On the spectrum of hypergraphs

Here we study the spectral properties of an underlying weighted graph of a non-uniform hypergraph by introducing different connectivity matrices, such as adjacency, Laplacian and normalized Laplacian matrices. We show that different structural properties of a hypergrpah, can be well studied using spectral properties of these matrices. Connectivity of a hypergraph is also investigated by the eigenvalues of these operators. Spectral radii of the same are bounded by the degrees of a hypergraph. The diameter of a hypergraph is also bounded by the eigenvalues of its connectivity matrices. We characterize different properties of a regular hypergraph characterized by the spectrum. Strong (vertex) chromatic number of a hypergraph is bounded by the eigenvalues. Cheeger constant on a hypergraph is defined and we show that it can be bounded by the smallest nontrivial eigenvalues of Laplacian matrix and normalized Laplacian matrix, respectively, of a connected hypergraph. We also show an approach to study random walk on a (non-uniform) hypergraph that can be performed by analyzing the spectrum of transition probability operator which is defined on that hypergraph. Ricci curvature on hypergraphs is introduced in two different ways. We show that if the Laplace operator, $\Delta$, on a hypergraph satisfies a curvature-dimension type inequality $CD (\mathbf{m}, \mathbf{K})$ with $\mathbf{m}>1$ and $\mathbf{K}>0$ then any non-zero eigenvalue of $- \Delta$ can be bounded below by $\frac{ \mathbf{m} \mathbf{K}}{ \mathbf{m} -1 }$. Eigenvalues of a normalized Laplacian operator defined on a connected hypergraph can be bounded by the Ollivier's Ricci curvature of the hypergraph

Fluorite Topology in Lanthanoid Coordination Polymers with Di- and Trimetallic Building Blocks

Reaction of the pseudotetrahedral tetracarboxylic acid proligand tetrakis­(4-carboxyphenyl)­silane (H₄L) with salts of various lanthanoid metals has afforded three new lanthanoid coordination polymers: {[H₃O]₂[Ce₂(L)₂(H₂O)₂]·2DMF·2H₂O}_∞ (<b>1</b>), {[Eu₃L₂(NO₃)­(DMF)₄(H₂O)]·0.5DMF·6H₂O}_∞ (<b>2</b>), and {Eu­(HL)­(DMF)₂(H₂O)]·DMF·2.5H₂O}_∞ (<b>3</b>) (DMF = <i>N</i>,<i>N</i>′-dimethylformamide), which have been structurally characterized by single-crystal X-ray diffraction. Compounds <b>1</b> and <b>2</b> are both noninterpenetrated three-dimensional networks that display the rare fluorite (CaF₂) topology. In compound <b>1</b>, the fully deprotonated L^4– ligands act as pseudotetrahedral 4-connecting nodes and dinuclear cerium-carboxylate building blocks act as 8-connecting nodes, while in compound <b>2</b> the L^4– ligands and trinuclear europium-carboxylate units are 4- and 8-connecting, respectively. In contrast, compound <b>3</b> exhibits a two-dimensional layered structure with triply deprotonated HL^3– ligands acting as 3-connecting units, linking single europium centers within each layer. Gas sorption studies of <b>2</b> show a high affinity of the pretreated microcrystalline solid for carbon dioxide gas

