Calculation Of Nuclear Magnetic Shieldings Research Articles

Density-functional calculations of relativistic spin-orbit effects on nuclear magnetic shielding in paramagnetic molecules

Terms arising from the relativistic spin-orbit effect on both hyperfine and Zeeman interactions are introduced to density-functional theory calculation of nuclear magnetic shielding in paramagnetic molecules. The theory is a generalization of the former nonrelativistic formulation for doublet systems and is consistent to O(alpha4), the fourth power of the fine structure constant, for the spin-orbit terms. The new temperature-dependent terms arise from the deviation of the electronic g tensor from the free-electron g value as well as spin-orbit corrections to hyperfine coupling tensor A, the latter introduced in the present work. In particular, the new contributions include a redefined isotropic pseudocontact contribution that consists of effects due to both the g tensor and spin-orbit corrections to hyperfine coupling. The implementation of the spin-orbit terms makes use of all-electron atomic mean-field operators and/or spin-orbit pseudopotentials. Sample results are given for group-9 metallocenes and a nitroxide radical. The new O(alpha4) corrections are found significant for the metallocene systems while they obtain small values for the nitroxide radical. For the isotropic shifts, none of the three beyond-leading-order hyperfine contributions are negligible.

Calculation of nuclear magnetic shieldings using an analytically differentiated relativistic shielding formula

Two expressions for nuclear-magnetic-shielding tensor components based on analytically differentiating the electronic energy of a system are presented. The first is based on a second-order Douglas-Kroll-Hess approach, in which the off-diagonal block terms of the transformed Dirac Hamiltonian are diminished to second order with respect to both the electrostatic nuclear attraction potential V and the magnetic vector potential A. The second expression is based on the method of Barysz-Sadlej-Snijders, in which the off-diagonal block terms in the transformed Dirac Hamiltonian are completely eliminated with respect to purely V terms, while they are diminished to second order with respect to terms including A. The two approaches are applied to the calculation of nuclear magnetic shieldings of HX (X=F, Cl, Br, I), H2X (X=O, S, Se, Te), and noble gas X (X =He,Ne,Ar,Kr,Xe) systems with common gauge origins. The results show that relativistic corrections of higher than second order are negligibly small, except for the paramagnetic parts of I, Te, and Xe shieldings. The present calculations yield very large positive values for the anisotropy of proton shielding, deltasigma(H) = sigmaparallel(H)-sigmaperpendicular(H), of HI compared to previous reports. Unfortunately, no experimental values for the anisotropy of proton shielding in HI are available for verification.

Calculation of nuclear magnetic shieldings: infinite-order Foldy–Wouthuysen transformation

A scheme for calculating nuclear magnetic shieldings using the infinite-order Foldy–Wouthuysen (FW) transformation proposed by Barysz and Sadlej (BS) is presented. The nuclear magnetic shieldings of hydrogen halides are calculated by three variant BS schemes; a double finite perturbation method for the external magnetic flux density (B 0) and the nuclear magnetic dipole moment ( μ M ) (BS/FPT-2), a single finite perturbation method for B 0 with analytical differentiation of energy with respect to ( μ M ) (BS/FPT-1), and an approximate analytical differentiation method with respect to both B 0 and μ M (BS/CHF). Although the BS/FPT-2 method is exact theoretically, the actual computation for heavy nuclei includes large error due to reduction of the number of significant figures. The BS/FPT-1 and BS/CHF approaches, on the other hand, yield reasonable values for all of the shieldings. Although several results could not be obtained by the BS/FPT-2 method, no serious contradictions were recognized among these three results. From a comparison of our results with values in the literature, our shieldings for the halogen nuclei are lower than those determined by the four-component relativistic random phase approximation (4-RPA), but the reason for this is not obvious.

Calculations of nuclear magnetic shielding in paramagnetic molecules

We propose and evaluate first principles methods for calculating the nuclear shielding tensor in open-shell, paramagnetic molecules, dealing with the case of small spin–orbit coupling that, in turn, implies the best applicability to light, organic compounds. The formalism is consistent up to second order in the fine structure constant, and includes orbital, fully anisotropic dipolar, and isotropic contact contributions to the tensor. The proposed method is implemented within the ab initio single- and multiconfiguration self-consistent field as well as density functional theory frameworks. The applications include small main-group radicals and larger nitroxide radicals. The analysis of the results and comparison with the experimental nuclear magnetic resonance data, which are available for the latter compounds, indicate promising accuracy and applicability of the density functional theory method to chemically interesting problems.

