Spin-orbit Coupling and Intersystem Crossing in 4H-Pyran-4-thione： CASSCF//TD-B3LYP Study

The intersystem crossing channels of gaseous 4H-pyran-4-thione were investigated using the CASSCF//TD-B3LYP methods and group theory. Using the effective one-electron spin-orbit Harniltonian, the strengths of spin-orbit coupling were estimated, which plays an essential role in the spin transitions between different spin states. Calculated results show that phosphorescence and non-radiative decay via intersystem crossing to the So state are concurrent processes occurring at the T1 state. A rapid depletion of the S1 state via intersystem crossing to the T1 state can be mediated by the T2 state, if spin relaxation is fast within the triplet levels. Our calculated results are in close agreement with experimental observations.     

On the catalytic role of Re+ in the oxygen transport activation of N2O by CO

The mechanism of the cyclic reaction N₂O(¹∑) + CO(¹∑⁺) → N₂(¹∑ _g ⁺ ) + CO₂(¹∑ _g ⁺ ) catalyzed by Re⁺ has been investigated on quintet and septet potential energy surfaces (PES). The reactions were studied by the B3LYP density functional method and the CCSD(T) theory. The calculated results of different PES show that the reaction proceeds in a two-step manner and spin crossing between different PES occurs. The involving crossing points (CPs) between the quintet and septet PES have been discussed by means of the intrinsic reaction coordinate approach. And the O-atom affinities testified that Re⁺ can capture O from N₂O and transfer O atom to CO in the two spin state, which are thermodynamically allowed. Furthermore, the spin–orbit coupling (SOC) is calculated between electronic states of different multiplicities at the CPs. For CP1 and CP2, the computed SOC constants are 8.34 and 10.09 cm^?1, respectively, obtained by using one-electron spin–orbit Hamiltonian in GAMESS. Therefore, the intersystem crossing at CP1 and CP2 occurs with a little probability because of the small SOC involved.     

Theoretical studies of the reactions of M+ (M = Ta,W, Re) with CS2

Gas-phase CS₂ activation by M⁺ (M = Ta, W, Re) was studied by the B3LYP density functional method.The geometries for reactants, transition states, and products were completely optimized. CS₂ activation mediated by M⁺ (M = Ta, W, Re) were found to be a spin-forbidden process as a result of the crossing among the multistate energetic profiles. On the basis of the Hammond postulate, this was a typical two-state reactivity reaction. Among the different potential energy surfaces, the crossing points had been explored. The spin–orbit coupling (SOC) was also calculated between the electronic states of different multiplicities at the crossing point to estimate the intersystem crossing probability. For CP1, CP2, and CP4, the computed SOC constants were 80.28, 128.65, and 526.77 cm^?1, which obtained by using one-electron spin–orbit Hamiltonian in Gaussian 09.     

New ab initio Potential Energy Surfaces for Cl（^2P3/2,^2P1/2）＋H2 Reaction

New global three dimensional potential energy surfaces for the Cl＋H2 reactive system have been constructed using accurate multireference configuration interaction calculations with a large basis set. The three lowest adiabatic potential energy surfaces correlating asymptotically with Cl（^2p）＋H2 have been transformed to adiabatic representation, which leads to a fourth coupling potential for non-linear geometries. In addition, the spin-orbit coupling surfaces have also been computed using the Breit-Pauli Hamiltonian. Properties of the new potential are described. Reaction dynamics based on the new potential agrees with the recent experimental results quite well.     

Renner-Teller and spin-orbit vibronic coupling effects in linear triatomic molecules with a half-filled pi shell

