On the existence of a natural common gauge-origin for the calculation of magnetic properties of atoms and molecules via gaugeless basis sets

It is proven that, within the conventional approach using a common origin and gaugeless basis sets for the calculation of atomic magnetizability and Larmor current density induced by an external magnetic field, the natural gauge origin coincides with the nucleus. Recipes for defining an optimal gauge origin for the calculation of magnetizability and magnetic shielding at the nuclei of a molecule are given. Within the common origin approach, the paramagnetic contributions to the components of magnetic tensors of a molecule are represented by a minimum number of non-vanishing parameters if the gauge origin is chosen at a point characterized by the total molecular symmetry, e.g., the center of electronic charge for magnetizabilities. It is shown that total values of diagonal components of the magnetic shielding tensor σ(I) at a nucleus I in a molecule, as well as separate diamagnetic σ(dI) and paramagnetic σ(pI) contributions, calculated via the common origin method, are origin independent for a number of local point group symmetries. The diagonal components (and the average value) of σ(I) depend on the gauge origin only for nuclear site symmetries C(1), C(s), C(n), C(nv), n = 2, 3.... Group-theoretical methods show interesting features, e.g., for S(4) local symmetry, in a coordinate transformation, the paramagnetic contribution to the zz component and to the trace of the shielding tensor is origin independent, whereas the xx and yy components mix into one another, in such a way that their sum remains constant.     

The calculation of indirect nuclear spin-spin coupling constants in large molecules

We present calculations of indirect nuclear spin-spin coupling constants in large molecular systems, performed using density functional theory. Such calculations, which have become possible because of the use of linear-scaling techniques in the evaluation of the Coulomb and exchange-correlation contributions to the electronic energy, allow us to study indirect spin-spin couplings in molecules of biological interest, without having to construct artificial model systems. In addition to presenting a statistical analysis of the large number of short-range coupling constants in large molecular systems, we analyse the asymptotic dependence of the indirect nuclear spin-spin coupling constants on the internuclear separation. In particular, we demonstrate that, in a sufficiently large one-electron basis set, the indirect spin-spin coupling constants become proportional to the inverse cube of the internuclear separation, even though the diamagnetic and paramagnetic spin-orbit contributions to the spin-spin coupling constants separately decay as the inverse square of this separation. By contrast, the triplet Fermi contact and spin-dipole contributions to the indirect spin-spin coupling constants decay exponentially and as the inverse cube of the internuclear separation, respectively. Thus, whereas short-range indirect spin-spin coupling constants are usually dominated by the Fermi contact contribution, long-range coupling constants are always dominated by the negative diamagnetic spin-orbit contribution and by the positive paramagnetic spin-orbit contribution, with small spin-dipole and negligible Fermi contact contributions.     

Calculation of magnetic properties of HF,H2O,NH3, and CH4 molecules using a longitudinal gauge for the vector potential

A transformation of the transverse Coulomb vector potential was implemented to calculate molecular magnetic properties via the random-phase approximation (RPA) within the framework of a “longitudinal gauge.” In this gauge, the diamagnetic contribution to magnetic susceptibility is a tensor with equal diagonal components as in atoms, irrespective of molecular symmetry, whereas diagonal and average diamagnetic contributions to the nuclear magnetic shielding are the same as in the Coulomb gauge. Near-Hartree–Fock magnetic susceptibility and nuclear magnetic shielding tensors were evaluated for a set of small molecules, HF, H₂O, NH₃, and CH₄, employing extended Gaussian basis sets. The peculiar features of the longitudinal gauge, and the fulfillment of a series of sum rules involving the virial operator, which must be satisfied to guarantee gauge invariance of total magnetic tensors, were exploited to check the degree of convergence of theoretical values and to estimate the corresponding Hartree–Fock limit for the properties. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 31–45, 1998     

Calculations of the orbital diamagnetic contribution to nuclear spin-spin coupling constants for molecules with multiple bonds

Ab initio SCF and Cl calculations of the orbital diamagnetic contribution to nuclear spin-spin coupon constants have been performed for a series of molecules containing multiple bonds. A striking feature of the results is the prediction of consistently large contributions to vicinal (trans) and geminal proton-proton couplings which oppose and dominate the corresponding orbital paramagnetic contributions.     

