Design of Spin-Frustrated Monomer-Type C60•− Mott Insulator

Spin-frustrated monomer-type Mott insulator C60 solids are discussed in this review article. For the C60 solids, the interfullerene center-to-center distance (r) is the key parameter that controls the competition between covalent bond-formation, itinerancy, and spin frustration. Eight C60 salts with various compositions and dimensionalities are reviewed. In all of these C60 salts except one, neither bond-formation nor long-range magnetic ordering was observed down to low temperatures. A plot of Weiss temperature (|ΘCW|) against r shows that |ΘCW| grows rapidly below r = 10.0 Å.

Strong geometrical spin frustration suppresses the classical long-range magnetic ordering of the Néel state, and allows the novel quantum states such as the quantum spin liquid (QSL) state for two-dimensional (2D) S = 1/2 antiferromagnets (AFs), as proposed by Anderson [1,2].The QSL phase is a quantum-disordered insulating phase, which has been theoretically predicted to have a ground state with many degenerate states [3], and, hence, to exhibit large spin entropy, even at 0 K.To obtain spin-frustrated materials, the geometries of spin lattice subject to contradictory constraints are crucial.The spin lattices-triangle (Figure 1a), kagome (Figure 1b), zigzag (Figure 1c), tetrahedron (Figure 1d), honeycomb (Figure 1e), and hyperkagome (Figure 1f)-have been discussed with the basic building block being the triangle [4].The magnetic exchange interaction J of spins is proportional to the square of transfer interaction t (Equation ( 1) ), where U is the on-site Coulomb repulsion energy, A small distance between two spins (r) induces a large t and eventually, a metallic band with itinerant electrons is formed.Equation (1) suggests that the QSL state may be adjacent to such an itinerant (metallic or superconducting (SC)) phase.When the distance between the spin sites in the triangular spin lattice becomes extremely large, all of the spins may act as Curie spins.With decreasing distance between the spins, spins feel frustration to each other and form a QSL or a compromised spin configurations which do not have spin frustration with geometry of spin lattice of 120 • (triangle, Figure 1g), 109 • (tetrahedron, Figure 1h), or collinear AF (AFC) (Figure 1i) structures [5][6][7][8][9][10][11].When the triangular spin lattice is distorted, the AF phase may get preferentially stabilized.Therefore, the QSL, AF, 120 • structure (Figure 1g), AFC, and metallic (or SC) phases compete with each other in a triangular spin lattice based on the parameters r, t, W, U, J, outer stimuli (temperature T, pressure P, magnetic field H, etc.), and the geometry of the spin lattice, where W is the bandwidth.
Catalysts 2018, 8, x FOR PEER REVIEW 2 of 37 decreasing distance between the spins, spins feel frustration to each other and form a QSL or a compromised spin configurations which do not have spin frustration with geometry of spin lattice of 120° (triangle, Figure 1g), 109° (tetrahedron, Figure 1h), or collinear AF (AFC) (Figure 1i) structures [5][6][7][8][9][10][11].When the triangular spin lattice is distorted, the AF phase may get preferentially stabilized.Therefore, the QSL, AF, 120° structure (Figure 1g), AFC, and metallic (or SC) phases compete with each other in a triangular spin lattice based on the parameters r, t, W, U, J, outer stimuli (temperature T, pressure P, magnetic field H, etc.), and the geometry of the spin lattice, where W is the bandwidth.In (j), t and t′ are interdimer transfer interactions between parallel dimers and perpendicular dimers, respectively, and t′/t represents the shape of the isosceles triangular spin lattice.In (k), t, t′, and t″ are interfullerene transfer interactions.Red arrows indicate spins.(j,k) were reproduced from [12].
The Curie-Weiss temperature ΘCW given by, is a parameter showing the easiness to access the QSL state, where, z, S, and kB are the number of the nearest neighbour sites, spin quantum number, and Boltzmann constant, respectively.The frustration index f defined by Equation ( 3), was proposed by Ramirez as a measure of the spin frustration [13,14].Strong spin-frustrated systems are thought to be those associated with f > 10.In Equation (3), Tm is the temperature at which magnetic ordering occurs.
However, QSL systems are scarce in materials with spin quantum numbers S > 1/2, even in the triangular spin lattices and kagome lattices with f ≥ 10 2 [30][31][32].Even for the triangular S = 1/2 spin systems with large |ΘCW| and f, no QSL materials have been prepared due to the difficulty in maintaining the precise geometry of spin-frustrated lattices at low temperatures [33][34][35].
The first real QSL candidate was a charge-transfer (CT) salt of a dimer-type Mott insulator κ-(ET)2Cu2(CN)3 [36], where ET is bis(ethylenedithio)tetrathiafulvalene (chemicals in this review are ellipsoid in (j) is an ET molecule and the black circle represents one spin site (ET) 2 .In (j), t and t are interdimer transfer interactions between parallel dimers and perpendicular dimers, respectively, and t /t represents the shape of the isosceles triangular spin lattice.In (k), t, t , and t" are interfullerene transfer interactions.Red arrows indicate spins.(j,k) were reproduced from [12].
The Curie-Weiss temperature Θ CW given by, is a parameter showing the easiness to access the QSL state, where, z, S, and k B are the number of the nearest neighbour sites, spin quantum number, and Boltzmann constant, respectively.The frustration index f defined by Equation ( 3), was proposed by Ramirez as a measure of the spin frustration [13,14].Strong spin-frustrated systems are thought to be those associated with f > 10.In Equation ( 3), T m is the temperature at which magnetic ordering occurs.
However, QSL systems are scarce in materials with spin quantum numbers S > 1/2, even in the triangular spin lattices and kagome lattices with f ≥ 10 2 [30][31][32].Even for the triangular S = 1/2 spin systems with large |Θ CW | and f, no QSL materials have been prepared due to the difficulty in maintaining the precise geometry of spin-frustrated lattices at low temperatures [33][34][35].The first real QSL candidate was a charge-transfer (CT) salt of a dimer-type Mott insulator κ-(ET) 2 Cu 2 (CN) 3 [36], where ET is bis(ethylenedithio)tetrathiafulvalene (chemicals in this review are shown in Figure 2) and [Cu 2 (CN) 3 − ] ∞ is a diamagnetic polymeric anion.The planar tridentate coordination of diamagnetic Cu(I) ions in [Cu 2 (CN) 3 − ] ∞ is the main driving force for the twodimensional (2D) triangular magnetic lattice that is composed of partially charged (ET) 2 •+ , since [Cu 2 (CN) 3 − ] ∞ has openings, and the arrangement of the anion openings is triangular due to the planar tridentate coordination of Cu(I) ions.The geometrical fit between a spin-site (ET) 2 •+ and the anion opening results in a triangular magnetic lattice (Figure 1j) according to a key-keyhole relation, where the key is the spin-site, (ET) 2 •+ , and the keyhole is the anion opening.
Catalysts 2018, 8, x FOR PEER REVIEW 3 of 37 shown in Figure 2) and [Cu2(CN)3 − ]∞ is a diamagnetic polymeric anion.The planar tridentate coordination of diamagnetic Cu(I) ions in [Cu2(CN)3 − ]∞ is the main driving force for the two-dimensional (2D) triangular magnetic lattice that is composed of partially charged (ET)2 •+ , since [Cu2(CN)3 − ]∞ has openings, and the arrangement of the anion openings is triangular due to the planar tridentate coordination of Cu(I) ions.The geometrical fit between a spin-site (ET)2 •+ and the anion opening results in a triangular magnetic lattice (Figure 1j) according to a key-keyhole relation, where the key is the spin-site, (ET)2 •+ , and the keyhole is the anion opening.
The features of the Mott insulator κ-(ET) 2 Cu 2 (CN) 3 are: (1) it has a nearly equilateral triangular lattice (t /t = 1.09) with very strong electron correlation (U/W = 0.93), ( 2) the QSL state is experimentally confirmed down to 20 mK [36,45,46], (3) the anisotropic SC state resides directly next to the QSL state without passing through the spin ordered AF state under pressure [47-49], (4) the transition from the QSL state to the metallic state shows positive pressure dependence, indicating that the residual spin entropy is in the QSL state [47][48][49][50], and (5) 13 C NMR measurements under hydrostatic pressure on the salt of ET with 13 C enriched at the central C=C bond ( 13 C-ET) indicate d-wave SC symmetry [51].Thus, a competition among the localized (and frustrated), itinerant, and exotic pairing of spins is manifested in this salt [36,[45][46][47][48][49][50][51][52][53].We have proposed designing principles for a QSL candidate residing next to the SC state based on the crystal, electronic, and spin structures for selected κ-(ET) 2 X (X = anion) including X = Cu 2 (CN) 3 as follows [54].
The requirements for a QSL state next to an itinerant state for κ-(ET) 2 X are: (1) the system has a low spin state (S = 1/2  6) the material must maintain weak energy dispersion along the weakest direction for the magnetic interactions of the 2D system, i.e., negligibly weak magnetic interaction perpendicular to the 2D magnetic layer in order to keep the geometry of spin-frustrated spin lattice down to low temperatures.In order to have such competition between localized and itinerant spins in a monomer-type Mott insulator, the best candidate will be the C 60 CT materials having triangular or hexagonal packing of C 60 •− molecules (Figure 1k) among several kinds of such monomer-type Mott insulators based on TTF, TSF, ET, TCNQ, etc. [67-69], where TTF, TSF, and TCNQ are well known donor or acceptor molecules in the CT salts [70].In these monomer-type Mott insulators, the upper-HOMO band for the CT solid of D  [71,72].In this review, the preparation, crystal and electronic structures, and the physical properties of spin-frustrated monomer-type Mott insulators of C 60 •− CT solids, namely (MDABCO + )(C 60 (7), and (DMI + ) 3 (C 60 •− )(I − ) 2 (8), and the design of the QSL systems will be discussed.