The vibronic and spin-orbit-induced interactions among the (3)Sigma(-), (1)Delta, and (1)Sigma(+) electronic states arising from a half-filled pi orbital of a linear triatomic molecule are considered, employing the microscopic (Breit-Pauli) spin-orbit coupling operator. The 6 x 6 Hamiltonian matrix is derived in a diabatic spin-orbital electronic basis set, including terms up to fourth order in the expansion of the molecular Hamiltonian in the bending normal coordinate about the linear geometry. The symmetry properties of the Hamiltonian are analyzed. Aside from the nonrelativistic fourth-order Renner-Teller vibronic coupling within the (1)Delta state and the second-order nonrelativistic vibronic coupling between the (1)Sigma(+) and (1)Delta states, there exist zeroth-order, first-order, as well as third-order vibronic coupling terms of spin-orbit origin. The latter are absent when the phenomenological expression for the spin-orbit coupling operator is used instead of the microscopic form. The effects of the nonrelativistic and spin-orbit-induced vibronic coupling mechanisms on the (3)Sigma(-), (1)Delta, and (1)Sigma(+) adiabatic potential energy surfaces as well as on the spin-vibronic energy levels are discussed for selected parameter values.     

Spin-inversion mechanisms in the reactions of transition metal cations (Sc+, Ti+, V+, Cr+, Mn+, Fe+, Co+, Ni+, and Cu+) with OCS in the gas phase: A perspective from automated reaction path search calculations

The mechanisms including spin-inversion have been systematically studied for the M⁺ + OCS → MS⁺ + CO/MO⁺ + CS (M denotes a transition metal from Sc to Cu) ion-molecule reactions using the automated reaction path search method. We used the lowest mixed-spin potential energy surface obtained from the diagonalization of the spin-coupled Hamiltonian matrix, whose diagonal elements are taken to be the lowest two spin states. This scheme can effectively locate approximate minimum energy crossing points between the two potential energy surfaces with different spin multiplicities. The spin-orbit couplings at spin-inversion points have been calculated to understand the efficiencies of nonadiabatic transitions. The obtained reaction pathways and the calculated spin-orbit couplings are employed to interpret previous experimental studies.     

Effects of geometric distortions and substituents on rates of intersystem crossing; diazomethane and pyrazoline

We report a qualititive perturbation-theoretic analysis of the effects of substituents on the spin-orbit coupling and on the density of states in diazomethane and pyrazoline systems. The importance of the form of the distortion which accompanies approach to the singlet-triplet crossing zone is also discussed with the aim of determining when reactions involving a spin multiplicity change can compete in rate with spin-conserving reactions.     

Theoretical Investigation for Two-state Reactivity of CO_2 Hydrogenation Catalyzed by Ru in Gas Phase

Gas-phase CO_2 catalyzed activation hydrogenation by Ru atoms was studied with density functional theory. Based on the structure optimization of different potential energy surfaces,there are two crossing points between singlet and triplet potential energy surfaces and there is a crossing point between quintet and triplet potential energy surfaces in the whole catalytic cycle. Spin transition probabilities in the vicinity of the intersections have been calculated by the Landau-Zener model theory. There are three minimum energy crossing points(MECPs) with strong spin-orbital coupling effect and higher spin transition probability,and all spin inversion occurred in s orbital and different d orbitals of ruthenium,indicating this is a typical two-state reactivity(TSR) reaction. Finally,the lowest energy reaction path is ensured.     

The Electronic Properties of Diradicals

A review of the various possible definitions of diradicals leads the authors to describe these systems as having two odd electrons in degenerate or nearly-degenerate molecular orbitals. A study of the wave-function for the two odd electrons shows that its form depends entirely on whether the diradical is homo- or heterosymmetric. Energy schemes are given in these two cases, as well as in the intermediate “non-symmetric” case. The extent of zwitterionic character in diradical states is also investigated. This is followed by a discussion of intersystem crossing between singlet and triplet diradical states via spin-orbit coupling and other mechanisms. The electronic matrix elements for spin-orbit coupling are calculated and evaluated numerically for various model cases. It is then possible to establish general rules for favorable (electronic) intersystem crossing. In 1,3 or 1,4 diradicals its efficiency is estimated to be comparable with that in aromatics. The role of the electron-nuclear hyperfine interaction in mixing singlet and triplet states, particularly in CIDNP, is explained. Finally the question of whether diradicals actually occur as secondary minima on potential energy surfaces is examined. Recent quantum-mechanical calculations, in contradiction to some thermochemical and kinetic evidence, lead to flat singlet surfaces without significant minima.     