Valence Tautomerism in One-Dimensional Coordination Polymers

The combination of the divergent bis-pyridyl linking ligands 1,2-bis­(4-pyridyl)­ethane (1,2-bpe), 4,4′-<i>trans</i>-azopyridine (azpy), and 1,3-bis­(4-pyridyl)­propane (1,3-bpp) with cobalt and 3,5-di-<i>tert-</i>butyldioxolene (3,5-dbdiox) ligands has afforded the complexes [Co­(3,5-dbdiox)₂(1,2-bpe)]_∞ (<b>1</b>), [Co­(3,5-dbdiox)₂(azpy)]_∞ (<b>2</b>), [<i>trans</i>-Co­(3,5-dbdiox)₂(1,3-bpp)]_∞ (<b>3a</b>), and [<i>cis</i>-Co­(3,5-dbdiox)₂(1,3-bpp)]_∞ (<b>3b</b>). All species are 1D coordination polymers that crystallize as solvated forms; the geometric isomers <b>3a</b>,<b>b</b> cocrystallize. Complexes <b>1</b>, <b>2</b>, and <b>3a</b> exhibit around the Co centers a trans disposition of the N-donor atoms from the pyridyl linkers, while an unusual cis disposition is evident in <b>3b</b>. Single-crystal X-ray structural analysis at 100 or 130 K of solvated forms of these complexes indicates that all complexes possess the {Co^III(3,5-dbcat)­(3,5-dbsq)} (3,5-dbcat = 3,5-di-<i>tert</i>-butylcatecholate; 3,5-dbsq = 3,5-di-<i>tert</i>-butylsemiquinonate) charge distribution at the temperature of data collection. Variable-temperature magnetic susceptibility studies reveal that <b>1</b>, <b>1</b>·1.5MeCN·2H₂O, <b>2</b>·2EtOH, and <b>3</b>·MeCN·H₂O (<b>3</b> = <b>3a</b>·<b>3b</b>) all exhibit thermally induced valence tautomeric (VT) transitions above 200 K. Multiple heating and cooling cycles indicate that in some cases the behavior is strongly dependent on desolvation processes. Most notably, further desolvation of <b>1</b>·1.5MeCN·2H₂O above 340 K affords χ_m<i>T</i> values that suggest unusual ferromagnetic coupling in the {<i>hs</i>-Co^II(3,5-dbsq)₂} valence tautomer. Compound <b>3</b>·MeCN·H₂O exhibits a two-step VT transition that may be ascribed to the presence of the cis and trans geometric isomers. Compounds <b>1</b>, <b>1</b>·1.5MeCN·2H₂O, <b>2</b>·2EtOH, and <b>3</b>·MeCN·H₂O all also exhibit a single photoinduced VT transition, comparable to those generally observed for nonpolymeric cobalt–dioxolene complexes

Valence Tautomerism in One-Dimensional Coordination Polymers

The combination of the divergent bis-pyridyl linking ligands 1,2-bis­(4-pyridyl)­ethane (1,2-bpe), 4,4′-<i>trans</i>-azopyridine (azpy), and 1,3-bis­(4-pyridyl)­propane (1,3-bpp) with cobalt and 3,5-di-<i>tert-</i>butyldioxolene (3,5-dbdiox) ligands has afforded the complexes [Co­(3,5-dbdiox)₂(1,2-bpe)]_∞ (<b>1</b>), [Co­(3,5-dbdiox)₂(azpy)]_∞ (<b>2</b>), [<i>trans</i>-Co­(3,5-dbdiox)₂(1,3-bpp)]_∞ (<b>3a</b>), and [<i>cis</i>-Co­(3,5-dbdiox)₂(1,3-bpp)]_∞ (<b>3b</b>). All species are 1D coordination polymers that crystallize as solvated forms; the geometric isomers <b>3a</b>,<b>b</b> cocrystallize. Complexes <b>1</b>, <b>2</b>, and <b>3a</b> exhibit around the Co centers a trans disposition of the N-donor atoms from the pyridyl linkers, while an unusual cis disposition is evident in <b>3b</b>. Single-crystal X-ray structural analysis at 100 or 130 K of solvated forms of these complexes indicates that all complexes possess the {Co^III(3,5-dbcat)­(3,5-dbsq)} (3,5-dbcat = 3,5-di-<i>tert</i>-butylcatecholate; 3,5-dbsq = 3,5-di-<i>tert</i>-butylsemiquinonate) charge distribution at the temperature of data collection. Variable-temperature magnetic susceptibility studies reveal that <b>1</b>, <b>1</b>·1.5MeCN·2H₂O, <b>2</b>·2EtOH, and <b>3</b>·MeCN·H₂O (<b>3</b> = <b>3a</b>·<b>3b</b>) all exhibit thermally induced valence tautomeric (VT) transitions above 200 K. Multiple heating and cooling cycles indicate that in some cases the behavior is strongly dependent on desolvation processes. Most notably, further desolvation of <b>1</b>·1.5MeCN·2H₂O above 340 K affords χ_m<i>T</i> values that suggest unusual ferromagnetic coupling in the {<i>hs</i>-Co^II(3,5-dbsq)₂} valence tautomer. Compound <b>3</b>·MeCN·H₂O exhibits a two-step VT transition that may be ascribed to the presence of the cis and trans geometric isomers. Compounds <b>1</b>, <b>1</b>·1.5MeCN·2H₂O, <b>2</b>·2EtOH, and <b>3</b>·MeCN·H₂O all also exhibit a single photoinduced VT transition, comparable to those generally observed for nonpolymeric cobalt–dioxolene complexes

Magnetic Excitations in Polyoxotungstate-Supported Lanthanoid Single-Molecule Magnets: An Inelastic Neutron Scattering and ab Initio Study