Characteristic Features of C 60 : Superconductors and Other Functions for C 60 Charge-Transfer Materials
Superconductivity (SC) is one of the most remarkable features of C 60 CT materials.An icosahedral C 60 molecule with I h symmetry has triply degenerate LUMO and LUMO + 1 orbitals with t 1u and t 1g symmetries, respectively.Such multiple degeneracy (N) contributes to the relaxation of the Mott criterion [73,74], i.e., U/W ~√N or an upper limit of U/W ~2.5, and the enhancement of the density of states at the Fermi level (D(ε F )) inducing high T c for SC [75], when C 60 is placed in the highly symmetric, i.e., cubic crystal field.Typical SC materials are represented as A 3 C 60 (A: alkali metal), e.g., Rb 3 C 60 (T c = 29 K [76]), Rb 2 CsC 60 (T c = 31 K [77]), and RbCs 2 C 60 (T c = 33 K [77]), with a face-centered cubic (fcc) structure.The structural and physical properties of A 3 C 60 and the related fullerene compounds were reviewed [73,[78][79][80].The critical temperature T c varies monotonously with the lattice constant, independent of the type of the alkali dopant [77,81].Thus far, about 40 SC materials have been synthesized with the highest T c of 33 K (RbCs 2 C 60 ) at normal pressure and 38 K (A15 or body-centered cubic (bcc) Cs 3 C 60 ) under a pressure of approximately 0.7 Gpa [82].The fcc phase of Cs 3 C 60 also shows SC (T c = 35 K) under an applied hydrostatic pressure of approximately 0.8 Gpa [83].
A 3 C 60 SCs show a dome-shaped curve of normalized T c versus lattice volume [82, 83,91], which may be a hallmark of the competition between electron-phonon attractive and electron-electron repulsive interactions.The observation of a Hebel-Slichter peak in the relaxation rate just below T c in NMR and µSR indicate a BCS-type isotopic gap [92,93].

Requirements for Spin-Frustrated Spin Lattice of C 60
•−

Competition among Bond-Formation, Itinerancy, Localization, and Frustration in Fulleride Solids
It can be easily seen that the close packing of C 60 •− leads to triangular (Figure 1k) or honeycomb (Figure 1e) spin lattices.In the C 60 system, there is another factor, namely, the bond-formation between C 60 •− molecules, which leads to competition among itinerancy, localization, and frustration.
When the interfullerene distance of the C 60 solids is small (r < 9.4 Å, which is indicated by blue dotted line in Figure 3a,b), the C 60 •− molecules tend to form dimers (9 and 10 in Figure 2b) [37,38] or polymers (11)(12)(13)(14)(15) in Figure 2b) by single or double bonds.Polymer 11 with K + is a 3D metal [39], while those with Rb + and Cs + are one-dimensional (1D) metals and become spin-density-wave insulators at low temperatures [40,100].Single carbon-carbon bonded polymer 12 in Na 2 CsC 60 seems to be an SC with T c about 3 K lower than that for the starting non-polymeric phase.Analogous lowering of T c at the formation of polymer 12 in Na 2 RbC 60 leads to very low T c or the absence of SC [41,101].The compound 13 is an insulator with high ionic conductivity [42], 14 is a diamagnetic semiconducting polymer [43], and 15 is a highly correlated metal [43].
Even when r > 9.4 Å, C 60 As can be seen in Figure 3a, C 60 compounds having an interfullerene distance in the range 9.4 Å < r < 10.0 Å at RT have not been extensively explored.It is highly plausible that the shorter interfullerene distance that is less than 9.7 Å instantaneously leads to bond-formation resulting in r ≤ 9.4 Å.
Above results indicate that, even at r ~9.9-10.1 Å, C 60 •− anion molecules dimerize, when the C 60 •− molecules are not properly protected against bond-formation.
The effective ) owing to the high polarizability of organic cation molecules, which is similar to that proposed for TCNQ CT solids [135].
The effect of orbital degeneracy on the Mott-Hubbard criterion leads to a conclusion that the Mott transition takes place at U/W = √ 3 or an upper limit of U/W ~2.5, attributed to triple degeneracy or negligible splitting of t 1u orbitals [73,94].Therefore, this relaxed Mott criterion is effective as long as the splitting of t 1u orbitals by Jahn-Teller distortion is not large enough, and forms one LUMO band.The upper limit of U/W requires W to be 0.27 eV, by considering the lowest value of U eff (calculated value-0.68 eV), which is about 2-3 times the calculated W value (0.10-0.15 eV for (TPC 0 )(MDABCO + )(C 60 •− )) by the AM1 method [12].A preliminary DFT calculation for this CT solid indicates a total W of about 0.48 eV at 160 K [12], which is still smaller than the estimated U eff and is not able to account for the metallic nature.Since the calculated W values in these systems are not reliable at present, only the ratio of overlap integrals or transfer interactions will be discussed in the following.

Packing of C 60 and Magnetic Interactions in Fulleride Solids
So far, no structure-magnetic property relationship, especially concerning the geometrical spin frustration and the QSL state has been studied for the triangular or hexagonally packed C 60