Automated reaction path searches for spin‐forbidden reactions

Many catalytic and biomolecular reactions containing transition metals involve changes in the electronic spin state. These processes are referred to as “spin‐forbidden” reactions within nonrelativistic quantum mechanics framework. To understand detailed reaction mechanisms of spin‐forbidden reactions, one must characterize reaction pathways on potential energy surfaces with different spin states and then identify crossing points. Here we propose a practical computational scheme, where only the lowest mixed‐spin eigenstate obtained from the diagonalization of the spin‐coupled Hamiltonian matrix is used in reaction path search calculations. We applied this method to the ^6,4FeO⁺ + H₂ → ^6,4Fe⁺ + H₂O, ^6,4FeO⁺ + CH₄ → ^6,4Fe⁺ + CH₃OH, and ⁷Mn⁺ + OCS → ⁵MnS⁺ + CO reactions, for which crossings between the different spin states are known to play essential roles in the overall reaction kinetics. © 2018 Wiley Periodicals, Inc.     

Theoretical study of spin-orbit coupling and intersystem crossing in the two-state reaction between Nb(NH2)3 and N2O

The two-state mechanism of the reaction of Nb(NH2)3 with N2O on the singlet and triplet potential energy surfaces has been investigated at the B3LYP level.Crossing points between the potential energy surfaces have been located using different methods.Analysis of the strain model shows that the singlet state of the four-coordinate(N2O)Nb(NH2)3 complex with N2O bonded via terminal N atom coordination(12) is more stable in the initial stage of reaction,since the bending of the N2O fragment [Edef(N2O) = 86.1 kcal mol-1] results in an energy splitting of the doubly degenerate LUMO;the low-energy LUMO can now strongly couple with the occupied Nb-localized d orbitals,forming a back-bond and transferring charge(q = 0.82 e) from Nb(NH2)3 to the N2O ligand.Going from 32 to 12,the reacting system changes spin multiplicity near the MECP(minimal energy crossing point) region,which takes place with a spin crossing barrier of 9.6-10.0 kcal mol-1.Analysis of spin-orbit coupling(SOC) indicates that MECP will produce a significant SOC matrix element.The value of SOC is 111.52 cm-1,due to the electron shift between two perpendicular φ orbitals with the same rotation direction,and the magnitude of the spin-multi-plicity mixing increases in the small energy gap between high-and low-spin states,greatly enhancing the probability of intersystem crossing.The probabilities of single(P1 ISC) and double(P2 ISC) passes estimated at MECP(SOC = 111.52 cm-1) are approximately 1.17×10-2 and 2.32×10-2,respectively.     

Influence of multi-atom bridging ligands on the electronic structure and magnetic properties of homodinuclear titanium molecules

The electronic structure and magnetic properties of homodinuclear titanium(III) molecules with bridging ligands from groups 14, 15, and 16 are examined. Single- and multireference methods with triple-zeta plus polarization basis sets are employed. Dynamic electron correlation effects are included via second-order multireference perturbation theory. Isotropic interaction parameters are calculated, and two of the complexes studied are predicted to be ferromagnetic based on multireference second-order perturbation (MRMP2) theory, using the TZVP(fg) basis set. Zero-field splitting parameters are determined using spin-orbit coupling obtained from complete active space (CAS) self-consistent field (SCF) and multiconfigurational quasi-degenerate perturbation theory (MCQDPT) wave functions. Three Breit-Pauli-based spin coupling methods were employed: full Breit-Pauli (HSO2), the partial two-electron method (P2E), and the semiempirical one-electron method (HSO1).     

Spin-orbit coupling in oxygen containing diradicals

Intermediate diradicals which occur in the Paterno-Büchi photocycloaddition and in the Norrish type I photoreactions have been calculated taking into account the spin-orbit coupling (SOC) between the singlet (S) and triplet (T) states. Reaction paths for the photocycloaddition of formaldehyde to ethene and the diradical products of the -cleavage of cyclohexanone have been optimized by the MNDO CI method for a number of different singlet and triplet states. SOC integrals are calculated by an effective one-electron approximation. Intermediate diradicals in the Paterno-Büchi reaction and the SOC effects are also studied ab initio with CAS SCF geometry optimization in a TZV basis set. Both methods predict a large SOC matrix element between the S and T states in the course of the C-C attack, while the SOC integral is two orders of magnitude smaller for the diradical produced in the C-O attack. In the Norrish type I photoreaction the oxygen atom also produces some nonzero contribution to the SOC integral which governs intersystem crossing in a ·C-C· diradical. For the diradicals produced by the -cleavage of cyclohexanone a vibronic interaction is responsible for the SOC mixing between the lowest S and T states. The importance of one-center versus two-center SOC contributions in diradicals is briefly discussed.     

