Second-order Spin-orbit Coupling Research Articles

Slow magnetic relaxation in a europium(II) complex.

Single-ion anisotropy is vital for the observation of Single-Molecule Magnet (SMM) properties (i.e., a slow dynamics of the magnetization) in lanthanide-based systems. In the case of europium, the occurrence of this phenomenon has been inhibited by the spin and orbital quantum numbers that give way to J = 0 in the trivalent state and the half-filled population of the 4f orbitals in the divalent state. Herein, by optimizing the local crystal field of a quasi-linear bis(silylamido) EuII complex, the [EuII(N{SiMePh2}2)2] SMM is described, providing an example of a europium complex exhibiting slow relaxation of its magnetization. This behavior is dominated by a thermally activated (Orbach-like) mechanism, with an effective energy barrier of approximately 8 K, determined by bulk magnetometry and electron paramagnetic resonance. Ab initio calculations confirm second-order spin-orbit coupling effects lead to non-negligible axial magnetic anisotropy, splitting the ground state multiplet into four Kramers doublets, thereby allowing for the observation of an Orbach-like relaxation at low temperatures.

Spin-vibronic coupling in the quantum dynamics of a Fe(III) trigonal-bipyramidal complex.

The presence of a high density of excited electronic states in the immediate vicinity of the optically bright state of a molecule paves the way for numerous photo-relaxation channels. In transition-metal complexes, the presence of heavy atoms results in a stronger spin-orbit coupling, which enables spin forbidden spin-crossover processes to compete with the spin-allowed internal conversion processes. However, no matter how effectively the states cross around the Franck-Condon region, the degree of vibronic coupling, of both relativistic and non-relativistic nature, drives the population distribution among these states. One such case is demonstrated in this work for the intermediate-spin Fe(III) trigonal-bipyramidal complex. A quantum dynamical investigation of the photo-deactivation mechanism in the Fe(III) system is presented using the multi-configurational time-dependent Hartree approach based on the vibronic Hamiltonian whose coupling terms are derived from the state-averaged complete active space self-consistent field/complete active space with second-order perturbation theory (CASPT2) calculations and spin-orbit coupling of the scalar-relativistic CASPT2 states. The results of this study show that the presence of a strong (non-relativistic) vibronic coupling between the optically bright intermediate-spin state and other low-lying states of the same spin-multiplicity overpowers the spin-orbit coupling between the intermediate-spin and high-spin states, thereby lowering the chances of spin-crossover while exhibiting ultrafast relaxation among the intermediate-spin states. In a special case, where the population transfer pathway via the non-relativistic vibronic coupling is blocked, the probability of the spin-crossover is found to increase. This suggests that a careful modification of the complex by incorporation of heavier atoms with stronger relativistic effects can enhance the spin-crossover potential of Fe(III) intermediate-spin complexes.

Magneto-optical spectra of the split nickel-vacancy defect in diamond

Nickel is a common impurity in high-pressure high-temperature diamond and may contaminate chemical vapor deposited diamond used for high-power electronics or quantum technology applications. Magneto-optical fingerprints of nickel have been known since decades, however, no consensus has been reached about the microscopic origins of nickel-related electron paramagnetic resonance, photoluminescence, and optically detected magnetic resonance spectra. The unknown nickel-related defect structures in diamond make it difficult to control them or harness them for a given application. As a consequence, nickel is considered as an impurity in diamond that should be avoided or its concentration should be minimized. Recent advances in the development of ab initio magneto-optical spectroscopy have significantly increased its accuracy and predictive power that can be employed for identification and in-depth characterization of paramagnetic color centers in diamond. In this study, we extend the accuracy of the ab initio magneto-optical spectroscopy tools towards self-consistent calculation of second-order spin-orbit coupling for paramagnetic color centers in solids. We apply the full arsenal of the ab initio magneto-optical spectroscopy tools to characterize the split nickel-vacancy defect in diamond which is one of the most stable nickel-related defect configurations. As a result, electron paramagnetic resonance and optical centers are positively identified in various charge states of the nickel-vacancy defect in diamond. In particular, the 1.40-eV optical center and the NIRIM-2 electron paramagnetic resonance center are identified as the single negative charge state of the split nickel-vacancy center. The defect possesses $S=\frac{1}{2}$ spin state with an orbital doublet ground state. We find that the coherence time of the ground-state spin is about 0.1 ms at cryogenic temperatures which can be optically initialized and readout by a $\mathrm{\ensuremath{\Lambda}}$-scheme protocol. Since the defect has inversion symmetry the optical signal is insensitive to the stray electric fields, which is an advantage for creating indistinguishable solid-state single-photon sources. We predict that the negatively charged nickel-vacancy defect has similar optical properties to those of the well-known silicon-vacancy defect in diamond but is superior in terms of electron spin coherence times. Our study resolves a few decades controversy about the nickel-related spectroscopy centers in diamond and turns nickel from an impurity to a resource in quantum technology applications.

