Group Theoretical Treatment Research Articles

Jahn-Teller states mixed by spin-orbit coupling in an electromagnetic field

Spin-orbit coupling plays a pivotal role in condensed matter physics. For instance, spin-orbit interactions affect the magnetization and transport dynamics in solids, while spins and momenta are locked in topological matter. Alternatively, spin-orbit entanglement may play an important role in exotic phenomena, like quantum spin liquids in 4d and 5d systems. An interesting question is how electronic states mixed by spin orbit coupling interact with electromagnetic fields, which may hold potential to tune their properties and reveal interesting physics. Motivated by our recent discovery of large gyrotropic signals in some Jahn-Teller manganites, here we explore the interaction of light with spin-mixed states in a d4 transition metal. We show that spin-orbit mixing enables electronic transitions that are sensitive to circularly polarized light, giving rise to a gyrotropic response that increases with spin-orbit coupling. Interestingly, photoexcited transitions that involve spin reversal are behind such gyrotropic resonances. Additionally, we find that the interaction with the electromagnetic field depends strongly on the relative orientation of the propagation of light with respect to Jahn-Teller distortions and spin quantization. We suggest that such interactions offer the opportunity to use electromagnetic waves at optical wavelengths to entangle orbital and spin degrees of freedom. Our approach, which includes a group-theoretical treatment of spin-orbit coupling, has wide applicability and provides a versatile tool to explore the interaction of electromagnetic fields with electronic states in transition metals with arbitrary spin-orbit coupling strength and pointgroup symmetries.

Practical treatment of singlet oxygen with density-functional theory and the multiplet-sum method

Singlet oxygen (O2) comes in two flavors -- namely the dominant lower-energy a 1 Delta g state and the higher-energy shorter-lived b 1 Sigma + g state -- and plays a key role in many photochemical and photobiological reactions. For this reason, and because of the large size of the systems treated, many papers have appeared with density-functional theory (DFT) treatments of the reactions of 1 O 2 with different chemical species. The present work serves as a reminder that the common assumption that it is enough to fix the spin multipicity as unity is not enough to insure a correct treatment of singlet oxygen. We review the correct group theoretical treatment of the three lowest energy electronic states of O 2 which, in the case of 1 O 2 is often so badly explained in the relevant photochemical literature that the explanation borders on being incorrect and prevents, rather than encourages, a correct treatment of this interesting and important photochemical species. We then show how many electronic structure programs, such as a freely downloadable and personal-computer compatible Linux version of deMon2k, may be used, together with the multiplet sum method (MSM), to obtain a more accurate estimation of the potential energy curves (PECs) of the two 1 O 2 states. Various strengths and weaknesses of different DFAs emerge through our application of the MSM method. In particular, the quality of the a 1 Delta g excitation energy reflects how well functionals are able to describe the spin-flip energy in DFT while the quality of the b 1 Sigma + g excitation energy reflects how well functionals are able to describe the spin-pairing energy in DFT. Finally we note that improvements in DFT-based excited-state methods will be needed to describe the full PECs of 1 O 2 including both the equilibrium bond lengths and dissociation behavior.

The unitary representations of the Poincar\'e group in any spacetime dimension

An extensive group-theoretical treatment of linear relativistic field equations on Minkowski spacetime of arbitrary dimension D\geqslant 3D≥3 is presented. An exhaustive treatment is performed of the two most important classes of unitary irreducible representations of the Poincar'e group, corresponding to massive and massless fundamental particles. Covariant field equations are given for each unitary irreducible representation of the Poincar'e group with non-negative mass-squared.

