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arXiv:2603.24467

Second-Order Magnetic Properties in Paramagnetic Molecules From a Current Density Formulation Including Scalar Relativistic Effects

Francesco Ferdinando Summa, Sonia Coriani, Andre Severo Pereira Gomes

quant-phSubmitted Mar 25, 2026• Announced Mar 26, 2026• Revised Sep 23, 2026 (v4)
v1 · Mar 25, 2026v2 · Jul 2, 2026v3 · Aug 23, 2026v4 · Sep 23, 2026
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Abstract

This work presents the theoretical background for the computation of nuclear magnetic shielding, nuclear hyperfine and magnetizability tensors of paramagnetic molecules, using a magnetically induced current density framework to account for both orbital and spin contributions. The resulting magnetizability tensor is fully consistent with the general Van Vleck formulation, recovering the temperature-dependent Curie contribution through the explicit integration of the magnetically induced spin current density. The methodology proposed herein provides a straightforward computational route that bypasses the complex evaluation of g-tensors and Zero-Field Splitting (ZFS) Hamiltonians. While the theoretical framework is general, we present applications rooted on physically motivated approximations where scalar relativistic effects are incorporated through corrections based on the Zeroth-Order Regular Approximation (ZORA) Hamiltonian within the ground-state spin density. This approach combines a relativistic self-consistent field (SCF) calculation for the ground-state spin density with a non-relativistic, origin-independent current density calculation for the orbital contribution. This hybrid strategy is shown to capture the Heavy-Atom Light-Atom (HALA) effect in 1H and 13C shieldings, particularly in paramagnetic molecular systems containing transition metals up to the 3d series. By restricting the relativistic treatment to the spin density, where scalar relativistic effects are dominant, and neglecting such effects on the orbital contribution of light atoms, this method offers a good compromise between computational efficiency and accuracy for the characterization of large open-shell molecular systems.

AI Summary

The paper addresses computation of nuclear magnetic shielding, hyperfine coupling and magnetizability tensors in paramagnetic molecules using magnetically induced current densities. It derives orbital and spin-current expressions within the zeroth-order regular approximation (ZORA), with thermal averaging over a spin multiplet recovering the Curie contribution proportional to $S(S+1)/T$. The aim is to evaluate these properties without explicitly calculating $g$ tensors or zero-field-splitting Hamiltonians. The implementation combines relativistic ground-state spin densities obtained from Douglas–Kroll–Hess calculations with nonrelativistic, gauge-origin-independent orbital currents. A separate approximate spin–orbit-coupling (SOC) treatment uses field-free noncollinear densities rather than explicit magnetic-field response. Tests cover five 3d metallocenes, rhodocene, a nitroxide radical, and magnetizabilities of $\mathrm{O_2}$, $\mathrm{NO}$ and $\mathrm{NO_2}$. Scalar-relativistic corrections change ligand proton shieldings by up to about 7 ppm and carbon shieldings by roughly 14–24 ppm in several metallocenes. Comparisons with published X2C calculations give $R^2=0.9999$ for proton shieldings, whereas carbon residuals remain approximately 5–10%, without systematic improvement from scalar corrections. Calculated gas-phase magnetizabilities are 3379.3, 1326.9 and 1259.3 ppm cm$^3$ mol$^{-1}$, respectively, versus experimental values of 3330, 1461 and 1249. Stated limitations include neglected relativistic orbital corrections, picture-change effects and explicit spin-dependent field response. The SOC ansatz lacks formal rigor and gives unacceptable accuracy for rhodocene; pseudocontact shifts are omitted. Experimental agreement is mixed, with particularly large discrepancies for manganocene, including the wrong sign of its proton chemical shift. Full coupled-perturbed relativistic response is deferred.

Some signs of AI-generated textIncrementalStrong assumptionsOverclaimsPossible error