Effect of spin-dependent interactions on the two-body loss rate in ultracold Rb85 collisions

The spin-dependent interactions (SDIs), consisted of second-order spin-orbit coupling (SOC) and magnetic dipole-dipole interactions (MDDI), play an important role in ultracold two-body collisions for heavy atoms. We survey the inelastic two-body loss induced by SDIs in ultracold collisions of $^{85}\mathrm{Rb}$ in Zeeman sublevel $|2,\ensuremath{-}2\ensuremath{\rangle}$ by using time-independent first-order perturbation theory (PT) and close-coupling (CC) approaches. The two-body loss rates from the lowest channel in ${M}_{F}=\ensuremath{-}4$ to outgoing channel ${M}_{F}=\ensuremath{-}3$ at three typical magnetic field ranges are discussed. Our calculations show PT can describe the two-body loss process well by considering the full scattering wave function even though in the region around Feshbach resonances where loss is significantly enhanced. The second-order SOC, which takes the same angular momentum coupling with MDDI, is shown to be important to $^{85}\mathrm{Rb}$ two-body loss, while this term was usually omitted in previous PT calculations in some systems. Besides, our numerical results indicate the pronounced dip of two-body loss rate around 571 G mainly comes from the superposition of losses from the two lower outgoing channels in ${M}_{F}=\ensuremath{-}2$ with close minima. We also demonstrated an interesting phenomenon: the interference of two-body loss induced by the second-order SOC and MDDI. It paves an alternative way to suppress inelastic collisions through the interference between different loss causes.

Ferromagnetic ordering of linearly coordinated Co ions in LiSr2[CoN2]

LiSr$_2$[CoN$_2$] single crystals were successfully grown out of Li-rich flux. Temperature- and field-dependent measurements of the magnetization in the range of $T = 2 - 300$ K and up to $\mu_{0}\textit{H} = 7$ T as well as measurements of the heat capacity are presented. Ferromagnetic ordering emerges below $T_C = 44$ K and comparatively large coercivity fields of $\mu_0H = 0.3$ T as well as pronounced anisotropy are observed upon cooling. Polycrystalline samples of the Ca analog LiCa$_2$[CoN$_2$] were obtained and investigated in a similar way. In both compounds Co manifests orbital contributions to the magnetic moment and large single-ion anisotropy that is caused by second-order Spin-orbit coupling. Quantum chemistry calculations reveal a magnetic anisotropy energy of 7 meV, twice as large as the values reported for similar Co $d^{8}$ systems.

Zero-Magnetic-Field Splitting in the Excited Triplet States of Octahedral Hexanuclear Molybdenum(II) Clusters: [{Mo6X8}Y6]2- (X, Y = Cl, Br, I).

