Approach of Electronic Structure Calculations to Crystal

Abstract

Nowadays, the importance of molecular crystals and solids with regular structures is increasing in both basic chemistry and applied fields. However, theoretical studies of those systems based on electronic structure theories have been limited. Although density functional theory (DFT) calculations using generalized gradient approximation type functional under periodic boundary condition is effective for such theoretical studies, we need some improvements for calculating the dispersion interaction and the excited state of crystals. Accordingly, in this chapter, two methods for calculating the electronic structures of molecular crystals are discussed: cluster-model/periodic-model (CM/PM)-combined method and quantum mechanics/periodic-molecular mechanics (QM/periodic-MM) method. In the CM/PM-combined method, an infinite crystal system is calculated by the DFT method under periodic boundary condition, and important moieties, which are represented by CMs, are calculated by either DFT method with hybrid-type functionals or wave function theories such as the Møller–Plesset second-order perturbation theory (MP2), spin-component-scaled-MP2, and coupled-cluster singles and doubles theory with perturbative triples (CCSD(T)). This method is useful for gas adsorption into crystals such as metal–organic frameworks. In the QM/periodic-MM method, an important moiety is calculated using a QM method such as the DFT method with hybrid-type functionals and wave function theories, where the effects of the crystal are incorporated into the QM calculation via the periodic MM method using a classical force field. This method is useful for theoretical studies of excited states and chemical reactions. The applications of these methods in the following processes are described in this chapter: adsorption of gas molecules on metal–organic frameworks, chemical reactions in crystals, and luminescence of the crystals of transition metal complexes. To the best of our knowledge, the theoretical calculations conducted in this chapter show one of the successful approaches of electronic structure theories to molecular crystals, because of the reasonable and practical approximations.

1 Introduction

In this section, we initially discuss the current stage of electronic structure theory from the viewpoint of its applications. Geometries of the molecules in the gas phase can be appropriately reproduced via electronic structure calculations. Difference between the relative energies of two compounds can be calculated with chemical accuracy (less than approximately 1 kcal mol⁻¹) at least using sufficiently suitable (nearly complete) basis sets and a high-quality wave function theory (for example, coupled-cluster singles and doubles theory with perturbative triples (CCSD(T)) [1] except for compounds possessing strong non-dynamical electron correlation effects (or static electron correlation effects), such as Cr dimer (Cr₂) and multinuclear Fe complexes with O₂ molecule(s). However, in experimental chemistry, target compounds very often exist in the solution phase. For the theoretical study of a compound and its chemical reaction in the solution phase, incorporation of solvent effects into electronic structure calculations is necessary. In many cases, the polarizable continuum model is used [2]. In this model, the solvent is considered as a continuous dielectric medium, and the electronic structure of the molecule(s) placed in its cavity is evaluated considering the polarization effect of the solvent. Solvent characteristics are represented by the sizes and polarizabilities of solvent molecules. In the reference interaction site model (RISM) [3], the solvent molecule is divided into interaction sites bearing charges, and the distribution of solvent molecules is determined based on statistical mechanics. In the RISM-self-consistent field (SCF) theory (RISM-SCF theory) [4,5,6], electronic structure of solute molecule is calculated by the quantum mechanics (QM) method taking into account the distribution of solvent molecules obtained by the RISM theory in a self-consistent manner between the electronic structure of the solute and the solvation structure. In the quantum mechanics/molecular mechanics (QM/MM) method, an important moiety is evaluated by the QM method, and steric and electrostatic effects of solvent molecules are taken into account in the QM calculation by representing the solvent molecules using the MM method. Although this method was proposed for theoretical calculations of biological systems such as proteins [7, 8], this method is effective to take into account solvent effects in electronic structure calculations. Thus, using this method, electronic structure calculations involving solvent effects can be performed in an approximate way.

Nowadays, electronic structure calculations of infinite systems, including crystals, with periodic structures are being actively conducted. Typically, the density functional theory (DFT) method with plane-wave basis sets under periodic boundary condition is employed for these calculations. For this type of calculation, several useful programs, for instance, the Vienna ab initio simulation package [9, 10] and Quantum ESPRESSO [11, 12], are already available. The main advantage of this type of computation method is that entire infinite systems can be evaluated by the QM method. Nevertheless, it has some limitations: when the unit cell is very small, hybrid DFT-type functionals and wave function theories can be used; however, when the unit cell is medium to large, the use of the hybrid-type functionals in the DFT calculations becomes difficult. Also, the use of the wave function theories is nearly impossible except for the system with very small unit cell. As most of the systems of interest in chemistry have large unit cells, DFT calculations have been performed using a generalized gradient approximation type (GGA-type) functional because of its low cost; various DFT functionals and their comparisons have been discussed in recent reviews [13, 14]. Although hybrid-type functionals can be used for single-point calculations without structural optimization, the applications of hybrid-type functionals in the geometry optimization of infinite systems with realistic unit cells are limited by their high costs. The GGA-type functional underestimates the activation barriers of chemical reactions and the energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) and overestimates the stabilities of low-spin states compared to high-spin states. Therefore, hybrid or range-separated type functionals are needed in the DFT calculations of molecules [13, 14]. Additionally, it should be noted that the DFT method is not very reliable for describing dispersion interaction and empirical parameters are used to evaluate the dispersion interaction, as described in the Sect. 11.2.1.

Recently, crystals, such as soft crystals and metal–organic frameworks (MOFs), have attracted significant interest [15]. They are considered molecular crystals; therefore, DFT calculations using at least a hybrid-type functional need to be conducted. Excited states and emission spectra of MOFs and soft crystals have been actively investigated experimentally. However, the application of electronic structure calculations under the periodic boundary condition in the calculation of excited states is impeded by the following shortcoming: as the population of molecules in the excited state is substantially smaller than that in the ground state, a unit cell consisting of a few molecules in the excited state and many molecules in the ground state surrounding the molecules in the excited state must be employed. This unit cell is very large for realistic computations of today. Even if we consider a minimal model in which an excited molecule is placed at the center of a cubic box and surrounded by the molecules in the ground state, the unit cell will comprise a total of 27 molecules. In the cases of transition metal complex, the size of such a unit cell becomes considerably large. Therefore, DFT calculations using GGA-type functionals under periodic boundary condition are not very sufficient for theoretical studies of molecular crystals and the use of better method is required.

In this chapter, we outline two methods involving crystal effects for the theoretical studies of molecular crystals, and discuss how to evaluate dispersion interactions and how to use hybrid DFT functional and/or post-Hartree–Fock (post-HF) methods. Then, we have examined the applications of these methods for the electronic structure calculations of molecular crystals. In the first method (namely, cluster-model/periodic-model combined method; abbreviated as CM/PM-combined method hereinafter), an infinite crystal system is evaluated via the DFT method under the periodic boundary condition, and the target moiety is calculated using wave function theories such as second-order Møller–Plesset perturbation theory (MP2), spin-component-scaled (SCS)-MP2, and coupled-cluster singles and doubles theory with perturbative triples (CCSD(T)). In the second method, the unit cell containing the main part under study is evaluated by the QM method, and the crystal effects from other unit cells are included in the calculation using classical MM method under periodic boundary condition. This method is named QM/periodic-MM method hereinafter. The CM/PM-combined method is suitable for the theoretical investigation of chemical events in the ground state and has been applied to the adsorption of gas molecules to MOFs or porous coordination polymers (PCPs). To apply the QM/MM method to a polymer, the problems of how to separate the QM and MM regions and how to connect them must be carefully solved. In contrast, these problems do not exist in the application of the QM/MM method to molecular crystals. Thus, the QM/periodic-MM method is appropriate for the theoretical study of molecular crystals. Particularly, this method is useful for geometry optimization of the transition states and intermediates of chemical reactions and molecules in excited states and evaluation of activation energy, reaction energy, absorption and emission spectra. As an example of the application of the CM/PM-combined method, we have described electronic structure calculations for the adsorption of gas molecules to MOFs/PCPs. Subsequently, as an example of the application of the QM/periodic-MM method, we have reported theoretical studies on the geometry optimization of the crystals of transition metal complexes, changes in the geometries and energies of the crystals of transition metal complexes induced by chemical reactions, and emission spectra of the crystals of transition metal complexes.

2 Computational Methods Based on Electronic Structure Theory for Molecular Crystals

For molecular crystals with flexible structures, such as soft crystals, both long-range periodic and short-range intermolecular interactions must be considered in a balanced manner. If the long-range periodic interactions are significantly stronger than the short-range intermolecular interactions, the crystal becomes stiff and inflexible. However, if the long-range periodic interactions are considerably weak, the long-range order is lost and the crystal becomes amorphous. Therefore, in the electronic structure calculations of soft crystals, both long-range periodic and short-range intermolecular interactions should be appropriately evaluated. Long-range periodic interactions primarily originate from Coulombic interactions and nuclear repulsions, both of which decrease proportionately to the r⁻¹ term (where r is the interatomic distance). Short-range interactions are mainly produced by nuclear repulsions and electrostatic, exchange-repulsion, and dispersion interactions, and charge-transfer (CT), and polarization interactions may also be involved, depending on the crystal type. Although electrostatic interactions, including point-charge–point-charge, point-charge–dipole-moment, and dipole-moment–dipole-moment interactions, are respectively correlated with r⁻¹, r⁻² and r⁻³ terms, exchange-repulsion, dispersion, CT, and polarization interactions are approximately correlated with exp(−r). Thus, these interactions become important in the short distance case. Estimation of these interactions is difficult in a classical manner and requires quantum chemical calculations. We discuss the CM/PM-combined, QM/periodic-MM, and related methods in the following sections.

2.1 Cluster-Model/Periodic-Model Combined (CM/PM-Combined) Method

As abovementioned in Sect. 11.1, the DFT method was employed to calculate the electronic structures of crystals under periodic boundary condition. In this chapter, we have termed this method periodic-DFT. This method facilitates the treatment of periodic interactions of an infinite system in a quantum chemical manner. To the best of our knowledge, periodic-DFT calculations based on the Perdew–Burke–Ernzerhof (PBE) functional [16] with empirical dispersion correction [17, 18] reproduce experimental structures in many cases.

MOFs or PCPs, which are considered a type of molecular crystals, are expected to be applied for the separation, storage, and trap** of gas molecules [19,20,21,22]. In the adsorption of gas molecules on MOFs, the gas molecules form non-covalent interactions with the MOFs, and therefore, dispersion interactions are expected to play important roles. Specifically, for MOFs without unsaturated open metal sites, dispersion interactions are very crucial. Simultaneously, electrostatic interactions arising from infinite crystal effects should be appropriately considered in the electronic structure calculations because electrostatic interactions can also contribute to the interaction energies of gas molecules when the gas molecules have dipole and/or quadrupole moments. In the DFT calculations, the dispersion interactions are generally calculated using the empirical parameters proposed by Grimme et al. [17, 18]. Nevertheless, the dispersion interaction should be evaluated according to the electronic structures of the target gas molecule and the MOF. For this purpose, the post-HF methods, such as MP2, SCS-MP2, and CCSD(T), must be employed. However, when the unit cell is composed of several transition metal complexes, the use of post-HF methods in the electronic structure calculations under periodic boundary condition is extremely difficult and impractical, as abovementioned, owing to the very long computation time and the need of huge work files.

To evaluate the interactions between gas molecules and MOFs with high accuracy, a hybrid approach has been proposed, in which the infinite system and the CM representing the important moiety of MOF are assessed by periodic-DFT and a post-HF method, respectively [23,24,25]. This method is similar to “our own N-layered integrated molecular orbital and molecular mechanics” (ONIOM) method [26] proposed by Morokuma et al. In this method, the infinite structure of a MOF (abbreviated as “IS”) is calculated by period-DFT with the PBE functional [16] using the empirical correction for the dispersion interactions [17, 18]  proposed by Grimme et al., and the CM (denoted as “C” in Eqs. 11.1 and 11.2) consisting of crucial moieties of the MOF and the target gas molecule(s) is evaluated using the post-HF methods such as MP2, SCS-MP2, and CCSD(T). For example, the entire MOF system comprising the unit cell shown by Scheme 11.1a was calculated by the periodic-DFT method with empirical dispersion correction, and the interaction between the gas molecules and the important moiety of the MOF shown by Scheme 11.1b was calculated by the post-HF method. In such a case, the binding energy (BE) of the gas molecule is represented by Eqs. (11.1) and (11.2):

Scheme 11.1

Schematic representation of gas adsorption into MOF (a), Periodic model (one unit cell of MOF with adsorbed gas molecule) for DFT calculation (b), and cluster model for post-Hartree–Fock calculations (c)

$$BE({\text{post-HF:PBE}} + {\text{D}}) = BE_{{\text{pbc}}}^{\text{PBE}{+}\text{D}}({\text{IS}}) + \Delta B{E^{\text{post-HF}}}({\text{C}})$$

(11.1)

$$\Delta BE^{\text{post-HF}}({\text{C}}) = BE^{\text{post-HF}}({\text{C}}) - BE^{\text{PBE}{+}\text{D}}({\text{C}})$$

(11.2)

where $BE({\text{post-HF:PBE}}{+}{\text{D}})$ is the binding energy (BE) of the gas molecule with the MOF; the subscript “pbc” of BE indicates that the calculation was performed under the periodic boundary condition; the superscripts “PBE+D” and “post-HF” represent DFT calculation using the PBE functional with dispersion correction and the post-HF calculation, respectively; and $\Delta B{E^{\text{post-HF}}}({\text{C}})$ is the post-HF correction calculated using CM. These equations are essentially the same as those in the case of the ONIOM method. However, there is some difference; in the CM/PM-combined method, the correction term $\Delta B{E^{\text{post-HF}}}({\text{C}})$ is evaluated as the sum of the correction terms calculated by several CMs (Scheme 11.1c), whereas in the two-layered ONIOM method, the correction term $\Delta B{E^{\text{post-HF}}}({\text{C}})$ is evaluated using one model. Thus, the accuracy of the CM/PM-combined method is lower than that of the ONIOM method. However, the accuracy of Eq. 11.2 is adequate when the dispersion interaction is corrected by Eq. 11.2 [27]. As Eq. 11.2 is used to correct the dispersion interaction in the CM/PM-combined method, the accuracy of this method is considered sufficient.