Inelastic neutron scattering (INS) has been used to investigate the crystal field (CF) magnetic excitations of the analogs of the most representative lanthanoid–polyoxometalate single-molecule magnet family: Na₉[Ln­(W₅O₁₈)₂] (Ln = Nd, Tb, Ho, Er). Ab initio complete active space self-consistent field/restricted active space state interaction calculations, extended also to the Dy analog, show good agreement with the experimentally determined low-lying CF levels, with accuracy better in most cases than that reported for approaches based only on simultaneous fitting to CF models of magnetic or spectroscopic data for isostructural Ln families. In this work we demonstrate the power of a combined spectroscopic and computational approach. Inelastic neutron scattering has provided direct access to CF levels, which together with the magnetometry data, were employed to benchmark the ab initio results. The ab initio determined wave functions corresponding to the CF levels were in turn employed to assign the INS transitions allowed by selection rules and interpret the observed relative intensities of the INS peaks. Ultimately, we have been able to establish the relationship between the wave function composition of the CF split Ln^III ground multiplets and the experimentally measured magnetic and spectroscopic properties for the various analogs of the Na₉[Ln­(W₅O₁₈)₂] family

Magnetic Excitations in Polyoxotungstate-Supported Lanthanoid Single-Molecule Magnets: An Inelastic Neutron Scattering and ab Initio Study

Inelastic neutron scattering (INS) has been used to investigate the crystal field (CF) magnetic excitations of the analogs of the most representative lanthanoid–polyoxometalate single-molecule magnet family: Na₉[Ln­(W₅O₁₈)₂] (Ln = Nd, Tb, Ho, Er). Ab initio complete active space self-consistent field/restricted active space state interaction calculations, extended also to the Dy analog, show good agreement with the experimentally determined low-lying CF levels, with accuracy better in most cases than that reported for approaches based only on simultaneous fitting to CF models of magnetic or spectroscopic data for isostructural Ln families. In this work we demonstrate the power of a combined spectroscopic and computational approach. Inelastic neutron scattering has provided direct access to CF levels, which together with the magnetometry data, were employed to benchmark the ab initio results. The ab initio determined wave functions corresponding to the CF levels were in turn employed to assign the INS transitions allowed by selection rules and interpret the observed relative intensities of the INS peaks. Ultimately, we have been able to establish the relationship between the wave function composition of the CF split Ln^III ground multiplets and the experimentally measured magnetic and spectroscopic properties for the various analogs of the Na₉[Ln­(W₅O₁₈)₂] family

Carbonate-Bridged Lanthanoid Triangles: Single-Molecule Magnet Behavior, Inelastic Neutron Scattering, and Ab Initio Studies

Optimization of literature synthetic procedures has afforded, in moderate yield, homogeneous and crystalline samples of the five analogues Na₁₁[{RE­(OH₂)}₃CO₃(PW₉O₃₄)₂] (<b>1-RE</b>; RE = Y, Tb, Dy, Ho, and Er). Phase-transfer methods have allowed isolation of the mixed salts (Et₄N)₉Na₂[{RE­(OH₂)}₃CO₃(PW₉O₃₄)₂] (<b>2-RE</b>; RE = Y and Er). The isostructural polyanions in these compounds are comprised of a triangular arrangement of trivalent rare-earth ions bridged by a μ₃-carbonate ligand and sandwiched between two trilacunary Keggin {PW₉O₃₄} polyoxometalate ligands. Alternating-current (ac) magnetic susceptibility studies of <b>1-Dy</b>, <b>1-Er</b>, and <b>2-Er</b> reveal the onset of frequency dependence for the out-of-phase susceptibility in the presence of an applied magnetic field at the lowest measured temperatures. Inelastic neutron scattering (INS) spectra of <b>1-Ho</b> and <b>1-Er</b> exhibit transitions between the lowest-lying crystal-field (CF) split states of the respective <i>J</i> = 8 and ¹⁵/₂ ground-state spin–orbit multiplets of the Ho^III and Er^III ions. Complementary ab initio calculations performed for these two analogues allow excellent reproduction of the experimental magnetic susceptibility and low-temperature magnetization data and are in reasonable agreement with the experimental INS data. The ab initio calculations reveal that the slight difference in coordination environments of the three Ln^III ions in each complex gives rise to differences in the CF splitting that are not insignificant. This theoretical result is consistent with the observation of multiple relaxation processes by ac magnetic susceptibility and the broadness of the measured INS peaks. The ab initio calculations also indicate substantial mixing of the <i>M</i>_<i>J</i> contributions to the CF split energy levels of each Ln^III ion. Calculations indicate that the CF ground states of the Ho^III centers in <b>1-Ho</b> are predominantly comprised of contributions from small <i>M</i>_<i>J</i>, while those of the Er^III centers in <b>1-Er</b> are predominantly comprised of contributions from large <i>M</i>_<i>J</i>, giving rise to slow magnetic relaxation. Although no direct evidence for intramolecular RE···RE magnetic coupling is observed in either magnetic or INS studies, on the basis of the ab initio calculations, we find noncollinear magnetic axes in <b>1-Er</b> that are coplanar with the erbium triangle and radially arranged with respect to the triangle’s centroid; thus, we argue that the absence of magnetic coupling in this system arises from dipolar and antiferromagnetic superexchange interactions that cancel each other out

