Spin Multiplets Research Articles

Spin-Orbit Coupling of Color Centers for Quantum Applications

Color centers in 4H-SiC and diamond are candidates for single photon sources and qubits for the implementation of quantum applications. The CI-cRPA method is able to correctly reproduce the structure of the underlying highly correlated wave functions for low and high spin multiplets. Aiming for a more complete understanding of spin-orbit mediated intersystem crossing between such states, we have extended CI-cRPA by a first order treatment of the spin-orbit interactions. We present preliminary results for intercrossing matrix elements and contributions of spin-orbit coupling to the zero field splitting of the silicon vacancy and di-vacancy in 4H-SiC, as well as the NV center in diamond using this approach.

Spectrum of molecular hexaquarks

We investigate the mass spectra of molecular-type hexaquark states in the dibaryon systems. These systems are composed of the charmed baryons [Σc(*),Ξc(′,*)], doubly charmed baryons [Ξcc(*)], and hyperons [Σ(*),Ξ(*)]. We consider all possible combinations of particle-particle and particle-antiparticle pairs, including the S-wave spin multiplets in each combination. We establish the underlying connections among the molecular tetraquarks, pentaquarks, and hexaquarks with the effective quark-level interactions. We find that the existence of molecular states in DD*, DD¯*, and ΣcD¯(*) systems leads to the emergence of a large number of deuteronlike hexaquarks in the heavy flavor sectors. Currently, there have been several experimental candidates for molecular tetraquarks and pentaquarks. The experimental search for near-threshold hexaquarks will further advance the establishment of the underlying dynamical picture of hadronic molecules and deepen our understanding of the role of spin-flavor symmetry in near-threshold residual strong interactions. Published by the American Physical Society 2024

Magnetism and electronic properties of ConMoP (n = 1 ~ 5) cluster: a DFT study.

Hydrogen has emerged as a promising clean energy carrier, underscoring the imperative need to comprehend its adsorption mechanisms. This study delves into the magnetic and electronic properties of Co-Mo-P clusters, aiming to unveil their catalytic potential in hydrogen production. Employing density functional theory (DFT), we optimized cluster configurations and scrutinized their magnetic behaviors. Our investigation unveiled 16 stable configurations of the ConMoP (n = 1 ~ 5) cluster, predominantly in steric forms. The magnetic attributes were primarily ascribed to the d orbitals of Co metal atoms, with Co3MoP exhibiting exceptional magnetic characteristics. Analysis of density of state diagrams revealed the prevalence of spin-up α-electrons in d orbitals, while spin-down β-electrons attenuated overall magnetic properties. Localized orbital (LOL) analysis highlighted stable covalent bonds within the clusters, affirming their catalytic potential. Orbital delocalization index (ODI) analysis revealed diverse spatial distribution ranges for orbitals across different configurations, suggesting a progressive attenuation of off-domain properties with increasing cluster size. Furthermore, infrared spectroscopy unveiled distinct vibrational peaks in various configurations, indicative of unique infrared activities. These findings contribute to a nuanced theoretical understanding of Co-Mo-P clusters and pave the path for future research aimed at augmenting their catalytic efficiency in hydrogen production. This study underscores the viability of Co-Mo-P clusters as alternatives to conventional Pt catalysts, offering insights into the design of novel materials for sustainable energy applications. Further research is warranted to explore the behavior of the Co-Mo-P system under diverse reaction conditions, fostering advancements in materials and energy science. In this study, we harnessed the ConMoP (n = 1 ~ 5) cluster as a simulation platform for probing the local structure of the material. Our aim was to scrutinize the magnetism, electronic characteristics influenced by the varying metal atoms within these clusters. A systematic exploration involved incrementing the number of metal atoms and expanding the cluster size to elucidate the corresponding property variations. Density functional theory (DFT) calculations were pivotal to our methodology, employing the B3LYP hybrid functional implemented in the Gaussian16 software package. The ConMoP (n = 1 ~ 5) cluster underwent optimization calculations and vibrational analysis at the def2-tzvp quantization level, yielding optimized configurations with diverse spin multiplet degrees. To comprehensively characterize and visually represent the stability, electronic features, and catalytic attributes of these configurations, we employed a suite of computational tools. Specifically, quantum chemistry software GaussView and wave function analysis software Multiwfn played integral roles. Through the integrated use of these computational tools, we acquired valuable insights into the magnetism, electronic characteristics of the ConMoP (n = 1 ~ 5) cluster, shedding light on their dependency on distinct metal atoms.