Ab initio study of dynamical E × e Jahn-Teller and spin-orbit coupling effects in the transition-metal trifluorides TiF3, CrF3, and NiF3

Multiconfiguration ab initio methods have been employed to study the effects of Jahn-Teller (JT) and spin-orbit (SO) coupling in the transition-metal trifluorides TiF(3), CrF(3), and NiF(3), which possess spatially doubly degenerate excited states ((M)E) of even spin multiplicities (M = 2 or 4). The ground states of TiF(3), CrF(3), and NiF(3) are nondegenerate and exhibit minima of D(3h) symmetry. Potential-energy surfaces of spatially degenerate excited states have been calculated using the state-averaged complete-active-space self-consistent-field method. SO coupling is described by the matrix elements of the Breit-Pauli operator. Linear and higher order JT coupling constants for the JT-active bending and stretching modes as well as SO-coupling constants have been determined. Vibronic spectra of JT-active excited electronic states have been calculated, using JT Hamiltonians for trigonal systems with inclusion of SO coupling. The effect of higher order (up to sixth order) JT couplings on the vibronic spectra has been investigated for selected electronic states and vibrational modes with particularly strong JT couplings. While the weak SO couplings in TiF(3) and CrF(3) are almost completely quenched by the strong JT couplings, the stronger SO coupling in NiF(3) is only partially quenched by JT coupling.     

Molecular integrals over the gauge-including atomic orbitals. II. The Breit-Pauli interaction

Each accompanying coordinate expansion (ACE) formula is derived for each of the orbit-orbit interaction, the spin-orbit coupling, the spin-spin coupling, and the contact interaction integrals over the gauge-including atomic orbitals (GIAOs) by the use of the solid harmonic gradient (SHG) operator. Each ACE formula is the general formula derived at the first time for each of the above molecular integrals over GIAOs. These molecular integrals are arising in the Breit-Pauli two-electron interaction for a relativistic calculation. We may conclude that we can derive a certain ACE formula for any kind of molecular integral over solid harmonic Gaussian-type orbitals by using the SHG operator. The present ACE formulas will be useful, for example, for a calculation of a molecule in a uniform magnetic field, for a relativistic calculation, and so on, with the GIAO as a basis function.     

气相中Co+催化N2O与C2H6循环反应催化剂活性的理论研究

采用密度泛函理论UB3LYP方法对Co⁺在三重态及五重态势能面上催化N₂O与C₂H₆进行循环反应的两态反应机理进行了研究. 运用Harvery方法优化了两自旋态势能面5个最低能量交叉点（MECP）,计算了MECP处自旋-轨道耦合作用. 采用Landau-Zener公式计算了自旋翻转处的系间窜越几率,各MECP处均可发生有效系间窜越. 通过应用Kozuch提出的能量跨度模型,Co⁺催化N₂O与C₂H₆在298K下反应生成CH₃CHO时有最大的TOF值3.35×10^-21 s^-1.     

