Electric Field Gradient in Chiral and Tetrahedral Molecules within High-Order LRESC Formalism.

In this work, we present the electric field gradient (EFG) given by the linear response elimination of the small component (LRESC) scheme up to the 1/c4 order (c is the speed of light in vacuum) in CHFClX (X = Br, I, At) chiral molecules, together with CHF2Br and CH2FX (X = Br, I, At) tetrahedral systems. The former could be good candidates for further parity violation studies, especially when heavy atoms are surrounding. In this context, the LRESC scheme demonstrates effective applicability to large tetrahedral and chiral molecules that incorporate heavy elements, with relativistic effects playing a crucial role. The LRESC results of EFG exhibit an excellent agreement with those calculated at the four-component level, giving differences of only hundredths order in a.u. (atomic units) for the bromine nucleus and less than 0.1 a.u. for the iodine nucleus. Regarding the other nuclei, for the chiral molecules, there is a heavy atom effect on the light atom (HALA) for chlorine and fluorine atoms as the substituent halogen atom becomes heavier. Furthermore, the electronic part of the EFG for the central carbon and the fluorine nuclei presents an important dependence with the environment in the molecules under study. With accurate calculations of the EFG and tabulated nuclear quadrupole moment, the nuclear quadrupole coupling constant is obtained within the LRESC scheme, including for the first time correlation effects on the spin-dependent corrections with this methodology, providing results close to the experimental ones for Cl, Br, and I atoms. At the Hartree-Fock level, the differences are around 6% for Br and I nuclei, and at the density functional theory level with the LDA and PBE0 functionals, the differences are no more than 2%.

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

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

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

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

High order relativistic corrections on the electric field gradient within the LRESC formalism.

In this work, we present relativistic corrections to the electric field gradient (EFG) given by the Linear Response Elimination of the Small Component (LRESC) scheme at 1/c2 order and including for the first time spin-dependent (SD) corrections at 1/c4 order. We show that these new terms improve the performance of LRESC as results with this methodology are very close to those calculated at the four-component Dirac-Hartree-Fock (4c-DHF) level. We assess the new corrections in BrY and AtY di-halogen (Y = F, Cl, Br, I, and At) and XZY bi-linear molecules (Z = Zn, Cd, and Hg; X, Y = F, Cl, Br, I, and At). At the 4c-DHF level, we analyze the contributions coming from the large and small components of the relativistic 4c wave function to the electronic part of EFG and compare them with the LRESC corrections to find their electronic origin. For the HgX2 (X = Cl, Br, and I) subset, when the SD correcting terms are included, LRESC calculations match very well with 4c-DHF ones and those from the literature, with differences less than 1% for molecules containing up to three heavy atoms. We show that LRESC gives accurate values of EFG, allowing the analysis of the electronic origin of relativistic effects in terms of well-known nonrelativistic operators.

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

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

Microsolvation of methylmercury: structures, energies, bonding and NMR constants ((199)Hg, (13)C and (17)O).

Hartree-Fock (HF) and second order perturbation theory (MP2) calculations within the scalar and full relativistic frames were carried out in order to determine the equilibrium geometries and interaction energies between cationic methylmercury (CH3Hg(+)) and up to three water molecules. A total of nine structures were obtained. Bonding properties were analyzed using the Quantum Theory of Atoms In Molecules (QTAIM). The analyses of the topology of electron densities reveal that all structures exhibit a partially covalent HgO interaction between methylmercury and one water molecule. Consideration of additional water molecules suggests that they solvate the (CH3HgOH2)(+) unit. Nuclear magnetic shielding constants σ((199)Hg), σ((13)C) and σ((17)O), as well as indirect spin-spin coupling constants J((199)Hg-(13)C), J((199)Hg-(17)O) and J((13)C-(17)O), were calculated for each one of the geometries. Thermodynamic stability and the values of NMR constants correlate with the ability of the system to directly coordinate oxygen atoms of water molecules to the mercury atom in methylmercury and with the formation of hydrogen bonds among solvating water molecules. Relativistic effects account for 11% on σ((13)C) and 14% on σ((17)O), which is due to the presence of Hg (heavy atom on light atom, HALA effect), while the relativistic effects on σ((199)Hg) are close to 50% (heavy atom on heavy atom itself, HAHA effect). J-coupling constants are highly influenced by relativity when mercury is involved as in J((199)Hg-(13)C) and J((199)Hg-(17)O). On the other hand, our results show that the values of NMR constants for carbon and oxygen, atoms which are connected through mercury (C-HgO), are highly correlated and are greatly influenced by the presence of water molecules. Water molecules introduce additional electronic effects to the relativistic effects due to the mercury atom.

