Spin–orbit coupling of spin‐frustrated systems

We have investigated the effects of spin–orbit (SO) interactions on noncollinear molecular magnetism by combining the classical Dzyaloshinsky–Moriya (DM) model and ab initio generalized spin orbital (GSO) method. We have derived an estimation scheme of the magnetic anisotropy energy (MAE) and the Dzyaloshinsky vector based on the SO first‐order perturbation theory (SOPT1) for GSO Hartree–Fock (GHF) solutions. We found that the fundamental results of GHF‐SOPT1 method can be reproduced by diagonalizing the core Hamiltonian plus SO terms, and that the spin topologies of odd‐ring systems can be determined by the topological indices of the singly occupied molecular orbitals. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Magnitude of Finite‐Nucleus‐Size Effects in Relativistic Density Functional Computations of Indirect NMR Nuclear Spin–Spin Coupling Constants

A spherical Gaussian nuclear charge distribution model has been implemented for spin‐free (scalar) and two‐component (spin–orbit) relativistic density functional calculations of indirect NMR nuclear spin–spin coupling (J‐coupling) constants. The finite nuclear volume effects on the hyperfine integrals are quite pronounced and as a consequence they noticeably alter coupling constants involving heavy NMR nuclei such as W, Pt, Hg, Tl, and Pb. Typically, the isotropic J‐couplings are reduced in magnitude by about 10 to 15 % for couplings between one of the heaviest NMR nuclei and a light atomic ligand, and even more so for couplings between two heavy atoms. For a subset of the systems studied, viz. the Hg atom, Hg₂²⁺, and Tl? X where X=Br, I, the basis set convergence of the hyperfine integrals and the coupling constants was monitored. For the Hg atom, numerical and basis set calculations of the electron density and the 1s and 6s orbital hyperfine integrals are directly compared. The coupling anisotropies of TlBr and TlI increase by about 2 % due to finite‐nucleus effects.     

Spin‐orbit ab initio and density functional theory investigation of bismuth monoboronyl,BiBO

The molecular properties of bismuth monoboronyl, BiBO, were investigated using high‐level ab initio and density functional theory calculations by including the effect of spin‐orbit coupling (SOC). SOC does not cause any change in the Bi? B bond length of BiBO, by contrast it causes significant elongation of the Bi? B bond of BiBO^?, by ～0.03 Å. The Bi? B bond length of BiBO^? that is calculated by considering SOC is almost identical to that of BiBO; this result is consistent with a recent experimental study. The term values of excited states of BiBO calculated by including SOC are in good agreement with the experimental results. One excited state which was not assigned in the previous experimental study is the Ω = 0⁺ state generated by strong SOC. In the theoretical calculations on molecules containing 6p‐block elements, including SOC is crucial for obtaining results that are consistent with the corresponding experimental results.     

An animated visualization of orbital angular momentum and spin‐orbit coupling

This paper presents an approach toward visualizing a complex orbital based on animation using a time‐dependent phase factor. This makes orbital angular momentum clearly visible, in a way that reflects the nature of the orbital angular momentum wavefunction. Visualization of this quantity is also useful for examining the effects of spin‐orbit coupling (SOC), in which higher orbital angular momentum states are admixed into the orbital; in this case, however, scaling of one phase‐component is needed. The phase orientation of a complex orbital, which is generally not guaranteed by the SCF procedure, must be considered when doing this. The method of visualization presented here may prove useful when analyzing properties where SOC is important, such as magnetic resonance parameters. Animated visualizations are performed, and compared with the method of phase‐colored isosurfaces, first for a model p‐orbital to explain the idea, and then for the singly‐occupied molecular orbitals of two small doublet radicals.     