Conformational analysis of singlet-triplet state mixing in Paternò-Büchi diradicals

Conformational dependence of spin-orbit coupling (SOC) in flexible Paternò-Büchi (PB) diradicals has been studied with high-level ab initio methods using both (i) one-electron spin-orbit Hamiltonian with parametrized (effective) nuclear charges in conjunction with a state-averaged MCSCF wave function as implemented by Robb in Gaussian 98 and (ii) complete one- and two-electron SOC with a fully optimized MCSCF triplet wave function and frozen core singlet as implemented by Furlani in the GAMESS computational package. The ab initio results revealed two distinct areas of elevated SOC values, one corresponding to the region whereby a cisoid conformation in the C-C-O-C fragment brings the two odd-electron orbitals closer to each other, and the other area corresponding to the partially eclipsed conformation lacking direct overlap between the spin centers. In this second region the 1,4-electronic communication is mediated by the oxygen's 2p-lone pair, which is suitably oriented to play the role of a "relay-antenna". The other critical factor affecting the rate of intersystem crossing (ISC)--singlet-triplet energy separation--was computed utilizing a multireference CASSCF-MP2 method to include dynamic correlation effects. The largest singlet-triplet energy gap, approximately 2 kcal/mol, was found for a gauche conformer (also a minimum SOC conformation). Rotation about the central C-O bond either toward the fully eclipsed (0 degrees ) or the partially eclipsed (120 degrees ) conformations decreases the singlet-triplet gap while increasing the value of the SOC matrix element. These computational findings support the Griesbeck model for stereochemistry of triplet PB reactions and provide a rigorous basis for predicting the probability of ISC in diradicals separated by a partially conjugated spacer.     

A quantum wave-packet study of intersystem crossing effects in the O(3P2,1,0,1D2)+H2 reaction

We present for the first time an exact quantum study of spin-orbit-induced intersystem crossing effects in the title reaction. The time-dependent wave-packet method, combined with an extended split operator scheme, is used to calculate the fine-structure resolved cross section. The calculation involves four electronic potential-energy surfaces of the 1A' state [J. Dobbyn and P. J. Knowles, Faraday Discuss. 110, 247 (1998)], the 3A' and the two degenerate 3A" states [S. Rogers, D. Wang, A. Kuppermann, and S. Wald, J. Phys. Chem. A 104, 2308 (2000)], and the spin-orbit couplings between them [B. Maiti, and G. C. Schatz, J. Chem. Phys. 119, 12360 (2003)]. Our quantum dynamics calculations clearly demonstrate that the spin-orbit coupling between the triplet states of different symmetries has the greatest contribution to the intersystem crossing, whereas the singlet-triplet coupling is not an important effect. A branch ratio of the spin state Pi32 to Pi12 of the product OH was calculated to be approximately 2.75, with collision energy higher than 0.6 eV, when the wave packet was initially on the triplet surfaces. The quantum calculation agrees quantitatively with the previous quasiclassical trajectory surface hopping study.     

The singlet and triplet states of phenyl cation. A hybrid approach for locating minimum energy crossing points between non-interacting potential energy surfaces

The phenyl cation is known to have two low-energy minima, corresponding to ¹ A ₁ and ³ B ₁ states, the first of which is more stable by ca. 25 kcal/mol. The minimum energy crossing point between these two surfaces, located at various levels including a hybrid method first described here, lies just above the minimum of the triplet, 0.12 kcal/mol at the CCSD(T)/cc-pVDZ//B3LYP/SV level, and there is significant spin-orbit coupling between the surfaces at this point. On the basis of these results, the lifetime of the triplet is expected to be very short. Received: 24 October 1997 / Accepted: 25 November 1997     

Density functional perturbational orbital theory of spin polarization in electronic systems. II. Transition metal dimer complexes

We present a theoretical scheme for a semiquantitative analysis of electronic structures of magnetic transition metal dimer complexes within spin density functional theory (DFT). Based on the spin polarization perturbational orbital theory [D.-K. Seo, J. Chem. Phys. 125, 154105 (2006)], explicit spin-dependent expressions of the spin orbital energies and coefficients are derived, which allows to understand how spin orbitals form and change their energies and shapes when two magnetic sites are coupled either ferromagnetically or antiferromagnetically. Upon employment of the concept of magnetic orbitals in the active-electron approximation, a general mathematical formula is obtained for the magnetic coupling constant J from the analytical expression for the electronic energy difference between low-spin broken-symmetry and high-spin states. The origin of the potential exchange and kinetic exchange terms based on the one-electron picture is also elucidated. In addition, we provide a general account of the DFT analysis of the magnetic exchange interactions in compounds for which the active-electron approximation is not appropriate.