Molecular Crystal Calculation Prospects for Structural Phase Transitions

Abstract

To establish the theory of soft crystals, computational chemistry must be applied to analyze the structural phase transitions of molecular crystals and develop new methodologies. The accuracy of first-principles calculations for molecular crystals has rapidly improved over the last decade with the contribution of the Cambridge Crystallographic Data Centre blind test, which predicts the crystal structure from the structural formula. However, it is often difficult to apply first-principles calculations to large molecular crystals, such as typical soft crystals, because of the computational cost. In this chapter, we review the applicability of crystal force field calculations as an alternative method for theoretically analyzing molecular crystals. We also introduce some examples of our previous collaborations and discuss the promising methodologies to elucidate the soft crystal phenomena.

1 Introduction

Soft crystals are a group of molecular materials that exhibit structural phase transitions from one stable single crystal to another single crystal (or amorphous state) upon weak external stimuli [1]. They are typically metallic complexes with photochemical changes of vapochromism and mechanochromism [2, 3] or organic compounds that exhibit unique properties of superelasticity [4, 5]. Many soft crystals can return to their original crystalline phase when the external stimuli is removed, or a different stimulus is added. These soft crystals are expected to develop into new functional materials and thus, it is important to understand the properties and phenomena observed in experiments and to elucidate the principles of their physical properties. This can be accomplished by applying computational chemistry and performing detailed analyses. Thus, how should we approach each of the various phenomena exhibited by these rare soft crystals using computational chemistry? Before we discuss our methodology and examples of our work, we will provide some information about computational chemistry for molecular crystals, including soft crystals.

1.1 Contribution of Crystal Structure Prediction to the Computational Chemistry of Molecular Crystals

Computational approaches to soft crystals present several major challenges that arise from the fact that soft crystals are molecular crystals, which have several characteristics that make them unsuited to computational analysis approaches. First, molecules composed of rigid covalent bonds have complex three-dimensional structures. Second, molecules can flexibly adopt multiple conformations by intramolecular interactions. In other words, the molecules as the crystalline fillings are not spherical like atoms but have complex three-dimensional structures and can be deformed. Third, the three-dimensional order is maintained by various types of intermolecular interactions that are much weaker than covalent interactions, such as isotropic interactions like van der Waals forces and electrostatic forces, strongly anisotropic interactions like hydrogen bonds, and coordination bonds in the case of metallic complexes. These factors cause a property called polymorphism, which is the formation of multiple metastable crystal structures with different molecular orientations and conformations. This property has made computational chemical approaches to molecular crystals difficult.

Computational chemistry methods for the analysis of molecular crystals have been rapidly developed in the last decade. In particular, the crystal structure prediction (CSP) blind tests [6,7,8,9,10,11] organized by the Cambridge Crystallographic Data Centre (CCDC) have undoubtedly made significant contributions to this development [12]. Through the blind tests, the accuracy of the dispersion force corrected density functional theory (DFT-D) method, which combines first-principles calculations of DFT with dispersion force correction by two-body potentials, has been considerably improved.

In the CSP blind test, only the structural formula and crystallization conditions of a molecule whose crystal structure has not yet been published are provided as questions, and participants predict the crystal structure using their prediction techniques. There are two major prediction techniques that participants must develop. One is the search for crystal polymorphs by generating a large number of possible crystal structures of the target molecule and ensuring that the crystal structure generated is the same as the experimentally observed crystal structure. Another is to rank the generated polymorphs and provide the highest evaluation of the crystal structure that can be regarded as one corresponding to the experimentally observed crystal structure. Ideally, the polymorphic structure with the lowest crystal energy should be the most thermodynamically stable. In the fourth blind test reported in 2009, the Neumann group was the first to use the DFT-D method with periodic boundary conditions [9]. In the sixth blind test reported in 2016, which included five different molecules, such as co-crystals, salt hydrates, and molecules with high conformational flexibility, eight of the nine crystal structures were correctly obtained using the improved DFT-D method [11]. Although the evaluation by the improved DFT-D method agrees with the experimental structure within or below the experimental error for the crystal structure, room for improvement in the crystal polymorph ranking by crystal energy has been identified.

