Supplementary Figure 1:

Transcription

1 a) b) C A O B O A B c) d) C A O B O C Supplementary Figure 1: Different views of the crystal packing of the X- ray- resolved LT polymorph of TTTA at 300 K (CCDC refcode = SAXPOW06). For a given 2D layer: (a) a plane of TTTA radicals perpendicular to the π stacks, viewed along c; (b) lateral view of the plane displayed in (a) to show the single molecular plane orientation of TTTAs. For a set of 2D layers: (c) π stacks viewed along b; (d) π stacks viewed along a. 1

2 a) b) O C A C O B A c) d) B C O A O B Supplementary Figure 2: Different views of the crystal packing of the X- ray- resolved HT polymorph of TTTA at 300 K (CCDC refcode = SAXPOW05). For a given 2D layer: (a) a plane of TTTA radicals perpendicular to the π stacks, viewed along b; (b) lateral view of the plane displayed in (a). For a set of 2D layers: (c) π stacks viewed along a; (d) π stacks viewed along c. 2

3 a) b) Supplementary Figure 3: For a given 2D layer, stacking of radicals in the X- ray resolved LT- 300 (a) and HT- 300 (b) structures. The grey and red cylinders mark the lateral S S contacts of TTTA molecules between neighboring stacks; the green and yellow cylinders identify the lateral six- center N S contacts. The two types of lateral contacts that are highlighted are those associated with distances shorther than the sum of the corresponding van der Waals radii. 3

4 A O a) B O C b) A Supplementary Figure 4: For a set of 2D layers, top view of the stacks of TTTA radicals in the X- ray resolved LT- 300 (a) and HT- 300 (b) structures. The grey and red cylinders mark the lateral S N or S S contacts between neighboring TTTA molecules; the green and yellow cylinders identify the lateral four- center N S contacts. The only lateral contacts that are highlighted are those associated with distances shorther than the sum of the corresponding van der Waals radii. 4

5 a) b) O A O B A C c) C B O Supplementary Figure 5: Three different views of the supercell used in the optimizations and simulations of the LT polymorph of TTTA. The supercell for LT has been constructed by duplicating the unit cell in all three spatial directions. 5

6 A a) b) C O O B C c) B A O Supplementary Figure 6: Three different views of the supercell used in the optimizations and simulations of the HT polymorph of TTTA. The supercell for HT has been constructed by replicating the unit cell in two of the spatial directions. Specifically, the relations between the supercell (sc) vectors and the unit cell (uc) vectors are: asc = 2 auc; bsc = 4 buc; csc = cuc 6

7 HT-300 (X-ray) HT-0 LT-300 (X-ray) LT-0 Supplementary Figure 7: Comparison between the X- ray resolved crystal structures at 300 K of LT and HT (left) and the corresponding optimized structures at 0 K (right). 7

8 a) b) c) d) e) f) g) h) i) Supplementary Figure 8: The different spin configurations that were provided as initial guess for the variable- cell optimization of HT. In all the different spin configurations, the spins of adjacent TTTA radicals within a stack are antiferromagnetically aligned, as shown in a). Panels b) to i) show a top- view (along the stacking direction) of the HT supercell. In these panels, the z- projection of the spin (either spin up or spin down) of the top TTTA molecule in each stack is indicated. 8

9 a) b) Energy (ev) Γ X Γ Y Γ Z Supplementary Figure 9: a) The magnetic unit cell ( HT- 8 cell) and spin configuration employed to compute the band structure of HT. Note that in this particular spin configuration (which is the most stable one), the spins of adjacent radicals within the stacks are antiferromagnetically coupled. b) Calculated band structure of the HT phase of TTTA at the regular structure observed in the X- ray measurements at 300 K. Only the four higher occupied bands and the four lower unoccupied bands are shown. Energy zero is set in the middle of the band gap. This band structure has been obtained with a spin- polarized calculation. Since the sets of bands for spin up and spin down electrons are exactly the same, only one set of bands is displayed. 9

10 a) b) Supplementary Figure 10: Electronic density of states (DOS) of the HT phase for the regular structure observed in the X- ray measurements at 300 K. a) DOS computed for the HT- 8 cell (see Supplementary Figure S9) using a 4x4x4 Monkhorst- Pack grid of k- points. b) DOS computed for the supercell of 32 TTTA molecules shown in Supplementary Figure 6 using Γ- point sampling of the Brillouin zone. The energy zero in both graphics is set in the middle of the band gap. 10

11 Supplementary Figure 11: Spin density distribution for one of the stacks of HT- 300, evaluated after optimizing the electronic wavefunction of the supercell comprising 32 TTTA radicals. Only the spin density of one stack is displayed for the sake of clarity. 11

12 Supplementary Figure 12: Electronic density of states (DOS) of the LT phase corresponding to the X- ray recorded structure at 300 K (LT- 300). This DOS has been computed for the crystalline unit cell of LT- 300 (which contains eight different TTTA molecules) using a 4x4x4 Monkhorst- Pack grid of k- points. 12

13 eclipsed dimer LT-300 slipped dimer HT-300 Supplementary Figure 13: The simplest cluster model systems used to validate the computational methodology employed. These models include a dimer of TTTA radicals excised from the X- ray resolved LT- 300 phase (left) and from the HT- 300 phase (right). The difference in the single- point energy between the slipped TTTA dimer of HT- 300 and the eclipsed TTTA dimer of LT- 300 has been computed with different electronic structure methods (see Supplementary Table 2 and the first subsection of the Supplementary Note 5). 13

