Spin-orbit Coupling Calculations Research Articles

Extensive ab initio calculation on low-lying excited states of SiN+ cation including spin-orbit coupling

Electronic structures and transition properties of the SiN+ molecule are investigated by performing ab initio calculations based on scalar relativistic multireference configuration interaction (MRCI) method plus Davidson correction (+Q). The potential energy curves (PECs) for the 22 Λ-S electronic states of the SiN+ are obtained. The spectroscopic constants of the bound states are determined, where good agreements with the available spectroscopic data are achieved. Spin-orbit coupling (SOC) effect is introduced via the state interaction approach with the full Breit-Pauli Hamiltonian operator for the state X3 Σ −, a 3 Π, 5 Σ − and 5 Π. This is the first time that SOC calculation has been carried out on the SiN+. The four states split into 15 Ω states after the consideration of the SOC effect. The SOC effect, leading to some avoided crossings, is found to be substantial for the SiN+. Moreover, the transition dipole moments (TDMs) and Franck-Condon factors are derived. Finally, the corresponding radiative lifetimes of the a 3 Π 1 state are evaluated.

Theoretical investigation into the Co+ catalytic activity in the cycle reaction of N2O with C2H6 in the gas phase

The spin-forbidden mechanism of the reaction between N 2 O and C 2 H 6 catalyzed by Co + has been investigated using UB3LYP density functional theory. The Harvey method has been applied to optimize five minimum energy crossing points (MECP) on both triplet and quintet potential energy surfaces. Possible spin inversion processes are discussed by means of spin-orbit coupling calculations. According to the calculation of probability of electron hopping using the Landau-Zener formula, effective intersystem crossing may occur at each MECP. The energetic span model proposed by Kozuch has been applied to the catalytic cycles, and shows the turnover frequency reaches 3.35 × 10 –21 s –1 when Co + catalyzes the reaction to produce CH 3 CHO at 298K. The reaction of N 2 O and C 2 H 6 catalyzed by Co + is a typical two-state reaction. An energy span model has been applied to evaluate the catalytic performance of Co + .

Flexible nuclear screening approximation to the two-electron spin–orbit coupling based on ab initio parameterization

The derivation, implementation, and validation of a new approximation to the two-electron spin-orbit coupling (SOC) terms is reported. The approximation, referred to as flexible nuclear screening spin-orbit, is based on the effective one-electron spin-orbit operator and accounts for two-electron SOC effects by screening nuclear charges. A highly flexible scheme for the nuclear screening is developed, mainly using parameterization based on ab initio atomic SOC calculations. Tabulated screening parameters are provided for contracted and primitive Gaussian-type basis functions of the ANO-RCC basis set for elements from H to Cm. The strategy for their adaptation to any other Gaussian basis set is presented and validated. A model to correct for the effect of splitting of transition metal d orbitals on their SOC matrix elements is introduced. The method is applied to a representative set of molecules, and compared to exact treatment and other approximative approaches at the same level of relativistic theory. The calculated SOC matrix elements are in very good agreement with their "exact" values; deviation below 1% is observed on average. The presented approximation is considered to be generally applicable, simple to implement, highly efficient, and accurate.

A Neoteric Neodymium Model: Ground and Excited Electronic State Analysis of NdF2+

Neodymium monofluoride dication was studied as a model of the Nd-F bond in NdFx. Multiconfigurational self-consistent field (MCSCF) and second order multireference quasi-degenerate perturbation theory (MCQDPT2) methods were used with a variety of active spaces to elucidate the roles of the Nd 4f, 5d, and 6s orbitals. Spin-orbit coupling calculations were performed at the SO-MCQDPT2 level, and potential energy curves were obtained for the four lowest energy quartet states as well as for the four lowest doublet states and the lowest sextet state. Inclusion of spin-orbit coupling splits these states into 30 levels. Equilibrium bond lengths, dissociation energies, transition energies, and crossing points were determined.