In the CM/PM-combined method, the CM (for instance, it consists of gas molecule and important moiety of MOF) and the entire crystal (for example, it is the infinite MOF system including adsorbed gas molecules) are calculated by a quantum chemical method. The accuracy of this combined method in terms of the enthalpy of adsorption of the gas molecule on the MOF has been discussed in the next section. Nevertheless, this CM/PM-combined method has a limitation because the post-HF calculation of the CM does not include the crystal effects. This problem can be solved using a method, such as the QM/periodic-MM method, that incorporates the crystal effects using MM molecules, which is described in the next section. Despite the abovementioned limitation, the CM/PM-combined method is very useful and powerful for investigating the adsorption of gas molecules to MOFs, as explained in Sect. 11.4.

2.2 Quantum Mechanics/Periodic-Molecular Mechanics (QM/Periodic-MM) Method

As abovementioned in Sect. 11.1, the QM/MM method was proposed to calculate the electronic structures of biological systems including metalloenzymes and metalloproteins [7, 8], and many theoretical calculations have been conducted on the biological systems using the QM/MM method [28, 29]. Solvation effects can also be investigated using the QM/MM method. Additionally, the QM/MM method has been applied to crystals [30,31,32,33,34,35,36]. Herein, we have outlined a previously reported method [36] and explained similar related methods.

Briefly, in the QM/MM method for molecular crystals, one unit cell is quantum mechanically evaluated, whereas the other unit cells are calculated using a classical force field. In the quantum mechanical calculation of one unit cell, electrostatic interactions and steric effects of other unit cells are considered using the classical force field (MM). In the case of metalloenzymes, the target (that is, the metal and the important moiety of the protein, for instance) differs from other parts (namely, the protein moiety distant from the metal and solvent molecules). However, in molecular crystals, the target unit cell and other unit cells must have the same structure and the same charge distribution. For this MM part, a periodic boundary condition can be used. The QM/periodic-MM method [36] was developed based on this concept.

Energy of the crystal (${E_{{\text{Crystal}}}}$) is proportional to the sum of the energy ${E_{{\text{QM}}}}({\text{I}})$ of the unit cell I acquired via the QM method and half of the energy of interaction ${E_{\text{QM/MM}}}$ between the QM region (I) and the MM region (other unit cells) calculated by the MM method, as shown in Eqs. 11.3 and 11.4:

$${E_{{\text{Crystal}}}} \propto {E_{{\text{QM}}}}({\text{I}}) + \frac{1}{2}{E_{\text{QM/MM}}}$$

(11.3)

${E_{{\text{QM}}}}({\text{I}})$ is evaluated using the wave function of the unit cell I and the Hamiltonian in the gas phase.

$${E_{{\text{QM}}}}({\text{I}}) = \left\langle {{\Phi_{{\text{QM}}}}({\text{I}})} \right|{\hat H^{{\text{gas}}}}\left| {\Phi_{{\text{QM}}}^{{\text{gas}}}({\text{I}})} \right\rangle$$

(11.4)

where ${\hat H^{{\text{gas}}}}$ is the Hamiltonian in the gas phase. ${E_{\text{QM/MM}}}$ is calculated using the electrostatic interactions and the Lennard–Jones (LJ) potentials.

$${E_{\text{QM/MM}}} = {E_{{\text{ES}}}} + {E_{{\text{LJ}}}}$$

(11.5)

where the electrostatic energy ${E_{{\text{ES}}}}$ of the entire crystal is evaluated via the Ewald summation. As indicated in Eq. 11.6, the electrostatic interaction ${E_{{\text{ES}}}}$ is approximated as the sum of the electrostatic interactions between the point charges of the MM part and the wave function of the QM part in the case of a short distance (named inner region), the electrostatic interactions between the point charges of QM and MM parts in the case of a long distance (named outer region), and the value estimated by the switching function (Eq. 11.7) connecting the short- and long-distance values in the case of intermediate region (named switching region).

$$\begin{aligned}{E}_{\text{ES}} & = \sum_{a\in {\text{MM}}}{q}_{a}\left[-\sum_{\mu \nu }{P}_{\mu \nu }\int d{\varvec{r}}\frac{{\chi }_{\mu }\left({\varvec{r}}\right){\chi }_{\nu }\left({\varvec{r}}\right)}{\left|{{\varvec{r}}}_{a} - {\varvec{r}}\right|} + \sum_{i\in {\text{QM}}}\frac{{Z}_{i}}{\left|{{\varvec{r}}}_{a}-{{\varvec{r}}}_{i}\right|}\right]w\left({{\varvec{r}}}_{a}\right)\\ & \quad + \sum_{a\in {\text{MM}}}{q}_{a}\left[\sum_{i\in {\text{QM}}}\frac{{{Q}_{i} + Z}_{i}}{\left|{{\varvec{r}}}_{a} - {{\varvec{r}}}_{i}\right|}\right]\left(1 - w\left({{\varvec{r}}}_{a}\right)\right)\end{aligned}$$

(11.6)

$$\begin{array}{c}w\left({{\varvec{r}}}_{a}\right)= \left\{\begin{array}{ll}1& {\text{for}} \, {\text{inner}} \, {\text{region}} \, \text{(I)}\\ Sw\left(\left|{{\varvec{r}}}_{0}^{a} -{\boldsymbol{ }{\varvec{r}}}_{0}^{\text{QM}}\right|;{{\varvec{r}}}_{\text{min}}, {{\varvec{r}}}_{\text{max}}\right)& {\text{for}} \, {\text{switching}} \, {\text{region}} \, \text{(III)}\\ 0& {\text{for}} \, {\text{outer}} \, {\text{region}} \, \text{(II)}\end{array}\right.\end{array}$$

(11.7)

where ${P}_{\mu \nu }$ is the density matrix, ${q}_{a}$ is the point charge of the a-th site of the MM part, ${\chi }_{\mu }\left({\varvec{r}}\right)$ is the atomic orbital of the QM part, ${Z}_{i}$ is the nuclear charge of the atom i in the QM part, and ${{\varvec{r}}}_{a}$ represents the coordinates of the a-th site. Accordingly, the Fock matrix element ${F}_{\mu \nu }^{\text{Crystal}}$ is represented by Eq. 11.8:

$$\begin{aligned}{F}_{\mu \nu }^{{\text{Crystal}}} & ={F}_{\mu \nu }^{\text{gas}} - \frac{1}{2}\left[\sum_{a\in {\text{MM}}}{q}_{a}\int d{\varvec{r}}\frac{{\chi }_{\mu }\left({\varvec{r}}\right){\chi }_{\nu }\left({\varvec{r}}\right)}{\left|{{\varvec{r}}}_{a} - {\varvec{r}}\right|}w\left({{\varvec{r}}}_{a}\right)\right.\\ &\quad + \sum_{a\in {\text{MM}}}{Q}_{\mu \nu ,a}\left\{\sum_{\lambda \sigma }{P}_{\lambda \sigma }\int d{\varvec{r}}\frac{{\chi }_{\lambda }\left({\varvec{r}}\right){\chi }_{\sigma }\left({\varvec{r}}\right)}{\left|{{\varvec{r}}}_{a} - {\varvec{r}}\right|} - \sum_{i\in {\text{QM}}}\frac{{Z}_{i}}{\left|{{\varvec{r}}}_{a} -{\boldsymbol{ }{\varvec{r}}}_{i}\right|}\right\}w\left({{\varvec{r}}}_{a}\right)\\ &\quad -\left. \sum_{a\in {\text{MM}}}\sum_{i\in {\text{QM}}}\left\{{q}_{a}\frac{{Q}_{\mu \nu ,i}}{\left|{{\varvec{r}}}_{a} - {{\varvec{r}}}_{i}\right|} +{Q}_{\mu \nu ,a}\frac{{{Q}_{i} + Z}_{i}}{\left|{{\varvec{r}}}_{a} -{{\varvec{r}}}_{i}\right|}\right\}\left(1 - w\left({{\varvec{r}}}_{a}\right)\right)\right]\end{aligned}$$

(11.8)

The calculation is repeated until the charge of each atom in the target QM part becomes the same as the point charge assigned to each atom in the MM part. Equation 11.6 clearly implies that the differentiation of ${Q}_{\mu \nu ,i}$ is necessary for geometry optimization; ${Q}_{\mu \nu ,i}=\langle {\chi }_{\mu }\left({\varvec{r}}\right)|{\widehat{Q}}_{i}|{\chi }_{\nu }\left({\varvec{r}}\right)\rangle$ where ${\widehat{Q}}_{i}$ is a population operator acting on atom i. During geometry optimization, we must impose the condition that the geometry of the QM part is the same as that of the MM part. Although geometry optimization of crystals has rarely been performed except for that in one pioneering study [33], it is now feasible via the QM/periodic-MM method here [36].

Geometry optimization is initiated by hypothesizing a QM structure (r), charges Q_i, and lattice vectors (t). The structure and Q_i distribution of the MM part are created using those of the QM part. The first step is to conduct QM calculations. When the Q_i distribution of the QM part is different from that of the MM part, the QM calculation is repeated by assigning the newly obtained Q_i distribution of the QM part to the MM part. When the Q_i distributions of the QM and MM parts become the same, the next optimization of t is started. During this optimization, the Q_i distributions of the QM and MM parts must be equal. After the optimization of t, geometry optimization of the QM part is performed using the new lattice vector t. During this geometry optimization, the Q_i distributions and structures of the QM and MM parts must be equal. Thus, geometry optimization calculation must be conducted through three loops.

Accuracy of the QM/MM method for geometry optimization was confirmed by the crystal structures of Pt(IV) and Pt(II) complexes. [Pt(CN)₄(H₂O)₂] is a six-coordinate Pt(IV) complex in which H₂O molecules are coordinated to a Pt atom via an O atom using the lone pair orbital on the O atom (Fig. 11.1a) [37]. [PtCl₂(NH₃)(L-glycine)](H₂O) is a four-coordinate Pt(II) complex. Although the H₂O molecule is involved in the crystal structure, it is not coordinated to Pt(II) (Fig. 11.1b). The O atom of the H₂O molecule does not approach Pt(II); instead, the H atom with a δ+ charge unexpectedly approaches Pt(II) [38]. This approaching of H₂O molecules to Pt(II) is surprising because it is generally understood that the negatively charged O atom tends to approach the positively charged Pt(II) atom but the positively charged H atom does not like to approach Pt(II) in [PtCl₂(NH₃)(L-glycine)](H₂O). However, this kind of interaction of H₂O molecule has been reported for similar Pt(II) complexes [39, 40]. This type of H-approaching H₂O molecule has also been theoretically discussed for Pt(II) complexes in aqueous solution [41]. Generally, this type of coordination structure is called a metal–hydrogen (or metal-H) bond [42]. Experimental and theoretical works revealed that the dispersion interaction instead of electrostatic interaction mainly contributes to the metal–H bond [39, 41, 42].

Fig. 11.1

Cited from Ref. [36] with permission of Elsevier Co. Ltd.

Unit cells of crystals of [Pt^IV(CN)₄(H₂O)₂] and [Pt^IICl₂(NH₃)(L-glycine)].

These two Pt complexes were theoretically investigated via QM/periodic-MM calculations, where DFT calculations with the B3LYP functional were utilized for the QM part. In the MM part, the AMBER force field [43] was used for the LJ parameters, except for the Pt atom. Because the LJ parameters for transition metal elements are not very appropriate, these parameters for the Pt atom were determined by fitting the potential energy curve obtained from the LJ parameters to the MP2-calculated curve. The results for Pt(CN)₄(H₂O)₂ are provided in Table 11.1a. The Pt–Pt distances acquired via the QM/Periodic-MM calculations, which correspond to the intermolecular distances in the crystal, are in better agreement with the experimental results than those in the case of the calculation of CM composed of four Pt complexes. Particularly, the Pt(1)–Pt(1′) distance significantly improved. The Pt(1)–Pt(2) and Pt(1′)–Pt(2′) distances were not bad even in the CM. This agreement was probably caused by the formation of a hydrogen bond between the H atom of the coordinated H₂O molecule of the Pt(1) complex and the terminal N atom of the CN ligand of the Pt(2) complex and that between the H atom of the Pt(1′) complex and the CN ligand of the Pt(2′) complex. However, this direct bonding interaction was absent between Pt(1) and Pt(1′) complexes, suggesting that the crystal packing and weak intermolecular interactions contribute to determine the Pt(1)–Pt(1′) distance. Consequently, the Pt(1)–Pt(1′) distance could not be appropriately calculated using the CM. The considerably improved Pt(1)–Pt(1′) distance obtained by the QM/periodic-MM method clearly shows that this method is powerful for the geometry optimization of molecular crystals. The lattice constants determined by the QM/periodic-MM method are also in suitable agreement with the experimental results.