14 DFT-TTTA II I E (kcal/mol) 2(Å) I II (Å) CASPT2-TTTA 0.5 II I II E (kcal/mol) 0.0 2(Å) 2 I (Å) DFT-TTTA 0.6 E (kcal/mol) 2(Å) 1.2 II 0.3 III I III II I n (Å) Supplementary Figure 14: Potential energy surfaces (PESs) for two model systems of a stack of TTTA radicals in the subspace spanned by the two geometrical variables displayed in the left- most scheme (ξ1 and ξ2). These two variables describe a sliding motion of the radicals relative to each other. The interplanar distance between radicals (which has been chosen to be 3.27 Å) is preserved along the motions defined by these two variables. From top to bottom, the depicted PESs correspond to DFT calculations on an isolated TTTA4 aggregate, to CASPT2 calculations on the same isolated aggregate, and to DFT calculations on a stack of TTTA radicals (TTTAn). The minima configurations (I and II) of each PES are shown on the right. The configuration associated with the regular stack (III) is also shown for the TTTAn model system, and corresponds to ξ1 = ξ2 = 0. The distances between the nitrogen atoms of the S N S moieties of adjacent radicals in configurations I- III are shown in Å. See second subsection of Supplementary Note 5 for further information. 14

15 a) b) Supplementary Figure 15: The intrastack dimerization in HT- 0 takes place at the expense of weakening some the lateral S S non- covalent interactions between TTTA molecules of different columns (highlighted in grey and red cylinders). The distances associated with these contacts are given in Angstroms: (a) the X- ray resolved HT- 300 structure, and (b) the optimized HT- 0 structure. Supplementary Fig show that that the energetic stabilization associated with the formation of dimers exceeds the energy penalty associated with the weakening of the network of lateral S S interactions. See third subsection of Supplementary Note 5 for further details. 15

16 E(kcalmol -1 ) slipped pair eclipsed pair slipped pair (Å) Supplementary Figure 16: The formation of an eclipsed dimer brings about an important energy stabilization. Potential energy profile (computed at the CASPT2 level) of a TTTA pair along the coordinate ξ, which drives a relative sliding motion between two TTTA radicals, while preserving the interplanar distance between the two radicals. In the model system herein employed, this distance has been set to 3.27 Å because this is the value for the interplanar distance between paired radicals in HT- 0. The red circle marks the point of the profile associated with the completely eclipsed pair. The green circles mark the two points of the profile associated with the slipped pairs (the geometries corresponding to the two green circles are equivalent by symmetry). The degree of relative slippage between TTTA molecules in the slipped pairs is the same as the one observed in the X- ray structure of HT In the depicted structures of the insets, the distances between the nitrogen atoms of the S- N- S moieties of the two radicals are marked in black, while the interplanar distances between the two radicals are marked in red. 16

17 E(kcalmol -1 ) (Å) Supplementary Figure 17: Potential energy profile (computed at the CASPT2 level) of the two TTTA molecules shown in the inset along the coordinate ξ. Such coordinate defines a motion whereby each radical is shifted along the line that connects its two carbon atoms. As ξ increases (decreases), the S S distance, which is marked in red dots, increases (decreases). The green, red and blue circles mark the points where the S S distance is 3.39 Å, 3.78 Å and 3.11 Å, respectively. The first distance corresponds to the lateral S S distance of the regular zig- zag pattern observed in the X- ray resolved HT- 300 structure (see Supplementary Fig. S15a). The other two distances correspond to the largest and smallest lateral S S distances of those marked in Supplementary Fig. 15b. 17

18 3 E (kcal mol -1 ) = = ξ(å) Supplementary Figure 18: Potential energy profile (computed at the CASPT2 level) of the model system shown on the left- hand side along the coordinate ξ. Progress along this coordinate causes a dimerization within the stacks and a distortion of the lateral S S contacts. The insets show the geometry of the model system at ξ=0 and at ξ=0.25. The distances between the nitrogen atoms of the S- N- S moieties of the two radicals in the same stack are marked with green dashed lines in both geometries. The lateral S S distances are marked with black dashed lines in both geometries. 18

19 Supplementary Figure 19: Natural orbitals and corresponding occupation numbers for the TTTA4 aggregate in a distorted π- stack (left, configuration I of Supplementary Fig. 14) and in a regular stack (right, configuration III of Supplementary Fig. 14). These orbitals result from CASSCF calculations, in which the active space includes 4 electrons in 4 orbitals. The four orbitals included are the SOMOs of each TTTA radical. 19

20 Supplementary Figure 20: Lateral views of the HT polymorph (top) and the LT polymorph of TTTA (bottom). The average structures obtained from the AIMD simulations at 300 K (depicted in red) are superimposed on the respective X- ray crystal structures recorded at 300 K (depicted in blue). The average structures were obtained from ca. 10 picoseconds of AIMD trajectories for both polymorphs. 20

21 PDF T=300 K T=225 K T=200 K T=175 K T=150 K T=100 K d c (Å) Supplementary Figure 21: Temperature- dependence of the probability distribution functions (PDF) associated with the distance between centroids of adjacent radicals in the stacks of the HT (black) and LT (red) polymorphs. Each PDF has been computed taking into account the thermal fluctuations of all the possible pairs of TTTA radicals in a supercell containing 32 TTTA radicals over a simulation of ca. 10 picoseconds (this is equivalent to having sampled configurations). 21

