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

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Nonrelativistic Representation of NMR Parameters

**B**is the external magnetic flux density, ${\mathsf{\mu}}_{N}$ is the nuclear magnetic moment, ${\widehat{H}}^{(nl)}$ is the interaction operators containing the n-th power of

**B**and l-th power of ${\mathsf{\mu}}_{N}$. The wave function of a system is represented in the form of power series of

**B**and ${\mathsf{\mu}}_{N}$:

_{ref}, and that of the compound under question, σ

_{sample}[55]:

**A**, uniquely determines the magnetic field $B=\nabla \times A(r)$, however, the otherwise statement is not true, i.e., a given magnetic field

**B**gives a variety of vector-potentials

**A**. Suppose, one adds the gradient of any scalar function $\nabla f$ to a given vector-potential

**A**. This results in a zero change in the magnetic field of

**B**, because $\nabla \times \nabla f=0$. This ambiguity emerges in the expressions (7) and (8) as the dependence of the resulting values on the radius-vector of the center of the coordinate system. The multivariance in describing a physical property with different vector potentials leads to a natural requirement to the property to be independent from the selected coordinate center. This requirement is usually referred to as the gauge invariance principle. The gauge invariance is trivially satisfied by the exact solutions of the Schrödinger equation (for the proof, see, for example, [56]), however, for approximate solutions this is not the case. This is a serious problem for quantum chemistry, which, in fact, is built upon the approximate solutions. The violation of the gauge invariance principle in approximate approaches of quantum chemistry occurs for two reasons: (a) the use of finite basis sets; (b) the fact that some quantum chemical schemes do not obey the virial theorem [57]. The first reason is usually considered the most serious; the latter is mentioned in the literature much less frequently [58]. For methods that do not violate the virial theorem, for example, the Hartree–Fock method, it can be shown that they provide gauge invariance of the observed physical quantities in the complete basis set (CBS) limit [59].

**K**

_{MN}has four contributions:

**K**as one third of its trace multiplied by a coefficient containing the product of the gyromagnetic ratios of the nuclei under consideration:

_{i}and ε

_{a}, respectively.

#### 2.2. Relativistic Representation of NMR Parameters

^{2}/c

^{2})

^{−1/2}, which allows to determine the “relativistic” contraction of the inner 1s shells ([(γ − 1)/γ] × 100%). That is not enough for the NMR parameters, since the relativistic corrections to these are rather non-local properties, which are determined by the electronic structure of the entire electron system of the molecule.

^{2}, and the moieties $\mathsf{\beta}$ and $\overrightarrow{\alpha}$ are the 4 × 4 matrices, called the Dirac matrices:

^{2}< E < 0.

_{e}is the number of electrons. However, the Hamiltonian of this large dimension is not applicable in practice. As a rule, all standard relativistic methods reduce the multi-electron problem to single-particle equations, so that the resulting equations are very similar to Equation (28). The construction of any single-particle approximations relies upon the four-component many-particle Dirac–Coulomb–Breit Hamiltonian (DCB) [134]:

## 3. Quantum Chemical Methods for Calculating NMR Parameters

#### 3.1. Configuration Interaction Methods

**c**is the vector-column of the CI-coefficients.

_{2}molecule within the GIAO approach [154].

^{6}, with N designating the number of basis set functions).

^{1}H and

^{13}C shielding tensors in the hydrogen and methyl halides, considering relativistic spin-orbit (SO) effects. The SO corrections were calculated analytically from the quadratic response functions using self-consistent field and multiconfiguration self-consistent field reference wave functions.

#### 3.2. Coupled Clusters Methods

^{4}, N

^{6}, N

^{8}, N

^{9}. The CCSD scheme has received a significant attention, but is often too expensive to be useful for molecules with more than 10 atoms. The CCSD provides high accuracy for many challenging response properties and is usually considered as a very accurate method for calibration of the other inferior computational methodologies. The CCSDT and higher-ranking pure CC schemes are out of routine use for today due to their dramatic scaling.

^{5}with respect to the number of basis set functions N [141]. Another approximate model, which was built on the similar concept, is the CC3 model [238]. The CC3 represents an intermediate model between the CCSD and CCSDT schemes, so that the computational scalability of CC3 is N

^{7}[217]. In general, the main principle of building the approximate CC

_{n}(n > 1) models is based on reducing the cluster equations for the n-fold cluster excitation amplitudes to the lowest non-vanishing order in the perturbation theory [217].

^{7}. However, the CCSD(T) has not been adopted for the calculations of SSCCs, because it has the triplet instability issue [239,240], which occurs when calculating the triplet FC and SD contributions to SSCCs. As opposed to the CCSD(T) model, the second derivatives within the CC3 scheme can be computed in two different ways, namely either with orbital relaxation effects explicitly included (the so-called ‘‘relaxed’’ CC3) or with the orbital relaxation effects excluded (the so-called ‘‘unrelaxed’’ CC3). The ‘‘unrelaxed’’ CC3 scheme circumvents possible problems with the triplet instabilities [224] and can be successfully applied to the calculation of the triplet properties such as FC and SD terms in SSCCs. To reduce the computational costs of CC methods, the resolution of the identity (RI) approximation [241,242] for two-electron integrals was applied [243,244], however, in practice, this is relevant only for the calculations of equilibrium geometries, harmonic frequencies, energy gradients, and some other first-order properties for now.

