Symmetry-based Selection Rules Research Articles

Anatomy of ultrafast quantitative magnetoacoustics in freestanding nickel thin films

We revisit the quantitative analysis of the ultrafast magneto-acoustic experiment in a freestanding nickel thin film by Kim and Bigot [1] by applying our recently proposed approach of magnetic and acoustic eigenmodes decomposition by Vernik et al. [2]. We show that the application of our modeling to the analysis of time-resolved reflectivity measurements allows for the determination of amplitudes and lifetimes of standing perpendicular acoustic phonon resonances with unprecedented accuracy. The acoustic damping is found to scale as $\propto\omega^2$ for frequencies up to 80~GHz and the peak amplitudes reach $10^{-3}$. The experimentally measured magnetization dynamics for different orientations of an external magnetic field agrees well with numerical solutions of magneto-elastically driven magnon harmonic oscillators. Symmetry-based selection rules for magnon-phonon interactions predicted by our modeling approach allow for the unambiguous discrimination between spatially uniform and non-uniform modes, as confirmed by comparing the resonantly enhanced magneto-elastic dynamics simultaneously measured on opposite sides of the film. Moreover, the separation of time scales for (early) rising and (late) decreasing precession amplitudes provide access to magnetic (Gilbert) and acoustic damping parameters in a single measurement.

Transient Symmetry Controls Photo Dynamics near Conical Intersections.

Excited-state chemistry lacks generalized symmetry rules. With many femtochemistry studies focused on individual cases, it is hard to build up the same level of chemical intuition for excited states as that for ground states. Here, we unravel the degrees of freedom involved in ultrafast internal conversion (IC) by mapping the vibrational coherence of the initial wavepacket and the dependence on molecular symmetry in various cyclic tertiary amines. Molecular symmetry plays an important role in the preservation of vibrational coherence in the transit from one electronic state to another. We show here that it is sufficient for the molecule to simply have the possibility of a more symmetric structure to achieve the preservation of vibrational coherence. It can be transient and still lead to preservation. This finding provides an additional angle on how symmetry influences electronic transitions and an additional piece to the puzzle of establishing symmetry-based selection rules for excited-state processes.

Erratum: Crystal symmetry based selection rules for anharmonic phonon-phonon scattering from a group theory formalism [Phys. Rev. B 103 , 184302 (2021)

Received 30 July 2021DOI:https://doi.org/10.1103/PhysRevB.104.099902©2021 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAnharmonic lattice dynamicsLattice thermal conductivityPhononsCondensed Matter, Materials & Applied Physics

Crystal symmetry based selection rules for anharmonic phonon-phonon scattering from a group theory formalism

Anharmonic phonon-phonon scattering serves a critical role in heat conduction in solids. Previous studies have identified many selection rules for possible phonon-phonon scattering channels imposed by phonon energy and momentum conservation conditions and crystal symmetry. However, the crystal-symmetry-based selection rules have mostly been \textit{ad hoc} so far in selected materials, and a general formalism that can summarize known selection rules and lead to new ones in any given crystal is still lacking. In this work, we apply a general formalism for symmetry-based scattering selection rules based on the group theory to anharmonic phonon-phonon scatterings, which can reproduce known selection rules and guide the discovery of new selection rules between phonon branches imposed by the crystal symmetry. We apply this formalism to analyze the phonon-phonon scattering selection rules imposed by the in-plane symmetry of graphene, and demonstrate the significant impact of symmetry-breaking strain on the lattice thermal conductivity. Our work quantifies the critical influence of the crystal symmetry on the lattice thermal conductivity in solids and suggests routes to engineer heat conduction by tuning the crystal symmetry.

A technical analog of the cosmological constant problem and a solution thereof

The near vanishing of the cosmological constant is one of the most puzzling open problems in theoretical physics. We consider a system, the so-called framid, that features a technically similar problem. Its stress-energy tensor has a Lorentz-invariant expectation value on the ground state, yet there are no standard, symmetry-based selection rules enforcing this, since the ground state spontaneously breaks boosts. We verify the Lorentz invariance of the expectation value in question with explicit one-loop computations. These, however, yield the expected result only thanks to highly nontrivial cancellations, which are quite mysterious from the low-energy effective theory viewpoint.