Unexpected geometrical effects on paramagnetic spin-orbit and spin-dipolar 2J(FF) couplings

The second-rank tensor character of the paramagnetic spin-orbit and spin-dipolar contributions to nuclear spin-spin coupling constants is usually ignored when NMR measurements are carried out in the isotropic phase. However, in this study it is shown that isotropic (2)J(FF) couplings strongly depend on the relative orientation of the C-F bonds containing the coupling nuclei and the eigenvectors of such tensors. Predictions about such effect are obtained using a qualitative approach based on the polarization propagator formalism at the RPA, and results are corroborated performing high-level ab initio spin-spin coupling calculations at the SOPPA(CCSD)/EPR-III//MP2/EPR-III level in a model system. It is highlighted that no calculations at the RPA level were carried out in this work. The quite promising results reported in this paper suggest that similar properties are expected to hold for the second-rank nuclear magnetic shielding tensor.     

Molecular magnetic properties via formal annihilation of paramagnetic contribution to electronic current density

A novel procedure for calculating magnetic susceptibilities and nuclear magnetic shieldings in molecules is outlined, based on formal annihilation of transverse paramagnetic contribution to quantum mechanical current density induced within the electron cloud by an external homogeneous, static magnetic field. Within this method all the components of nuclear magnetic shielding are independent of a gauge translation, in any calculation relying on the algebraic approximation, irrespective of size and quality of the gaugeless basis set adopted; magnetic susceptibilities are invariant for center-symmetric molecules (virtual invariance is actually observed for molecules of arbitrary symmetry). Large basis set calculations of near-Hartree-Fock magnetic properties and maps describing a current density field of acetylene molecule carried out via the new procedure are compared with corresponding ones adopting the common origin-coupled Hartree-Fock approach and methods formally annihilating the diamagnetic contribution to the current density. © 1996 John Wiley & Sons, Inc.     

Methodological aspects in the calculation of parity-violating effects in nuclear magnetic resonance parameters

We examine the quantum chemical calculation of parity-violating (PV) electroweak contributions to the spectral parameters of nuclear magnetic resonance (NMR) from a methodological point of view. Nuclear magnetic shielding and indirect spin-spin coupling constants are considered and evaluated for three chiral molecules, H2O2, H2S2, and H2Se2. The effects of the choice of a one-particle basis set and the treatment of electron correlation, as well as the effects of special relativity, are studied. All of them are found to be relevant. The basis-set dependence is very pronounced, especially at the electron correlated ab initio levels of theory. Coupled-cluster and density-functional theory (DFT) results for PV contributions differ significantly from the Hartree-Fock data. DFT overestimates the PV effects, particularly with nonhybrid exchange-correlation functionals. Beginning from third-row elements, special relativity is of importance for the PV NMR properties, shown here by comparing perturbational one-component and various four-component calculations. In contrast to what is found for nuclear magnetic shielding, the choice of the model for nuclear charge distribution--point charge or extended (Gaussian)--has a significant impact on the PV contribution to the spin-spin coupling constants.     

Second-order polarization propagator calculations of indirect nuclear spin-spin coupling tensors in the water molecule

The contribution to indirect nuclear spin-spin coupling tensors provided by the Fermi contact, the spin-dipolar, the Fermi contact/spin-dipolar crossterm, and the paramagnetic spin-orbit interactions are investigated in a zeroth-, first- (the same as the coupled Hartree-Fock method), and second-order polarization propagator approach. Numerical applications to the water molecule show that the second-order results for both the HO and the HH coupling constants are in good agreement with experimental data - especially if vibrational corrections and the diamagnetic spin-orbit contributions are taken into account. We find that the correlation corrections beyond coupled Hartree-Fock are important. We also report how the second-order results are influenced by neglect of some of the most time-consuming steps in the calculation.     

Calculations of the indirect nuclear spin–spin coupling constants of PbH4

We report ab initio calculations of the indirect nuclear spin–spin coupling constants of PbH₄ using a basis set which was specially optimized for correlated calculations of spin–spin coupling constants. All nonrelativistic contributions and the most important part of the spin–orbit correction were evaluated at the level of the random phase approximation. Electron correlation corrections to the coupling constants were calculated with the multiconfigurational linear-response method using extended complete and restricted active space wavefunctions as well as with the second-order polarization propagator approximation and the second-order polarization propagator approximation with coupled-cluster singles and doubles amplitudes. The effects of nuclear motion were investigated by calculating the coupling constants as a function of the totally symmetric stretching coordinate. We find that the Fermi contact term dominates the Pb‐H coupling, whereas for the H‐H coupling it is not more important than the orbital paramagnetic and diamagnetic contributions. Correlation affects mainly the Fermi contact term. Its contribution to the one-bond coupling constant is reduced by correlation, independent of the method used; however, the different correlated methods give ambiguous results for the Fermi contact contribution to the H‐H couplings. The dependence of both coupling constants on the Pb‐H bond length is dominated by the change in the Fermi contact term. The geometry dependence is, however, overestimated in the random phase approximation. Received: 16 November 1998 / Accepted: 30 March 1999 / Published online: 14 July 1999     