Theoretical Views on Activation of Methane Catalyzed by Hf2+and Oxidation of CO (x1Σ+) by N2O (x1Σ+) Catalyzed by HfO2+and TaO2+

The mechanisms of activation of CH4 catalyzed by (1/3)Hf(2+) and oxidation of CO by N2O catalyzed by (1/3)HfO(2+) or (2/4)TaO(2+) have been investigated using the B3LYP level of theory. For the activation of methane, the TSR (two-state reactivity) mechanism has been certified through the spin-orbit coupling (SOC) calculation and the Landau-Zener-type model. In the vicinity of the minimum energy crossing point (MECP), SOC equals 900.23 cm(-1) and the probability of intersystem crossing is approximately 0.62. Spin inversion makes the activation barrier decline from 1.63 to 0.57 eV. NBO analysis demonstrates that empty 6s and 5d orbitals of the Hf atom play the major role for the activation of C-H bonds. Finally, CH4 dehydrogenates to produce Hf-CH2(2+). For oxidation of CO by N2O catalyzed by HfO(2+) or TaO(2+), the covalent bonds between transition metal atoms and the oxygen atom restrict the freedom of valence electrons. Therefore, they are all SSR (single-state reactivity). The oxygen atom is directly extracted during the course of oxygen transfer, and its microscopic essence has been discussed. The detailed kinetic information of two catalytic cycles has been calculated by referencing the "energetic span (δE)" model. Finally, TOF(HfO(2+))/TOF(TaO(2+)) = 2.7 at 298.15 K, which has a good consistency with the experimental result.

Spin-orbit coupling calculations with the two-component normalized elimination of the small component method

A new algorithm for the two-component Normalized Elimination of the Small Component (2cNESC) method is presented and tested in the calculation of spin-orbit (SO) splittings for a series of heavy atoms and their molecules. The 2cNESC is a Dirac-exact method that employs the exact two-component one-electron Hamiltonian and thus leads to exact Dirac SO splittings for one-electron atoms. For many-electron atoms and molecules, the effect of the two-electron SO interaction is modeled by a screened nucleus potential using effective nuclear charges as proposed by Boettger [Phys. Rev. B 62, 7809 (2000)]. The use of the screened nucleus potential for the two-electron SO interaction leads to accurate spinor energy splittings, for which the deviations from the accurate Dirac Fock-Coulomb values are on the average far below the deviations observed for other effective one-electron SO operators. For hydrogen halides HX (X = F, Cl, Br, I, At, and Uus) and mercury dihalides HgX2 (X = F, Cl, Br, I) trends in spinor energies and SO splittings as obtained with the 2cNESC method are analyzed and discussed on the basis of coupling schemes and the electronegativity of X.

Dimensionality-controlled Mott transition and correlation effects in single-layer and bilayer perovskite iridates

We studied Sr${}_{2}$IrO${}_{4}$ and Sr${}_{3}$Ir${}_{2}$O${}_{7}$ using angle-resolved photoemission spectroscopy, making direct experimental determinations of intra- and intercell coupling parameters as well as Mott correlations and gap sizes. The results are generally consistent with LDA+$U$+spin-orbit coupling calculations, though the calculations missed the momentum positions of the dominant electronic states and neglected the importance of intercell coupling on the size of the Mott gap. The calculations also ignore the correlation-induced spectral peak widths, which are critical for making a connection to activation energies determined from transport experiments. The data indicate a dimensionality-controlled Mott transition in these 5$d$ transition-metal oxides.

Theoretical study of the mechanism for the cycle reaction of N<sub>2</sub>O and CH<sub>4</sub> catalyzed by Fe<sup>+</sup>(<sup>6</sup>D) to yield CH<sub>3</sub>OH

The mechanism of the gas phase reaction between N₂O and CH₄ catalyzed by Fe⁺(⁶D) to yield CH₃OH has been investigated on the B3LYP and CCSD(T) levels. The potential energy surfaces (PESs) were characterized in detail using molecular orbital theory and natural bonding orbital (NBO). Crossing points between the different potential energy surfaces and the possibility of spin inversion were discussed based on calculation of spin-orbit coupling (SOC). The energetic span model developed by Kozuch was corrected in order to deal with non-adiabatic reactions. Finally, we calculated the turnover frequency (TOF) of the catalytic cycle and found the TOF determining transition state (TDTS) and the TOF determining intermediate (TDI) using the treated energetic span model.

Solvent Effect on the Stokes Shift and on the Nonfluorescent Decay of the Daidzein Molecular System

The flavonoids have been the target of several experimental works due to its influence in the human health as antioxidant elements. The fluorescence properties of these compounds have been widely studied due to the large Stokes shifts experimentally observed and the variety of processes that lead to the fluorescence. In the present work the role of the solvent in the large Stokes shift experimentally observed in the daidzein molecular system in water is theoretically studied. Also studied is the nonfluorescent decay mechanism in a polar aprotic solvent like acetonitrile. The solvent effect in the ground and in the low-lying excited electronic states is taken into account by using the sequential-QM/MM methodology. Excited state properties like equilibrium geometries and transition energies were studied by using multiconfigurational calculations, CASSCF and CASPT2. The excited electronic state responsible for the fluorescence spectrum in water was identified, and the large Stokes shift seems to be the result of the large interaction of the system in this electronic state with the solvent. On the other hand, spin-orbit coupling calculations, between the singlet and triplet electronic states, indicate favorable conditions for intersystem crossing, in agreement with the experimental result of nonfluorescence observation.