Phonon-phonon scattering selection rules and control: an application to nanofriction and thermal transport.

Phonon–phonon scattering processes are the crucial phenomena which account for phonon decay, thermal expansion, heat transfer, protein dynamics, spin relaxation and related quantities. In this work, we show how the symmetries of the system determine which scattering processes are allowed at any order of anharmonic approximation, irrespective of the chemical composition. We also discuss how to control the system symmetries to switch on and off any single scattering process. We apply the presented results to the study and control of nanoscale intrinsic friction and thermal transport in lamellar van der Waals transition metal dichalcogenides. Thanks to its general formulation, the presented framework expands the materials science tool set for the design of nanoengineered thermally-active materials, irrespective of the specific chemical composition and atomic topology.

Mapping momentum-dependent electron-phonon coupling and nonequilibrium phonon dynamics with ultrafast electron diffuse scattering

Despite their fundamental role in determining material properties, detailed momentum-dependent information on the strength of electron-phonon and phonon-phonon coupling (EPC and PPC, respectively) across the entire Brillouin zone (BZ) has proved difficult to obtain. Here we demonstrate that ultrafast electron diffuse scattering (UEDS) directly provides such information. By exploiting symmetry-based selection rules and time-resolution, scattering from different phonon branches can be distinguished even without energy resolution. Using graphite as a model system, we show that UEDS patterns map the relative EPC and PPC strength through their profound sensitivity to photoinduced changes in phonon populations. We measure strong EPC to the $K$-point transverse optical phonon of $A_1'$ symmetry ($K-A_1'$) and along the entire longitudinal optical branch between $\Gamma-K$, not only to the $\Gamma-E_{2g}$ phonon as previously emphasized. We also determine that the subsequent phonon relaxation pathway involves three stages; decay via several identifiable channels to transverse acoustic (TA) and longitudinal acoustic (LA) phonons (1-2 ps), intraband thermalization of the non-equilibrium TA/LA phonon populations (30-40 ps) and interband relaxation of the LA/TA modes (115 ps). Combining UEDS with ultrafast angle-resolved photoelectron spectroscopy will yield a complete picture of the dynamics within and between electron and phonon subsystems, helping to unravel complex phases in which the intertwined nature of these systems have a strong influence on emergent properties.

Two-photon absorption spectroscopy of stilbene and phenanthrene: Excited-state analysis and comparison with ethylene and toluene.

Two-photon absorption (2PA) spectra of several prototypical molecules (ethylene, toluene, trans- and cis-stilbene, and phenanthrene) are computed using the equation-of-motion coupled-cluster method with single and double substitutions. The states giving rise to the largest 2PA cross sections are analyzed in terms of their orbital character and symmetry-based selection rules. The brightest 2PA transitions correspond to Rydberg-like states from fully symmetric irreducible representations. Symmetry selection rules dictate that totally symmetric transitions typically have the largest 2PA cross sections for an orientationally averaged sample when there is no resonance enhancement via one-photon accessible intermediate states. Transition dipole arguments suggest that the strongest transitions also involve the most delocalized orbitals, including Rydberg states, for which the relative transition intensities can be rationalized in terms of atomic selection rules. Analysis of the 2PA transitions provides a foundation for predicting relative 2PA cross sections of conjugated molecules based on simple symmetry and molecular orbital arguments.

Two-photon absorption spectroscopy of trans-stilbene, cis-stilbene, and phenanthrene: Theory and experiment.

Two-photon absorption (2PA) spectroscopy provides complementary, and sometimes more detailed, information about the electronic structure of a molecule relative to one-photon absorption (1PA) spectroscopy. However, our understanding of the 2PA processes is rather limited due to technical difficulties in measuring experimental 2PA spectra and theoretical challenges in computing higher-order molecular properties. This paper examines the 2PA spectroscopy of trans-stilbene, cis-stilbene, and phenanthrene by a combined experimental and theoretical approach. The broadband 2PA spectra of all three compounds are measured under identical conditions in order to facilitate a direct comparison of the absolute 2PA cross sections in the range 3.5-6.0 eV. For comparison, the theoretical 2PA cross sections are computed using the equation-of-motion coupled-cluster method with single and double substitutions. Simulated 2PA spectra based on the calculations reproduce the main features of the experimental spectra in solution, although the quantitative comparison is complicated by a number of uncertainties, including limitations of the theoretical model, vibronic structure, broadening of the experimental spectra, and solvent effects. The systematic comparison of experimental and theoretical spectra for this series of structurally similar compounds provides valuable insight into the nature of 2PA transitions in conjugated molecules. Notably, the orbital character and symmetry-based selection rules provide a foundation for interpreting the features of the experimental 2PA spectra in unprecedented detail.