Some mathematical properties of gauge transformations with respect to the Coulomb gauge: Variational analysis of an energy functional

As gauge invariance of computed magnetic properties, usually partitioned into diamagnetic and paramagnetic terms, is not achieved within the algebraic approximation, unless ad hoc techniques are adopted, a general variational treatment is analyzed, attempting to minimize the term more difficult to evaluate accurately, i.e., the paramagnetic contribution to magnetic susceptibility, by means of a gauge transformation. It is shown that an absolute minimum in a variational sense cannot be determined a priori. However, a “local” minimum of the paramagnetic contribution to magnetic susceptibility can be arrived at by employing general gauge transformations of polynomial form. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 599–606, 2000     

One-bond 1J(15N─19F) spin–spin coupling constants of cationic fluorinating reagents: Insights from DFT calculations

We have investigated, by means of density functional theory protocols, the one-bond ¹J(¹⁵N─¹⁹F) spin–spin coupling constants in a series of fluorinating reagents, containing the N─F bond, recently studied experimentally. The results of the calculations show a very good linear relationship with the experimental values, even though only the M06-2X(PCM)/pcJ-2//B3LYP/6-311G(d,p) level affords a very low mean absolute error. The calculations allow to analyze the various molecular orbitals contributions to the J coupling and to rationalize the observed positive sign, corresponding to a negative sign of the reduced spin–pin coupling constant K(N─F). Moreover, of the four Ramsey contributions, only the diamagnetic spin orbit is negligible, whereas the paramagnetic spin orbit and spin dipole terms decrease the magnitude of the Fermi contact (FC) term by an amount that goes from a minimum of 35% up to more than 60% of the FC term itself. Several effects have been investigated, namely, the contribution of the long-range solvent reaction field, relativistic corrections, and conformational and vibrational effects.     

Binding energies and 19F nuclear magnetic deshielding in paramagnetic halogen-bonded complexes of TEMPO with haloperfluorocarbons

19F NMR measurements and theoretical calculations were performed to study paramagnetic complexes of iodoperfluorocarbons with stable nitroxide radicals. Contrary to what is usually measured for diamagnetic halogen-bonded complexes involving iodoperfluorocarbons, it was found that the formation of complexes with the 2,2,6,6-tetramethyl(piperidin-1-yloxyl) (TEMPO) radical determines downfield shifts in the 19F NMR spectra. The experimental finding was confirmed by calculating nuclear shielding using density functional theory and correcting the isotropic diamagnetic (19)F chemical shift with contact interactions evaluated from the hyperfine coupling tensor. The computational analysis of the interaction between CF3I and TEMPO, by using DFT and MP2 theories, showed that the occurrence of the halogen bond between the interacting partners is associated with a significant charge transfer to CF3I and that the measured downfield shift is due to the occurring spin transfer.     

DFT study of the NMR properties of xenon in covalent compounds and van der waals complexes-implications for the use of 129Xe as a molecular probe

The NMR properties (chemical shift and spin-spin coupling constants) of (129)Xe in covalent compounds and weakly bound complexes have been investigated by DFT methods including relativistic effects. For covalent species, a good agreement between experimental and calculated results is achieved without scalar relativistic effects, but their inclusion (with a triple-zeta, double-polarization basis set) leads to some improvement in the quality of the correlation. The spin-orbit coupling term has a significant effect on the shielding constant, but makes a small contribution to the chemical shift. Coupling constants contain substantial contributions from the Fermi contact and paramagnetic spin-orbit terms; unlike light nuclei the spin-dipole term is also large, whereas the diamagnetic spin-orbit term is negligible. For van der Waals dimers, the dependence of the xenon chemical shift and anisotropy is calculated as a function of the distance. Small (<1 Hz) but non-negligible through-space coupling constants between (129)Xe and (13)C or (1)H are predicted. Much larger couplings, of the order of few Hz, are calculated between xenon and (17)O in a model silicate residue.     