Large amplitude inversion tunneling motion in ammonia, methylamine, hydrazine, and secondary amines: From structure determination to coordination chemistry

• Molecular structures of ammonia derivatives captured by spectroscopy. • Tunneling motion as essential information to understand coordination complexes. • Structures and internal dynamics from experimental and theoretical points of view. • Using the structures of isolated ligand to determine the structure of complexes. Inversion tunneling in a symmetric double minimum potential is a challenging large amplitude motion problem which causes all rotational energy levels to split into a symmetric and an antisymmetric level. Not like other types of large amplitude motion such as internal rotation, inversion tunneling appears much rarer, but the fundamental knowledge gained from this motion is essential to understand the complex structures in coordination chemistry. A double minimum potential needed for inversion tunneling requires the symmetry of the frame to which the inversion object is attached to be C s or higher, while there is no restriction on the symmetry of the frame for internal rotation. This review summarizes four most classic examples of molecules featuring inversion tunneling motion and their coordination complexes. The textbook example, ammonia, with its umbrella motion during which the nitrogen atom passes from one side of the plane formed by the three hydrogens to the other, will be presented with information on its group theoretical considerations, theoretical studies, electronic ground state spectra in the infrared range, and its coordination complexes, especially those formed with calcium and copper. In the second classic example, methylamine, CH 3 NH 2 , the inversion motion of the amino group –NH 2 is coupled with a methyl internal rotation –CH 3 . The spectrum of methylamine can be treated using a tunneling formalism applying the G 12 permutation-inversion symmetry group. Extensive studies on the ground state, the first and second excited torsional states ν 15 , the inversion wagging state ν 9 , and the C-N stretching band ν 8 exist along with their combinations. Pertubations and blends have made the analysis of methylamine very challenging. The third subject is on hydrazine with complex internal dynamics governed by three large amplitude motions: two –NH 2 tunneling motions and an internal rotation (torsion) of the two amino groups around the N-N bond. Regarding the spectrum of hydrazine, the vibrational ground state lies in the microwave region. The first, second, and third excited torsional band ν 7 , symmetric wagging ν 6 , and asymmetric wagging ν 12 are found in the infrared range. Group theoretical treatment and tunneling formalism can only be used to fit sub-bands individually, while global fits still remain a difficult task. Metal complexes with hydrazine and methylamine are introduced, proving the importance of spectroscopic understanding of molecular structures for the knowledge of coordination chemistry. Finally, secondary amines feature inversion tunneling of the hydrogen atom attached to the nitrogen, which is accompanied by two methyl internal rotations but not coupled with them so that they can be treated separately. For this class of molecules, only spectra of the vibrational ground state in the microwave domain will be considered.

Contributions to polarization and polarization switching in antiphase boundaries of SrTiO3 and PbZrO3

We use a recently developed method—based on layer group analysis combined with the Landau theory—to investigate the polar properties of antiphase boundaries (APBs) in SrTiO3 and PbZrO3. For SrTiO3, we find that, in addition to the biquadratic, Houchmandazeh-Laizerowicz-Salje (HLS) coupling bijklPiPjϕkϕl in the Landau-Ginzburg free energy expansion, additional rotopolar terms of the form WijklPiϕk∂ϕl∂xj contribute considerably to the polarization of antiphase boundaries in these materials. The rotopolar terms can be split into a symmetric flexoelectric part and an antisymmetric one. The antisymmetric Lifshitz term leads to a macroscopic polarization of APBs, which can be switched by application of an external electric field. For PbZrO3, the observed polarization profiles [Wei et al., Mater. Res. Bull. 62, 101 (2015)] are fully compatible with the symmetries of the corresponding layer groups. Unlike in SrTiO3, there exists no Lifshitz invariant WijklPiηk∂ηl∂xj for the order parameter ηi(i=1,…,12) describing the displacements of lead atoms. However, a detailed group theoretical treatment indicates that the polarity of APBs in PbZrO3 is driven by higher order interactions between polarization Pi, order parameter ηk, and order parameter gradients ∂ηl∂xj.

Morin-type transition in 5C pyrrhotite

Abstract We report the discovery of a low-temperature spin-flop transition in 5C pyrrhotite at ~155 K that is similar to those seen in hematite at 260 K and FeS (troilite) at 440 K. The 5C crystal was produced by annealing a 4C pyrrhotite crystal at 875 K to produce a change in the vacancy-ordering scheme that developed during cooling. The 5C structure is confirmed by single-crystal X-ray diffraction and the stoichiometry and homogeneity by electron microprobe and SEM BSE mapping. Resonant ultrasound spectroscopy (RUS), heat capacity, and magnetization measurements from room temperature down to 2 K are reported. The transition is marked by a steep change in elastic properties at the transition temperature, a peak in the heat capacity, and weak anomalies in measurements of magnetization. Magnetic hysteresis loops and comparison with the magnetic properties of 4C pyrrhotite suggest that the transition involves a change in orientation of moments between two different antiferromagnetic structures, perpendicular to the crystallographic c-axis at high temperatures and parallel to the crystal-lographic c-axis at low temperatures. The proposed structures are consistent with a group theoretical treatment that also predicts a first-order transition between the magnetic structures.

