Treatment Of Spin-orbit Coupling Research Articles

Low-lying [formula omitted] electronic states of BiF: Perturbative versus variational approaches of spin–orbit coupling

The low-lying Ω states of bismuth monofluoride (BiF) below 50000 cm−1 have been theoretically studied using the multi-reference configuration interaction and the equation-of-motion coupled-cluster methods, where the spin–orbit coupling (SOC) effects are respectively considered perturbatively or variationally. Since the perturbative treatment of SOC is not a good approximation for the heavy atom Bi, the latter method exhibits higher accuracy and better agreements with the experimental spectroscopic constants. Our results show that the second 3Π state of BiF is due to the occupation on the Rydberg 7s-shell of Bi, which gives rise to the so-called “triplet C” state and its sub-states reported in the early literatures. With the help of our theoretical results, some experimentally assigned spectral bands with considerable confusion have also been clarified and reassigned.

Relativistic Effects in Modeling the Ligand K-Edge X-ray Absorption Near-Edge Structure of Uranium Complexes.

Accurate modeling of the complex electronic structure of actinide complexes requires full inclusion of relativistic effects. In this study, we examine the effect of explicit inclusion of spin-orbit coupling (SOC) versus scalar relativistic effects on the predicted spectra for heavy-element complexes. In this study, we employ a relativistic two-component Hamiltonian in the X2C form with all of the electrons in the system being considered explicitly to compare and contrast with previous studies that included the relativistic effects by means of relativistic effective core potentials (RECPs). A few uranium complexes are chosen as model systems. Comparison of the computed Cl K-edge X-ray absorption spectra with experimental data shows significantly improved agreement when a variational relativistic treatment of SOC is performed. In particular, we note the importance of SOC terms to obtain not only correct transition energies but also correct intensities for these heavy-element complexes because of the redistribution of ligand bonding character among the valence MOs. While RECPs generally agree well with all-electron scalar relativistic calculations, there are some differences in the predicted spectra of open-shell systems. These methods are still suitable for broad application to analyze the qualitative nature of transitions in X-ray absorption spectra, but caution is recommended for quantitative analysis, as SOC can be non-negligible for both open- and closed-shell heavy-element systems.

Perturbation Theory Treatment of Spin-Orbit Coupling, Part I: Double Perturbation Theory Based on a Single-Reference Initial Approximation.

We develop a perturbation theory for solving the many-body Dirac equation within a given relativistic effective-core potential approximation. Starting from a scalar-relativistic unrestricted Hartree-Fock (SR UHF) solution, we carry out a double perturbation expansion in terms of spin-orbit coupling (SOC) and the electron fluctuation potential. Computationally convenient energy expressions are derived through fourth order in SOC, second order in the electron fluctuation potential, and a total of third order in the coupling between the two. Illustrative calculations on the halogen series of neutral and singly positive diatomic molecules show that the perturbation expansion is well-converged by taking into account only the leading (nonvanishing) term at each order of the electron fluctuation potential. Our perturbation theory approach provides a computationally attractive alternative to a two-component self-consistent field treatment of SOC. In addition, it includes coupling with the fluctuation potential through third order and can be extended (in principle) to multireference calculations, when necessary for both closed- and open-shell cases, using quasi-degenerate perturbation theory.

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.

Beyond chemical accuracy in the heavy p-block: The first ionization potentials and electron affinities of Ga-Kr, In-Xe, and Tl-Rn.

A relativistic coupled-cluster version of the Feller-Peterson-Dixon composite method has been used to accurately calculate the first ionization potentials (IPs) and electron affinities (EAs) of the post-d, p-block elements Ga-Rn. Complete basis set extrapolations including outer-core correlation at the CCSD(T) level of theory were combined with contributions from higher order electron correlation up to CCSDTQ, quantum electrodynamic effects (Lamb shift), and spin-orbit (SO) coupling including the Gaunt contribution. Several methods for including SO were investigated, in which all involved the four-component (4c) Dirac-Coulomb (DC) Hamiltonian. The treatment of SO coupling was the contribution that limited the final accuracy of the present results. In the cases where 4c-DC-CCSD(T) could be reliably used for the SO contributions, the final composite IPs and EAs agreed with the available experimental values to within an unsigned average error of just 0.16 and 0.20 kcal/mol, respectively. In all cases, the final IPs and EAs were within 1 kcal/mol of the available experimental values, except for the EAs of the group 13 elements (Ga, In, and Tl), where the currently accepted experimental values appear to be too large by as much as 4 kcal/mol. The values predicted in this work, which have estimated uncertainties of ±0.5 kcal/mol, are 5.25 (Ga), 7.69 (In), and 7.39 (Tl) kcal/mol. For the EAs of Po and At, which do not have experimental values, the current calculations predict values of 34.2 and 55.8 kcal/mol with estimated uncertainties of ±0.6 and ±0.3 kcal/mol, respectively.

