16 }
NMR characterization of antibiotics has been often supported by molecular modeling of their structure and spectroscopic properties. Theoretical studies on ciprofloxacin, a widely used fluoroquinolone antibiotic, were reported by Koch et al.^{143 } Using MP2 and DFT methodology combined with a range of basis sets, they tested the quality of predictions of ^{13}C NMR chemical shifts and ^{19}F–^{13}C coupling constants of ciprofloxacin. DMSO solvent was included within the PCM approach in both geometry optimization and GIAO calculations of nuclear magnetic shieldings. Better reproduction of experimental chemical shifts was observed for a neutral ciprofloxacin than for its zwitterion form, with the combination of B3LYP and augccpVDZ performing the best. We should keep in mind, however, that this basis set produces large errors^{125,144 } and one should not rely on accidental error cancellation.
NMR parameters are sensitive to molecular geometry and the presence and strength of intermolecular hydrogen bonds. Evaluation of hydrogen bond energies in solution is a challenging task. Tolstoy et al.^{145 } propose the use of ^{1}H NMR chemical shift difference of NH protons for aniline as a proton donor, complexed with proton acceptors containing nitrogen, and correlating it with the Hbond energy and the structure of the complex. 21 complexes were studied and critically analyzed. It was important that signals of free and bound aniline protons were observed as separate peaks in the experimental studies. Thus, the position of H(free) could serve as an internal standard. Structures of the model complexes were optimized at the B3LYP/augccpVTZ level of theory, harmonic vibrational frequencies calculated and the complexation energies were corrected for basis set superposition error (BSSE) using the Boys–Bernardi counterpoise approach^{146,147 } (CP). GIAOpredicted ^{1}H nuclear magnetic shieldings were calculated at the same level of theory and referenced to TMS. The authors observed an approximate monoexponential decrease of complexation energy with the separation of donor and acceptor nitrogens, r(N_{d}…N_{a}). The latter parameter could be measured from Xray diffraction studies. A similar exponential dependence on r(N_{d}…N_{a}) was observed for H(bound) chemical shift change upon complexation. This change was described by a simple formula and allowed the corresponding interatomic separations to be derived from the position of the complexed proton. In addition, linear correlations between complexation energies and the change of H(bound) chemical shift were observed. In a similar work, Tolstoy et al.^{148 } reported on characterization of the hydrogen bond formed between phosphinic acids and substituted pyridines, by careful analysis of their ^{1}H and ^{31}P NMR chemical shifts changes.
^{15}N NMR has been often used for characterizing nitrogencontaining organic compounds. Thus, accurate prediction of theoretical ^{15}N nuclear magnetic shieldings and the corresponding chemical shifts is of great practical interest. Krivdin et al.^{149 } reported on the accuracy factors, including the selection of functional, basis set, as well as the importance of solvent effect inclusion for computing ^{15}N NMR chemical shifts in a sets of condensed nitrogencontaining heterocycles. Initial tests were performed with eight density functionals (B3LYP, BHLYP, OLYP, OPBE, OPW91, PBE0, KT2 and KT3). The authors observed the best performance of the OLYP functional in combination with augpcS3(N)//pc2, combined with a locally dense basis set scheme. The mean absolute error was 5.2 ppm for this calculation scheme (the range of chemical shifts was about 300 ppm). Interestingly, worse performance of Keal and Tozer's KT2 and KT3 density functionals, optimized for nuclear shielding prediction, was noticed. Solvent effects on predicted nuclear shieldings were included via PCM approach. However, for a difficult case of phenanthroline in methanol, two explicit solvent molecules were added to the first solvation sphere and the resulting complex with two CH_{3}O–H…N hydrogen bonds was embedded in polarizable continuum.