Reduced dimension rovibrational variational calculations of the S1 state of C2H2. I. Methodology and implementation

The bending and torsional degrees of freedom in S1 acetylene, C2H2, are subject to strong vibrational resonances and rovibrational interactions, which create complex vibrational polyad structures even at low energy. As the internal energy approaches that of the barrier to cis-trans isomerization, these energy level patterns undergo further large-scale reorganization that cannot be satisfactorily treated by traditional models tied to local minima of the potential energy surface for nuclear motion. Experimental spectra in the region near the cis-trans transition state have revealed these complicated new patterns. In order to understand near-barrier spectroscopic observations and to predict the detailed effects of cis-trans isomerization on the rovibrational energy level structure, we have performed reduced dimension rovibrational variational calculations of the S1 state. In this paper, we present the methodological details, several of which require special care. Our calculation uses a high accuracy ab initio potential surface and a fully symmetrized extended complete nuclear permutation inversion group theoretical treatment of a multivalued internal coordinate system that is appropriate for large amplitude bending and torsional motions. We also discuss the details of the rovibrational basis functions and their symmetrization, as well as the use of a constrained reduced dimension rovibrational kinetic energy operator.

Torsion–Inversion Tunneling Patterns in the CH-Stretch Vibrationally Excited States of the G12 Family of Molecules Including Methylamine

Two torsion-inversion tunneling models (models I and II) are reported for the CH-stretch vibrationally excited states in the G12 family of molecules. The torsion and inversion tunneling parameters, h(2v) and h(3v), respectively, are combined with low-order coupling terms involving the CH-stretch vibrations. Model I is a group theoretical treatment starting from the symmetric rotor methyl CH-stretch vibrations; model II is an internal coordinate model including the local-local CH-stretch coupling. Each model yields predicted torsion-inversion tunneling patterns of the four symmetry species, A, B, E1, and E2, in the CH-stretch excited states. Although the predicted tunneling patterns for the symmetric CH-stretch excited state are the same as for the ground state, inverted tunneling patterns are predicted for the asymmetric CH-stretches. The qualitative tunneling patterns predicted are independent of the model type and of the particular coupling terms considered. In model I, the magnitudes of the tunneling splittings in the two asymmetric CH-stretch excited states are equal to half of that in the ground state, but in model II, they differ when the tunneling rate is fast. The model predictions are compared across the series of molecules methanol, methylamine, 2-methylmalonaldehyde, and 5-methyltropolone and to the available experimental data.

Crystallization in the Three-Component Systems Rb<sub>2</sub>SO<sub>4-</sub>MSO<sub>4-</sub>H<sub>2</sub>O (M = Mg, Co, Ni, Cu, Zn) at 298 K

The crystallization in the three-component systems Rb2SO4-MSO4-H2O (M = Mg, Co, Ni, Cu, Zn) is studied by the method of isothermal decrease of supersaturation. It has been established that isostructural double compounds, Rb2M(SO4)2·6H2O (M = Mg, Co, Ni, Cu, Zn), , crystallize from the ternary solutions within wide concentration ranges. The infrared spectra are discussed with respect to the normal vibrations of the sulfate ions and water molecules. The unit-cell group theoretical treatment of the double salts is presented. The extent of energetic distortions of guest ions (about 2 mol%) matrix-isolated in the respective selenates, (M' = K, Rb, ; M" = Mg, Co, Ni, Cu, Zn), is commented.

