Spin-orbit Coupling Operator Research Articles

Understanding the multiplet fine structures of MgO: Cr3+ with combination method of first−principles calculations and ligand field theory

The interpretative and predictive capabilities on the energy fine structures of transition metal ions in solids are explored by integrating the first -principles calculations with ligand-field (LF) theory. The charge-compensated site of MgO: Cr3+ is chosen as a case and the aim is to determine the energy-level fine structure and EPR g factor g// and g⊥. The approach involves initially computing all the required parameters for LF theory through first-principles calculations, and these parameters are subsequently fed into the LF model Hamiltonian, which fully incorporates interactions between different spectrum terms through the LF operator, spin-orbit coupling (SOC) operator and other relevant factors. The calculated zero-field splitting parameter D and g factors of the ground term A24(t23) are reasonably good agreement with the experimental observations. However, the energy levels of excited states are slightly underestimated compared to experimental values due to the underestimation of Racah parameters in the first-principles calculations, except the T24[t22(T13)e] term which is insensitive to Racah parameters for it is equal to 10Dq relative to ground term A24(t23) with possible minor corrections due to perturbations.

Consistent Second-Order Treatment of Spin-Orbit Coupling and Dynamic Correlation in Quasidegenerate N-Electron Valence Perturbation Theory.

We present a formulation and implementation of second-order quasidegenerate N-electron valence perturbation theory (QDNEVPT2) that provides a balanced and accurate description of spin-orbit coupling and dynamic correlation effects in multiconfigurational electronic states. In our approach, the energies and wave functions of electronic states are computed by treating electron repulsion and spin-orbit coupling operators as equal perturbations to the nonrelativistic complete active-space wave functions, and their contributions are incorporated fully up to the second order. The spin-orbit effects are described using the Breit-Pauli (BP) or exact two-component Douglas-Kroll-Hess (DKH) Hamiltonians within spin-orbit mean-field approximation. The resulting second-order methods (BP2- and DKH2-QDNEVPT2) are capable of treating spin-orbit coupling effects in nearly degenerate electronic states by diagonalizing an effective Hamiltonian expanded in a compact non-relativistic basis. For a variety of atoms and small molecules across the entire periodic table, we demonstrate that DKH2-QDNEVPT2 is competitive in accuracy with variational two-component relativistic theories. BP2-QDNEVPT2 shows high accuracy for the second- and third-period elements, but its performance deteriorates for heavier atoms and molecules. We also consider the first-order spin-orbit QDNEVPT2 approximations (BP1- and DKH1-QDNEVPT2), among which DKH1-QDNEVPT2 is reliable but less accurate than DKH2-QDNEVPT2. Both DKH1- and DKH2-QDNEVPT2 hold promise as efficient and accurate electronic structure methods for treating electron correlation and spin-orbit coupling in a variety of applications.

A route to high-accuracy abinitio description of electronic excited states in high-spin lanthanide-containing species: A case study of GdO.

Accurate description of electronic excited states of high-spin molecular species is a yet unsolved problem in modern electronic structure theory. A composite computational scheme developed in the present work contributes to solving this task for a challenging case of lanthanide-containing molecules. In the scheme, the highest-spin states whose wavefunctions are dominated by a single Slater determinant are described at the single-reference (SR) CCSD(T) level, whereas the lower-spin states, being inherently multiconfigurational in their nature, are treated with multireference (MR) methods, MRCI and/or CASPT2. An original technique which scales MR results against SR CCSD(T) ones to improve the accuracy in the former is proposed and examined, taking the example of 12 electronic states of gadolinium monoxide, X9Σ-, Y7Σ-, A'9Δ, A1'7Δ, A9Π, A17Π, B9Σ-, B17Σ-, C9Π, C17Π, D9Σ-, and D17Σ-, up to 35 000cm-1. A multitude of the corresponding Ω (spin-coupled) states was then studied within the state-interacting approach employing the full Breit-Pauli spin-orbit coupling operator with CASSCF-generated ΛS states as a basis. For all ΛS and Ω states, the Gd-O bond lengths, spectroscopic constants ωe, ωexe, αe, and adiabatic excitation energies are obtained. The theoretical predictions are in good agreement with the experimental data, with deviations in excitation energies not exceeding 350cm-1 (1 kcal/mol). The spectroscopic properties of the yet unobserved electronic states, A'9Δ, A1'7Δ, C9Π, C17Π, D9Σ-, and D17Σ-, are evaluated for the first time.