Table 11.1 Optimized geometries (in Angstrom) of Pt(CN)₄(H₂O) and PtCl₂(NH₃)(L-Glycine)(H₂O) by QM/Periodic-MM method^a

In [PtCl₂(NH₃)(L-glycine)](H₂O), H₂O molecules are involved in the crystal structure; however, these H₂O molecules do not directly interact with Pt complexes. In the case of crystal H₂O molecules, the positions and orientations of H₂O molecules are flexible. Therefore, the optimization of these geometrical parameters is usually difficult. As presented in Table 11.2b, the Pt(1)–Pt(1′) distance in the case of the CM exhibits a substantial error; nevertheless, the distance optimized by the QM/periodic-MM method is in appropriate agreement with the experimental results. This appropriate agreement results from the facts that the relative positions of Pt(1) and Pt(1′) complexes depend on the overlap between the doubly occupied d_z² orbitals of these complexes, and in such a case, the exchange repulsion, dispersion interaction, short and long-range electrostatic interactions, and packing effects of the crystal play important roles in determining the relative positions of Pt and Pt′ complexes; in the CM case, the long-range electrostatic interaction and the packing effects are not evaluated well. The Pt(1)–H_w distance, where “Hw” represents the H atom of the H₂O molecule, should be the same as the Pt(2)–H_w distance. However, the Pt(1)–H_w distance is significantly different from the Pt(2)–H_w distance in the CM. There are two types of Pt(1)–H_w distances (Table 11.1b), which depend on the orientation of the H₂O molecule. Although these distances are not considerably different from the experimental values in the case of the CM consisting of two Pt complexes, they are significantly different for the CM comprising four Pt complexes. These results imply that the CM does not accurately determine the orientations of H₂O molecules in a crystal. In contrast, the QM/periodic-MM method appropriately reproduces the two Pt (1)–H_w distances, which indicates that this method can reproduce the positions and orientations of the molecules adsorbed to a molecular crystal.

As above mentioned, using the QM/periodic-MM method, the structures of molecular crystals can be precisely determined. Note that the LJ parameters for transition metal elements are not very suitable and need to be reoptimized to reproduce the potential energy surface; for re-optimization, the potential energy surface obtained by MP2 calculation or DFT calculation including dispersion correction can be used as a reference.

2.3 Embedding Method Similar to QM/Periodic-MM Method

Several embedding methods similar to the QM/periodic-MM method have been proposed, where the main part is calculated by the quantum mechanics and the other moiety is represented by MM or density matrix. Herein, we introduced a method to embed the QM region in the distribution of point charges.

The method of embedding QM in point charges has been reported in numerous studies [41,42,43,44,45,46,47,48,49,50,51,52,53]. This method is beneficial for investigating chemical species adsorbed on metal oxide surfaces and the chemical events occurring on solid surfaces. Electronic structure calculations of solid surfaces are typically performed under periodic boundary conditions using a slab model (Scheme 11.2a). Nevertheless, the periodic-DFT calculation of the slab model using a hybrid-type functional is challenging because of the long computation time. Furthermore, the post-HF calculation is practically impossible when a unit cell is not very small. Thus, the CM (Scheme 11.2b) must be utilized for DFT calculations with a hybrid-type functional or post-HF calculations. However, a major issue in the application of a simple CM is that the electrostatic fields of other atoms on the surface are not considered in this CM. In metal oxides, each metal atom has a positive charge and each O atom has a negative charge. Therefore, these positive and negative charges must be included in the calculation of the CM. This electronic structure calculation can be conducted using a CM embedded in the distribution of the point charge placed on each atom of the solid surface (Scheme 11.2c). By arranging the slab model in three dimensions (Scheme 11.2d), the electrostatic field of the solid surface can be analytically represented. Thereafter, the electrostatic potential can be included in the Hamiltonian of the electronic structure calculation of CM. This potential is called the periodic electrostatic potential. This method was not employed for geometry optimization, but it was applied to single-point calculations using the geometry optimized by DFT calculations under periodic boundary condition.

Scheme 11.2

This scheme is cited from Ref. [53] with permission of American Chemical Society

Model for electronic structure calculations of metal clusters supported on the surface of a crystalline solid.

Comparison of the interaction energies of Rh₂ supported on Al₂O₃ and AlPO₄ surfaces calculated via different methods is presented in Table 11.2. As expected, the interaction energy calculated using the bare CM is significantly different from that evaluated by the slab model, indicating that the calculation without point charges results in a large error. When approximately one-thousand point charges were placed on the surface, the error considerably decreased. When one-hundred-thousand point charges were put on the surface, the evaluated interaction energy converged to the value obtained using the periodic electrostatic interaction. In the case of Rh₂ on the AlPO₄ surface, the SCF convergence was not obtained in DFT calculations of the bare CM and the CM with a small number of point charges (<7.55 × 10⁴). Nevertheless, it is noted that the SCF convergence of the DFT calculation was obtained when a large number of point charges were added to the CM. The final interaction energy was overestimated, compared with that in the case of the slab model. A possible reason for this difference is the difficulty in determining the distribution of point charges in the boundary region of the CM to reproduce the electrostatic potential from an infinite system. Table 11.2 also demonstrates that the interaction energy differs between the GGA-type PBE and hybrid-type B3LYP functionals. These results indicate that the CM and DFT functional should be carefully selected for interaction energy calculation.

Table 11.2 Interaction energy^a (in eV unit) of Rh₂ with Al₂O₃ and AlPO₄ surfaces

2.4 Quantum Embedding Theories

In this section, we have introduced two recent advances in embedding methods, which incorporate non-classical interactions between the main and other parts. In the abovementioned CM/PM-combined method, the effect of the ambient environment is not considered in the CM calculation of the main part. In the QM/periodic-MM method described in Sects. 11.2.2 and 11.2.3, the effects of periodic electrostatic interactions on the QM part can be treated within the scope of classical theory; in other words, it cannot capture the quantum interaction between the main part (that is the QM part) and other parts. To solve this problem, a new embedding method, collectively called quantum embedding theory, has recently attracted attention.

At first, we have presented the density functional embedding theory (DFET), which is based on the DFT viewpoint and treats the density of the entire system by dividing it into major and other parts. The original form of the DFET was proposed by Cortona [54, 55] in 1991 and by Wesolowski and Warshel [56] in 1993, and it was primarily developed by Carter et al. [57].

The DFT method is based on the concept that the energy of a system is a function of electron density $\rho (r)$ (denoted as ${E}_{\text{DF}}\left[\rho \right]$). Herein, ${E}_{\text{DF}}\left[\rho \right]$ is divided into three parts, each of which depends on the $\rho (r)$ of the main part ${\rho }_{\text{A}}(r)$ (represented as ${E}_{\text{A}}\left[{\rho }_{\text{A}}\right]$), $\rho (r)$ of other parts ${\rho }_{\text{B}}(r)$ (denoted as ${E}_{\text{B}}\left[{\rho }_{\text{B}}\right]$), and the interaction energy $\Delta E[{\rho }_{\text{A}},{\rho }_{\text{B}}]$,

$$\begin{array}{c}{E}_{\text{DF}}\left[\rho \right] ={E}_{\text{A}}\left[{\rho }_{\text{A}}\right] +{E}_{\text{B}}\left[{\rho }_{\text{B}}\right] + \Delta E\left[{\rho }_{\text{A}},{\rho }_{\text{B}}\right].\end{array}$$

(11.9)

Using this expression, the energy of the main part is replaced by, for example, that of the post-HF method ${E}_{\text{WF}}\left[{\rho }_{\text{A}}\right]$, which leads to the following expression of embedding energy:

$$\begin{array}{c}{E}_{\text{WF/DF}}\left[\rho \right] ={E}_{\text{WF}}\left[{\rho }_{\text{A}}\right] +{E}_{\text{DF}}\left[\rho \right] -{E}_{\text{A}}\left[{\rho }_{\text{A}}\right] + \Delta E\left[{\rho }_{\text{A}},{\rho }_{\text{B}}\right].\end{array}$$

(11.10)

To date, the DFET is substantially similar to the abovementioned CM/PM-combined and ONIOM methods; however, the most notable feature of the DFET is that ${E}_{\text{WF}}\left[{\rho }_{\text{A}}\right]$ is self-consistently solved under the potential provided by the surrounding part (${V}_{\text{B}}$), where ${V}_{\text{B}}$ is defined by Eq. 11.11 according to the concept of the DFT method.

$$\begin{array}{c}{V}_{\text{B}} = \frac{\delta\Delta E\left[{\rho }_{\text{A}},{\rho }_{\text{B}}\right]}{\delta {\rho }_{\text{A}}}\end{array}$$

(11.11)

If the Hamiltonian of the main part is defined by ${\widehat{H}}_{\text{A}}$, the energy of the main part under ${V}_{\text{B}}$ is presented as a solution of the following optimization problem:

$$\begin{array}{c}{E}_{\text{WF}}\left[{\rho }_{\text{A}}\right] =\underset{{\Psi }_{\text{A}}}{\mathrm{min}}\langle {\Psi }_{\text{A}}|{\widehat{H}}_{\text{A}} +{V}_{\text{B}}|{\Psi }_{\text{A}}\rangle =\underset{{\Psi }_{\text{A}}\to {\rho }_{\text{A}}}{\mathrm{min}}\langle {\Psi }_{\text{A}}|{\widehat{H}}_{\text{A}}|{\Psi }_{\text{A}}\rangle .\end{array}$$

(11.12)

In this optimization problem, ${V}_{\text{B}}$ is determined such that the electron density ${\rho }_{\text{A}}$ by the post-HF method for the main part is equal to that by the DFT method for the whole system. Consequently, the quantum interactions from the other parts can be at least qualitatively incorporated into the calculation of the main part.

The DFET, in combination with DFT calculations of periodic systems, enables natural incorporation of the quantum effects in the bulk of solids and periodic electrostatic interactions into the CM calculations of the main part of the system. Numerous applications of DFET to ionic and metal oxide crystals [54, 57] and to molecular adsorption on metal surfaces and metal nanoparticles [58,59,60,61,62] have been reported. Although DFET can be easily applied to molecular crystals, to date, few applications of DFET in the electronic structure calculations of molecular crystals have been reported. The number of applications of DFET in molecular crystals may increase in the future with an increase in the practicality of DFET.

The disadvantage of DFET is that the interaction between the main part and other parts is treated only at the DFT level. Thus, the accuracy of the calculation degrades, when interactions, such as dispersion interactions, charge-transfer interactions, and higher-order electron correlation effects are involved, because dispersion interaction and higher-order electron correlation effect cannot be described well within the theoretical framework of the DFT method, and because charge-transfer interaction between the main part and the remaining is only approximately evaluated in the DFET method. However, similar to the DFT method, DFET is certainly a powerful tool for most molecular systems and materials.

Herein, we have introduced the density matrix embedding theory (DMET) [63, 64], which was proposed by Knizia and Chan in 2012 as a method similar to DFET. DMET is based on the concept of decomposing the wave function of the entire system into many-body basis functions of the main part and bath, which has the same dimensions as the main part;

$$\begin{array}{c}\left|\Psi \rangle \right. = \sum_{i}{\lambda }_{i}\left|{\alpha }_{i}\rangle \right.\left|{\beta }_{i}\rangle \right.\end{array}$$

(11.13)

where ${\alpha }_{i}$ is the i-th state of the main part and ${\beta }_{i}$ is the i-th state of the bath, which couples with ${\alpha }_{i}$. The coefficient ${\lambda }_{i}$ is a positive real number (singular value) related to the weight of the $i$-th product state. In this case, regardless of the size of the non-main part, the dimension of ${\beta }_{i}$ is equal to that of ${\alpha }_{i}$ (that is, states in the main part). Therefore, the size of a problem can be reduced to at most twice that for the main part.

Nevertheless, to precisely calculate ${\beta }_{i}$, solving an exact wave function of the whole system is necessary, and thus, ${\beta }_{i}$ can only be obtained approximately in practice. In the original study of DMET, mean-field approximation was utilized to construct the wave function of the entire system, which was used as a reference system to approximately compute ${\beta }_{i}$. That is, similar to the case of DFET, the many-body wave function can be optimized so that the one-electron ${P}_{\mu \nu }$ of the main part reproduces the one-particle ${P}_{\mu \nu }$ of the main part obtained from the mean-field wave function of the entire system.