^{1}H,

^{13}C,

^{15}N,

^{17}O, and

^{19}F nuclei. The CCSD method applied in conjunction with the uncontracted aug-ccJ-pVTZ basis set was found to be very accurate for calculating the

^{1}J

_{CF}. This follows from the fact that the estimated errors for the spin–spin coupling constants

^{1}J

_{CF}turned out to be about 2.0 Hz, given that the values of this type of coupling constants usually exceed 200 Hz. For the two- and three-bond couplings involving 1–2 row elements, it was found that it is quite important to add diffuse functions. With the diffuse functions added to the ccJ-pVXZ and pcJ-n basis sets, the CCSD method gives very good results. In particular, for the

^{2}J

_{CF}SSCCs, calculated at the CCSD/aug-ccJ-pVTZ level, the typical error was found to be only 0.6 Hz. It was also shown that if the higher accuracy is needed within the CCSD framework, the basis set error can be reduced by roughly a factor of two by going to the quadruple zeta basis set.

^{1}J

_{CN}in HCN and the SSCCs of F

_{2}CO, where the results of Del Bene deviated significantly from those of Faber and Sauer and the experimental data.

_{CC3}-J

_{CCSD}) and the residual triples correction (J

_{CCSDT}-J

_{CC3}) to various one-bond nuclear spin–spin coupling constants are illustrated in Figure 1.

^{1}J

_{CN}in FCN and

^{1}J

_{OF}in OF

_{2}, where the correlation corrections to the CCSD results due to the triple excitations (triples) inclusion effect occurred to be as much as 15.7% and 6.4%, correspondingly. The changes in geminal and vicinal SSCCs due to the triples effect were found to be rather more significant as compared to the one-bond SSCC, with the differences of up to 10%, and even more—13.6%—for the

^{3}J

_{FH}in fluoroacetylene. In these calculations, it was established that the most important contributions arising from the connected triple excitations in the coupled cluster expansion are accounted for at the CC3 level. Thus, the CC3 method is expected to become a standard approach for the calculation of reference values of the nuclear spin–spin coupling constants.

_{2}F

_{2}), which represents an extremely challenging test for modern quantum chemical methods, within the CCSDT and CC3 models, respectively. For adequate comparison with the experiment, vibrational and relativistic corrections were calculated. Coupled cluster methods were used with very large basis sets and complete basis set (CBS) extrapolations. Namely, for the calculation of NMR shielding constants, basis sets as large as aug-cc-pCV7Z were used. Spin–spin coupling constants have been determined with specialized versions of the correlation consistent basis sets ccJ-pVXZ, further augmented with diffuse functions. Calculated values of spin–spin coupling constants turned out to be in very good agreement with the experiment. To be more precise, for the trans isomer of dinitrogen difluoride, the final calculated values of

^{1}J

_{NN}(−18.25 Hz),

^{1}J

_{NF}(172.98 Hz) and

^{2}J

_{NF}(−61.97 Hz) differ from the experimental values (−18.5, 172.8, and −62.8 Hz, respectively) by less than 1 Hz, while the calculated three-bond fluorine-fluorine spin–spin coupling constant,

^{3}J

_{FF}(−303.61 Hz), was found to deviate from the experimental datum (−316.4 Hz) by only 12.79 Hz (that is 4% of the experimental value). For the cis isomer, the differences occurred to be consistently larger: between 2 and 4 Hz for the one- and two-bond couplings,

^{1}J

_{NN},

^{1}J

_{NF},

^{2}J

_{NF}, and about 19 Hz for the three-bond coupling,

^{3}J

_{FF}. The deviation between calculated (gas phase) and experimental (solvated)

^{19}F shielding constants of the cis and trans isomers was found to be 15.7 and 11.1 ppm, respectively.

#### 3.3. Density Functional Theory

**r**). From this theorem it follows that, for a given many-particle system, the external potential ν(

**r**) and, thus, the Hamiltonian and thereby every ground state property of this system are determined only by the electron density ρ(

**r**). The second Hohenberg–Kohn theorem establishes a variational principle of quantum mechanics, which states that the electron density that minimizes the energy of the overall functional E[ρ(

**r**)] is the true electron density. This can be rephrased as follows: the energy of a given N-electron system, E[ρ(

**r**)], has a minimum equal to the ground state energy E

_{0}, which implies that for any trial electron density function, such that ∫ρ

_{trial}(

**r**)d

^{3}

**r**= N, the energy of the system, E[ρ

_{trial}(

**r**)], must satisfy the inequality E[ρ

_{trial}(

**r**)] ≥ E

_{0}. The Kohn–Sham (KS) computational scheme [272] is based on the Hohenberg–Kohn theorems. In that scheme, the exact function of the ground state electron density of a given many-particle system is replaced by a function of non-interacting particles. In KS theory, the total energy is written as:

_{s}), the interaction energy with the external field (E

_{ext}), the Hartree (E

_{H}) and the exchange-correlation (XC), E

_{xc}, energies. Thus, E

_{xc}is nothing more but the sum of errors originating from using the approximation of non-interacting particles to describe the kinetic energy term and E

_{H}instead of a real interelectronic interaction energy:

_{xc}[ρ] = (T[ρ] − T

_{s}[ρ]) + (E

_{ee}[ρ] − E

_{H}[ρ])

^{2}ρ).