22 A vib (kcal mol -1 ) A vib-anharmonic A vib-harmonic T (K) Supplementary Figure 22: Comparison of the temperature- dependence of ΔAvib obtained by means of Equation 1 in Supplementary Note 10 (black) with that obtained within the harmonic approximation (red). The lines connect the points as a guide to the eye. This graphic demonstrates that the anharmonic effects of the HT phase (which are taken into account when using Equation 1 in Supplementary Note 10) do indeed play a major role in the temperature- dependence of ΔAvib. 22

23 Supplementary Figure 23: Single crystal X- ray diffraction ω oscillation photographs for HT The panels (a) and (b) are for Crystal I, and (c) and (d) are for Crystal II. The vertical directions in (a), (b), (c) and (d) are along the b* (the stacking direction), a*, b* and c* axes, respectively. Due to a limited movable range of the goniometer, it was necessary to use two crystals to obtain the comprehensive data for all axes. The shape of every spot is slightly elongated along both the horizontal and vertical directions. The former is caused by the continuous X- ray. The latter, which is significant especially in (a) and (c) (note that (a) and (c) correspond to a photograph along the same axis for two different crystals), is caused by a diffuse scattering due to an active thermal vibration or to structural disorder. Since there is no superlattice diffraction from a periodicity, which is twice as large as the unit cell, dimerization does not occur in the short period. This allows us to exclude the structural disorder as the origin of the observed diffuse scattering. Therefore, the elongation of spots is likely to be caused by an active thermal vibration. The temperature- dependence of the X- ray diffraction ω oscillation images presented in Supplementary Fig. 24 proves that this is indeed the case. 23

24 Supplementary Figure 24: Temperature dependence of the single crystal X- ray diffraction ω oscillation photographs for Crystal I, recorded at (a) 320, (b) 300, (c) 260, (d) 240, (e) 215, and (f) 205 K, respectively. The insets show the magnifications of the (106) diffraction peak. It is observed that the Bragg s peaks for the vertical or stacking direction exhibit significant changes upon cooling the crystal from 320 K. This means that the elongation of the spots in the ω oscillation photographs is caused by an active thermal vibration. This in turn is consistent with the probability distribution functions of Figure 4 in the main text and Supplementary Fig. 21. Overall, the X- ray diffuse scattering measurements support the observation in the AIMD simulations that the TTTA radicals in HT (for temperatures higher than ca. 200 K) stand most likely at the midpoint of the double well potential (see Figure 3 in the main text) and that they feature important oscillations around that point. At 205 K, the streak spots appear between (h0l) and (h±1l), due to the dimerization, which indicates the coexistence of HT and LT. To make this change easily to see and evaluate the reproducibility, we carried out the temperature dependence measurement for Crystal II (see Supplementary Figure 25), in which the exposure time was three times as long as that for Crystal I. 24

25 Supplementary Figure 25: Temperature dependence of the single crystal X- ray diffraction ω oscillation photographs for Crystal II, recorded at (a) 300, (b) 215, and (c) 150 K, respectively. At 300 K, all the spots are assignable to those from the HT phase. Then at 215 K, where the phase transition occurs partially, there are the spots of the HT phase and the streak of the LT phase. It is notable that the feature of this figure is significantly different from that of Supplementary Fig. 24e even though they are the data at the same temperature. This is because the temperature at which the phase transition begins depends on the crystals. Upon further cooling to 150 K, TTTA is in the LT phase completely. The streak peaks for the LT phase, which look like partial Debye- Scherrer rings, indicate that the mosaicity of the crystal becomes more significant after the phase transition. This phenomenon is often observed for various phase transitions with drastic structural changes. 25

26 Supplementary Figure 26: Scheme of one TTTA radical. The atom labels allow one to identify which atom the atomic displacement parameters collected in Supplementary Tables 3 4 correspond to. 26

27 LT-300 (X-ray) LT-300 (AIMD) HT-300 (X-ray) HT-300 (AIMD) Supplementary Figure 27: Thermal ellipsoids of the atoms of the TTTA molecules in the LT- 300 and HT- 300 phases. Left: experimental thermal ellipsoids obtained from X- ray measurements; Right: thermal ellipsoids obtained from the ab initio molecular dynamics (AIMD) simulations. The displayed ellipsoids correspond to a probability of 50%. These thermal ellipsoids have been plotted on the basis of the data collected in Supplementary Tables 3 and 4. 27

28 Intensity (a.u.) Frequency (THz) Supplementary Figure 28: Vibrational density of states (VDOS) for HT (red curve) and LT (black curve) phases of TTTA. These VDOS were computed by means of a Fourier transform of the total velocity autocorrelation function (i.e., the mass weighted sum of the atom velocity autocorrelation functions) obtained from the molecular dynamics trajectories of HT and LT at 300 K. The spectra were computed after 10 ps of simulation time. 28

29 Supplementary Table 1: Supercell parameters of both the LT and HT polymorphs before (Xray) and after (opt) the variable- cell geometry optimizations. The (a,b,c) parameters are given in Angstrom, and (α,β,γ) in degrees. a b c α β γ LTXray a LTopt HTXray a HTopt a The (a,b,c) supercell parameters of LT and HT are multiples of the (a,b,c) parameters defining the unit cells of the LT- 300 and HT- 300 X- ray resolved structures, respectively. The relations between the supercell (sc) parameters and the unit cell (uc) parameters are the following ones: alt,sc = 2 alt,uc; blt,sc = 2 blt,uc; clt,sc = 2 clt,uc aht,sc = 2 aht,uc; bht,sc = 4 bht,uc; cht,sc = cht,uc 29