Akhmedov et al.^{150 } reported on the synthesis of new cyanosubstituted analogues of Tröger's bases from bromo derivatives and their careful characterization using several NMR techniques. Experimental data were supported by DFT modeling of their structure at B3LYP/631G**, B3LYP/631+G**, B3LYP/6311+G**, B3LYP/6311++G(2d,2p), B3LYP/ccpVTZ, and B3LYP/augccpVTZ levels of theory. The authors were also interested in determining longrange ^{1}H–^{1}H and ^{13}C–^{1}H couplings for the Tröger's bases. They also reported on the complete and unambiguous assignment of the ^{1}H, ^{13}C and ^{15}N NMR spectra. GIAOcalculated nuclear shieldings were obtained with B3LYP hybrid density functional combined with several Pople and Dunningtype basis sets. It is important to note the potential of GIAODFT methodology for differentiating proton chemical shifts of the stereospecific exo and endo methylene protons in the studied compounds. However, no method was able to predict carbon chemical shifts with high accuracy. Longrange proton–proton and carbon–proton coupling constants using the valenceoriented basis sets 6311+G**, 6311++G(2d,2p), ccpVTZ, and augccpVTZ, were predicted in reasonable agreement with experiment. However, for better performance, dedicated basis sets for coupling constants should be applied in these calculations.
Inclusion complexes are promising drug delivery systems, and cyclodextrins are potential host molecules capable of forming inclusion complexes with small molecules. As result, there is a growing interest to apply α, β and γcyclodextrins as drug carriers in modern medical treatment. Theoretical DFT studies on a possible inclusion complex between βcyclodextrin and 8anilinonaphthalene1sulfonate was reported by Leila et al.^{151 } Structure, interaction energies and selected spectroscopic parameters in vacuum and water, applying the PCM approach, were calculated with B3LYP, M062X and WB97XD density functionals using the 631G* basis set. They observed that the orientation of guest molecule pointing toward the lower rim –OH groups of the βCD, which has an inner hydrophobic cavity, was energetically preferred. This agreed with experiment. GIAO B3LYP/631G calculations were performed on WB97XD/631G*optimized geometries. The obtained nuclear chemical shieldings were converted to chemical shifts using TMS as reference, and the theoretically calculated changes in chemical shifts of solute protons upon interaction with host molecule were compared with experimental data. In general, both experimental spectral changes and predicted values were very small. Such results highlight the importance of proper referencing of experimental spectra (both TMS and DSS^{152 } form inclusion complexes with βCD and cannot be used as references).
3.2 Biological applications
Fonseca et al.^{153 } calculated nuclear magnetic shieldings of nucleic acid components (adenine, cytosine, guanine and thymine) in liquid water. Initially, adenine, guanine, thymine and cytosine molecules were optimized at the MP2/augccpVDZ level of theory in water, modeled by the PCM method. Next, the authors used classical Monte Carlo simulation to create different solute–solvent configurations. Subsequent calculations, employing the supermolecule model with the first microsolvation sphere included, revealed a very high sensitivity of the amino nitrogen magnetic shieldings to hydrogen bonding (shifts from −5.48 to −33.42 ppm). In addition, a strong deshielding effect for the hydrogens of the protonated nitrogen atoms was observed. The 6311++G(2d,2p), as well as pcS2, pcS3 and augpcS2 Jensen's polarizationconsistent basis sets, were used for predicting the nuclear magnetic shieldings. The ^{1}H and ^{15}N shielding constants of the studied nucleic acids in water were calculated using the GIAOB3LYP approach in the presence of water. The calculated ^{1}H and ^{15}N chemical shifts well reproduced experimental results.