solids. Hexagonal packing of C 60
•− has been suggested for (tetramethylammonium)(C 60 polycrystals based on the postulated structure, from both the calculation of total energy for various arrangements of the component molecules and the observed powder diffraction pattern [136].The estimated r was 10.13 Å.The conductivity of the pellet sample was 10 −2 S•cm −1 and the effective magnetic moment of the complex was ~1.75 µ B in good agreement with the value for the system containing one S = 1/2 spin per formula unit.However, the susceptibility did not indicate AF interactions.(Na + )(C 60 were obtained by the diffusion method.C 60 , reductant CH 3 CH 2 SNa, and MDABCO•I were stirred in a PhCl 2 /PhCN mixture.TPC was dissolved in the obtained solution and n-hexane was layered.The diffusion was carried out over a period of two months to give black hexagonal prisms on the walls of the tube of sizes up to 0.5 × 2 × 2 mm 3 (Figure 4c) [146].
The formation of 3 can be well interpreted by two kinds of key-keyhole relation (Figure 4).In the first step, three TPC molecules (D  Figure 5 shows the size of MDABCO + (Figure 5a,b), the height of TPC (Figure 5c), and some parts of Supramolecule 1 (Figure 5d) at 200 K.The neighboring TPC molecules are separated by 9.82 Å, so that it is able to prevent the bond-formation between C60 •− molecules when they pack on the template in a hexagonal lattice structure, where the intermolecular distance of MDABCO + molecules corresponds to N•••N (blue points in Figure 4(a-2)) with separation distance of 9.99 Å.The MDABCO + molecule is thicker (7.27 Å) than the thickness of the holder, which is composed of three TPC molecules (7.05 Å).Consequently, one side of Supramolecule 1 has a concave shape where N atoms are centered and the Me of N-Me + group of MDABCO + extrudes from the holder on the opposite side.Therefore, the C60 •− hexagonal layer on the top of the polycationic template in Figure 4(a-2), namely Layer A (Figure 6a,b) has different steric and electronic environment than the C60 •− hexagonal layer at the bottom of the polycationic template (Layer B, Figure 6a,c).
Figure 7b shows one Supramolecule 4, (TPC 0 )3(MDABCO + )3(C60 •− ).The 2D assembly of     At 300 K, C60 •− molecules are ordered in Layer A (r = 10.07Å, overlap integral s = 1.91 × 10 −3 , Figure 8a), while C60 •− molecules in Layer B are disordered.At the same temperature, half of the MDABCO + cations are disordered between three orientations that are linked by their rotation about the lattice threefold axis.On Layer A, the r value decreases monotonously to 9.97 Å at 183 K at which temperature a transition from rhombohedral to triclinic occurs.Assuming linear shrinkage of the interfullerene distance along the a axis below 160 K, r = 9.54 Å is evaluated at around 4 K, where no dimerization was experimentally detected.The calculated Fermi surfaces of both C60 •− layers have a closed 2D pocket at Γ point and suggest 2D metallic nature in the ab plane (Figure 8b).At 300 K, C60 •− molecules are ordered in Layer A (r = 10.07Å, overlap integral s = 1.91 × 10 −3 , Figure 8a), while C60 •− molecules in Layer B are disordered.At the same temperature, half of the MDABCO + cations are disordered between three orientations that are linked by their rotation about the lattice threefold axis.On Layer A, the r value decreases monotonously to 9.97 Å at 183 K at which temperature a transition from rhombohedral to triclinic occurs.Assuming linear shrinkage of the interfullerene distance along the a axis below 160 K, r = 9.54 Å is evaluated at around 4 K, where no dimerization was experimentally detected.The calculated Fermi surfaces of both C60 •− layers have a closed 2D pocket at Γ point and suggest 2D metallic nature in the ab plane (Figure 8b).With decreasing temperature, ordering of the orientations of C60 •− and MDABCO + started below 200 K.A complete ordering of all three of the component molecules was found in the crystal structure at 160 K (Figure 6).These observations suggest that the orientation disorder of C60    With decreasing temperature, ordering of the orientations of C60 •− and MDABCO + started below 200 K.A complete ordering of all three of the component molecules was found in the crystal structure at 160 K (Figure 6).These observations suggest that the orientation disorder of C60   With decreasing temperature, ordering of the orientations of C 60 •− and MDABCO + started below 200 K.A complete ordering of all three of the component molecules was found in the crystal structure at 160 K (Figure 6).These observations suggest that the orientation disorder of C 60 The resistivity measurements carried out using the four probe method indicate that salt 3 is metallic within the ab plane from 360 (1.8 S•cm −1 ) to 200 K (14 S•cm −1 ), after which a rapid enhancement of the metallic nature occurs from 200 K to 185 K (33 S•cm −1 ) (Figure 9a).The resistivity measurements carried out using the four probe method indicate that salt 3 is metallic within the ab plane from 360 (1.8 S•cm −1 ) to 200 K (14 S•cm −1 ), after which a rapid enhancement of the metallic nature occurs from 200 K to 185 K (33 S•cm −1 ) (Figure 9a).The temperature range of this anomaly between 200 K and 185 K, indicated by two red lines, coincides well with that of an ordering of C60 •− in Layer B, showing that the ordered C60 •− radical anions in Layer B start to participate in the metallic transport below 200 K.The resistivity could not be measured correctly below 70 K because of a large increase in the contact resistance, due to its air-sensitivity, even though it was measured in an inert atmosphere.The contactless microwave and optical measurements revealed that the conductivity increased down to 100-25 K and the metallic state is preserved down to 5 K (Figure 9b) [12].The optical conductivity is nearly flat below 150 K down to 5 K.
The asymmetry ratio of the Dysonian EPR line shape between the maximum and minimum of the absorption derivative (A/B, Figure 9c inset) of 3 is compared with that of β-(ET)2I3 (Figure 9c) [151], which is highly metallic and shows SC with Tc = 1.5 K [152].For β-(ET)2I3, the EPR signal is Lorentzian (A/B = 1) at RT and the peak ratio A/B increases below 130 K to about A/B = 3.0 (at approximately 50 K) followed by a gradual decrease down to 5 K.In comparison, the A/B values of 3 are considerably large even at RT (A/B = 2.4).It exhibits an abrupt increase below 200 K reaching a maximum of 4.2 at 183 K, which coincides well with the rapid conductivity increase, then falls to 2.3-2.4 below 183 K.The A/B ratio slowly decreases below 100 K, but the Dysonian shape is observed even The temperature range of this anomaly between 200 K and 185 K, indicated by two red lines, coincides well with that of an ordering of C 60 •− in Layer B, showing that the ordered C 60 •− radical anions in Layer B start to participate in the metallic transport below 200 K.The resistivity could not be measured correctly below 70 K because of a large increase in the contact resistance, due to its air-sensitivity, even though it was measured in an inert atmosphere.The contactless microwave and optical measurements revealed that the conductivity increased down to 100-25 K and the metallic state is preserved down to 5 K (Figure 9b) [12].The optical conductivity is nearly flat below 150 K down to 5 K.
The asymmetry ratio of the Dysonian EPR line shape between the maximum and minimum of the absorption derivative (A/B, Figure 9c inset) of 3 is compared with that of β-(ET) 2 I 3 (Figure 9c) [151], which is highly metallic and shows SC with T c = 1.5 K [152].For β-(ET) 2 I 3 , the EPR signal is Lorentzian (A/B = 1) at RT and the peak ratio A/B increases below 130 K to about A/B = 3.0 (at approximately 50 K) followed by a gradual decrease down to 5 K.In comparison, the A/B values of 3 are considerably large even at RT (A/B = 2.4).It exhibits an abrupt increase below 200 K reaching a maximum of 4.2 at 183 K, which coincides well with the rapid conductivity increase, then falls to 2.3-2.4 below 183 K.The A/B ratio slowly decreases below 100 K, but the Dysonian shape is observed even at 4 K (A/B = 1.64), thus confirming the existence of a highly conducting state down to 4 K.
At 230-330 K, molar magnetic susceptibility (χ M ) can be fitted by a combination of the Pauli and Curie-Weiss terms: A major difference between MQ + (Figure 10a-c) and MDABCO + is that the vertical size of MQ + (7.87 Å) is larger than that of MDABCO + cation (7.27 Å) since the carbon atom with hydrogen in MQ + instead of uncharged nitrogen atom in MDABCO + that is caused some kind of distortion in the layered packing of C 60 molecules.The size of MQ + is 5.86 Å × 6.40 Å (Figure 10c), which is very close to that of MDABCO + .Figure 10d,e show the sizes of the fundamental units at 250 K. Similar to the MDABCO + salt, three TCP and one MQ + molecules form Supramolecule 5; (TPC 0 ) 3 (MQ + ), with a periodicity of TPC molecules of an average of 10.15 Å (Figure 10d) and periodicity of N-site in MQ + molecules of an average of 10.04 Å (Figure 10e).
Crystal 4 has lower symmetry (triclinic unit cell) than 3 at RT. Similar to 3, crystal 4 has layered packing in which hexagonal fullerene layers alternate with the (TPC 0 )(MQ + ) layers along the c axis (Figure 11a) in the sequence of (TPC 0 )(MQ + ) layer/C 60    From a different point of view, C60 •− molecules fit into the concaves in the polycationic template of layered unit of [(TPC 0 )(MQ + )]∞ (Figure 12c) to form Layer A in 4 (Figure 12d).Figure 12e,f illustrate how one C60 •− molecule in Layer A is embedded between two layers of (TPC 0 )(MQ + ) where the TPC molecules in the upper and lower layers are shown in different colors.C60 •− molecules in Layer A are well fitted in the TPC hole formed by the six TPC molecules in the upper and lower (TPC 0 )(MQ + ) layers (Figure 12e).In Supramolecule 5 [(TPC 0 )3(MQ + )] (Figure 10d), the extra hydrogen atom (red circle in Figure 10a) in MQ + prevents the MQ + cation from arranging vertically relative to the fullerene layers resulting in a lowered crystal symmetry.Figure 13g,h show the calculated Fermi surfaces by the tight-binding method combined with the semiempirical (AM1) molecular orbital calculations based on crystal structures at 250 K and 100 K, respectively.The band calculation of Layer A at 250 K indicates that the salt has 2D Fermi surfaces.However, due to the doubling of the unit cell along the b axis, a semi-metallic Fermi surface was estimated at 100 K.
Surprisingly, Layer B also has a hexagonal packing of C60 •− molecules in spite of the use of longer cation molecule MQ + by 0.60 Å than MDABCO + .Figure 13 shows the formation of Layer B similar to that in Figure 7. Figure 13a shows Supramolecule 7 by the first key-keyhole relation made of three TPC and three MQ + molecules.The C60 molecule fit into the hollow site where three MQ + molecules formed that corresponds to the crossing points of black lines in Figure 13c to form  From a different point of view, C60 •− molecules fit into the concaves in the polycationic template of layered unit of [(TPC 0 )(MQ + )]∞ (Figure 12c) to form Layer A in 4 (Figure 12d).Figure 12e,f illustrate how one C60 •− molecule in Layer A is embedded between two layers of (TPC 0 )(MQ + ) where the TPC molecules in the upper and lower layers are shown in different colors.C60 •− molecules in Layer A are well fitted in the TPC hole formed by the six TPC molecules in the upper and lower (TPC 0 )(MQ + ) layers (Figure 12e).In Supramolecule 5 [(TPC 0 )3(MQ + )] (Figure 10d), the extra hydrogen atom (red circle in Figure 10a) in MQ + prevents the MQ + cation from arranging vertically relative to the fullerene layers resulting in a lowered crystal symmetry.Figure 13g,h show the calculated Fermi surfaces by the tight-binding method combined with the semiempirical (AM1) molecular orbital calculations based on crystal structures at 250 K and 100 K, respectively.The band calculation of Layer A at 250 K indicates that the salt has 2D Fermi surfaces.However, due to the doubling of the unit cell along the b axis, a semi-metallic Fermi surface was estimated at 100 K.
Surprisingly, Layer B also has a hexagonal packing of C60 •− molecules in spite of the use of longer cation molecule MQ + by 0.60 Å than MDABCO + .Figure 13 shows the formation of Layer B similar to that in Figure 7. Figure 13a    The 2D assembly of supramolecular units [(TPC 0 )3(MQ + )3(C60 •− )] leads to Layer B on the (TPC 0 )(MQ + ) layer (Figure 13d).From another view, the C60 •− molecules assemble using polycationic template of 2D assembly of the first supramolecular units (Figure 13c) and leads to Layer B on the (TPC 0 )(MQ + ) layer (Figure 13d).
The relatively large distances between C60 •− prevent their dimerization but allow for the manifestation of a magnetic interaction between them.Reciprocal molar magnetic susceptibility is described well by the Curie-Weiss law in the 30-300 K range with negative Weiss temperature of ΘCW = −27 K (Figure 14c), indicating AF interaction of spins in the fullerene layers.The |ΘCW| is small, owing to the weaker AF interactions than that in 3 because of the larger interfullerene distance.In spite of the strong AF interaction of spins, magnetic ordering is not observed down to 1.9 K in this distorted triangular spin lattice system (f > 14).The resistivity measurements were The calculated overlap integrals at 100 K are s = 0.78 × 10 −3 (//a), 1.82 × 10 −3 (//b), and 2.24 × 10 −3 (//a + b) for Layer A and 2.81 × 10 −3 (//a), 1.97 × 10 −3 (//b), and 1.51 × 10 −3 (//a + b) for Layer B. The calculated bandwidths are 0.103 (0.112) and 0.097 (0.113) eV for Layer A and Layer B at 250 K (100 K), respectively.Similar to 3, the ratios 2t b /(t a + t a+b ) and 2(t a + t b )/(t a + t b ) are 0.85 and 1.17 for Layer A and 0.91 and 0.64 for Layer B at 100 K. Since the calculated Fermi surface shows 1D properties, it is more appropriate to use 2t a /(t a + t a+b ) instead of 2t b /(t a + t a+b ) and 2t a+b /(t a + t b ).The calculated anisotropy of the transfer interactions at 250 K is t a :t b :t a+b = 1.04:1:1 (t /t = 1.04) and 1:1.40:1.23 (t /t = 0.76) for Layer A and Layer B, respectively.The anisotropy changed to 1:0.90:1.11(t /t = 0.99) for Layer A and 1:0.70:0.54(t /t = 1.61) for Layer B at 100 K.The anisotropy of Layer A is close to that of κ-(ET) 2 Cu 2 (CN) 3 , and the geometrical spin frustration is comparable to that of 3.