30 Supplementary Table 2: Difference in the single- point energy (ΔE) between the slipped TTTA dimer of HT- 300 and the eclipsed TTTA dimer of LT- 300 (see Supplementary Fig. 13) computed with different electronic structure methods. Electronic structure method ΔE (kcal mol - 1 ) a PBE- D2 / 25 Ry b 2.80 PBE- D2 / 35 Ry b 2.98 CAS(2,2)- PT2 / aug- cc- pvdz c 4.39 a ΔE is defined as Edimer- HT Edimer- LT. b Plane wave pseudopotential calculations carried out with the CPMD code c CAS(2,2)- PT2 means that second order perturbation theory is applied to a Complete Active Space wavefunction, whose active space includes 2 electrons in 2 orbitals. The two orbitals included in the CAS are the SOMOs (Singly Occupied Molecular Orbitals) of each TTTA radical of the pair. 30

31 Supplementary Table 3: Computed / Experimental a atomic displacement parameters (Uij, in Å 2 ) for the TTTA radicals in the HT polymorph at 300 K. These atomic displacement parameters give rise to the thermal ellipsoids shown in Supplementary Figure 27. Atom Label b U11 c U22 d U33 U12 U13 U23 C / / / / / / C / / / / / / N / / / / / / N / / / / / / N / / / / / / S / / / / / / S / / / / / / S / / / / / / a Experimental U ij as obtained from the X- ray diffraction measurements b See Supplementary Figure 26 for the labels that identify each atom of the TTTA molecule c Note that the 1,2,3 subindices refer to the three spatial directions. In particular, U 11=U xx, U 22=U yy, U 33=U zz, U 12=U xy, U 13=U xz and U 23=U yz. d The highlighted column corresponds to the motions along the stacking direction 31

32 Supplementary Table 4: Computed / Experimental a atomic displacement parameters (Uij, in Å 2 ) for the TTTA radicals in the LT polymorph at 300 K. These parameters give rise to the thermal ellipsoids shown in Supplementary Figure 27. Atom Label a U11 b U22 U33 c U12 U13 U23 TTTA- 1 d C / / / / / / C / / N / / / / / / / / / / N / N / S / S / S / / / / / / / / / / / TTTA / / / / / / / / / / / / / / / C / C / N / N / N / S / S 7 S / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / a Experimental U ij as obtained from the X- ray diffraction measurements. b See Supplementary Figure 26 for the labels that identify each atom of TTTA. c Note that the 1,2,3 subindices refer to the three spatial directions. In particular, U 11=U xx, U 22=U yy, U 33=U zz, U 12=U xy, U 13=U xz and U 23=U yz. d The highlighted column corresponds to the motions along the stacking direction. e The atomic displacement parameters are reported for two TTTA molecules because each stack of radicals in LT comprises two crystallographically independent molecules. 32

33 Supplementary Note 1: Structural description of the two polymorphs of the TTTA crystal The packing of the triclinic ( P1 space group) LT and monoclinic (P21/c space group) HT crystal structures of TTTA recorded at 300 K 1 (LT- 300 and HT- 300) is displayed in Supplementary Fig. 1 and 2. Both crystals comprise 2D layers formed by π- stacks of TTTA radicals that are laterally linked by a series of intermolecular S S contacts and six- center S N contacts (see Supplementary Fig. 3). The 2D layers in LT- 300 contain one single molecular plane orientation (Supplementary Fig. 1b and 3a), whereas the 2D layers in HT- 300 contain two distinct molecular plane orientations (Supplementary Fig. 2b and 3b). The interstack S S contacts in the 2D layers of both LT and HT phases define a zigzag pattern (Supplementary Fig. 3). Yet the zigzag pattern of LT is different from that of HT: while one every four S S contacts is broken in LT, the zigzag pattern of HT is completely regular. The different 2D layers of both phases of the TTTA crystal are linked by a series of interstack S N and S S contacts (see Supplementary Fig. 4). Some of the S N contacts give rise to 4- center bridges. As explained in the main text, the stacks of radicals present in the LT phase are distorted π- stacks that comprise slipped pairs of nearly eclipsed radicals (see Supplementary Fig. 1b). On the contrary, the π- stacks of HT are regular stacks of radicals, where each TTTA molecule exhibits a slipped overlap with its two adjacent molecules along the stacking direction (see Supplementary Fig. 2b). 33

34 Supplementary Note 2: Structure of the LT polymorph at 0 K The LT 0 structure is almost identical to the X- ray resolved crystal structure of LT 300. The computed supercell parameters are in excellent agreement with the experimental ones (see Supplementary Fig. 7 and Supplementary Table 1). The most noticeable difference between these two sets of parameters is found in the c vector, which is the vector associated with the stacking direction. Upon optimization, the c vector decreases (the optimized c vector at 0 K is ca. 2% shorter than the experimental one at 300 K) and this results in slightly shorter distances between adjacent radicals along the stacks, as compared to the crystal structure (see Figure 2 in the main text). This is due to the thermal expansion of the crystal at finite temperatures. Finite- difference normal mode analyses of LT 0 confirm that this structure is a minimum. 34

