Hund's Rule Research Articles

Design of Anti-Hund Organic Emitters Based on Heptazine.

Hund's rule, which is powerful in governing the first excited states of closed-shell organic materials, can hardly be violated to get inverted singlet-triplet gap (INVEST) molecules with negative singlet-triplet energy gaps (ΔEST), although INVEST materials have shown extraordinary photophysical properties and promising device performance especially in light-emitting diodes. Here, we propose a facile strategy to construct emissive INVEST molecules by introducing different types of substituents to heptazine in various modes, which can effectively tune the ΔEST to be negative with the enlarged oscillator strength (f) for the high fluorescence rate of the heptazine derivatives. Systematic computational studies show that the double substitution of electron-donating units with another nonconjugated substituent in hybrid substitution mode is the most favorable way in achieving slightly negative ΔEST and large f values; the conjugated substituent will compete with heptazine to make the molecule deviate from the INVEST feature. Especially, a series of high-performance heptazine-based INVEST emitters were constructed, exhibiting ΔEST low to -0.362 eV, f up to 0.0436, as well as a wide range emission color from 339 to 716 nm. Also, the designed molecules were predicted to have fluorescence radiative rates up to 106 s-1, along with efficient reverse intersystem crossing rates reaching 108 s-1. Importantly, the figure of merit (FM) was first proposed as a parameter to wholly evaluate the performance of INVEST emitters, and the highest FM of 0.198 was found in the triazine and double nonconjugated amine-substituted heptazine. These results highlight the great potential of the heptazine chromophore in constructing INVEST emitters, revealing fundamental structure-property understandings for the material design of efficient anti-Hund organic molecules with improved emission properties.

Influence of pseudo-Jahn-Teller activity on the singlet-triplet gap of azaphenalenes.

We analyze the possibility of symmetry-lowering induced by pseudo-Jahn-Teller interactions in six previously studied azaphenalenes that are known to have their first excited singlet state (S1) lower in energy than the triplet state (T1). The primary aim of this study is to explore whether Hund's rule violation is observed in these molecules when their structures are distorted from C2v or D3h point group symmetries by vibronic coupling. Along two interatomic distances connecting these point groups to their subgroups Cs or C3h, we relaxed the other internal degrees of freedom and calculated two-dimensional potential energy subsurfaces. The many-body perturbation theory (MP2) suggests that the high-symmetry structures are the energy minima for all six systems. However, single-point energy calculations using the coupled-cluster method (CCSD(T)) indicate symmetry lowering in four cases. The singlet-triplet energy gap plotted on the potential energy surface also shows variations when deviating from high-symmetry structures. A full geometry optimization at the CCSD(T) level with the cc-pVTZ basis set reveals that the D3h structure of cyclazine (1AP) is a saddle point, connecting two equivalent minima of C3h symmetry undergoing rapid automerization. The combined effects of symmetry lowering and high-level corrections result in a nearly zero singlet-triplet gap for the C3h structure of cyclazine. Azaphenalenes containing nitrogen atoms at electron-deficient sites - 2AP, 3AP, and 4AP - exhibit more pronounced in-plane structural distortion; the effect is captured by the long-range exchange-interaction corrected DFT method, ωB97XD. Excited state calculations of these systems indicate that in their low-symmetry energy minima, T1 is indeed lower in energy than S1, upholding the validity of Hund's rule. Jahn-Teller analysis predicts the symmetries of the in-plane distortion vibrational modes as or B2: C2v → Cs agreeing with the vibrational frequencies of the saddle-points.

Singlet-Triplet Inversions in Through-Bond Charge-Transfer States.

Molecules where the lowest excited singlet state is lower in energy than the lowest triplet are highly promising for a number of organic materials applications as efficiency limitations stemming from spin statistics are overcome. All molecules known to possess such singlet-triplet inversions exhibit a pattern of spatially alternating but nonoverlapping HOMO and LUMO orbitals, meaning the lowest excited states are of a local character. Here, we demonstrate that derivatives of the bicyclic hydrocarbon calicene exhibit Hund's rule violations in charge-transfer (CT) states between its rings. These CT states can be tuned with substituents, so that the first excited singlet and triplet state are energetically inverted. This provides a conceptual connection between the emerging fields of inverted gap molecules and existing molecular design rules for state-of-the-art thermally activated delayed fluorescence materials.