Fully relativistic calculation of nuclear magnetic shieldings and indirect nuclear spin-spin couplings in group-15 and -16 hydrides

Fully relativistic calculations of the isotropic and anisotropic parts of both indirect nuclear spin–spin couplings J1(X-H) and J2(H-H) and nuclear magnetic shieldings σ(X) and σ(H) for the group-15 and -16 hydrides are presented. Relativistic calculations were performed with Dirac–Fock wave functions and the random phase approximation method. Results are compared to its nonrelativistic counterpart. Paramagnetic and diamagnetic contributions to the nuclear magnetic shielding constants are also reported. We found very large relativistic corrections to both properties in the sixth-row hydrides (BiH3 and PoH2). Our calculations of the relativistic corrections to the isotropic part of σ at the heavy nucleus X show that it is roughly proportional to Z3.2 in both series of molecules. Paramagnetic term σp is more sensitive to the effects of relativity than the diamagnetic one σd, even though both have a behavior proportional to third power of the nuclear charge Z.

Calculation of nuclear magnetic shielding constants using variation-perturbation method with moderate size basis sets

Within a framework of variation-perturbation formalism of Karplus–Kolker, an algorithm and a program for the nuclear magnetic shielding calculations with ground-state wave functions were developed. The calculations of nuclear magnetic shielding σ in some diatomic molecules and also in methane, water and CH3F molecules with 6-311G ab initio wave functions were carried out. The results were compared with the experimental data and the results obtained by other theoretical methods. This method can be considered as an alternative approach for the calculations of the magnetic anisotropic effects on chemical shifts and can be used for predicting the tendencies in magnetic shielding alteration due to the different fragments in a big molecular systems.

Calculation of the structure and nuclear magnetic shielding constants of some H-bonded carbon acid complexes

Electronic and spatial structure of some H-bonded complexes ((HCOOH)n (n=1–4), formic, acrylic acids and their complexes with water and dimethylsulfoxide) were studied by ab initio methods using MP2/RHF//6-31G∗, RHF/6-31G∗ basis sets, and also by using density functional theory (DFT) within the B3LYP approximation. The possibility of formation of chain and cyclic complexes of formic and acrylic acids with dimethylsulfoxide is shown. In a cyclic complex two hydrogen bonds are formed. One is as usually with the O–H bond (SO⋯H–O), the other is between the acid carbonyl group and one hydrogen atom of DMSO methyl groups (C–H⋯OC). The calculations of nuclear magnetic shielding were carried out within a framework of the coupled Hartree–Fock method using gauge invariant atomic orbitals. Analysis of the results for various basis sets and comparison of the calculated and experimentally obtained chemical shifts show that the calculations predict a significant downfield shift for 1H and 13C nuclei for H-bonded systems. The results of calculations of 17O magnetic shielding constants strongly depend on the used basis set and theoretical approach.

Nuclear magnetic shieldings in solution: Gauge invariant atomic orbital calculation using the polarizable continuum model

We present the gauge invariant atomic orbital (GIAO) calculation of nuclear magnetic shieldings for solvated molecules described within the polarizable continuum model (PCM). The performance of the PCM-GIAO approach is tested in a benchmark calculation of isotropic C13, N15, and O17 shielding constants for CH3CN and CH3NO2 in vacuo and in water, both at the Hartree–Fock and density functional levels of theory. Various aspects of the calculation of solvent effects on these properties, such as the dependence on the basis set, the electron correlation, and the size of the molecular cavity embedding the solute, are taken into account and discussed. An interpretation of the gas-to-solution shielding variations in terms of a combined action of the solvent reaction field and the shielding polarizabilities is also given.

Density functional calculations of nuclear magnetic shieldings using the zeroth-order regular approximation (ZORA) for relativistic effects: ZORA nuclear magnetic resonance

We present a new relativistic formulation for the calculation of nuclear magnetic resonance (NMR) shielding tensors. The formulation makes use of gauge-including atomic orbitals and is based on density functional theory. The relativistic effects are included by making use of the zeroth-order regular approximation. This formulation has been implemented and the Hg199 NMR shifts of HgMe2, HgMeCN, Hg(CN)2, HgMeCl, HgMeBr, HgMeI, HgCl2, HgBr2, and HgI2 have been calculated using both experimental and optimized geometries. For experimental geometries, good qualitative agreement with experiment is obtained. Quantitatively, the calculated results deviate from experiment on average by 163 ppm, which is approximately 3% of the range of Hg199 NMR. The experimental effects of an electron donating solvent on the mercury shifts have been reproduced with calculations on HgCl2(NH3)2, HgBr2(NH3)2, and HgI2(NH3)2. In addition, it is shown that the mercury NMR shieldings are sensitive to geometry with changes for HgCl2 of approximately 50 ppm for each 0.01 Å change in bond length, and 100 ppm for each 10° change in bond angle.