The temperature ( T) dependences of the emissions from the tetra- n-butylammonium salts of [{Mo6X8}Y6]2- (X, Y = Cl, Br, and I) in optically transparent polyethylene glycol dimethacrylate matrixes were studied in the T range of 3-300 K. [{Mo6Cl8}Y6]2-, [{Mo6Br8}Y6]2-, and [{Mo6I8}I6]2- showed the T-dependent emission characteristics similar to those of other hexanuclear Mo(II), Re(III), and W(II) clusters reported previously, while [{Mo6I8}Br6]2- and [{Mo6I8}Cl6]2- exhibited the emission properties different from those of other [{Mo6X8}Y6]2- clusters. The photophysical behavior of these clusters was explained by the excited triplet state spin-sublevel ( Φn, n = 1-4) model irrespective of the nature of X and Y. The zero-magnetic-field splitting energies between the lowest energy (Φ1) and the higher energy spin sublevels (Φ4 or Φ3) caused by the first- or second-order spin-orbit coupling, Δ E14 or Δ E13, were evaluated to be 620-870 or 50-99 cm-1, respectively. We found the linear correlation between the Δ E14 or Δ E13 value and the fourth power of the atomic number ( Z) of the inner halide X: Δ E14 or Δ E13 vs { Z(X)}4 (correlation coefficient: cc = ∼ 0.999). Furthermore, we also found the correlation between Δ E14 or Δ E13 and the 95Mo NMR chemical shift of the cluster. These findings gave very important insight into the spin-orbit coupling and zero-magnetic-field splitting in the excited triplet states of transition metal complexes.

Phase Locking between Different Partial Waves in Atom-Ion Spin-Exchange Collisions

We present a joint experimental and theoretical study of spin dynamics of a single ^{88}Sr^{+} ion colliding with an ultracold cloud of Rb atoms in various hyperfine states. While spin exchange between the two species occurs after 9.1(6) Langevin collisions on average, spin relaxation of the Sr^{+} ion Zeeman qubit occurs after 48(7) Langevin collisions, which is significantly slower than in previously studied systems due to a small second-order spin-orbit coupling. Furthermore, a reduction of the endothermic spin-exchange rate is observed as the magnetic field is increased. Interestingly, we find that while the phases acquired when colliding on the spin singlet and triplet potentials vary largely between different partial waves, the singlet-triplet phase difference, which determines the spin-exchange cross section, remains locked to a single value over a wide range of partial waves, which leads to quantum interference effects.

Synthesis, characterization and magnetism of homoleptic bis(5-aryl-2-iminopyrrolyl) complexes of iron(II) and cobalt(II)

Four sterically congested and highly electronically unsaturated bis(5-aryl-2-iminopyrrolyl) ML2 complexes of Fe(II) and Co(II), [M{κ2N,N-5-(2,6-R2-C6H3)-NC4H2-2-C(H)N(2,6-iPr2-C6H3)}2] (M = Fe, R = H (1a); M = Fe, R = Me (1b); M = Co, R = H (2a); M = Co, R = Me (2b)), were synthesized by metathetic reaction of the in situ prepared sodium salts of the new 5-aryl-2-(N-2,6-diisopropylformimino)pyrrole HL type ligand precursors (R = H (Ia), Me (Ib)) with FeCl2 or CoCl2, in moderate to good yields. The four-coordinate complexes were characterized by elemental analysis, X-ray diffraction, magnetic susceptibility measurements in solution and in solid state, and 57Fe Mössbauer spectroscopy. All complexes have a high-spin electronic configuration, as measured both in solution, through the Evans method, and in solid state, through SQUID magnetic measurements, and Mössbauer spectroscopy for the case of Fe(II) complexes, displaying a tetracoordinated distorted tetrahedral coordination geometry about the respective metal center, as elucidated by X-ray diffraction. The inclusion of ortho-methyl groups in the 5-phenyl substituent of the 2-iminopyrrolyl ring in complexes 1b and 2b has an important influence in the structure of the complexes, giving rise to pronounced geometry distortions in particular for 1b. This complex exhibits a see-saw-like geometry whereas a pyramidalization of the pyrrolyl nitrogen is observed for 2b. Overall, the magnetic properties of the complexes can be rationalized by taking into account the effect of the magnetic anisotropy arising from the second-order spin-orbit coupling and the geometrical distortion. The simulation of the magnetic data using a zero-field splitting (ZFS) Hamiltonian led to values of the axial anisotropy, D, that are coherent for Co(II) and Fe(II) tetrahedral complexes, D ≈ −50 cm−1 (2a, 2b) and 4 cm−1 (1a). In the case of 1b the simulation evidenced a strong magnetic anisotropy with both axial and rhombic contributions (D ≈ −50 cm−1, E ≈ 10 cm−1) which are in line with the strongly distorted coordination sphere of the complex.