Singlet-Triplet Inversion in Triangular Boron Carbon Nitrides.

The discovery of singlet-triplet (ST) inversion in some π-conjugated triangle-shaped boron carbon nitrides is a remarkable breakthrough that defies Hund's first rule. Deeply rooted in strong electron-electron interactions, ST inversion has garnered significant interest due to its potential to revolutionize triplet harvesting in organic LEDs. Using the well-established Pariser-Parr-Pople model for correlated electrons in π-conjugated systems, we employ a combination of CISDT and restricted active space configuration interaction calculations to investigate the photophysics of several triangular boron carbon nitrides. Our findings reveal that ST inversion in these systems is primarily driven by a network of alternating electron-donor and electron-acceptor groups in the molecular rim, rather than by the triangular molecular structure itself.

Orbital magnetization in two-dimensional materials from high-throughput computational screening

Abstract We calculate the orbital magnetization of 822 two-dimensional magnetic materials from the Computational 2D Materials Database (C2DB). For compounds containing 5$d$ elements we find orbital moments of the order of 0.3--0.5 $\mu_\mathrm{B}$, which points to the necessity of including these in any type of magnetic modeling and comparison with experiments. It is also shown that the alignment of orbital moments with respect to the spin largely follows the predictions from Hund's rule and that deviations may be explained by the $d$-band splitting originating from the crystal field - for example in the important case of CrI$_3$. Finally, we show that for certain insulators, Hubbard corrections may lead to large and fully unquenched orbital moments that are pinned to the lattice rather than the spin and that these moments can lead to enormous magnetic anisotropies. Such unquenched ground states are only found from density functional theory calculations that include both Hubbard corrections and self-consistent spin--orbit coupling and largely invalidates the use of the magnetic force theorem for calculating magnetic anisotropies.

Erratum: Roles of Hund's rule coupling in excitonic density-wave states [Phys. Rev. B 90 , 245144 (2014)

Erratum: Roles of Hund's rule coupling in excitonic density-wave states [Phys. Rev. B <b>90</b> , 245144 (2014)

Spectroscopic Quantification of the Inverted Singlet-Triplet Gap in Pentaazaphenalene.

We report the direct and accurate spectroscopic quantification of the inverted singlet-triplet gap in 1,3,4,6,9b-pentaazaphenalene. This measurement is achieved by directly probing the lowest singlet and triplet states via high-resolution cryogenic anion photoelectron spectroscopy. The assignment of the first excited singlet state is confirmed by visible absorption spectroscopy in an argon matrix at 20 K. Our measurements yield an inverted singlet-triplet gap with ΔEST= -0.047(7) eV. The accurate quantification of the singlet-triplet gap presented here allows for direct evaluation of various computational electronic structure methods and highlights the critical importance of the proper description of the double excitation character of these electronic states. Overall, this study validates the idea that despite Hund's multiplicity rule, useful organic chromophores can have inherently inverted singlet-triplet gaps.

Exotic Magnetism in Perovskite KOsO_{3}.

A new perovskite KOsO_{3} has been stabilized under high-pressure and high-temperature conditions. It is cubic at 500K (Pm-3m) and undergoes subsequent phase transitions to tetragonal at 320K (P4/mmm) and rhombohedral (R-3m) at 230K as shown from refining synchrotron x-ray powder diffraction (SXRD) data. The larger orbital overlap integral and the extended wave function of 5d electrons in the perovskite KOsO_{3} allow to explore physics from the regime where Mott and Hund's rule couplings dominate to the state where the multiple interactions are on equal footing. We demonstrate an exotic magnetic ordering phase found by neutron powder diffraction along with physical properties via a suite of measurements including magnetic and transport properties, differential scanning calorimetry, and specific heat, which provide comprehensive information for a system at the crossover from localized to itinerant electronic behavior.

Interlayer-Coupling-Driven High-Temperature Superconductivity in La_{3}Ni_{2}O_{7} under Pressure.

