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     

The modified all-valence INDO method with the inclusion of spin–orbit coupling

The all-valence INDO method has been modified for the inclusion of spin–orbit coupling effects. In the method presented, the Hamiltonian includes spin–orbit coupling and the basis set constitutes the singlet and triplet determinental wave functions constructed from molecular orbitals resulting from nonrelativistic calculations. Eigenvectors obtained are later used for the evaluation of transition probabilities among different states. The results presented include lifetimes of different states of organic molecules and transition energies for halogen molecules and they are in a good agreement with experimental results. © 1992 John Wiley & Sons, Inc.     

Development of spin‐orbit coupling for stochastic configuration interaction techniques

To perform spin‐orbit coupling calculations on atoms and molecules, good zeroth‐order wavefunctions are necessary. Here, we present the software development of the Monte Carlo Configuration Interaction (MCCI) method, to enable calculation of such properties, where MCCI iteratively constructs a multireference wavefunction using a stochastic procedure. In this initial work, we aim to establish the efficacy of this technique in predicting the splitting of otherwise degenerate energy levels on a range of atoms and small diatomic molecules. It is hoped that this work will subsequently act as a gateway toward using this method to investigate singlet‐triplet interactions in larger multireference molecules. We show that MCCI can generate very good results using highly compact wavefunctions compared to other techniques, with no prior knowledge of important orbitals. Higher‐order relativistic effects are neglected and spin‐orbit coupling effects are incorporated using first‐order degenerate perturbation theory with the Breit‐Pauli Hamiltonian and effective nuclear charges in the one‐electron operator. Results are obtained and presented for B, C, O, F, Si, S, and Cl atoms and OH, CN, NO, and C₂ diatomic radicals including spin‐orbit coupling constants and the relative splitting of the lowest energy degenerate state for each species. Convergence of MCCI to the full configuration interaction result is demonstrated on the multireference problem of stretched OH. We also present results from the singlet‐triplet interaction between the and both the and states of the O₂ molecule. © 2017 Wiley Periodicals, Inc.     

Ab initio study of the electronic states of BS including spin–orbit coupling

The entire 30 Ω states generated from the 12 valence and two Rydberg Λ–S states of the BS radical have been studied at the MR-CISD + Q level of theory for the first time. The effects of spin–orbit coupling and the avoided crossing rule between electronic states of the same symmetry were analyzed. Spectroscopic constants of several excited states that have never been observed in experiment were obtained. The transition properties of several low-lying bound excited states to ground state transitions, including the transition dipole moments and the Franck–Condon factors, were also calculated, from which the corresponding single-channel radiative lifetimes were derived.     

Revisiting the nature of the ZnO ground state: Influence of spin–orbit coupling

Relativistic calculations of the low-lying electronic states of the ZnO molecule are performed for the Λ–Σ states, ¹Σ⁺, ¹Π, ¹Δ, ³Π and ³Σ⁻, at the CCSD(T) or MRCI level, using scalar relativistic energy-consistent pseudopotentials, and the EPCISO method for spin–orbit CI coupling. The ZnO ground state is assigned to 0⁺ symmetry and has ¹Σ⁺ character around the equilibrium region. The spectroscopic constants (r_e, ω_e) of the 0⁺ ground state are in good agreement with experimental results. Interpenetration of the vibrational levels of the two lowest 0⁺ states is also shown.     

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.     

On the evaluation of CI matrix elements for a canonically ordered basis

Using the unitary group approach it is shown that the amount of storage needed for the construction of symbolic CI matrix element lists for N-electron basis functions with large numbers of open shells and arbitrary multiplicities may substantially be reduced compared to methods currently available in the literature.     

Electronic transition moment with spin‐orbit coupling of the molecular ion KRb+

By using the electronic wave functions obtained from an ab initio calculation, including the spin‐orbit coupling, the electronic transition moments have been investigated for two bound states of symmetry Ω = 1/2 and Ω = 3/2 of the molecular ion KRb⁺. Based on a canonical functions approach for the determination of the vibrational wave functions, the matrix elements have been calculated for the bound states considered for v = 0, 10, 20 with v′‐ v = 0, 1, 2, …, 6; by using the same canonical approach, the eigenvalues and abscissas of the corresponding turning points (r_min and r_max) have been investigated for these states that obtained from a theoretical ab initio calculation up to v = 105. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Electronic spectra and intersystem spin‐orbit coupling in 1,2‐ and 1,3‐squaraines

The main photophysical properties of a series of recently synthetized 1,2‐ and 1,3‐squaraines, including absorption electronic spectra, singlet‐triplet energy gaps, and spin‐orbit matrix elements, have been investigated by means of density functional theory (DFT) and time‐dependent DFT approaches. A benchmark of three exchange‐correlation functionals has been performed in six different solvent environments. The investigated 1,2 squaraines have been found to possess two excited triplet states (T₁ and T₂) that lie below the energy of the excited singlet one (S₁). The radiationless intersystem spin crossing efficiency is thus enhanced in both the studied systems and both the transitions could contribute to the excited singlet oxygen production. Moreover, they have a singlet‐triplet energy gap higher than that required to generate the cytotoxic singlet oxygen species. According to our data, these compounds could be used in photodynamic therapy applications that do not require high tissue penetration. © 2014 Wiley Periodicals, Inc.     