The advantage of DMET is that it facilitates exact calculations “in principle” regardless of how the main part is defined. Certainly, the exact solution cannot be acquired because of the approximation used in the construction of the reference system; however, even a fragment where the covalent bond is cleaved can be possibly defined, for instance, by considering the two C atoms (the ethylene fragment) of the benzene ring as the main part (Scheme 11.3).

Scheme 11.3

Schematic explanation of the density matrix embedding theory (DMET) in quantum chemistry. An ethylene fragment of the benzene molecule can be accurately solved by considering that the bath size of the DMET wave function is only twice the sizes of the quantum states in the ethylene fragment

Thus, basically, DMET can be used to treat all quantum interactions in a partitioned manner and has attracted substantial attention as a beneficial tool to treat strongly correlated electron systems, including the two-dimensional Hubbard model of high-T_c superconductivity, in condensed matter physics [63, 65, 66]. DMET in combination with response theory has also been applied to the calculation of bulk spectra of solids and has successfully reproduced the bulk spectra of solids with a few major sites in a demonstrative calculation using the Hubbard model [67]. Recently, DMET has been gradually linked with the molecular orbital method, and the resulting method has been utilized for two-dimensional crystal systems of boron nitride and nickel oxide [68].

DFET and DMET discussed in this section are still under development and have few practical applications. Nevertheless, the non-classical treatment of the interaction between the main part and the other parts had not been considered in the conventional embedding methods. Thus, these quantum embedding methods are expected to be promising calculation methods for complex systems including crystals and solid surfaces.

3 Application of Cluster Model/Periodic-Model (CM/PM) Combined Method to Metal-Organic-Frameworks (MOFs)

3.1 Gas Adsorption and Lateral Interaction

Methane (CH₄), the main component of natural gas, has attracted significant attention as a fuel because of low CO₂ production during its combustion. However, due to the very low boiling point of CH₄, efficient transportation and storage of CH₄ are necessary for its application as a fuel. MOFs can be used for the transportation and storage of CH₄ because of their large specific surface areas [69]. Nevertheless, unlike olefins and other substances, CH₄ weakly interacts with materials due to its nonpolar electronic structure, spherical shape, and the absence of any functional group useful for intermolecular interaction. To effectively adsorb CH₄ to MOFs, knowledge of the strength and character of the interaction of CH₄ with MOFs and the mechanism of adsorption of CH₄ to MOFs is required. For this purpose, a theoretical study with the correct estimation of the dispersion interaction is necessary. This is true because CH₄ does not have any functional groups that can interact with transition metal complexes and the dispersion interaction seems to play important roles in the interaction of CH₄ with other materials.

MOFs, for example, CPO-27-Mg and CPO-27-Ni, consisting of 2,5-dihydroxyterephthalate (hdtp) and divalent metal ions demonstrate high adsorption performances for CH₄ [70]. Sauer et al. theoretically investigated the adsorption of CH₄ to CPO-27-Mg [23]. Considering the importance of the dispersion interaction, they used periodic-DFT for a whole infinite system and the post-HF method for CMs, as discussed in Sect. 11.2.1. Neutron diffraction results reveal that CPO-27-Mg has a honeycomb-like structure with large one-dimensional pores of 11–12 Å diameter (Fig. 11.2a). There are two types of CH₄ adsorption sites in CPO-27-Mg:Mg²⁺ site (denoted as Mg-site) and linker site (that is, hdtp ligand site, abbreviated as L-site) (Fig. 11.2b). In addition to these two sites, the vacant space above these sites, which is called the second-layer site (denoted as 2’nd L-site), can be useful for CH₄ adsorption. To calculate the BEs of CH₄ molecules at these sites on CPO-27-Mg, periodic-DFT calculations using the PBE+D functional were performed as the first step, followed by correction at the post-HF level with MP2 calculations using the CMs shown in Fig. 11.2c, d and CCSD(T) calculations using the smaller CM shown in Fig. 11.2e. The calculations indicated that totally six CH₄ molecules were adsorbed (one CH₄ molecule per site) on the Mg-, L-, and 2’nd L-sites in one unit cell (Fig. 11.2b) comprising six Mg-, L-, and 2’nd L-sites.

Fig. 11.2

Cited from Ref. [23] with permission of the American Chemical Society

Model systems used in the calculations. Conventional unit cell of CPO-27-Mg viewed along the c direction (a), part of the MOF with six CH₄ molecules at Mg-site and one (yellow) at a linker (L) site (b), cluster model 6B adopted for MP2 calculations for CH₄ adsorption on the Mg- and L-sites (c), cluster model Mg₂dhtp adopted for MP2 calculations (d), where CH₄ adsorption occurs on the second layer site, cluster models adopted for CCSD(T) calculations for CH₄ adsorption on the Mg- and L-sites (e).

Table 11.3 presents the adsorption energies of CH₄ molecules at the three sites of CPO-27-Mg calculated by several methods. At the Mg-site, the periodic-DFT-calculated BE of the infinite model was slightly different from the DFT-calculated BE of the CM (6B, Fig. 11.2c). Corrections at the MP2 and CCSD(T) levels using the CMs were also small. At the L-site, the BE of the CM (6B) was different from the periodic-DFT-calculated BE by 5.2 kJ mol⁻¹. Additionally, the BEs calculated for the CM by the DFT and MP2 methods differed by 1.3 kJ mol⁻¹. At the 2’nd L-site, the BE of the CM (Mg₂dhtp, Fig. 11.2d) and the periodic-DFT-calculated BE differed by 3.1 kJ mol⁻¹. The BE obtained by the periodic-DFT calculations using the PBE+D functional increased in the following order: 2’nd L-site ≤ L-site < Mg-site; this trend was the same as that of the BE corrected by the post-HF calculations. However, the difference between the BEs at the 2’nd L- and L-sites increased to 3.2 kJ mol⁻¹ upon correction by the post-HF calculations. This implies that the post-HF correction increases the BE at the L-site as compared to that at the 2’nd L-site, suggesting the need for the correct evaluation of dispersion interactions at these sites. This seems reasonable because the dispersion interaction substantially contributes to the BEs at these sites. The computational result demonstrating that the adsorption energy at the Mg site is the largest is consistent with the general understanding that metal ions exhibit strong binding interactions with gas molecules. Moreover, the BEs determined by the dispersion (D) part of the PBE+D functional are considerably larger than those evaluated by the PBE functional for all the three sites, indicating that the dispersion interaction significantly contributes to the adsorption energy. The large BE at the Mg-site and small BEs at the L- and 2’nd L-sites imply that CH₄ initially adsorbs on the Mg-site of CPO-27-Mg followed by on the L- and 2’nd L-sites.

Table 11.3 Binding energy (in kJ mol⁻¹) of CH₄ with CPO-27-Mg at different sites calculated with different methods on periodic PBE+D structures and cluster models cut out from these periodic PBE+D structures

Because a large number of CH₄ molecules are adsorbed in one unit cell, the intermolecular interactions between CH₄ molecules (termed the lateral interaction in Ref. [23]) are expected to affect the adsorption energy and enthalpy. Thus, in the theoretical study, the interaction energies between two CH₄ molecules in the gas phase and between two CH₄ molecules adsorbed on different sites of CPO-27-Mg were compared. In the gas phase, the distance between two CH₄ molecules was optimized to be 3.64 Å by MP2/aug-cc-pVTZ, which was moderately smaller than that optimized by PBE+D (3.82 Å). The CH₄–CH₄ intermolecular interaction energy calculated by CCSD(T)/aug-cc-pVTZ was −1.7 and −2.3 kJ mol⁻¹ at these distances, respectively. When all six Mg-sites of this MOF are occupied by CH₄ molecules, the distances between CH₄ molecules on two adjacent Mg-sites are 5.17 (a/b direction) and 6.89 Å (c direction). As the distance between CH₄ molecules adsorbed on these sites is slightly longer than that in the gas phase, the interaction energies of CH₄ molecules become weaker (−0.5 kJ mol⁻¹). The interaction energies of the two CH₄ molecules in the gas phase and the two CH₄ molecules adsorbed on the Mg- and L-sites (−1.7 kJ mol⁻¹) are comparable. The distances and interaction energies between the CH₄ molecules adsorbed on adjacent L-sites are 4.44 Å and −1.2 kJ mol⁻¹, respectively. Because the adsorbed CH₄ molecules are adjacent to other CH₄ molecules, the contributions of the lateral interaction energy to the adsorption energies of CH₄ molecules at the Mg- and L-sites are −2.1 and −2.7 kJ mol⁻¹, respectively. These values indicate that the effect of lateral interaction cannot be neglected in the calculation of adsorption isotherm.

The adsorption energies of CH₄ at the Mg-, L-, and 2’nd L-sites and the enthalpy change of CH₄ adsorption can be evaluated using Eq. 11.14.

$$\frac{{1}}{{6}}{\left( {{\text{C}}{{\text{H}}_{4}}} \right)_{{\text{6n}}}}{\text{/MOF}} + {\text{C}}{{\text{H}}_{4}} \to \frac{{1}}{{6}}{\left( {{\text{C}}{{\text{H}}_{4}}} \right)_{\text{6n + 6}}}{\text{/MOF}}$$

(11.14)

The thermodynamic quantities of CH₄ adsorption calculated by the DFT method with corrections by MP2 and CCSD(T) calculations are presented in Table 11.4. The CH₄ molecule in the gas phase has six degrees of freedom for translational and rotational movements. In contrast, the movements of CH₄ molecule adsorbed to MOFs with respect to the adsorption site is described as oscillations with six low frequencies. However, the adsorbed CH₄ molecule may retain the rotational degrees of freedom because of the weak adsorption interaction, whereas the translational degrees of freedom are lost. Table 11.4 provides a comparison between the results obtained via these two different calculations; in the former one, the six degrees of freedom are completely lost, and in the latter one, the three rotational degrees of freedom are retained. Note that not only the entropy, but also the zero-point energy (ZPE) and thermal energy (E_therm) vary with the change in the number of degrees of freedom. Consequently, the Gibbs energy change (ΔG) for adsorption becomes more negative by approximately 4 kJ mol⁻¹ when the rotational degrees of freedom are retained as compared to the case when they are lost. Using these changes in the thermodynamic quantities, the adsorption isotherm for CH₄ to CPO-27-Mg was calculated utilizing the multi-site Langmuir equation (Eq. 11.15):

$$\theta = \sum\limits_{{\text{site}}} {{{{K_{{\text{site}}}}P} \mathord{\left/ {\vphantom {{{K_{{\text{site}}}}P} {(1 + {K_{{\text{site}}}}P)}}} \right. \kern-0pt} {(1 + {K_{{\text{site}}}}P)}}}$$

(11.15)

Table 11.4 Adsorption enthalpy change, (ΔH) and zero-point energy change (ZPE), thermal energy change (ΔE_therm) and adsorption entropy term (TΔS) calculated for CH₄ adsorption at different sites using the PM/CM-Combined Method (all in kJ mol⁻¹ at 179 K)

where θ and P are the coverage and pressure, respectively, and K is the equilibrium constant for the adsorption of CH₄ at a site of CPO-27-Mg.

As shown by the dotted line in Fig. 11.3, if the degrees of freedom of the adsorbed CH₄ molecule are completely frozen (that is, all translational and rotational degrees of freedom are changed to vibrational degrees of freedom), the adsorption amount of CH₄ is significantly underestimated. If we consider that three rotational degrees of freedom remain in the adsorbed CH₄ molecule, the calculated adsorption amount of CH₄ is higher than the experimental value, as indicated by the solid line in Fig. 11.3. This calculation is for the case where all the sites of MOFs are available for the adsorption of CH₄ molecules (denoted as ideal MOF; solid line). Nevertheless, the experimentally determined availability of the adsorption sites of CPO-27-Mg is approximately 78%. Based on this availability, the calculated adsorption amount of CH₄ is in appropriate agreement with the experimental data acquired at both 179 and 298 K.

Fig. 11.3

This figure is cited from Ref. [23] with permission of American Chemical Society

Calculated excess adsorption isotherms for CH₄ adsorption in CPO-27-Mg for which experimentally determined availability (78%) of adsorption sites is assumed. Data points are experimentally determined amounts of adsorbed CH₄. Dashed lines: isotherms for adsorbed CH₄ with six vibrational motions of adsorbed CH₄ molecule. Solid lines: isotherm calculated with three rotational degrees of freedom for adsorbed CH₄ molecule.

Similarly, the adsorptions of CO and N₂ molecules on CPO-27-Mg have been theoretically investigated, and the adsorption isotherms have been calculated [24].

In conclusion, via the periodic-DFT calculation of the entire infinite system with post-HF correction using the cluster model the interaction between the adsorption sites of MOFs and the adsorbed gas molecules can be accurately evaluated. This method is expected to be a powerful tool for analyzing and understanding the gas adsorption behaviors of MOFs.