Calculation of nuclear magnetic shielding using a parametric gauge for the vector potential

A transformation of the transverse Coulomb vector potential was implemented to calculate magnetic shielding tensor via the random-phase approximation (RPA) within the framework of a `parametric gauge'. The main features of our formulation are focused in the fact that even keeping unaltered the expressions coming from the use of the Coulomb `gauge' for the magnetic susceptibility tensor, it provides alternative ones for the nuclear magnetic shielding tensor. These new expressions depend on an arbitrary parameter and on the gradient electric field exerted by the electrons on the position of the nuclei, in an explicit manner. Sum rules for charge and current conservation and `gauge' invariance are also derived. Near Hartee-Fock nuclear magnetic shielding tensors were evaluated for some small molecules: H 2O, NH 3 and CH 4, and the fulfilment of a series of sum rules, which must be satisfied to guarantee gauge invariance of total, nuclear magnetic shielding tensors, were exploited to check the degree of confidence and, or, the advantages of the proposed gauge.

Calculation of nuclear magnetic shieldings. XIII. Gauge-origin independent relativistic effects

The gauge-origin independent expression for the relativistic nuclear magnetic shieldings was derived from the Douglas-Kroll transformation of the no-pair equation and the use of the gauge including atomic orbitals (GIAOs) proposed by London. Using our expression the relativistic spin free effect on the nuclear magnetic shieldings was evaluated for the four hydrogen halide molecules, HF, HCl, HBr, and HI, at the coupled Hartree-Fock (CHF) level with uncontracted Cartesian Gaussian-type basis sets. It was found that the GIAO-CHF results are very similar to the shielding values calculated with the fixed gauge origins at the halogen nuclei. The calculated results showed that the spin independent relativistic effect produces high-field shifts at both the halogen nuclei and protons in the hydrogen halides. However, the computed spin free effect was too small to interpret the very large upshield proton shifts observed in HBr and HI molecules.

Calculation of nuclear magnetic shieldings. XII. Relativistic no-pair equation

A Schrödinger-Pauli type two-component perturbation theory has been presented for the calculation of relativistic effects of nuclear magnetic shieldings. The expression for the relativistic nuclear magnetic shieldings are derived from the Douglas-Kroll transformation of the no-pair equation for a molecule, which bears a nuclear magnetic dipole moment, and which is placed in an external magnetic field. The exact form of the relativistic kinetic energy is included in the eigenvalue equation which is solved variationally. We calculated the relativistic mass correction effect on the nuclear magnetic shieldings in the four hydrogen halide molecules, HF, HCl, HBr, and HI, at the coupled Hartree-Fock (CHF) level. It was shown that the mass correction effect increases the nuclear magnetic shieldings of the halogen nuclei. The increments in the shieldings are proportional to about the third power of the atomic numbers of the halogen nuclei. This increase in the shieldings results from the mass correction effect concentrating the electrons in the vicinity of the heavy nucleus, the so-called relativistic contraction.

Calculation of nuclear magnetic shieldings. XI. Vibrational motion effects

Nuclear magnetic shieldings in first- and second-row hydrides were calculated with electron correlation taken into account through third order. The calculation was performed using London’s gauge-invariant atomic orbitals (GIAOs) and finite-field Mo/ller–Plesset perturbation theory (FF-MPPT). Furthermore, the vibrational motion corrections to the magnetic shieldings were evaluated. It was shown that the calculated isotropic shielding constants at the experimental geometries are higher than the experimental values, but that vibrational corrections are generally negative and improve the calculated shielding constants.