Magnetism of metallacrown single-molecule magnets: From a simplest model to realistic systems

Electronic and magnetic properties of molecular nanomagnets are determined by competing energy scales due to the crystal field splitting, the exchange interactions between transition metal atoms, and relativistic effects. We present a comprehensive theory embracing all these phenomena based on first-principles calculations. In order to achieve this goal, we start from the ${\mathrm{FeNi}}_{4}$ cluster as a paradigm. The system can be accurately described on the ab initio level yielding all expected electronic states in a range of multiplicities from 1 to 9, with a ferromagnetic ground state. By adding the spin-orbit coupling between them we obtain the zero-field splitting. This allows to introduce a spin Hamiltonian of a giant spin model, which operates on a smaller energy scale. We compare the computed parameters of this Hamiltonian with the experimental and theoretical magnetic anisotropy energies of the monolayer Ni/Cu(001). In line with them, we find that the anisotropy almost entirely originates from the second-order spin-orbit coupling, the spin-spin coupling constitutes only a small fraction. Finally, we include the ligand atoms in our consideration. This component has a decisive role for the stabilization of molecules in experimental synthesis and characterization, and also substantially complicates the theory by bringing the superexchange mechanisms into play. Since they are higher-order effects involving two hopping matrix elements, not every theory can describe them. Our generalization of the corresponding perturbation theory substantiates the use of complete active space methods for the description of superexchange. At the same time, our numerical results for the $\left\{{\mathrm{CuFe}}_{4}\right\}$ system demonstrate that the Goodenough-Kanamori rules, which are often used to determine the sign of these exchange interactions, cannot deliver quantitative predictions due to the interplay of other mechanisms, e. g., involving multicenter Coulomb integrals. We conclude by comparing ab initio values of the exchange interaction constants for the $\left\{{\mathrm{CuCu}}_{4}\right\}$ and $\left\{{\mathrm{CuFe}}_{4}\right\}$ metallacrown magnetic molecules with experimental values determined by fitting of the magnetic susceptibility curves ${\ensuremath{\chi}}_{M}T(T)$, and attribute the remaining discrepancy between them to the role of virtual electron excitations into and out of the active space (dynamical correlations).

Theoretical evidence for the direct 3MLCT-HS deactivation in the light-induced spin crossover of Fe(ii)-polypyridyl complexes.

Spin-orbit couplings have been calculated in twenty snapshots of a molecular dynamics trajectory of [Fe(bpy)3]2+ to address the importance of geometrical distortions and second-order spin-orbit coupling on the intersystem crossing rate constants in the light-induced spin crossover process. It was found that the effective spin-orbit coupling between the 3MLCT and 5T2 state is much larger than the direct coupling in the symmetric structure, which opens the possibility of a direct 3MLCT-5T2 deactivation without the intervention of triplet metal-centered states. Based on the calculated deactivation times, we conclude that both the direct pathway and the one involving intermediate triplet states are active in the ultrafast population of the metastable HS state, bringing in agreement two experimental observations that advocate for either deactivation mechanism. This resolves a long-standing dispute about the deactivation mechanism of Fe(ii)-polypyridyl complexes in particular, and about light-induced magnetism in transition metal complexes in general.

Effect of Second-Order Spin-Orbit Coupling on the Interaction between Spin States in Spin-Crossover Systems.

The second-order spin-orbit coupling is evaluated in two transition-metal complexes to establish the effect on the deactivation mechanism of the excited low-spin state in systems that undergo spin transitions under the influence of light. We compare the standard perturbational approach to calculate the second-order interaction with a variational strategy based on the effective Hamiltonian theory and show that the former one can only be applied in some special cases and even then gives results that largely overestimate the interaction. The combined effect of geometry distortions and second-order spin-orbit coupling leads to sizeable interactions for states that are nearly uncoupled in the symmetric (average) structure of the complex. This opens the possibility of a direct deactivation from the singlet and triplet states of the metal-to-ligand charge-transfer manifold to the final high-spin state as suggested from the interpretation of experimental data but so far not supported by theoretical descriptions of the light-induced spin crossover.