Quantum Rabi Model forN-State Atoms

A tractable N-state Rabi Hamiltonian is introduced by extending the parity symmetry of the two-state model. The single-mode case provides a few-parameter description of a novel class of periodic systems, predicting that the ground state of certain four-state atom-cavity systems will undergo parity change at strong-coupling. A group-theoretical treatment provides physical insight into dynamics and a modified rotating wave approximation obtains accurate analytical energies. The dissipative case can be applied to study excitation energy transfer in molecular rings or chains.

Vacuum-type space-like string surfaces in AdS3 × S3

We construct and classify all space-like minimal surfaces in AdS3 × S3 which globally admit coordinates with constant-induced metric on both factors. Up to O(2, 2) × O(4) transformations all these surfaces, except one class, are parameterized by four real parameters. The classes of surfaces correspond to different regions in this parameter space and show quite different boundary behavior. Our analysis uses a direct construction of the string coordinates via a group theoretical treatment based on the map of AdS3 × S3 to . This is complemented by a cross check via standard Pohlmeyer reduction. After embedding in AdS5 × S5 we calculate the regularized area for solutions with a boundary spanned by a 4-point scattering s-channel momenta configuration.

Studies on X-ray Thomson Scattering from Antiferroquadrupolar Order in TmTe

We study Thomson scattering from the antiferroquadrupole ordering phase in TmTe. On the basis of the group theoretical treatment, we classify the selection rules of the scattering intensity governed by the orientation of the scattering vector G. Then, numerical verification is performed by invoking the ground states which are deduced from a J=7/2 multiplet model. The obtained intensity varies drastically depending on the magnitude and direction of G. We also calculate the scattering intensities under the applied field for H//(001) and (110). Their results behave differently when the orientation of G is changed, which is ascribed to the difference of their primary order parameters; O_{2}^{0} and O_{2}^{2} for H // (001) and (110), respectively. We make critical comparisons between our results for TmTe and the experimental ones for CeB_6. First, we assert that the intensities expected from TmTe at several forbidden Bragg spots are sufficient enough to be experimentally detected. Second, their intensities at (7/2,1/2,1/2) differ significantly and may be attributed to the difference of the order parametersbetween the \Gamma_3-type (O_{2}^{2} and O_{2}^{0}) and \Gamma_5-type (O_{yz}, O_{zx}, and O_{xy}) components, respectively.

Group theoretical treatment of Jahn–Teller versus spin-orbit effects on geometries, rovibronic levels and nuclear spin species of bismuth and antimony clusters

Clusters of very heavy group 15 elements, particularly those of Sb and Bi are intriguing due to the interplay of vibronic and relativistic effects, which quench the Jahn–Teller distortion, and thus resulting in the retention of higher symmetry in the double group. Group theoretical treatment of Bi n clusters and comparisons with Sb n clusters as a function of cluster size and coordination polyhedra has been carried out. The spin-orbit and Jahn–Teller effects have dual roles, thus resulting in interesting rovibronic and nuclear spin species arising from spin-9/2 209Bi. The rovibronic and nuclear spin species of 121Sb and 123Sb are also quite interesting due to their equal relative abundance and high nuclear spin. It has been shown by using group theoretical and combinatorial methods that these clusters exhibit interesting nuclear spin multiplets and distributions of rovibronic levels as a function of polyhedral geometry, which is influenced by the quenching of Jahn–Teller pseudorotation by spin-orbit effects in heavier Bi n clusters. The results are contrasted with Sb n clusters which undergo Jahn–Teller pseudorotations. It is also pointed out how the techniques considered here can be applied to Tl n and In n clusters.

Raman scattering by Hg2F2 polycrystals

The Raman spectra of Hg2F2 polycrystals are experimentally measured for the first time. The spectra of Hg2F2 are interpreted using X-ray diffraction analysis and group-theoretical treatment. The results obtained are discussed in comparison with the spectra of Hg2Hal2 crystals.

A group theoretic treatment of f-electrons in endohedral fullerenes

The most abundant fullerene molecule, C60, has just the right size, chemical stability and activity to host a large variety of atoms. Endohedrally doped fullerenes have been synthesised in the past. A rare earth doped fullerene shows an icosahedral symmetry. A group theoretic treatment of f-electrons in a strong icosahedral crystal field has been discussed in the present study. This method shows a convenience over the conventional structures given by Racah for the treatment of f electrons in strong crystal fields. It is expected that it will make the perturbation calculations easier. These calculations will lead to an understanding of magnetic properties of nano devices.