Calculation of nuclear magnetic shieldings. X. Relativistic effects

The lowest order relativistic effect theory for nuclear magnetic shieldings was derived from a two-component positive energy Hamiltonian. It was shown that the previous relativistic shielding theory based on the two-component Hamiltonian is not gauge invariant and the new terms have to be added to make a result gauge invariant. The presented theory is gauge invariant to the order of (Z/137)4 where Z is the atomic number of the heaviest atom in the molecule. One of the new contributions to the relativistic magnetic shieldings is a second-order perturbation term due to the combination of the spin-orbit interaction and the Fermi-contact interaction. A numerical estimation for this term was performed for the four hydrogen halides, HF, HCl, HBr, and HI. The computational results showed that the contribution of this term to the hydrogen shieldings is negligibly small, but the contribution to the halogen atoms is considerable.

Effects of the order of the energy asymptotes on the calculations of nuclear magnetic shieldings and static polarizabilities

The second-order polarization propagator approximation (SOPPA) has been applied to the calculation of the nuclear magnetic shielding constants and static polarizabilities of CO, N2, F2, and CH4 in order to investigate the effect of decreasing the order (in perturbation theory) of the poles of the propagator, i.e., the energy asymptotes, but still requiring that all response terms to second order must be included. Our results show that the higher than second-order contributions from the poles are of vital importance for the nuclear magnetic shieldings and of less, but not negligible, importance for the static polarizabilities. As the order of the poles is decreased the SOPPA isotropic shieldings approach the results obtained within second-order perturbation theory (MP2), especially for CO, N2, and F2. This behavior is not as pronounced for the C shielding of CH4 and for polarizabilities. For the shieldings we obtain the best agreement with MP2 (and experiments) when the poles are calculated as Hartree–Fock energy differences using frozen ground state orbitals, and for the polarizabilities when the poles are calculated in the random phase approximation and Tamm–Dancoff approximation.

Calculation of nuclear magnetic shieldings. IX. Electron correlation effects

Nuclear magnetic shieldings were calculated with electron correlations through third order. The calculation was performed on the use of London’s gauge invariant atomic orbitals (GIAO) and the finite field many-body perturbation theory (FF-MBPT), in which the two kinds of perturbation fields, i.e., the external magnetic field and the field due to the nuclear magnetic moment, were introduced to get the perturbed one-electron states. The Hartree–Fock (HF) values and the second- and third-order Mo/ller–Plesset (MP) correlation corrections of the nuclear magnetic shieldings were calculated in (i) four first-row hydrides HF, H2O, NH3, and CH4; and (ii) three linear molecules with a multiple bond N2, CO, and HCN. The calculations showed that the post-HF correlations are important and much improve the calculated shielding values, and that the second-order corrections are positive, but the third-order ones are negative. Furthermore, the rovibrational corrections to the shielding constants in the HF molecule at 300 K were estimated to be −9.9 ppm for F and −0.66 ppm for H.

Calculation of nuclear magnetic shieldings. VIII. Gauge invariant many-body perturbation method

The calculation of nuclear magnetic shieldings σ was performed for four first-row hydrides, i.e., HF, H2O, NH3, and CH4 using the finite-field many-body perturbation theory through second order in the electron correlation correction with London’s gauge invariant atomic orbitals. The calculated isotropic shieldings σ̄ for the heavy atoms were larger than the experimental values. The rotational and vibrational correction for the σ̄ of the F atom in HF at 300 K was estimated to be −4.0 ppm.

Nuclear magnetic shielding in molecules. The application of GIAO's in LCAO-X ?-calculations

A nonempirical method for the calculation of nuclear magnetic shielding based on the four current density functional formalism is presented. Using SCF-LCAO-Xα-calculations with application of GIAO's effective one particle equations are solved. The results for nuclear magnetic shielding in diatomic molecules are of good quality, compared with other theoretical and experimental data.

A comparison of the IGLO and LORG methods for the calculations of nuclear magnetic shieldings

AbstractThe individual gauge for localized orbitals (IGLO) and localized orbital/local origin (LORG) methods for the calculation of chemical shieldings are compared from their theoretical and computational viewpoints. A detailed analysis of the fluorine α‐substituent effect in a series of fluoromethanes is given in terms of the IGLO and LORG bond contributions. The performance of both methods is discussed for molecular systems of fairly different sizes.

Iterative LORG calculations of nuclear magnetic shieldings. 13C shielding tensors in 2-norbornenone

A computationally efficient new implementation of the LORG (localized orbital-local origin) method for nuclear magnetic shieldings is described, in which the paramagnetic response term is obtained by means of a reduced linear equations method instead of by explicit diagonalization of the A–B matrix. Comparative timings for calculations on a number of molecules are presented, and nuclear shieldings are calculated for 2-norbornenone, a computationally demanding case.