**GGAs:****Hybrid GGA functionals:****Meta-GGA functionals:****Hybrid meta-GGA functionals:**- and
**Long-range corrected hybrid functionals:**

_{σ}. The parameters γ and δ were optimized so as to reproduce the NMR shielding constants of a wide range of molecules as accurately as possible. As a result, the final values of the parameters γ and δ are −0.006 and 0.1, respectively. The functionals KT2 and KT3 have more complicated forms and depend on more variational parameters as compared to the KT1. They were created on the basis of KT1 in order to improve the description of other properties, such as ionization potentials, electron affinity, proton affinity, bond angles, bond lengths, electronic polarizability, thermodynamic properties, etc. Extensive testing carried out by Keal and Tozer [324] showed that, despite significant modifications, the KT2 and KT3 functionals are not inferior in accuracy relative to KT1 functional in the calculations of shielding constants.

^{13}C,

^{15}N,

^{17}O and

^{19}F nuclei. At that, the OPBE was found to be the best method among the considered functionals, namely the B3LYP, PBE0, BLYP, PBE, OLYP, and OPBE. Moreover, since the parameters of OPTX functional were optimized by fitting to the unrestricted HF energies of the first- and second-row atoms, this functional (and so as the other functionals based on it) can be expected to give reliable excited state triplet properties in the sense of triplet instability issue.

^{1}H and

^{13}C NMR chemical shifts of six neutral and protonated alkylpyrroles by Zahn et al. [312] very recently. The considered functionals included 9 GGA functionals, 16 hybrid GGA functionals, 5 meta-GGA functionals, 4 hybrid meta-GGA, and 5 long-range corrected hybrid functionals. As stated by the authors, the aim of their study consisted in finding a DFT method that can outperform the MP2 and B3LYP methods in accuracy at predicting the chemical shifts for neutral and protonated alkylpyrroles. For

^{13}C chemical shifts, it was found that most of the functionals perform better than B3LYP, with the hybrid meta-GGA functional TPSSh demonstrating the best performance. At that, there was no single functional found to outperform the MP2 method. For the

^{1}H chemical shifts, it was found that all the considered functionals outperformed the MP2 and B3LYP methods, with the best performance demonstrated by TPSSh, likewise in the case of

^{13}C chemical shifts.

_{FC}SSCCs, it was shown that meta-GGA functional M06-L demonstrates surprisingly high accuracy, outperforming any other investigated functional, including the PBE0, otherwise considered one of the most reliable for this type of SSCCs. Although the computation of nuclear magnetic resonance (NMR) parameters involving F is known to be a challenging task [38], even with a rather small basis, such as pcJ-1, M06-L provided the results with a MAD = 11.7 Hz, whereas the MAD for the PBE0 results was assessed as much as 60.0 Hz. Providing that the J

_{FC}cover the range of about 300 Hz, the achieved accuracy can be considered as particularly remarkable. It was found that the accuracy of the M06-L/pcJ-1 scheme does not stem from a well-suited exchange or correlation part of the functional. Instead, Sauer et al. assumed that that high accuracy can arise from a fortuitous cancellation of errors, as revealed by investigating the convergence of the basis set. Their findings also indicated that

^{1}J

_{FC}constants are highly dependent on the amount of exact exchange included in the expression of the functional, with large fractions being critically important to achieving satisfactory results. Sauer et al. have also studied the effects of the geometry on the

^{1}J

_{FC}and found that optimizing the geometry at the same level of theory as used for the calculation of SSCCs generally improves the quality of the results.

_{λ}with the expansion coefficients ${C}_{\lambda i}^{L(\overrightarrow{X},\overrightarrow{Y})}$ depending on two magnetic perturbations, $\overrightarrow{X}$ and $\overrightarrow{Y}$. The small components can be expressed as the linear combination of the magnetically balanced (restricted condition, in the simplest approximation) basis functions ${\chi}_{\lambda}^{S(\overrightarrow{X},\overrightarrow{Y})}$ with the field-dependent coefficients ${C}_{\lambda i}^{S(\overrightarrow{X},\overrightarrow{Y})}$. Thus, the four-component molecular orbitals depend on the magnetic perturbations via both MO coefficients and basis functions. Due to this fact, the linear responses of MOs in Equation (73) can be expressed as follows:

_{p}through the usual linear-response expansion coefficients:

_{uv}can be presented as follows:

_{λ}. The unperturbed MO coefficients for the large and small components are usually obtained during the self-consistent-field (SCF) procedure. The linear-response coefficients of the occupied molecular orbitals are derived from the normalization condition, and that for the unoccupied molecular orbitals are obtained within the perturbation theory.

#### 3.4. Polarization Propagator Methods

**M**

^{−1}using the spectral theorem, which assumes solving the generalized eigenvalue problem for the matrix

**M**. In general, the calculation of NMR properties within the framework of the polarization propagator approach can briefly be described as a sequence of several steps: (a) solving the unperturbed Hartree–Fock problem in order to find the molecular coefficients, orbital energies, and the ground state energy; (b) calculating the elements of the effective matrices; (c) solving the generalized eigenvalue problem for the matrix

**M**; (d) evaluating the polarization propagator through Equation (88).

^{4}).

**M**in the SOPPA approach, are replaced with the coupled cluster amplitudes from the corresponding CC scheme, which makes it imperative to solve the system of cluster equations within the CC-modified SOPPA models. This complicates calculations, and, in the case of the SOPPA(CCSD) scheme, this even increases the scalability of the method by an order of magnitude in terms of the number of basis set functions N, i.e., from N

^{5}for SOPPA to N

^{6}for SOPPA(CCSD). The SOPPA method and its modifications have become very popular approaches in the calculations of SSCCs of various types [406,407,412,413,414,415].