Theoretical studies of nuclear magnetic resonance parameters for the proton-exchange pathways in porphyrin and porphycene

The nuclear magnetic resonance (NMR) parameters in porphyrin and porphycene have been calculated to investigate their changes during the process of proton exchange, using density-functional theory (DFT) for both the spin-spin coupling constants and the shielding constants. In addition, in calculations on the smaller 1,3-bis(arylimino)isoindoline molecule, we have tested the performance of our computational approach against experimental data. The calculated nuclear spin-spin coupling constants and shielding constants have been analyzed as functions of the progress of the proton transfer between two nitrogen atoms. The one-bond couplings between proton and nitrogen, dominated by the Fermi-contact term, decay steeply as the internuclear distance increases. The small changes in the intramolecular J(HH) coupling between two inner protons are mainly determined by the sum of relatively large spin-orbit terms. The isotropic shielding constant shows a strong deshielding of the nitrogen nuclei as the proton migrates away. Both the isotropic shielding of the exchanged protons and the shielding anisotropy exhibit a minimum close to the transition states.     

Perturbational calculations of parity-violating effects in nuclear-magnetic-resonance parameters

We investigate the effects of the parity-violating electroweak interaction in the spectral parameters of nuclear magnetic resonance. Perturbational theory of parity-violating effects in the nuclear magnetic shielding is presented to the order of G(F)alpha, and in the indirect spin-spin coupling, to the order of G(F)alpha3. These leading-order parity-violating corrections are evaluated using analytical linear-response theory methods based on Hartree-Fock and density-functional theory reference states. Parity-violating contributions to spin-spin couplings are evaluated for the first time at the first-principles level. Calculations are carried out for two chiral halomethanes, bromochlorofluoromethane and bromofluoroiodomethane.     

Complete prediction of the 1H NMR spectrum of organic molecules by DFT calculations of chemical shifts and spin-spin coupling constants

1H NMR chemical shifts and coupling constants for several aromatic and aliphatic organic molecules have been calculated with DFT methods. In some test cases (furan, o-dichlorobenzene and n-butyl chloride) the performance of several functionals and basis sets has been analyzed, and the various contributions to spin-spin coupling (Fermi-contact, diamagnetic and paramagnetic spin-orbit) have been evaluated. The latter two components cancel each other, so that the calculation of the contact term only is sufficient for an accurate evaluation of proton-proton couplings. Such calculated values are used to simulate the 1H NMR spectra of organic molecules with complicated spin systems (e.g. naphthalene, o-bromochlorobenzene), obtaining a generally very good agreement with experimental spectra with no prior knowledge of the involved parameters.     

Correlated and gauge origin independent calculations of magnetic properties

Summary We have applied a gauge origin invariant method for calculations of nuclear magnetic shielding constants to the singly bonded molecules BF, F₂, BH₃, CH₄, NH₃, H₂O, and HF as well as to the¹H shielding constants of HCN and C₂H₂. The calculations were performed at the RPA and second order polarization propagator (SOPPA) level of theory. For most molecules the correlation contribution in SOPPA is less diamagnetic than in the comparable MP2 calculations. For F₂, SOPPA gives a large paramagnetic correlation correction whereas the MP2 method gives a very small correlation contribution. For all molecules agreement with experimental results is generally improved at the SOPPA level compared to RPA. We have also demonstrated that second order gauge origin invariant, common and local origin (SOLO) methods do not necessarily give the same shielding even in the limit of a converged basis set.     

Four-component relativistic theory for nuclear magnetic shielding constants: critical assessments of different approaches

Both formal and numerical analyses have been carried out on various exact and approximate variants of the four-component relativistic theory for nuclear magnetic shielding constants. These include the standard linear response theory (LRT), the full or external field-dependent unitary transformations of the Dirac operator, as well as the orbital decomposition approach. In contrast with LRT, the latter schemes take explicitly into account both the kinetic and magnetic balances between the large and small components of the Dirac spinors, and are therefore much less demanding on the basis sets. In addition, the diamagnetic contributions, which are otherwise "missing" in LRT, appear naturally in the latter schemes. Nevertheless, the definitions of paramagnetic and diamagnetic terms are not the same in the different schemes, but the difference is only of O(c(-2)) and thus vanishes in the nonrelativistic limit. It is shown that, as an operator theory, the full field-dependent unitary transformation approach cannot be applied to singular magnetic fields such as that due to the magnetic point dipole moment of a nucleus. However, the inherent singularities can be avoided by the corresponding matrix formulation (with a partial closed summation). All the schemes are combined with the Dirac-Kohn-Sham ansatz for ground state calculations, and by using virtually complete basis sets a new and more accurate set of absolute nuclear magnetic resonance shielding scales for the rare gases He-Rn have been established.