Spin–Orbit Contributions in High‐Spin Nitrenes/Carbenes: A Hybrid CASSCF/MRMP2 Study of Zero‐Field Splitting Tensors

Zero‐field splitting (ZFS) tensors ( D tensors) of organic high‐spin oligonitrenes/oligocarbenes up to spin‐septet are quantitatively determined on the basis of quantum chemical calculations. The spin–orbit contributions, D ^SO tensors are calculated in terms of a hybrid CASSCF/MRMP2 approach, which was recently proposed by us. The spin–spin counterparts, D ^SS tensors are computed based on McWeeny–Mizuno’s equation in conjunction with the RODFT spin densities. The present calculations show that more than 10 % of ZFS arises from spin–orbit interactions in the high‐spin nitrenes under study. Contributions of spin‐bearing site–site interactions are estimated with the aid of a semi‐empirical model for the D tensors and found to be ca. 5 % of the D ^SO tensor. The analysis of intermediate states reveal that the largest contributions to the calculated D ^SO tensors are attributed to intra‐site spin flip excitations and delocalized π and π* orbitals play an important role in the inter‐site spin–orbit interactions.     

An Exceptionally Close,Non‐Bonded Hydrogen–Hydrogen Contact with Strong Through‐Space Spin–Spin Coupling

Condensation of 1,8,13‐tris(mercaptomethyl)triptycene and tris(bromomethyl)methane yields an in,in‐cyclophane with two inwardly directed methine groups. Based on X‐ray analysis and DFT and MP2 calculations, the hydrogen–hydrogen non‐bonded contact distance is estimated to be 1.50–1.53 Å. Furthermore, the two in‐hydrogen atoms show obvious spin–spin coupling with J=2.0 Hz.     

Spin–orbit effects on magnetically induced current densities in the () clusters

This study reports the spin–orbit effects on the aromaticity of the , , , , , and anionic clusters via the magnetically induced current‐density method. All‐electron density functional theory (DFT) calculations were carried out using the four‐component Dirac‐Coulomb (DC) hamiltonian, including scalar and spin–orbit relativistic effects. The magnetic index of aromaticity was calculated by numerical integration over the current flow between two atoms in the pentagonal ring. These values were compared to the spin‐free values (spin–orbit coupling switched off), in order to assess the spin–orbit effect on aromaticity. It was found that in the heavy anions, and , there is a significant influence of the spin–orbit coupling. © 2018 Wiley Periodicals, Inc.     

Choice of Spin–Orbit Correction Terms in Gaussian Model Chemistries

Spin–orbit correction terms for use in Gaussian‐2 theory and other model chemistries for third‐row atoms and molecules are calculated by several methods with the objective of finding a reliable method that can be applied in a routine and economical manner in the spirit of Gaussian model chemistries. The results are evaluated for the test set of molecules and ions used in the original extension of Gaussian‐2 theory to third‐row atoms. Further results are presented for systems where Gaussian‐2 results are reported in the literature without spin–orbit correction terms and for some larger molecules. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1552–1556, 2001     

Spin–Orbit Coupling in Phosphorescent Iridium(III) Complexes

We study the excited states of two iridium(III) complexes with potential applications in organic light‐emitting diodes: fac‐tris(2‐phenylpyridyl)iridium(III) [Ir(ppy)₃] and fac‐tris(1‐methyl‐5‐phenyl‐3‐n‐propyl‐[1,2,4]triazolyl)iridium(III) [Ir(ptz)₃]. Herein we report calculations of the excited states of these complexes from time‐dependent density functional theory (TDDFT) with the zeroth‐order regular approximation (ZORA). We show that results from the one‐component formulation of ZORA, with spin–orbit coupling included perturbatively, accurately reproduce both the results of the two‐component calculations and previously published experimental absorption spectra of the complexes. We are able to trace the effects of both scalar relativistic correction and spin–orbit coupling on the low‐energy excitations and radiative lifetimes of these complexes. In particular, we show that there is an indirect relativistic stabilisation of the metal‐to‐ligand charge transfer (MLCT) states. This is important because it means that indirect relativistic effects increase the degree to which SOC can hybridise singlet and triplet states and hence plays an important role in determining the optical properties of these complexes. We find that these two compounds are remarkably similar in these respects, despite Ir(ppy)₃ and Ir(ptz)₃ emitting green and blue light respectively. However, we predict that these two complexes will show marked differences in their magnetic circular dichroism (MCD) spectra.     

