Quantifying the symmetry content of the electronic structure of molecules: molecular orbitals and the wave function

A scheme to quantify the symmetry content of the electronic wave function and molecular orbitals for arbitrary molecules is developed within the formalism of Continuous Symmetry Measures (CSMs). After defining the symmetry operation expectation values (SOEVs) as the key quantity to gauge the symmetry content of molecular wavefunctions, we present the working equations to be implemented in order to carry out real calculations using standard quantum chemistry software. The potentialities of a symmetry analysis using this new method are shown by means of some illustrative examples such as the changes induced in the molecular orbitals of a diatomic molecule by an electronegativity perturbation, the breaking of orbital symmetry along the dissociation path of the H(2) molecule, the changes in the molecular orbitals upon a geometrical distortion of the benzene molecule, and the inversion symmetry content in the different spin states of the [Fe(CH(3))(4)](2-) complex.     

Mechanism on two-electron oxidation of ubiquinol at the Qp site in cytochrome bc1 complex: B3LYP study with broken symmetry

The molecular and electronic structures of the Rieske iron-sulfur [2Fe-2S] cluster with an imidazolate and imidazole were investigated by using usual unrestricted and broken symmetry B3LYP methods for the highest and lowest spin states, respectively. The electronic structures of the lowest spin states were determined by the spin contaminations and natural orbital analyses. It was shown that the spin contamination presents the number of pairs of the antiferromagnetic spin couplings. The oxidation mechanism of the ubquinol at the Q(p) site of the cytochrome bc(1) complex was also examined by the broken symmetry B3LYP methods. In the [2Fe-2S] clusters with an imidazolate, the oxidized and lowest spin state, [(Imz(-))FeS](ox)LS, was lowest in energy among four possible states, consistent with experimental observations. In the examination of the mechanism of the ubquinol oxidation, it was confirmed that the ubiquinol docks between the imidazolate of [2Fe-2S] clusters and Glu272(-) of cytochrome b by the hydrogen bonds before the oxidation proceeds, consistent with the experimental proposals. Our results support a "Glu272-first" mechanism that Glu272 serves as an acceptor of the first proton from the ubiquinol and subsequently the proton-coupled electron transfer (PCET) occurs from the ubisemiquinone anion to the Rieske iron-sulfur [2Fe-2S] cluster.     

On the triplet ground state of tetrahedral X4 clusters (X = Li, Na, K, Cu)

The lowest electronic state of distorted tetrahedral X(4) clusters (with X = Li, Na, K, Cu) is studied at coupled-cluster level using high-quality atomic basis sets. The ground state is found to have a triplet spin symmetry for this kind of geometry and for all the considered atomic species. The equilibrium geometries correspond to Jahn-Teller-distorted oblate tetrahedra having D(2d) symmetry, and tetrahedric structures are local minima on the potential-energy surfaces for the triplet states. Their energies lie between 0.2 eV (for the K(4) cluster) and 0.9 eV (for Cu(4)) above the absolute minimum of the corresponding systems, which is a spin singlet having a rhombus geometry.     

Structural and electronic properties of graphene nanotube-nanoribbon hybrids

The structural and electronic properties of a hybrid of an armchair graphene nanotube and a zigzag graphene nanoribbon are investigated by first-principles spin-polarized calculations. These properties strongly depend either on the nanotube location or on the spin orientation. The interlayer spacing, the transverse distance from the center of the ribbon and the stacking configuration affect the electronic structures. The antiferromagnetic configuration has a lower total energy than the ferromagnetic one. The interlayer atomic interactions between the two subsystems would change the low energy dispersions, open subband spacings, and induce more band-edge states. Moreover, such interactions create an energy gap and break the spin degeneracy in the antiferromagnetic configuration. The band-edge-state energies are sensitive to the nanotube location.     

A unified view of the theoretical description of magnetic coupling in molecular chemistry and solid state physics

The magnetic interactions in organic diradicals, dinuclear inorganic complexes and ionic solids are presented from a unified point of view. Effective Hamiltonian theory is revised to show that, for a given system, it permits the definition of a general, unbiased, spin model Hamiltonian. Mapping procedures are described which in most cases permit one to extract the relevant magnetic coupling constants from ab initio calculations of the energies of the pertinent electronic states. Density functional theory calculations within the broken symmetry approach are critically revised showing the contradictions of this procedure when applied to molecules and solids without the guidelines of the appropriate mapping. These concepts are illustrated by describing the application of state-of-the-art methods of electronic structure calculations to a series of representative molecular and solid state systems.     