The relatively large distances between C 60
•− prevent their dimerization but allow for the manifestation of a magnetic interaction between them.Reciprocal molar magnetic susceptibility is described well by the Curie-Weiss law in the 30-300 K range with negative Weiss temperature of Θ CW = −27 K (Figure 14c), indicating AF interaction of spins in the fullerene layers.The |Θ CW | is small, owing to the weaker AF interactions than that in 3 because of the larger interfullerene distance.
In spite of the strong AF interaction of spins, magnetic ordering is not observed down to 1.9 K in this distorted triangular spin lattice system (f > 14).The resistivity measurements were impossible owing to small size of the crystals.In summary, concerning the geometry of spin lattice of 4, the C 60  .43Å at 120 K) are essential within the C 60 layer and the ratio of the transfer interactions are t 1 :t 2 :t 3 = 0.9:1:0.5 for the major C 60 orientation.In spite of the hexagonal environment of the fullerenes in the layers, vdW C•••C contacts are formed with only three fullerene neighbors.Therefore, the geometry of model spin lattice is not triangular owing to weak magnetic interaction shown by dashed green lines (for r = 10.43 Å, s 3 = 0.66 × 10 −3 ) in Figure 15c.Blue (for r = 9.90 Å, s 1 = 1.25 × 10 −3 ) and red (for r = 10.20 Å, s 2 = 1.41 × 10 −3 ) lines represent the main interactions J 1 :J 2 :J 3 = 0.79:1:0.22.Even though the r value for blue line is much shorter than that of the red line, s 2 is larger than s 1 owing to more favorable orientation of C 60 for s 2 .The main magnetic interactions indicated by red lines extend along the b axis, and such 1D zigzag magnetic chains are connected by magnetic interactions along the c axis by blue lines.The unit of the spin lattice is edge-shared hexagonal, which is composed of four red lines and two blue lines (Figure 15c) and forms a 2D layer.
The temperature dependence of the reciprocal magnetic susceptibility for salt 5 is linear in the 70-300 K range with Θ CW = −11 K (f = 5.5, no dimerization of C 60 •− occurred).The resistivity at RT is symmetry, the C60 •− layer shows distorted hexagonal packing (Figure 15b).Further, the TMP + cations form pairs and are deeply embedded in the C60 •− layers to deform the C60 •− packing and are found near the center of the hexagonally arranged C60 •− molecules.Methyl groups of TMP + and PhCN molecules work to prevent the close approach of C60 •− molecules and no bond-formation between C60 •− was detected down to 2 K.The shortest r value along the interlayer direction is 10.39 Å at 120 K, indicating weak interlayer interactions, where the spin lattice should be 2D.Insoluble precipitates obtained by the reduction of C 60 with sodium fluorenone ketyl in PhCl 2 in the presence of Ph 3 PMeBr were dissolved by the addition of PhCN.n-Hexane was layered on the filtered solution to grow single crystals of (PhCN 0 )(Ph 3 MeP + )(C 60 •− ) (6) where Ph 3 MeP + has a threefold symmetry [105].The crystal structure at 250 K (Figure 16) shows hexagonal packing of C 60 and supramolecules (PhCN 0 )(Ph 3 PMe + ) are located in the centers of fullerene hexagon.Hence, C 60 •− has only three negatively charged fullerene neighbors similar to that observed in 5 (Figure 15b).The somewhat low value of r = 9.92, 9.96, and 10.07 Å may result in higher spin frustration than for 5.The lowest interlayer r is 10.15 Å suggesting strong 2D nature within the C 60 layer in Figure 16.The C 60 •− molecules a 1 -a 4 in Figure 16 form a flat layer, where the overlap integrals are s(a 1 −a 2 ) = 2.14 × 10 −3 , s(a 1 −a 3 ) = 2.29 × 10 −3 , s(a 1 −a 4 ) = 0.70 × 10 −3 , and s(a 2 −a 3 ) ~s(a 3 -a 4 ) ~s(a 2 -a 4 ) ~s(a 1 -a 1 ) = 0.The spin lattice geometry (Figure 16b) is approximated as the 1D nonuniform zigzag chain along the b axis with alternating red and blue lines and the lines are connected by weak magnetic interactions by green lines (ratio of J values = 1:0.87:0.09).Salt 6 shows much stronger 1D properties than those of salt 5.
The EPR intensity decreases from 295-220 K, smoothly followed by a rapid decrease due to reversible dimerization of C 60 •− below 220 K. Upon cooling down to 120 K, the C 60 •− radical anion pairs, which has r = 9.92 Å at 250 K, form singly bonded (C 60 − ) 2 dimers with r = 9.28 Å.Therefore, no Θ CW value is determined in this system.The steric protection to avoid the bond-formation is not sufficient in this solid.The solvent free crystal (Ph 3 MeP + )(C 60 •− ) (2) exhibits completely different structural and physical properties (vide infra) without the dimerization down to 1.9 K.
Insoluble precipitates obtained by the reduction of C60 with sodium fluorenone ketyl in PhCl2 in the presence of Ph3PMeBr were dissolved by the addition of PhCN.n-Hexane was layered on the filtered solution to grow single crystals of (PhCN 0 )(Ph3MeP + )(C60 •− ) (6) where Ph3MeP + has a threefold symmetry [105].The crystal structure at 250 K (Figure 16) shows hexagonal packing of C60 •− and supramolecules (PhCN 0 )(Ph3PMe + ) are located in the centers of fullerene hexagon.Hence, C60 •− has only three negatively charged fullerene neighbors similar to that observed in 5 (Figure 15b).The somewhat low value of r = 9.92, 9.96, and 10.07 Å may result in higher spin frustration than for 5.The lowest interlayer r is 10.15 Å suggesting strong 2D nature within the C60 layer in Figure 16.The C60 •− molecules a1-a4 in Figure 16  The EPR intensity decreases from 295-220 K, smoothly followed by a rapid decrease due to reversible dimerization of C60 •− below 220 K. Upon cooling down to 120 K, the C60 •− radical anion pairs, which has r = 9.92 Å at 250 K, form singly bonded (C60 − )2 dimers with r = 9.28 Å.Therefore, no ΘCW value is determined in this system.The steric protection to avoid the bond-formation is not sufficient in this solid.The solvent free crystal (Ph3MeP + )(C60 •− ) (2) exhibits completely different structural and physical properties (vide infra) without the dimerization down to 1.9 K.
The C 60 layer and {(PhCl 2 0 )[(Ph 3 P) 3 Au + ] 2 } layer, which is indicated by black arrows in Figure 17b, alternate along the c axis.C 60 molecules move from the layers toward planar PhCl 2 molecules to form strongly corrugated C 60 layers.
The fullerene layer consists of different charged C 60 molecules with −1 and 0 denoted as I and II, respectively, in Figure 17a-c).Interestingly, C 60 •− molecules are sandwiched between a (Ph 3 P) 3  the supramolecule {(PhCl2 0 )[(Ph3P)3Au + ]2} is a cationic template that accommodates C60 molecules hexagonally [149].The crystal structure was solved for a crystal slowly cooled down to 100 K. Hexagonal corrugated C60 layer is sandwiched between the layers of {(PhCl2 0 )[(Ph3P)3Au + ]2} along the c axis (Figure 17a).Fullerenes and PhCl2 molecules located on the C3v symmetry axes are statistically disordered between three orientations.The (Ph3P)3Au + cations are ordered and located on the C3v symmetry axis.The (Ph3P)3Au + cations are too large in size to fit into the size of a C60 molecule, but the size of supramolecule {(PhCl2 0 )[(Ph3P)3Au + ]2} approximately corresponds to that of three C60 molecules.The C60 layer and {(PhCl2 0 )[(Ph3P)3Au + ]2} layer, which is indicated by black arrows in Figure 17b, alternate along the c axis.C60 molecules move from the layers toward planar PhCl2 molecules to form strongly corrugated C60 layers.
The fullerene layer consists of different charged C60 molecules with −1 and 0 denoted as I and II, respectively, in Figure 17a-c).Interestingly, C60 •− molecules are sandwiched between a (Ph3P)3Au + cation molecule and a PhCl2 molecule while C60 0 molecules are sandwiched between two (Ph3P)3Au + molecules along the c axis, as shown in Figure 17b.Negatively charged and neutral C60 molecules are closely packed within hexagonal layers with r (I•••II) = 10.02Å, while between C60 •− , it is long Hexagonal packing of C 60 8) [145], where cationic template is delivered by a three-dimensional (3D) network of [(DMI + ) 3 (I − ) 2 ] (Figure 18), where DMI + is N,N'-dimethylimidazolium cation with no threefold symmetry.This system is not within the (D 1 0 )(D 2 + )(C 60 •− ) scheme, but the supramolecular cationic template [(DMI + ) 3 (I − ) 2 ] has threefold symmetry.The single crystals were obtained by a diffusion method.C 60 , an excess of DMI•I and reductant CH 3 CH 2 SNa were stirred in a PhCl 2 /PhCN mixture.The mixture was cooled and n-hexane was layered over the solution.The diffusion was carried out during one month to give the single crystals on the wall of the tube with the size of 1 × 1 × 0.5 mm 3 .
The 3D network of [(DMI + ) 3 (I − ) 2 ] is held together by the hydrogen (H)-bonds between the H atoms of DMI + cation (Figure 18a) and I − anion.The crystal structures at 100 K are shown in Figure 18b,c The top and bottom of Supramolecule 11 are capped by Supramolecule 10 along the c axis, yielding Supramolecule 13, [(Supramolecule 9)3(Supramolecule 10)(C60 •− )], as schematically shown in Figure 18(h-1), where C60 •− molecules labeled 1-7 form one layer (Layer 1).Similar capping As a result, geometry of the unit of model spin lattice is not triangular, but distorted tetragonal, i.e., tetrahedron with t /t ~0.The AF spin configuration is expected within a 1D Mott insulating C 60 •− chain of black-blue-black-blue-fullerenes along the c axis, while within a layer of blue or black fullerenes in the ab plane parallel spin configurations are expected.Such spin units form a column along the c axis and the columns are arranged in the bc plane (Figure 19a).The other spin unit composed of C 60 ] has the same distorted tetrahedral geometry and form similar 2D packing to that in Figure 19a.These 2D spin sheets are connected, for example, through 7 and 7 to form 3D network of magnetic interactions.It is emphasized here that according to the 3D template network composed of DMI + and I − molecules, the C 60 •− radical anions provide the 2H-hexagonal 3D packing by a 3D key-keyhole relation between cation [(DMI + ) 3 (I − ) 2 ] + and anion C 60 The geometry of the spin lattice of 8 is expected to be 3D apex sharing triangular bipyramid based on the r values.We calculated overlap integrals based on the two main orientations of C 60 molecules at 100 K and the mean square of each is s = 0.12 × 10 −3 .Based on the overlap integrals, the model spin lattice is shown in Figure 19b, which are arranged in the ab plane to form 3D apex sharing triangular bipyramid.
The effective magnetic moment of 8 is 1.64 µ B , slightly smaller than the value of 1.73 µ B for the system containing one S = 1/2 spin per formula unit.The Curie-Weiss temperature of −9.6 K in the 40-300 K range indicates AF coupling of spins (Figure 19d).Effective magnetic moment also decreases below about 50 K (Figure 19e) due to AF coupling of spins.Because of the large interfullerene distance, 11.05 Å at 100 K, and small overlap integrals, the AF interaction is not strong.The long-range magnetic ordering is not observed down to 1.9 K.
In summary, concerning the geometry of the spin lattice of 8, the cationic supramolecule [(DMI + ) 3 (I − ) 2 ] forms threefold assemblies that act as a template for the C 60 •− molecules to achieve hexagonal stacking of (C 60 The geometry of the spin lattice of 8 is expected to be 3D apex sharing triangular bipyramid based on the r values.We calculated overlap integrals based on the two main orientations of C60 molecules at 100 K and the mean square of each is s = 0.12 × 10 −3 .Based on the overlap integrals, the model spin lattice is shown in Figure 19b, which are arranged in the ab plane to form 3D apex sharing triangular bipyramid.Since the structure at 100 K is complicated by the appearance of one and a half independent molecules both for C60 •− and MDABCO + molecules, at first the schematic of a possible spin lattice is examined based on the crystal structure at 250 K. Figure 20(  Since the structure at 100 K is complicated by the appearance of one and a half independent molecules both for C 60 •− and MDABCO + molecules, at first the schematic of a possible spin lattice is examined based on the crystal structure at 250 K.At low temperatures, the C 60 •− environment is strongly anisotropic in terms of r and overlap integrals s.The r values inside corrugated Layer A 2 or Layer 1 3 , namely bonds A − 2 and 1 −, show shrinkage by 0.3-0.7%,while those bonds that are connecting neighboring corrugated layers, namely bonds 1 − 2 and A − 3 , show larger shrinkage by 0.5-1.0%.Therefore, though the corrugated square nature shown by Figure 20(e-2) is important to account for the spin interactions, the spin interactions between corrugated layers along the a axis becomes more significant at low temperatures.The model spin lattice geometry keeps the distorted bipyramidal one down to low temperatures.No dimerization was detected down to 1.9 K.The temperature dependence of the molar magnetic susceptibility χ M of 1 showed a maximum at 46 K, followed by a decrease but then an increase below 10 K owing to the Curie impurity of about 2.7% of total amount of C 60 •− .Figure 21  Catalysts 2018, 8, x FOR PEER REVIEW 26 of 37 the spin interactions between corrugated layers along the a axis becomes more significant at low temperatures.The model spin lattice geometry keeps the distorted bipyramidal one down to low temperatures.No dimerization was detected down to 1.9 K.The temperature dependence of the molar magnetic susceptibility χM of 1 showed a maximum at 46 K, followed by a decrease but then an increase below 10 K owing to the Curie impurity of about 2.7% of total amount of C60 •− .Figure 21    The overlap integrals between C 60  The small size of the MDABCO + cations with threefold symmetry and the absence of solvent molecules induced a densely packed 3D bipyramidal C60 •− packing in 1 resulting in strong AF interactions.However, the strong anisotropic packing of C60 •− may reduce the AF interaction considerably.The key-keyhole relation between MDABCO + and C60 •− is not clear in 1, however, it should be emphasized that MDABCO + cations work to prevent the bond-formation even though the r values became small (○ 1 -○ 2 = 9.91 Å, ○ A -○ 3 = 9.96 Å at 100 K). , where Ph3MeP + is a triphenylmethylphosphonium cation with threefold symmetry, were prepared by the reduction of C60 by (Ph3MeP + )(vanadyl(IV) phthalocyanine) in PhCl2 and slow mixing of the obtained PhCl2 solution in n-hexane [105].
The temperature dependence of the molar magnetic susceptibility χM shows an increase below 10 K, owing to the Curie impurity of about 1.2% of total amount of C60 •− .Figure 23d showed the temperature dependence of χM and 1/χM after the correction of Curie impurity.No peak of χM was Several zigzag-chain spin systems, which are the simplest frustrated magnet and treated by J 1 -J 2 model or zigzag chain model, have been developed, such as CaV 2 O 4 (S = 1, V 3+ , T N = 69 K) [160], Cu[2-(2-aminomethyl)pyridine]Br 2 (S = 1/2, Cu 2+ ) [161,162], (VO)(µ 3 -MoO 4 )(BPY) (S = 1/2, V 4+ ) [163,164], and F 2 PIMNH [165].A theoretical study predicted that the S = 1/2 zigzag chain has a gapless phase for J 1 /J 2 < 0.241 [166,167], as exemplified for Cu[2-(2-aminomethyl)pyridine]Br 2 (J 1 /J 2 = 0.2, J 1 /k B = 8.5 K) [162] and (VO)(µ 3 -MoO 4 )(BPY) (J /J = 0.2, J 1 /k B = 51 K) [164].It is known that the interchain magnetic interactions are critical for the spin-ladder system either to manifest a Néel ordered or disordered spin-frustrated state, and the critical value is reported to be J/J = 0.11, where J and J are intraladder and interladder interactions, respectively [168].Even though the actual J 3 values for the above zigzag systems were not estimated in these reports, it is likely that the J 3 values are very small according to their crystal structures.If we are able to expand the separation between the zigzag chains for 2 using more bulky cation molecules than Ph 3 MeP + , we may have a real zigzag system of C 60 •− .
The temperature dependence of the molar magnetic susceptibility χ M shows an increase below 10 K, owing to the Curie impurity of about 1.2% of total amount of C 60 •− .Figure 23d showed the temperature dependence of χ M and 1/χ M after the correction of Curie impurity.No peak of χ M was observed even though 2 has strong spin frustration.The magnetic behavior is described well by the Curie-Weiss law in the 30-300 K range with Θ CW = −60 K, and no AF ordering was observed down to 1.9 K (f > 30).Between C 60 •− molecules arranged along the a axis, Me groups of Ph 3 MeP + cation molecules penetrate and prevent the dimerization even at r = 10.08-10.10Å.