Conformational Dependence of Through‐Space Tellurium–Tellurium Spin–Spin Coupling in Peri‐Substituted Bis(Tellurides)

Three related series of peri‐substituted bis(tellurides) bearing naphthalene, acenaphthene and acenaphthylene backbones (Nap/Acenap/Aceyl(TeY)₂ (Nap=naphthalene‐1,8‐diyl N ; Acenap=acenaphthene‐5,6‐diyl A ; Aceyl=acenaphthylene‐5,6‐diyl Ay ; Y=Ph 1 ; Fp 2 ; Tol 3 ; An‐p­ 4 ; An‐o­ 5 ; Tp 6 ; Mes 7 ; Tip 8 ) have been synthesised and their solid‐state structures determined by X‐ray crystallography. Molecular conformations were classified as a function of the two C9‐C‐Te‐C(Y) dihedral angles (θ); in the solid all members adopt AB or CCt configurations, with larger Te(aryl) moieties exclusively imposing the CCt variant. Exceptionally large J(¹²⁵Te,¹²⁵Te) spin–spin coupling constants between 3289–3848 Hz were obtained for compounds substituted by bulky Te(aryl) groups, implying these species are locked in a CCt‐type conformation. In contrast, compounds incorporating smaller Te(aryl) moieties are predicted to be rather dynamic in solution and afford much smaller J values (2050–2676 Hz), characteristic of greater populations of AB conformers with lower couplings. This conformational dependence of through‐space coupling is supported by DFT calculations.     

Use of the normal coordinates in variational and perturbative ab initio handling of the vibronic and spin–orbit couplings in Π electronic states of linear tetra‐atomic molecules

Avariational and a perturbative approach are developed to handle the combined effect of the vibronic and spin–orbit couplings in Π electronic states of tetra‐atomic molecules with linear equilibrium geometry. Both of them are based on the use of the normal vibrational bending coordinates. The perturbative treatment is carried out via two schemes for partition of the model Hamiltonian: In the first, the spin–orbit coupling term is treated as a perturbation; in the second, it is included in the zeroth‐order Hamiltonian. It is demonstrated that both perturbative approaches lead to the same second‐order formulae when the spin–orbit coupling constant is small compared to the bending frequency, but much larger than the splitting of potential surfaces upon bending. These approaches are used to calculate the vibronic and spin–orbit structure in the X²Π electronic state of HCCS by employing the ab initio‐computed potential energy surfaces. Complete numerical equivalence of the results obtained with the present variational approach and those generated by the algorithms employing internal vibrational coordinates is demonstrated. The restrictions concerning the applicability of the perturbative approaches are discussed in terms of the agreement between the results obtained by means of them with those generated in the corresponding variational computations. The general reliability of the model employed is checked by comparing the theoretical results with the available experimental data. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

First example of the correlated calculation of the one‐bond tellurium–carbon spin–spin coupling constants: Relativistic effects,vibrational corrections,and solvent effects

This work reports on the comprehensive calculation of the NMR one‐bond spin–spin coupling constants (SSCCs) involving carbon and tellurium, ¹J(¹²⁵Te,¹³C), in four representative compounds: Te(CH₃)₂, Te(CF₃)₂, Te(C?CH)₂, and tellurophene. A high‐level computational treatment of ¹J(¹²⁵Te,¹³C) included calculations at the SOPPA level taking into account relativistic effects evaluated at the 4‐component RPA and DFT levels of theory, vibrational corrections, and solvent effects. The consistency of different computational approaches including the level of theory of the geometry optimization of tellurium‐containing compounds, basis sets, and methods used for obtainig spin–spin coupling values have also been discussed in view of reproducing the experimental values of the tellurium–carbon SSCCs. Relativistic corrections were found to play a major role in the calculation of ¹J(¹²⁵Te,¹³C) reaching as much as almost 50% of the total value of ¹J(¹²⁵Te,¹³C) while relativistic geometrical effects are of minor importance. The vibrational and solvent corrections account for accordingly about 3–6% and 0–4% of the total value. It is shown that taking into account relativistic corrections, vibrational corrections and solvent effects at the DFT level essentially improves the agreement of the non‐relativistic theoretical SOPPA results with experiment. © 2016 Wiley Periodicals, Inc.     