Toward Chemical Accuracy in ab Initio Thermochemistry and Spectroscopy of Lanthanide Compounds: Assessing Core-Valence Correlation, Second-Order Spin-Orbit Coupling, and Higher Order Effects in Lanthanide Diatomics.

The higher order (HO) correlation beyond the coupled-cluster single double (triple) CCSD(T) level of theory, second-order spin-orbit coupling (SOC), and core-valence (CV) correlation effects on bond length, re, vibrational frequency, ωe, and dissociation energy, De, are studied for a set of 17 lanthanide containing diatomics, including lanthanum, europium, ytterbium, and lutetium oxides and halides. Convergence in the magnitudes of the SOC, CV, and HO corrections with respect to basis set size is examined using a sequence of double, triple, and quadruple-ζ basis sets, with the complete basis set (CBS) limit estimates provided in most cases. The CV effects on De, re, and ωe are calculated to amount up to 1.3 kcal·mol-1, 0.008 Å, and 5 cm-1, respectively. A detailed analysis of the origin of the CV effect with a particular accounting for various subvalence shells reveals that, generally, the Ln 4d correlation makes a major contribution, although in some instances the lower-lying 4sp shells contribute largely and even more substantially than 4d. The second-order SOC effect evaluated via two-component and four-component relativistic techniques proves to be non-negligible, especially for heavier species, e.g., LaI, in which it is as large as 0.8 kcal·mol-1 in De, 0.002 Å in re, and 1.3 cm-1 in ωe. Higher order correlation effects assessed through the CCSDT(Q) level are mostly less than 0.8 kcal·mol-1, 0.004 Å, and 5 cm-1; however, in species with prominence of nondynamical correlation, e.g., EuO, YbF, and LuO, the HO correction can amount to 1.2-1.6 kcal·mol-1, 0.005-0.008 Å, and 8-30 cm-1 in De, re, and ωe, respectively. In general, the [CCSD(T)+CV]/CBS + SOC + HO composite results are in good agreement with the available experimental data, exhibiting a mean absolute deviation of 1.8 kcal·mol-1 in De, 0.0023 Å in re, and 3.5 cm-1 in ωe. A significant experimental outlier, the bond length in YbI, is revealed, implying the need for re-examination of the experimental data.

The missing halide: Millimeter-wave spectroscopy of the MgI radical (X2Σ+)

The pure rotational spectrum of the MgI radical in its ground electronic state (X2Σ+) has been measured using millimeter/submillimeter wave direct absorption techniques in the region of 200–300GHz. The molecule was created in a DC discharge by the reaction of magnesium vapor, produced in a Broida-type oven, with CH3I. Between five to twelve transitions were recorded for the v=0, 1, and 2 vibrational states of 24MgI, as well as the v=0 state for the isotopologues 25MgI and 26MgI, measured in their natural magnesium abundance. All observed transitions exhibited large spin-rotation splittings. Rotational, centrifugal distortion, and spin-rotation constants were determined for each isotopologue and the excited vibrational states of 24MgI. Equilibrium parameters Be, αe, De, βe, γe and γe′ were derived for the main isotopologue, as well as the equilibrium bond length, re=2.5730 (1)Å. The large spin-rotation constant of γ∼300MHz in MgI is thought to arise from second-order spin-orbit coupling originating in the nearby A2Π state, with possible contributions from other excited 2Π states.

Strong Spin-Phonon Coupling Mediated by Single Ion Anisotropy in the All-In-All-Out Pyrochlore Magnet Cd_{2}Os_{2}O_{7}.

Spin-phonon coupling mediated by single ion anisotropy was investigated using optical spectroscopy and first-principles calculations in the all-in-all-out pyrochlore magnet Cd_{2}Os_{2}O_{7}. Clear anomalies were observed in both the phonon frequencies and linewidths at the magnetic ordering temperature. The renormalization of the phonon modes was exceptionally large, signifying the presence of an unconventional magnetoelastic term from large spin-orbit coupling. In addition, the relative phonon frequency shifts show a strong correlation with the modulation of noncubic crystal field by the corresponding lattice distortion. Our observation establishes a new type of spin-phonon coupling through single ion anisotropy, a second-order spin-orbit coupling term, in Cd_{2}Os_{2}O_{7}.