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

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

Core-dependent and ligand-dependent relativistic corrections to the nuclear magnetic shieldings in MH4-n Y n (n = 0-4; M = Si, Ge, Sn, and Y = H, F, Cl, Br, I) model compounds.

The nuclear magnetic shieldings of Si, Ge, and Sn in MH(4-n) Y(n) (M = Si, Ge, Sn; Y = F, Cl, Br, I and n = 1-4) molecular systems are highly influenced by the substitution of one or more hydrogens by heavy-halogen atoms. We applied the linear response elimination of small components (LRESC) formalism to calculate those shieldings and learn whether including only a few of the leading relativistic correction terms is sufficient to be able to quantitatively reproduce the full relativistic value. It was observed that the nuclear magnetic shieldings change as the number of heavy halogen substituents and their weights vary, and the pattern of σ(M) generally does not exhibit the normal halogen dependence (NHD) behavior that can be seen in similar molecular systems containing carbon atoms. We also analyzed each relativistic correction afforded by the LRESC method and split them in two: core-dependent and ligand-dependent contributions; we then looked for the electronic mechanisms involved in the different relativistic effects and in the total relativistic value. Based on this analysis, we were able to study the electronic mechanism involved in a recently proposed relativistic effect, the "heavy atom effect on vicinal heavy atom" (HAVHA), in more detail. We found that the main electronic mechanism is the spin-orbit or σ p (T(3)) correction, although other corrections such as σ p (S(1)) and σ p (S(3)) are also important. Finally, we analyzed proton magnetic shieldings and found that, for molecules containing Sn as the central atom, σ(H) decreases as the number of heavy halogen substituents (of the same type: either F, Cl, or Br) increases, albeit at different rates for different halogens. σ(H) only increase as the number of halogen substituents increases if the halogen is iodine.

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

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

Relativistic effects on nuclear magnetic shieldings of CH(n)X(4-n) and CHXYZ (X, Y, Z = H, F, Cl, Br, I).

Nuclear magnetic shieldings of both carbon and hydrogen atoms of haluro methyl molecules are highly influenced by the substitution of one or more hydrogen by halogen heavy atoms. We applied the linear response elimination of small components, LRESC, formalism to calculate such shieldings and learn whether including only few terms is enough for getting quantitative reproduction of the total shieldings or not. First, we discuss the contribution of all leading relativistic corrections to σ(C), in CHX(2)I molecular models with X = H, F, and Cl, and show that spin-orbit (SO) effects are the main ones. After adding the SO effects to the non-relativistic (NR) results, we obtain ~ 97% (93%) of the total LRESC values for σ(C) (σ(H)). The magnitude of SO terms increases when the halogen atom becomes heavier. In this case, such contributions to σ(C) can be extrapolated as a function of Z, the halogen atomic number. Furthermore, when paramagnetic spin-orbit (PSO) contributions are also considered, we obtain results that are within 1% of the total LRESC value. Then we study in detail the main electronic mechanisms involved to contribute C and H shieldings on CH(n)X(4 - n) (n = 1, 3), and CHXYZ (X, Y, Z = F, Cl, Br, I) model compounds. The pattern of σ(C) for all series of compounds follows a normal halogen dependence (NHD), though with different rate of increase. A special family of compounds is that of CHF(2)X for which σ(nr)(C) follows an inverse halogen dependence though the total shielding have a NHD due to the SO contributions. For the series CH(3)X (X = F, Cl, Br and I), we found that σ(SO) ~ Z(X) (2.53). Another important finding of this work is the logarithmic dependence of σ(SO)(C) with the substituent atomic number: ln σ(SO)(C) = A(X) + a(X) Z(Y) for both family of compounds CH(2)XY and CHX(2)Y. We also performed four-component calculations using the spin-free Hamiltonian to obtain SO contributions within a four-component framework.