^{4}), and that is quite an advantage as compared to SOPPA methods and its modifications. However, it neglects the electron correlation effects, which is problematic, especially for the triplet NMR properties resulting in triplet instabilities [416,417]. In this way, there is a need for the correlated wavefunction-based approaches to the calculation of NMR molecular properties, which would have less computational demands as compared to SOPPA or its modifications and yet would be more reliable than the RPA method. Recent developments of Sauer’s group pointed at resolving the issue. In particular, the higher RPA (HRPA) model [418,419,420] could have become one of the plausible extensions of the RPA method. It includes the second-order correction to the matrix

**M**of SOPPA, while the contributions of the double excitations found in the SOPPA model are omitted. This turned out to be an obstacle, as the importance of double excitations has already been acknowledged by that moment and methods such as CIS(D) [158,421,422], RPA(D) [420,423], and HRPA(D) [420] have been developed for the excitation energy calculations with promising results [420,421,422,423,424,425,426,427]. Thus, Schnack–Petersen et al. [428] presented two new modifications of the RPA method for the calculations of NMR indirect nuclear spin–spin coupling constants recently, namely the RPA(D) and HRPA(D). In these models, the double excitation contribution is treated noniteratively as a correction to the results of RPA or HRPA levels. The idea of the D-extended approaches consists in the solution of the generalized eigenvalue problem using the pseudoperturbation theory. That means expansion of the matrix

**M**and the vectors

**P**and

**Q**in a kind of perturbation series, based on the deficiency in relation to the corresponding SOPPA matrices and vectors. Thus, the SOPPA matrices and vectors can be partitioned into the contributions of different (pseudo-)order terms, where the zeroth-order matrices are chosen as those corresponding to a smaller problem, namely RPA or HRPA approximations.

^{4}O

^{1}due to the fact that both methods require transformation with the SOPPA matrices. In O-N-V terms, the RPA equations scale as N

^{4}, while the leading terms in the SOPPA approach scale as a partial two-electron integrals transformation, N

^{4}O

^{1}. According to the authors’ statement, within the RPA(D) model, only the RPA equations are solved iteratively and the N

^{4}O contribution are calculated once using the converged RPA vectors. Thus, the savings of RPA(D) relative to SOPPA for a large system is proportional to the number of iterations required to converge the SOPPA equations. The HRPA(D) method, on the other hand, requires the same amount of N

^{4}O terms as a SOPPA iteration, though the calculation of some mostly demanding terms can be avoided. The savings of HRPA in terms of the computational cost of an iteration is thus quite small in the typical case of V >> O. The test calculations of different types of SSCCs in a number of small inorganic molecules performed in [428] showed that both the RPA(D) and HRPA(D) models yield the results of good accuracy compared to the SOPPA model with noticeable time savings.

**Ω**is the diagonal matrix of excitation energies:

**x**is the matrix of “spectroscopic amplitudes”:

**M**and

**f**are found by comparison of Equation (91) with the Feynman–Goldstone diagrammatic series for

**Π**(ω) through a given order n of the perturbation theory. For this purpose, the matrices

**M**and

**f**are expanded in the power series in the fluctuation potential:

**M**=

**M**

^{(0)}+

**M**

^{(1)}+

**M**

^{(2)}+ …,

**f**=

**f**

^{(0)}+

**f**

^{(1)}+

**f**

^{(2)}+ …

^{5}.

_{2}, CO, H

_{2}O, HCN, NH

_{3}, CH

_{4}, C

_{2}H

_{2}, PH

_{3}, SiH

_{4}, CH

_{3}F, and C

_{2}H

_{4}. The calculated indirect nuclear spin–spin coupling constants occurred to be in a good agreement with the experimental data, which means that this method may happen to be promising for applications to larger molecules.

^{1}H) and σ(

^{127}I) in the HI molecule [130]. It should be noted that this was preceded by the developments of Quiney et al. [453,454,455], who showed the applicability of the London orbitals to the CP-DHF theory. Further, the developments of the 4RPA method were concentrated on the correct inclusion of the magnetic balance condition in the formalism of relativistic localized atomic orbitals, which is necessary for all relativistic four-component methods for calculating NMR shielding constants. Cheng et al. [104] presented the theory of the so-called magnetically balanced gauge-including atomic orbitals theory (MB-GIAOs), in which each “magnetically balanced” atomic orbital has its own local origin located in its center. Cheng and co-authors refuted an accepted statement that the magnetic balance (MB) condition is incompatible with the four-component relativistic polarization propagator theory 4c-PPT [456]. They believed that the statement is rather counterintuitive as its random phase approximation is, in the static limit, fully equivalent to the CP-DHF theory, and, in its turn, the CP-DHF theory is known to be compatible with the formalism of magnetically balanced localized orbitals. Thus, they showed that the MB-GIAO scheme can be combined with any four-component electronic structure calculation method, in particular, with the 4PPT methods. However, the first demonstration of the efficiency of the theory of relativistic MB-GIAO was carried out on the example of the shielding constants calculated within the framework of the Dirac–Kohn–Sham theory (DKS) or four-component DFT (4DFT) method [457,458]. The calculations of nuclear shielding constants within the 4RPA formalism were presented in a number of papers [112,452,459], however, this method has not received as much popularity as the Dirac–Kohn–Sham approach.