Far-Infrared Graphene Plasmonic Crystals for Plasmonic Band Engineering

We introduce far-infrared graphene plasmonic crystals. Periodic structural perturbation-in a proof-of-concept form of hexagonal lattice of apertures-of a continuous graphene medium alters delocalized plasmonic dynamics, creating plasmonic bands in a manner akin to photonic crystals. Fourier transform infrared spectroscopy demonstrates band formation, where far-infrared irradiation excites a unique set of plasmonic bands selected by phase matching and symmetry-based selection rules. This band engineering may lead to a new class of graphene plasmonic devices.

Communication: Selection rules for tunneling splitting of vibrationally excited levels

Five symmetry-based selection rules are formulated that relate the tunneling splitting of a vibrationally excited level to that of the ground level in molecules with a symmetric double-minimum potential. The rules, which explain why excited levels frequently have smaller splittings than zero-point levels, are used to interpret the observed and calculated splittings in malonaldehyde.

Diagnosis of the σ-, π- and (σ+π)-Aromaticity by the Shape of the NICSzz-Scan Curves and Symmetry-Based Selection Rules

The NICSzz-scan curves of aromatic organic, inorganic and “all-metal” molecules in conjunction with symmetry-based selection rules provide efficient diagnostic tools of the σ-, π- and/or double (σ + π)-aromaticity. The NICSzz-scan curves of σ-aromatic molecules are symmetric around the z-axis, having half-band widths approximately less than 3 Å with the induced diatropic ring current arising from Tx,y-allowed transitions involving exclusively σ-type molecular orbitals. Broad NICSzz-scan curves (half-band width approximately higher than 3 Å) characterize double (σ + π)-aromaticity, the chief contribution to the induced diatropic ring current arising from Tx,y-allowed transitions involving both σ- and π-type molecular orbitals. NICSzz-scan curves exhibiting two maxima at a certain distance above and below the molecular plane are typical for (σ + π)-aromatics where the π-diatropic ring current overwhelms the σ-type one. In the absence of any contribution from the σ-diatropic ring current, the NICSzz(0) value is close to zero and the molecule exhibits pure π-aromaticity.

Diagnosis of magnetoresponsive aromatic and antiaromatic zones in three‐membered rings of d‐ and f‐block elements

Magnetoresponsive three-membered rings of d- and f-block elements have been thoroughly investigated with the help of electronic structure calculation methods. The magnetic response of the clusters was evaluated by the Nucleus Independent Chemical Shifts (NICS)(zz)-scan curves, which in conjunction with symmetry-based selection rules for the most significant translationally and rotationally allowed transitions helped rationalize and predict the orbital-type of aromaticity/antiaromaticity of the clusters. The magnetoresponsive early (Groups 3, 4, and 5) transition metal M(3) rings exhibit successive aromatic and antiaromatic zones separated by a nodal plane. The magnetoresponsive late (Groups 11 and 12) transition metal M(3) rings exhibit long-range aromatic zone with the NICS(zz)(R) values decaying rapidly and monotonically with respect to R. The magnetic response of Group 10 transition metal M(3) rings is similar to that of the early transition metal M(3) rings, but it is long-range antiaromatic only for the [c-Ni(3)] cluster. The NICS(zz)-scan curve of the [(H(t)La)(3)(mu(2)-H)(6)] cluster is indicative of weak pure sigma-aromaticity due to the induced diatropic ring current from the translationally allowed a(1)(') --> e' and e' --> a(1)(') transitions. The aromatic-antiaromatic behavior of the [(H(t)Ce)(3)(mu(2)-H)(6)](+) and [(H(t)Tm)(3)(mu(2)-H)(6)](2-) clusters is similar to that of the early d-block elements. The magnetic response of [(H(t)Yb)(3)(mu(2)-H)(6)](3-) is similar to that of [c-Hg(3)](2-). The [(H(t)Lu)(3)(mu(2)-H)(6)] cluster can be considered as a doubly (sigma + pi) aromatic system, with the sigma-aromatic component being much stronger than the pi-aromatic one. Finally, the [(X(t)Re)(3)(mu(2)-X)(6)] and [(X(t)Ru)(3)(mu(2)-X)(6)](+) (X = Cl, Br, I) clusters exhibit significant aromatic character with the greatest contribution to the induced diatropic ring currents coming from pi-type transitions.

