**Shunsuke Kitou 1,\*, Yuto Hosogi 1, Ryo Kitaura 2, Toshio Naito 3,4,5, Toshikazu Nakamura <sup>6</sup> and Hiroshi Sawa 1,\***


Received: 15 October 2020; Accepted: 30 October 2020; Published: 3 November 2020

**Abstract:** The physical properties of molecular crystals are governed by the frontier orbitals of molecules. A molecular orbital, which is formed by superposing the atomic orbitals of constituent elements, has complicated degrees of freedom in the crystal because of the influence of electron correlation and crystal field. Therefore, in general, it is difficult to experimentally observe the whole picture of a frontier orbital. Here, we introduce a new method called "core differential Fourier synthesis" (CDFS) using synchrotron X-ray diffraction to observe the valence electron density in materials. By observing the valence electrons occupied in molecular orbitals, the orbital state can be directly determined in a real space. In this study, we applied the CDFS method to molecular materials such as diamond, C60 fullerene, (MV)I2, and (TMTTF)2*X*. Our results not only demonstrate the typical orbital states in some materials, but also provide a new method for studying intramolecular degrees of freedom.

**Keywords:** X-ray diffraction; single crystal; electron density; molecular orbital

#### **1. Introduction**

The degree of freedom of electron orbitals in materials is closely related to physical properties, such as electrical conductivity, magnetic, dielectric, and optical properties, and crystal structure [1–3]. An electron orbital corresponds to the existence probability of electrons in a real space. In other words, the orbital state can be correctly understood by observing the spatial distribution of electrons in materials. Among the three degrees of freedom in electrons (i.e., charge, spin, and orbital), because the charge and spin can readily respond to external electric and/or magnetic fields, their properties are relatively easy to measure. On the other hand, several experimental methods, such as electron spin resonance (ESR), nuclear magnetic resonance (NMR), high harmonics generated from femtosecond laser pulses [4], ultraviolet angle-resolved photoelectron spectroscopy [5], polarized neutron diffraction [6], resonant X-ray scattering [7], and multipole analysis [8] using X-ray diffraction (XRD), are known to extract orbital information that control the anisotropy of the physical properties. However, in these methods, the measurable substances are limited, and the results may depend on the applied model. Furthermore, as the molecular substance formed by the bonding of multiple atoms has several independent parameters that describe the crystal structure, it is very difficult to estimate the frontier orbital state of the entire system from quantum chemical or first-principles calculations.

The spatial distribution of electrons observed experimentally will be useful for understanding the orbital states in materials. In this sense, XRD is one of the most powerful experimental probes to observe electrons around atoms in a solid. XRD is based on the scattering phenomenon of electrons in a crystal, where the periodic structure of electrons and diffraction intensities are linked by a Fourier transform. In principle, a complete electron density (ED) distribution can be reproduced by performing the inverse Fourier transform on the 'infinite' number of diffraction intensities corresponding to the Fourier series. However, in reality, because the number of diffraction reflections obtained by the XRD experiment is 'finite', it is impossible to reproduce the complete ED distribution owing to the mathematical truncation effects of the Fourier synthesis. Moreover, of all the electrons around an atom, only valence electrons, which are very small in number compared to the total number of electrons in the crystal, are responsible for their physical properties. Therefore, to extract the frontier orbital information from the ED, only the valence electrons need to be extracted. Due to these difficulties, instead of observing the ED, the crystal structure is usually determined from XRD experiments by refining the internal parameters (e.g., atomic positions, atomic displacement parameters, and site occupancies) of the assumed crystal structural model through the least squares method to match the diffraction intensities. This method is called a crystal structural analysis. To access the orbital state (i.e., the distribution of electrons) with high accuracy in materials from the XRD, a new method of analysis is required.