#### 3.5. Methods Based on the Many-Body Perturbation Theory

^{5}, with N being the number of basic functions). The correlation energy in MP2 is determined by the following equation:

_{pp}. Indexes i, j, k, l, … refer to occupied spin orbitals, while a, b, c, d designate the vacant (unoccupied) spin-orbitals. The general indices p, q, r, s, … are the spin-orbitals that can be either occupied or vacant. The MP2 expressions for the shielding tensor are obtained by differentiating the Equation (97) with respect to the nuclear magnetic moment, and then with respect to the magnetic field. The resulting expression has the form [463]:

^{6}and N

^{7}, respectively, against N

^{5}), but if applied with the approximation of the resolution-of-the-identity (RI) [241,482], the calculations with these models could become more feasible. In practice, Gauss suggested the GIAO-SDQ-MBPT(4) approximation for the GIAO-MP4 approach [476], specifically for the cases when triple correlation effects play a minor role in a given system. The computational cost of the GIAO-SDQ-MBPT(4) approach scales as N

^{6}.

^{125}Te nucleus in various tellurium compounds. It was shown that the calculated magnetic shielding constants and NMR chemical shifts are well reproduced by the relativistic MP2 theory as compared to the experimental values.

## 4. Computational Factors Influencing the Accuracy of NMR Spectrum Modeling

#### 4.1. Specialized Basis Sets

#### 4.1.1. Specialized Basis Sets for Calculating Spin–Spin Coupling Constants

_{i}= αβ

^{i}, where α is the last exponent in the original basis set (ζ

_{n}), and β is the ratio of the two last exponents β = ζ

_{n}/ζ

_{n}

_{-1}. Such a way of building additional exponential sets leads to the even-tempered functional series, for which the ratio between two neighboring exponents is a constant: ζ

_{i}/ζ

_{i}

_{-1}= β = const. In most cases, dealing with the design of the nonrelativistic J-oriented basis sets, the energy-optimized correlation consistent Dunning’s basis sets [266,495,496,497,498] are usually taken as starting basis sets, though, there are several examples of using medium-size polarized Sadley’s basis sets MSP [499,500,501,502] or small Huzinaga basis sets [62,503], although the latter two are much less popular.

^{1}J(

^{13}C,

^{1}H) and

^{2}J(

^{1}H,

^{1}H) in the CH

_{4}molecule. It was proposed to use fully uncontracted aug-cc-pVTZ basis set, augmented by four additional tight s-type functions, minus one f-function, on the carbon and hydrogen atoms. Helgaker et al. [200] analyzed the basis set dependence of

^{1}J(H,F) and

^{1}J(O,H) in two simple molecules, HF and H

_{2}O, respectively, using the complete active space self-consistent field (CASSCF) method with a large active space. The Dunning’s basis sets cc-pVXZ (X = D, T, Q, 5, 6) were decontracted completely in the s-space, resulting in cc-pVXZ-su0 sets, then a sequence of n tight s functions with the exponents forming a geometric progression was added, resulting in a series of basis sets cc-pVXZ-sun. His group also conducted the investigation [302] on how the expansion of the s-space of Huzinaga’s basis sets, HIII [503], affects the SSCCs with leading FC contribution calculated at the DFT-B3LYP level. Overall, it was shown that the sequential saturation of the s shell provides a gradual improvement of the description of the FC contribution to the coupling constants.

^{1}H,

^{13}C,

^{15}N, and

^{17}O nuclei with leading Fermi-contact contribution within the framework of nonrelativistic DFT approach.

^{1}H,

^{13}C,

^{15}N, and

^{19}F. It was mentioned that the PEC method is aimed at the generation of very efficient small property-oriented basis sets, which provide more accurate results as compared to the other property-oriented basis sets of similar sizes, while, for the larger basis sets, the accuracy of the results is expected to be comparable to that provided by the other property-oriented commensurate basis sets. The first calculations of SSCCs involving the nuclei

^{1}H,

^{13}C,

^{15}N, and

^{19}F in a wide row of small molecules, performed with the pecJ-n (n = 1, 2), ccJ-pVXZ (X = D, T) and pcJ-n (n = 1, 2) basis sets at the CCSD level of theory with taking into account solvent and vibrational corrections, confirmed the above-mentioned statement. In particular, the accuracy of

^{1}J(C,C) SSCCs, calculated with the pecJ-1 and pec-2 basis sets was characterized by the MAPE of 3.1% and 2.6%, respectively, against the experiment. At the same time, the MAPEs of the theoretical corresponding data, obtained with the ccJ-pVDZ and pcJ-1 basis sets were found to be approximately 9.8% and 10.6%, and for the ccJ-pVTZ and pcJ-2 basis sets, these figures occurred to be 3.7% and 2.3%, correspondingly.