The structure of natural products depends on the nature of the solvent and is often probed by ^{1}H NMR spectroscopy, while the signal assignment is supported by DFT calculation. Gerothanassis et al.^{154 } continued their research on natural products by combining an NMR experiment and DFT calculations.^{155 } The authors reported on hydrogen bonding, solvation and conformation of chrysophanol, emodin and physcion in nonpolar and polar environments. They observed very strong intramolecular Hbonds which were reflected by very deshielded narrow signals at chemical shifts>10 ppm in proton spectra of the studied molecules. Solvents differing in dielectric constant (chloroform, acetone and DMSO) were used for NMR measurements. It is very important to notice that addition of trace amounts of trifluoroacetic acid changed very broad, and almost invisible peaks into sharp, diagnostic OH signals. Geometries of the studied compounds were optimized at B3LYP/631+G*, B97XD/631+G*, APFD/631+G*, M062X/Def2TZVP and TPSSh/TZVP levels of theory in the above selected solvents using the PCM approach and GIAO nuclear magnetic shieldings, predicted in solution at the B3LYP/6311+G(2d,p) level of theory. They reported a very good linear correlation between experimental and theoretical ^{1}H NMR chemical shifts for the studied compounds (R^{2}>0.99). They also tested a discrete solvent model by placing one solvent molecule in the vicinity of the OH group. This allowed one conformer with predicted ^{1}H chemical shift differing significantly from experiment to be rejected. Note, the predicted OH chemical shifts for Xray structures deviated markedly from experiment (theoretical structures nicely reproduced the observed peak positions).
The importance of the hydrogen bond network and the relation between ^{1}H NMR chemical shift and interaction energies in selected biphenyls, alkanes, azaalkanes, and oxaalkanes was studied theoretically by Lomas.^{156 } All structures were optimized at the B3LYP/6311+G** level of theory and GIAO ^{1}H nuclear magnetic shieldings were predicted with the PBE0 density functional combined with Dunning's ccpVTZ correlationconsistent basis set. Aromatic and aliphatic proton chemical shifts were obtained with respect to benzene and TMS, and a perfect linear correlation between theoretical proton chemical shifts in the gas phase and experimental data in chloroform was observed (R^{2}>0.998). He observed an interesting relation in a form of decaying exponential curve between predicted NMR shift of “in” CH proton versus CH⋯HC exchange energy for several compounds. HerbertPucheta et al.^{157 } combined modified experimental NMR protocols with theoretical GIAO calculations of the active component of Mexican moss (Pilotrichella flexilis), which shows promising biological and therapeutic activity. The RHFoptimized structure of C_{30}H_{50} lup20(29)ene and the crystal structure of betulin diacetate were obtained in the Gaussian 09 program^{10 } and refined at the DFTB3LYP/6311G(2d,p) level of theory. The GIAO approach was used for predicting proton and carbon nuclear magnetic shieldings in chloroform within the PCM approximation. Chemical shifts were calculated with respect to nuclear shieldings of TMS at the same level of theory. Linear correlations between the predicted and experimental ^{1}H and ^{13}C chemical shifts demonstrated good performance of the theoretical protocol. A new triquinane sesquiterpenoid, presilphiperfolane7α,8αdiol, was recently isolated from Pulicaria vulgaris by Radulović et al.^{158 } The authors combined experimental NMR studies with GIAODFT calculations to determine the structure of the extracted compound. The authors recorded proton spectra in ten deuterated solvents (CDCl_{3}, CCl_{4}, CS_{2}, methanold_{4}, DMSOd_{6}, acetoned_{6}, CD_{3}CN, 1,4dioxane, benzened_{6}, and pyridined_{5}) and analyzed the coupling constants. The ^{1}H^{1}H multiplet patterns were sensitive to applied solvent, and the solventinduced shifts, ΔδA, for nonpolar and polar H atoms differed significantly (from −0.25 to 0.45 ppm and from −0.05 to 2.84 ppm). In particular, large positive solventinduced shifts were observed in pyridine. The structures of eight diastereomeric presilphiperfolane7,8diols were fully optimized at the B3LYP/6311++G** level of theory and the dominating, i.e. the lowest energy structures determined. GIAO nuclear magnetic shieldings were calculated using the B3LYP/6311++G** level of theory and the WP04 density functional.^{159 } The latter functional was reparametrized to produce accurate ^{1}H chemical shifts in chloroform. In addition, more demanding calculations were conducted at the WP04/augccpVDZ level of theory. Solvent effects were included via the PCM model, and the agreement with experimental proton data was good (R^{2}>0.94). The ^{13}C chemical shifts were also calculated at the same level of theory. However, the agreement between predicted and experimental Jcouplings was not so satisfactory. One reason could be the use of standard basis sets, not modified for calculating indirect spin–spin coupling constants.