Magnetic Monopoles and Synthetic Spin-Orbit Coupling in Rydberg Macrodimers

We show that sizable Abelian and non-Abelian gauge fields arise in the relative motion of two dipole-dipole interacting Rydberg atoms. Our system exhibits two magnetic monopoles for adiabatic motion in one internal two-atom state. These monopoles occur at a characteristic distance between the atoms that is of the order of one micron. The deflection of the relative motion due to the Lorentz force gives rise to a clear signature of the effective magnetic field. In addition, we consider nonadiabatic transitions between two near-degenerate internal states and show that the associated gauge fields are non-Abelian. We present quantum mechanical calculations of this synthetic spin-orbit coupling and show that it realizes a velocity-dependent beam splitter.

Extensive ab initio study of the electronic states of S2 molecule including spin-orbit coupling

The potential energy curves (PECs) of 15 Λ-S states and 24 Ω states generated from the 13 Λ-S bound states of the S2 molecule are investigated in detail using an ab initio quantum chemical method. The PECs are calculated for internuclear separations from 0.12 to 1.10 nm by the complete active space self-consistent field method, which is followed by the internally contracted multireference configuration interaction approach with the Davidson modification (MRCI + Q). The spin-orbit (SO) coupling effect is accounted for by the Breit–Pauli Hamiltonian. To discuss the effect on the energy splitting by the core-electron correlations, the all-electron basis set, cc-pCVTZ with and without 2s2p correlations, is used for the SO coupling calculations of the A3 and B′3Πg Λ-S states since their measurements can be found in the literature. By comparison, the cc-pCVTZ basis set with 2s2p correlations is chosen for the SO coupling calculations of 13 Λ-S bound states. To improve the quality of PECs, core-valence correlation and scalar relativistic corrections are included. Scalar relativistic correction calculations are made using the third-order Douglas–Kroll Hamiltonian (DKH3) approximation at the level of a cc-pV5Z basis set. Core-valence correlation corrections are taken into account with a cc-pCVTZ basis set. The spectroscopic parameters of 13 Λ-S bound states and 24 Ω states are calculated. With the PECs obtained by the MRCI + Q/aug-cc-pV6Z + CV + DK + SO calculations, the SO coupling splitting energies are 379.25 cm−1 between the A′3 and A′2 Ω state, 83.40 cm−1 between the A1 and A0− Ω state and 210.91 cm−1 between the B′2 and B′1 Ω state, which agree well with the corresponding measurements of 383, 77.51 and 209 cm−1, respectively. Moreover, other spectroscopic parameters are also in excellent agreement with the measurements. It demonstrates that the spectroscopic parameters of 24 Ω states reported here for the first time can be expected to be reliable predicted ones.

Spin–orbit effect and magnetic anisotropy in Pt clusters

Based on the non-collinear (NCL) and spin–orbit coupling (SOC) implementation of the density-functional theory (DFT) calculations, isomeric structures, spin and orbital moments, magnetic anisotropies, as well as magnetic anisotropy energy (MAE) are investigated for Ptn clusters (n=2–8, 13). Our studies show that the planar structures are superior to the three-dimensional structures in stabilizing Pt clusters up to n=7, favored by both scalar relativistic calculations and SOC calculations. For each cluster size, relative stabilities of different low-lying energy isomers are not alternated by the SOC effects. Strong magnetic anisotropies and remarkable orbital moments (0.1–0.4μB/atom) are found in Pt clusters. The easy axes usually prefer the orientations that are in the basal planes of the structures and parallel to the principal axes of symmetry. There is always collinearity between total spin moments and total orbital moments, favoring a strict ferromagnetic coupling. It is revealed that small Pt clusters mostly exhibit large MAEs.

Spin-orbit coupling effects in the gas-phase reaction Se + O2

The potential energy surface sections of the ground and excited states of the SeOO (Cs) superoxo complex have been constructed on the basis of spin-orbit coupling calculations. The activation barriers to the forward and reverse reactions—superoxide SeOO ↔ peroxide Se(O2) ↔ dioxide SeO2—have been calculated. Changes in the spectral and spin properties of oxygen (zero-field splitting) and possible channels of formation of active forms of 1O2 oxygen (1Δg, 1Σg+) upon dissociation of excited states of selenium oxo complexes have been scrutinized.