Exciting flavins: Absorption spectra and spin–orbit coupling in light–oxygen–voltage (LOV) domains

Flavins are central molecular chromophores for many photobiological processes. In this paper, several aspects of the photophysics and photochemistry of lumiflavin in a (protein) environment will be studied with the help of quantum chemical methods. In a first part of the paper, we present vertical singlet excitation energies for lumiflavin (a molecule of the isoalloxazin type), using time-dependent density functional theory (TD-DFT) in conjunction with the B3LYP hybrid functional. When calculated for isolated species, TD-DFT excitation energies are generally blue-shifted relative to the experimental absorption spectra of isoalloxazines in solution, or in a protein environment. We develop four different models to account for environmental effects, with special emphasis on the LOV1 domain of Chlamydomonas reinhardtii. It is found that the two lowest, allowed singlet excitations are sensitive to the polarizability of an environment, to hydrogen bonds, and to geometrical constraints imposed by the surrounding protein. All of this brings theory and experiment in better agreement.

In the second part of the paper the light-induced adduct formation in LOV domains, between the chromophore and a neighbouring cystein unit is investigated. Energies along a model “reaction path” are calculated on the DFT/B3LYP and MCQDPT2 level of theories. A transition state for a H-transfer between the mercapto (SH–) group of cystein, and the N(5) position of flavin is found. The reaction requires spin–orbit coupling between the S₀ and the T₁ states of the system. Spin–orbit coupling constants between S₀ and T₁ are calculated, and found to be in the range of several tens of cm⁻¹ after the transition state was passed. A biradical intermediate was observed along the reaction path.     

NMR spin–spin coupling constants in hydrogen‐bonded glycine clusters

The influence of the hydrogen bond formation on the NMR spin–spin coupling constants (SSCC), including the Fermi contact (FC), the diamagnetic spin‐orbit, the paramagnetic spin‐orbit, and the spin dipole term, has been investigated systematically for the homogeneous glycine cluster, in gas phase, containing up to three monomers. The one‐bond and two‐bond SSCCs for several intramolecular (through covalent bond) and intermolecular (across the hydrogen‐bond) atomic pairs are calculated employing the density functional theory with B3LYP and KT3 functionals and different types of extended basis sets. The ab initio SOPPA(CCSD) is used as benchmark for the SSCCs of the glycine monomer. The hydrogen bonding is found to cause significant variations in the one‐bond SSCCs, mostly due to contribution from electronic interactions. However, the nature of variation depends on the type of oxygen atom (proton‐acceptor or proton‐donor) present in the interaction. Two‐bond intermolecular coupling constants vary more than the corresponding one‐bond constants when the size of the cluster increases. Among the four Ramsey terms that constitute the total SSCC, the FC term is the most dominant contributor followed by the paramagnetic spin‐orbit term in all one‐bond interaction.     

Semiempirical spin–orbit coupling calculations. I. Theory and method. Benzophenone as a test case

The molecular spin–orbit coupling operator is brought into a simplified form through a convenient choice of origin for the orbital angular momentum operator. The eigenvalue problem of the Hamiltonian that includes the spin–orbit (SOC ) operator as a perturbation is solved by means of a linear variational procedure in the basis of the spin-pure molecular eigenstates. Test calculations on benzophenone are presented and the results are compared to experiment. We discuss the minimal size of the spin-pure variational basis needed to achieve stable results as well as the amount of single-excitation configurational mixing needed to describe the spin-pure molecular eigenstates.     

Relativistic effects in the one‐bond spin–spin coupling constants involving selenium

One‐bond spin–spin coupling constants involving selenium of seven different types, ¹ J(Se,X), X = ¹H, ¹³C, ¹⁵ N, ¹⁹ F, ²⁹Si, ³¹P, and ⁷⁷Se, were calculated in the series of 14 representative compounds at the SOPPA(CCSD) level taking into account relativistic corrections evaluated both at the RPA and DFT levels of theory in comparison with experiment. Relativistic corrections were found to play a major role in the calculation of ¹ J(Se,X) reaching as much as almost 170% of the total value of ¹ J(Se,Se) and up to 60–70% for the rest of ¹ J(Se,X). Scalar relativistic effects (Darwin and mass‐velocity corrections) by far dominate over spin–orbit coupling in the total relativistic effects for all ¹ J(Se,X). Taking into account relativistic corrections at both random phase approximation and density functional theory levels essentially improves the agreement of theoretical results with experiment. The most ‘relativistic’ ¹ J(Se,Se) demonstrates a marked Karplus‐type dihedral angle dependence with respect to the mutual orientation of the selenium lone pairs providing a powerful tool for stereochemical analysis of selenoorganic compounds. Copyright © 2014 John Wiley & Sons, Ltd.     

