*3.3. (MV)I2*

In (MV)I2 (space group: *P*21/*c*) [26], the valence of an MV molecule is 2+ because of two I − ions. The crystal structure of (MV)I2 and the molecular structure of MV2<sup>+</sup> are shown in Figure 5a,b, respectively. We performed the structural analysis and CDFS analysis of (MV)I2 at 100 K. Figure 5d shows the VED of an MV2<sup>+</sup> molecule, in which H, C, and N atoms have 1*s*1, 2*s*22*p*2, and 2*s*22*p*<sup>3</sup> valence electrons, respectively. Although C and N<sup>+</sup> have the same number of valence electrons, the distribution states of valence electrons on the C and N atoms in the MV2<sup>+</sup> molecule are quite different from each other. The ED around C is spatially spread, whereas that around N<sup>+</sup> is localized on the atom. This result corresponds to the difference in electronegativities between the C and N atoms, where the N atoms are more electronegative than the C atoms. Furthermore, there is a difference in the EDs around the two C atoms bonded to N in the six-membered ring of the MV2<sup>+</sup> molecule. The ED around the C(1) atoms close to the I <sup>−</sup> anions (surrounded by red dotted circles) on the MV2<sup>+</sup> molecular plane, which is in the direction of the hydrogen bond, is lower than that of the C(2) atoms located at the opposite side. This result suggests that the anisotropic VED distribution on the MV2<sup>+</sup> molecule is realized by the anisotropic electrostatic interactions with the surrounding the I − ions. A CDFS analysis directly visualized the distribution state of complicated molecular orbitals composed of different types of atoms.

### *3.4. (TMTTF)2X*

Finally, we applied the CDFS analysis to quasi-1D molecular conductors (TMTTF)2*X* [27], which shows various electronic properties in the pressure–temperature phase diagram [28–31]. Figure 6a,b shows the crystal structure of (TMTTF)2PF6 and the molecular structure of TMTTF, respectively. For the charge ordering phase transition of (TMTTF)2PF6 (*T*CO ∼ 67 K) at ambient pressure, although responses associated with the charge ordering are confirmed by the dielectric constant [32,33], NMR [34,35], ESR [36], infrared, and Raman spectroscopies [37,38], no evidence of charge ordering has been observed from the crystal structure [39,40]. Thus, this transition from the dimer Mott phase to the charge ordering phase (Figure 6c), which is associated with the lack of an inversion center, has been called a mysterious 'structureless transition' [41–43]. We investigated the crystal structure and VED distribution of the charge ordering phase in (TMTTF)2PF6 using synchrotron XRD [9].

**Figure 5.** (**a**) Crystal structure of (MV)I2; (**b**) Molecular structure of MV2+; (**c**) Relationship between a MV2<sup>+</sup> molecule and surrounding I − ions. There is an inversion center at the central C–C bond in the MV2<sup>+</sup> molecule. The least square (LS) plane, on which the central six C atoms are located, is shown in green. The I − ions surrounded by the red dotted circles exist near the green plane. (**d**) VED distribution of the MV2<sup>+</sup> molecule. This density plane corresponds to the green plane in (**c**). The orientations of the MV2<sup>+</sup> molecules are the same between (**c**,**d**).

**Figure 6.** (**a**) Crystal structure of (TMTTF)2PF6; (**b**) Molecular structure of a TMTTF; (**c**) Schematic configuration of the dimer Mott and charge ordering (CO) phases; (**d**,**e**) Bond length in the (**d**) hole-rich and (**e**) hole-poor TMTTF molecules at 30 K in the charge ordering phase of (TMTTF)2PF6. The red (blue) values indicate that the bonds are shorter (longer) than the others at the same positions. The black values indicate that bonds at the same position are consistent within the error range.

