Rh₁₉⁻: A high-spin super-octahedron cluster

Abstract

Probing atomic clusters with magic numbers is of supreme importance but challenging in cluster science. Pronounced stability of a metal cluster often arises from coincident geometric and electronic shell closures. However, transition metal clusters do not simply abide by this constraint. Here, we report the finding of a magic-number cluster Rh₁₉⁻ with prominent inertness in the sufficient gas-collision reactions. Photoelectron spectroscopy experiments and global-minimum structure search have determined the geometry of Rh₁₉⁻ to be a regular O_h‑[Rh@Rh₁₂@Rh₆]⁻ with unusual high-spin electronic configuration. The distinct stability of such a strongly magnetic cluster Rh₁₉⁻ consisting of a nonmagnetic element is fully unveiled on the basis of its unique bonding nature and superatomic states. The 1-nanometer–sized O_h-Rh₁₉⁻ cluster corresponds to a fragment of the face-centered cubic lattice of bulk rhodium but with altered magnetism and electronic property. This cluster features exceptional electron-spin state isomers confirmed in photoelectron spectra and suggests potential applications in atomically precise manufacturing involving spintronics and quantum computing.

A magic-number metal cluster Rh19 is discovered with regular Oh-[Rh@Rh12@Rh6] and unusual high-spin electronic configuration.

INTRODUCTION

Developing functional materials with anticipated stability, regular structures, and well-defined components is one of the most important subjects in chemistry. Metal-atomic clusters hold promise in a variety of applications because of their versatile properties that can be tailored by cluster structure, size, and composition. Ongoing efforts in this field have been devoted to synthesizing monolayer-protected metal clusters and exploring pure metal clusters at reduced sizes with atomic precision (1–3). By solving the Schrödinger equation for a spherical nucleus in average central potentials, many stable clusters have been predicted with magic numbers of delocalized valence electrons (4–6). Meanwhile, metal clusters also find enhanced stability for those of favorable geometry within Mackay icosahedrons (7), i.e., $N=1+{\displaystyle \sum _{i=1}^n}(10i^2+2)$, where n refers to the number of geometric shells, while N corresponds to magic numbers of atoms at 13, 55, 147···. Similar situations also allow metal clusters to exhibit magic numbers of atoms within Marks decahedra (8) and Leary tetrahedra (9, 10). However, it remains elusive to fully understand the diversity of atomic packing and stability of transition metal (TM) clusters.

Recent advances have led to the superatom concept (11–16), which enables us to better understand itinerant electrons with delocalized orbitals in metal clusters akin to isolated atoms. For example, Al₁₃ was found to exhibit an electronic configuration as a halogen atom (17, 18); in addition, the VNa₈ cluster was demonstrated as a magnetic superatom showing a filled d subshell with a magnetic moment at 5.0 Bohr magneton (μ_B) (19). Because of the relative activity and less-contracted radial extension of the outermost d and f electrons, heavy TM clusters may undergo a shape relaxation without restriction to spherical symmetry (20), and the filling of superatomic orbitals may not simply mimic atoms in terms of Pauli exclusion principle and Hund’s rule (21), showing a diversity of structures and electronic/magnetic properties (10).

Rhodium, with an unfilled d electron shell configuration of 4d⁸5s¹, allows for diverse coordination numbers and valence states (e.g., +1, +3, and + 4). As the Rh─O bond dissociation energy is much larger than the Rh─Rh bond (~405 versus ~235 kJ mol⁻¹), and Rh_n clusters can readily coordinate with O₂, it is challenging to prepare pure Rh_n clusters, especially the anions (22). It is therefore difficult to identify magic-number rhodium clusters by the oxygen-etching reactions (18, 23, 24). Among the few studies devoted to understanding the reactivity of pure rhodium clusters (25), size effects in the reactions with N₂O and alkanes were noted under single-collision conditions (26, 27), showing local minima of reaction rates at Rh₅⁺, Rh₁₉⁺, and Rh₂₈⁺. Inasmuch as rhodium and its compounds are ubiquitous in chemical synthesis and petroleum engineering, radiology, and dentistry, as well as electrodes and electrical devices, understanding the structural stability and size-dependent properties is crucial for their applications.