Analysis of Tcc and Tbb based on the hadronic molecular model and their spin multiplets

Tcc(ccu¯d¯)+ has been reported by the LHCb experiment in 2022. The analysis using the Breit-Wigner parametrization found the small binding energy, 0.273 MeV, which is measured from the threshold of D*+D0. In this paper, we consider Tcc as a DD* hadronic molecule as a deuteronlike state. The one-boson exchange model is employed as for the heavy meson interactions, where we determine the cutoff parameter Λ to reproduce the reported binding energy of Tcc with I(JP)=0(1+). We discuss the properties of the bound state and also search for Tcc with quantum numbers other than 0(1+). Furthermore, we analyze Tbb as a bottom counterpart of Tcc, which involves two bottom quarks, and obtain several bound states. Finally, we consider the light-cloud basis for wave functions of the doubly heavy tetraquarks in the heavy quark limit. Using the basis, we find the spin multiplets of their bound states, indicating the spin structures of diquarks in Tcc and Tbb with the finite quark masses. Published by the American Physical Society 2024

First spectroscopic study of odd-odd Cu78

Nuclei in the vicinity of Ni78 are important benchmarks for nuclear structure, which can reveal changes in the shell structure far from stability. Spectroscopy of the odd-odd isotope Cu78 was performed for the first time in an experiment with the EURICA setup at the Radioactive Isotope Beam Factory at RIKEN Nishina Center. Excited states in the neutron-rich isotope were populated following the β decay of Ni78 produced by in-flight fission and separated by the BigRIPS separator. A level scheme based on the analysis of γ−γ coincidences is presented. Tentative spin and parity assignments were made when possible based on the β-decay feeding intensities and γ-decay properties of the excited states. Time correlations between β and γ decay show clear indications of an isomeric state with a half-life of 3.8(4) ms. Large-scale Monte Carlo shell-model calculations were performed using the A3DA-m interaction and a valence space comprising the full fp shell and the 1g9/2 and 2d5/2 orbitals for both protons and neutrons. The comparison of the experimental results with the shell-model calculations allows interpreting the excited states in terms of spin multiplets arising from the proton-neutron interaction. The results provide further insight into the evolution of the proton single-particle orbitals as a function of neutron number, and quantitative information about the proton-neutron interaction outside the doubly magic Ni78 core.Received 6 October 2022Revised 3 January 2023Accepted 9 January 2023DOI:https://doi.org/10.1103/PhysRevC.107.044301©2023 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBeta decayEnergy levels & level densitiesIsomer decaysNuclear spin & parityNuclear structure & decaysNucleon-nucleon interactionsShell modelProperties59 ≤ A ≤ 89Nuclear Physics

Possible Violation of the Spin-Hamiltonian Concept in Interpreting the Zero-Field Splitting.

The majority of experimental data in electron spin resonance and molecular magnetism are interpreted in terms of the spin-Hamiltonian (SH) formalism. However, this is an approximate theory that requires a proper testing. In the older variant, the multielectron terms are used as a basis in which the D-tensor components are evaluated by employing the second-order perturbation theory (PT) for nondegenerate states; here, the spin-orbit interaction expressed via the spin-orbit splitting parameter λ serves for the perturbation. The model space is restricted only to the fictitious spin functions |S, M⟩. In the case of the orbital (quasi) degeneracy of the ground term, the PT tends to diverge and the subtracted D, E, and g parameters are false. In the second variant working in the "complete active space" (CAS), the spin-orbit coupling operator is involved by the variation method resulting in the spin-orbit multiplets (energies and eigenvectors) The multiplets can be evaluated either by applying ab initio CASSCF + NEVPT2 + SOC calculations or by using semiempirical generalized crystal-field theory (with the one-electron SOC operator depending upon ξ). The resulting states can be projected onto the subspace of the spin-only kets in the way that the eigenvalues stay invariant. Such an effective Hamiltonian matrix can be reconstructed using six independent components of the symmetric D-tensor from which the D and E values are obtained by solving linear equations. The eigenvectors of the spin-orbit multiplets in the CAS allow determining the dominating composition of the spin projection─cumulative weights of |±M⟩. These are conceptually different from those generated by the SH alone. It is shown that in some cases, the SH theory works satisfactorily for a series of transition-metal complexes; however, sometimes it fails. The ab initio calculations on the SH parameters are compared with the approximate generalized crystal-field theory conducted at the experimental geometry of the chromophore. In total, 12 metal complexes have been analyzed. One of the criteria that assesses the validity of SH is the projection norm N for spin multiplets (this has not to be far from 1). Another criterion is the gap in the spectrum of the spin-orbit multiplets that separates the hypothetical (fictitious) spin-only manifold from the rest of the states.