1.2 Challenges of Molecular Crystal Calculations Using the Force Field Method

The results of the CSP blind test suggest that the methodology currently employed in crystal structure prediction can predict unknown crystal structures, possibly with higher accuracy than results obtained experimentally. However, it suffers from computational costs. The DFT-D method is a relatively high-precision method among first-principles calculations but it requires a computer with high computing power. Therefore, it is not a good strategy to adopt the first-principles calculation to determine polymorphs more efficiently.

Crystal force field calculations based on classical mechanics are an important potential solution. In general, the crystal energies and optimized crystal structures of the input crystal structures can be obtained 100–1000 times faster than the first-principles calculations. In other words, the structure optimization of several trial crystal structures generated by the crystal polymorphism search can be completed in a realistic calculation time. More accurate structures, energies, or properties based on electronic states can be obtained in a relatively short time by performing re-evaluation or structure optimization again using the DFT-D method. This is a more realistic and practical strategy.

The problem with the crystal force field calculation is its accuracy. The accuracy of the crystal force field, which is an extension of the molecular force field for a single molecule to the crystal structure, is almost the same or slightly worse than the experimental error [11]. The crystal energies do not correlate well with those of the DFT-D method. Although the DFT-D method can accurately be used to search for crystal polymorphs, the same accuracy cannot be obtained if the crystal force field used to optimize the trial crystal structure does not reproduce the appropriate structure and energy. In fact, the success of the Neumann group in the CSP blind test is largely due to the force field construction function (tailor-made force field) [13] introduced in their crystal structure prediction program GRACE. In their method, the crystal structure of a target molecule is optimized in advance by the corrected DFT method and the crystal force field is automatically adjusted to reproduce the obtained crystal structure. Their success indicates that it is very effective in establishing molecular crystallography to obtain a crystal force field that better resembles the result obtained from the DFT method.

Crystal force field calculations have some challenges inherent to soft crystals. Since most of the molecules exhibiting soft crystal phenomena are metallic complexes, crystal force field calculations are not always available because of missing the force field parameters. In addition, as the force field method, in principle, describes molecules according to the valence bond rule, it may not be able to describe metal complexes properly. It is also difficult to geometrically define the coordination bonds (interactions) between the central metal ion and the organic ligands of metallic complexes and to describe how they change depending on the oxidation number of the central metal. Therefore, traditional molecular mechanics has not been actively treated for the coordinate compounds as one of the available methodologies, and it was not clear how to apply them in practice.

In Sect. 10.2, we review the classical mechanics-based crystalline force field calculations employed by us to analyze soft crystals. In Sect. 10.3, we present examples of our computational chemistry research on soft crystals and show how we have solved some of the aforementioned challenges. Section 10.4 outlines the approaches we are developing to analyze structural phase transitions, as exemplified by soft crystal phenomena, and describes new methodologies that will be developed in the future.

2 Crystal Force Field Calculation

Crystal force field calculations are an extension of molecular force fields, which have been developed mainly to reproduce three-dimensional structures, energies, and vibrational frequencies of isolated molecules, that make it possible to calculate crystal structures under periodic boundary conditions. Unlike first-principles calculations, molecular force fields do not treat electrons in an exposed manner but describe molecules using classical mechanical formulas for the “fields” that electrons rapidly form around atoms. This is called “molecular mechanics” [14]. To extend the molecular force field to a crystal force field, it is necessary to accurately calculate the crystal structure, crystal energies, and lattice vibrations, especially if the goal is to accurately reproduce the relative stability between crystal polymorphs [15]. Therefore, in crystal force field calculations, the force field functions for intermolecular interactions in crystals are usually the same as those for molecular force fields, even though some modifications are made to treat intermolecular interactions more rigorously [16]. Various force fields have been employed in crystal calculations under the requirement that they reproduce the crystal structure within less than the experimental error and allow the evaluation of the relative crystal energies of the stable and metastable phases. Furthermore, it should also be able to reproduce and analyze the observed crystal polymorphic structures and their behavior up to matching lattice vibrations (k = 0 phonons) [17].