We have recently developed an integrated instrument of time-of-flight mass spectrometry (TOF-MS) and photoelectron spectroscopy (PES) (figs. S1 and S2) that provides experimental techniques for preparing pure metal clusters and studying their reactivity and electronic structures (28, 29). Here, we report the formation and reactions of anionic Rh_n⁻ (n = 3 to 33) clusters with a few typical gases (including O₂, CO₂, CH₄, and CH₃Br). A magic-number cluster Rh₁₉⁻ is identified, and its photoelectron spectra are measured. Through computational chemistry modeling, a high-spin super-octahedron structure of O_h Rh₁₉⁻ is determined, and the agreement between experiment and theory suggests the existence of a series of electron-spin state isomers (ESSIs) of the Rh₁₉⁻ cluster.

RESULTS

The finding of magic-number Rh₁₉⁻ cluster

The pure Rh_n⁻ (n = 3 to 33) clusters were prepared via a laser evaporation (LaVa) source and reacted with O₂, CO₂, etc. in the downstream flow tube reactor combined with the TOF-MS instrument. It is found that the reactions of Rh_n⁻ with O₂ are too reactive to identify magic-number species (fig. S3), but the reactions of Rh_n⁻ with CO₂ exhibit interesting size dependence, as shown in Fig. 1A. Notably, most of these Rh_n⁻ clusters react with CO₂ to form Rh_n(CO₂)_m⁻ adducts, whereas Rh_8,9⁻ and Rh₁₉⁻ are relatively inert. Especially noteworthy is the Rh₁₉⁻ cluster, which shows up progressively with an increased dose of the CO₂ reactant and becomes dominant in the mass distribution (for details, see fig. S4, ESI). To verify the size-dependent reactivity and distinct stability of the Rh₁₉⁻, we carried out additional tests for the reactions with CH₄ and CH₃Br. It turns out that the reaction of Rh_n⁻ with CH₃Br (Fig. 1B) exhibits a similar phenomenon as Rh_n⁻ reacting with CO₂, although the reactivity with CH₄ is too weak (fig. S5). To our surprise, Rh₁₉⁻ emerges again as the only dominant species in the mass spectra upon introducing a large dose of CH₃Br gas reactant (fig. S6). In contrast, the neighboring counterparts Rh_16–18⁻ are rather reactive with CH₃Br to form the products of Rh_nCHBr⁻ and Rh_nBr⁻. While several previous studies have been devoted to understanding the reactivity of rhodium clusters (22), a magic number of rhodium clusters have not been identified. Our experimental observations are reminiscent of the previous findings of C₆₀ and Al₁₃⁻ (17, 30) and reveal the prominent inertness of Rh₁₉⁻ in surviving sufficient gas collision reactions.

Fig. 1. MS observation.

Nascent mass distribution of rhodium cluster anions observed via TOF-MS, and after exposure to CO₂ (A) and CH₃Br (B) reactants in the downstream flow tube reactor, with different doses of the reactant gas (5% in helium) controlled by a 10 Hz pulsed valve, respectively. A 532-nm laser with a pulse energy of 15 to 35 mJ was used to generate the Rh_n⁻ clusters. The Rh_n⁻ are labeled with the number N.

Structure determination of octahedron Rh₁₉⁻

Figure 2A presents the PES of Rh₁₉⁻ taken at 355- and 266-nm laser excitation, respectively. The peaks at 2.72 and 3.45 eV are assigned to the vertical detachments of ground-state Rh₁₉⁻ to yield the ground state (X) and the first excited state (A) of Rh₁₉, respectively. As shown below, the distinct spectral bands of the experimental PES spectra are well consistent with the simulated photoelectron detachment energies (fig. S10 and table S4). The profile with a broadband is essentially associated with the dense electronic energy levels of the neutral Rh₁₉ cluster, especially the existence of ESSIs of Rh₁₉⁻ (infra vide).

Fig. 2. Determination of geometric structure.

(A) The experimental photoelectron spectra of the Rh₁₉⁻ clusters with a 355- and 266-nm laser, respectively. The peaks marked with X, A, B, and C are assigned to the electronic transition from the O_h Rh₁₉⁻ anion to the electron-detached Rh₁₉ neutral, while those marked with x₋₁, x₊₁, and x₊₂ are associated with the ESSIs of Rh₁₉⁻ and Rh₁₉ (table S2). (B) The lowest-energy structure of Rh₁₉⁻ determined by USPEX and TGMin, along with the van der Waals radius and (C) the Hirshfeld charge population. (D) Calculated spin density population of the Rh₁₉⁻ cluster. (E) Structural evolution sketch of Rh₁₉⁻ based on an Rh@Rh₁₂ coordination, cubic Rh@Rh₈ kernel coordination, octahedral Rh₆ fusing, and tetrahedron Rh₄ interpenetration.