#### 4.1.2. Specialized Basic Sets for Calculating NMR Chemical Shifts

_{4}, NH

_{3}, H

_{2}O, SiH

_{4}, PH

_{3}, H

_{2}S, CO

_{2}, C

_{2}H

_{4}, and C

_{2}H

_{2}calculated at the GIAO-CPHF level with five different families of basis sets, namely, the Pople’s basis sets 6-31G, 6-311G with and without polarization and diffuse functions [513,514,533,534,535,536,537,538], Ahlrich basis sets (Karlsruhe-XZP, X = D, T, Q) [268,539], basis sets of Schindler and Kutzelnigg (IGLO II, III, IV) [61,62], Widmark basis sets (ANO(Lund)) [540,541,542], and correlation-consistent basis sets of Dunning, (aug)cc-pVXZ (X = D, T, Q) [266,495,496,497,498] against the GIAO-CPHF CBS limit. The fastest convergence was demonstrated within the family of IGLO basis sets, with the mean relative error less than 3% for the two largest representatives of this family, IGLO III, IV. However, they are not apt for the medium and large molecules due to their large sizes. The important conclusion made by the authors consists of the following: for an accurate calculation of nuclear shielding constants, a basis set of at least valence triple-ζ quality and with at least one set of polarization functions is needed. This conclusion is in accordance with Carmichael’s observation [543] that, for the reliable calculations of the shielding constants, there is a need for flexibility in the outer-core inner-valence regions. Basis sets with tightly contracted core orbitals, such as the correlation-consistent basis sets and the small ANO sets, have little flexibility in the core region and perform poorly in the calculations of nuclear shielding constants. In general, just as in the case of SSCCs, standard, energy-optimized, single-electron basis sets are ineffective for the nuclear shielding constants, since in order to approach the CBS limit within the framework of a particular method, it is necessary to use rather large, non-specialized basis sets.

^{1/4}between its exponents) were determined for a large set of more than 250 molecules. For the heavy elements, functions with larger exponents were added to flexibilize the description of the density in the outer-core region. The exponents were optimized in several cycles reducing the mean absolute error in the NMR shielding constants. To improve the description of the inner-most shells, tight p functions were introduced, namely a single additional p function was added for each element. Therefore, the exponent of the inner-most primitive function of the parent x2c-SVPall or x2c-TZVPall sets was scaled with a factor of 6.5 and the outer-most primitive was excluded from the segment and utilized as augmenting function to increase the flexibility. The contraction coefficients of the new segment were re-optimized at the X2C level of theory in atomic calculations resorting to quasi-Newton algorithm based on the variational principle. The x2c-SVPall-s and x2c-TZVPall-s basis sets were further compared to the segmented-contracted Jensen’s basis sets pcSseg-n (n = 0–4) based on the percent-wise error [545] measured against the large reference even-tempered basis set.

#### 4.2. Vibrational Corrections

_{exp}as follows:

_{i}is a mass-weighted displacement of the nuclei from the expansion geometry along the normal coordinate i (q

_{i}= r

_{i}− r

_{exp,i}), and N is the number of normal coordinates, namely, 3K–6 in general case, or 3K–5 for linear molecules, with K being the number of atoms in the molecule. ${P}_{\mathrm{exp},{i}_{1},{i}_{2},\dots ,{i}_{n}}^{(n)}$ represents the nth derivative of the property at the expansion point with respect to normal coordinates; ${P}_{\mathrm{exp}}^{(0)}$ is the property at the expansion point. To find the vibrational wave function Ψ, a standard Rayleigh–Schrödinger vibrational perturbation theory is applied. The vibrational perturbation theory uses the harmonic oscillator Hamiltonian as the zeroth-order Hamiltonian. The latter represents the sum of the nuclear kinetic energy operator and the quadratic term of the potential energy surface expansion in terms of deviations q

_{i}:

^{(n)}and Ψ

^{(n)}to E

^{(0)}and Ψ

^{(0)}. In the first order, E

^{(1)}= <Ψ

^{(0)}|H

^{(1)}| Ψ

^{(0)}> equals to zero, as H

^{(1)}(see Equation (108)) contains only the odd terms with respect to at least one geometrical displacement q

_{i}. The second-order energy can be regarded as an energy functional:

_{i}-depending multipliers (for the exact equation, see [587]). Thus, in the second-order of vibrational perturbation theory, the ZPV correction to a property can be written as [590,591]:

#### 4.2.1. Vibrational Corrections to Spin–Spin Coupling Constants

_{eq}, and zero-point vibrational correction (ZPVC), ΔJ

_{vib}, as follows [561,590,591]:

_{i}; ω

_{i}are the harmonic vibrational frequencies, and F

_{eq,ijj}are the semi-diagonal cubic force constants. Thus, the calculation of vibrational corrections requires the calculation of the geometric derivatives of the potential energy surface and the SSCCs by nuclear coordinates. This results in the large number of repeated calculations of the SSCCs for different molecular geometries. In general, this is a very time-consuming procedure, which requires large computational effort. At that, due to the particular form of the of the operators representing the interaction between the nuclear spins and the electronic spin and angular momentum, SSCCs require specialized J-oriented basis sets and computational approaches that circumvent the triplet instability problems manifesting in the calculations of the FC and SD contributions. Therefore, the calculation of the vibrational corrections to the NMR spin–spin coupling constants is generally considered to be more challenging task than the calculation of those to any other linear response property.

_{4}molecules (X = C, Si, Ge, Sn) at different temperatures at the RPA, SOPPA, and CASSCF levels. According to their results, the value of the vibrational corrections to the considered SSCCs, was found to be about 1–3%, in average, of the total values of SSCCs. Yachmenev et al. [597] estimated the vibrational corrections to nitrogen-proton and proton-proton SSCCs at different temperatures, namely 0 and 300 K, in ammonia isotopomers. For nitrogen-proton and deuterium-proton SSCCs, it was shown that the total vibrational corrections are about 0.6% and 5% of the total values, respectively. The effect of non-zero temperature was found to be insignificant, amounting in hundredths of Hz. Jordan et al. [598] calculated the vibrational corrections for nitrogen–nitrogen SSCC through the hydrogen bond in model hydrogen-bonded complex, CNH:NCH. In that paper, the expectation values of <