Summary
Geometrical spin frustration is discussed for monomer-type Mott insulators of C 60 CT solids.When compared with the ET QSL system, bond-formation between C 60 molecules and disorder of C 60 molecule additionally participate in the competition among the itinerancy, localization, and spin frustration.The donor ability, size, shape, and symmetry of donor molecules in multi-component approach provide suitable geometrical space and spatial regulation for C 60 The former has a distorted edge-shared honeycomb 2D spin lattice with J 1 :J 2 :J 3 = 0.79:1:0.zigzag chains with t /t = 1.50, J 1 :J 2 :J 3 = 1:0.44:0.10,and Θ CW = −60 K (f > 30).
Even though the overlap integrals between C 60 molecules depend on the molecular orientation of C 60 , the center-to-center distance between C 60 •− molecules r is the key parameter that determines the competition among the bond-formation, itinerancy, and spin frustration.Figure 24 shows the relation between |Θ CW | values and interfullerene distances r in this study.In the following, the summary and the perspective developed are presented. 1.
|Θ CW | seems to increase rapidly when r < 10 Å and magnetic dimensionality is 3D.Such low values of r were realized for two-component CT solids with cation molecules of small size with threefold symmetry.However, it is difficult to find a good key-keyhole relation to provide uniform triangular or hexagonal packing of C 60 •− for the two-component case.Furthermore, single crystals of CT solids were not always obtainable.For example, a quinuclidinium cation, which is smaller than MDABCO + and MQ + , gave no CT solids so far.