Electronic and magnetic properties of monolayer MnS2

First-principles calculations are performed to study the electronic and magnetic properties of monolayer MnS2. Based on the electronic structure, a half-metallic state is predicted for monolayer MnS2. The magnetic moment is 3.0 [Formula: see text] per formula unit, and the main contribution is localized at the transition metal site Mn with a local moment of 3.733 [Formula: see text]. The magnetic anisotropy energy (MAE) is 0.056 meV per formula unit with an easy axis perpendicular to the plane, and it indicates that monolayer MnS2 belongs to the category of Ising magnets. The positive MAE of nanosheets mainly stems from the area around [Formula: see text] in the reciprocal space. To find the microscopic origin, we take the method of the second-order spin orbit coupling. The occupied spin-up dz2 state and the unoccupied spin-down dyz states in the [Formula: see text] point through the [Formula: see text] operator make positive contributions to the MAE.

Large Easy-Plane Magnetic Anisotropy in a Three-Coordinate Cobalt(II) Complex [Li(THF)4 ][Co(NPh2 )3

Magnetic anisotropy is the key element in the construction of single-ion magnets, a kind of nanomagnets for high-density information storage. This works describes an unusual large easy-plane magnetic anisotropy (with a zero-field splitting parameter D of +40.2 cm-1 ), mainly arising from the second-order spin-orbit coupling effect in a trigonal-planar CoII complex [Li(THF)4 ][Co(NPh2 )3 ], revealed by combined studies of magnetism, high frequency/field electron paramagnetic resonance spectroscopy, and ab initio calculations. Meanwhile, the field-induced slow magnetic relaxation in this complex was mainly attributed to the Raman process.

Composite vibrational spectroscopy of the group 12 difluorides: ZnF2, CdF2, and HgF2.

The vibrational spectra of group 12 difluorides, MF2 (M = Zn, Cd, Hg), were investigated via coupled cluster singles, doubles, and perturbative triples, CCSD(T), including core correlation, with a series of correlation consistent basis sets ranging in size from triple-zeta through quintuple-zeta quality, which were then extrapolated to the complete basis set (CBS) limit using a variety of extrapolation procedures. The explicitly correlated coupled cluster method, CCSD(T)-F12b, was employed as well. Although exhibiting quite different convergence behavior, the F12b method yielded the CBS limit estimates closely matching more computationally expensive conventional CBS extrapolations. The convergence with respect to basis set size was examined for the contributions entering into composite vibrational spectroscopy, including those from higher-order correlation accounted for through the CCSDT(Q) level of theory, second-order spin-orbit coupling effects assessed within four-component and two-component relativistic formalisms, and vibrational anharmonicity evaluated via a perturbative treatment. Overall, the composite results are in excellent agreement with available experimental values, except for the CdF2 bond-stretching frequencies compared to spectral assignments proposed in a matrix isolation infrared and Raman study of cadmium difluoride vapor species [Loewenschuss et al., J. Chem. Phys. 50, 2502 (1969); Givan and Loewenschuss, J. Chem. Phys. 72, 3809 (1980)]. These assignments are called into question in the light of the composite results.

The electronic structure of vanadium monochloride cation (VCl(+)): tackling the complexities of transition metal species.