Possible Violation of the Spin-Hamiltonian Concept in Interpreting the Zero-Field Splitting.

The majority of experimental data in electron spin resonance and molecular magnetism are interpreted in terms of the spin-Hamiltonian (SH) formalism. However, this is an approximate theory that requires a proper testing. In the older variant, the multielectron terms are used as a basis in which the D-tensor components are evaluated by employing the second-order perturbation theory (PT) for nondegenerate states; here, the spin-orbit interaction expressed via the spin-orbit splitting parameter λ serves for the perturbation. The model space is restricted only to the fictitious spin functions |S, M⟩. In the case of the orbital (quasi) degeneracy of the ground term, the PT tends to diverge and the subtracted D, E, and g parameters are false. In the second variant working in the "complete active space" (CAS), the spin-orbit coupling operator is involved by the variation method resulting in the spin-orbit multiplets (energies and eigenvectors) The multiplets can be evaluated either by applying ab initio CASSCF + NEVPT2 + SOC calculations or by using semiempirical generalized crystal-field theory (with the one-electron SOC operator depending upon ξ). The resulting states can be projected onto the subspace of the spin-only kets in the way that the eigenvalues stay invariant. Such an effective Hamiltonian matrix can be reconstructed using six independent components of the symmetric D-tensor from which the D and E values are obtained by solving linear equations. The eigenvectors of the spin-orbit multiplets in the CAS allow determining the dominating composition of the spin projection─cumulative weights of |±M⟩. These are conceptually different from those generated by the SH alone. It is shown that in some cases, the SH theory works satisfactorily for a series of transition-metal complexes; however, sometimes it fails. The ab initio calculations on the SH parameters are compared with the approximate generalized crystal-field theory conducted at the experimental geometry of the chromophore. In total, 12 metal complexes have been analyzed. One of the criteria that assesses the validity of SH is the projection norm N for spin multiplets (this has not to be far from 1). Another criterion is the gap in the spectrum of the spin-orbit multiplets that separates the hypothetical (fictitious) spin-only manifold from the rest of the states.

Relativistic multimode Jahn-Teller and pseudo-Jahn-Teller effects in tetrahedral systems

The relativistic pseudo-Jahn-Teller effect 2T×(t2+e)=(E1/2+G3/2)×(t2+e) and the Jahn-Teller effect 2E×(t2+e)=G3/2×(t2+e) are considered, employing the microscopic (Breit-Pauli) spin–orbit coupling operator. The orbital and spin-orbital parts of electronic Hamiltonian are expanded up to the second order terms in normal modes. Five-dimensional (t2+e) potential energy surfaces are obtained in analytical form as functions of invariants of the Td group.

Spin–orbit configuration interaction study of spectral properties of PbO

Relativistic calculations of the structural and spectral properties of the PbO molecule can provide fundamental information about the importance of a proper treatment of angular momentum coupling among electrons in order to achieve accurate computational results for spectral properties. Specifically, the nature of these couplings in PbO is expected to be intermediate between the LS- and jj-coupling limits because of its light/heavy element composition. This article reports potential energy curves, transition energies, electric dipole transition moments, permanent dipole moments and spectroscopic constants of PbO calculated using a multireference single plus double excitations spin–orbit configuration interaction approach in the context of relativistic effective core potentials and their concomitant spin–orbit coupling operators. The calculated results are in general agreement with both available experimental results as well as earlier calculations. New values for properties of excited states are also reported. It is noteworthy that certain properties show larger deviations from previous calculations. These deviations are attributed to direct and indirect relativistic effects resulting from diatomic electron–electron angular momentum coupling effects, which are included consistently in the calculations reported herein.

Theoretical analysis of the long-distance limit of NMR chemical shieldings.