To determine the structure and unveil the origin of distinct inertness of the Rh₁₉⁻ cluster, we conducted first-principles global-minimum (GM) search for the ground-state structure using the ab initio evolutionary algorithm of Universal Structure Predictor-Evolutionary Xtallography (USPEX) (31) embedded in Vienna Ab Initio Simulation Package (VASP) (32) and then optimized the structure of Rh₁₉⁻ with Gaussian 09 software (fig. S8). The global lowest energy structure of Rh₁₉⁻ corresponds to a regular octahedron, with a central Rh atom (Rh_c), 12 edge atoms (Rh_e), and 6 apex Rh atoms (Rh_a) anchored at the six square-surface centers of the Rh@Rh₁₂ moiety. Meanwhile, an independent basin-hopping GM search by using TGMin code (33) confirmed the lowest energy structure of Rh₁₉⁻ within O_h symmetry and located a number of low-lying structural isomers. Notably, the density functional theory calculations indicate a ground state of 10 spin-unpaired electrons (figs. S9 and S10) with a calculated spin magnetic moments of 10.95 μ_B for the ground-state Rh₁₉⁻ (S = 10/2 with 0.58 μ_B per atom), which agrees well with the previously measured magnetic moments of 11.59 μ_B for the Rh₁₉ neutral (i.e., S = 11/2 with 0.61 ± 0.08 μ_B per atom) (34). The geometric structure (Fig. 2B) of the O_h Rh₁₉⁻ shows a van der Waals radius of 11.2 Å, suggesting a 1 nm sized magnet of the Rh₁₉⁻ cluster.

From the energetics analysis (figs. S14 and S16), the highest occupied molecular orbital–lowest unoccupied molecular orbital (LUMO) gap and average Rh-Rh binding energy of the O_h Rh₁₉⁻ do not correspond to maximal values; however, the O_h Rh₁₉⁻ bears the third largest vertical electron detachment energy (VDE) and second largest adiabatic electron detachment energy among all the Rh_n⁻ (n = 2 to 20) clusters. Furthermore, the O_h Rh₁₉⁻ shows the largest effective coordination number among these Rh_n⁻ (n = 2 to 20) clusters (fig. S17); in addition, the O_h Rh₁₉⁻ shows the smallest energy gain for a CH₃Br molecule adsorbed on the Rh_n⁻ (n = 6, 17 to 20) clusters, which is consistent with the prominent inertness of Rh₁₉⁻ in the experimental observation. From the Hirshfeld population analysis (Fig. 2C and table S9), Rh₁₉⁻ shows positive charge distribution on the core Rh_c atom, while negative fractional charges are evenly distributed on the 12 Rh₁₂ shell atoms, and the 6 Rh_e peripheral atoms are almost electroneutral. Further, the spin density analysis (Fig. 2D) shows that the unpaired spins mainly locate on the core-shell Rh@Rh₁₂ instead of the six peripheral Rh_a atoms. The balanced electrostatic interactions and spin density between the Rh_c atom and the shell Rh_e atoms contribute to the stability of the [Rh@Rh₁₂]⁻ frame, while the capped six Rh_a atoms resemble octahedral coordination ligands, thus rendering enhanced stability of such a high-spin octahedron cluster without being subjected to Jahn-Teller distortion. This is in contrast to the other Rh_n⁻ clusters of lower symmetry (fig. S13).

From Born-Oppenheimer molecular dynamics simulations, the regular octahedral structure of Rh₁₉⁻ is dynamically stable up to 900 K (fig. S21). Whereas a few other 19-atom clusters were found to prefer a D_5h double-icosahedron structure (fig. S18) (35), the D_5h Rh₁₉⁻ is 1.88 eV higher in energy than the O_h structure (fig. S19). The O_h Rh₁₉⁻ has a smaller edge length (O_h 61.40 versus D_5h 123.75 Å), also a smaller surface area (92.36 versus 98.25 Ų) and a smaller volume (63.26 versus 68.05 ų) in geometry. The formation of a super-octahedron structure is consistent with the stacking principle based on generalized Wulff construction (36), which rationalizes the stability of such a magic-number cluster within O_h symmetry (37, 38).