The newly discovered high-temperature superconductivity in La_{3}Ni_{2}O_{7} under pressure has attracted a great deal of attention. The essential ingredient characterizing the electronic properties is the bilayer NiO_{2} planes coupled by the interlayer bonding of 3d_{z^{2}} orbitals through the intermediate oxygen atoms. In the strong coupling limit, the low-energy physics is described by an intralayer antiferromagnetic spin-exchange interaction J_{∥} between 3d_{x^{2}-y^{2}} orbitals and an interlayer one J_{⊥} between 3d_{z^{2}} orbitals. Taking into account Hund's rule on each site and integrating out the 3d_{z^{2}} spin degree of freedom, the system reduces to a single-orbital bilayer t-J model based on the 3d_{x^{2}-y^{2}} orbital. By employing the slave-boson approach, the self-consistent equations for the bonding and pairing order parameters are solved. Near the physically relevant 1/4-filling regime (doping δ=0.3∼0.5), the interlayer coupling J_{⊥} tunes the conventional single-layer d-wave superconducting state to the s-wave one. A strong J_{⊥} could enhance the interlayer superconducting order, leading to a dramatically increased T_{c}. Interestingly, there could exist a finite regime in which an s+id state emerges.

Resilience of Hund's rule in the chemical space of small organic molecules.

We embark on a quest to identify small molecules in the chemical space that can potentially violate Hund's rule. Utilizing twelve TDDFT approximations and the ADC(2) many-body method, we report the energies of S1 and T1 excited states of 12 880 closed-shell organic molecules within the bigQM7ω dataset with up to 7 CONF atoms. In this comprehensive dataset, none of the molecules, in their minimum energy geometry, exhibit a negative S1-T1 energy gap at the ADC(2) level while several molecules display values <0.1 eV. The spin-component-scaled double-hybrid method, SCS-PBE-QIDH, demonstrates the best agreement with ADC(2). Yet, at this level, a few molecules with a strained sp3-N center turn out as false-positives with the S1 state lower in energy than T1. We investigate a prototypical cage molecule with an energy gap <-0.2 eV, which a closer examination revealed as another false positive. We conclude that in the chemical space of small closed-shell organic molecules, it is possible to identify geometric and electronic structural features giving rise to S1-T1 degeneracy; still, there is no evidence of a negative gap. We share the dataset generated for this study as a module, to facilitate seamless molecular discovery through data mining.

Cage doping of Ti, Zr, and Hf-based 13-atom nanoclusters: two sides of the same coin.

Transition metal nanoclusters can exhibit unique and tunable properties which result not only from their chemical composition but also from their atomic packing and quantized electronic structures. Here, we introduce a promising family of bimetallic TM@Ti12, TM@Zr12, and TM@Hf12 nanoclusters with icosahedral geometry, where TM represents an atom from groups 3 to 12. Density functional theory calculations show that their stability can be explained with familiar concepts of metal cluster electronic and atomic shell structures. The magnetic properties of these quasispherical clusters are entirely consistent with superatom electronic shells and Hund's rules, and can be tuned by the choice of the TM dopant. The computed cluster atomization energies were analyzed in terms of the elements' cohesive energy, Ecoh, and contributions from geometric distortion, Edis, surface energy, Es, and ionic bonding, Ei. Some clusters have anomalous stability relative to Ecoh + Edis + Es + Ei. We attribute this to superatomic character associated with a favorable atomic and electronic shell structure. This raises the possibility of designing stable superatoms and materials with tailored electronic and magnetic properties.

Extension of molecules with an inverted singlet-triplet gap with conjugated branches to alter the oscillator strength.