Efficiency of the NICSzz-scan curves to probe the antiaromaticity of organic and inorganic rings/cages

In the present study it was demonstrated that the NICS(zz)-scan patterns along with symmetry-based selection rules can unequivocally probe the antiaromaticity in a wide range of antiaromatic organic and inorganic rings/cages. The NICS(zz)-scan profiles typical of antiaromaticity correspond to symmetric curves around the axis perpendicular to the ring plane with the positive NICS(zz)(R) values decaying rapidly and monotonically with respect to the distance R from the ring center. The magnitude of the induced paratropic ring currents is determined by the magnitude of the excitation energies of the rotationally (R(z)) allowed HOMO --> LUMO transitions. The appearance of the NICS(zz)-scan curves in conjunction with the symmetry-based selection rules constitute a powerful magnetic criterion of antiaromaticity capable of predicting the antiaromaticity in the realm of antiaromatic organic, inorganic and "all-metal" molecules lifting up all controversies with respect to the aromaticity/antiaromaticity for some peculiar aromatic/antiaromatic molecules.

Supercycled homonuclear dipolar decoupling sequences in solid-state NMR

We compare the performance of the windowed phase-modulated Lee–Goldburg ( wPMLG) and the windowed decoupling using mind boggling optimisation ( wDUMBO) sequences at various magic-angle spinning rates and nutation frequencies of the pulses. Additionally, we introduce a supercycled version of wDUMBO and compare its efficiency with that of the non-supercycled implementation of wDUMBO. The efficiency of the supercycled version of wPMLG, denoted wPMLG-S2, is compared with a new supercycled version of wPMLG that we notate as wPMLG-S3. The interaction between the supercycled homonuclear dipolar decoupling sequences and the sample rotation is analysed using symmetry-based selection rules.

Implications of Molecular Orbital Symmetries and Energies for the Electron Delocalization of Inorganic Clusters

Isostructural clusters exhibit contrasting magnetic properties when the number of electrons differs. Surprisingly, the same is true even for isoelectronic cages (e.g. O(h) B6H6(2-) is diatropic, whereas O(h) Si6(2-) is paratropic) or for those with different substitutents (e.g. T(d) B4H4 is paratropic, whereas T(d) B4F4 is diatropic). Indeed, the total nucleus-independent chemical shift (NICS) values, based on shieldings computed at cluster centers, may range considerably in magnitude and even change from diatropic (up-field shifted) to paratropic (down-field shifted). Similarly, individual dissected canonical molecular orbital contributions to the total NICS values computed at the "gauge-including atomic orbitals" (GIAO) level vary greatly. This contrasting behavior arises from molecular orbital energy differences, from the extent of orbital overlap, as well as from symmetry-based selection rules derived from group theory. Differences in magnetic properties may originate from the symmetry of the orbitals; specifically from the forbidden nature of the highest occupied molecular orbital --> lowest unoccupied molecular orbital (HOMO --> LUMO) electronic excitation weighted by the occupied-unoccupied orbital energy difference. Thus, HOMO-NICS values are generally highly paratropic if the HOMO --> LUMO rotational transition is allowed by symmetry selection rules.

Selection Rules for the High Harmonic Generation Spectra

For three-dimensional many-electron time periodic Hamiltonians which are invariant under dynamical symmetry of order $N$ we prove that only the $(\mathrm{nN}\ifmmode\pm\else\textpm\fi{}1)\mathrm{th}$, $n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,2,\dots{},$ harmonics are generated. We discuss the application of the dynamical symmetry based selection rules to the generation of high harmonics by thin crystals. The derived selection rules are demonstrated numerically for a one-dimensional model, showing that the dynamically symmetric systems can be used not only as ``filters'' of the very high harmonics but also as their ``amplifiers.''