Ab initio spin-orbit calculations on the lowest states of the nickel dimer

Potential energy curves of the lowest electronic states of the Ni(2) dimer are calculated near the equilibrium using the multireference ab initio methods including the spin-orbit interaction. Scalar-relativistic results fully confirm previous qualitative interpretations based on the correlation with atomic limits and the symmetry of vacancies in the atomic 3d(9) shells. Spin-orbit calculations firmly establish the symmetry of the ground state as 0(+)(g) and give the excitation energies 70 ± 30 cm(-1) and 200 ± 80 cm(-1) for the lowest 0(-)(u) and 5(u) states, respectively. The model electronic spectrum of the Ni(2) shows some trends that might be observed in matrix isolation far-infrared and electron spin resonance spectra.     

Analogies between Jahn–Teller and Rashba spin physics

In developing physical theories, analogical reasoning has been found to be very powerful, as attested by a number of important historical examples. An analogy between two apparently different phenomena, once established, allows one to transfer information and bring new concepts from one phenomenon to the other. Here, we discuss an important analogy between two widely different physical problems, namely, the Jahn–Teller distortion in molecular physics and the Rashba spin splitting in condensed matter physics. By exploring their conceptual and mathematical features and by searching for the counterparts between them, we examine the orbital texture in Jahn–Teller systems, as the counterpart of the spin texture of the Rashba physics, and put forward a possible way of experimentally detecting the orbital texture. Finally, we discuss the analogy by comparing the coexistence of linear Rashba + Dresselhaus effects and Jahn–Teller problems for specific symmetries, which allow for nontrivial spin and orbital textures, respectively.     

On the photomagnetism of nitronyl nitroxide, imino nitroxide, and verdazyl-substituted azobenzene

The cis- and trans-azobenzenes are known as photochromic isomers with the trans- converting into the cis-form and vice versa upon irradiation with specific wavelengths. We have quantum chemically investigated the cis- and trans-forms of substituted azobenzene diradicals, with two nitronyl nitroxides, imino nitoxides, or verdazyls at para positions and serving as monoradical centers, to determine whether they can exhibit a photoassisted magnetic crossover. Geometries of both substituted and unsubstituted molecules have been optimized by density functional (DF) method UB3LYP using the 6-311G(d,p) basis set. Optimization of the geometry of the cis isomers has required special care. Single point singlet, triplet, and broken symmetry calculations have been done using 6-311++G(3df, 3pd) basis set. The magnetic exchange coupling constants have been estimated from the broken symmetry calculations. Absorption wavelengths have been estimated for both substituted and unsubstituted species from time-dependent DF treatment using restricted spin-polarized methodology RB3LYP and 6-311++G(3df, 3pd) basis set. From the similarity in the calculated absorption wavelengths for the unsubstituted and substituted azobenzenes, and the increased oscillator strengths (f) for the substituted species, we predict that the diradical isomers would be strongly photochromic. From our triplet state and broken symmetry calculations, we predict that both the cis- and the trans-diradicals are antiferromagnetically coupled. This prediction is consistent with the spin alternation rule, and the possibility of a magnetic crossover is nonexistent for these species.     

Electronic structure investigations of quasicrystals

We present a review of the determination of density of states (DOS) of quasicrystals using valence band photoemission spectroscopy. The absence of fine or spiky structure in the angle-integrated DOS of quasicrystals suggests the possibility of delocalized electronic states. These were confirmed with angle-resolved photoemission studies, which clearly establish the presence of dispersing features attributed to momentum-dependent bandstructure. Such dispersing states are observed not only for deeper-lying sp states, but also for d-derived bands near the Fermi level. Data from three different high symmetry surfaces of decagonal Al–Ni–Co, an ideal model system, are presented. We find that only a few dominant reciprocal lattice vectors are sufficient to describe the quasiperiodic potential, and the implications for electronic properties are discussed.     

α-SrMnO3电子结构的第一性原理研究

采用平面波赝势方法对钙钛矿型锰酸盐氧化合物α-SrMnO3的电子结构进行了第一性原理研究. 六方钙钛矿型结构α-SrMnO3化合物为磁性绝缘体, 磁基态对应于共面八面体及共顶点八面体间的磁性交换作用均为反铁磁性(AFM), 其禁带宽度为1.6 eV; 费米能级附近的Mn3d态与O2p态存在很强的杂化作用, 属于共价绝缘体, 这种强共价性使得Mn4+的自旋磁矩偏离理想值. 采用Noodleman的对称性破缺方法, 根据α-SrMnO3不同磁有序态的总能量拟合出α-SrMnO3中的自旋交换耦合常数. α-SrMnO3的局部微结构(Mn—O—Mn)决定了整个体系的特殊磁性交换作用. 共面及共顶点的八面体间均存在AFM交换作用, 并且共顶点八面体间的AFM作用比较强.     

Orbital forms of gap equations and instability conditions

Previously derived forms of instability conditions at the spin orbital level are specialized to orbital levels of different kinds. Similarly the special forms of the gap equation that are obtained for different types of broken symmetry solutions are analyzed and discussed.Supported by the Swedish Natural Sciences Research Council.     