^{2}J(N,N)>

_{0,vib}were obtained from the two-dimensional potential energy surface for CN

_{a}H:N

_{b}CH, generated in the N

_{a}-H and N

_{b}-H distances at the MP2 level and the global SSCC surface, which was calculated at the EOM-CCSD level of theory. Despite the fact that only two vibrational modes were considered, the property calculations were performed for 108 single-point molecular geometries, which illustrates the extremely high computational cost of the vibrational problem. Del Bene et al. [599] carried out the calculations of the vibrational effects on the F-F SSCCs (

^{2h}J

_{F-F}) for the FHF

^{-}molecule. The coupling constant surface was generated at the EOM-CCSD level, and two-dimensional wavefunctions for the symmetric and asymmetric stretching vibrations were obtained from the potential energy surface evaluated at the CCSD(T) level of theory. The effect of FHF

^{-}bending mode was also investigated using the one-dimensional calculations along the bending normal coordinate. In the ground vibrational state, the expectation value of F-F SSCC, <

^{2h}J(F,F)>

_{0,vib}, was found to be 212.7 Hz, which is significantly less, namely, by 41.7 Hz, than that at the equilibrium geometry (254.4 Hz). At the same time, the effect of the bending mode was found to be unessential, namely, in the ground vibrational state of the bending mode, the average value of

^{2h}J(F,F) is 253.2 Hz, which is very similar to the equilibrium value of 254.4 Hz. This small effect was explained by the large difference in masses of the hydrogen and fluorine atoms, which leads to the fact that the bending vibration principally involves the motion of the hydrogen atom, leaving the F-F distance practically unchanged.

_{2}, HF, CO, N

_{2}, H

_{2}O, HCN, NH

_{3}, CH

_{4}, C

_{2}H

_{2}) on the basis set at the DFT(B3LYP) level. They considered two series of Huzinaga basis sets. The first sequence consisted of the Huzinaga sets HII, HIII, and HIV [503] with the polarization functions and contraction patterns of van Wüllen and Kutzelnigg et al. [600], and the second series included the Huzinaga’s basis sets, possessing enlarged flexibility in the inner core region, namely the HX-sun basis sets (X = II, n = 2; X = III, n =3; X = IV, n = 4). They have found that the HIV-su4 basis set gives the vibrational corrections close to the CBS limit, achievable within the DFT model. Therefore, it was recommended to use the HIV-su4 basis set only in very precise calculations of vibrational corrections to SSCCs in very small systems. At the same time, smaller basis sets such as HIII-su3, also give very good accuracy, therefore, it was recommended to use them in the routine calculations of vibrational effects on the SSCCs of larger systems. The ZPVCs obtained in ref. [591] were compared with those calculated in previous works [404,405,407,412,592,594,601,602,603]. It was found that the DFT vibrational corrections are in good agreement with those calculated using the other correlated non-empirical methods, except for two cases of striking differences—the

^{1}J

_{NN}in N

_{2}and

^{3}J

_{HH}in C

_{2}H

_{2}. Although the obtained B3LYP results turned out to be close to experiment, the authors did not attach much significance to this fact since, for the particular systems, the B3LYP might predict much too low equilibrium coupling constants. In general, it is well known that restricted Kohn–Sham theory is known to manifest the triplet instability problem [316], which leads to an unbalanced description of the ground state and the most important excited states of a given symmetry, thus, providing a poor description of the molecular property of interest that depends on these states. In the case of SSCCs, the triplet instability results in incorrect calculation of the SD and FC terms (which are the triplet second-order properties) and may sometimes give an error of several orders of magnitude.

#### 4.2.2. Vibrational Corrections to NMR Shielding Constants

_{2}, HF, N

_{2}, CO, and F

_{2}at the coupled cluster singles and doubles level augmented by a perturbative correction for triple excitations, CCSD(T). The shielding constants for the lowest rovibrational states of the considered diatomic molecules were obtained by solving the rovibrational Schrödinger equation with the finite-element techniques followed by evaluating appropriate expectation values. Temperature effects have been accounted for by applying the Boltzmann averaging. The total calculated shielding constants were in good agreement with the available experimental values, except for the F

_{2}molecule. The deviations observed for the F

_{2}molecule turned out to be about 4 ppm smaller than the experimental value.

_{2}. The shielding tensors of

^{13}C and

^{77}Se nuclei, with and without taking into account the vibrational and rotational degrees of freedom, were calculated using several ab initio methods in conjunction with the GIAO formalism. The results obtained at the CHF, MCSCF (RAS, CAS), and DFT (LDA, BLYP and BPW91) levels were compared with theoretical data taken from other different sources. Thus, according to the results, obtained within the CAS and DFT(BPW91) methods, the vibrational-rotational corrections (T = 300 K) to σ(

^{77}Se) in the

^{77}SE=

^{13}C=

^{80}Se isotopomer is about 2–2.5% relative to the full values. The vibrational-rotational corrections to the σ(

^{13}C) were found to be about 5.5–6%. Overall, it was found that the effect of rotational degrees of freedom does not exceed hundredths of a percent of the total value of both shielding constants. In the particular case of the CSe

_{2}molecule, the authors have come to a conclusion that taking into account vibrational and rotational degrees of freedom worsens the agreement of theoretical values with the experiment. This observation has been explained by the lack of relativistic and solvent corrections.