A very large window of ^{17}O NMR chemical shifts enables good separation of its resonances in experimental spectra. Thus, ^{17}O NMR is extremely sensitive to hydrogen bonding and weak interactions in solution. On the other hand, it exhibits very short spin–lattice relaxation times, T_{1}, and suffers from broad signals due to its quadrupolar nature (spin=5/2). In addition, its very low natural abundance (0.038%) and low NMR receptivity limits its use to concentrated solutions or enriched molecules. An updated review on the application of ^{17}O NMR to numerous organic, as well as biological, molecules in water was reported by Wu.^{160 } The author discusses a very large number of studies, supported by 286 references. Hundreds of millions of patients are affected by very small dust particles (PM_{2.5}) produced by burning fossil coal in electric hydro plants in China (larger particles are stopped by electro filters). According to Yang et al.,^{161 } about 40% of patients with lung problems caused by PM_{2.5} particles in China are treated with traditional Chinese medicine (TCM). However, contemporary approaches are supported using modern analytical tools, including sophisticated NMR studies. Extracts of medicinal plants with antioxidant properties have been characterized by Yang et al.^{161 } using 1D and 2D ^{1}H and ^{13}C NMR measurements. It is worth emphasizing that, instead of determining chemical shifts, the solution measurements were conducted by direct measurement of nuclear magnetic shieldings using deuterium lock signal, previously referenced to the helium3 resonance frequency. This way the experimentally measured proton and carbon nuclear magnetic resonances of two flavonoids, daidzein and puerarin, were directly compared with DFT (B3LYP, M062X and PBE0 density functionals combined with augccpVTZ and 6311G(2d,p) basis sets) predicted parameters in vacuum and methanol. The experimental data were reproduced particularly well using PBE0 density functional and the 6311G(2d,p) basis set. This suggests some accidental error calculation in case of the smaller basis set. Interestingly, the authors also showed some correlation between calculated ^{13}C chemical shifts and antiradical activity of the studied compounds and considered a singlestep hydrogen atom transfer mechanism. Another naturally occurring flavonoid is kaempferol. Milenković et al.^{162 } characterized theoretically kaempferol at B3LYPD3/6311++G** level of theory, and compared the obtained parameters with experimental IR, Raman and ^{1}H and ^{13}C NMR data. Solvent effects were included using the PCM approach and the vibrational parameters were scaled by 0.9873 for easier comparison with experimental wavenumbers. No experimental structure of kaempferol is available, so the calculated one could fairly accurate reproduce its geometry. For better matching GIAOpredicted proton chemical shifts with experiment, the authors also used a scaling factor of 0.932, determined by linear regression. Theoretical proton and carbon chemical shifts closely reproduced the experimental spectra.
Studies^{163 } of solvent effects on the anticancer trans(NHC)PtI_{2}Py complex were reported using DFT calculations, modeled by MPW1PW91 density functionals and the def2TZVPPD basis set with an effective core potential (ECP) for platinum. The environment was modeled by ethyl alcohol, methyl alcohol, acetonitrile, nitromethane and DMSO (dielectric constant increasing from 1 in vacuum to 47) and the calculated platinum shielding dropped from 2598 to 2234 ppm. At the same time, the ^{13}C isotropic shielding increased from 6.4 to 8.4 ppm. The authors also observed changes of onebond Pt–C and Pt–N indirect spin–spin coupling constants due to the presence of different solvent. However, the lack of relativistic contribution due to the presence of fairly heavy platinum atom makes this study of limited accuracy.
4 Modeling nuclear magnetic shieldings for the solid state
Alkorta et al.^{164 } reported on the good performance of the gaugeincluding projector augmented waves method^{165 } (GIPAW) in predicting ^{13}C and ^{15}N chemical shifts in polymorphs of azoles and benzazoles. However, they omitted a carbon signal directly bonded to bromine from the linear correlation between experimental and calculated ^{13}C data, as the HALA effect was not included in their nonrelativistic calculations.