Six electronic states (X (4)Σ(-), A (4)Π, B (4)Δ, (2)Φ, (2)Δ, (2)Σ(+)) of the vanadium monochloride cation (VCl(+)) are described using large basis set coupled cluster theory. For the two lowest quartet states (X (4)Σ(-) and A (4)Π), a focal point analysis (FPA) approach was used that conjoined a correlation-consistent family of basis sets up to aug-cc-pwCV5Z-DK with high-order coupled cluster theory through pentuple (CCSDTQP) excitations. FPA adiabatic excitation energies (T0) and spectroscopic constants (re, r0, Be, B0, D¯e, He, ωe, v0, αe, ωexe) were extrapolated to the valence complete basis set Douglas-Kroll (DK) aug-cc-pV∞Z-DK CCSDT level of theory, and additional treatments accounted for higher-order valence electron correlation, core correlation, and spin-orbit coupling. Due to the delicate interplay between dynamical and static electronic correlation, single reference coupled cluster theory is able to provide the correct ground electronic state (X (4)Σ(-)), while multireference configuration interaction theory cannot. Perturbations from the first- and second-order spin orbit coupling of low-lying states with quartet spin multiplicity reveal an immensely complex rotational spectrum relative to the isovalent species VO, VS, and TiCl. Computational data on the doublet manifold suggest that the lowest-lying doublet state ((2)Γ) has a Te of ∼11 200 cm(-1). Overall, this study shows that laboratory and theoretical rotational spectroscopists must work more closely in tandem to better understand the bonding and structure of molecules containing transition metals.

Perturbative treatment of spin-orbit coupling within spin-free exact two-component theory.

This work deals with the perturbative treatment of spin-orbit-coupling (SOC) effects within the spin-free exact two-component theory in its one-electron variant (SFX2C-1e). We investigate two schemes for constructing the SFX2C-1e SOC matrix: the SFX2C-1e+SOC [der] scheme defines the SOC matrix elements based on SFX2C-1e analytic-derivative theory, hereby treating the SOC integrals as the perturbation; the SFX2C-1e+SOC [fd] scheme takes the difference between the X2C-1e and SFX2C-1e Hamiltonian matrices as the SOC perturbation. Furthermore, a mean-field approach in the SFX2C-1e framework is formulated and implemented to efficiently include two-electron SOC effects. Systematic approximations to the two-electron SOC integrals are also proposed and carefully assessed. Based on benchmark calculations of the second-order SOC corrections to the energies and electrical properties for a set of diatomic molecules, we show that the SFX2C-1e+SOC [der] scheme performs very well in the computation of perturbative SOC corrections and that the "2eSL" scheme, which neglects the (SS|SS)-type two-electron SOC integrals, is both efficient and accurate. In contrast, the SFX2C-1e+SOC [fd] scheme turns out to be incompatible with a perturbative treatment of SOC effects. Finally, as a first chemical application, we report high-accuracy calculations of the (201)Hg quadrupole-coupling parameters of the recently characterized ethylmercury hydride (HHgCH2CH3) molecule based on SFX2C-1e coupled-cluster calculations augmented with second-order SOC corrections obtained at the Hartree-Fock level using the SFX2C-1e+SOC [der]/2eSL scheme.

Ultracold, radiative charge transfer in hybrid Yb ion–Rb atom traps

Ultracold hybrid ion–atom traps offer the possibility of microscopic manipulation of quantum coherences in the gas using the ion as a probe. However, inelastic processes, particularly charge transfer can be a significant process of ion loss and has been measured experimentally for the ion immersed in a Rb vapour. We use first-principles quantum chemistry codes to obtain the potential energy curves and dipole moments for the lowest-lying energy states of this complex. Calculations for the radiative decay processes cross sections and rate coefficients are presented for the total decay processes; and . Comparing the semi-classical Langevin approximation with the quantum approach, we find it provides a very good estimate of the background at higher energies. The results demonstrate that radiative decay mechanisms are important over the energy and temperature region considered. In fact, the Langevin process of ion–atom collisions dominates cold ion–atom collisions. For spin-dependent processes [] the anisotropic magnetic dipole–dipole interaction and the second-order spin–orbit coupling can play important roles, inducing coupling between the spin and the orbital motion. They measured the spin-relaxing collision rate to be approximately five orders of magnitude higher than the charge-exchange collision rate []. Regarding the measured radiative charge transfer collision rate, we find that our calculation is in very good agreement with experiment and with previous calculations. Nonetheless, we find no broad resonances features that might underly a strong isotope effect. In conclusion, we find, in agreement with previous theory that the isotope anomaly observed in experiment remains an open question.