3.2 Gas Adsorption to Flexible MOF with Gate-Opening Mechanism

Flexible MOFs are a type of soft crystals. The flexibilities of MOFs were predicted by Kitagawa et al. in 1998 [71] and experimentally reported in 2002 [72] by the same group. Thereafter, many efforts have been devoted to creating flexible MOFs and develo** their applications. Gate-opening adsorption of gas molecules is one of the characteristic features of flexible MOFs, in which structural transformations induced by adsorbed gas molecules accelerate the subsequent adsorption of gas molecules. This is contrary to our intuitive expectation that adsorbed gas molecules generally suppress the successive adsorption of gas molecules. Interestingly, a unique sigmoidal isotherm has been observed for the gate-opening adsorption of gas molecules, which is different from the typical Langmuir-type isotherm. Various theoretical approaches have been applied to the adsorption of gas molecules to flexible MOFs to understand the dynamic behaviours of flexible MOFs and the sigmoidal adsorption isotherm at the molecular level [73,74,75,76]. However, precise knowledge of the gate-opening adsorption mechanism is limited.

In a recent theoretical study [25], the CM/PM-combined method (Sect. 11.2.1) was applied to the gate-opening adsorption of gas molecules to a flexible MOF to elucidate the characteristic features of gate-opening gas adsorption and the origin of the sigmoidal adsorption isotherm. As examples, CO₂ adsorptions to [Fe(ppt)₂]_n (PCP-N, Hppt = 3-(2-pyrazinyl)-5-(4-pyridyl)-1,2,4-triazole) and its isostructural analogue [Fe(dpt)₂]_n (PCP-C, Hdpt = 3-(2-pyridinyl)-5-(4-pyridyl)-1,2,4-triazole) [77] were investigated. Although these two MOFs have similar structures (Fig. 11.4a, b), they exhibit completely different adsorption features (Fig. 11.4c), as summarized below: (i) gate-opening CO₂ adsorption occurs in the case of PCP-N, (ii) CO₂ adsorption to PCP-N takes place with a sigmoidal adsorption isotherm, and (iii) non-gate-opening CO₂ adsorption with the Langmuir-type isotherm occurs in the case of PCP-C.

Fig. 11.4

Cited from Refs. [25, 77] with permission of the American Chemical Society

Experimental crystal structures of flexible MOF PCP-N (a), rigid MOF PCP-C (b), and adsorption isotherms of PCP-N and PCP-C at 195 K (c).

Figure 11.5 depicts one unit cell of the periodic structure of PCP-N and the cluster models (CMs) employed in the study. Single-crystal X-ray diffraction study reported that CO₂ adsorption occurred at three different sites: a large cage (site I), a small aperture next to site I (site II), and a small pore around Fe²⁺ (site III) (Fig. 11.5a) [77]. The positions and orientations of CO₂ molecules in one unit cell could not be clearly observed experimentally. Therefore, their positions and orientations were refined by periodic-DFT calculations using plane-wave basis sets. The number of adsorbed CO₂ molecules was calculated to be six, six, and three for sites I, II, and III, respectively. BEs of the CO₂ molecule at these sites of PCP-N were evaluated using periodic-DFT with the PBE functional [16], where the Grimme’s “D3” [17, 18] dispersion correction was used. Then, the post-HF correction at the SCS-MP2 level was performed using the CMs shown in Fig. 11.5b–d (CM^A, CM^B, and CM^C for sites I, II, and III, respectively). The geometries of the CMs were obtained from the PBE-D3-optimized infinite system (IS), where the dangling bonds of CMs were capped by H atoms. Subsequently, the CCSD(T) correction was conducted using the smaller CMs (SCMs) such as SCM^A1–A18, SCM^B1–B7, and SCM^C1–C4 (Fig. 11.5e–g).

Fig. 11.5

Cited from Ref. [25] with permission of the American Chemical Society

Periodic model of MOF with adsorbed CO₂ molecules at three different sites (a), cluster models used in SCS-MP2 correction (b–d) and small cluster models (e) used in CCSD(T) correction for CO₂ adsorptions at sites I, II, and III, respectively.

Table 11.5 presents a comparison between the BEs of CO₂ on the different sites of PCP-N calculated by periodic-DFT and CM/PM-combined methods. The BE decreases in the order site I > site III > site II in both the PBE-D3 calculations of the infinite (IS) system and the CM/PM-combined calculations. The BEs calculated using these two methods were slightly different; apparently, the PBE-D3 overestimated the BEs (Table 11.5a), whereas the SCS-MP2 correction underestimated the BEs. However, the underestimation of BEs by the SCS-MP2 calculation was moderately overcome by the CCSD(T) correction. These results demonstrate that the post-HF correction using SCS-MP2 and CCSD(T) is needed to quantitatively calculate the BEs of the gas molecules adsorbed to MOFs.

Table 11.5 Comparison between CO₂ binding energy (BE, kcal mol⁻¹) with PCP-N at different sites calculated by periodic DFT and hybrid methods

Herein, we have compared the calculated ΔH_ads values with the experimental values. The $\Delta {H_{{\text{ads}}}}$ value was experimentally evaluated to be −5.71 to −6.45 kcal mol⁻¹ for the adsorption of one to four CO₂ molecules at site I. As presented in Table 11.5b, the $\Delta {H^{\text{PBE-D3}}}$ value is larger than the experimental value, and $\Delta {H^{\text{SCS-MP2:PBE-D3}}}$ is smaller than the experimental value; However, the CCSD(T):SCS-MP2 corrected value, $\Delta {H^{\text{SCS-MP2:PBE-D3}}} + \Delta B{E^{\text{CCSD(T)}}}$, is in agreement with the experimental value. This comparison clearly shows that the CP/PM-combined method is effective for calculating $\Delta {H_{{\text{ads}}}}$ and a post-HF correction is required.

The results in Table 11.5a indicate that CO₂ initially adsorbs on site I followed by on sites II and III. In the case of continuous CO₂ adsorption on site I, the BE gradually increased from −5.41 to −6.10 kcal mol⁻¹ at the SCS-MP2:PBE-D3 level with an increase in the number of adsorbed CO₂ molecules from one to six (Table 11.5b). The interaction energy E_INT(H–G) between CO₂ and PCP-N slightly changed with an increase in the number of adsorbed CO₂ molecules from one to six, where G and H represent a guest molecule (gas molecule) and a host system (MOF), respectively. However, the CO₂−CO₂ interaction energy E_INT(G–G) slowly increased (became more negative) from 0.0 to −1.05 kcal mol⁻¹. Additionally, the framework deformation energy E_DEF per one CO₂ molecule gradually decreased from 0.35 to 0.17 kcal mol⁻¹ with an increase in the number of adsorbed CO₂ molecules from one to six. Consequently, the BE per CO₂ molecule at site I increased with an increase in the number of adsorbed CO₂ molecules.

Because a total of fifteen CO₂ molecules were experimentally observed in PCP-N, theoretical study investigated if CO₂ molecules further adsorbed on PCP-N after site I was completely occupied by six CO₂ molecules. Notably, the BE of the seventh CO₂ molecule was positive (repulsive) at site II and very small at site III (Table 11.6), suggesting that the seventh CO₂ molecule could not adsorb on sites II and III when site I was occupied by six CO₂ molecules. This is reasonable because the six CO₂ molecules at site I suppress the geometric deformations of sites II and III, which are required for CO₂ adsorption at sites II and III. However, the BEs increased (that is, became more negative) from 1.20 to –4.77 kcal mol⁻¹ at site II and from –0.62 to –5.26 kcal mol⁻¹ at site III with an increase in the number of adsorbed CO₂ molecules from one to six at site II and from one to three at site III. This increase in the BE is rational because E_DEF per one CO₂ molecule considerably decreases with an increase in the number of adsorbed CO₂ molecules, whereas E_INT per CO₂ molecule slightly changes (E_DEF and E_INT are defined in footnotes b and c of Table 11.5). Therefore, the sum of the BEs of the six CO₂ molecules at site II and those of the three CO₂ molecules at site III becomes negative. These results show that one CO₂ molecule cannot adsorb at sites II and III when site I is occupied by six CO₂ molecules, but totally nine (six + three) CO₂ molecules can simultaneously adsorb at sites II and III even when the site I is occupied by six CO₂ molecules. Note that the BEs of CO₂ molecules at sites II and III were similar, strongly suggesting the simultaneous adsorption of a total of nine CO₂ molecules occurs on sites II and III. This phenomenon corresponds to the gate-opening adsorption mechanism, as discussed below.

Table 11.6 Binding energy (BE, kcal mol⁻¹) of CO₂ at the sites II and III of PCP-N in the presence six CO₂ molecules at the site I

Summarising the abovementioned results, the equilibrium equations for CO₂ adsorption are presented as follows:

PCP-N:

$$\begin{aligned}{\text{At site I:}}\quad\,\, & {\text{CO}}_{2} + \mathbf{PCP}\text{-}\mathbf{N} \cdot {({\text{CO}}_{2})}_{\text{n}} \overset {} \rightleftarrows \mathbf{PCP}\text{-}\mathbf{N} \cdot {({\text{CO}}_{2})}_{\text{n} + 1}\\ & \qquad \text{n} = 0{-}5,\end{aligned}$$

(11.16)

$${\text{At sites II and III: }}\quad\,\,{\text{9 CO}}_{2} + \mathbf{PCP}\text{-}\mathbf{N} \cdot {({\text{CO}}_{2})}_{6} \rightleftarrows \mathbf{PCP}\text{-}\mathbf{N} \cdot {({\text{CO}}_{2})}_{15}$$

(11.17)

The equilibrium constant K of these equations was evaluated using the Gibbs energy change by adsorption of gas molecule ΔG^o calculated by the CM/PM-combined method:

PCP-N:

$${\text{At}}\,{\text{site}}\,{\text{I:}}\,\,{K_1} = \exp \left( { - {{\Delta G_{{\text{ads}},1}^{\text{o}}} \mathord{\left/ {\vphantom {{\Delta G_{{\text{ads}},1}^{\text{o}}} {RT}}} \right. \kern-0pt} {RT}}} \right)$$

(11.18)

$${\text{At}}\,{\text{sites}}\,{\text{II}}\,{\text{and}}\,{\text{III:}}\,\,{K_2} = \exp \left( { - {{9\Delta G_{{\text{ads}},2}^{\text{o}}} \mathord{\left/ {\vphantom {{9\Delta G_{{\text{ads}},2}^{\text{o}}} {RT}}} \right. \kern-0pt} {RT}}} \right)$$

(11.19)

As PCP-C is isostructural to PCP-N, PCP-C also has three possible sites for CO₂ adsorption. The calculated BE for the adsorption of CO₂ on PCP-C decreases in the following order: site I (−4.55) > III (−2.12) > II (−2.00), where the values in parentheses are SCS-MP2:PBE-D3-calculated BEs in kcal mol⁻¹. This result indicates that CO₂ initially adsorbs at site I of PCP-C, similar to the case of PCP-N. After site I is fully occupied by six CO₂ molecules, further adsorption of CO₂ molecules at sites II and III of PCP-C is difficult because the BEs for the adsorption of CO₂ molecules at sites II and III significantly decrease (–0.27 and –0.15 kcal mol⁻¹, respectively) after the adsorption of six CO₂ molecules at site I. These small BE values arise from the large E_DEF values for CO₂ adsorptions at sites II and III of PCP-C; actually, the E_DEF value is 6.23 and 7.83 kcal mol⁻¹ for sites II and III, respectively. Even when six and three CO₂ molecules simultaneously adsorb at sites II and III, these large E_DEF values cannot be overcome by the sum of E_INT values. Thus, in the case of PCP-C, CO₂ adsorption occurs only at site I and does not take place at sites II and III.

For CO₂ adsorption on PCP-C, the equilibrium equation and K are represented by Eqs. 11.20 and 11.21, which are the same as those at site I in the case of PCP-N (Eqs. 11.16 and 11.18):

$$\begin{aligned}{\text{PCP-C}}\,{\text{site}}\,{\text{I:}}\,\,\quad & {\text{CO}}_{2} + \mathbf{PCP}\text{-}\mathbf{C} \cdot {{\text{(CO}}_{2})_{\text{n}}}\overset {} \leftrightarrows \mathbf{PCP}\text{-}\mathbf{C} \cdot {({\text{CO}_{2}})_{\text{n} + 1}},\\ &\qquad \quad n = 0{-}5\end{aligned}$$

(11.20)

$$K = \exp \left( { - {{\Delta G_{{\text{ads}}}^{\text{o}}} \mathord{\left/ {\vphantom {{\Delta G_{{\text{ads}}}^{\text{o}}} {RT}}} \right. \kern-0pt} {RT}}} \right)$$

(11.21)

At sites I of PCP-N and PCP-C, CO₂ adsorption occurs according to Eqs. 11.16 and 11.20, respectively. In this case, the adsorption coverage θ under a certain pressure (P) can be expressed by the Langmuir equation (Eq. 11.22):

$$\theta = K\left( {{p \mathord{\left/ {\vphantom {p {p_0}}} \right. \kern-0pt} {p_0}}} \right)\left[ {1 + K\left( {{p \mathord{\left/ {\vphantom {p {p_0}}} \right. \kern-0pt} {p_0}}} \right)} \right]$$

(11.22)

where p₀ is the standard pressure and K is the equilibrium constant of Eqs. 11.18 and 11.21. At sites II and III of PCP-N, nine CO₂ molecules instead of one CO₂ molecule can be simultaneously adsorbed because the sum of nine E_INT values is sufficient to overcome the sum of the E_DEF values of nine CO₂ molecules; in contrast, the BE of one CO₂ molecule is not enough to overcome the E_DEF value. Therefore, the adsorption isotherm can be described by Eq. 11.23:

$$\theta = K{\left( {{p \mathord{\left/ {\vphantom {p {p_0}}} \right. \kern-0pt} {p_0}}} \right)^9}\left[ {1 + K{{\left( {{p \mathord{\left/ {\vphantom {p {p_0}}} \right. \kern-0pt} {p_0}}} \right)}^9}} \right]$$

(11.23)

This equation is essentially the same as that of the Langmuir–Freundlich adsorption isotherm, where the order “9” of the $\left( {{p \mathord{\left/ {\vphantom {p {p_0}}} \right. \kern-0pt} {p_0}}} \right)$ term is valuable depending on the adsorption equilibrium; it must be larger than 1.0 for the Langmuir–Freundlich equation, whereas it must be 1.0 for the usual Langmuir equation.