How to calculate the rate constants for nonradiative transitions between the MS components of spin multiplets?

Predicting the rates of spin-dependent processes characterised by nonradiative transitions between electronic states with different spin multiplicities is important for understanding the mechanisms of many photochemical and catalytic reactions. To calculate these rates, it is necessary to define the spin state representation and the couplings between these states that drives the interstate transitions. In this work, we describe three different approaches to calculating the spin–orbit coupling (SOC), transition probabilities, and rate constants between the MS components of the electronic states with different spin multiplicities. We implemented these approaches in our nonadiabatic statistical theory (NAST) software package, which predicts the transition probabilities and rate constants of spin-dependent processes using information obtained from electronic structure calculations. We discuss the advantage and drawbacks of each approach and, as an example, calculate the rate constants for transitions between the spin states in the active site model of the protein rubredoxin.

Quantum embedding methods for correlated excited states of point defects: Case studies and challenges

A quantitative description of the excited electronic states of point defects and impurities is crucial for understanding materials properties, and possible applications of defects in quantum technologies. This is a considerable challenge for computational methods, since Kohn-Sham density functional theory (DFT) is inherently a ground-state theory, while higher-level methods are often too computationally expensive for defect systems. Recently, embedding approaches have been applied that treat defect states with many-body methods, while using DFT to describe the bulk host material. We implement such an embedding method, based on Wannierization of defect orbitals and the constrained random-phase approximation approach, and perform systematic characterization of the method for three distinct systems with current technological relevance: a carbon dimer replacing a B and N pair in bulk hexagonal BN (${\mathrm{C}}_{\text{B}}{\mathrm{C}}_{\text{N}}$), the negatively charged nitrogen-vacancy center in diamond (${\mathrm{NV}}^{\ensuremath{-}}$), and an Fe impurity on the Al site in wurtzite AlN (${\text{Fe}}_{\text{Al}}$). In the context of these test-case defects, we demonstrate that crucial considerations of the methodology include convergence of the bulk screening of the active-space Coulomb interaction, the choice of exchange-correlation functional for the initial DFT calculation, and the treatment of the ``double-counting'' correction. For ${\mathrm{C}}_{\text{B}}{\mathrm{C}}_{\text{N}}$ we show that the embedding approach gives many-body states in agreement with analytical results on the Hubbard dimer model, which allows us to elucidate the effects of the DFT functional and double-counting correction. For the ${\mathrm{NV}}^{\ensuremath{-}}$ center, our method demonstrates good quantitative agreement with experiments for the zero-phonon line of the triplet-triplet transition. Finally, we illustrate challenges associated with this method for determining the energies and orderings of the complex spin multiplets in ${\text{Fe}}_{\text{Al}}$.

Tunneling theory for a bilayer graphene quantum dot’s single- and two-electron states

The tuneability and control of quantum nanostructures in two-dimensional materials offer promising perspectives for their use in future electronics. It is hence necessary to analyze quantum transport in such nanostructures. Material properties such as a complex dispersion, topology, and charge carriers with multiple degrees of freedom, are appealing for novel device functionalities but complicate their theoretical description. Here, we study quantum tunnelling transport across a few-electron bilayer graphene quantum dot. We demonstrate how to uniquely identify single- and two-electron dot states’ orbital, spin, and valley composition from differential conductance in a finite magnetic field. Furthermore, we show that the transport features manifest splittings in the dot’s spin and valley multiplets induced by interactions and magnetic field (the latter splittings being a consequence of bilayer graphene’s Berry curvature). Our results elucidate spin- and valley-dependent tunnelling mechanisms and will help to utilize bilayer graphene quantum dots, e.g., as spin and valley qubits.