After some years of controversy, it was recently demonstrated how to obtain the correct long-distance limit [point-dipole approximation (PDA)] of pseudo-contact nuclear magnetic resonance chemical shifts from rigorous first-principles quantum mechanics [Lang et al., J. Phys. Chem. Lett. 11, 8735 (2020)]. This result confirmed the classical Kurland-McGarvey theory. In the present contribution, we elaborate on these results. In particular, we provide a detailed derivation of the PDA both from the Van den Heuvel-Soncini equation for the chemical shielding tensor and from a spin Hamiltonian approximation. Furthermore, we discuss in detail the PDA within the approximate density functional theory and Hartree-Fock theories. In our previous work, we assumed a relatively crude effective nuclear charge approximation for the spin-orbit coupling operator. Here, we overcome this assumption by demonstrating that the derivation is also possible within the fully relativistic Dirac equation and even without the assumption of a specific form for the Hamiltonian. Crucial ingredients for the general derivation are a Hamiltonian that respects gauge invariance, the multipolar gauge, and functional derivatives of the Hamiltonian, where it is possible to identify the first functional derivative with the electron number current density operator. The present work forms an important foundation for future extensions of the Kurland-McGarvey theory beyond the PDA, including induced magnetic quadrupole and higher moments to describe the magnetic hyperfine field.

SOiCI and iCISO: combining iterative configuration interaction with spin–orbit coupling in two ways

The near-exact iCIPT2 approach for strongly correlated systems of electrons, which stems from the combination of iterative configuration interaction (iCI, an exact solver of full CI) with configuration selection for static correlation and second-order perturbation theory (PT2) for dynamic correlation, is extended to the relativistic domain. In the spirit of spin separation, relativistic effects are treated in two steps: scalar relativity is treated by the infinite-order, spin-free part of the exact two-component (X2C) relativistic Hamiltonian, whereas spin–orbit coupling (SOC) is treated by the first-order, Douglas–Kroll–Hess-like SOC operator derived from the same X2C Hamiltonian. Two possible combinations of iCIPT2 with SOC are considered, i.e., SOiCI and iCISO. The former treats SOC and electron correlation on an equal footing, whereas the latter treats SOC in the spirit of state interaction, by constructing and diagonalizing an effective spin–orbit Hamiltonian matrix in a small number of correlated scalar states. Both double group and time reversal symmetries are incorporated to simplify the computation. Pilot applications reveal that SOiCI is very accurate for the spin–orbit splitting (SOS) of heavy atoms, whereas the computationally very cheap iCISO can safely be applied to the SOS of light atoms and even of systems containing heavy atoms when SOC is largely quenched by ligand fields.

Air-Stable Dy(III)-Macrocycle Enantiomers: From Chiral to Polar Space Group

Air-Stable Dy(III)-Macrocycle Enantiomers: From Chiral to Polar Space Group

Magnetic Anisotropy in a Cubane-like Ni4 Complex: An Ab Initio Perspective.

Magnetic anisotropy, in the absence of an external magnetic field, relates to the degeneracy lift of energy levels. In the standard case of transition metal complexes, this property is usually modeled by an anisotropic spin Hamiltonian and one speaks of "zero-field splitting" (ZFS) of spin states. While the case of mononuclear complexes has been extensively described by means of ab initio quantum mechanical calculations, the literature on polynuclear complexes studied with these methodologies is rather scarce. In this work, advanced multiconfigurational wave function theory methods are applied to compute the ZFS of the ground S = 4 state of an actual tetranickel(II) complex, displaying a magnet behavior below 0.5 K. First, the isotropic couplings are computed in the absence of the spin-orbit coupling operator, in the full complex and also in clusters with only two active nickel(II) centers, confirming the occurrence of weak ferromagnetic couplings in this system. Second, the single-site magnetic anisotropies are computed on a cluster bearing only one active nickel(II) site, showing that the single-site anisotropy axes are not oriented in an optimal fashion for generating a large uniaxial molecular anisotropy. Furthermore, the possibility for involving only a few local orbital excited states in the calculation is assessed, actually opening the way for a consistent and manageable treatment of the ZFS of the ground S = 4 state. Third, multiconfigurational calculations are performed on the full complex, confirming the weak uniaxial anisotropy occurring for this state and also, interestingly, revealing a significant contribution of the lowest-lying orbitally excited S = 3 states. Overall, by comparison with the experiment, the reported results question the common habit of using only one structure, in particular derived from a crystallography experiment, to compute magnetic anisotropy parameters.