In this paper, we introduce the core differential Fourier synthesis (CDFS) method [9,10] for directly observing the valence electron density (VED) in a crystal from XRD data. The details of the method are also explained in [10]. CDFS can efficiently extract only the valence electron information, which corresponds to the physical properties. By using this method, the orbital state that governs the anisotropy of the physical properties can be directly observed in a real space. Here, we show the results of applying the CDFS method to diamond, C60 fullerene, (MV)I2 (MV = methyl viologen), and (TMTTF)2*X* (TMTTF = tetramethyl tetrathiafulvalene; *X* = PF6, and AsF6).

#### **2. Methods**

The XRD experiments using single crystals were performed on the BL02B1 beamline at the synchrotron radiation facility SPring-8 in Japan [11]. A helium-gas-blowing device (Japan Thermal Engineering Co., Ltd., Japan) was employed to cool the sample. A two-dimensional (2D) imaging plate (FUJIFILM Co., Japan) and a CdTe PILATUS (DECTRIS Ltd., Switzerland) detectors, which had a dynamic range of ~106, were installed in the diffractometer (Rigaku Co., Japan). Diffraction intensity averaging and a structural analysis were performed using SORTAV [12] and Jana2006 [13], respectively. The crystal structure and ED distribution figures were visualized using VESTA [14].

The diffraction intensity *I*(*K*) obtained from the XRD experiment can be described as a Fourier transform of the ED ρ(*r*) using the scattering vector *K* (Equation (1)).

$$I(\mathbf{K}) = S \left| \int\_{\text{all}} \rho(r) e^{-i\mathbf{K}r} dr \right|^2 \tag{1}$$

Here, *S* is a scaling factor. The integration range ( all in Equation (1)) corresponds to the range in which X-rays interfere in the crystal. In the case of a crystal, *I*(*K*) can be described as Equation (2) using Laue functions *L*(*s*), *L*(*t*), and *L*(*u*), where *K* = *sa*∗ + *tb*∗ + *uc*∗ .

$$\begin{array}{rcl}I(\mathsf{K}) & = & S \, L(\mathsf{s})L(t)L(u) \Big| \int\_{\text{unif}} \rho(\mathsf{r}) \mathrm{e}^{-i\mathsf{K}\mathsf{r}} d\mathsf{r} \Big|^{2} \\ & & \propto \left| F\_{\mathsf{obs}}(\mathsf{K}) \right|^{2} \end{array} \tag{2}$$

Here, *F*obs(*K*) is the absolute value of the experimentally observed crystal structure factor. Generally, if the number of unit cells *N* in a crystal is sufficiently large, then the Laue function becomes

*N*<sup>2</sup> only when all of the *s*, *t*, *u* are integers; otherwise, *I*(*K*) = 0. In principle, the ED distribution can be reproduced by the inverse Fourier transform of the diffraction intensity according to

$$\rho(r) = \frac{1}{V} \sum\_{\mathbf{K}}^{\infty} F\_{\text{obs}}(\mathbf{K}) \mathbf{e}^{\mathbf{i}\mathbf{K}r} \,, \tag{3}$$

if the infinite diffraction data were observed. However, the calculation of the ED generally has three problems. (i) To extract the VED with anisotropic information, a sufficiently wide dynamic range of intensity is required. (ii) *F*obs(*K*) is obtained from the experimentally diffraction intensity because of the relationship of *I*(*K*) ∝ *F*obs(*K*) 2 . In this case, *F*obs(*K*) does not include information of the phase term *P* = *ei*φ(*K*) as *F*obs(*K*) = *F*obs(*K*) *<sup>P</sup>* <sup>=</sup> *F*obs(*K*) *ei*φ(*K*), which is necessary for the calculation of ρ(*r*). (iii) Because the number of the *F*obs(*K*) data is finite, the mathematical truncation effect seriously affects the ED distribution.