Evaluating quantum alchemy of atoms with thermodynamic cycles: Beyond ground electronic states.

Due to the sheer size of chemical and materials space, high-throughput computational screening thereof will require the development of new computational methods that are accurate, efficient, and transferable. These methods need to be applicable to electron configurations beyond ground states. To this end, we have systematically studied the applicability of quantum alchemy predictions using a Taylor series expansion on quantum mechanics (QM) calculations for single atoms with different electronic structures arising from different net charges and electron spin multiplicities. We first compare QM method accuracy to experimental quantities, including first and second ionization energies, electron affinities, and spin multiplet energy gaps, for a baseline understanding of QM reference data. Next, we investigate the intrinsic accuracy of "manual" quantum alchemy. This method uses QM calculations involving nuclear charge perturbations of one atom's basis set to model another. We then discuss the reliability of quantum alchemy based on Taylor series approximations at different orders of truncation. Overall, we find that the errors from finite basis set treatments in quantum alchemy are significantly reduced when thermodynamic cycles are employed, which highlights a route to improve quantum alchemy in explorations of chemical space. This work establishes important technical aspects that impact the accuracy of quantum alchemy predictions using a Taylor series and provides a foundation for further quantum alchemy studies.

Global Natural Orbital Functional: Towards the Complete Description of the Electron Correlation.

The current work presents a natural orbital functional (NOF) for electronic systems with any spin value independent of the external potential being considered, that is, a global NOF (GNOF). It is based on a new two-index reconstruction of the two-particle reduced density matrix for spin multiplets. The emergent functional describes the complete intrapair electron correlation, and the correlation between orbitals that make up both the pairs and the individual electrons. The interorbital correlation is composed of static and dynamic terms. The concept of dynamic part of the occupation numbers is introduced. To evaluate the accuracy achieved with the GNOF, calculation of a variety of properties is presented. They include the total energies and energy differences between the ground state and the lowest-lying excited state with different spin of atoms from H to Ne, ionization potentials of the first-row transition-metal atoms (Sc-Zn), and the total energies of a selected set of 55molecular systems in different spin states. The GNOF is also applied to the homolytic dissociation of selected diatomic molecules in different spin states and to the rotation barrier of ethylene, both paradigmatic cases of systems with significant multiconfigurational character. The values obtained agree with those reported at high level of theory and experimental data.

Boosting the resolution of multidimensional NMR spectra by complete removal of proton spin multiplicities

Over decades multidimensional NMR spectroscopy has become an indispensable tool for structure elucidation of natural products, peptides and medium sized to large proteins. Heteronuclear single quantum coherence (HSQC) spectroscopy is one of the work horses in that field often used to map structural connectivity between protons and carbons or other hetero nuclei. In overcrowded HSQC spectra, proton multiplet structures of cross peaks set a limit to the power of resolution and make a straightforward assignment difficult. In this work, we provide a solution to improve these penalties by completely removing the proton spin multiplet structure of HSQC cross peaks. Previously reported sideband artefacts are diminished leading to HSQC spectra with singlet responses for all types of proton multiplicities. For sideband suppression, the idea of restricted random delay (RRD) in chunk interrupted data acquisition is introduced and exemplified. The problem of irreducible residual doublet splitting of diastereotopic CH2 groups is simply solved by using a phase sensitive JRES approach in conjunction with echo processing and real time broadband homodecoupling (BBHD) HSQC, applied as a 3D experiment. Advantages and limitations of the method is presented and discussed.

Construction strategies for high-nitrogen M8N60 complexes with high detonation heat and controllable stability

Due to electronic structure of conventional primary explosives with CHNO, such systems present a significant challenge to improving energy utilization. In this work, high-nitrogen multicore metal Complex M8N60 (M = Cr, Mo, W, Mn, Re) were built as primary explosive by density functional theory (DFT) to contain metal–metal multiple bonds (M≡M) and multipoint coordination ability. Metal-metal(M−M) and metal–ligand (M−N) fuzzy bond orders were calculated by DFT at the PBE/LanL2DZ level, which suggests M8N60 is stable and a high detonation heat compound. Moreover, the stability of spin state of transition metals can be changed with the spin multiplet because of spin orbitals. Thus, M8N60 may be a class of new high-energy primary explosive and spintronic material because of high electronic polarizability, which can realize information storage dependent on electron spins.