Perturbative treatment of spin-orbit-coupling within spin-free exact two-component theory using equation-of-motion coupled-cluster methods.

A scheme is reported for the perturbative calculation of spin-orbit coupling (SOC) within the spin-free exact two-component theory in its one-electron variant (SFX2C-1e) in combination with the equation-of-motion coupled-cluster singles and doubles method. Benchmark calculations of the spin-orbit splittings in 2Π and 2P radicals show that the accurate inclusion of scalar-relativistic effects using the SFX2C-1e scheme extends the applicability of the perturbative treatment of SOC to molecules that contain heavy elements. The contributions from relaxation of the coupled-cluster amplitudes are shown to be relatively small; significant contributions from correlating the inner-core orbitals are observed in calculations involving third-row and heavier elements. The calculation of term energies for the low-lying electronic states of the PtH radical, which serves to exemplify heavy transition-metal containing systems, further demonstrates the quality that can be achieved with the pragmatic approach presented here.

Intersystem-crossing and phosphorescence rates in fac-Ir(III)(ppy)3: a theoretical study involving multi-reference configuration interaction wavefunctions.

We have employed combined density functional theory and multi-reference configuration interaction methods including spin-orbit coupling (SOC) effects to investigate the photophysics of the green phosphorescent emitter fac-tris-(2-phenylpyridine)iridium (fac-Ir(ppy)3). A critical evaluation of our quantum chemical approaches shows that a perturbational treatment of SOC is the method of choice for computing the UV/Vis spectrum of this heavy transition metal complex while multi-reference spin-orbit configuration interaction is preferable for calculating the phosphorescence rates. The particular choice of the spin-orbit interaction operator is found to be of minor importance. Intersystem crossing (ISC) rates have been determined by Fourier transformation of the time correlation function of the transition including Dushinsky rotations. In the electronic ground state, fac-Ir(ppy)3 is C3 symmetric. The calculated UV/Vis spectrum is in excellent agreement with experiment. The effect of SOC is particularly pronounced for the metal-to-ligand charge transfer (MLCT) band in the visible region of the absorption spectrum which does not only extend its spectral onset towards longer wavelengths but also experiences a blue shift of its maximum. Pseudo-Jahn-Teller interaction leads to asymmetric coordinate displacements in the lowest MLCT states. Substantial electronic SOC and a small energy gap make ISC an ultrafast process in fac-Ir(ppy)3. For the S1↝T1 non-radiative transition, we compute a rate constant of kISC = 6.9 × 10(12) s(-1) which exceeds the rate constant of radiative decay to the electronic ground state by more than six orders of magnitude, in agreement with the experimental observation of a subpicosecond ISC process and a triplet quantum yield close to unity. As a consequence of the geometric distortion in the T1 state, the T1 → S0 transition densities are localized on one of the phenylpyridyl moieties. In our best quantum chemical model, we obtain phosphorescence decay times of 264 μs, 13 μs, and 0.9 μs, respectively, for the T1,I, T1,II, and T1,III fine-structure levels in dichloromethane (DCM) solution. In addition to reproducing the correct orders of magnitude for the individual phosphorescence emission probabilities, our theoretical study gives insight into the underlying mechanisms. In terms of intensity borrowing from spin-allowed transitions, the low emission probability of the T1,I substate is caused by the mutual cancellation of contributions from several singlet states to the total transition dipole moment. Their contributions do not cancel but add up in case of the much faster T1,III → S0 emission while the T1,II → S0 emission is dominated by intensity borrowing from a single spin-allowed process, i.e., the S2 → S0 transition.