Molecules that violate Hund's rule and possess negative singlet-triplet gaps (ΔEST) have been actively studied for their potential usage in organic light emitting diodes without the need for thermal activation. However, the weak oscillator strength from the symmetry of such molecules has been recognized as their shortcoming for their application in optoelectronic devices. A group of molecules with a common structural motif involving the original molecule with an inverted gap having branches consisting of conjugated molecules of varied structures and extent of conjugation have been predicted to have desirable oscillator strength, but only few detailed and comprehensive studies regarding the form of excited states and the reason behind the improved oscillator strength have been carried out. We show in this work a series of analyses that suggest that the increase of oscillator strength is correlated with the nature of the excited state changing from a localized excitation to a delocalized excitation involving the central molecule and the branches. The resulting oscillator strength thus depends on the energetic matching of the branching molecule and the central molecule, rather than solely the oscillator strength of the central molecule. From the ΔEST inversion point of view, the static correlation with low-lying doubly excited configurations, the key mechanism behind the inversion in the localized excited state, weakens as the excited states delocalize. As a consequence, the dynamic correlation has a more decisive effect in determining the singlet-triplet gap.

Heptazine, Cyclazine, and Related Compounds: Chemically-Accurate Estimates of the Inverted Singlet-Triplet Gap.

Molecules that violate Hund's rule and exhibit an inverted gap between the lowest singlet S1 and triplet T1 excited states have attracted considerable attention due to their potential applications in optoelectronics. Among these molecules, the triangular-shaped heptazine, and its derivatives, have been in the limelight. However, conflicting reports have arisen regarding the relative energies of S1 and T1. Here, we employ highly accurate levels of theory, such as CC3, to not only resolve the debate concerning the sign but also quantify the magnitude of the S1-T1 gap. We also determined the 0-0 energies to evaluate the significance of the vertical approximation. In addition, we compute reference S1-T1 gaps for a series of 10 related molecules. This enables us to benchmark lower-order methods for future applications in larger systems within the same family of compounds. This contribution can serve as a foundation for the design of triangular-shaped molecules with enhanced photophysical properties.

Role of Spin Polarization and Dynamic Correlation in Singlet-Triplet Gap Inversion of Heptazine Derivatives.

The new generation of proposed light-emitting molecules for organic light-emitting diodes (OLEDs) has raised considerable research interest due to its exceptional feature─a negative singlet-triplet (ST) gap violating Hund's multiplicity rule in the excited S1 and T1 states. We investigate the role of spin polarization in the mechanism of ST gap inversion. Spin polarization is associated with doubly excited determinants of certain types, whose presence in the wave function expansion favors the energy of the singlet state more than that of the triplet. Using a perturbation theory-based model for spin polarization, we propose a simple descriptor for prescreening of candidate molecules with negative ST gaps and prove its usefulness for heptazine-type molecules. Numerical results show that the quantitative effect of spin polarization decreases linearly with the increasing highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO-LUMO) exchange integral. Comparison of single- and multireference coupled-cluster predictions of ST gaps shows that the former methods provide good accuracy by correctly balancing the effects of doubly excited determinants and dynamic correlation. We also show that accurate ST gaps may be obtained using a complete active space model supplemented with dynamic correlation from multireference adiabatic connection theory.

What can lattice DFT teach us about real-space DFT?

In this paper we establish a connection between density functional theory (DFT) for lattice models and common real-space DFT. We consider the lattice DFT description of a two-level model subject to generic interactions in Mermin's DFT formulation in the grand canonical ensemble at finite temperature. The case of only density-density and Hund's rule interaction studied in earlier work is shown to be equivalent to an exact-exchange description of DFT in the real-space picture. In addition, we also include the so-called pair-hopping interaction which can be treated analytically and, crucially, leads to non-integer occupations of the Kohn-Sham (KS) levels even in the limit of zero temperature. Treating the hydrogen molecule in a minimal basis is shown to be equivalent to our two-level lattice DFT model. By means of the fractional occupations of the KS orbitals (which, in this case, are identical to the many-body ones) we reproduce the results of full configuration interaction, even in the dissociation limit and without breaking the spin symmetry. Beyond the minimal basis, we embed our HOMO-LUMO model into a standard DFT calculation and, again, obtain results in overall good agreement with exact ones without the need of breaking the spin symmetry.

Scaling theory of intrinsic Kondo and Hund's rule interactions in magic-angle twisted bilayer graphene

Scaling theory of intrinsic Kondo and Hund's rule interactions in magic-angle twisted bilayer graphene

The inverted singlet-triplet gap: a vanishing myth?