The precise structural parameters at 30 K in the charge ordering phase are obtained by a high-angle analysis. Figure 6d,e shows the bond length in the hole-rich and hole-poor TMTTF molecules at 30 K, respectively. The C=C bonds at the center and both sides in the hole-rich molecule are longer than those in the hole-poor molecule. The central C–S bonds in the hole-rich molecule are shorter than those in the hole-poor molecule. These results correctly reflect the tendency of the charge ordering state in the TMTTF dimer. The amount of charge transfer δco is estimated from the bond length in the TMTTF molecule. Two types of formulas, *q* = −15.55 + 20.42*r* (set 1) and *q* = −26.88 + 34.98*r* (set 2), are given for empirically calculating the valence of a TMTTF molecule in [44]. Here, *q* is the valence of the TMTTF, and *r*(= *a*/*b*) is a ratio of the central (*a*) C=C bond and (*b*) C–S bonds length. The amount of charge transfer δco in the TMTTF dimer is calculated as δco = - *<sup>q</sup>*hole-poor <sup>−</sup> *<sup>q</sup>*hole-rich /2. By using these formulas, δco = 0.10*e* (set 1) and 0.17*e* (set 2) are obtained from our structural analysis results at 30 K. The difference of the highest occupied molecular orbital levels between the hole-poor and hole-rich TMTTF molecules, ∼ 21.7 meV, was confirmed by the extended Hückel calculation [45]. These results show that our crystal structural analysis revealed the charge ordering state in (TMTTF)2PF6.

Figure 7a,b shows the VED of the hole-rich and hole-poor TMTTF molecules in the charge ordering phase of (TMTTF)2PF6 obtained from the CDFS analysis. In this case, the valence electrons of H, C, and S constituting the TMTTF molecule correspond to 1*s*1, 2*s*22*p*2, and 3*s*23*p*4, respectively. The ED corresponding to the bonding orbital on the C=C bonds and the node of ED corresponding to the antibonding orbital on the C–S bonds are clearly shown. Moreover, the ED reflecting the isotropic *s* orbitals is observed on the S and C atoms of the methyl groups. Almost no difference is observed between the appearance in the two VED distributions (Figure 7a,b), because the amount of charge transfer (δco ≤ 0.5*e*) is too small compared to the number of valence electrons of a TMTTF molecule (64*e*). Therefore, we compared the number of valence electrons between the two molecules in the dimer. By comparing the number of valence electrons in the atomic basin of the respective atoms in the TMTTF molecule calculated by Bader's topological analysis [46], the amount of charge transfer was determined as δco = 0.10*e* [9], which is consistent with the estimation from the bond length (δco = 0.10*e*) (set 1), from the Raman (δco = 0.055*e* [38]), from the infrared (δco = 0.075*e* [31]), and from the NMR spectroscopies (δco = 0.14*e* [35]).

To investigate the intramolecular degrees of freedom, we focus on (TMTTF)2AsF6, which has the common crystal structure (TMTTF)2PF6 and undergoes a charge ordering phase transition at *T*CO ∼ 100 K [47]. A larger charge transfer in the dimer than that of (TMTTF)2PF6 was confirmed in the charge ordering phase of (TMTTF)2AsF6 by the Raman (δco = 0.09*e* [38]), infrared (δco = 0.105*e* [31]), and NMR spectroscopies (δco = 0.25*e* [48], δco = 0.17*e* [49]). We investigated whether the difference in the amount of charge transfer affects the VED distribution of the TMTTF molecule. As a result of the high-angle analysis of (TMTTF)2AsF6 at 30 K, δco = 0.34*e* (set 1), which is larger than that of *X* = PF6, is determined from the bond lengths in the TMTTF molecules using the formulas in [44]. This tendency is consistent with those described in previous reports of *X* = PF6 and AsF6 [31,35,38,48,49].

Figure 7c,d shows the VED of the hole-rich and hole-poor TMTTF molecules in the charge ordering phase of (TMTTF)2AsF6 obtained from the CDFS analysis, respectively. Almost no difference is observed in the appearance between the hole-rich and hole-poor TMTTF molecules in *X* = AsF6, and the VED distributions of *X* = AsF6 are also approximately identical to those of *X* = PF6. In this case, the difference in the magnitude of the absolute value of the ED between *X* = PF6 and AsF6 is of little significance because the CDFS analysis is based on the inverse Fourier transform of finite data (Equation (7)). However, when calculating the amount of the charge transfer in the dimer from the VED of *X* = AsF6, the charge transfer δco = 0.43*e* is obtained, which is larger than that of *X* = PF6. This tendency is consistent with that in our structural analysis.