^{−}and H

_{3}O

_{2}

^{−}were investigated by means of ab initio calculations at the RPA and SOPPA levels by Sauer et al. [617]. The effective shielding constants were obtained by averaging of the property over the rovibrational wavefunctions, which are the solutions of the one-dimensional radial Schrödinger equation, solved numerically [618]. The dependence of the nuclear magnetic shielding constants in H

_{3}O

_{2}

^{−}on the anharmonic symmetric and antisymmetric O

^{…}H

^{…}O stretching motions and on the internal rotation motion of the outer hydrogens was studied with the non-rigid bender model Hamiltonian [619] at the RPA level. The dependence of the shielding constants in OH

^{−}on the bond length was investigated at RPA and SOPPA levels. For all atoms of H

_{3}O

_{2}

^{−}, with the exception of the outer hydrogen atoms, a strong dependence on the vibrational quantum number was found for the nuclear magnetic shielding constants. Namely, the NMR shielding constant of the oxygen atom in H

_{3}O

_{2}

^{−}and OH

^{−}were found to increase and decrease, respectively, with the vibrational quantum number. For the hydrogen shielding constants, the opposite behavior was found.

^{1}H shielding constant manifests strong dependence on the internuclear separation. The zero-point vibrationally averaged shielding constant was calculated as the averaged Taylor series expansion:

_{e}. The first and second derivatives of the shielding constant, σ

_{r}and σ

_{rr}were determined by fitting the fourth-order polynomial to the shielding constants calculated in the bond-distance interval 1.4–1.8 Å, with points separated by 0.05 Å. The averages of the bond length extension and its square were calculated using the harmonic and cubic force constants, F

_{rr}and F

_{rrr}by fitting the fourth-order polynomial to the total SOC corrected energies. It was found that the total nonrelativistic vibrationally averaged

^{1}H shielding constant for HI is slightly increased compared to the value calculated at the equilibrium distance (tenth of ppm). At that, a heavy atom SOC induced effect was shown to reverse the sign of the vibrational contribution to the

^{1}H shielding.

_{2}O molecule, the vibrational effects on nuclear shielding constants were studied thoroughly [561,574,588,615,621,622,623,624]. A pioneering study was performed by Fowler and Raynes [574], using an empirical force-field with HF shielding surfaces to obtain a ZPVC of −13.1 ppm to σ(

^{17}O) of H

_{2}

^{17}O. Then, the correlated study of rovibrational effects on the nuclear shielding constants in the water molecule was performed by Vaara et al. [625]. The restricted active space self-consistent field (RASSCF) wave function with large RAS and large basis sets were used to calculate the rovibrational corrections and the related temperature and isotope dependencies with high accuracy. It was shown that the rovibrational effects are as important as those of electron correlation. In particular, the rovibrational corrections were found as 3.7% and 1.8% for the isotropic oxygen and hydrogen shielding constants, respectively, in the

^{1}H

_{2}

^{17}O isotopomer at 300 K. On the basis of the calculations presented in ref. [625] and the CCSD(T) results of Gauss et al. [259,626], a new absolute shielding scale for the

^{17}O nucleus was proposed, namely the value of the oxygen shielding constant of H

_{2}

^{17}O isotopomer in the gas phase at 300 K was established as 324.0 ± 1.5 ppm. Wigglesworth et al. [621] have also performed ab initio calculations of hydrogen and oxygen shielding surfaces for the water molecule at the MCSCF level. The rovibrationally averaged shielding constants of the various isotopomers of water and their temperature dependences were obtained. To determine the relevant coefficients for the expansions of the σ(H) and σ(O) around the equilibrium geometry in terms of the symmetry coordinates, describing the displacements from equilibrium geometry, the calculations of shielding constants at 49 distinct locations on the proton surface and 37 distinct locations on the oxygen surface were carried out. Wigglesworth et al. obtained ZPVC to σ(

^{17}O) of H

_{2}

^{17}O as −9.9 ppm.

_{2}

^{16}O and σ(O) in H

_{2}

^{17}O were simulated as follows:

_{e}(

^{17}O) for equilibrium value occurred to be substantially different. The value of σ

_{e}(

^{17}O) proposed by Wigglesworth et al. was 333.723 ppm, while that of Vaara et al. was 324.0 ± 1.5 ppm. Wigglesworth et al. [605] continued the investigation of the rovibrational effects on carbon and hydrogen NMR shielding constants on example of acetylene molecule. The calculations were performed at the correlated MCSCF level of theory using gauge-including atomic orbitals and a large basis set.

^{17}O. They used the CCSD(T) method with a large basis set and an accurate numerical description of the vibrational problem to compute the ZPVC of −11.7 ppm. Komorovsky et al. [627] combined previously published high-quality experimental spin–rotation data, accurate coupled cluster calculations of Puzzarini and personal relativistic four-component Kohn–Sham density functional calculations of the shielding and spin–rotation constants of H

_{2}

^{17}O, and revised the absolute shielding value for the

^{17}O nucleus of H

_{2}

^{17}O as 328.4(3) ppm at 300 K.

^{1}H) and σ(

^{17}O), whereas aug-pc-n predicted a slightly smaller absolute value.

^{125}Te NMR chemical shifts were recently calculated by Rusakova et al. [629]. In that paper, the main factors affecting the accuracy and computational cost of the calculation of

^{125}Te NMR chemical shifts in medium-size organotellurium compounds were analyzed at the GIAO−DFT level. The LDBS schemes, relativistic corrections, solvent effects and vibrational corrections were considered as the primary accuracy factors.

#### 4.3. Solvation Models

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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