Central claim may not hold: The claimed general Van Vleck consistency and avoidance of g/ZFS calculations are established only for a restricted spin-multiplet response, while the implementation neglects rather than reproduces general SOC- and ZFS-dependent magnetic response. The numerical evidence supports conditional accuracy rather than uniform reliability through the 3d series, with manganocene showing a wrong-sign proton shift and a carbon shift approximately 33.3% above experiment. The proposed spin current has divergence proportional to grad K · curl Q, and neither the atomic argument in the appendix nor the separately computed nonrelativistic orbital current establishes conservation of the implemented molecular hybrid current. The statistical-mechanics appendix confuses fixed-field saturation with zero-field response when concluding that spin shielding and magnetizability vanish at zero temperature.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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This work presents the theoretical background for the computation of nuclear magnetic shielding, nuclear hyperfine and magnetizability tensors of paramagnetic molecules, using a magnetically induced current density framework to account for both orbital and spin contributions. The resulting magnetizability tensor is fully consistent with the general Van Vleck formulation, recovering the temperature-dependent Curie contribution through the explicit integration of the magnetically induced spin current density. The methodology proposed herein provides a straightforward computational route that bypasses the complex evaluation of g-tensors and Zero-Field Splitting (ZFS) Hamiltonians. While the theoretical framework is general, we present applications rooted on physically motivated approximations where scalar relativistic effects are incorporated through corrections based on the Zeroth-Order Regular Approximation (ZORA) Hamiltonian within the ground-state spin density. This approach combines a relativistic self-consistent field (SCF) calculation for the ground-state spin density with a non-relativistic, origin-independent current density calculation for the orbital contribution. This hybrid strategy is shown to capture the Heavy-Atom Light-Atom (HALA) effect in 1H and 13C shieldings, particularly in paramagnetic molecular systems containing transition metals up to the 3d series. By restricting the relativistic treatment to the spin density, where scalar relativistic effects are dominant, and neglecting such effects on the orbital contribution of light atoms, this method offers a good compromise between computational efficiency and accuracy for the characterization of large open-shell molecular systems.
The paper addresses computation of nuclear magnetic shielding, hyperfine coupling and magnetizability tensors in paramagnetic molecules using magnetically induced current densities. It derives orbital and spin-current expressions within the zeroth-order regular approximation (ZORA), with thermal averaging over a spin multiplet recovering the Curie contribution proportional to S(S+1)/TS(S+1)/TS(S+1)/T. The aim is to evaluate these properties without explicitly calculating ggg tensors or zero-field-splitting Hamiltonians.
The implementation combines relativistic ground-state spin densities obtained from Douglas–Kroll–Hess calculations with nonrelativistic, gauge-origin-independent orbital currents. A separate approximate spin–orbit-coupling (SOC) treatment uses field-free noncollinear densities rather than explicit magnetic-field response. Tests cover five 3d metallocenes, rhodocene, a nitroxide radical, and magnetizabilities of O2\mathrm{O_2}O2​, NO\mathrm{NO}NO and NO2\mathrm{NO_2}NO2​. Scalar-relativistic corrections change ligand proton shieldings by up to about 7 ppm and carbon shieldings by roughly 14–24 ppm in several metallocenes. Comparisons with published X2C calculations give R2=0.9999R^2=0.9999R2=0.9999 for proton shieldings, whereas carbon residuals remain approximately 5–10%, without systematic improvement from scalar corrections. Calculated gas-phase magnetizabilities are 3379.3, 1326.9 and 1259.3 ppm cm3^33 mol−1^{-1}−1, respectively, versus experimental values of 3330, 1461 and 1249.
Stated limitations include neglected relativistic orbital corrections, picture-change effects and explicit spin-dependent field response. The SOC ansatz lacks formal rigor and gives unacceptable accuracy for rhodocene; pseudocontact shifts are omitted. Experimental agreement is mixed, with particularly large discrepancies for manganocene, including the wrong sign of its proton chemical shift. Full coupled-perturbed relativistic response is deferred.
Absolute nuclear magnetic shielding constants and relative chemical shifts are second-derivative response properties that underpin the interpretation of Nuclear Magnetic Resonance (NMR) spectra and provide critical insights into the structural and electronic environments of diverse chemical systems. However, their accurate computation is often constrained by the need for method-specific, analytical response implementation, and therefore is particularly challenging for non-variational, correlated wavefunction methods where analytical second derivatives are frequently unavailable. Here we present a scalable framework that computes NMR shielding tensors through finite magnetic field differentiation of complex-valued, gauge-including atomic-orbital (GIAO) based self-consistent field (SCF) calculations. By combining hybrid MPI and OpenMP parallelization and resolution-of-identity (RI) approximation-based Coulomb (J) and Exchange (K) implementation and mixed numerical/analytical derivatives computational scheme, we achieve very good efficiency for hybrid density functional theory (DFT) NMR shielding, relative to existing state-of-the-art analytic implementations across realistic chemical systems of various sizes. Forward differentiation with optimal step sizes derived from rigorous error analysis retains favorable numerical errors much smaller than experimental uncertainty or intrinsic DFT errors. The results demonstrate that RI-accelerated finite magnetic field calculations can obtain DFT-level NMR shielding constants very precisely, providing a scalable foundation for extensions to more advanced quantum chemistry methods in future work.