Using Eqs. 11.22 and 11.23, the isotherms for the adsorptions of CO₂ on PCP-N and PCP-C were calculated (Fig. 11.6). Evidently, the calculated amount of adsorbed CO₂ molecules was smaller than the experimental value under low pressure. This is appropriate because the $\Delta G_{{\text{ads}}}^{\text{o}}$ value for the adsorption of gas molecules on the MOF is often underestimated by the assumption that the rotational and translational movements of the gas molecules adsorbed to the MOF are completely frozen, as discussed in Sect. 11.3.1. However, the gas molecules adsorbed to the MOF easily move. Thus, it is likely that one to two freedoms of rotational movement remain [23]. In this case, the $\Delta G_{{\text{ads}}}^{\text{o}}$ value for CO₂ adsorption may increase by 1.0–1.5 kcal mol⁻¹ from the value obtained for the case of totally frozen rotational movements. The CO₂ adsorption isotherms for PCP-N and PCP-C calculated using the modified $\Delta G_{{\text{ads}}}^{\text{o}}$ values are in suitable agreement with the experimentally observed results (Fig. 11.6a, b). Experiments revealed that the adsorption of CO₂ on PCP-N occurred with a stepwise isotherm, whereas that on PCP-C took place with a non-stepwise isotherm (black and red lines, respectively, in Fig. 11.4). This stepwise adsorption isotherm for PCP-N was appropriately reproduced via calculation. As comprehensively shown in Fig. 11.6c, the adsorption of CO₂ at site I (red line) starts at p₀ = 0.0, which is a normal feature observed in the Langmuir-type isotherm without a sigmoidal curve. In contrast, the adsorptions of CO₂ at sites II and III (blue line) do not start at p = 0.0; instead, they start at approximately p = 3 kPa with sigmoidal feature. This isotherm indicates that CO₂ adsorption does not occur at sites II and III when p < 3 kPa, whereas it suddenly starts at approximately p = 3 kPa. That is, gate-opening adsorption with a sigmoidal adsorption isotherm originates from the equilibrium represented by Eq. 11.23. The Eq. 11.23 is presented when two or more gas molecules simultaneously adsorb to the MOF rather than only one gas molecule. This leads to gate-opening adsorption with a sigmoidal adsorption isotherm.

Fig. 11.6

Cited from Ref. [25] with permission of the American Chemical Society

Calculated CO₂ adsorption isotherms at 195 K by PCP-N (a), that by PCP-C (b) and that by PCP-N at low pressure (c). The black lines represent experimental isotherms; blue and red lines represent isotherms calculated by increasing the adsorption Gibbs energy change by 1.0 and 1. 5 kcal mol⁻¹, respectively. Inset represents the expansion at low pressure region.

In summary, the CM/PM-combined method is a powerful computational method for calculating the BE of a gas molecule with a flexible MOF. Using this combined method, the mechanism and key factors of gate-opening adsorption with the sigmoidal adsorption isotherm have been clearly elucidated; adsorption of one gas molecule cannot occur because the large E_DEF cannot be overcome by the E_INT of one gas molecule. Nevertheless, there is a possibility that the sum of E_INT values overcomes the total E_DEF, when an E_DEF value per one gas molecule decreases but an E_INT value per one gas molecule decreases little as the number of adsorbed gas molecules increases. These are true in the case of flexible MOFs. In this case, simultaneous adsorption of more than one gas molecule can occur. This is the case for the adsorptions of gas molecules on sites II and III of PCP-N. On the other hand, if E_DEF is significantly large such that the simultaneous adsorption of more than one gas molecule cannot overcome the total E_DEF, gas adsorption does not take place. This is the case for the adsorptions of gas molecules on sites II and III of PCP-C. These findings show that E_DEF must be accurately calculated to better understand the gate-opening gas adsorption. Therefore, a simple cluster model is not useful; instead, the CM/PM-combined method must be employed in theoretical calculations of infinite system.

Here, we need to mention one weak point of the CM/PM-combined method. Although E_INT of a gas molecule is evaluated using the post-HF correction, E_DEF is calculated using the DFT method without the post-HF correction. If the post-HF correction is applied to the calculation of E_DEF using a reasonable CM, the quality of the CP/PM-combined method will improve.

4 Application of QM/Periodic-MM Method to Molecular Crystal

4.1 Isomerization of Ruthenium(II) Sulfur Dioxide Complex in Crystal

Photoinduced linkage isomerization occurs when a ligand coordinates to a metal atom in several metastable structures different from an equilibrium structure. This linkage isomerization is expected to be useful for optical data storage [78,79,80] because the ground state of the isomer can be assigned as “0” and the metastable state can be designated as “1”. Metastable structures can be experimentally detected by photocrystallography [81,82,83]. Thus, photoinduced linkage isomerization has attracted considerable interest as a prototype of molecular memories. A suitable example is the photoinduced linkage isomerization of the crystals of the ruthenium(II) sulfur dioxide complexes [Ru(NH₃)₄(SO₂)L]ⁿ⁺ 1 (n = 1 for L = Cl⁻ and n = 2 for L = H₂O (w) or pyridine (py)) [84,85,86,87,88,89,90] (Scheme 11.4). In the ground state, the SO₂ ligand coordinates to the Ru atom via the S atom (η¹-S form; GS). Under photoirradiation, GS isomerizes to a metastable η²-SO-coordinated form (η²-SO form; MS2) and then to another metastable η¹-O-coordinated form (η¹-O form; MS1). After photoirradiation, MS1 thermally returns to GS in the ground state via MS2. To apply this material to molecular memory, controlling the thermal isomerization of MS1 to GS is necessary. However, to date, the details of thermal isomerization in crystal are unclear. As we do not have an appropriate understanding of the chemical reactions occurring in crystals, currently, theoretical knowledge of the chemical reactions taking place in crystals is primarily important.

Scheme 11.4

Cited from Ref. [91] with permission of the American Chemical Society

Ground state of [Ru(NH₃)₄(SO₂)L]ⁿ⁺ (n = 1 for L = Cl; n = 2 for L = H₂O or pyridine) and its meta-stable states produced by photo-irradiation.

Thermal isomerization occurs when the crystal is not extremely rigid. If the crystal is highly rigid, thermal isomerization is difficult to occur. In contrast, if the crystal is extremely flexible, thermal isomerization quickly takes place as in solution, which is not suitable for application in molecular memory. Therefore, the crystal must be appropriately rigid (or flexible). This indicates that theoretical calculations must be performed by accurately considering the crystal effects.

Because the isomerization of a Ru complex in a crystal is different from the phase transition of the crystal, the transition state must be optimized for one Ru complex in the presence of other Ru complexes that do not simultaneously undergo isomerization. This theoretical study has recently been reported using the QM/periodic-MM method [91].

The QM/periodic-MM calculation was conducted using a QM region composed of one target complex ([Ru(NH₃)₄(SO₂)Py]²⁺, 1_Py (QM0) (Py = pyridine)), two nearest-neighboring Ru complexes (1Py (QM1)), five Cl⁻ counter ions, and one crystal H₂O molecule (QM2). A similar QM region was employed for [Ru(NH₃)₄(SO₂)(H₂O)]²⁺ (1_W) and [Ru(NH₃)₄(SO₂)Cl]⁺ (1_Cl). For the MM moiety, the LJ parameters of the AMBER force field were used, except for the Ru atom. As discussed in Sect. 11.2.2, the LJ parameters for the Ru atom were readjusted such that the acquired LJ parameters reproduced the B3PW91-calculated potential energy curve.

At first, the geometries of the GS-, MS1-, and MS2-crystals were optimized. Thereafter, the transition states and intermediates were optimized for the GS-, MS1-, and MS2-crystals to explore the influence of the crystal structure on the transition state and activation barrier of isomerization. Because it is interesting to know differences in geometry among gas phase, solution, and crystal, the geometry changes were compared among gas phase, solution, and crystal (Fig. 11.7), where δr = R(Ru-S) – R(Ru-O) is taken as an approximate reaction coordinate. This δr is a suitable measure representing the extent of the progress of isomerization as it is highly negative for GS, close to zero for MS2, and highly positive for MS1. Interestingly, the geometries change in similar manner among gas phase, aqueous solution, and crystal. These moderate differences suggest that the chemical reaction occurs in a similar cavity in both the solution and the crystal. Nevertheless, the energy changes were moderately different, as shown in Fig. 11.8, where the energy zero was taken for the equilibrium geometry of GS in the gas phase. As the orientations of ligands are slightly different in the crystal and gas phases, the geometries in this figure were optimized starting from those in the crystal phase. Particularly, the energy changes in the gas phase differed from those in the solution and crystal phases. In the gas phase, MS1A is at a lower energy than those of MS2A in 1_W and 1_Py. This relative energy implies that isomerization does not occur from MS1A to MS2A in the gas phase, which is contrary to the experimentally observed result for the crystal phase, showing that MS1A → MS2A isomerization takes place in the crystal phase. Only MS1A of 1_Cl is at a higher energy than that of MS2A, indicating that MS1A → MS2A isomerization can occur in 1_Cl. These results suggest that the reactions taking place in the crystal phase cannot be investigated in the gas phase, as expected.

Fig. 11.7

Cited from the SI of Ref. [91] with permission of the American Chemical Society

Geometry changes in thermal isomerization of [Ru(NH₃)₄(SO₂)L]ⁿ⁺ (n = 1 for L = Cl; n = 2 for L = H₂O or pyridine) from meta-stable state (η¹-O coordinated form MS1) to ground state (η¹-S coordinated form GS) through the second meta-stable state (η²-SO coordinated form MS2). (Red); Gas phase, (Green); Aqueous solution, (Blue); Crystal, and (Black bold); experimental values.

Fig. 11.8

Cited from Ref. [91] with permission of the American Chemical Society

Energy changes (in kcal/mol) in thermal isomerization of [Ru(NH₃)₄(SO₂)L]ⁿ⁺ (n = 1 for L = Cl; n = 2 for L = H₂O or pyridine) from meta-stable state (η¹-O coordinated form MS1) to ground state (η¹-S coordinated form GS) through the second meta-stable state (η²-SO coordinated form MS2) in gas phase and aqueous solution, where the most stable structure in gas phase was taken as reference (energy zero) and GSA was optimized starting from the orientations of NH₃ and L ligands taken in the crystal. Values without parenthesis are relative energy to the most stable species. In parenthesis is activation energy.

In the aqueous solution phase, the geometry and energy changes were calculated using the three-dimensional (3D)-RISM-SCF method by combining the solvation effects obtained by the 3D-RISM method [92, 93] with the electronic structure theory in a self-consistent manner [4,5,6]. The 3D-RISM-SCF method is effective for incorporating solvation effects when the target solute is large, highly polarizable, and charged like a transition metal complex bearing molecular charge(s) [94]. In the aqueous solution phase, MS1A of all the complexes is at a higher energy than that of MS2A, and MS2A is at a higher energy than that of GSA (Fig. 11.8b). Furthermore, the activation energy of MS1A → MS2A isomerization is smaller than that of MS2A → GSA isomerization. These results indicate that MS1A → MS2A isomerization occurs at low temperatures, whereas MS2A → GSA isomerization takes place at higher temperatures, as experimentally observed in the crystal phase.

For GS-, MS1-, and MS2-crystals, unexpectedly, the relative energies of GS, MS1A, and MS2A and the activation energies for the two isomerization steps were not different so much, as shown in Fig. 11.9, where the results of MS2-geometries have been omitted to save space because the energy changes for the MS2-geometries are similar to those for the MS1-geometries. However, the energy levels of all isomers and transition states depend on lattice vectors. Their energy levels relative to GS are lower in the crystal phases with lattice vectors at lower temperatures, whereas they become higher in the crystal phases with lattice vectors at higher temperatures. These results imply that the differences in crystal packing between at low and high temperatures lead to differences in relative energies. In all the lattice vectors, however, MS1 of all the complexes is at a higher energy than that of MS2A, and MS2 is at a higher energy than that of GSA. The activation energy of MS1A → MS2A isomerization is smaller than that of MS2A → GSA isomerization in all the lattice vectors. These results are in agreement with the experimentally observed results, demonstrating that MS1A → MS2A isomerization occurs at low temperatures, whereas MS2A → GSA isomerization takes place at higher temperatures in the crystal phase.

