Alejandro F. Maldonado

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

Magnetic molecular spectroscopic properties, like NMR J-coupling and magnetic shielding σ, have been studied by non-relativistic quantum methods since their discovery. When they were found to depend strongly on relativistic effects in molecules containing heavy atoms, this started a new area of intensive research into the development of methods that include such effects. In most cases non-relativistic concepts were extended to the new field though keeping the previous non-relativistic point of view. Quantum mechanics can be formulated by two different formal approaches. Molecular physics and quantum chemistry were developed mostly within the Schrödinger or Heisenberg approaches. The path integral formalism of Feynman is less well known. This may be the reason why propagators are not broadly known in this field of research. Polarization propagators were developed in the early 1970s. Since that time they have been successfully applied to calculate NMR spectroscopic parameters. They are special theoretical devices from which one can do a deep analysis of the electronic mechanisms that underly any molecular response property from basic theoretical elements, like molecular orbitals, electronic excitation energies, coupling pathways, entanglement, contributions within different levels of theory, etc. All this is obtained in a natural way in both regimes: relativistic and non-relativistic. Its relativistic generalization in the early 1990s and the finding of a quantum electrodynamic (QED)-based theory for them, has given us the opportunity to improve our understanding of the physics behind such parameters. In this paper we give a presentation of polarization propagators that start in non-relativistic quantum physics and end up with the introduction of QED effects. The same and powerful basic quantum ideas are applied throughout this review, so that coherence and beauty arise in a natural way. We will give a new understanding that comes from the three levels of theory: non-relativistic, relativistic and QED. We will be cautious to highlight what one can learn when working in any of these regimes, being aware that the same concepts may have different meanings. Starting with a general overview of polarization propagators we develop both the theory and applications in each of the three levels mentioned above. We show how some of the most subtle properties of the NMR spectroscopic parameters can be explained in a simple and beautiful way: the sign of J, the Karplus rule, cooperative effects, diamagnetic and paramagnetic contributions; we also show new rules that appear within the relativistic regime and the way previously valid rules within the non-relativistic theory are broken. We highlight the fact that within the relativistic theory of polarization propagators the whole set of electronic mechanisms that appears within the NR regime and within its quasi-relativistic extension is unified. Furthermore the usual notion of diamagnetism and paramagnetism are now nonsense as such because they arise only after as an approximation from expressions which include a unique type of electronic (electron–positron) mechanism.

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.

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.

The appearance of an interval of energies that contain the whole diamagnetic contribution to NMR magnetic shieldings.

Working within relativistic polarization propagator approach, it was shown in a previous article that the electronic origin of diamagnetic contributions to NMR nuclear magnetic shielding, sigmad, are mostly excitations that fit in a well defined interval of energies such that 2mc2<or==(epsiloni-epsilons)<4mc2. That interval of energies does not have, in principle, any physical reason to be so well defined, and gives a large amount of the total contribution to sigmad, e.g., close to 98% of it. Then a further study is given in this article, where we show some of the main characteristics of that interval of energy, such as its universal appearance and basis set independence. Our main result is the finding that sigmad is completely described by that interval of excitation energies, i.e., there is no contribution arising from outside of it. Most of the contributions belonging to that interval arise from virtual electronic energies larger than -3mc2. For heavier atoms, there are few contributions from states with virtual negative energies smaller than -3mc2. The model systems under study were noble gases, XH (X=Br, I, and At), XH2 (X=O, S, Se, Te, and Po), XH3 (X=N, P, As, Sb, and Bi); XH4 (X=Sn and Pb), and SnXH3 (X=Br and I). The pattern of contributions of occupied molecular orbitals (MOs) is also shown, where the 1s1/2 is the most important for excitations ending in the bottom half part of the above mentioned interval. On the other hand, the contribution of the other occupied MOs are more important than that of 1s1/2 for the other part of such interval. We also show that sigmad is electron correlation independent within both relativistic and nonrelativistic domain. In the case of sigmap, we find out a clear dependence of electron correlation effects with relativistic effects, which is of the order of 30% for Pb in PbH4.

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 MH4−nYn (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 σpT(3) correction, although other corrections such as σpS(1) and σpS(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.

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 nuclear magnetic shieldings of CHnX4−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.

Relativistic and electron correlation effects on NMR J-coupling of Sn and Pb containing molecules

We studied the influence of relativistic and electron correlation effects on NMR J-couplings in the following set of heavy-atom containing molecules:

Relativistic effects on the nuclear magnetic shieldings of rare-gas atoms and halogen in hydrogen halides within relativistic polarization propagator theory

XY_4