Zeolites form well defined aluminosilicate porous structures, often used as support in heterogenous catalysis. Their structures, important for catalytic processes, can be differentiated by solidstate magicangle spinning^{166 } (MAS) NMR spectroscopy. It is known that the alumina ion can be located in several different crystallographic positions in the framework of zeolites. The presence of Al in various tetrahedral sites (Tsites) controls the catalytic properties of such systems, and theoretical modeling of ^{27}Al and ^{29}Si nuclear magnetic shieldings could significantly support experimental characterization of zeolites. Svelle et al.^{167 } performed combined experimental solidstate NMR studies on siliconrich zeolite ZSM23 and conducted theoretical calculations of ^{27}Al shieldings. First, using the plane wave basis sets^{168 } with a kineticenergy cutoff at 700 eV and a chargedensity cutoff at 7000 eV they modeled the zeolite structure with BEEFvdw exchange−correlation functional.^{169 } DFT nuclear magnetic shieldings were calculated both for selected finite size molecular clusters, as well as with the GIPAW approach for a periodic systems. According to their tests, at least four coordination layers around the investigated site should be used in the former approach to obtain results comparable to periodic calculations. The studied density functionals (PBE, BLYP and B3LYP) were combined with TZVP, 6311+G** and augpcSseg2 basis sets. Comparing the experimental and DFT calculated nuclear magnetic shieldings the authors proposed an assignment of the ^{27}Al isotropic chemical shifts for the five identified resonances from various tetrahedral Al sites. Ferreira^{170 } reported on DFTGIPAW calculations on seven Alcontaining intermetallic compounds in order to model the available experimental chemical shift range of ^{27}Al (from about −200 to 1600 ppm). All NMR calculations were performed on crystal structures of the selected compounds. A rough linear correlation was shown for a plot of theoretical vs. experimental ^{27}Al chemical shifts. He also demonstrated a failure of generalized gradient approximation (GGA) XC functionals in such calculations.
Water still remains a mysterious molecule and substance. Its behavior in confined spaces is also surprising. Recently Carnevale, Pelupessy, and Bodenhausen^{171 } reported on variable temperature studies of water trapped in barium chlorate monohydrate using solidstate (MAS) NMR. From the analysis of the exchangeinduced broadening, coalescence, and narrowing of the crossterm splitting in MAS spectra they estimated the rate of exchange of the two protons between 140 and 190 K. The theoretically predicted nuclear magnetic shielding tensor, calculated at the PBEDFT level of theory within the GIPAW approximation, enabled explanation of the observed experimental NMR pattern. Contributions due to incoherent quantum tunneling of water in the lowtemperature regime were postulated.
Jiao and Bauer^{172 } studied experimentally cyclopentadienyl lithium in the solid state using ^{6}Li CPMAS NMR. GIAO B3LYP/TZVP calculations produced ^{6}Li and ^{13}C nuclear magnetic shielding tensors with inclusion of THF solvent effects, modeled using the SMD approach. TMS and lithium cation in water, [Li(H_{2}O)_{4}]^{+}, were calculated as the corresponding references, allowing determination of theoretical chemical shifts. In addition, GIAO calculations were also performed using B3PW91, BP86, PBEPBE and M062X density functionals. It was stressed that finite model GIAOB3LYP calculations of lithium and carbon nuclear magnetic shielding tensors well reproduce the experiment in the solid state. They also pointed out that the lithium tensor component perpendicular to cyclopentadienyl ring was extremely shielded (δ_{33}=−43.0 ppm), which had not been previously observed.