Effect of spin–orbit coupling on strong field ionization simulated with time-dependent configuration interaction

Time-dependent configuration interaction with a complex absorbing potential has been used to simulate strong field ionization by intense laser fields. Because spin-orbit coupling changes the energies of the ground and excited states, it can affect the strong field ionization rate for molecules containing heavy atoms. Configuration interaction with single excitations (CIS) has been employed for strong field ionization of closed shell systems. Single and double excitation configuration interaction with ionization (CISD-IP) has been used to treat ionization of degenerate states of cations on an equal footing. The CISD-IP wavefunction consists of ionizing single (one hole) and double (two hole/one particle) excitations from the neutral atom. Spin-orbit coupling has been implemented using an effective one electron spin-orbit coupling operator. The effective nuclear charge in the spin-orbit coupling operator has been optimized for Ar+, Kr+, Xe+, HX+ (X = Cl, Br, and I). Spin-orbit effects on angular dependence of the strong field ionization have been studied for HX and HX+. The effects of spin-orbit coupling are largest for ionization from the π orbitals of HX+. In a static field, oscillations are seen between the 2Π3/2 and 2Π1/2 states of HX+. For ionization of HX+ by a two cycle circularly polarized pulse, a single peak is seen when the maximum in the carrier envelope is perpendicular to the molecular axis and two peaks are seen when it is parallel to the axis. This is the result of the greater ionization rate for the π orbitals than for the σ orbitals.

A Molecular Crystal Shows Multiple Correlated Magnetic and Ferroelectric Switchings

Simultaneous control of the magnetic and electric properties of materials is crucial for their application in next-generation memory and sensor devices. Herein, we report a single-crystal Co(II) co...

Spin-orbit coupling in periodic systems with broken time-reversal symmetry: Formal and computational aspects

We discuss the treatment of spin-orbit coupling (SOC) in time-reversal-symmetry-broken periodic systems for relativistic electronic structure calculations of materials within the generalized noncollinear Kohn-Sham density functional theory (GKSDFT). We treat SOC self-consistently and express the GKS orbitals in a two-component spinor basis. Crucially, we present a methodology (and its corresponding implementation) for the simultaneous self-consistent treatment of SOC and exact nonlocal Fock exchange operators. The many advantages of the inclusion of nonlocal Fock exchange in the self-consistent treatment of SOC, as practically done in hybrid exchange-correlation functionals, are both formally derived and illustrated through numerical examples: (i) it imparts a local magnetic torque (i.e., the ability of the two-electron potential to locally rotate the magnetization with respect to a starting guess configuration) that is key to converge to the right solution in noncollinear DFT regardless of the initial guess for the magnetization; (ii) because of the local magnetic torque, it improves the rotational invariance of noncollinear formulations of the DFT; (iii) it introduces the dependence on specific pieces of the spinors (i.e., those mapped onto otherwise missing spin blocks of the complex density matrix) into the two-electron potential, which are key to the correct description of the orbital- and spin-current densities and their coupling with the magnetization; and (iv) when space-inversion symmetry is broken, it allows for the full breaking of time-reversal symmetry in momentum space, which would otherwise be constrained by a sum rule linking the electronic band structure at opposite points in the Brillouin zone ($\underline{\mathbf{k}}$ and $\ensuremath{-}\underline{\mathbf{k}}$). The presented methodology is implemented in a developmental version of the public crystal code. Numerical tests are performed on the model system of an infinite radical chain of ${\mathrm{Ge}}_{2}\mathrm{H}$ with both space-inversion and time-reversal symmetries broken, which allows to highlight all the above-mentioned effects.

Mechanism of L2,3-edge x-ray magnetic circular dichroism intensity from quantum chemical calculations and experiment-A case study on V(IV)/V(III) complexes.

