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.     

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 α-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 α-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.     

Comparative study of the Hartree–Fock and the Hartree–Fock–Slater method

The Hartree–Fock method (standard Roothaan closed-shell HF –LCAO theory) and the Hartree–Fock–Slater method (restricted HFS –LCAO –DV method developed by Baerends and Ros) have been compared with emphasis on the respective one-electron equations and on the matrix elements of the respective Fock operators. Using the same STO basis in the two cases, the matrix elements of the Fock operators and of their separate one-electron, Coulomb, and exchange contributions have been calculated for the same orbitals and density of the ground state of the diatomic molecule ZnO. The effects of methodical (exchange potential) and numerical (DV method, density fit) differences between the HF and HFS methods on the various matrix elements have been analyzed. As expected the methodical effect prevails and is responsible for the higher (less negative) values of the matrix elements of the HFS Fock operator compared to those of the HF Fock operator. Numerical effects are observable also and are caused by the difference in integration procedures (DV method), not by the density fit.     

Spin ground state and magnetic properties of cobalt(II): relativistic DFT calculations guided by EPR measurements of bis(2,4-acetylacetonate)cobalt(II)-based complexes

The spin ground state of the core ion and structure of the bis(2,4-acetylacetonate)cobalt(II) model complex and its synthetic aqua and ethanol derivatives, Co(acac)(2)L(n), (L = EtOH, H(2)O), were examined by means of density functional theory (DFT) calculations supported by electron paramagnetic resonance (EPR) measurements. Geometry optimizations were carried out for low-spin (doublet) and high-spin (quartet) states. For the Co(acac)(2) complex two possible conformations, a square-planar and a tetrahedral one, were taken into account. For all structures relative energies were calculated with both "pure" and hybrid functionals. The calculated data were complemented with the results of the EPR investigations carried out at liquid helium temperature, allowing for definite assignment of the high-spin state for the Co(acac)(2)(EtOH)(2) complex. However, because of the unresolved spectral features, only effective g-values could be assessed, whereas the zero-field splitting parameters (ZFS) were calculated by means of the spin-orbit mean field (SOMF) relativistic DFT method for which direct spin-spin (SS) and spin-orbit coupling (SOC) contributions were quantified.     

First-principles calculations of zero-field splitting parameters

In this work, an implementation of an approach to calculate the zero-field splitting (ZFS) constants in the framework of ab initio methods such as complete active space self-consistent field, multireference configuration interaction, or spectroscopy oriented configuration interaction is reported. The spin-orbit coupling (SOC) contribution to ZFSs is computed using an accurate multicenter mean-field approximation for the Breit-Pauli Hamiltonian. The SOC parts of ZFS constants are obtained directly after diagonalization of the SOC operator in the basis of a preselected number of roots of the spin-free Hamiltonian. This corresponds to an infinite order treatment of the SOC in terms of perturbation theory. The spin-spin (SS) part is presently estimated in a mean-field fashion and appears to yield results close to the more complete treatments available in the literature. Test calculations for the first- and second-row atoms as well as first-row transition metal atoms and a set of diatomic molecules show accurate results for the SOC part of ZFSs. SS contributions have been found to be relatively small but not negligible (exceeding 1 cm(-1) for oxygen molecule). At least for the systems studied in this work, it is demonstrated that the presented method provides much more accurate estimations for the SOC part of ZFS constants than the emerging density functional theory approaches.     

Self-consistent treatment of spin-orbit interactions with efficient Hartree-Fock and density functional methods

Efficient self-consistent field (SCF) schemes including both scalar relativistic effects and spin-orbit (SO) interactions at Hartree-Fock (HF) and density functional (DFT) levels are presented. SO interactions require the extension of standard procedures to two-component formalisms. Efficiency is achieved by using effective core potentials (ECPs) and by employing the resolution-of-the-identity approximation for the Coulomb part (RI-J) in pure DFT calculations as well as also for the HF-exchange part (RI-JK) in the case of HF or hybrid-DFT treatments. The procedures were implemented in the program system TURBOMOLE; efficiency is demonstrated for comparably large systems, such as Pb54. Relevance of SO effects for electronic structure and stability is illustrated by treatments of small Pb and Po clusters with and without accounting for SO effects.     

Approximate Relativistic calculations within the one-center approximation for the series CH4 TO PbH4

Using one-component relativistic theory combined with an Xαβ local exchange, we have carried out calculations for the series CH₄ to PbH₄ within the spherically symmetric one-center expansion approximation. Bond lenghts, breathing force constant, ionization potentials and spin-orbit splitting of the filled valence p shell thus obtained, are found to be in excellent agreement with Dirac-Hartre-Fock results.     