Molecules with an inverted singlet-triplet gap (STG) between the first excited singlet and triplet states, for example, heptazine, have recently been reported and gained substantial attention since they violate the famous Hund's rule. Utilizing state-of-the-art high-level ab initio methods, the singlet-triplet gap vanishes and approaches zero from below whatever is improved in the theoretical description of the molecules: the basis set or the level of electron correlation. Seemingly, the phenomenon of inverted singlet-triplet gaps tends to vanish the closer we observe.

Kondo Lattice Model of Magic-Angle Twisted-Bilayer Graphene: Hund's Rule, Local-Moment Fluctuations, and Low-Energy Effective Theory.

We apply a generalized Schrieffer-Wolff transformation to the extended Anderson-like topological heavy fermion (THF) model for the magic-angle (θ=1.05°) twisted bilayer graphene (MATBLG) [Phys. Rev. Lett. 129, 047601 (2022)PRLTAO0031-900710.1103/PhysRevLett.129.047601], to obtain its Kondo lattice limit. In this limit localized f electrons on a triangular lattice interact with topological conduction c electrons. By solving the exact limit of the THF model, we show that the integer fillings ν=0,±1,±2 are controlled by the heavy f electrons, while ν=±3 is at the border of a phase transition between two f-electron fillings. For ν=0,±1,±2, we then calculate the Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions between the f moments in the full model and analytically prove the SU(4) Hund's rule for the ground state which maintains that two f electrons fill the same valley-spin flavor. Our (ferromagnetic interactions in the) spin model dramatically differ from the usual Heisenberg antiferromagnetic interactions expected at strong coupling. We show the ground state in some limits can be found exactly by employing a positive semidefinite "bond-operators" method. We then compute the excitation spectrum of the f moments in the ordered ground state, prove the stability of the ground state favored by RKKY interactions, and discuss the properties of the Goldstone modes, the (reason for the accidental) degeneracy of (some of) the excitation modes, and the physics of their phase stiffness. We develop a low-energy effective theory for the f moments and obtain analytic expressions for the dispersion of the collective modes. We discuss the relevance of our results to the spin-entropy experiments in TBG.

Inverted singlet–triplet emitters for organic light-emitting diodes

Theoretical and experimental studies since the 1980s have pointed to the existence of organic molecules that violate Hund's rule of maximum multiplicity, with the lowest singlet excited state having lower energy than the lowest triplet excited state. With the rising prevalence of organic light-emitting diodes (OLEDs) in display technology, these types of molecules are being investigated as a new class of organic emitters. The singlet–triplet inversion implies that thermal activation is not necessary to achieve fast triplet harvesting, providing potential benefits over conventional thermally activated delayed fluorescence emitters. Here, we overview prominent studies regarding inverted singlet and triplet excited states in the context of OLEDs.

Enhanced spin-orbit coupling and orbital moment in ferromagnets by electron correlations

In atomic physics, the Hund's rule states that the largest spin and orbital state is realized due to the interplay of spin-orbit coupling (SOC) and Coulomb interactions. Here, we show that in ferromagnetic solids the effective SOC and the orbital magnetic moment can be dramatically enhanced by a factor of $1/[1\ensuremath{-}(2{U}^{\ensuremath{'}}\ensuremath{-}U\ensuremath{-}{J}_{H}){\ensuremath{\rho}}_{0}]$, where $U$ and ${U}^{\ensuremath{'}}$ are the on-site Coulomb interaction within the same orbitals and between different orbitals, respectively, ${J}_{H}$ is the Hund's coupling, and ${\ensuremath{\rho}}_{0}$ is the average density of states. This factor is obtained by using the two-orbital as well as five-orbital Hubbard models with SOC. We also find that the spin polarization is more favorable than the orbital polarization, being consistent with experimental observations. The theory is also extended to study the spin fluctuations and long-range Coulomb interactions, and can be applied to understand the enhanced orbital magnetic moment and giant Faraday effect in ferromagnetic nanogranules in recent experiments. This present paper provides a fundamental basis for understanding the enhancements of SOC and orbital moment by Coulomb interactions in ferromagnets, which would have wide applications in spintronics.