Ab initio calculations of the ground and excited states of the ZrN molecule including spin‐orbit effects

The electronic structures with spin‐orbit effects of the zirconium nitride ZrN molecule are investigated by the methods of multireference single and double configuration interaction. The potential energy curves are calculated along with the spectroscopic constants for the lowest‐lying 34 spin‐orbit states Ω in ZrN. A good agreement is displayed by comparing the calculated spectroscopic constants with those available experimentally. The permanent dipole moments are calculated along with the vibrational energies. New results are obtained in this work for 29 spin‐orbit states and their spectroscopic constants calculated. © 2015 Wiley Periodicals, Inc.     

GRECP/5e‐MRD‐CI calculation of the electronic structure of PbH

The correlation calculation of the electronic structure of PbH is carried out with the generalized relativistic effective core potential (GRECP) and multireference single‐ and double‐excitation configuration interaction (MRD‐CI) methods. The 22‐electron GRECP for Pb is used and the outer core 5s, 5p, and 5d pseudospinors are frozen using the level‐shift technique, so only five external electrons of PbH are correlated. A new configuration selection scheme with respect to the relativistic multireference states is employed in the framework of the MRD‐CI method. The [6, 4, 3, 2] correlation spin–orbit basis set is optimized in the coupled cluster calculations on the Pb atom using a recently proposed procedure, in which functions in the spin–orbital basis set are generated from calculations of different ionic states of the Pb atom and those functions are considered optimal that provide the stationary point for some energy functional. Spectroscopic constants for the two lowest‐lying electronic states of PbH (²Π_1/2, ²Π_3/2) are found to be in good agreement with the experimental data. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Indirect relativistic bridge and substituent effects from the ‘heavy’ environment on the one‐bond and two‐bond 13C1H spin–spin coupling constants

Indirect relativistic bridge effect (IRBE) and indirect relativistic substituent effect (IRSE) induced by the ‘heavy’ environment of the IV‐th, V‐th and VI‐th main group elements on the one‐bond and geminal ¹³C? ¹H spin–spin coupling constants are observed, and spin‐orbit parts of these two effects were interpreted in terms of the third‐order Rayleigh–Schrödinger perturbation theory. Both effects, IRBE and IRSE, rapidly increase with the total atomic charge of the substituents at the coupled carbon. The accumulation of IRSE for geminal coupling constants is not linear with respect to the number of substituents in contrast to the one‐bond couplings where IRSE is an essentially additive quantity. Copyright © 2015 John Wiley & Sons, Ltd.     

Ab initio and DFT study of the electronic structures and spectroscopic properties of pyrene ligands and their cyclometalated complexes

Two ligands 1‐diphenylphosphinopyrene (1‐PyP) ( L ₁ ), 1,6‐bis(diphenylphosphino)‐pyrene (1,6‐PyP) ( L ₂ ) and their cyclometalated complexes [Pt(dppm)(1‐PyP‐H)]⁺ ( 1 ), [Pt₂(dppm)₂(1,6‐PyP‐H₂)]²⁺ (dppm = bis(diphenylphosphino)methane ( 2 ), and [Pd(dppe)(1‐PyP‐H)⁺ (dppe = bis(diphenylphosphino)ethane) ( 3 ) are investigated theoretically to explore their electronic structures and spectroscopic properties. The ground‐ and excited‐state structures are optimized by the density functional theory (DFT) and single‐excitation configuration interaction method, respectively. At the time‐dependent DFT (TDDFT) and B3LYP level, the absorption and emission spectra in solution are obtained. As revealed from the calculations, the lowest‐energy absorptions of 1 and 3 are attributed to the mixing ligand‐to‐metal charge transfer (CT)/intraligand (IL)/ligand‐to‐ligand CT transitions, while that of 2 is attributed to the IL transition. The lowest‐energy phosphorescent emissions of the cyclometalated complexes are attributed to coming from the ³ILCT transitions. With the increase of the spin‐orbit coupling effect, the phosphorescence intensities and the emissions wavelength are correspondingly increased. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Singlet and Triplet Excited States and Intersystem Crossing in Free‐Base Porphyrin: TDDFT and DFT/MRCI Study