Figure 2E shows the structure evolution of Rh₁₉⁻ based on different hypothetical growth modes by assuming that it begins with an Rh₁₃, Rh₉, Rh₆, or Rh₄ moiety and then grows up via vertex sharing, capping, or fusing. First, the octahedral Rh₁₉⁻ is viewed as an Rh₁₃⁻ motif capped by six Rh atoms (Rh₁₃@Rh₆), where the anchoring of six Rh_a atoms functions as protection to stabilize the cluster. The anchoring of six Rh_a atoms not only lowers the surface energy of square surfaces but also helps to balance the charge distribution thus passivating the Rh@Rh₁₂ moiety (39, 40). Note that a metallic core of 13 atoms is also involved in many ligand-protected nanoclusters (41), showing diverse ligand accommodation and preferential growth on such a primitive cluster unit. In addition, Rh₁₉⁻ can be viewed as Rh₉@Rh₆@Rh₄, for which a body-centered cubic Rh@Rh₈ is formed first, and then the six faces are capped by six Rh atoms (two Rh_a and four Rh_e atoms), followed by the addition of other four Rh_a atoms on the waist. This channel appears to be feasible by noting the relatively larger mass abundances of Rh₈⁻ and Rh₉⁻ in the MS experiments (Fig. 1). On the other hand, the O_h-Rh₁₉⁻ may also be viewed as the fusing of six Rh₆ units, or tetrahedral Rh₄ interpenetration, showing the likely multiple pathways in forming such an octahedral cluster.

Electronic configuration analysis

We further studied the electronic configuration of Rh₁₉⁻ to fully understand the origin of its distinctive stability. Figure 3A presents the projected densities of states (PDOS) for the α-orbitals (majority spin) of Rh₁₉⁻ (details in fig. S24), where the contributions by the 5s and 4d orbitals of the three types of atoms (Rh_c, Rh_a, and Rh_e) are displayed. It is seen that the 4d orbitals show a dominant contribution to the total DOS, while the Rh_e-5s (red line) and Rh_a-5s (red dot line) contribute to the low-energy states and frontier states. Some of the frontier orbitals exhibit diffused superatomic orbital features (fig. S26), which is consistent with the results of the adaptive natural density partitioning (42) and electron localization function analyses (figs. S20 and S25). Among the delocalized superatomic orbitals in Rh₁₉⁻, the dominant contribution to the superatomic 1S orbital is made by the 5s electrons of the Rh_c atom, while the 4d-based orbitals contribute to the superatomic 2S, 2P, 3S, and 3P orbitals (table S8). Especially interesting is that the superatomic 2S orbital is mainly contributed by the d_z² orbitals of the 12 Rh_e atoms, while the superatomic 3S is dominated by the d_z² orbitals of the 6 Rh_a atoms in a local coordinate system (table S8). It is inferred that the electronic configuration within unique superatomic orbitals significantly reinforces the structure stability of the Rh₁₉⁻ cluster.

Fig. 3. Electronic structure.

PDOS and orbitals of Rh₁₉⁻, with different colors corresponding to the total density of states (TDOS) and contributions by the Rh_c-5s, Rh_a-5s, Rh_e-5s, Rh_c-4d, Rh_a-4d, and Rh_e-4d orbitals, respectively. Insets: Typical orbitals with superatomic characters corresponding to jellium 1S, 2S, 1P, 3S, and 2P shells for the Rh₁₉⁻ clusters. Iso value = 0.02 unless otherwise stated. HOMO, highest occupied molecular orbital.

In addition to the distinct superatomic feature (11–13), the novelty of Rh₁₉⁻ also lies in its remarkable spin-unpaired electrons pertaining to a magnetic superatom of a cluster consisting of fully nonmagnetic Rh atoms (43–45). Notably, bulk rhodium is also nonmagnetic. The origin of high-spin state of Rh₁₉⁻ is supposed to be a unity of opposites of superatomic itinerant electrons and localized d electrons. Such a high-spin magic-number cluster (S = 10/2) may have intriguing properties for spintronics and quantum information processing (e.g., photon or magnetic-field manipulated spin decoherency) (46). This cluster with 10 spin unpaired electrons could be a mimic of the predicted element Utn (Z = 130) (47).