Variabletemperature ^{13}C MAS NMR was applied to investigate disorder and local mobility in diethylcarbamazine citrate by Venâncio et al.^{173 } The experimental NMR data, measured for this drug used for the treatment of filariasis in tropical countries, were supported by GIPAW calculations of nuclear magnetic shieldings. GIPAW calculations allowed the authors to monitor the carbon atoms involved in structural disorder. Duarte et al.^{174 } studied hydrogen bonds in the supramolecular arrangement of crystalline gabapentin protic salts. This anticonvulsant drug crystallizes in three polymorphic forms. The solidstate NMR studies were supported by in silico modeling of HB formation in these pharmaceutics. The structures were optimized with the PBE density functional using the GIPAW approach. Nuclear magnetic shieldings were obtained with M062X density functional and the calculated ^{1}H NMR chemical shift changes varied from 0.4 to ∼5.8 ppm.
Xray diffraction (XRD) is a powerful technique enabling localization of heavier atoms in crystals. However, it cannot precisely determine hydrogen atom positions (and in general elements with low atomic number, Z). For example, a typical C–H interatomic separation of about 1.09 Å, determined in the gas phase, and is “measured” to be significantly shorter by Xray (typically 0.9 to 0.96 Å). A combination of specific solidstate NMR experiments (in socalled “NMR crystallography”) with DFT calculations could alleviate this problem. Hayes et al.^{175 } reported on refining positions of hydrogen atoms in the hydrated carbonate mineral, hydromagnesite [4MgCO_{3}·Mg(OH)_{2}·4H_{2}O]. Thus, atomic positions are refined by DFT and the theoretical calculated nuclear magnetic shielding tensor is compared with experimental NMR data. In the experimental NMR spectra the authors observed two carbon signals originating from nonequivalent positions in the crystal lattice. The lineshape of static ^{13}C signal was influenced by their proximity to oxygen and hydrogen atoms in the crystal lattice. The atomic positions were optimized using PBE density functional, and the nuclear magnetic shieldings calculated. This procedure allowed for more accurate positioning of atoms in the studied crystal. An important issue in NMR crystallography is graphical presentation of nuclear magnetic shielding (or chemical shift) tensor which is sensitive to anisotropy of atomic and molecular neighborhood. A practical tool allowing visualization of nuclear shielding tensors in solid state NMR, written in Mathematica program,^{176 } was proposed by Mueller et al.^{177 }
Dračínský et al.^{178 } proposed a simple way leading to higher accuracy in nuclear shielding prediction in crystals. Periodic calculations with planewave basis sets are limited to generalgradientapproximation (GGA) density functionals and are not so accurate as typical calculations on finite models, using better methods and basis sets. Corrections to nuclear magnetic shieldings predicted by GIPAW methods are obtained by calculating nuclear magnetic shielding tensor in the gas phase for a single isolated molecule at the geometry directly taken from the crystal structure. 631G*, 6311+G(2d,p) and pcSsegn (n=1–3) basis sets were used in combination with the PBE0 hybrid density functional. In addition a highly accurate CCSD/6311+G(2d,p) method was employed. The authors added corrections, calculated with a hybrid density functional, and improved the accuracy of ^{13}C, ^{15}N and ^{17}O chemical shifts predicted for crystalline isocytosine, methacrylamide and testosterone.
Charpentier et al.^{179 } proposed to employ machine learning (ML) as a fast and efficient method alternative to DFTGIPAW calculations for nuclear magnetic shielding tensor prediction for aluminosilicate glasses. The authors constructed the reference database by performing DFTGIPAW NMR calculations on hundreds of oxide glass structures generated through classical molecular dynamics (MD). As result, 500 atomic structures and their NMR parameters were generated for the database as a reference set. Detailed explanation of ML procedure was given and the superior performance of the popular smooth overlap of atomic positions (SOAP) descriptors for predicting NMR isotropic magnetic shieldings in sodoaluminosilicate glass demonstrated.