In this work, we present a combined experimental and theoretical study on the V L2,3-edge x-ray absorption (XAS) and x-ray magnetic circular dichroism (XMCD) spectra of VIVO(acac)2 and VIII(acac)3 prototype complexes. The recorded V L2,3-edge XAS and XMCD spectra are richly featured in both V L3 and L2 spectral regions. In an effort to predict and interpret the nature of the experimentally observed spectral features, a first-principles approach for the simultaneous prediction of XAS and XMCD spectra in the framework of wavefunction based ab initio methods is presented. The theory used here has previously been formulated for predicting optical absorption and MCD spectra. In the present context, it is applied to the prediction of the V L2,3-edge XAS and XMCD spectra of the VIVO(acac)2 and VIII(acac)3 complexes. In this approach, the spin-free Hamiltonian is computed on the basis of the complete active space configuration interaction (CASCI) in conjunction with second order N-electron valence state perturbation theory (NEVPT2) as well as the density functional theory (DFT)/restricted open configuration interaction with singles configuration state functions based on a ground state Kohn-Sham determinant (ROCIS/DFT). Quasi-degenerate perturbation theory is then used to treat the spin-orbit coupling (SOC) operator variationally at the many particle level. The XAS and XMCD transitions are computed between the relativistic many particle states, considering their respective Boltzmann populations. These states are obtained from the diagonalization of the SOC operator along with the spin and orbital Zeeman operators. Upon averaging over all possible magnetic field orientations, the XAS and XMCD spectra of randomly oriented samples are obtained. This approach does not rely on the validity of low-order perturbation theory and provides simultaneous access to the calculation of XMCD A, B, and C terms. The ability of the method to predict the XMCD C-term signs and provide access to the XMCD intensity mechanism is demonstrated on the basis of a generalized state coupling mechanism based on the type of the excitations dominating the relativistically corrected states. In the second step, the performance of CASCI, CASCI/NEVPT2, and ROCIS/DFT is evaluated. The very good agreement between theory and experiment has allowed us to unravel the complicated XMCD C-term mechanism on the basis of the SOC interaction between the various multiplets with spin S' = S, S ± 1. In the last step, it is shown that the commonly used spin and orbital sum rules are inadequate in interpreting the intensity mechanism of the XAS and XMCD spectra of the VIVO(acac)2 and VIII(acac)3 complexes as they breakdown when they are employed to predict their magneto-optical properties. This conclusion is expected to hold more generally.

Vibronic and Spin-Orbit Couplings of 3Π and 3Σ+ Electronic States in Linear Triatomic Molecules

A two-electron model describing vibronic and spin-orbit couplings of 3Π and 3Σ+ electronic states in linear triatomic molecules is proposed. The analysis is based on the use of an ab initio Breit–Pauli form of the spin-orbit coupling operator. Electronic Hamiltonian symmetry operators are shown to involve both spatial operations (acting on electronic coordinates) and matrix operations (acting on electron spins). In our analysis, we take into account only deformation π-modes, and the resulting 9 × 9 vibronic matrix actually describes the relativistic Renner pseudo-effect (3Π + 3Σ+) × π. The eigenvalues of the vibronic matrix (i.e., potential energy surfaces) have an axial symmetry, however, they cannot be represented by analytical expressions. The calculated vibronic Hamiltonian contains six electrostatic and 16 spin-orbit real parameters.

Spin-orbit coupling from a two-component self-consistent approach. II. Non-collinear density functional theories.

We revise formal and numerical aspects of collinear and noncollinear density functional theory (DFT) in the context of a two-component self-consistent treatment of spin-orbit coupling (SOC). While the extension of the standard one-component theory to a noncollinear magnetization is formally well-defined within the local density approximation, and therefore results in a numerically stable theory, this is not the case within the generalized gradient approximation (GGA). Previously reported formulations of noncollinear DFT based on GGA exchange-correlation potentials have several limitations: (i) they fail at reducing (either formally or numerically) to the proper collinear limit (i.e., when the magnetization is parallel or antiparallel to the z axis everywhere in space); (ii) they fail at ensuring a quantitative rotational invariance of the total energy and even a qualitative rotational invariance of the spatial distribution of the magnetization when a SOC operator is included in the Hamiltonian; (iii) they are numerically very unstable in regions of small magnetization. All of the above-mentioned problems are here shown (both formally and through test examples) to be solved by using instead a new formulation of noncollinear DFT for GGA functionals, which we call the "signed canonical" theory, as combined with an effective screening algorithm for unstable terms of the exchange-correlation potential in regions of small magnetization. All methods are implemented in the CRYSTAL program and tests are performed on simple molecules to compare the different formulations of noncollinear DFT.

Fundamental Role of Fock Exchange in Relativistic Density Functional Theory.