Spin—orbit and vibronic perturbations on the chiroptical spectra of dissymmetric pseudo-tetragonal metal complexes

The influence of spin—orbit and vibronic interactions upon the chiroptical properties of nearly degenerate dd transitions in metal complexes of pseudo-tetragonal symmetry is investigated. A model system is considered in which three nearly degenerate dd excited states are coupled via both spinorbit and vibronic interactions. Vibronic interactions among the three nearly degenerate dd electronic states are assumed to arise from a pseudo-Jahn—Teller (PJT) mechanism involving three different vibrational modes (each nontotally symmetric in the point group of the undistorted model system).A vibronic hamiltonian is constructed (for the excited states of the model system) which includes linear coupling terms in each of the three PJT-active vibrational modes as well as a linear coupling term in one totally symmetric mode of the system and a spin—orbit interaction term. Wavefunctions and eigenvalues for the spin—orbit/vibronic perturbed excited states. of the model system are obtained by diagonalizing this hamiltonian in a basis constructed of uncoupled vibrational and electronic (orbital and spin) wavefunctions.Rotatory strengths associated with transitions to vibronic levels of the perturbed system are calculated and “rotatory strength spectra” are computed assuming gaussian shaped vibronic spectral components. Calculations are carried out for a number of vibronic and spin—orbit coupling parameters and for various splitting energies between the interacting electronic states. The calculated results suggest that chiroptical spectra associated with transitions to a set of nearly degenerate dd excited states of a chiral transition metal complex cannot be interpreted directly without some consideration of the effects introduced by spin—orbit and vibronic perturbations. These perturbations can lead to substantial alterations in the sign patterns and intensity distributions of rotatory strength among vibronic levels derived from the interacting electronic states and it is generally not valid to assign specific features in the observed circular dichroism spectra to transitions between states with well-defined electronic (orbital and spin) identities.Our theoretical model is conservative with respect to the total (or net) rotatory strength associated with transitions to levels derived from the three interacting electronic states; the vibronic and spin—orbit coupling operators are operative only within this set of states. That is, the total (or net) rotatory strength associated with these transitions remains invariant to the vibronic and spin—orbit coupling parameters of the model.     

Electronic structure and magnetism at the active site in ferredoxin: Ab initio approach to (Fe2S2)2+ complex with the 1st peptide shell

We have studied the iron–sulfur cluster systems which model an active site of ferredoxin proteins by using the first-principles electronic structure calculation. The modeled molecule is a complex between the (Fe₂S₂)²⁺ core and the amino acid residues which surround the core. The electronic structure of oxidized state for the molecules is presented. The antiferromagentic arrangement for Fe atomic magnetizations was obtained as the ground state. The spin polarized state of the half-filled Fe 3d orbitals is consistent with the formal valence of Fe³⁺. The induced spin density on the cysteine S atoms was found to be parallel to the direction of magnetization on the nearest Fe atom. The hybridized states consisting of N 2p and C 2p orbitals at the side chain of Arg residue appeared just above the highest occupied molecular orbital level for the free standing peptide.     

The any particle molecular orbital approach: A short review of the theory and applications

The any particle molecular orbital (APMO) approach extends regular electronic structure methods to study atomic and molecular systems in which electrons and other particles are treated simultaneously as quantum waves. A number of electronic structure methodologies have been extended under the APMO framework and applied to investigate nuclear quantum effects including isotope effects and nuclear delocalization and to calculate proton binding energies and affinities. In addition, APMO methodologies have been employed to analyze physical and chemical properties of atomic and molecular systems containing exotic subatomic particles.     

On the identification of symmetry‐forbidden spin subspaces for configurations employing nonorthogonal orbitals

The implications of orbital symmetry for a configuration's spin function are considered from the fairly general viewpoint required for multiconfiguration wave functions employing nonorthogonal orbitals. A general procedure is derived for the imposition of the spin constraints that may be necessary to ensure a given configuration, or a set of configurations as a whole, exhibit required symmetry properties, thus preventing symmetry contamination of the overall electronic wave function. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 24–31, 2000     

A re-evaluation of the two-step spin crossover in the trinuclear cation [Co3(dipyridylamido)4Cl2]+