Fig. 11.9

Cited from Ref. [91] with permission of the American Chemical Society

Energy changes (in kcal/mol) in thermal isomerization of [Ru(NH₃)₄(SO₂)L]ⁿ⁺ (n = 1 for L = Cl; n = 2 for L = H₂O or pyridine) from meta-stable state (η¹-O coordinated form MS1) to ground state (η¹-S coordinated form GS) through the second meta-stable state (η²-SO coordinated form MS2) in crystal. Values without parenthesis are relative energy to the most stable species. In parenthesis is activation energy.

Interestingly, these results are qualitatively similar to those obtained for the solution phase, indicating that the reaction behaviors of [Ru(NH₃)₄(SO₂)L]ⁿ⁺ in the solution phase are not significantly different from those in the crystal phase. This is one of the surprising results. The similar reaction behaviors of [Ru(NH₃)₄(SO₂)L]ⁿ⁺ in the solution and crystal phases suggest that the reaction in the solution phase occurs in the solvation cage, whereas that in the crystal phase takes place in the crystal cavity. Nevertheless, all the metastable intermediates and transition states are at higher energies relative to that of GSA in the crystal phase than in the solution phase. These results imply that the metastable intermediates and transition states suffer from larger steric repulsion with crystal than the ground state GSA does. This is consistent with our general understanding that unstable intermediates and transition states have longer Ru–SO₂ distances. Accordingly, in the crystal phase, the activation barrier of isomerization is not always larger than that in the solution phase.

In summary, the QM/periodic-MM calculations revealed that the chemical reaction in the crystal phase occurs in a crystal cavity similar to the solvation cage in the solution phase; however, the crystal effects are larger in the metastable intermediate and transition state of the crystal reaction than the solvation effects in the solution reaction. Although the achieved results are not surprising, they are important because to date, no clear theoretical result of chemical reactions in crystals has been reported with evidence. These results demonstrate that the QM/periodic-MM method is a powerful tool for the theoretical study of chemical reactions in crystals.

4.2 Meta-Metal to Ligand Charge-Transfer (MMLCT) Excited State of Transition Metal Complexes in Crystal

Recently, theoretical studies of the excited states in the molecular crystals of a gold(I) phenyl(phenyl isocyanide)gold(I), (Ph)Au^I(NCPh), and a platinum(II) dicyanobipyridine complex, Pt^II(CN)₂(bpy), have been performed using the QM/periodic-MM method [95, 96]. As these studies have been summarized in a recent review [97], we have only provided a brief overview of these studies here.

(1) Excited states and emission spectra in the crystal of gold(I) phenyl phenylisocyanide complex: This complex has attracted considerable interest ever since Ito et al. have reported that the absorption and emission spectra were changed by mechanical stimulation [98]. This phenomenon is attributed to the single-crystal-to-single-crystal transition induced by mechanical stimulation. Nevertheless, it is not easy to investigate experimentally the influences of the changes in the crystal phase on the excited state of the complex, the mechanism behind the changes in the excited state with the occurrence of phase transition, and the extent of changes in the excited state because of the very small population of excited molecules.

When the emission spectra of metal complexes in the crystal phases or in highly concentrated solution phases significantly change from those in dilute solution, it is generally understood that a metal–metal (M–M) bond is formed in the excited state and CT excitation occurs from the antibonding orbital of the M–M bond to the π*-orbital of the ligand. When Au(I) complexes with d¹⁰ electron configuration approach each other, d–d bonding and antibonding molecular orbitals are formed between those two Au(I) complex (Scheme 11.5). Pt(II) has a d⁸ electron configuration. When four ligands are located on the x and y axes in the Pt(II) complex, the d_z² orbital is doubly occupied; thus, the d_z² orbital of one Pt(II) complex forms bonding and antibonding molecular orbitals with that of the neighboring Pt(II) complex, as in the case of Au(I), and a similar CT transition occurs. This CT transition is called metal-metal to ligand CT (MMLCT) excitation because CT occurs from the M–M bond to the ligand. Consequently, an M–M bond with a formal bond order of 0.5 is produced between these two M atoms in the MMLCT excited state; that is, exciplexes of Au(I) and Pt(II) complexes are generated by photoexcitation in the crystal phase and/or the concentrated solution phase. The emission from this MMLCT excited state should be considerably different from that of a single molecule. However, there are several open questions: Is this exciplex actually produced in the crystal phase? And can the MMLCT emission be observed? Evaluating the geometries of the excited states and emission energies in molecular crystals via theoretical calculations can provide answers to those questions but challenging.

Scheme 11.5

Metal–metal to ligand charge-transfer (MMLCT) excitation

(Ph)Au^I(NCPh) has two crystal phases: 1y, which demonstrates yellow luminescence, and 1b, which exhibits blue luminescence [98]. The optimized geometries of these phases are depicted in Fig. 11.10. In 1y, the Au–Au distance is long (4.73 and 5.73 Å), and CH–π interactions are developed between Au complexes, whereas π–π interactions do not exist. In 1b, the Au–Au distance is short (3.18 and 3.79 Å), and both CH–π and π–π interactions are formed between Au complexes. Moreover, the 1b-to-1y phase transition occurs. The differential scanning calorimetry (DSC) analysis has revealed that 1b is more stable than 1y by 7 kJ mol⁻¹, consistent with the experimental fact showing that 1b-to-1y phase transition occurs.

Fig. 11.10

Two crystal phases of phenyl(phenyl isocyanide)gold(I) complex. Cited from Ref. [95] with permission of the American Chemical Society

The relative stabilities of 1b and 1y were theoretically analyzed by the QM/periodic-MM method using four Au complexes as the QM part in 1b and six Au complexes as the QM part in 1y (Scheme 11.6). These models were employed for the following reasons: In 1y, which has alternating short and long Au–Au intervals, the MMLCT excited state is possibly formed by the two Au complexes with short interval. When the two central Au complexes participate in exciplex formation, the other two Au complexes with short Au–Au intervals exist above the exciplex and the remaining two Au complexes with short Au–Au interval exist below the exciplex. That is, the exciplex contacts with two pairs of two Au complexes in the ground state, when six Au(I) complexes are involved in the QM region (Scheme 11.6b). In 1b, the distance between all Au complexes is long. When the two Au complexes at the central position form the MMLCT excited state, one Au complex in the ground state is located above the exciplex and the other Au complex in the ground state is located below the exciplex such that the exciplex is sandwiched with the Au complexes in the ground state. When four Au complexes are involved in the QM region, therefore, the exciplex is surrounded by two Au complexes at the ground state in the QM region (Scheme 11.6a). Using these models, the relative stabilities of 1b and 1y have been investigated by considering the energy per molecule of the Au complex. Using the positions of the Au nuclei determined in the experiment, the rotations of the Ph and PhNC ligands, orientations of the Au complexes, and geometries of the ligand moieties have been optimized. The optimized geometries reveal that the CH–π interaction is indeed present in 1b and the π–π interaction does not exist (Fig. 11.10a), whereas the CH–π and π–π interactions are present in 1y (Fig. 11.10b).

Scheme 11.6

Cited from Ref. [95] with permission of the American Chemical Society

Models of crystal structures of phenyl(phenyl isocyanide)gold(I) complex employed for QM/Periodic-MM calculation of 1b (a) and 1y (b).

Additionally, the calculations indicated that 1b was more stable than 1y and the relative stabilities of 1b and 1y were governed by the QM–MM and MM–MM interactions but were not by the QM part. These results suggest that long-range interactions are critical for determining the relative stabilities of molecular crystals. The QM–QM interaction has been corrected by the MP2 method; nevertheless, the stability of the molecular crystal of this Au(I) complex is not changed very much by the MP2 correction. Using these ground-state structures, QM/periodic-MM calculations were conducted by applying time-dependent (TD)-DFT with the B3LYP-D3 functional to evaluate the excitation energies. The lowest-energy excitation was an intramolecular CT excitation from the π-orbital of the Ph ligand to the π*-orbital of the PhNC ligand. The excitation energy was obtained with an error of about 0.5 eV from the experimental value. Adsorption energies of the surface Au complex and the internal Au complex were also compared using the slab model and were found to be significantly different.

Geometry optimization of the excited state was performed by the QM/periodic-MM method using the B3LYP functional. 1b does not form an exciplex consisting of two Au complexes. This is reasonable because of the long Au–Au distances of 4.69 and 5.87 Å at the ground state (Fig. 11.11a). Instead, the lowest-energy excited state is formed through triplet π–π* excitation in one PhNC ligand. In contrast, for 1y, the Au–Au distance is shorter at the ground state (2.85 Å), suggesting that d_z²–d_z² bonding and antibonding molecular orbitals are formed between two Au complexes and one-electron excitation occurs from the d_z²–d_z² antibonding molecular orbital to the π* orbital of PhNC ligand. In fact, these two molecular orbitals are singly occupied in the excited state (Fig. 11.11b). Table 11.7 presents a comparison of the emission spectra and the experimental results. In the case of 1b, the energy of the emission spectrum is in appropriate agreement with the experimental value, with an error of 0.2 eV. The emission spectra of the Au(I) complex on the surface and the interior of the crystal have been compared, which exhibit a slight difference as compared to the cases of the adsorption spectra. This slight difference originates from the fact that the adsorption spectrum is induced by CT excitation from the Ph ligand to the PhNC ligand, as discussed above, whereas emission occurs from the π–π* excited state in one PhNC ligand; in other words, the Ph and PhNC ligands are influenced in a different manner by the atmosphere, leading to the difference in the absorption spectrum between at the surface and interior, whereas both π and π* orbitals of the PhNC ligand are affected to a similar extent by the crystal atmosphere, leading to the similar emission spectrum between at the surface and interior. In 1y, ³MMLCT1 is the first excited state (Fig. 11.11b), and ³MMLCT2 is observed at different positions (Fig. 11.11b). The emission energies of ³MMLCT1 and ³MMLCT2 are moderately different. This is an artificial error originating from the small size of the QM region: ³MMLCT1 is produced at one place in the QM region, whereas ³MMLCT2 is generated in two different Au complexes. If the size of the QM region is sufficiently increased, both ³MMLCT1 and ³MMLCT2 should have the same structure and the same energy. Because of the small QM region, the circumstances of the ³MMLCT1 state moderately differ from those of the ³MMLCT2. For ³MMLCT, almost no difference was observed between the energies of the emission spectra of the complex at the surface and the interior of the crystal. Furthermore, the theoretically calculated emission energy is in suitable agreement with the experimental results for ³MMLCT. In these calculations of emission spectra, the ΔSCF method was used instead of TD-DFT. Certainly, although the emission spectra can be calculated by the TD-DFT using the excited-state structure, the results of the ΔSCF method are in better agreement with the experimental results in our experience.

Fig. 11.11

The optimized geometries of the triplet excited state of phenyl(phenyl isocyanide)gold(I) complex in crystal phases 1b and 1y. Cited from Ref. [95] with permission of the American Chemical Society

Table 11.7 Emission energies of Au(Ph)(PhNC) in crystal phases 1b and 1y

(2) Temperature effects on the excited state structure and emission spectrum of [Pt^II(CN)₂(bpy)] (bpy = 2,2′-bipyridine) complex in crystals: Planar four-coordinate Pt(II) complexes exhibit weak or no emission in dilute solution phase; however, they often demonstrate strong emission spectra in the concentrated solution or crystal phase. This phenomenon is called aggregation-induced phosphorescence or assembly-induced phosphorescence (abbreviated as AIP) [99,100,101]. Generally, when metal complexes are located in close proximity to each other in the high-concentration solution or crystal phase, MMLCT (Scheme 11.5) is produced, and thus, a new emission spectrum is observed. This emission spectrum becomes broad with an increase in temperature, and the emission peak shifts to a higher energy in many cases. A typical example is depicted in Fig. 11.12 [102]. The emission spectrum of the [Pt^II(CN)₂(bpy)] complex in the crystal phase was measured at various temperatures ranging from low (15 K) to room temperature. At 15 K, a sharp emission spectrum was noticed at approximately 660 nm, whereas at room temperature, a broad spectrum was observed at approximately 620–630 nm [102]. Figure 11.12b shows the Pt–Pt distances at different temperatures. Clearly, the Pt–Pt distance increased with an increase in temperature, indicating that the d_z2–d_z2 antibonding molecular orbital between two Pt(II) complexes shifted to a lower energy and the MMLCT excitation energy shifted to a higher energy with an increase in the Pt–Pt distance in MMLCT. Nevertheless, whether the crystal structure in the ground state is reflected in the structure of the excited state is still unclear. Additionally, the reason for the broadening of the spectrum has not yet been clarified.