Extensive ab initio study of the electronic states of CP radical including spin–orbit coupling

The potential energy curves (PECs) of 18 Λ–S states and 13 Ω states generated from six Λ–S states of the CP radical are studied in detail using an ab initio quantum chemical method. All the PEC calculations are performed for internuclear separations from 0.10 to 1.11nm by the complete active space self-consistent field method, which is followed by the internally contracted multireference configuration interaction approach with the Davidson modification. The spin–orbit (SO) coupling effect is accounted for by the Breit-Pauli Hamiltonian. To discuss the effect on the energy splitting by the core-electron correlations, two all-electron basis sets, cc-pCVTZ and cc-pVTZ, are used for the SO coupling calculations. In order to improve the quality of the PECs, core–valence correlation and scalar relativistic corrections are included. Core–valence correlation corrections are taken into account with a cc-pCVTZ basis set. Scalar relativistic correction calculations are carried out using the third-order Douglas-Kroll Hamiltonian approximation at the level of a cc-pV5Z basis set. The spectroscopic parameters of 17 Λ–S bound states and 13 Ω states are calculated. When the core-electron correlations are used to calculate the SO coupling splitting by the all-electron cc-pCVTZ basis set and when the core–valence correlation and scalar relativistic corrections are included in the PECs, the SO coupling splitting energy between the A2Π3/2 and A2Π1/2 Ω states is 153.63cm−1, which agree well with the measurements of 157.87cm−1. Moreover, other spectroscopic parameters are also in excellent agreement with the available experimental ones. It demonstrates that the spectroscopic parameters of 14Σ+, 14Δ, 14Π, 22Π, 16Σ+, 14Σ−, 32Π, 12Δ, 12Σ−, 22Δ, 22Σ−, 24Δ, 24Σ− and 26Σ+ Λ–S states and 13 Ω states reported here can be expected to be reliable predicted ones.

Investigations on spectroscopic parameters and molecular constants of SO+ (b4∑-) cation

The potential energy curve (PEC) of b4Σ- electronic state of the SO+ cation is calculated using the internally contracted multireference configuration interaction approach with the Davidson modification (MRCI+Q) for internuclear separations from 0.103 to 1.083 nm. The basis set used is a correlation- consistent aug-cc-pV5Z basis set. The spin-orbit coupling effect on the spectroscopic parameters is taken into account by the state interaction approach with the full Breit-Pauli operator with all-electron basis set, aug-cc-pCVTZ. To improve the quality of PEC and spin-orbit coupling constant, core-valence correlation and relativistic corrections are included. Core-valence correlation correction is calculated using a cc-pCVTZ basis set. Relativistic corrections are included by the third-order Douglas-Kroll Hamiltonian approximation at the level of a cc-pV5Z basis set. At the MRCI+Q/aug-cc-pV5Z+CV+DK level, the spin-orbit coupling constant of the SO+ (b4Σ-1/2,3/2) is 1 cm-1 when the aug-cc-pCVTZ basis set is used for the spin-orbit coupling calculations The spectroscopic parameters are determined and compared with those reported in the literature. Excellent agreement is found between the present results and the measurements. The vibrational level G(v) inertial rotation constant Bv and centrifugal distortion constant Dv are predicted for each vibrational state of the b4Σ- electronic state by solving the ro-vibrational Schrödinger equation of nuclear motion using Numerov's method and those of the first 2 vibrational states are reported for the non-rotation SO+ cation. Comparison with the measurements demonstrates that the present vibrational manifolds are both reliable and accurate. They should be good predictions for future experimental or theoretical research.

A matrix method for the calculation of nonlocal exchange potential and spin-orbit coupling in the first-principles calculation method

Adequate treatment of exchange interaction and inclusion of spin-orbit coupling are important in obtaining reliable electronic structures of semiconductors and magnetic materials. Here, a matrix method has been derived to calculate nonlocal exchange potential and spin-orbit coupling within a muffin-tin-orbital based first-principles calculation method. Good agreement between calculated and experimental band gaps for various zinc-blende and wurtzite semiconductors demonstrated that direct calculation of nonlocal exchange potential is feasible. This method may be an improvement over the currently used LSDA + U or LSDA + GGA + U method, which uses an adjustable U parameter, for wide-band-gap semiconductors and magnetic materials. In this method, the use of an extended Hamiltonian matrix, which contains off-diagonal matrix elements between up- and down-spin states due to spin-orbit coupling, enables mutual adjustment of the occupation of up- and down-spin states through atomic orbitals. In contrast, in the conventional spin-polarized calculation method up- and down-spin sub-matrices are diagonalized separately. The calculated spin-orbit splitting for the triplet states at the Γ point for GaAs and ZnS are in good agreement with experimental data.