A full‐pivoting algorithm for the Cholesky decomposition of two‐electron repulsion and spin‐orbit coupling integrals

A significant reduction in the computational effort for the evaluation of the electronic repulsion integrals (ERI) in ab initio quantum chemistry calculations is obtained by using Cholesky decomposition (CD), a numerical procedure that can remove the zero or small eigenvalues of the ERI positive (semi)definite matrix, while avoiding the calculation of the entire matrix. Conversely, due to its antisymmetric character, CD cannot be directly applied to the matrix representation of the spatial part of the two‐electron spin‐orbit coupling (2e‐SOC) integrals. Here, we present a computational strategy to achieve a Cholesky representation of the spatial part of the 2e‐SOC integrals, and propose a new efficient CD algorithm for both ERI and 2e‐SOC integrals. The proposed algorithm differs from previous CD implementations by the extensive use of a full‐pivoting design, which allows a univocal definition of the Cholesky basis, once the CD δ threshold is made explicit. We show that is the upper limit for the errors affecting the reconstructed 2e‐SOC integrals. The proposed strategy was implemented in the ab initio program Computational Emulator of Rare Earth Systems (CERES), and tested for computational performance on both the ERI and 2e‐SOC integrals evaluation. © 2017 Wiley Periodicals, Inc.     

Two‐component calculations of spin–orbit effects for a van der Waals molecule Rn2

Electronic structures of the weakly bound Rn₂ were calculated by the two‐component Møller–Plesset second‐order perturbation and coupled‐cluster methods with relativistic effective core potentials including spin–orbit operators. The calculated spin–orbit effects are small, but depend strongly on the size of basis sets and the amount of electron correlations. Magnitudes of spin–orbit effects on D_e (0.7–3.0 meV) and R_e (−0.4∼−2.2 Å) of Rn₂ are comparable to previously reported values based on configuration interaction calculations. A two‐component approach seems to be a promising tool to investigate spin–orbit effects for the weak‐bonded systems containing heavy elements. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 139–143, 1999     

Nuclear spin–spin coupling and nuclear motion

The vibration and rotation of molecules affects nuclear spin–spin coupling constants. This manifests itself as a temperature dependence of the coupling and also as an isotope effect (after allowing, where necessary, for differing magnetogyric ratios of the two nuclei involved in the isotopic substitution). Within the Born–Oppenheimer approximation, a nuclear spin–spin coupling surface can be defined for each pair of coupled nuclei. This surface is sampled by the nuclei as they undergo the excursions about equilibrium geometry that are governed by the force field. An accurate ab initio carbon–proton spin–spin coupling surface for the methane molecule has been calculated. This was obtained by summing the surfaces for each of the four contributions—Fermi contact, spin–dipolar, orbital paramagnetic, and orbital diamagnetic—expressed as power series in terms of symmetry coordinates. Preliminary calculations for ¹³CH₄ and ¹³CD₄ give a difference of only 6% between the calculated and observed nuclear motion contributions. The observed temperature dependence is also accounted for by the calculations. For these isotopomers, bond stretching plays the dominant role. © 1994 John Wiley & Sons, Inc.     

Through‐space 19F–19F spin–spin coupling in ortho‐fluoro Z‐azobenzene

We report through‐space (TS) ¹⁹F–¹⁹F coupling for ortho‐fluoro‐substituted Z ‐azobenzenes. The magnitude of the TS‐coupling constant (^TSJ_FF) ranged from 2.2–5.9 Hz. Using empirical formulas reported in the literature, these coupling constants correspond to non‐bonded F–F distances (d_FF) of 3.0–3.5 Å. These non‐bonded distances are significantly smaller than those determined by X‐ray crystallography or density functional theory, which argues that simple models of ¹⁹F–¹⁹F TS spin–spin coupling solely based d_FF are not applicable. ¹H, ¹³C and ¹⁹F data are reported for both the E and Z isomers of ten fluorinated azobenzenes. Density functional theory [B3YLP/6‐311++G(d,p)] was used to calculate ¹⁹F chemical shifts, and the calculated values deviated 0.3–10.0 ppm compared with experimental values. Copyright © 2015 John Wiley & Sons, Ltd.     

Spin–other–orbit and spin–orbit interactions in ƒ2 and ƒ3 electron configurations

Expressions of the matrix elements of the spin–other–orbit and spin–orbit interactions for the various multiplets of all the states of ?²- and ?³-electron configurations are reported and used to evaluate the Hartree–Fock values of these interactions in the neutral atoms Ce(4?²), Pr(4?³), Ho(4?¹¹) and Er(4?¹²). The required values of the spin–spin parameters M, and the spin-orbit parameter ζ for these atoms were obtained using numerical Hartree–Fock wave functions.