Experimental determination of accurate values of ^{1}H chemical shielding anisotropy (CSA) is challenging. On the other hand, this parameter reflects the local environment of protons in the solid state. Czernek and Brus^{180 } reported on plane wave DFT calculations using two functionals (GIPAWPBE and GIPAWrevPBE) to characterize the structures and the ^{1}H NMR chemical shift tensors of solid maleic, malonic, and citric acids, as well as Lhistidine hydrochloride monohydrate. Geometries of molecular clusters were obtained from periodic calculations using the PBE density functional, and GIAOMP2 and GIAOB3LYP chemical shift were calculated with the 6311++G(2d,2p) basis set. Using a linear regression model, they significantly improved the accuracy of proton chemical shielding tensor prediction/determination (final RMS about 2 ppm).
The quality and properties of olefinbased polymers are of utmost importance for industry, and the polymerization process depends on the quality of catalysts.^{181 } Standard catalysts used in polyolefin production are based on the Ziegler–Natta formulation,^{182 } and the main components of these catalysts are titanium tetrachloride (TiCl_{4}), used as precursor on magnesium dichloride support (MgCl_{2}) in the presence of specific organic ligands with oxygen atoms (aromatic esters and diethers, acting as Lewis acids). The two magnetically active titanium isotopes are halfinteger quadrupolar nuclei (^{47}Ti and ^{49}Ti with S=5/2 and S=7/2 respectively^{183 }), which makes experimental measurements particularly difficult as MAS only partially decreases the width of the central transition peak. Iijima et al.^{184 } reported on experimental and theoretical studies on model titanium system on a magnesium support. The performance of selected density functionals and basis sets was initially tested against the experimental chemical shift of Ti(CH_{3})_{4} at 1325 ppm. They selected the TZVP basis set and three density functionals for geometry optimization (PBEh1PBE, PBEPBE and B3PW91). Next GIAO calculations were performed using B3LYP and B3PW91 functionals combined with the basis set used for structure modeling. Isolated molecules of TiCl_{4} and Ti(CH_{3})_{4} with T_{d} symmetry were considered good (symmetrical) models with nuclear magnetic shielding/chemical shift tensors less affected by solvent. The best performance was observed for the B3PW91/TZVP//B3PW91/TZVP approach, and this was used in subsequent calculation of ^{49}Ti NMR parameters for TiCl_{4}/MgCl_{2} systems. The models studied were created by different arrangements of the guest TiCl_{4} molecule adsorbed onto the (110), (104) and (104)step defect surfaces of MgCl_{2}. The authors observed a dependence between Ti–Cl distance in the formed complex and the calculated NMR parameters (they analyzed several components of nuclear shielding tensor including the isotropic term and anisotropy). The DFTpredicted^{47,49 } Ti NMR spectra of TiCl_{4} adsorbed weakly onto the (104) surface of MgCl_{2} closely reproduced experimental data.
5 Calculation of nuclear shieldings in the presence of “exotic” halogen, chalcogen, pnicogen, and tetrel interactions
Hydrogen bonding has been studied for over century.^{185,186 } Its importance is well appreciated, for the presence of life, liquid water at normal conditions, and in material sciences. Hbond interactions are dominated by electrostatic forces.^{187 } Recently, experimental and theoretical studies pointed out to specific, but controversial, attractive interactions between two negatively charged atoms.