Quasiclassical Trajectory Calculations of the Rate Constant of the OH + HBr → Br + H2O Reaction Using a Full-Dimensional Ab Initio Potential Energy Surface Over the Temperature Range 5 to 500 K.

We report a permutationally invariant, ab initio potential energy surface (PES) for the OH + HBr → Br + H2O reaction. The PES is a fit to roughly 26 000 spin-free UCCSD(T)/cc-pVDZ-F12a energies and has no classical barrier to reaction. It is used in quasiclassical trajectory calculations with a focus on the thermal rate constant, k(T), over the temperature range 5 to 500 K. Comparisons with available experimental data over the temperature range 23 to 416 K are made using three approaches to treat the OH rotational and associated electronic partition function. All display an inverse temperature dependence of k(T) below roughly 160 K and a nearly constant temperature dependence above 160 K, in agreement with experiment. The calculated rate constant with no treatment of spin-orbit coupling is overall in the best agreement with experiment, being (probably fortuitously) within 20% of it.

Ab-initio study of free standing TiO2 clusters: Stability and magnetism

We report the structural behavior of nanoscale Titanium Dioxide (TiO2) clusters as well as their magnetic properties by varying the cluster size with the help of ground state geometries. The clusters of atomic scale rutile (TiO2)n, where n = 1-11, have been considered and geometrically stabilized through the Density Functional Theory as implemented in Vienna ab-initio Simulation Package. It is being observed that as the size of cluster increases from n = 2 to 11, the total energy decreases. The results of formation energy reveal the fact that as the cluster grows, it moves towards the stability and it is observed that n = 11 is the most stable structure. The stabilized clusters are different in geometries and co-ordination numbers. Finally, all the clusters have been investigated with self consistent treatment of spin orbit coupling for magnetism studies. The magnetic properties of free clusters depict oscillatory behavior for magnetic moment with respect to the cluster size.

Plane-wave pseudopotential study of the light actinides

Norm conserving pseudopotentials have been generated for the light actinides (Th-Np) and the plane waves + pseudopotential formalism has been used to study their crystal structures at zero temperature as a function of pressure. The often complex alpha phases of these elements have been fully relaxed, and we have used a thorough treatment of spin-orbit coupling. The zero-pressure zero-temperature equilibrium volumes and bulk moduli are consistent with previous all-electron full-potential calculations, and, up to uranium, in excellent agreement with experiment. This is also the case for cell parameters and pressure-induced phase transitions. It is likely that, from neptunium on, a more careful treatment of electronic correlations and/or relativistic effects is necessary to reproduce the experimental data with the same precision.

Plane-wave pseudopotential study on mechanical and electronic properties for IV and III-V crystalline phases with zinc-blende structure

Norm conserving pseudopotentials have been generated for the light actinides (Th-Np) and the plane waves + pseudopotential formalism has been used to study their crystal structures at zero temperature as a function of pressure. The often complex alpha phases of these elements have been fully relaxed, and we have used a thorough treatment of spin-orbit coupling. The zero-pressure zero-temperature equilibrium volumes and bulk moduli are consistent with previous all-electron full-potential calculations, and, up to uranium, in excellent agreement with experiment. This is also the case for cell parameters and pressure-induced phase transitions. It is likely that, from neptunium on, a more careful treatment of electronic correlations and/or relativistic effects is necessary to reproduce the experimental data with the same precision.

Theoretical studies on excited states of Ne2

Previously reported potential curves for 0 − (3 P 0) and 0 − (3 P 0) of Ne2 obtained from large-scale configuration-interaction calculations supplemented by semiempirical spin-orbit calculations are compared with potential curves deduced from experimental studies by Beyer and Haberland. Although the agreement between curves is very good, the assignment of states is opposite.Ab initio spin-orbit coupling matrix elements were calculated based on large CI wavefunctions for states dissociating to Ne + Ne*(3s). It was found that they hardly change between 5 and 20a0.Ab initio spin-orbit corrected potential curves differ little from previous curves relying on the semiempirical treatment of spin-orbit coupling, and all former conclusions remain essentially unchanged.