Extensive time-dependent DFT (TDDFT) and DFT/multireference configuration interaction (MRCI) calculations are performed on the singlet and triplet excited states of free-base porphyrin, with emphasis on intersystem crossing processes. The equilibrium geometries, as well as the vertical and adiabatic excitation energies of the lowest singlet and triplet excited states are determined. Single and double proton-transfer reactions in the first excited singlet state are explored. Harmonic vibrational frequencies are calculated at the equilibrium geometries of the ground state and of the lowest singlet and triplet excited states. Furthermore, spin–orbit coupling matrix elements of the lowest singlet and triplet states and their numerical derivatives with respect to nuclear displacements are computed. It is shown that opening of an unprotonated pyrrole ring as well as excited-state single and double proton transfer inside the porphyrin cavity lead to crossings of the potential energy curves of the lowest singlet and triplet excited states. It is also found that displacements along out-of-plane normal modes of the first excited singlet state cause a significant increase of the 2|H_so|S₁>, 1|H_so|S₁>, and 1|H_so|S₀> spin–orbit coupling matrix elements. These phenomena lead to efficient radiationless deactivation of the lowest excited states of free-base porphyrin via intercombination conversion. In particular, the S₁→T₁ population transfer is found to proceed at a rate of ≈10⁷ s⁻¹ in the isolated molecule.     

A theoretical structural analysis of the factors that affect 1JNH, 1hJNH and 2hJNN in N–H···N hydrogen‐bonded complexes

Calculations of ¹ J_NH, ^1h J_NH and ^2h J_NN spin–spin coupling constants of 27 complexes presenting N–H·N hydrogen bonds have allowed to analyze these through hydrogen‐bond coupling as a function of the hybridization of both nitrogen atoms and the charge (+1, 0, ? 1) of the complex. The main conclusions are that the hybridization of N atom of the hydrogen bond donor is much more important than that of the hydrogen bond acceptor. Positive and negative charges (cationic and anionic complexes) exert opposite effects while the effect of the transition states ‘proton‐in‐the‐middle’ is considerable. Copyright © 2008 John Wiley & Sons, Ltd.     

Structural studies on aryl‐substituted enaminoketones and their thio analogues. Part II. Experimental and DFT calculated carbon–carbon coupling constants,nJ(CC)'s (n = 1–3)

Spin–spin carbon–carbon coupling constants across one, two and three bonds, J(CC), have been measured for a series of aryl‐substituted Z‐s‐Z‐s‐E enaminoketones and their thio analogues. As a result, a large set, altogether 178, of J(CC)s has been obtained. It consists of 82 couplings across one bond, 31 couplings across two bonds and 65 couplings across three bonds. Independently, the DFT calculations at the B3PW91/6‐311++G(d,p)//B3PW91/6‐311++G(d,p) level yielded a set of theoretical J(CC) values. A comparison of these two sets of data gave an excellent linear correlation with parameters a and b close to ideal; a = 0.9978 which is not far from unity and b = 0.22 Hz which is close to zero. The ¹J(CC) couplings determined for the crucial fragment of the molecules, i.e. ? C?C? C?O (or ? C?C? C?S), are: ¹J(C?C) ≈ 68 Hz (67 Hz) and ¹J(C? C) = 60.5 Hz (60.0 Hz). The corresponding couplings found for the Z‐s‐Z‐s‐E isomer of the parent enaminoketone, 4‐methylamino‐but‐3‐en‐2‐one are 64.1 and 59.3 Hz, respectively. The most sensitive towards substitution of the oxygen atom by sulfur are two‐bond couplings between the α‐vinylic and aromatic C_ipso carbon atoms, which attain 12 Hz in the enaminoketone derivatives and decrease to 5 Hz in their thio analogues. Copyright © 2009 John Wiley & Sons, Ltd.