In this section, we review the crystal energies, underlying crystal force fields, and their crystal structural predictions using the crystal force field calculation method [18,19,20,21] implemented in the computational chemistry program CONFLEX [22,23,24].

2.1 Crystal Energy

Given a molecular structure, space group, and lattice constant in an asymmetric unit, a crystal structure can be constructed on a computer. The crystal energy corresponding to the internal energy of the thermodynamic state for the crystal structure ${E}_{\mathrm{crystal}}$ can be defined as the energy per asymmetric unit in Eq. (10.1) [11, 21].

$$E_{{{\text{crystal}}}} = E_{{{\text{intra}}}} + E_{{{\text{lattice}}}}$$

(10.1)

where ${E}_{\mathrm{intra}}$ is the sum of the intramolecular interaction energies in the asymmetric unit and ${E}_{\mathrm{lattice}}$ is defined as

$$E_{{{\text{lattice}}}} = E_{{{\text{inter}}}}^{{{\text{AU}}}} + \frac{1}{2}\mathop \sum \limits_{i}^{N} \mathop \sum \limits_{S}^{M} \mathop \sum \limits_{J}^{N} E_{{{\text{inter}}}} \left( {i;S,J} \right)$$

(10.2)

In Eq. (10.2), the first term ${E}_{\mathrm{inter}}^{\mathrm{AU}}$ is the sum of the intermolecular interaction energies in the asymmetric unit, which is zero if there is only one molecule in the asymmetric unit. The second term is the sum of the intermolecular interaction energies between the molecules in the asymmetric unit and the molecules replicated by the translational symmetry. In the CONFLEX crystal calculation model, the range for this sum is pre-specified by the nearest-neighbor interatomic distance between the molecules in the asymmetric unit and the replicated molecules (Fig. 10.1). In the crystal structure optimization, the coordinates of the atoms in the asymmetric unit and the lattice constants are used as parameters for structural optimization while maintaining the symmetry of the crystal to obtain the energy-minimal structure. For the evaluation of intra- and intermolecular interactions in the crystal, we employ the existing force field, MMFF94s [25,26,27,28,29,30,31,32].

Fig. 10.1

Molecular crystal calculation model of CONFLEX (Reprinted (adapted) with permission from Ref. [42]. Copyright (2022) American Chemical Society.). The asymmetric unit is depicted in red and replicated molecules that the nearest-neighbor interatomic distance from molecules in asymmetric unit is less than R_crystal are in dark gray. The molecules depicted in light grey are not including in the crystal energy calculation

In CONFLEX, we can calculate the crystal energy by inputting the crystal structure determined by X-ray crystallography, that is, the molecular structure and space group in the asymmetric unit and the lattice constant. In the same way, either or both the molecular structure and the lattice constant in the asymmetric unit of the crystal structure can be optimized. However, only the space group is kept constant during optimization. The crystal structure obtained by the optimization with minimal crystal energy agrees with the crystal structure determined by X-ray crystallography to the same extent as the experimental error [11]. Empirically, it is less than approximately 1.0 Å in root-mean-square deviation (RMSD) for a superposition of 15 molecules (RMSD₁₅) in a crystal. If the deviation from the experimental structure is large, likely, the crystal force field applied to the molecule is not appropriate. In such cases, it may be necessary to reconsider the force field parameters.