35 Supplementary Note 3: Structure of the HT polymorph at 0 K As explained in the main text, the stacks of the HT 0 structure differ substantially from the stacks observed in HT 300. In particular, the stacks of HT 0 comprise slipped radical pairs with alternate short and long distances between the centroids of adjacent radicals. This distorted π stack is at odds with the HT 300 experimental structure, where the separation between adjacent radicals of the same stack is always uniform. Finite- difference normal mode analyses of HT 0 confirm that this structure is a minimum. It is worth mentioning that eight different variable- cell optimizations have been performed for the HT phase. In each one of these optimizations, a different initial spin configuration was provided as initial guess. The eight different spin configurations that were tried are displayed in Supplementary Fig. 8. In all these initial spin configurations, the spins of adjacent radicals within a stack are antiferromagnetically coupled because it has already been proven 2 that the magnetic exchange coupling constant between adjacent radicals within the stack in HT 300 is antiferromagnetic (ca. 155 cm - 1 ). The eight spin configurations shown in Supplementary Fig. 8 thus differ from each other in the relative spin arrangement of neighboring TTTA radicals located in different stacks. The variable- cell optimization of five of the configurations depicted in Supplementary Fig. 8 (configurations c), d), e), h) and i)) led to an intrastack dimerization process, thereby furnishing the HT 0 structure of Supplementary Fig. 7 (i.e. a structure with distorted π- stacks, as shown in Figure 2b in the main text). On the contrary, the variable- cell optimizations of configurations b), f) and g) (shown in Supplementary Fig. 8) preserved the regular structure of the π- stacks observed in HT 300. However, the energies of these regular structures lie ca kcal mol - 1 higher in energy than the dimerized HT 0 structure shown in Supplementary Fig 7. In addition, when running AIMD simulations at 30 K with these regular structures as initial configurations, the intrastack dimerization process was observed to occur in less than 1 picosecond. These AIMD runs thus prove that the regular structures initially obtained in the variable- cell optimizations of configurations b), f) and g) are not stable configurations. The variable- cell optimization became trapped in the regular structures because the potential energy surface of the system is very flat around that region. Despite the intrastack dimerization process upon geometry optimization of HT, the HT 0 structure is completely different from LT 0. In fact, HT 0 preserves not only the monoclinic symmetry of HT 300 (see Supplementary Table 1) but also its two distinct molecular- plane orientations (see Supplementary Fig. 7). In this sense, the overall structure of HT 0 resembles that of HT 300. Similarly to what is observed for LT, the major difference between the computed lattice parameters and the experimental ones for HT is found in the vector associated with the stacking direction (the b vector). Specifically, the computed parameter for HT 0 is ca. 6% smaller than that for HT 300 due to the thermal expansion of the crystal at finite temperatures. 35

36 Supplementary Note 4 Validation of the computational methodology: Band structure calculations It is well known that both phases of TTTA behave as insulators: the LT phase is a Peierls insulator, while the HT phase is a Mott insulator. 3,4 Therefore, the very first thing that needs to be checked concerning the computational methodology is whether the PBE functional is able to properly describe this behavior. To this end, we performed band structure and electronic DOS (density of states) calculations carried out for both phases of TTTA. These calculations were done using the X- ray recorded structures of HT- 300 (i.e. the structure with the regular stacking motif) and LT Concerning the analysis for the HT phase, we first did the electronic structure calculations using a supercell containing eight different TTTA molecules. This supercell, which we will denote by HT- 8, comprises four different stacks of radicals (each stack containing two radicals) and is a fourth part of the supercell displayed in Supplementary Figure 6c. The HT- 8 cell is actually the magnetic unit cell of the HT phase of TTTA, which, in turn, is the crystalline unit cell doubled along the stacking direction. The HT- 8 cell is the magnetic unit cell of HT because it is the smallest cell that allows one to describe the antiferromagnetic interactions between neighboring TTTA radicals within the stacks. The correct description of these antiferromagnetic interactions is crucial because it is well established that the magnetic exchange coupling constant between adjacent radicals within a stack in HT is antiferromagnetic. 2 Note that using the crystalline unit cell (i.e. with only one radical per column) would result in a ferromagnetic coupling between spins of adjacent radicals within a stack, in contradiction with the experimental data and previous computational studies. 2 The crystalline unit cell of HT can thus not be employed for electronic structure calculations of HT. The spin configuration chosen to carry out the band structure and DOS calculations of the HT- 8 cell is displayed in Supplementary Fig. 9a. This particular spin configuration corresponds to the spin configuration shown in Supplementary Fig. 8b, which is the most stable spin configuration among all the configurations displayed in Supplementary Figure 8. The band structure and DOS of the HT- 8 cell have been obtained using a 4x4x4 Monkhorst- Pack grid of k- points. As clearly displayed in Supplementary Fig. 9b, the dispersion of the bands in the HT- 8 is not very pronounced in any direction. The band structure also indicates that there is a significant gap between the occupied and unoccupied bands. This is confirmed by the electronic DOS (see Supplementary Fig. 10a), which reveals that the band gap in HT- 8 is around 0.6 ev. It is important to stress that this computed band gap is in line with the experimental observation that the HT phase behaves as an insulator. The fact that the gap computed at the PBE level is lower than the experimentally observed optical gap (1.5 ev 3 ) is hardly a surprise, given the well- known tendency of GGA functionals to underestimate band gaps. We have also computed the electronic DOS of the HT phase for the supercell that has been employed in the AIMD simulations and the variable- cell optimizations, that is, 36