2.
The cationic supramolecular template with threefold symmetry leads to uniform triangular or hexagonal packing of C 60 •− for three-component case based on the key-keyhole relation.
For three-component case, it is critical to decrease the r value and increase the magnetic dimensionality.

Figure 1 .
Figure 1.Geometries of spin lattices having strong spin-frustration (a-f), non-frustrated spin-configurations (g) 120°, (h) 109°, and (i) collinear AF (AFC): (j,k) are schematic view of triangular spin lattice of dimer-type Mott insulator of κ-(ET)2X (j) and monomer-type Mott insulator of C60• − (k).Purple ellipsoid in (j) is an ET molecule and the black circle represents one spin site (ET)2.In (j), t and t′ are interdimer transfer interactions between parallel dimers and perpendicular dimers, respectively, and t′/t represents the shape of the isosceles triangular spin lattice.In (k), t, t′, and t″ are interfullerene transfer interactions.Red arrows indicate spins.(j,k) were reproduced from[12].

Figure 5
Figure5shows the size of MDABCO + (Figure5a,b), the height of TPC (Figure5c), and some parts of Supramolecule 1 (Figure5d) at 200 K.The neighboring TPC molecules are separated by 9.82 Å, so that it is able to prevent the bond-formation between C 60•− molecules when they pack on the template in a hexagonal lattice structure, where the intermolecular distance of MDABCO + molecules corresponds to N•••N (blue points in Figure4(a-2)) with separation distance of 9.99 Å.The MDABCO + molecule is thicker (7.27 Å) than the thickness of the holder, which is composed of three TPC molecules (7.05 Å).Consequently, one side of Supramolecule 1 has a concave shape where N atoms are centered and the Me of N-Me + group of MDABCO + extrudes from the holder on the opposite side.Therefore, the C 60 •− hexagonal layer on the top of the polycationic template in Figure4(a-2), namely Layer A (Figure6a,b) has different steric and electronic environment than the C 60 •− hexagonal layer at the bottom of the polycationic template (Layer B, Figure6a,c).Figure7demonstrates the key-keyhole relation for layer B. Three TPC 0 and three MDABCO + molecules constitute Supramolecule 3, [TPC 0 ] 3 [MDABCO + ] 3 (Figure7a), according to the first

Figure 6 .
Figure 6.Crystal structure of (TPC 0 )(MDABCO + )(C60 •− ) at 160 K: The (TPC 0 )(MDABCO + ) layer and C60 •− layer stack along the c axis with the sequence of the Layer A of C60 •− /(TPC 0 )(MDABCO + ) layer/Layer B of C60 •− .(a) C60 •− molecules in Layer A are arranged between the N atoms of MDABCO (drawn in red and N atoms and methyl groups are drawn in dark and bright blue, respectively) when viewed along the c axis.The methyl groups of MDABCO molecules are arranged towards the C60 molecules in Layer B and outline an octopore around C60. Configuration of the molecules in the slabs A (b) and B (c).The thin lines show the geometry (triangular lattice) connected between the centers of C60 molecules.

Figure 6 .
Figure 6.Crystal structure of (TPC 0 )(MDABCO + )(C60 •− ) at 160 K: The (TPC 0 )(MDABCO + ) layer and C60 •− layer stack along the c axis with the sequence of the Layer A of C60 •− /(TPC 0 )(MDABCO + ) layer/Layer B of C60 •− .(a) C60 •− molecules in Layer A are arranged between the N atoms of MDABCO (drawn in red and N atoms and methyl groups are drawn in dark and bright blue, respectively) when viewed along the c axis.The methyl groups of MDABCO molecules are arranged towards the C60 molecules in Layer B and outline an octopore around C60. Configuration of the molecules in the slabs A (b) and B (c).The thin lines show the geometry (triangular lattice) connected between the centers of C60 molecules.

Figure 6 . 37 Figure 7 .Figure 8 .
Figure 6.Crystal structure of (TPC 0 )(MDABCO + )(C 60 •− ) at 160 K: The (TPC 0 )(MDABCO + ) layer and C 60 •− layer stack along the c axis with the sequence of the Layer A of C 60 •− /(TPC 0 )(MDABCO + ) layer/Layer B of C 60 •− .(a) C 60 •− molecules in Layer A are arranged between the N atoms of MDABCO (drawn in red and N atoms and methyl groups are drawn in dark and bright blue, respectively) when viewed along the c axis.The methyl groups of MDABCO molecules are arranged towards the C 60 molecules in Layer B and outline an octopore around C 60 .Configuration of the molecules in the slabs A (b) and B (c).The thin lines show the geometry (triangular lattice) connected between the centers of C 60 molecules.At 300 K, C 60 •− molecules are ordered in Layer A (r = 10.07Å, overlap integral s = 1.91 × 10 −3 , Figure 8a), while C 60 •− molecules in Layer B are disordered.At the same temperature, half of the MDABCO + cations are disordered between three orientations that are linked by their rotation about the lattice threefold axis.On Layer A, the r value decreases monotonously to 9.97 Å at 183 K at which − is closely linked with that of MDABCO + .The calculated overlap integrals at 160 K are s = 2.57 × 10 −3 (//a), 2.03 × 10 −3 (//b), and 2.76 × 10 −3 (//a + b) for Layer A and 2.45 × 10 −3 (//a), 2.21 × 10 −3 (//b), and 1.61 × 10 −3 (//a + b) for Layer B. The calculated bandwidth W is 0.103 eV at 300 K for Layer A, 0.150 eV and 0.133 eV for Layers A and B at 160 K, respectively.The calculated anisotropy of the transfer interactions ta:tb:ta+b = 1:1:1 for Layer A above 183 K changed to ta:tb:ta+b = 1.27:1:1.36for Layer A and ta:tb:ta+b = 1.52:1.37:1for Layer B at 160 K.The ratio of the triangular spin lattice is defined as 2ta/(tb + ta+b), 2tb/(ta + ta+b), and 2ta+b/(ta + tb).However, the last two definitions provide inadequate t′/t values of 0.60-0.76for layer A and 0.54-0.69for Layer B at 160 K that suggests a much enhanced 2D nature than that at RT.So using the ratio t′/t = 2ta/(tb + ta+b), the calculated anisotropy is 1.00 (300 K) and 1.07 (160 K) for Layer A and 0.99 (185 K) and 1.28 (160 K) for Layer B. The spin frustration of this spin lattice is comparable to that of κ-(ET)2Cu2(CN)3 (t′/t = 1.09 at RT, 1.07 at 100 K) [54] and stronger than a Mott insulator κ-(ET)2B(CN)4 [150] (t′/t = 1.42 at RT, 1.61 at 100 K, ground state is valence-bond solid).However, the increase of W together with the ordering of C60 •− molecules in