With regard to (i), the relationship between the number of focused electrons and the number of electrons in the unit cell is important. In the case of a quasi-one-dimensional (1D) molecular conductor (TMTTF)2*X*,a3/4-filled system, because *X*<sup>−</sup> is a monovalent anion, such as PF<sup>−</sup> <sup>6</sup> , AsF<sup>−</sup> <sup>6</sup> , SbF<sup>−</sup> <sup>6</sup> , Br−, or ClO− <sup>4</sup> , there is one hole for two TMTTF molecules forming a dimer. The number of electrons in the (TMTTF)2*X* unit is approximately 350. To estimate the amount of charge transfer of less than 1*e* in the TMTTF dimer in the charge ordering state, a dynamic range that can accurately extract 12/3502 = 1/122500 <sup>∼</sup> 10−<sup>5</sup> of the measured maximum intensity is necessary. Here, the 10−<sup>5</sup> signal information should be observed with a sufficient signal-to-noise (*S*/*N*) ratio. This requirement is sufficiently fulfilled using the current synchrotron radiation facility, in which the dynamic range of 10<sup>6</sup> is guaranteed.

With regard to (ii), because the phase term can be assigned from the calculated crystal structure factor

$$F\_{\rm cal}(\mathbf{K}) = \sum\_{j} f\_{j} T\_{j} e^{-i\mathbf{K}r\_{j}} \tag{4}$$

as *P* = *F*cal(*K*)/ *F*cal(*K*) , Equation (3) can be rewritten as

$$\rho(r) = \frac{1}{V} \sum\_{\mathbf{K}}^{\infty} \left| F\_{\text{obs}}(\mathbf{K}) \right| \mathcal{P} e^{i\mathbf{K}r}.\tag{5}$$

Here, *r<sup>j</sup>* is the *j*th atomic position, *Tj* is the *j*th atomic displacement parameter, and *fj* is the *j*th atomic scattering factor, which is described as

$$f\_{\hat{\jmath}}(\mathbf{K}) = \int\_{\text{atom}} \rho\_{\hat{\jmath}}(\mathbf{r}) e^{-i\mathbf{K}\mathbf{r}} d\mathbf{r}.\tag{6}$$

Figure 1 shows the atomic scattering factor of carbon as a function of sin θ/λ [15]. Here, λ is the wavelength of the incident X-ray, and θ is the XRD angle. While the contribution of the core electrons extends to the high-angle region, the contribution of the valence electrons exists only in the low-angle region (sin <sup>θ</sup>/<sup>λ</sup> <sup>≤</sup> 0.5 Å−1). When calculating the ED, it is necessary to determine *<sup>F</sup>*cal(*K*) with high accuracy to obtain the correct phase term *P*. For this purpose, the structural refinement is performed using only the high-angle *I*(*K*) (i.e., high-angle analysis), where the contribution of the spatially spread valence electron is very small.

**Figure 1.** Atomic scattering factor of carbon [15]. Black, blue, and orange lines indicate the contribution of the total, core, and valence electrons, respectively.

With regard to (iii), the truncation effect of the Fourier synthesis can be suppressed using the CDFS method. The equation of the inverse Fourier transform by the CDFS method is described as

$$\rho\_{\upsilon}(r) = \frac{1}{V} \sum\_{\mathbf{K}} \left[ \left| F\_{\text{obs}}(\mathbf{K}) \right| \mathbf{P} - \left| \sum\_{j} f\_{j}^{\text{core}} T\_{j} e^{-i\mathbf{K}r\_{j}} \right| \mathbf{P}^{\text{core}} \right] \mathbf{e}^{i\mathbf{K}r} \right] + \frac{n\_{\upsilon}}{V}. \tag{7}$$