We perform a formal analysis of relativistic density functional theory for the treatment of spin-orbit coupling (SOC), noncollinear magnetization (NCM), and orbital current density (OCD). We identify specific components of the spinors (namely, those mapped onto imaginary diagonal spin-blocks of the density matrix) that arise from the SOC operator and define the OCD. We show that these pieces of the spinors only enter in the bielectronic part of the potential through the exact Fock exchange (FE) operator. The lack of FE therefore leads to a correspondingly incorrect physical description of SOC, NCM, and OCD. This analysis is complemented with an illustrative example, where we show that, while in the absence of FE, the theory fails even at reproducing the expected right-hand relationship between the NCM and OCD, its inclusion provides results that match those from a reference SOC configuration-interaction calculation.

Disentangling Electronic and Vibrational Effects in the Prediction of Band Shapes for Singlet–Triplet Transitions

The band shape of phosphorescent emission of benzophenone has been computed by using the first order perturbative expansion of singlet and triplet states with the spin–orbit coupling operator as perturbation and by evaluating Franck–Condon integrals with an efficient strategy for handling the whole set of vibrational coordinates. The computed band shape compares well with the experimental one, showing that modern computational tools yield reliable spin–orbit couplings to be used for evaluating the rates of singlet–triplet transitions in modern optoelectronic devices.

High-pressure insulator-to-metal transition in Sr3Ir2O7 studied by x-ray absorption spectroscopy

High-pressure x-ray absorption spectroscopy was performed at the Ir $L_3$ and $L_2$ absorption edges of Sr$_3$Ir$_2$O$_7$. The branching ratio of white line intensities continuously decreases with pressure, reflecting a reduction in the angular part of the expectation value of the spin-orbit coupling operator, $\left\langle {\bf L} \cdot {\bf S} \right\rangle$. Up to the high-pressure structural transition at 53 GPa, this behavior can be explained within a single-ion model, where pressure increases the strength of the cubic crystal field, which suppresses the spin-orbit induced hybridization of $J_{\text{eff}} = 3/2$ and $e_g$ levels. We observe a further reduction of the branching ratio above the structural transition, which cannot be explained within a single-ion model of spin-orbit coupling and cubic crystal fields. This change in $\left\langle {\bf L} \cdot {\bf S} \right\rangle$ in the high-pressure, metallic phase of Sr$_3$Ir$_2$O$_7$ could arise from non-cubic crystal fields or a bandwidth-driven hybridization of $J_{\text{eff}}=1/2,\,3/2$ states, and suggests that the electronic ground state significantly deviates from the $J_{\text{eff}}=1/2$ limit.

A Restricted Open Configuration Interaction with Singles Method To Calculate Valence-to-Core Resonant X-ray Emission Spectra: A Case Study.

In this work, a new protocol for the calculation of valence-to-core resonant X-ray emission (VtC RXES) spectra is introduced. The approach is based on the previously developed restricted open configuration interaction with singles (ROCIS) method and its parametrized version, based on a ground-state Kohn–Sham determinant (DFT/ROCIS) method. The ROCIS approach has the following features: (1) In the first step approximation, many-particle eigenstates are calculated in which the total spin is retained as a good quantum number. (2) The ground state with total spin S and excited states with spin S′ = S, S ± 1, are obtained. (3) These states have a qualitatively correct multiplet structure. (4) Quasi-degenerate perturbation theory is used to treat the spin–orbit coupling operator variationally at the many-particle level. (5) Transition moments are obtained between the relativistic many-particle states. The method has shown great potential in the field of X-ray spectroscopy, in particular in the field of transition-metal L-edge, which cannot be described correctly with particle–hole theories. In this work, the method is extended to the calculation of resonant VtC RXES [alternatively referred to as 1s-VtC resonant inelastic X-ray scattering (RIXS)] spectra. The complete Kramers–Dirac–Heisenerg equation is taken into account. Thus, state interference effects are treated naturally within this protocol. As a first application of this protocol, a computational study on the previously reported VtC RXES plane on a molecular managanese(V) complex is performed. Starting from conventional X-ray absorption spectra (XAS), we present a systematic study that involves calculations and electronic structure analysis of both the XAS and non-resonant and resonant VtC XES spectra. The very good agreement between theory and experiment, observed in all cases, allows us to unravel the complicated intensity mechanism of these spectroscopic techniques as a synergic function of state polarization and interference effects. In general, intense features in the RIXS spectra originate from absorption and emission processes that involve nonorthogonal transition moments. We also present a graphical method to determine the sign of the interference contributions.