Jahn-Teller effect in tetrahedral symmetry: Large-amplitude tunneling motion and rovibronic structure of CH4+ and CD4+

The energy level structures of the ground vibronic states of 12CH4+, 13CH4+, and 12CD4+ have been measured by pulsed-field-ionization zero-kinetic-energy photoelectron spectroscopy. The nuclear spin symmetries of the tunneling-rotational levels have been determined in double-resonance experiments via selected rotational levels of the v3=1 and v3=2 vibrational levels of the X 1A1 ground state of CH4. The energy level structures of 12CH4+, 13CH4+, and 12CD4+ have been analyzed with an effective tunneling-rotational Hamiltonian. The analysis together with a group theoretical treatment of the Tx(e+t2) Jahn-Teller effect in the Td(M) group prove that the equilibrium geometry of 12CH4+, 13CH4+, and 12CD4+ has C2v symmetry and characterize the pseudorotational dynamics in these fluxional cations. The tunneling behavior is discussed in terms of the relevant properties of the potential energy surface, some of which have been recalculated at the CCSD(T)/cc-pVTZ level of ab initio theory.

Symmetry of magnetic structures: the case of CeAl2

This article presents a group-theoretical investigation of an incommensurate magnetic structure in the case where the crystal structure is centrosymmetric while the structure vector k is not conserved by inversion. A first method starts with the conventional group-theoretical treatment taking into account the transformations which conserve k and then combines the solutions to obtain a centrosymmetric and real magnetization. A second method includes inversion in the group-theoretical treatment, for instance by introducing the complex conjugation operator and corepresentations. The general theory is applied to CeAl 2 with the scope of improving the knowledge of its magnetic structure.

Group theoretical treatment of photoelectron spectra of high- T c superconductors: hidden symmetry and colour pairs

A group theoretical technique for the construction of two-electron wavefunctions with zero total momentum in solids is developed and applied to high-T c superconductors. Making use of experimental photoelectron data, the eventual structure of pair states is proposed. Group theoretical analysis of experimental data shows that the interaction between pairs connected by time reversal is significant.

Wavelet transform on the circle and the real line: A unified group-theoretical treatment

We present a unified group-theoretical derivation of the continuous wavelet transform (CWT) on the circle S 1 and the real line R , following the general formalism of coherent states (CS) associated to unitary square integrable (modulo a subgroup, possibly) representations of the group SL ( 2 , R ) . A general procedure for obtaining unitary representations of a group G of affine transformations on a space of signals L 2 ( X , d x ) is described, relating carrier spaces X to (first- or higher-order) “polarization subalgebras” P X . We also provide explicit admissibility and continuous frame conditions for wavelets on S 1 and discuss the Euclidean limit in terms of group contraction.

Exciton states and optical properties of CdSe nanocrystals

The optical spectra of CdSe nanocrystals up to 55 A in diameter are analyzed in a wide range of energies from the fine structure of the low-energy excitations to the so-called high-energy transitions. We apply a symmetry-based method in two steps. First we take the tight-binding (TB) parameters from the bulk sp^{3}s^{*} TB model, extended to include the spin-orbit interaction. The full single-particle spectra are obtained from an exact diagonalization by using a group-theoretical treatment. The electron-hole interaction is next introduced: Both the Coulomb (direct) and exchange terms are considered. The high-energy excitonic transitions are studied by computing the electric dipole transition probabilities between single-particle states, while the transition energies are obtained by taking into account the Coulomb interaction. The fine structure of the lowest excitonic states is analyzed by including the electron-hole exchange interaction and the wurtzite crystal-field terms in the exciton Hamiltonian. The latter is diagonalized in the single electron-hole pair excitation subspace of progressively increasing size until convergence. The peaks in the theoretical transition spectra are then used to deduce the resonant and nonresonant Stokes shifts, which are compared with their measured values in photoluminescence experiments. We find that the final results depend on the crystal-field term, the relative size of the surface and the degree of saturation of the dangling bonds. The results show a satisfactory agreement with the available experimental data.