Controlling Spin Crossover in a Family of Dinuclear Fe(III) Complexes via the Bis(catecholate) Bridging Ligand

Spin crossover (SCO) complexes can reversibly switch between low spin (LS) and high spin (HS) states, affording possible applications in sensing, displays, and molecular electronics. Dinuclear SCO complexes with access to [LS–LS], [LS–HS], and [HS–HS] states may offer increased levels of functionality. The nature of the SCO interconversion in dinuclear complexes is influenced by the local electronic environment. We report the synthesis and characterization of [{FeIII(tpa)}2spiro](PF6)2 (1), [{FeIII(tpa)}2Br4spiro](PF6)2 (2), and [{FeIII(tpa)}2thea](PF6)2 (3) (tpa = tris(2-pyridylmethyl)amine, spiroH4 = 3,3,3′,3′-tetramethyl-1,1′-spirobi(indan)-5,5′,6,6′-tetraol, Br4spiroH4 = 3,3,3′,3′-tetramethyl-1,1′-spirobi(indan)-4,4′,7,7′-tetrabromo-5,5′,6,6′-tetraol, theaH4 = 2,3,6,7-tetrahydroxy-9,10-dimethyl-9,10-dihydro-9,10-ethanoanthracene), utilizing non-conjugated bis(catecholate) bridging ligands. In the solid state, magnetic and structural analysis shows that 1 remains in the [HS–HS] state, while 2 and 3 undergo a partial SCO interconversion upon cooling from room temperature involving the mixed [LS–HS] state. In solution, all complexes undergo SCO from [HS–HS] at room temperature, via [LS–HS] to mixtures including [LS–LS] at 77 K, with the extent of SCO increasing in the order 1 2 3. Gas phase density functional theory calculations suggest a [LS–LS] ground state for all complexes, with the [LS–HS] and [HS–HS] states successively destabilized. The relative energy separations indicate that ligand field strength increases following spiro4– 4spiro4– 4–, consistent with solid-state magnetic and EPR behavior. All three complexes show stabilization of the [LS–HS] state in relation to the midpoint energy between [LS–LS] and [HS–HS]. The relative stability of the [LS–HS] state increases with increasing ligand field strength of the bis(catecholate) bridging ligand in the order 1 2 3. The bromo substituents of Br4spiro4– increase the ligand field strength relative to spiro4–, while the stronger ligand field provided by thea4– arises from extension of the overlapping π-orbital system across the two catecholate units. This study highlights how SCO behavior in dinuclear complexes can be modulated by the bridging ligand, providing useful insights for the design of molecules that can be interconverted between more than two states

Controlling Spin Crossover in a Family of Dinuclear Fe(III) Complexes via the Bis(catecholate) Bridging Ligand