Nuclear charge-distribution effects on the NMR spectroscopy parameters.

We present here a systematic study about the influence of the size and type of nuclear charge-distribution models (Gaussian and point-like) on the NMR spectroscopic parameters, the nuclear magnetic shielding σ and the indirect nuclear spin J-coupling. We found that relativistic effects largely enhance the nuclear charge-distribution effects (NChDE) on those parameters being them quite sensitive to the nuclear model adopted for calculations. Results for two rare gas atoms (Kr, Rn) and few molecular systems like HX, (X = Br, I, At), CH(4), SnH(4), SnIH(3), SnI(2)H(2), and PbIH(3) are presented. J-couplings are more sensitive than shieldings in both, relativistic and non-relativistic (NR) regimes. The highest effect (close to 11% of variation in relativistic calculations with that two different nuclear models) is observed for J(Pb-I) in PbIH(3). A similar effect is found for J(Pb-H) in the same molecule, close to 9%. The NChDE for σ(Sn) in SnI(4-n)H(n) with n = 1, 2 is as large as few ppm (between 3 and 8.56 ppm). For J(Sn-H) in this set of molecules, it goes from 37 Hz for SnH(4) to 54 Hz for SnI(2)H(2). Furthermore, we found that the vicinal NChDE is very small though not zero. For (1)J(Sn-H) in SnIH(3), the NChDE of iodine is close to 2 Hz (0.1%). We also studied the NChDE on the ground state electronic energies of atoms and molecules. We found that these effects are only important within the relativistic regime but not within the NR one. They are in good agreement with previous works.

Relativistic effects on group-12 metal nuclear shieldings.

The leading-order perturbation theory approach to relativistic effects on the nuclear magnetic shielding provides an economic method for obtaining the chemical shifts in heavy-element containing systems. The method features detailed analysis potential in terms of the different physical mechanisms affecting the shielding tensors of heavy nuclei. The perturbative nature, however, results in an increasing error with increasingly heavy elements in the system. In this work, we investigate the performance of the Breit-Pauli perturbation theory (BPPT) against fully relativistic four-component theory in computing the nuclear shielding constants as well as the chemical shifts with respect to corresponding atomic ions of group-12 metals, M = Zn, Cd, and Hg, in dimethyl M(CH(3))(2) and aqueous M(H(2)O)(6)(2+) complexes. It is shown that five out of the total of sixteen BPPT correction terms are responsible for most of the relativistic corrections for the chemical shift of studied metals. The relativity is important already for Cd and BPPT is proven to work well up to Hg for the chemical shift, as calibrated with the fully relativistic method.

The UKB prescription and the heavy atom effects on the nuclear magnetic shielding of vicinal heavy atoms.

Fully relativistic calculations of NMR magnetic shielding on XYH3 (X = C, Si, Ge and Sn; Y = Br, I), XHn (n = 1-4) molecular systems and noble gases performed with a fully relativistic polarization propagator formalism at the RPA level of approach are presented. The rate of convergence (size of basis set and time involved) for calculations with both kinetic balance prescriptions, RKB and UKB, were investigated. Calculations with UKB makes it feasible to obtain reliable results for two or more heavy-atom-containing molecules. For such XYH3 systems, the influence of heavy vicinal halogen atoms on sigma(X) is such that heavy atom effects on heavy atoms (vicinal plus their own effects or HAVHA + HAHA effects) amount to 30.50% for X = Sn and Y = I; being the HAHA effect of the order of 25%. So the vicinal effect alone is of the order of 5.5%. The vicinal heavy atom effect on light atoms (HALA effect) is of the order of 28% for X = C and Y = I. A similar behaviour, but of opposite sign, is observed for sigma(Y) for which sigmaR-NR (I; X = C) (HAHA effect) is around 27% and sigmaR-NR(I; X = Sn) (HAVHA + HAHA effects) is close to 21%. Its electronic origin is paramagnetic for halogen atoms but both dia- and paramagnetic for central atoms. The effect on two bond distant hydrogen atoms is such that the largest variation of sigma(H) within the same family of XYH3 molecules appears for X = Si and Y = I: around 20%. In this case sigma(H; X = Sn, Y = I) = 33.45 ppm and sigma(H; X = Sn, Y = H) = 27.82 ppm.