37 the supercell that contains 32 TTTA radicals (see Supplementary Fig.6). The electronic DOS for this specific supercell has been calculated using Γ- point sampling of the Brillouin zone and the spin configuration shown in Supplementary Fig. 8b. As previously stated, in this spin configuration, all the spins of adjacent radicals within a stack are antiferromagnetically coupled. As displayed in Supplementary Fig. 10b, the electronic DOS obtained for this supercell is quite similar to that obtained with the HT- 8 cell. The most relevant aspect of the DOS for the larger supercell is that it also features a noticeable band gap of ca. 0.6 ev. It thus follows that the electronic structure associated with the large supercell used in the simulations is in agreement with the experimental observation that HT is not metallic. The evaluation of the spin density distribution within a TTTA stack of HT after wavefunction optimization of the crystal structure of HT- 300 (using the supercell of Supplementary Fig. 6) indicates that the antiferromagnetic coupling between adjacent TTTA is preserved. As displayed in Supplementary Fig. 11, there is an alternance of spin- up and spin- down distribution within any given stack. This spin alternance can be unambiguously assigned to an overall antiferromagnetic spin coupling between any two adjacent TTTA radicals, in agreement with the measured magnetic data. The resulting AFM pattern of interactions with well- localized spins is not compatible with a metallic ground state. Instead it is consistent with an insulator ground state, as experimentally observed for TTTA. This thus provides further evidence of the non- metallic character predicted by PBE for the HT- phase. With reference to the electronic structure of the LT phase, the electronic DOS displayed in Supplementary Fig. 12 demonstrates that PBE predicts that this phase is not metallic, in line with the experimental observations. For this particular phase, the band gap obtained at the PBE level is ca. 0.8 ev. Overall, the results presented in this subsection clearly demonstrate that the PBE functional is able to properly describe the electronic structure of both phases of the strongly correlated TTTA material. Validation of the computational methodology employed: DFT- D2 all- electron calculations In order to verify the results obtained with the plane wave pseudopotential calculations done with the QUANTUM ESPRESSO code, the variable- cell optimizations of both LT and HT were also performed with the CRYSTAL09 code 5,6. For these all- electron calculations, an Aug- cc- pvtz basis set 7,8 was employed. Like in the plane wave calculations, the optimizations with CRYSTAL09 were done with the PBE exchange correlation functional supplemented with the Grimme correction for the dispersion energy (DFT- D2 parametrization). The supercells employed in the CRYSTAL09 calculations are smaller than the supercells used for the calculations with QUANTUM ESPRESSO and CPMD. In particular, the supercells employed in CRYSTAL09 contain 8 TTTA molecules (they comprise 4 π- stacks of radicals, each of them containing 2 radicals). Concerning the Brillouin zone sampling, the variable- cell optimizations with CRYSTAL09 were done with a 4 x 4 x 4 Monkhorst- Pack grid of k- points. 37

38 The results obtained with CRYSTAL09 are in complete accordance with those obtained with QUANTUM ESPRESSO. Indeed, the optimization of HT with CRYSTAL09 led to the very same intrastack dimerization process observed with QUANTUM ESPRESSO. Several spin configurations were provided as initial guess and all the configurations with an intrastack antiferromagnetic coupling between adjacent radicals underwent the dimerization process. 38

39 Supplementary Note 5: Validation of the computational methodology employed: CASPT2 benchmark results The validity of the DFT- D2 results was assessed against CASPT2 calculations 9,10 on a series of model systems. All the CASPT2 calculations were carried out with the MOLCAS code 11 and an all- electron Aug- cc- pvdz basis set 7,8. The Cholesky decomposition 12,13 was used to treat the two- electron integrals in all the CASPT2 calculations. Benchmark results on a dimer model The first validation tests were performed on TTTA dimers, whose geometries were taken from the X- ray resolved LT- 300 and HT- 300 structures. The difference in the single point energies between the slipped dimer of HT- 300 and the eclipsed dimer of LT- 300 (see Supplementary Fig. 13) computed at the PBE- D2 (PBE + Grimme correction) level and the CAS(2,2)- PT2 level are collected in Supplementary Table 2. The values of this table reveal that the results obtained with PBE- D2 are in fair agreement with those obtained with CASPT2. Both methods predict that the eclipsed dimer is more stable than the slipped dimer, in line with the experimental observation that the LT polymorph is the thermodynamically stable polymorph at low temperatures. Benchmark results on a stack of four TTTA radicals The second model employed for validation purposes consists of an isolated stack of four TTTA radicals (TTTA4, see Supplementary Fig. 14). Given a regular arrangement of the radicals in the TTTA4 aggregate, two distinct distorted stacks are conceivable if one considers a dimerization process along the stacking direction. The spin pairing process can thus lead to either configuration I, characterized by an eclipsed- slipped- eclipsed sequence of TTTA pairs, or configuration II, characterized by a slipped- eclipsed- slipped sequence of TTTA pairs. Note that the configuration with a uniform separation between the nitrogen atoms of the S N S moieties of adjacent radicals will be referred to as configuration III. The validity of the PBE+D2 results was assessed against CASPT2 calculations by computing the potential energy surface (PES) of the TTTA4 model in the subspace spanned by ξ1 and ξ2, which drive a relative sliding motion between TTTA radicals along the stack (see Supplementary Fig. 14). Note that configurations I and II can be smoothly transformed into each other by changing these two geometrical variables. According to Supplementary Fig. 14, the computed PES at the PBE- D2 level is quite similar to that computed at the CASPT2 level. What is even more important is that the topology of the computed PES at the CASPT2 and DFT levels is the same. In particular, both PESs show that configurations I and II are minima, while the regular arrangement of configuration III is not a stationary point. Therefore, the CASPT2 calculations strongly support the validity of the PBE- D2 calculations, thereby proving that the intrastack dimerization process observed upon optimization of HT 0 is not a mere artifact of PBE- D2 but a phenomenon that reflects the true physics of the TTTA system. 39