Spin crossover (SCO) complexes can reversibly switch between low spin (LS) and high spin (HS) states, affording possible applications in sensing, displays, and molecular electronics. Dinuclear SCO complexes with access to [LS–LS], [LS–HS], and [HS–HS] states may offer increased levels of functionality. The nature of the SCO interconversion in dinuclear complexes is influenced by the local electronic environment. We report the synthesis and characterization of [{FeIII(tpa)}2spiro](PF6)2 (1), [{FeIII(tpa)}2Br4spiro](PF6)2 (2), and [{FeIII(tpa)}2thea](PF6)2 (3) (tpa = tris(2-pyridylmethyl)amine, spiroH4 = 3,3,3′,3′-tetramethyl-1,1′-spirobi(indan)-5,5′,6,6′-tetraol, Br4spiroH4 = 3,3,3′,3′-tetramethyl-1,1′-spirobi(indan)-4,4′,7,7′-tetrabromo-5,5′,6,6′-tetraol, theaH4 = 2,3,6,7-tetrahydroxy-9,10-dimethyl-9,10-dihydro-9,10-ethanoanthracene), utilizing non-conjugated bis(catecholate) bridging ligands. In the solid state, magnetic and structural analysis shows that 1 remains in the [HS–HS] state, while 2 and 3 undergo a partial SCO interconversion upon cooling from room temperature involving the mixed [LS–HS] state. In solution, all complexes undergo SCO from [HS–HS] at room temperature, via [LS–HS] to mixtures including [LS–LS] at 77 K, with the extent of SCO increasing in the order 1 2 3. Gas phase density functional theory calculations suggest a [LS–LS] ground state for all complexes, with the [LS–HS] and [HS–HS] states successively destabilized. The relative energy separations indicate that ligand field strength increases following spiro4– 4spiro4– 4–, consistent with solid-state magnetic and EPR behavior. All three complexes show stabilization of the [LS–HS] state in relation to the midpoint energy between [LS–LS] and [HS–HS]. The relative stability of the [LS–HS] state increases with increasing ligand field strength of the bis(catecholate) bridging ligand in the order 1 2 3. The bromo substituents of Br4spiro4– increase the ligand field strength relative to spiro4–, while the stronger ligand field provided by thea4– arises from extension of the overlapping π-orbital system across the two catecholate units. This study highlights how SCO behavior in dinuclear complexes can be modulated by the bridging ligand, providing useful insights for the design of molecules that can be interconverted between more than two states

Carbonate-Bridged Lanthanoid Triangles: Single-Molecule Magnet Behavior, Inelastic Neutron Scattering, and Ab Initio Studies

Optimization of literature synthetic procedures has afforded, in moderate yield, homogeneous and crystalline samples of the five analogues Na₁₁[{RE­(OH₂)}₃CO₃(PW₉O₃₄)₂] (<b>1-RE</b>; RE = Y, Tb, Dy, Ho, and Er). Phase-transfer methods have allowed isolation of the mixed salts (Et₄N)₉Na₂[{RE­(OH₂)}₃CO₃(PW₉O₃₄)₂] (<b>2-RE</b>; RE = Y and Er). The isostructural polyanions in these compounds are comprised of a triangular arrangement of trivalent rare-earth ions bridged by a μ₃-carbonate ligand and sandwiched between two trilacunary Keggin {PW₉O₃₄} polyoxometalate ligands. Alternating-current (ac) magnetic susceptibility studies of <b>1-Dy</b>, <b>1-Er</b>, and <b>2-Er</b> reveal the onset of frequency dependence for the out-of-phase susceptibility in the presence of an applied magnetic field at the lowest measured temperatures. Inelastic neutron scattering (INS) spectra of <b>1-Ho</b> and <b>1-Er</b> exhibit transitions between the lowest-lying crystal-field (CF) split states of the respective <i>J</i> = 8 and ¹⁵/₂ ground-state spin–orbit multiplets of the Ho^III and Er^III ions. Complementary ab initio calculations performed for these two analogues allow excellent reproduction of the experimental magnetic susceptibility and low-temperature magnetization data and are in reasonable agreement with the experimental INS data. The ab initio calculations reveal that the slight difference in coordination environments of the three Ln^III ions in each complex gives rise to differences in the CF splitting that are not insignificant. This theoretical result is consistent with the observation of multiple relaxation processes by ac magnetic susceptibility and the broadness of the measured INS peaks. The ab initio calculations also indicate substantial mixing of the <i>M</i>_<i>J</i> contributions to the CF split energy levels of each Ln^III ion. Calculations indicate that the CF ground states of the Ho^III centers in <b>1-Ho</b> are predominantly comprised of contributions from small <i>M</i>_<i>J</i>, while those of the Er^III centers in <b>1-Er</b> are predominantly comprised of contributions from large <i>M</i>_<i>J</i>, giving rise to slow magnetic relaxation. Although no direct evidence for intramolecular RE···RE magnetic coupling is observed in either magnetic or INS studies, on the basis of the ab initio calculations, we find noncollinear magnetic axes in <b>1-Er</b> that are coplanar with the erbium triangle and radially arranged with respect to the triangle’s centroid; thus, we argue that the absence of magnetic coupling in this system arises from dipolar and antiferromagnetic superexchange interactions that cancel each other out