^{186 } Lu and Scheiner^{188 } created several noncovalent A…N complexes with ammonia, where A is the central atom of FX, FHY, FH_{2}Z, and FH_{3}T molecules (X=Cl, Br, I; Y=S, Se, Te; Z=P, As, Sb; T=Si, Ge, Sn). Structures and spectroscopic parameters for these complexes were obtained from DFT calculations. M062X/augccpVDZ calculations were conducted on “light atoms” and relativistic effects were incorporated for I, Te, Sb and Sn via the augccpVDZPP pseudopotential. The calculated AN bond was longer due to electron density transfer into the antibonding σ*(A–F) orbital with the corresponding order of bond lengths: halogen>halcogen>tetrel∼pnicogen. GIAO nuclear magnetic shieldings were calculated and dramatic changes upon complexation observed. A positive change of nuclear shielding is observed for central atom A. Thus, for I, Cl and Br its nuclear shielding changes by 72, 478 and 1675 ppm, respectively. Larger shift changes were calculated for pnicogen than for tetrel bonds. NMR parameters are very sensitive to relativistic effects, but pseudopotentials include them in a very approximate way. Unfortunately, no two or four component relativistic corrections were included in these calculations. These studies were extended by Scheiner et al.^{189 } to noncovalent interactions of small molecules forming σhole bond with Nmethylacetamide (NMA) as a simple model of proteins. The following Lewis acids were studied: FX, HFY, H_{2}FZ, and H_{3}FT (X=Cl, Br, I; Y=S, Se, Te; Z=P, As, Sb; T=Si, Ge, Sn). The structures were optimized at MP2/augccpVDZ level of theory (augccpVTZPP pseudopotential was used for heavier atoms). GIAO NMR calculations were performed with MP2 method, combined with allelectron SapporoDKH3DZP2012diffuse basis set, to account for relativistic effects. Interestingly, they observed an increase of shielding for the bridging atom A in the complexes (an opposite effect is observed for a proton in a hydrogen bond). In addition, the NMR shielding of the O atom, directly involved in the bond, increases, and an opposite effect is observed for C and N atoms. Linear changes between MP2/augccpVDZ calculated interaction energies and changes in NMR shielding of O, N and C atoms of NMA were reported.
Alkorta et al.^{190 } reported on estimation of NMR parameters in 60 halogenbonded complexes formed between dihalogen molecules (XY with X, Y= F, Cl, Br, I and At) and NH_{3}, H_{2}O, PH_{3} and SH_{2} at the MP2/augccpVTZ (augccpVTZPP pseudopotential used for I and At) level of theory. The structures were optimized at the MP2/augccpVDZ level of theory with ECP for I and At. Nuclear shieldings were obtained using relativistic twocomponent spin–orbit (SO) ZORA calculations at the PB86/QZ4P level of theory conducted in the ADF program. For comparison, nonrelativistic PB86/QZ4P calculations were also completed. The NMR calculations were conducted using Slatertype basis sets,^{101 } not used in other program packages. The relativistic correction to ^{15}N nuclear shielding constant due to the presence of Br and I heavy atom in X_{2}…NH_{3} complexes was about 9 and 15 ppm respectively. In case of PH_{3}, the corresponding changes of ^{13}P nuclear shieldings were larger (34 and 48 ppm).
Abbreviations
 ADF
Amsterdam Density Functional program
 BSSE
Basis Set Superposition Error
 CBS
complete basis set
 CCSD
Coupled Cluster method with Singles and Doubles
 CCSD(T)
Coupled Cluster with Singles and Doubles and perturbatively included Triple excitations
 CFOUR
CoupledCluster techniques for Computational Chemistry computer program
 CP
counterpoise method
 CP MAS
Cross Polarization Magic Angle Spinning
 CV
CoreValence
 DFT
Density Functional Theory
 DIRAC
Program for Atomic and Molecular Direct Iterative Relativistic Allelectron Calculations
 ECP
Effective Core Potential basis sets
 Gaussian
computer program
 GIAO
GaugeIncluding Atomic Orbital
 GIPAW
GaugeIncluding Projector Augmented Waves
 GTO
GaussianType Orbital
 HALA
Heavy Atom on Light Atom
 HF SCF
Hartree–Fock SelfConsistent Field
 MP2
second order MollerPlesset Perturbation Theory
 NICS
Nuclear Independent Chemical Shift
 NMR
Nuclear Magnetic Resonance
 NR
NonRelativistic
 PCM
Polarizable Continuum solvent Model
 ReSpect
Relativistic Spectroscopy DFT Program
 RHF
Restricted Hartree–Fock
 RMS
Root Mean Square deviation
 SD
SpinDipole
 SO
Spin–Orbit
 SOPPA
SecondOrder Polarization Propagator Approximation
 SSCC
Indirect Spin–Spin Coupling Constant
 STO
SlaterType Orbital
 QED
Quantum ElectroDynamic
 ZORA
Zeroth Order Regular Approximation
This work was partly supported by the University of Opole.