Methane conversion by transition metal-doped vanadium oxide clusters

Efficient catalysis of CH4 conversion to liquid oxygenates has long been pursued but remains challenging. A deep understanding of the reaction mechanism and fundamental rule for tailoring the activity of catalysts is highly imperative. By ab initio calculations, we show that magnetic [V2O5]n (n = 2, 3, 4) clusters doped with Ti/Zr/Hf possess unique activity for CH4 oxidation to CH3COOH and CH3OH. The active O atoms bonded with dopants have spin multiplets that govern electron transfer and C−H cleavage. The adsorption energy of CH4 is linearly correlated with the p band centre difference between the majority and minority spin channels of the active O atoms.

Point-group selection rules and universal momentum-transfer dependencies for inelastic neutron scattering on molecular spin clusters

Recent significant progress in inelastic neutron scattering (INS) has rendered this technique even more useful for the characterization of magnetic systems, including molecular spin clusters. By so-called four-dimensional INS on single-crystal probes, excitation spectra can be recorded in large portions of momentum-transfer (Q) and energy-transfer (E) space. Spin-selection rules permit $\mathrm{\ensuremath{\Delta}}S=0,\ifmmode\pm\else\textpm\fi{}1$ transitions between different spin multiplets. Additional selection rules can be imposed by point-group symmetry but were not discussed yet. As most synthetic spin clusters with interesting magnetic properties have high molecular symmetry, a clear understanding of this issue will be helpful for interpreting INS spectra. Here we discuss point-group INS selection rules for magnetically isotropic or anisotropic spin clusters. Rings and a number of spin polyhedra with cubic or icosahedral symmetry are chosen as illustrative and relevant examples. These systems exhibit a significant number of point-group selection rules in isotropic spin models, and most of them maintain a smaller number of selection rules in anisotropic spin models. We also explain how the Q dependence of certain excitations depends exclusively on the point-group symmetry of the states involved in the transition, an aspect that had thus far only been detailed for spin rings. We provide the universal Q-dependent intensity functions (and their powder-averaged forms) for a set of polyhedra (cube, icosahedron, truncated tetrahedron, cuboctahedron, dodecahedron, icosidodecahedron, and truncated icosahedron). Overall, these results help to disentangle the relevant dynamical information contained in INS spectra from those features that are entirely determined by molecular symmetry.

Temperature-dependent dynamics of endohedral fullerene Sc2@C80(CH2Ph) studied by EPR spectroscopy.

Endohedral fullerenes are promising materials for the quantum information and quantum processing due to the unique properties of the electron-nuclear spin system well isolated from the environment inside the fullerene cage. The endofullerene Sc2@C80(CH2Ph) features a strong hyperfine interaction between one electron spin 1/2 localized at the Sc2 dimer and two equivalent 45Sc nuclear spins 7/2, which yields 64 well resolved EPR transitions. We report a comprehensive analysis of the temperature dependence of the EPR spectrum of Sc2@C80(CH2Ph) dissolved in d-toluene measured in a wide temperature range above and below the melting point. The nature of the electron spin coherence phase memory is investigated. The properties of all resonance lines in a liquid phase were treated within the model of the free rotational diffusion. Both, analytical expressions and numerical examination provide an excellent agreement between the experimental and simulated spectra. A detailed study of the experimental data confirms the assumption of the independent motions of the fullerene cage and the Sc2 core. The data obtained show three regimes of molecular motion detected at different temperatures: the free rotation of both the fullerene cage and its bi-metal core, the motion of the core in the frozen fullerene cage, and, finally, a state with a fixed structure of both parts of the metallofullerene molecules. The data analysis reveals a significant nuclear quadrupole interaction playing an important role for the mixing of the different nuclear spin multiplets.

SU(2|1) supersymmetric spinning models of chiral superfields

We construct SU(2|1), d = 1 supersymmetric models based on the coupling of dynamical and semi-dynamical (spin) multiplets, where the interaction term of both multiplets is defined on the generalized chiral superspace. The dynamical multiplet is defined as a chiral multiplet (2, 4, 2), while the semi-dynamical multiplet is associated with a multiplet (4, 4, 0) of the mirror type.

Calculation of spin-orbit couplings using RASCI spinless one-particle density matrices: Theory and applications.