Chemical bonding analysis

Figure 4 depicts Kohn-Sham molecular orbital energy–level correlations between [Rh@Rh₁₂]⁻ and [Rh₆] to unveil the bonding nature of O_h Rh₁₉⁻ (figs. S27 to S31). As the symmetry changes, the 5s-based orbitals with a_1g + t_1u + t_2g + e_g + t_2u symmetries are split into two series. That is, the occupied bonding orbitals with a_1g and t_1u symmetries and vacant antibonding orbitals with t_2g, a_1g, e_g, and t_2u symmetries. Among them, the bonding 5s-based orbitals (6a_1g + 11t_1u) are dominated by the core-shell [Rh@Rh₁₂]⁻ moiety, of which the stability can be reinforced by strong interactions with capped Rh_a atoms, pertaining to superatom-atom bonding nature analogous to that in Au₂₀ (10, 48). Compared to the small energy-level splitting of a_1g, t_1u and e_g in the shell [Rh₆] unit as a result of weak orbital overlaps between the Rh_a atoms, the bonding and antibonding orbitals in the [Rh@Rh₁₂@Rh₆]⁻ cluster display a wide energy region with an obvious energy gap indicative of strong Rh_e-Rh_c and Rh_e-Rh_e interactions. All the 5s-based bonding orbitals are fully occupied within an eight-electron configuration of (a_1g)²(t_1u)⁶, whereas all the antibonding orbitals are vacant for Rh@Rh₁₂, which obeys the 6n + 2 rule of cubic aromaticity (49, 50). In this regard, we have calculated the nucleus-independent chemical shift (NICS) values and found that the NICS(1)_zz is up to −38.7 parts per million (table S11), showing decent aromaticity of this cluster.

Fig. 4. Bonding analysis.

Kohn-Sham MO energy-level correlation diagram of the O_h [Rh@Rh₁₂@Rh₆]⁻ cluster. The insets show the contour plots (Iso value = 0.03) of the 5s-based orbitals (6t_1u and 4a_1g) in the kernel O_h-[Rh@Rh₁₂]⁻ unit (left) in comparison with the related orbitals (11t_1u and 6a_1g) in the [Rh@Rh₁₂@Rh₆]⁻ cluster (right). An illustration in the upper right corner shows the superatom-atom interaction pattern of Rh₁₃@Rh₆. SOMO, singly occupied molecular orbital; ADE, adiabatic electron detachment energy.

Apart from the 5s manifold of orbitals, the 4d-based orbitals of [Rh@Rh₁₂@Rh₆]⁻ can be divided into three series (fig. S28): 77 fully occupied orbitals, 8 vacant orbitals, and 10 singly occupied molecular orbitals (SOMOs) that mainly derive from the [Rh@Rh₁₂]⁻ moiety. The ground state has a stable half-filled open-shell configuration of (10t_2g)³(15t_1u)³(4e_u)²(11e_g)² with S = 10/2. The energy levels of unoccupied orbitals provided by the outer atoms all increase during the linear superposition, accompanied by degressive occupied orbital energy levels, thereby reducing the total energy of the system. Compared to the vacant antibonding orbitals in [Rh@Rh₁₂@Rh₆]⁻, the SOMO also corresponds to an antibonding orbital but with an enlarged energy gap (0.92 eV between the α-SOMO and LUMO at the B3LYP level) compared with the kernel O_h-[Rh@Rh₁₂]⁻ moiety. Notably, there is strong spin polarization in this system, leading to, for instance, the splitting of α- and β-MO of 7t_1g as large as 0.86 eV.