Accurate prediction of nuclear magnetic resonance (NMR) shielding constants for strongly correlated systems remains challenging. In this context, we present a density matrix renormalization group self-consistent-field (DMRG-SCF) implementation for NMR shielding calculations with large active spaces. Within a gauge-including atomic orbital DMRG-SCF (GIAO-DMRG-SCF) framework, we formulate and solve the coupled-perturbed DMRG-SCF (CP-DMRG-SCF) equations using local-site effective Hamiltonians. This approach enables the direct evaluation of the wave function's response to an external magnetic field. In the limit of small active spaces, the computed NMR shieldings are in agreement with conventional complete active space self-consistent-field (CASSCF) results, while the present approach naturally extends to larger active spaces inaccessible to standard multiconfigurational methods. Applications to a series of molecular systems, particularly strongly correlated transition metal complexes, demonstrate that the proposed implementation can be both reliable and accurate.
Abstract The electronic structure of magnetic lanthanide atoms is fascinating from a fundamental perspective. They have electrons in a submerged open 4f shell lying beneath a filled 6s shell with strong relativistic correlations leading to a large magnetic moment and large electronic orbital angular momentum. This large angular momentum leads to strong anisotropies, i. e. orientation dependencies, in their mutual interactions. The long-ranged molecular anisotropies are crucial for proposals to use ultracold lanthanide atoms in spin-based quantum computers, the realization of exotic states in correlated matter, and the simulation of orbitronics found in magnetic technologies. Short-ranged interactions and bond formation among these atomic species have thus far not been well characterized. Efficient relativistic computations are required. Here, for the first time we theoretically determine the electronic and ro-vibrational states of heavy homonuclear lanthanide Er 2 and Tm 2 molecules by applying state-of-the-art relativistic methods. In spite of the complexity of their internal structure, we were able to obtain reliable spin–orbit and correlation-induced splittings between the 91 Er 2 and 36 Tm 2 electronic potentials dissociating to two ground-state atoms. A tensor analysis allows us to expand the potentials between the atoms in terms of a sum of seven spin–spin tensor operators simplifying future research. The strengths of the tensor operators as functions of atom separation are presented and relationships among the strengths, derived from the dispersive long-range interactions, are explained. Finally, low-lying spectroscopically relevant ro-vibrational energy levels are computed with coupled-channels calculations and analyzed.
Spin Orbit Coupling (SOC) is introduced to molecular ab initio density matrix renormalization group (DMRG) calculations. In the presented scheme, one first approximates the electronic ground state and a number of excited states of the Born-Oppenheimer (BO) Hamiltonian with the aid of the DMRG algorithm. Owing to the spin-adaptation of the algorithm, the total spin S is a good quantum number for these states. After the non-relativistic DMRG calculation is finished, all magnetic sublevels of the calculated states are constructed explicitly, and the SOC operator is expanded in the resulting basis. To this end, spin orbit coupled energies and wavefunctions are obtained as eigenvalues and eigenfunctions of the full Hamiltonian matrix which is composed of the SOC operator matrix and the BO Hamiltonian matrix. This treatment corresponds to a quasi-degenerate perturbation theory approach and can be regarded as the molecular equivalent to atomic Russell-Saunders coupling. For the evaluation of SOC matrix elements, the full Breit-Pauli SOC Hamiltonian is approximated by the widely used spin-orbit mean field operator. This operator allows for an efficient use of the second quantized triplet replacement operators that are readily generated during the non-relativistic DMRG algorithm, together with the Wigner-Eckart theorem. With a set of spin-orbit coupled wavefunctions at hand, the molecular g-tensors are calculated following the scheme proposed by Gerloch and McMeeking. It interprets the effective molecular g-values as the slope of the energy difference between the lowest Kramers pair with respect to the strength of the applied magnetic field. Test calculations on a chemically relevant Mo complex demonstrate the capabilities of the presented method.
We present an implementation of the relativistic quantum-chemical density matrix renormalization group (DMRG) approach based on a matrix-product formalism. Our approach allows us to optimize matrix product state (MPS) wave functions including a variational description of scalar-relativistic effects and spin–orbit coupling from which we can calculate, for example, first-order electric and magnetic properties in a relativistic framework. While complementing our pilot implementation ( Knecht, S. J. Chem. Phys. 2014, 140, 041101 ), this work exploits all features provided by its underlying nonrelativistic DMRG implementation based on an matrix product state and operator formalism. We illustrate the capabilities of our relativistic DMRG approach by studying the ground-state magnetization, as well as current density of a paramagnetic f 9 dysprosium complex as a function of the active orbital space employed in the MPS wave function optimization.
The properties of the electronic ground state of the polar and paramagnetic chromium--closed-shell-atom molecules have been investigated. State-of-the-art ab initio techniques have been applied to compute the potential energy curves for the chromium--alkaline-earth-metal-atom, CrXXX (X=BeX=\mathrm{Be}X=Be, Mg, Ca, Sr, Ba), and chromium--ytterbium, CrYb, molecules in the Born-Oppenheimer approximation for the high-spin ${X}^{7}{\ensuremath{\Sigma}}^{+}$ electronic ground state. The spin restricted open-shell coupled cluster method restricted to single, double, and noniterative triple excitations, RCCSD(T), was employed and the scalar relativistic effects within the Douglas-Kroll-Hess Hamiltonian or energy-consistent pseudopotentials were included. The permanent electric dipole moments and static electric dipole polarizabilities were computed. The leading long-range coefficients describing the dispersion interaction between the atoms at large interatomic distances C6{C}_{6}C6​ are also reported. The molecules under investigation are examples of species possessing both large magnetic and electric dipole moments making them potentially interesting candidates for ultracold many-body physics studies.