Density functional theory studies on structures and absorption spectra of [Au(tpy)Cl]2+ and its derivatives: Role of basis set,functional, solvent effect,and spin orbit effect

The geometries of [Au(tpy)Cl]²⁺ (tpy = 2,2′:6′,2″‐terpyridine) and its derivatives ( 1 – 4 ) were optimized using relativistic density functional theory (DFT) at both scalar and two‐component spin orbit coupling (SOC) level of theory via zero order regular approximation (ZORA). The combination of OPTX exchange, PW91c correlation functional (denoted as OP91), all‐electron ZORA TZ2P basis set was found to be the optimal combination for geometry. The results reveal that both SOC and substituents have little effect on the geometry of complexes 1 – 4 . Then, their absorption spectra were investigated by scalar relativistic time dependent DFT (TDDFT)/SAOP/TZ2P in vacuum, in CH₂Cl₂, CH₃CN solvents by means of conductor like screening model. The calculations indicate that the nature of the low‐lying spin‐allowed excited states is gold‐perturbed intraligand transition, namely charge reorganization. This fact also demonstrates that the influence of the polarity of solvent on absorption spectra of 1 – 4 is negligible. The spin orbit TDDFT was also performed to get further insight into the effect of SOC on the absorption spectra. It is found that the SOC has little influence on the simulation of electronic spectrum of complexes 1 – 4 due to no significant involvement of d‐orbitals during electronic transition. Our conclusions are reliable and are in good agreement with the previous experimental results and theoretical investigations. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Irreducible tensor operator methods in intermediate-field coupling

The algebra of irreducible tensor operators is developed in the intermediate-field coupling case. The Wigner-Eckart theorem is formulated for a simple irreducible tensor operator as well as for the Kronecker and scalar products of these operators. The expressions required for the calculation of Coulomb repulsion, crystal field splitting, spin-orbit interaction, and Zeeman effect are given in detail. Recent applications to various problems in spectroscopy and magnetism of transition metal compounds are referred to.     

Spin-spin contributions to the zero-field splitting tensor in organic triplets, carbenes and biradicals-a density functional and ab initio study

An evaluation study for the direct dipolar electron spin-spin (SS) contribution to the zero-field splitting (ZFS) tensor in electron paramagnetic resonance (EPR) spectroscopy is presented. Calculations were performed on a wide variety of organic systems where the SS contribution to the ZFS dominates over the second-order spin-orbit coupling (SOC) contribution. Calculations were performed using (hybrid) density functional theory (DFT), as well as complete active space self-consistent field (CASSCF) wave functions. In the former case, our implementation is an approximation, because we use the two-particle reduced spin-density matrix of the noninteracting reference system. In the latter case, the SS contribution is approximated by a mean-field method which, nevertheless, gives accurate results, compared to the approximation free computation of the SS part in a CASSCF framework. For the case of the triplet dioxygen molecule, it was shown that restricted open-shell density functional theory (RODFT), as well as CASSCF, can provide accurate spin-spin couplings while spin-unrestricted DFT leads to much larger errors. Furthermore, 15 organic radicals, including several 1,3 and 1,5 diradicals, dinitroxide biradicals, and even a chlorophyll a model system, were examined as test cases to demonstrate the accuracy and efficiency of our approach within a DFT framework. Accurate D values with root-mean-square deviations of 0.0035 cm(-1) were obtained. Furthermore, all trends, including those due to substituent effects, were correctly reproduced. In a different set of calculations, the polyacenes benzene, naphthalene, anthracene, and tetracene were studied. Applying DFT, the absolute D values were noticeably underestimated, but it was possible to correctly reproduce the trend to smaller D values with larger size of the systems. Finally, it was demonstrated that our approach is also well-suited for the study of carbenes. The smaller organic radicals of this work were also studied, through the use of CASSCF wave functions. This was a special advantage in the case of the triplet polyacenes, where the CASSCF approach gave better results than the DFT method. In comparing spin-restricted and spin-unrestricted results, it was shown through a natural orbital analysis and comparison to high-level ab initio calculations that even small amounts of spin polarization introduced by the unrestricted calculations lead to large deviations between the unrestricted Kohn-Sham (UKS) and restricted open-shell Kohn-Sham (ROKS) approaches. It is challenging to understand why the ROKS results show much better correlation with the experimental data.     

Relativistic corrections in the LCGTO–local-spin-density method. I. Atomic bases for transition metals

Compact orbital GTO basis sets optimized with a nonrelativistic Hamiltonian, then decontracted, are utilized in an SCF treatment with a quasi-relativistic scalar Hamiltonian including the mass–velocity and one-electron Darwin operators. Ionization and excitation energies, orbital energies, and radial mean values obtained from different expansion patterns have been tested in atomic calculations for Ag and generalized for Cu and Au atoms. The one-electron spin–orbit operator is also used in an SCF treatment. Spin–orbit coupling energies are calculated for Cu, Ag, and Au atoms.     