Fig. 11.12

Emission spectrum of Pt(CN)₂(bpy) (bpy = 2,2′-bipyridine) at various temperatures (a) and the lattice vectors with Pt–Pt distance a to m in (b) represent 292, 260, 240, 220, 180, 160, 140, 120, 100, 60, 45, 30, 15 K, respectively. λ_ex = 514.5 nm. Cited from Ref. [102] with permission of the American Chemical Society

A QM/periodic-MM theoretical study of [Pt^II(CN)₂(bpy)] molecular crystal was conducted [96], in which the temperature dependences of the excited-state structure and emission spectrum were discussed. A unit cell comprising n QM and (225-n) MM molecules was used for calculation, and three kinds of QM regions were considered. The first type of QM region included the central two molecules, and the second and third kinds of QM regions comprised the central three and four molecules, respectively. Using these models, QM/periodic-MM calculations were performed. The excited-state structures of [Pt^II(CN)₂(bpy)] in the gas and crystal phases were different; for instance, in the crystal phase, the orientations of bpy and CN in the ground and excited states were the same. However, in the gas phase, bpy and CN were twisted around each other and formed a staggard structure. Originally, the staggard structure of the ligands was stable due to steric repulsion between the two ligands; on the other hand, in the crystal phase, the eclipsed orientation was probably induced by the crystal packing effect, and the orientation change upon excitation was suppressed by the crystal effect. When two Pt complexes were used in the QM region, the MMLCT excited state (denoted as ³[Pt-MMLCT]₂) was formed by these two Pt complexes. When three Pt complexes were used in the QM region, the excited state (denoted as ³[Pt-MMLCT]₃) was delocalized into three Pt(II) complexes. When four Pt complexes were used in the QM region, the excited state (denoted as ³[Pt-MMLCT]₄) was delocalized into four QM Pt(II) complexes, where the excited state character was stronger and the spin density was larger in the central two Pt molecules than those in the terminal two Pt molecules. Because of this delocalization, the energies of the three excited states, namely, ³[Pt-MMLCT]₂, ³[Pt-MMLCT]₃, and ³[Pt-MMLCT]₄, were different (Table 11.8). Therefore, the populations of these excited states varied with respect to temperature.

Table 11.8 Relative energies (in kcal/mol) of dimer, trimer, and tetramer MMLCT excited states and their populations (in %)

The potential energy curve with respect to the Pt–Pt distance in the excited state was shallow; thus, the vibrational frequency was small. This implies that the populations of vibrationally excited states are large. In this study, the vibrational wave function was obtained by solving the Schrödinger equation for the Pt–Pt vibration, and the population of each vibrationally excited state was calculated at low and room temperatures based on the Boltzmann distribution law. For vibrational wave functions with higher quantum numbers, the probability of a structure deviating from the equilibrium nuclear configuration is large. At low temperatures, the populations of the vibrationally excited states are small, which leads to a small probability of structure deviated from the equilibrium nuclear configuration and therefore results in a sharp spectrum. At room temperature, the populations of the vibrationally excited states are large; therefore, the probability of deviating structure is large and a broadened spectrum is presented at room temperature, compared with that at low temperature (Fig. 11.13a). This feature resembles the phenomenon that the symmetry-forbidden d–d transition can have an oscillator strength at room temperatures owing to the probabilities of the vibrationally excited states [103]. However, the peak of the spectrum was almost equal to that obtained at a low temperature when only the probability of deviating structure was considered. As abovementioned, the populations of ³[Pt-MMLCT]₂, ³[Pt-MMLCT]₃, and ³[Pt-MMLCT]₄ change with an increase in temperature because of the variation in their energies; their populations were evaluated using the Boltzmann distribution law (Table 11.8): the population of ³[Pt-MMLCT]₄ was 80% at 10 K, whereas that of ³[Pt-MMLCT]₃ was 74% at room temperature. This change is induced by not only the Boltzmann distribution, but also the fact that the Pt–Pt distance in the crystal phase becomes shorter at low temperature and longer at room temperature (Fig. 11.12b). In fact, the tetramer exciplex becomes more stable at low temperature and less stable at room temperature than other exciplexes because the three Pt–Pt distances become shorter in the tetramer exciplex and the tetramer exciplex formation easily occurs when the intermolecular distance is short. Instead, the trimer exciplex becomes  more stable than the tetramer exciplex at room temperature because the intermolecular distance becomes longer at room temperature than that at low temperature. Even at room temperature, the trimer exciplex is more stable than the dimer exciplex because the delocalization of the electronic structure of the trimer is larger than that of the dimer and significantly stabilized the trimer compared to the dimer. Considering the change in the population of the exciplex and the deviation of the structure of the vibrationally excited state from the equilibrium structure, the emission spectrum was calculated to be sharp at a low energy (1.95 eV) when temperature is low, and broad at a high energy (2.01 eV) when room temperature (Fig. 11.13b). These computational results (Fig. 11.13b) are in appropriate agreement with the experimental results shown in Fig. 11.12a.

Fig. 11.13

Theoretically calculated emission spectrum of [Pt(CN)₂(bpy)] (bpy = 2,2′-bipyridine) from the MMLCT excited state consisting of dimer (a) and that calculated using the Boltzmann distribution of dimer, trimer, and tetramer excited states (b). Cited from Ref. [96] with permission of the American Chemical Society

In summary, the broadening of the emission spectrum induced by an increase in temperature is explained by the fact that the population of the vibrationally excited state increases at high temperature, and the high-energy shift of the emission spectrum is understood by the change in the populations of several exciplexes. These results indicate that the QM/periodic-MM method is promising for the theoretical study of the excited states in crystals.

5 Advantages and Disadvantages of CM/PM-Combined and QM/Periodic-MM Methods from the Viewpoint of Application

Herein, we have compared the CM/PM-combined and QM/periodic-MM methods from the viewpoints of their applications to molecular crystals. The CM/PM-combined method was applied to the adsorption of gas molecules to MOFs [23,24,25]; the enthalpy of adsorption could be evaluated with adequate accuracy and the adsorption isotherm could be reproduced well, as described in Sect. 11.3. In the theoretical studies, structural optimization was performed via periodic-DFT calculation of the entire crystal system, and the adsorption energy was evaluated using periodic-DFT calculation of the entire system and post-HF correction for appropriate cluster models.

This CM/PM-combined method is not suitable for geometry optimization of excited state and calculation of emission spectrum of excited state because a very large unit cell is required for the geometry optimization and the emission spectrum calculation owing to the considerably smaller population of molecules at excited state than that of molecules at ground state. In the theoretical study of chemical reactions in crystals, we must also utilize a large unit cell because of the following reasons: if we calculate the intermediate and transition state using one unit cell, the reaction simultaneously occurs in all other unit cells; these geometry and energy changes correspond to phase transitions. In a chemical reaction, the simultaneous occurrence of reactions in all the unit cells is improbable; that is, the chemical reaction takes place in one unit cell, whereas no reaction occurs in many other unit cells.

Considering these features of the excited state and chemical reaction, the CM/PM-combined method is appropriate for theoretical calculations of the chemical events, such as the adsorption of gas molecules to MOFs or zeolites and adsorption of molecules to solid surfaces with regular structures, occurring in the ground states of molecular crystals.

In contrast, in the QM/periodic-MM method, the main part of a molecular crystal is calculated by the QM method and the remaining parts are evaluated via the classical force field (MM). The QM part at excited state can possibly be assessed in the presence of MM molecules bearing the structure and charge distribution in the ground state, as discussed in Sect. 11.4.2. Similar calculations can be conducted to investigate chemical reactions in crystals (Sect. 11.4.1). When the target is a solid, including MOFs and zeolites, with an infinite series of bonds, defining the QM and MM parts by cutting some of the infinite series of bonds is necessary. Particularly, the application of this method is challenging when the cutting procedure cannot be easily performed; For instance, cutting procedure is very difficult for extended and/or conjugated systems such as graphite. On the other hand, the QM/periodic-MM method can be facilely applied to a molecular crystal because the molecular crystal is clearly and reasonably separated into the QM and MM parts without any problem. For instance, the QM/periodic-MM method is suitable for theoretical studies of absorption spectra, excited-state structures, emission spectra [95, 96], and the structural and energy changes induced by chemical reactions in molecular crystals because the cutting procedure is not required in the case of molecular crystals.

In summary, these methods can be improved as follows: In the CP/PM-combined method, cutting out appropriate cluster models (CMs) is necessary. Also, in the high-quality calculation of the CM, the crystal effect is not considered currently, which is a limitation of this method. This limitation can be overcome using the QM/periodic-MM method for the calculation of CM, as abovementioned. In this case, charge distribution in the boundary region between the QM and MM parts must be carefully set.

In the QM/periodic-MM method, the excited-state, intermediate, and transition-state in the QM part directly interact with MM molecules when the QM part is small. As well known, the QM–MM interaction is less accurately evaluated than the QM–QM interaction. Moreover, relaxations of the geometries and electronic structures of MM molecules are not considered in this method because they are taken to be the same as those for the molecular crystal in the ground state or before the chemical reaction. In fact, the relaxations of the geometries and electronic structures occur during chemical reactions and excitations. Therefore, we need to employ a QM region composed of an important moiety, in which chemical events, such as excitation and chemical reactions, occur, and other QM molecules that do not directly participate in the events but surround the important moiety. The idea of fragment molecular orbital [104] proposed by Kitaura et al. may be applicable to this case; however, its combination with the QM/periodic-MM method has not yet been carried out. Additionally, non-dynamical (or static) electron correlation often needs to be considered to calculate the excited state and conical intersection by this method. In this case, we employed the multireference wave function method. When exciplex consists of several transition-metal complexes, the active space becomes large and the usual complete active space self-consistent-field (CASSCF) calculation cannot cope with it. In such a case, the density matrix renormalization group (DMRG) method [105] can be applied to the multireference wave function method. If the quantum embedding theory outlined in Sect. 11.3.2 can be easily implemented, these theoretical calculations incorporating the non-dynamical electron correlation effects can be performed using a combination of DMRG and quantum embedding procedures.

Recently, the photochemistry of infinite system has attracted significant attention in the experimental field. Herein, we have discussed the theoretical study of the photochemistry of crystalline materials. This computational subject is gaining importance in connection with solar cells, photochemical reactions, and so on. Photochemistry is also of much interest in MOFs [106]. For theoretical studies of excited states in crystalline materials, at least the use of DFT methods with hybrid-type functionals is required, and in many cases, the use of post-HF methods, for example, CASSCF, symmetry-adapted-cluster/configuration interaction method (SAC-CI), and equation‐of‐motion coupled‐cluster (EOM-CC), is desirable. Nevertheless, the application of these methods for periodic-DFT calculations of infinite systems is difficult from the viewpoint of computational time, indicating that we need to improve the approach to photochemistry of infinite systems.

In this regard, the following practical procedure can be conducted: we can obtain the ground-state structure using periodic-DFT calculations, which is not difficult. Using the ground-state geometry, we can acquire the excitation energy through the TD-DFT calculation with the GGA-type functional, where an entire system is employed. Then, corrections using hybrid DFT functional or post-HF methods such as SAC-CI, EOM-CC, and state-averaged CASSCF, are added. These calculations can be performed using an appropriate CM. Then, the excitation energy can be corrected using the ONIOM procedure. These calculations can be conducted by placing the QM part in the MM part of the ground state obtained by periodic-DFT calculation of the infinite system. Although to date, theoretical calculations of this type have not been performed, this is a practical approximation and seems reasonable.

6 Conclusions and Perspectives

In this chapter, two methods, namely, the CM/PM-combined method and the QM/periodic-MM method, for theoretical studies of the electronic structures of molecular crystals are outlined and their applications are presented. The CM/PM-combined method was applied to the theoretical study of the adsorption of gas molecules to MOFs. In the study, the adsorption isotherms of gas molecules were appropriately reproduced. The QM/periodic-MM method was employed for the theoretical studies of the crystal structures of transition metal complexes, structural and energy changes induced by chemical reactions in crystals, and the excited-state structures and emission spectra of molecular crystals. Similar embedding methods proposed for the theoretical studies of solids and solid surfaces are discussed.

Nevertheless, these methods have several limitations, some of which can be overcome, as described in Sect. 11.5. The importance of molecular crystals and solids with regular structures is increasing in both basic chemistry and applied fields. Thus, theoretical studies of these materials are required to obtain theoretical knowledge and correct understanding of the various chemical phenomena occurring in solid systems and the properties of solid systems.

Furthermore, amorphous solids will become more important in a near future. Amorphous systems consist of several different microstructures. Theoretical calculations of this type of systems without periodic structures are very difficult. In such a case, considering their Boltzmann distributions is necessary. A hybrid Monte Carlo (MC)/MD reaction method (named Red-Moon method), recently proposed by Nagaoka et al. [107], has been applied to amorphous materials such as solid electrolyte interphase (SEI) film in secondary batteries. Thus, progress in the theoretical studies of large amorphous systems consisting of microstructures has also started.

Currently, knowledge of the structures, functions, and electronic properties of molecules in solids, whether crystalline or amorphous, is crucially important and will increase more in a near future. The electronic structure theory in solid-state chemistry needs to be further developed, which is challenging from the viewpoints of both method development and application.