Density functional theory is used to explore the electronic states involved in the remarkable two-step spin crossover (S = 0 --> S = 1 --> S = 2) in the cationic extended metal atom chain [Co(3)(dpa)(4)Cl(2)](+) (dpa = the anion of 2-dipyridylamine) (R. Clérac, F. A. Cotton, K. R. Dunbar, T. Lu, C. A. Murillo and X. Wang, J. Am. Chem. Soc., 2000, 122, 2272). The calculations are consistent with a model in which all three spin states share one common feature-a vacancy in the d(xy) orbital on the central cobalt atom which is stabilised by pi donation from four amide groups. As a result, all three can be considered to contain a Co(2+)-Co(3+)-Co(2+) chain. The singlet and triplet states arise from antiferromagnetic and ferromagnetic coupling, respectively, between the unpaired electron in this d(xy) orbital and another localised entirely on the terminal cobalt centres (the antisymmetric combination of Co d(z(2))). The singlet-triplet transition does not, therefore, populate any additional antibonding orbitals, and as a result the structure is almost invariant around the characteristic temperature of the singlet-triplet transition. In the most stable quintet, in contrast, the symmetry of the Co-Co-Co chain is broken, giving rise to a localised high-spin Co(II) centre (S = 3/2), ferromagnetically coupled to a Co(III)-Co(II) dimer (S = 1/2). The structural changes associated with this transition are apparent in the X-ray data in subtle changes in both Co-N and Co-Cl bond lengths, although their magnitude is damped by the relatively low population (18%) of the quintet even at 300 K.     

Capture of polarized electrons into excited terms of fast atoms during grazing surface collisions

Fast ions are scattered from magnetized surfaces under grazing angles of incidence. During the interaction with the surface charge exchange is effective and results in a population of stable and excited atomic terms. This capture of electrons is characterized by anisotropic distributions of atomic orbital angular momenta and in addition — for magnetized targets — by anisotropic distributions of electronic spins. We will discuss in some detail, how these anisotropic distributions can be studied via the analysis of the state of polarization of the fluorescent light, emitted in electronic transitions from excited terms of free atoms after the impact with the surface. We show that a defined variation of the magnetization of the target affects the polarization of the emitted light in a characteristic way, which allows to deduce the electronic spin polarization of the atoms. The method implies some perspectives with respect to the study of magnetic properties of the vacuum-solid interface well above the topmost layer of surface atoms.     

Orbital symmetry,orbital stability,and orbital pairing rules for organic reactions in the ground state

A generalization of the Hartree–Fock molecular orbital (MO) theory for treating diradical intermediates was explained pictorially by drawing molecular orbitals of diradical species such as ring-opened trimethylene. The generalized MO theory applied to elucidate electronic mechanisms of concerted, ionic, radical, and ion-radical reactions of organic reactants in the ground state. Generalized MO computations revealed the most essential characteristics of these reactions and mutal relationships between the worlds of Woodward–Hoffmann and Hughes–Ingold. Generalized MO studies supported our orbital symmetry, stability and pairing rules for concerted, ionic and radical reactions in the ground state, respectively. An extension of MO treatments to excited states reactions was briefly pointed out in relation to the density and spin correlation functions by the multireference CI wave functions.     

Effect of metal-oxygen covalent bonding on the competition between Jahn-Teller distortion and charge disproportionation in the perovskites of high-spin d(4) metal ions LaMnO(3) and CaFeO(3)

The perovskites LaMnO(3) and CaFeO(3) consisting of high-spin d(4) transition metal ions undergo different types of distortions, i.e., a Jahn-Teller distortion in LaMnO(3) and a charge disproportionation in CaFeO(3). We investigated the electronic factor causing this difference on the basis of first principles spin-polarized electronic band structure calculations for their ideal cubic structures and also tight-binding electronic band structure calculations for their ideal cubic and distorted structures. Our study shows that a charge disproportionation is favored over a Jahn-Teller distortion in CaFeO(3) because the covalent character is strong in the Fe-O bond, while the opposite is true for LaMnO(3) because the covalent character is weak in the Mn-O bond. In spin-polarized electronic band structure calculations, the covalency of the M-O (M = Fe, Mn) bond is enhanced in the up-spin bands but is reduced in the down-spin bands. Our analysis shows that electron-electron repulsion causes the energy gap between the metal 3d and the oxygen 2p orbitals to become larger for the down-spin than for the up-spin-orbital interactions. Thus in the d-block e(g) bands of both LaMnO(3) and CaFeO(3) the metal 3d orbital contribution is larger in the down-spin than in the up-spin bands.     

Spin-orbit effects on a gold-based superatom: a relativistic Jellium model

The inclusion of relativistic effects always brings to the scientific community great and stimulating surprises. To consider the spin-orbit term, which accounts for the interaction between the spatial and spin coordinates, requires the use of double point groups of symmetry in order to solve the Dirac equation or the two component approximation to it, leading to total angular momenta (j) functions, atomic or molecular spinors, instead of pure orbital angular momenta (l), atomic or molecular orbitals. Large and small components, derived from the Dirac treatment, depict wavefunctions corresponding to fermions, electrons, which are described for the first time for a superatom case. In addition, their behavior is revisited in order to clarify the effects of the inclusion of the spin-orbit coupling into the electronic structure calculations, which can be extended to other superatoms, clusters, molecules and atoms.