Figure 8 .
Figure 8.(a) View of the ab plane of fullerene Layer A in (TPC 0 )(MDABCO + )(C 60 •− ) at 300 K.The van der Waals C•••C contacts shorter than 3.42 Å are shown by dashed lines.Numbers 1-3 indicate the r values; (b) The calculated Fermi surface of Layer A at 200 K, (c) that of Layers A at 160 K, and (d) that of Layer B at 160 K. (a) was reproduced from [12] and (b−d) were from [146].
and Θ CW = −31 K (blue curve in Figure9d).C of 0.160 emu•K•mol −1 corresponds to the contribution of about 43% of the spins from the total amount of C 60 (C = 0.374 emu•K•mol −1 for 100% of spins).Consequently, the spins in one layer (Layer B) are treated as localized ones and they interact antiferromagnetically with Θ CW of −31 K.A reversible decrease in χ M is observed at 200-230 K. Below 200 K, the temperature-independent susceptibility (χ 0 ) of about 10.0 × 10 −4 emu•mol −1 is attributed to the Pauli paramagnetic contribution, implying a metallic state down to 1.9 K.The scenario is that ordering of both C 60 •− in layer B and MDABCO + triggered a transition from a non-metallic and antiferromagnetically frustrated state to a metallic state for spins of C 60 •− in Layer B, while ordered C 60 •− in Layer A kept its 2D itinerancy over the entire temperature range.The strong coupling between the ordering of C 60 and physical properties is intriguing and was previously observed in some fullerene salts [153-156].Salt 3 is the first 2D monomer-type C 60 •− organic metal composed of only light elements (C, H, N).Even though rapidly cooled AC 60 •− (A = Cs and Rb) were reported to be monomer-type metals below 150 K and 125 K, respectively, definitive information is needed concerning the stoichiometry, metallic behavior, dimensionality, and crystal structure to confirm a monomer-type metal [71,72,157].Summarizing the information concerning the geometry of spin lattice of 3, the C 60 •− molecules form hexagonal stacking according to the geometry of cationic template (TPC 0 )(MDABCO + ), both of the component molecules have threefold symmetry, by key-keyhole relation.Layer B has t /t = 0.99 (185 K), which indicates strong spin frustration and is close to those of QSL candidates κ-(ET) 2 M 2 (CN) 3 (M = Cu; t /t = 1.09,Ag; t /t = 0.97), |Θ CW | = 31 K is estimated in the range of 260−300 K, and r = 10.07Å (RT).No dimerization of C 60 •− occurred down to 1.9 K. 4.1.2.Only Frustrated Spins in 2D Hexagonal Packing of C 60 •− in (TPC 0 )(MQ + )(C 60 •− ) By a using MQ + instead of MDABCO + , where MQ + is N-methylquinuclidinium cation, both of which have threefold symmetry, an AF insulator (TPC 0 )(MQ + )(C 60 •− ) (4) was obtained [12].The C 60 molecule, a 10-fold molar excess of CH 3 CH 2 SNa, and a 5-fold molar excess of MQ•I were reacted in a PhCl 2 /PhCN mixture.Into a filtered solution, TPC was dissolved and filtered.n-Hexane was layered over the obtained solution.Black hexagonal prisms up to 0.2 × 0.5 × 0.5 mm 3 were harvested after two months.

Figure 12 . 37 Figure 12 .
Figure 12.Key-keyhole relation for the Layer A in (TPC 0 )(MQ + )(C 60 •− ): Asymmetric placing of MQ + in the (TPC 0 ) 3 hole to form Supramolecule 5 by first key-keyhole relation (a) and docking of C 60 •− into the pit of Supramolecule 5 gives Supramolecule 6 (second key-keyhole relation) (b); (c) A layer of [(TPC 0 )(MQ + )] ∞ composed of Supramolecule 5; (d) A layer of [(TPC 0 )(MQ + )] ∞ and C 60 •− molecules are assembled by fitting the C 60 molecules into the concaves in the layer of [(TPC 0 )(MQ + )] ∞ to form the Layer A of (TPC 0 )(MQ + )(C 60 •− ).Black lines are guides to the eye.C 60 molecules are arranged at the crossing points of black lines; (e,f) show the relation among the C 60 molecule in Layer A and upper and lower layers of (TPC 0 )(MQ + ).Upper and lower layers of (TPC 0 )(MQ + ) are drawn in different colors: top down view (e) and side view (f); (g,h) Calculated Fermi surface of C 60 assemble in Layer A at 250 K (g) and 100 K (h) [12].

Figure 14 .
Figure 14.Packing of C60 •− at 250 K and magnetic behavior of (TPC 0 )(MQ + )(C60 •− ) [12]: (a) Packing pattern of C60 in Layer A in the ab plane; (b) Packing pattern of C60 in Layer B in the ab plane.Van der Waals C•••C contacts shorter than 3.42 Å are shown by dashed lines.Numbers 1-3 indicate the center-to-center distances between C60 •− ; (c) Temperature dependence of molar magnetic susceptibility and reciprocal molar magnetic susceptibility of (TPC 0 )(MQ + )(C60 •− ).Red curve shows the fitting of the molar magnetic susceptibility data in the 50-300 K by the Curie-Weiss law with Weiss temperature of −27 K.The calculated overlap integrals at 100 K are s = 0.78 × 10 −3 (//a), 1.82 × 10 −3 (//b), and 2.24 × 10 −3 (//a + b) for Layer A and 2.81 × 10 −3 (//a), 1.97 × 10 −3 (//b), and 1.51 × 10 −3 (//a + b) for Layer B. The calculated bandwidths are 0.103 (0.112) and 0.097 (0.113) eV for Layer A and Layer B at 250 K (100K), respectively.Similar to 3, the ratios 2tb/(ta + ta+b) and 2(ta + tb)/(ta + tb) are 0.85 and 1.17 for Layer A and 0.91 and 0.64 for Layer B at 100 K. Since the calculated Fermi surface shows 1D properties, it is more appropriate to use 2ta/(ta + ta+b) instead of 2tb/(ta + ta+b) and 2ta+b/(ta + tb).The calculated anisotropy of the transfer interactions at 250 K is ta:tb:ta+b = 1.04:1:1 (t′/t = 1.04) and 1:1.40:1.23 (t′/t = 0.76) for Layer A and Layer B, respectively.The anisotropy changed to 1:0.90:1.11(t′/t = 0.99) for Layer A and 1:0.70:0.54(t′/t = 1.61) for Layer B at 100 K.The anisotropy of Layer A is close to that of κ-(ET)2Cu2(CN)3, and the geometrical spin frustration is comparable to that of 3.The relatively large distances between C60 •− prevent their dimerization but allow for the manifestation of a magnetic interaction between them.Reciprocal molar magnetic susceptibility is described well by the Curie-Weiss law in the 30-300 K range with negative Weiss temperature of ΘCW = −27 K (Figure14c), indicating AF interaction of spins in the fullerene layers.The |ΘCW| is small, owing to the weaker AF interactions than that in 3 because of the larger interfullerene distance.In spite of the strong AF interaction of spins, magnetic ordering is not observed down to 1.9 K in this distorted triangular spin lattice system (f >14).The resistivity measurements were

Figure 14 .
Figure 14.Packing of C 60 •− at 250 K and magnetic behavior of (TPC 0 )(MQ + )(C 60 •− ) [12]: (a) Packing pattern of C 60 in Layer A in the ab plane; (b) Packing pattern of C 60 in Layer B in the ab plane.Van der Waals C•••C contacts shorter than 3.42 Å are shown by dashed lines.Numbers 1-3 indicate the center-to-center distances between C 60 •− ; (c) Temperature dependence of molar magnetic susceptibility and reciprocal molar magnetic susceptibility of (TPC 0 )(MQ + )(C 60 •− ).Red curve shows the fitting of the molar magnetic susceptibility data in the 50-300 K by the Curie-Weiss law with Weiss temperature of −27 K.
Au + cation molecule and a PhCl 2 molecule while C 60 0 molecules are sandwiched between two (Ph 3 P) 3 Au + molecules along the c axis, as shown in Figure 17b.Negatively charged and neutral C 60 molecules are closely packed within hexagonal layers with r (I•••II) = 10.02Å, while between C 60 •− , it is long with r (I•••I) = 10.37 Å due to corrugation.The magnetic interactions between C 60 •− molecules in the neighboring fullerene layers are expected to be small based on its r value (r ~13.9 Å).As a result, the magnetic interactions are 2D.Each C 60 •− has only three negatively charged fullerene neighbors within a fullerene layer, namely C 60 •− molecule a 1 is surrounded by C 60 •− molecules a 2 -a 4 in Figure 17c.They form distorted tetrahedral spin lattice composed of C 60 •− molecules a 1 -a 4 (red lines in Figure 17d).The a 1 molecule projects out of the a 2 -a 4 plane in Figure 17d by only 3.03 Å.The tetrahedral units are arranged in the 2D plane by apex-sharing.The overlap integrals have not been obtained due to severe disorder of fullerene molecules.Owing to the very large center-to-center distance between the C 60 •− molecules, the AF interaction is weak (Θ CW = −5 K, f = 2.6).In order to enhance the magnetic interactions, smaller sized cationic supramolecules than [(PhCl 2 0 ){(Ph 3 P) 3 Au + } 2 ] well matched with two C 60 molecules would be preferable.The resistivity at RT is approximately ρ = 4 × 10 5 Ω•cm.EPR measurements confirmed no dimerization of C 60 •− down to 4.2 K.
e-1)  shows the packing of C60 •− viewed along the a axis, that corresponds to Figure20aat 100 K.The square corrugated C60 •− packing composed of C60 •− molecules (one ○ A and four ○ 2 ) form a pyramidal shape with r(○ A −○ 2 ) = 10.08 Å, which is schematically shown in Figure 20(e-2).The other combination of C60 •− molecules ○ 1 and ○ 3 form the equivalent pyramid and these two pyramids have short contacts with r values of 10.01 Å for ○ 1 −○ 2 and 10.11 Å for ○ 1 −○ A .As a consequence, the unit of intermolecular interactions is approximated as a distorted bipyramid, as shown in Figures 20(e-3).The plane of ○ 2 −○ 2 ′−○ 2 ′′−○ 2 ′′′ bisects the bond ○ 1 −○ A by 6.91 Å and 3.20 Å.The distorted bipyramids are connected to each other by sharing edge in the bc plane to form a 2D sheet.Figure 20(e-4) shows the part of the sheet