40 The different stability of the two minima found for TTTA4 (configurations I and II) is due to the lack of periodic boundary conditions along the stacking direction. Configuration I is more stable due to the presence of two eclipsed TTTA pairs against just one in configuration II. In fact, when periodic boundary conditions are imposed (TTTAn model), these two configurations become degenerate in energy, and configuration III becomes a transition state (see Supplementary Fig. 14). Benchmark resuts on lateral intermolecular interactions between TTTA radicals The intrastack dimerization process occuring in HT- 0 brings about some changes in the distances associated with the lateral contacts between neighboring TTTA radicals belonging to distinct columns. Most of those changes are insignificant, but one of them is quite noticeable and important. As shown in Supplementary Fig. 15, one set of S S lateral contacts in the X- ray resolved HT- 300 structure defines a regular zigzag pattern, with all the S S distances being Å. Such regular zigzag is disrupted in the optimized HT- 0 structure because one of the S S distances is significantly elongated (the S S distance of Å is larger than the sum of the van der Waals radii of two sulfur atoms). In view of this analysis, one can conclude that the intrastack dimerization in HT- 0 takes place at the expense of weakening some of the lateral non- covalent interactions between TTTA molecules of different columns. The fact that the dimerization is observed at the PBE- D2 level means that the energetic stabilization associated with the formation of dimers exceeds the energy penalty associated with the weakening of the network of lateral interactions. CASPT2/Aug- cc- pvdz calculations on model systems confirm that this is indeed the case, as shown below. Let us first consider the pair of TTTA radicals depicted in the left- hand part of Supplementary Fig. 16 and the geometrical variable ξ, which drives a relative sliding motion between the two radicals. By virtue of such sliding motion, such variable is able to smoothly transform a slipped pair of radicals into a completely eclipsed dimer and vice versa, while preserving the distance between the TTTA molecular planes (which has been chosen to be 3.27 Å). As shown in Supplementary Fig. 16, the eclipsed dimer is ca. 3.9 kcal mol - 1 more stable than the slipped pair at CAS(2,2)- PT2 level. Note that this CASPT2 result is slightly different from the one collected in Supplementary Table 2. This is due to the fact that the value of that Table was computed for the geometries of the radical pairs found in the LT- 300 and HT- 300 crystals, whereas the curve plotted in Supplementary Fig. 16 was computed for a model system in which only the relative degree of slippage was allowed to change. In addition, the distance between the TTTA planes in the model system of Supplementary Fig. 16 is slightly shorter than the interplanar distance in LT- 300 and HT Let us now consider the model system shown in the inset of Supplementary Fig. 17. This model also includes two TTTA molecules, but each molecule belongs to a different stack of radicals. Specifically, one TTTA belongs to the stack depicted on the left of Supplementary Fig. 15a, while the other TTTA radical belongs to the stack depicted on the right of Supplementary Fig. 15a. The two TTTA molecules were directly taken from the X- ray HT- 300 structure and they present a S S contact at 40

41 3.395 Å. By moving the two TTTA radicals along the geometrical variable ξ (see Supplementary Fig. 17), the S S distance is changed. In particular, the variable ξ moves each TTTA radical along the vector connecting its two carbon atoms. Notice that the relative intra- stack sliding motion depicted in Supplementary Fig. 16 results as well from moving the radicals along the vector defined by their C- C bond. Therefore, the motion driven by ξ in Supplementary Fig. 17 mimics the motion followed by the TTTA molecules during the intrastack dimerization process in HT- 0. The energy penalty in going from a S S distance of 3.39 Å to a S S distance of 3.78 Å along ξ is ca. 1.3 kcal mol - 1 at CAS(2,2)- PT2 level. This energy penalty is three times smaller than the stabilization of 3.9 kcal mol - 1 that accompanies the eclipsed pair formation in the model system of Supplementary Fig. 16. These results thus indicate that the formation of eclipsed TTTA dimers largely compensates the weakening of the lateral S S contacts caused by the intrastack dimerization in HT- 0. The results obtained with the model systems of Supplementary Fig. 16 and 17 were further corroborated with the tetramer model displayed in Supplementary Fig. 18. This new model system was built by taking two adjacent TTTA molecules of each column of Supplementary Fig. 15a. The interplanar distance between adjacent TTTA radicals in each stack was set to 3.27 Å, as done with the model system of Supplementary Fig. 16. The ξ coordinate drives a relative sliding motion of the adjacent radicals within each stack, while preserving their interplanar separation. Progress along ξ brings about a smooth transformation of slipped pairs into eclipsed pairs in each stack and a distortion of the network of lateral S S contacts between the two stacks. This model system thus takes simultaneously into account the two effects that were considered separately in the model systems of Supplementary Fig. 16 and 17. These are two antagonistic effects in energetic terms: the formation of eclipsed pairs is accompanied by an energy decrease, whereas the distortion of the lateral contacts involves an energy increase. According to the CAS(4,4)- PT2 potential energy profile displayed in Supplementary Fig. 18, progress along ξ is accompanied by a decrease of the energy of the system. This confirms that the formation of eclipsed dimers dominates over the weakening of the lateral contacts in the intrastack dimerization observed in HT- 0. All things considered, the CASPT2 calculations carried out on several model systems substantiate the results obtained at the PBE- D2 level. Not only do the CASPT2 calculations prove that the regular π- stack of HT- 300 is not a minimum- energy structure, but they also prove that the intrastack dimerization observed in HT- 0 occurs because the energy decrease accompanying the formation of TTTA dimers is larger (in absolute value) than the energy increase associated with the concomitant distortion of the network of lateral contacts between neighboring stacks. 41