Room Temperature Binding of CO to Cobaltous Porphyrin π Cation Radical: Spectroscopic Characterization of Mono and Bis CO Complexes with Cobaltic Porphyrin

When CO(g) is added to solutions of the cobaltous porphyrin π cation radical [Co(II)OEP•]ClO4, prepared by oxidation of CoOEP by AgClO4 in anhydrous CH2Cl2, room temperature binding of CO to the metal center occurs. Two distinct products result, [(CO)Co(III)OEP]ClO4 and [(CO)2Co(III)OEP]ClO4. These compounds exhibit Soret maxima at 366 and 414 nm, respectively. Resonance Raman (RR) spectra measured before the addition of CO are used to characterize the cobaltous porphyrin π cation radical, and, after the addition of CO, RR spectra obtained by using 363.8 nm excitation confirm the occurrence of the five-coordinate Co(III) porphyrin complex. This complex, along with halide ligated analogs, displays some structure-sensitive vibrational frequencies which suggest that an unusual distortion of the porphyrin core occurs in CH2Cl2 solution. RR spectra obtained by using 413.1 nm excitation after the addition of CO are used to identify the second product as a typical six-coordinate cobaltic porphyrin. Titrations with CO reveal that the five- and six-coordinate complexes, [(CO)Co(III)OEP]ClO4 and [(CO)2Co(III)OEP]ClO4, form with P1/2 values of 36 ± 3 and 4000 ± 300 Torr of CO, respectively. Isosbestic points in the UV−vis spectra occur at 368 nm for binding of the first CO and at 385 nm during binding of the second CO ligand. FTIR and RR spectra of [(CO)Co(III)OEP]ClO4 reveal vibrations at 2110 and 441 cm-1 which shift upon substitution of 13CO or C18O for the natural abundance CO. Isotope sensitive vibrations of [(CO)2Co(III)OEP]ClO4 were measured at 2137 and 468 cm-1. In each case the modes above 2100 cm-1 correspond to C⋮O stretching frequencies, while the lower frequency features are assigned to Co−C stretching motions. The resonance Raman enhancement of these isotope sensitive frequencies confirm CO binding to the metal center. The selective presence of these features in the Raman and FTIR data is consistent with that expected from symmetry based selection rules for five- and six-coordinate CO complexes. The relatively high C⋮O stretching frequencies observed are suggestive of weak metal dπ → ligand π* backbonding that results from the oxidation of Co(II) to Co(III).

Selection rules for two-channel Kondo models of U 4+ and Ce 3+ ions in metals

Symmetry-based selection rules are developed providing minimal criteria for the existence of two-channel Kondo interactions between conduction electrons and the low energy degrees of freedom on U 4+ and Ce 3+ in a metal host, assuming that the underlying microscopics are regulated by the Anderson Hamiltonian. An additional dynamic selection rule is imposed on Ce 3+ ions. The selection rules restrict the two-channel quadrupolar Kondo effect to U 4+ ions in cubic, tetragonal and hexagonal symmetry. For hexagonal and tetragonal symmetry, the Kondo effect for a U 4+ ion will always be quadrupolar. The selection rules for Ce 3+ ions restrict the two-channel magnetic Kondo effect to one of three possible doublet ionic ground states in hexagonal symmetry and the lone doublet ionic ground state in cubic symmetry. The dynamical selection rule apparently excludes two-channel Kondo behavior for Yb 3+ ions.

Isomerisations of the fullerenes

The Stone-Wales (pyracylene) rearrangement is a hypothetical mechanism for interconversion of fullerene isomers, accounting for mobility of the pentagons on the surfaces of these clusters. Two versions of the transformation, involving patches of 4 and 12 rings, respectively, can be defined. Energetic and group-theoretical aspects of the transformation are discussed, symmetry-based selection rules devised and complete isomerisation maps presented for small fullerenes (C 20 to C 40) and isolated-pentagon isomers (C 78 to C 88). Fullerenes that self-racemise under this transformation include D 2 C 28. Limitations on the usefulness of this mechanism as a means of rationalising experimental isomer distributions are briefly discussed.