DISCUSSION

It is worth noting that the PES of Rh₁₉⁻ cluster exhibits electronic hot bands before the first major electronic transition labeled as X (VDE₁). As revealed by the Perdew–Burke–Ernzerhof (PBE) functional calculations, while the Rh₁₉⁻ cluster has a high-spin ground state S = 10/2, the other spin states due to different electron spin coupling (S = 0, 1, …, 10/2, 12/2…,16/2) lie in a small energy range, indicating the existence of numerous nearly degenerate ESSIs of the Rh₁₉⁻ cluster. Therefore, the broad PES hot bands can be understood via the electronic transitions from thermally populated ESSI levels of the Rh₁₉⁻ isomers to the corresponding neutral Rh₁₉ following the PES selection rule. To illustrate the electronic hot band mechanism of the PES for such clusters, Fig. 5 sketches the electronic transitions from the O_h Rh₁₉⁻ ground state to the electron-detached Rh₁₉ neutral. Because of the near degeneracy of the spin states of the anion the neutral, the ESSIs of the anion are populated approximately on the basis of Boltzmann distribution (with an ESSI population ratio n_j/n_i = e^{−(ɛ_j−ɛ_i)/kT}). At room temperature (kT ~200 cm⁻¹), there is likely a comparable time scale for vibrational relaxation, spin conversion, and vertical detachment to form the ESSIs of electron-detached neutral Rh₁₉. The diversity of temperature-dependent population of the electronic and vibrational states of the high-spin clusters perplexes their PES measurements, reflected in the ESSIs of both cluster anion and the electron-detached neutral, as well as accelerative vibrations for a certain long-lived excited state. Notably, the super-octahedron Rh₁₉⁻ cluster has a spatial O_h symmetry but with different spin couplings at finite temperature. As the total wave function Ψ(R,S) = Ψ(R)Ψ(S) involves both electron-spin (S) isomers and likely low-lying spatial-structure (R) isomers, the PES feature of this cluster is rather complex, and concomitant electronic detachments from the populated ESSIs of the anion are inevitable for such high-spin clusters. It is therefore important to take ESSIs into account in the PES identification. In addition, the interactions between the spatial and spin coordinates are vital to understanding spin-orbit effects and scalar relativistic (SR) effects of TM clusters (51).

Fig. 5. PES mechanism involving ESSIs.

A sketch showing the electronic transition from the ESSIs of O_h Rh₁₉⁻ anion to the electron-detached Rh₁₉ neutral, both electronic and vibrational states involved. The curves labeled as S₀, S₀′, and S₁′ correspond to the electronic ground states and the first excitation state; the dot curves refer to ESSIs; and the horizon lines with sine wave mean vibrational states of the Rh₁₉⁻ anion and the Rh₁₉ neutral. All the drop arrow lines depict the electronic transition between two states.

In summary, a magic-number TM cluster Rh₁₉⁻ is found with a super-octahedron structure that features high stability and multiple nearly-degenerate ESSIs. Such a highly magnetic O_h Rh₁₉⁻ provides superatomic building blocks of 1 nm sizes for developing previously unidentified magnetic micromaterials; the [Rh@Rh₁₂@Rh₆]⁻ structure bears six unsaturated apex Rh atoms indicating ideal active sites for catalysis. This high-spin Rh₁₉⁻ cluster shows a perfect example of the coexistence of multiple ESSIs while keeping the same spatial structure. This kind of stable metal-atomic clusters may find interest in atomically precise manufacturing involving quantum computing and spintronics.

MATERIALS AND METHODS

Experimental

The experiments were carried out by using a customized reflection time-of-flight mass spectrometer combined with the newly developed pulsed LaVa source, a compact flow tube reactor, and a 45° ion-extraction photoelectron velocity map imaging apparatus (29, 52, 53). A translatable and rotating rhodium disk (99.95%, ɸ = 16 mm) was used as the target of laser ablation to generate the Rh_n⁻ clusters, with He (99.999%, 1 MPa) as a buffer gas. A 532-nm laser with a pulse energy of 15 to 35 mJ was used for the laser ablation. Downstream the cluster beam, different amounts of O₂, CO₂, CH₄, and CH₃Br (~5% in He, 0.1 MPa) were injected by a pulsed valve to react with rhodium clusters in the flow tube reactor.

Computational

Unbiased GM structure search to determine the ground-state structure of the Rh₁₉⁻ cluster was conducted by using both TGMin code (33) and ab initio evolutionary algorithm USPEX (31) embedded in the VASP software (32). The generalized gradient approximation with the PBE exchange-correlation functional was used for the energetics calculations, while orbital analysis and orbital energy–level correlation diagram are based on hybrid functional B3LYP in Gaussian and ADF software packages, respectively. Relativistic quantum chemical studies were performed, with SR effects considered by zero-order regular approximation to account for the mass velocity and Darwin effects.

Supplementary Materials

This PDF file includes:

Supplementary Text

Figs. S1 to S31

Tables S1 to S11

References