This work presents the formalism and implementation for calculations of spin-orbit couplings (SOCs) using the Breit-Pauli Hamiltonian and non-relativistic wave functions described by the restricted active space configuration interaction (RASCI) method with general excitation operators of spin-conserving spin-flipping, ionizing, and electron-attaching types. The implementation is based on the application of the Wigner-Eckart theorem within the spin space, which enables the calculation of the entire SOC matrix based on the explicit calculation of just one transition between the two spin multiplets. Numeric results for a diverse set of atoms and molecules highlight the importance of a balanced treatment of correlation and adequate basis sets and illustrate the overall robust performance of RASCI SOCs. The new implementation is a useful addition to the methodological toolkit for studying spin-forbidden processes and molecular magnetism.

DoNOF: An open-source implementation of natural-orbital-functional-based methods for quantum chemistry

The natural orbital functional theory (NOFT) has emerged as an alternative formalism to both density functional (DF) and wavefunction methods. In NOFT, the electronic structure is described in terms of the natural orbitals (NOs) and their occupation numbers (ONs). The approximate NOFs have proven to be more accurate than those of the density for systems with a significant multiconfigurational character, on one side, and scale better with the number of basis functions than correlated wavefunction methods, on the other side. A challenging task in NOFT is to efficiently perform orbital optimization. In this article we present DoNOF, our open source implementation based on diagonalizations that allows to obtain the resulting orbitals automatically orthogonal. The one-particle reduced-density matrix (1RDM) of the ensemble of pure-spin states provides the proper description of spin multiplets. The capabilities of the code are tested on the water molecule, namely, geometry optimization, natural and canonical representations of molecular orbitals, ionization potential, and electric moments. In DoNOF, the electron-pair-based NOFs developed in our group PNOF5, PNOF7 and PNOF7s are implemented. These JKL-only NOFs take into account most non-dynamic effects plus intrapair-dynamic electron correlation, but lack a significant part of interpair-dynamic correlation. Correlation corrections are estimated by the single-reference NOF-MP2 method that simultaneously calculates static and dynamic electron correlations taking as a reference the Slater determinant formed with the NOs of a previous PNOF calculation. The NOF-MP2 method is used to analyze the potential energy surface (PES) and the binding energy for the symmetric dissociation of the water molecule, and compare it with accurate wavefunction-based methods.

How to reveal the nature of three or more pentaquark states

Within the chiral unitary approach and with the constraints of heavy quark spin symmetry, we study the coupled channel interactions of ${\bar D}^{(*)}\Sigma_c^{(*)}$ channels, close to whose thresholds three pentaquark-like $P_c$ states have been reported by the LHCb Collaboration. In the present work, we take into account the contributions of pion exchanges via box diagrams to the interaction potentials, and therefore lift the degeneracy in the masses of ${\bar D}^*\Sigma_c^{(*)}$ spin multiplets. Fitting the $J/\psi p$ invariant mass distributions in the $\Lambda_b^0 \to J/\psi K^- p$ decay, we find that the LHCb pentaquark states can not be reproduced in the direct $J/\psi p$ production in the $\Lambda_b^0$ decay, and can only be indirectly produced in the final state interactions of the $\Lambda_b^0$ decay products, ${\bar D}^*\Sigma_c^{(*)}$, which further supports the nature of these states as $\bar{D}\Sigma_c$ molecules. Based on the fit results obtained, we study the partial decay widths/branching ratios to other decay channels, $\bar{D}^* \Lambda_c$, $\bar{D} \Lambda_c$, and $\eta_c N$, and the corresponding invariant mass distributions. The resonances with $J^P=\frac{1}{2}^-$, $P_c(4312)$, $P_c(4440)$ and the one of $\bar{D}^* \Sigma_c^*$ around 4500 MeV, have large partial decay width into $\eta_c N$, and thus, can be easily seen in the $\eta_c N$ invariant mass distributions. By contrast, the states with $J^P=\frac{3}{2}^-$, $P_c(4457)$, the (predicted) narrow $P_c(4380)$ and the bound state of $\bar{D}^* \Sigma_c^*$ with a mass of about 4520 MeV, do not decay into $\eta_c N$. Therefore, the $\eta_c N$ channel should be studied in future to provide further insights into the nature of these states, especially that of the $P_c(4440)$ and $P_c(4457)$.