Here, *f* core *<sup>j</sup>* is the *j*th atomic scattering factor with only the core electron contribution, which corresponds to the blue line in Figure 1. *P*core is the phase term with only the core electron contribution calculated as *P* = *F*core cal (*K*)/ *F*core cal (*K*) . *nv* is the total number of valence electrons contained in the unit cell. The 0 0 0 Bragg reflection intensity cannot be observed experimentally. When ignoring the second term *nv*/*V* in Equation (7), the total number of electrons in the unit cell becomes zero. Therefore, the VED distribution data are corrected by adding the *nv*/*V* term. In general, (sin θ/λ)max measured experimentally is <sup>∼</sup> 2.5Å−<sup>1</sup> (*d*min <sup>∼</sup> 0.2Å) even when using the short-wavelength X-ray obtained at the synchrotron radiation facility. In Figure 1, a nonzero value remains in *f*carbon- = *f* core carbon at sin θ/λ = 2.5 Å<sup>−</sup>1. Therefore, even if a short-wavelength X-ray is used, the mathematical truncation effect cannot be avoided from the calculation using Equation (5). Meanwhile, in Equation (7), the inverse Fourier transform is performed on the term obtained by subtracting the core electron contribution from *F*obs(*K*) , which contains the contribution of all electrons. Because most of the contribution of the remaining *f* valence carbon exists only in sin <sup>θ</sup>/<sup>λ</sup> <sup>≤</sup> 0.5 Å−1, the VED can be extracted with a little truncation effect of the Fourier synthesis by the CDFS analysis.

#### **3. Results and Discussion**

#### *3.1. Diamond*

First, we describe the VED of diamond, which is a typical compound that forms *sp*<sup>3</sup> hybridized orbitals. Diamond forms a crystal structure in which C atoms are arranged in three dimensional space (space group: *Fd*3*m*) (Figure 2a). Each C atom has six electrons, where the inner 1*s* orbital is occupied by two electrons, and the outer 2*s* and 2*p* orbitals are occupied by two electrons. In general, the valence electrons occupying the outer shell orbitals contribute to the physical properties. In the case of diamond, the four valence electrons of each C atom occupy the *sp*<sup>3</sup> hybridized orbitals, forming a very strong bond state called a covalent bond.

**Figure 2.** (**a**) Crystal structure of diamond; (**b**) Synchrotron X-ray diffraction (XRD) data of a diamond crystal at 30 K. (**c**,**d**) Total electron density (ED) distribution of diamond calculated by the general inverse Fourier transform of diffraction intensity using Equation (5); (**c**) Surface plot; (**d**) Sectional view of the (4 4 0) plane.

Because the d-glide (*h* + *k* + *l* - 4*n* reflections disappear) exists in the space group *Fd*3*m*, the diffraction intensity of a 2 2 2 reflection is originally zero. However, the 2 2 2 (and its equivalent reflections) intensity appears in the data obtained by the synchrotron XRD experiment at 30 K (Figure 2b). This finding corresponds to the fact that the extinction rule of *d*-glide is violated by the bonding electrons existing between C atoms. Using the high-resolution diffraction data obtained by the short-wavelength X-rays of synchrotron radiation, an ED analysis based on the usual inverse Fourier transform equation (Equation (5)) is performed, where reflections that satisfy the *d*-glide extinction rule are also included. Because the obtained total ED distribution of diamond is greatly disturbed (Figure 2c,d), the VED distribution corresponding to the covalent bonds cannot be confirmed. This is the mathematical truncation effect of the Fourier synthesis mentioned above.

Figure 3 shows the VED distribution of diamond obtained by the CDFS analysis using Equation (7). A very smooth ED distribution is observed, and the VED corresponding to the C–C covalent bonds is clearly visible. The density at the center point of the C–C covalent bonds is <sup>∼</sup> 1.3*e*/Å3. Compared with the VED distribution obtained by the first-principles calculation [16], this result not only qualitatively reproduces its shape, but also quantitatively reproduces its density well. As shown here, the VED distribution can be extracted with high accuracy through the CDFS analysis. In diamond, valence electrons contributed only to covalent bonds. In the following, we will introduce some systems in which the valence electrons play more roles. In particular, beginning with a small, simple molecule with a rather localized electronic system, we will progressively go into the details of more complicated molecular systems with delocalized electrons, concluding with discussion of those capable of phase transitions.

**Figure 3.** Valence electron density (VED) distribution of diamond calculated by the CDFS method using Equation (7): (**a**) Surface plot, and (**b**) Sectional view of the (4 4 0) plane.