**Figure 7.** (**a**–**d**) VED distribution of hole-rich and hole-poor TMTTF molecules of (TMTTF)2PF6 and (TMTTF)2AsF6 on LS planes, respectively. The LS planes are defined by the central two C and four S atoms of the TMTTF molecules. (**e**) H–F distances of 2.7 Å or less in the hole-rich TMTTF molecule of (TMTTF)2AsF6. The orientations of the TMTTF molecules are the same between (**c**,**e**). (**f**) Charge ordering patterns of hole-rich and hole-poor TMTTF molecules in (TMTTF)2*X* (*X* = PF6 and AsF6), which indicate a 2D Wigner crystal state. The region surrounded by the dotted square shows a TMTTF molecule dimer.

The importance of H–F interactions between the methyl group in the TMTTF molecule and *X* anions is pointed out in this system [50]. Thus, we investigated the distance between hydrogen in the methyl group and fluorine in the anion. Figure 7e shows the H–F distances of 2.7 Å or less in the hole-rich TMTTF molecule. In this regard, we focused on the anisotropy of the VED distribution in the TMTTF molecule in *X* = AsF6 (Figure 7c). The VED is concentrated around the methyl groups, but there seems to be no correlation between the distribution state and H–F distances. This tendency is the same for the hole-poor TMTTF in *X* = AsF6 (Figure 7d) and for the hole-rich and hole-poor TMTTF in *X* = PF6 (Figure 7a,b). Therefore, no clear effect of closed-shell anions was confirmed from the VED distribution of TMTTF.

From the above results, we directly revealed that the spatial charge ordering pattern formed a 2D Wigner crystal state (Figure 7f) from the molecular structure and VED distribution in the charge ordering phase in (TMTTF)2*X* (*X* = PF6 and AsF6). This pattern is also consistent with previous ESR experiment [36] and theoretical expectations [51–53].

#### **4. Conclusions**

We succeeded in directly observing the VED distribution of several molecular materials, i.e., diamond, C60 fullerene, (MV)I2, and (TMTTF)2*X*, using synchrotron XRD and the CDFS method. When a molecule is formed by bonds between atoms, various interactions work depending on the shapes and energy levels of the hybridized orbitals. As a result, even molecules with relatively simple structures produce complex molecular orbitals. Therefore, it is difficult to observe the whole picture of molecular orbitals with existing experimental methods. On the other hand, because the CDFS method using synchrotron XRD can directly observe the distribution state of valence electrons occupying the reconstructed molecular orbitals in a real space, it is possible to take an approach which differs from the existing methods for the study of molecular substances. Furthermore, the VED distribution obtained from the CDFS analysis corresponds to the square of the wave function, which provides essential information for quantum chemical and first-principles calculations. In fact, with complementary study of the CDFS analysis and first-principles calculations in transition metal oxides, we succeeded in clarifying the whole orbital state formed by the localized 3*d* orbitals on an atom and the metal–ligand hybridized orbitals [10]. The research methods we propose may signal a breakthrough in the study of the orbital states in materials.

**Author Contributions:** S.K. and H.S. designed and coordinated this study. R.K., T.N. (Toshio Naito) and T.N. (Toshikazu Nakamura) synthesized the samples. S.K., Y.H. and H.S. performed the XRD experiment; S.K. and Y.H. analyzed the XRD data. S.K. and H.S. wrote the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by a Grant-in-Aid for Scientific Research (No. JP19J11697) from JSPS. The synchrotron radiation experiments were performed at SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2011B0083, No. 2019A0070, and No. 2019B0070).

**Acknowledgments:** We thank Ken Niwa for providing materials used in the experiments, Takeshi Hara, Keita Kojima, Taishun Manjo, Naoyuki Katayama, and Kunihisa Sugimoto for their support in the synchrotron XRD experiments, Mariano De Souza, and Jean-Paul Pouget for fruitful discussions.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

CCDC 2036797 contains the supplemental crystallographic data of C60 fullerene at 30 K. The data is provided free of charge by The Cambridge Crystallographic Data Centre [54].