Figure 20 .
Figure 20.(a) View of crystal structure of (MDABCO + )(C 60 •− ) along the a axis at 100 K; (b) View along the b axis at 100 K.The van der Waals contacts between fullerenes are shown in green dashed lines in (a,b); (c) C 60 •− molecules stack along the b axis to form columns as shown with (c-1) or without MDABCO + (c-2, for simplicity).Yellow and red C 60 •− columns are arranged alternately along the c axis to form a corrugated sheet in the bc plane; (d) Environment of one C 60 •− radical anion (shown by red color) from eight neighboring C 60 •− and five MDABCO + cations at 250 K. Center-to-center distances between red fullerene (A) and surrounding C 60 •− (marked by numbers 1-3): A−1: 10.107, A−2: 10.081 and A−3: 10.008 Å. Van der Waals contacts between C 60 •− and nitrogen atom of the MDABCO + cation are shown by green dashed lines.Only the major orientation of C 60 •− is shown; (e) (e-1) Corrugated layer composed of molecules A and 2 at 250 K; (e-2) Geometry of a unit of possible spin lattice made of A and 2; (e-3) Schematic of bipyramidal spin lattice composed of molecules A, 1, and 2; (e-4) The unit shown by (e-3) forms 2D layer by sharing an edge.Only the ribbon extending along the c axis is shown; (e-5) The other unit of bipyramidal spin lattice composed of 1, A, and 3 forms equivalent 2D layer; (e-6) The model geometry of the possible spin lattice obtained by sharing apexes of the layer in (e-4,e-5) along the a axis to form a 3D distorted bipyramidal spin lattice (a,b,d), (e-1) from [147]).
Figure 20(e-1) shows the packing of C 60 •− viewed along the a axis, that corresponds to Figure 20a at 100 K.The square corrugated C 60 •− packing composed of C 60 •− molecules (one A and four 2 ) form a pyramidal shape with r( A − 2 ) = 10.08 Å, which is schematically shown in Figure 20(e-2).The other combination of C 60 •− molecules 1 and 3 form the equivalent pyramid and these two pyramids have short contacts with r values of 10.01 Å for 1−2 and 10.11 Å for 1 − A .As a consequence, the unit of intermolecular interactions is approximated as a distorted bipyramid, as shown in Figure 20(e-3).The plane of 2 − 2 − 2 − 2 bisects the bond 1 − A by 6.91 Å and 3.20 Å.The distorted bipyramids are connected to each other by sharing edge in the bc plane to form a 2D sheet.Figure 20(e-4) shows the part of the sheet extending along the c axis.The other unit of bipyramidal spin lattice composed of C 60 •− molecules 1 , A , and 3 forms equivalent 2D layer with different orientation (Figure 20(e-5)).The model geometry of the possible spin lattice of 1 is obtained by sharing apexes of the layers in Figure 20(e-4) and Figure 20(e-5) along the a axis to form 3D distorted bipyramidal spin lattice in which corrugated layers alternate as Layer A 2 /Layer 1 3 along the a axis (Figure 20(e-6)).
shows the temperature dependence of χ M and 1/χ M after the correction of Curie impurity.The magnetic susceptibility clearly indicates a characteristic peak near 50 K.Such peak in χ M has been usually detected in the low-D (1D-2D) Mott insulators with strong spin frustration, such as κ-(ET) 2 X (X = Cu 2 (CN) 3 , Ag 2 (CN) 3 , B(CN) 4 , CF 3 SO 3 )[36,45-51,54,65,66,128,150,158].The Θ CW temperature of −118 K was derived in the 70-300 K range (Figure21b).The temperature dependence of χ M was fitted by the Heisenberg model for square 2D AF coupling of spins[159] to give J/k B = −25.3K.The long-range magnetic ordering is not observed down to 1.9 K (f > 62).

Figure 21 .
Figure 21.dependence of molar magnetic susceptibility χ M (a) and 1/χ M (b) of (MDABCO + )(C 60 •− ) after correction of Curie impurity.Red curve in (a) is the fit by the Heisenberg model for square 2D AF coupling of spins with J/k B = −25.3K. Red line in (b) is the Curie-Weiss fit with Θ CW = −118 K [147].

Figure 22 shows 37 Figure 22 .
Figure 22 shows the network between the two kinds of C 60 •− radical anions at 100 K that are connected by several magnitudes of overlap integrals.The dominant interaction (red line) is 4.74 × 10 −3 between the molecules 1 with 9.96 Å distance that forms a pair of C 60 •− between the adjacent layers.The second largest interaction (purple line) is 2.66 × 10 −3 between the molecules 2 with 10.14 Å uniform distance that extends linearly along the a axis.The third one (black line) is 1.96 × 10 −3 between the molecules 1 with 10.05 Å distance which forms zig-zag path ways along the c axis.Note that the shortest distance of 9.91 Å (orange line) between the different kinds of molecules 1 and 2 resulted in only the fifth largest interaction of 1.25 × 10 −3 .As demonstrated also in 2 and 5, the magnitude of overlap integrals between C 60 •− anion radicals does not necessarily scale with the closeness between them.The relative orientation of the molecular orbitals in the nearest neighbors as well as the center-to-center distances between them plays an important role to characterize the molecular interactions in the crystal.Catalysts 2018, 8, x FOR PEER REVIEW 27 of 37

Figure 23 .
Figure 23.(a) Crystal structure of (Ph 3 MeP + )(C 60 •− ) showing the double chains containing triangles from C 60 •− ; (b) view along the a axis showing the arrangement of double chains in the bc plane; (c) Schematic of possible zigzag spin lattice with weak interchain interactions (J 1 :J 2 :J 3 = 1:0.44:0.10)together with r and s; (d) Temperature dependence of molar magnetic susceptibility χ M and 1/χ M after correction of Curie impurity.Red curve in (d) is the Curie-Weiss fit with Θ CW = −60 K. (a,b,d) were reproduced from [105].
), (2) the system should be a Mott insulator in ambient conditions, (3) its Mott insulating state has both a partial CT state close to the itinerant region and a small Mott gap, (4) the spin lattice should have a geometry that affords a strong geometrical frustration, i.e., t /t ~1 for a triangular spin lattice, (5) a high |Θ CW | or high |J| value to observe the QSL state at the experimentally available temperatures, and ( While for the C 60 CT solids, a partial CT state such as ET 1/2+ is not necessary and the completely ionized C 60 •− molecules are able to afford the itinerant state owing to the triply degenerate LUMO t 1u orbital, e.g., the compound CsC 60 quenched in liquid N 2 is reported to exhibit metallic behavior down to low temperatures •+ X − (D: TTF, TSF, ET) and the lower-LUMO band for the CT solid of M + A •− (A: TCNQ) are completely filled, where HOMO and LUMO are the highest occupied and lowest unoccupied molecular orbitals, respectively.The Mott gap being large makes it difficult for the competition between localization and itinerancy in the solids, thus requiring partial CT state or dimer-type Mott insulating state to satisfy the requirement 3 above mentioned.
will be 125]iding map for the search for C 60 functional materials.A phase diagram of C 60 3− solids indicates that the SC state is in the vicinity of spin ordered AF states while the critical temperature T c decreases as approaching to the AF phase[82,120,125].Therefore, the C 60 system is a potential candidate for the QSL state to be close to the metallic, SC, and bond-formed states.
[130][131][132]hbour Coulomb repulsion energy) values in solids are found to be 0.8-1.3eV[130][131][132],when an electron is added to a C 60 molecule surrounded by other C 60 molecules in the fcc lattice, which is due to polarization by the charged C 60 molecules.While, •− is closely linked with that of MDABCO + .The calculated overlap integrals at 160 K are s = 2.57 × 10 −3 (//a), 2.03 × 10 −3 (//b), and 2.76 × 10 −3 (//a + b) for Layer A and 2.45 × 10 −3 (//a), 2.21 × 10 −3 (//b), and 1.61 × 10 −3 (//a + b) for Layer B. The calculated bandwidth W is 0.103 eV at 300 K for Layer A, 0.150 eV and 0.133 eV for Layers A and B at 160 K, respectively.The calculated anisotropy of the transfer interactions t a :t b :t a+b = 1:1:1 for Layer A above 183 K changed to t a :t b :t a+b = 1.27:1:1.36forLayerAandta :t b :t a+b = 1.52:1.37:1forLayerBat160K.The ratio of the triangular spin lattice is defined as 2t a /(t b + t a+b ), 2t b /(t a + t a+b ), and 2t a+b /(t a + t b ).However, the last two definitions provide inadequate t /t values of 0.60-0.76forlayerAand0.54-0.69forLayerBat160K that suggests a much enhanced 2D nature than that at RT.So using the ratio t /t = 2t a /(t b + t a+b ), the calculated anisotropy is 1.00 (300 K) and 1.07 (160 K) for Layer A and 0.99 (185 K) and 1.28 (160 K) for Layer B. The spin frustration of this spin lattice is comparable to that of κ-(ET) 2 Cu 2 (CN) 3 (t /t = 1.09 at RT, 1.07 at 100 K)[54]and stronger than a Mott insulator κ-(ET) 2 B(CN)4[150] (t /t = 1.42 at RT, 1.61 at 100 K, ground state is valence-bond solid).However, the increase of W together with the ordering of C 60•− molecules in Layer B below 200 K gave rise to a superior itinerancy (metallic state) than localization (QSL state) for this salt.