激发态1，2-二硫环丁烯去活性和异构化机理的理论研究

运用完全活性空间多组态CASSCF方法研究了激发态1,2-二硫环丁烯（1,2-Dithiete）势能面交叉机理．自旋．轨道耦合（SOC）常数采用完全Pauli-Breit旋轨耦合算符（包括单电子和双电子项）进行计算,其强度为198．37或211．35cm^-1,对不同自旋态跃迂起着重要作用．研究结果表明：光激发1,3-dithiol-2-one导致形成主要产物trans—dithioglyoxal（Trans-MinS0）和次级产物thiolthioketene．计算与实验观察结果一致．     

Calculation of the zero-field splitting tensor on the basis of hybrid density functional and Hartree-Fock theory

The zero-field splitting (ZFS) (expressed in terms of the D tensor) is the leading spin-Hamiltonian parameter for systems with a ground state spin S>12. To first order in perturbation theory, the ZFS arises from the direct spin-spin dipole-dipole interaction. To second order, contributions arise from spin-orbit coupling (SOC). The latter contributions are difficult to treat since the SOC mixes states of different multiplicities. This is an aspect of dominant importance for the correct prediction of the D tensor. In this work, the theory of the D tensor is discussed from the point of view of analytic derivative theory. Starting from a general earlier perturbation treatment [F. Neese and E. I. Soloman, Inorg. Chem. 37, 6568 (1998)], straightforward response equations are derived that are readily transferred to the self-consistent field (SCF) Hartree-Fock (HF) or density functional theory (DFT) framework. The main additional effort in such calculations arises from the solution of nine sets of nonstandard coupled-perturbed SCF equations. These equations have been implemented together with the spin-orbit mean-field representation of the SOC operator and a mean-field treatment of the direct spin-spin interaction into the ORCA electronic structure program. A series of test calculations on diatomic molecules with accurately known zero-field splittings shows that the new approach corrects most of the shortcomings of previous DFT based methods and, on average, leads to predictions within 10% of the experimental values. The slope of the correlation line is essentially unity for the B3LYP and BLYP functionals compared to approximately 0.5 in previous treatments.     

A Spin-orbit Coupling Study on the Quinoline-and Porphyrin-based Photosensitizers

The spin-orbit coupling(SOC) of four porphyrin- and quinoline-based compounds has been studied using Pauli-Breit SOC operator with one- and two-electron terms. The results revealed that the yield of singlet oxygen is affected by the spin-orbit coupling matrix element involving the emitting triplet and the perturbing singlet state. Investigated quinoline-based compounds have more high SOC values than those porphyrin-based compounds due to spin parallel electron pairs of oxygen. The open shell d8 of metal Pt can induce the stronger exchange interactions than the closed shell p6 of metal Mg, resulting in bigger SOC matrix element in quinoline-based Pt complex than in the quinoline-based Mg complex. Simultaneously, potential energy curves of the first excited sate and the first triplet sate have been calculated, which proves that all investigated complexes can induce singlet oxygen. These computational findings support quinolin-based compounds have high singlet oxygen yields and provide a rigorous basis for predicting the probability of singlet oxygen yields in plane-type molecules.     

含Hg化合物的相对论赝势从头计算研究——HgX2(X=Cl,Br,I)的电子结构

本文应用相对论赝势从头计算方法, 在不同基组水平上, 系统地研究了卤化汞(HgX_2, X=Cl,Br,I)系列的电子结构。表明除Hg的6s主要参与成键外, 5dz~2也起了重要的作用。并且随卤素原子序的增加, π成键作用也增强。同时还应用单电子自旋-轨道耦合方法, 研究了旋-轨耦合效应的影响, 指定了该系列化合物的光电子能谱。     

AB initio effective spin—orbit operators for use in atomic and molecular structure calculations. Results for carbon and silicon

Ab initio effective spin-orbit operators for carbon and silicon are derived from relativistic effective core potentials based on Dirac—Fock wavefunctions. Transferability of these operators to electronic states other than the one used in the original derivation is treated by calculating spin—orbit splittings of various neutral and ionic atomic states.     

A CNDO/INDO molecular orbital formalism for the elements H to Br. theory

A CNDO and INDO formalism is presented that can be used for any atom combination up to bromine under inclusion of the first transition metal series. The semiempirical parameters were chosen to reproduce results ofab initio calculations on metalorganic compounds. The calculational results are invariant to rotations of the coordinate system but not to a general transformation into other basis functions. The one-center Coulomb-expressions were selected in order to include intraatomic correlation contributions. Within the CNDO model this could be achieved by the scaled monopole termF ₀, while in the INDO framework the one-center Coulomb integrals are given as a sum of the monopole-contributionF ₀ and higher multipole contributions expressed as a linear combination of Slater-Condon parameters. The invariance problem in the case of local rotations within the INDO approximation was solved by considering the combination of one-center Coulomb and exchange integrals as a function ofl but independent ofm. The two-center electron-electron interaction terms were calculated via the Dewar-Sabelli, Ohno-Klopman relation. Penetration effects were treated according to Fischer and Kollmar. For the resonance integralH _v ^AB parameters are used which carry information related to the directed nature of the chemical bond by using optimized Klondyke functions. The core-core repulsion is constructed as a superposition of a soft potential function, describing polarization effects of the atomic cores, and a hard repulsion function, avoiding the collapse of the atomic cores with decreasing distance.