Theoretical Investigation for the Cycle Reaction of N2O (x1∑+) with CO (1∑+) Catalyzed by IrOn+ (n = 1, 2) and Utilizing the Energy Span Model to Study Its Kinetic Information

The mechanisms of the reactions between N(2)O and CO catalyzed by IrO(n)(+) (n = 1, 2) have been investigated using B3LYP and CCSD(T) levels of theory. Spin inversion among three reaction profiles corresponding to the quintet, triplet, and singlet multiplicities was discussed by using spin-orbit coupling (SOC) calculations. The probability of electron hopping in the vicinity of the (MECP) has been calculated by the Landau-Zener-type model. The single P(1)(ISC) and double P(2)(ISC) passes estimated at MECP1(#) (SOC = 198.61 cm(-1)) are approximately 0.11 and 0.20, respectively. Important analysis and explanations were done using molecular orbital theory and natural bonding orbital (NBO). The energetic span (δE) model coined by Kozuch was applied in this cycle. The turnover frequency (TOF)-determining transition state (TDTS) and TDI (TOF-determining intermediate) were confirmed. Finally, TOF(IrO(+))/TOF(IrO(2)(+)) = 0.38 at 298 K.

Two-component natural spinors from two-step spin-orbit coupled wave functions

We developed an algorithm to obtain the natural orbitals (natural spinors) from the two-step spin-orbit coupled wave functions. These natural spinors are generally complex-valued, mixing two spin components, and they can have similar symmetry properties as the j-j spinors from the one-step spin-orbit coupling calculations, if the reduced density equally averages all the components of a multi-dimensional irreducible representation. Therefore, the natural spinors can serve as an approximation to the j-j spinors and any wave function analysis based on the j-j spinors can also be performed based on them. The comparison between the natural spinors and the j-j spinors of three representative atoms, Tl, At, and Lu, shows their close similarity and demonstrates the ability of the natural spinors to approximate the j-j spinors.

Theoretical study of the reactions of lanthanide ions (Ce+ , Pr+ ) with CO2 in the gas phase

The mechanisms of the reaction of lanthanide ions (Ce+ and Pr+) with the carbon dioxide were investigated at the B3LYP level of theory. The crossing points (CPs) between the different potential energy surfaces (PESs) have been located by means of the intrinsic reaction coordinate approach used by Yoshizawa et al., and corresponding minimum energy crossing points (MECPs) that we obtained by the mathematical algorithm proposed by Harvey et al. has also been used. In addition, possible spin inversion processes are discussed by means of spin-orbit coupling (SOC) calculations. The value of 61.68 cm−1 (for MECP1) and 69.17 cm−1 (for MECP2) for the SOC constants indicates that the spin crossing process in Ln+ + CO2 (1∑) reaction can be occurred efficiently because of the large SOC involved. And the O-atom affinities (OA) testified that the argumentation is thermodynamically allowed. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

On the Photophysics of 1,6‐Diphenyl‐1,3,5‐Hexatriene Isomers and Rotamers

Herein, the low-lying electronic states of various isomers and rotamers of 1,6-diphenyl-1,3,5-hexatriene (DPH) are studied by a combination of density functional theory and multireference configuration interaction. Starting from the all-trans nuclear arrangement, trans-cis isomerization pathways in the electronic ground state and in the first excited triplet state were determined. Further, spin-orbit coupling calculations were carried out at selected points where singlet-triplet energy gaps are small. The calculations reveal that the primarily excited, optically bright 1(1)B(u) state undergoes a curve crossing with the optically dark multiconfigurational 2(1)A(g) state upon geometry relaxation in the excited state. The strong vibronic coupling of the two singlets in the neighborhood of the conical intersection provides a conclusive explanation for the experimentally observed fast equilibration of the states and the appearance of delayed fluorescence. With regard to the trans-cis isomerization of the central CC bond, the perpendicular conformation is found to represent a maximum on the energy profile not only of the electronic ground state, but also of the low-lying excited states. The lack of a strong driving force along the torsional coordinate explains the low tendency of DPH for isomerization. Finally, the results of our spin-orbit coupling calculations suggest that the intramolecular formation of DPH molecules in the T(1)(1(3)B(u)) state proceeds from the 1(1)B(u) state and involves intermediately the T(2)(1(3)A(g)) state.