42 Supplementary Note 6: Nature of the binding interaction in TTTA dimers Here we will explore the electronic structure of the distorted π- stack (configuration I) and the regular stack (configuration III) of the TTTA4 aggregate employed as a model system for the CASPT2 calculations (see Supplementary Fig. 14). In configuration I, the TTTA4 aggregate can be viewed as two interacting eclipsed π- TTTA2 moieties. The two natural orbitals with larger (smaller) occupation numbers have bonding (antibonding) character within each π- TTTA2 moiety (see Supplementary Fig. 19). A similar picture is obtained for configuration III, but TTTA radicals are now evenly arranged and no π- TTTA2 moieties can be identified. The occupation numbers of the singlet ground state for both configurations I and III are indicative of the multireferent nature of their wavefunctions (see Supplementary Fig. 19). Accordingly, configuration I can be described as a mixture of closed shell (37%) and open- shell (63%) singlets. The first component is responsible for the bond- pairing, i.e. the covalent bonding. On the contrary, the occupation numbers of configuration III are consistent with a predominant open- shell singlet character. The larger contribution of closed shell singlet in the dimerized configuration thus partially explains why this configuration is more stable than the regular one. The other key factor that contributes to the relative stabilization of the dimerized structure is the dispersion interaction energy between adjacent radicals. Indeed, at the CASSCF(4,4) level, the regular aggregate is more stable than the dimerized one. This situation is reversed only when the dynamic correlation is included via CASPT2 calculations. All this data allows us to define the bonding within each eclipsed π- TTTA2 dimers as a long multicenter bond. 14,15,16,17 Therefore, the picture that emerges from this analysis is the following: in going from the dimerized aggregate to the regular aggregate the multicenter long- bond within each π- TTTA2 dimer is almost broken, and the open- shell singlet character of the wavefunction increases. This is in agreement with previous work 2 aimed at addressing the different magnetic properties of LT and HT phases of TTTA. Therein it was shown that the LT phase is diamagnetic due to the strongly antiferromagnetic coupling ( cm - 1 ) between the two spins of the radicals of the eclipsed π- TTTA dimer. Such large magnetic interaction can thus be ascribed to the multicenter long- bond character of the eclipsed π dimers. Overall, the analysis of the electronic structure herein presented provides a rationale for the intrastack dimerization observed in the optimization of HT 0. Indeed this dimerization is driven by the formation of long multicenter bonds between pairs of TTTA radicals. The presence of this kind of bonds also explains why the LT phase is energetically more stable than HT. 42

43 Supplementary Note 7: Dynamics of LT and HT at 300 K The average structures obtained from ca. 10 picoseconds of AIMD trajectories for the LT and HT polymorphs at 300 K (LT 300 and HT 300) are virtually identical to the respective X ray crystal structures at this temperature (see Supplementary Fig. 20). These AIMD simulations were performed in the NVT ensemble, using the experimental supercell parameters collected in Supplementary Table 1. 43

44 Supplementary Note 8: Dynamics of LT and HT at different temperatures Complementing Figure 4 in the main text, Supplementary Fig. 21 displays the temperature dependence of the probability distribution functions (PDFs) associated with the distance between centroids dc of adjacent radicals within a given stack of both LT and HT. The information embodied in both Figure 4 in main text and Supplementary Fig. 21 is essentially the same. The PDFs of the LT phase remain bimodal in the whole temperature range, whereas the unimodal PDFs of HT at room temperature gradually transform into bimodal PDFs as the temperature is lowered. The fact that the PDFs for HT are unimodal for temperatures higher than ca. 200 K is in consonance with the regular π- stacks of TTTA radicals observed in X- ray measurements and with the single crystal X- ray diffraction ω oscillation photographs of HT within the temperature range of K (see Supplementary Fig. 24). Besides, the unimodal PDF at 225 K is consistent with the regular stacking motif observed by X- ray diffraction measurements of the HT polymorph at 225 K in Ref. 18 (CCDC code: SAXPOW04). This temperature is one of the lowest temperatures at which the structure of HT has ever been resolved because for lower temperatures the HT phase collapses into the dimerized LT polymorph. It thus follows that the results of our AIMD simulations on HT agree with the experimental data in the whole range of temperatures in which the HT phase has been observed. The trajectories employed to compute the PDFs of Supplementary Fig. 21 and the 2D- PDFs of Figure 4 in the main text were generated by means of AIMD simulations performed in the NVT ensemble. All the generated trajectories span a time range of ca. 10 picoseconds. In the simulations of HT, a different supercell was used at every temperature. The supercell parameters employed at 300 K are the experimental parameters collected in Supplementary Table 1. For the simulations at 225 K, the supercell parameters were defined based on the unit cell parameters of the X- ray resolved HT- 225 structure (CCDC code: SAXPOW04). For the simulations at lower temperatures, the parameters defining the supercells were obtained by linear extrapolation on the basis of the variation of those parameters in going from the cell at 300 K to the cell at 225 K. 44