2.2 Crystal Force Field

In CONFLEX, the energy defined in Eq. (10.1) is calculated using the molecular force field based on classical mechanics. The MMFF94s potential energy function employed is constructed as described in Eq. (10.3) [25, 26, 32]:

$$E_{{{\text{FF}}}} = E_{{{\text{str}}}} + E_{{{\text{bend}}}} + E_{{{\text{tors}}}} + E_{{{\text{oop}}}} + E_{{{\text{str}} - {\text{bend}}}} + E_{{{\text{vdW}}}} + E_{{{\text{el}}}}$$

(10.3)

where ${E}_{\mathrm{str}}$ is the bond stretching, ${E}_{\mathrm{bend}}$ is the angle bending, ${E}_{\mathrm{tors}}$ is the bond torsion, ${E}_{\mathrm{oop}}$ is the out-of-plane angle, ${E}_{\mathrm{str}-\mathrm{bend}}$ is the coupling of stretching and bending, and ${E}_{\mathrm{vdW}}$ and ${E}_{\mathrm{el}}$ correspond to the van der Waals (vdW) and electrostatic interactions, respectively. In CONFLEX, the force field, which has been developed mainly for the evaluation of single molecule structures, is extended to evaluate crystal energies. That is, the first term in Eq. (10.1), namely ${E}_{\mathrm{intra}}$, is evaluated using Eq. (10.3). Furthermore, the last two terms in Eq. (10.3), namely ${E}_{\mathrm{vdW}}$ and ${E}_{\mathrm{el}}$, are not only the force field potential function forms for the intramolecular non-bonded interactions in the asymmetric unit but also the intermolecular interaction energies ${E}_{\mathrm{lattice}}$ in Eq. (10.2) in the crystal force field calculations.

As MMFF94s has been developed to calculate potential energies for organic compounds, few force field parameters are available for the elements constituting metallic and inorganic materials. To perform crystal calculations for many metal complexes that exhibit soft crystal phenomena, it is necessary to start by determining the force field parameters. In particular, the force field parameters for the vdW interactions are the most important factors that determine the accuracy of the crystal structure and energy optimized by the crystal force field.

The vdW interactions of the MMFF94s used for the crystal force field is obtained from Eq. (10.4):

$$E_{{vdW_{ij} }} = \varepsilon_{IJ} \left( {\frac{{1.07R_{IJ}^{*} }}{{R_{ij} + 0.07R_{IJ}^{*} }}} \right)^{7} \left( {\frac{{1.12R_{IJ}^{*7} }}{{R_{ij}^{7} + 0.12R_{IJ}^{*7} }} - 2} \right)$$

(10.4)

where ${R}_{ij}$ is the distance between atoms i-j in the crystal and ${R}_{IJ}^{*}$ is a parameter determined by the combination of atom types I-J, which is obtained from the following equations:

$$R_{IJ}^{*} = 0.5\left( {R_{II}^{*} + R_{JJ}^{*} } \right)\left( {1 + 0.2\left( {1 - {\text{exp}}\left( { - 12\gamma_{IJ}^{2} } \right)} \right)} \right)$$

(10.5)

$$\gamma_{IJ} = \left( {R_{II}^{*} - R_{JJ}^{*} } \right)/\left( {R_{II}^{*} + R_{JJ}^{*} } \right)$$

(10.6)

$$\varepsilon_{IJ} = \frac{{181.16G_{I} G_{J} \alpha_{I} \alpha_{J} }}{{\left( {\alpha_{I} /N_{I} } \right)^{1/2} + \left( {\alpha_{J} /N_{J} } \right)^{1/2} }}\frac{1}{{R_{IJ}^{* 6} }}$$

(10.7)

where ${N}_{I}$ of atom type I is the Slater–Kirkwood effective valence electron number, ${G}_{I}$ is a scale factor determined to reproduce the experimental values for rare gasses, and ${R}_{II}^{*}$ corresponds to the van der Waals radius of atom type I, which is defined using Eq. (10.8):

$$R_{II}^{*} = A_{I} \alpha_{I}^{1/4}$$

(10.8)

In Eq. (10.8), ${A}_{I}$ is a scale factor describing the outermost electron spread. Thus, many force field parameters of the vdW interactions depend on the outermost-shell electron configuration and are optimized with reference to rare gas experiments [26, 32]. Therefore, to add a new atom type, the atomic polarization factor α must be calibrated. In our series of studies on soft crystals, we determined the force field parameters for several metal ions that are central to them. Specific examples are outlined in Sect. 10.3.

2.3 Crystal Structure Prediction

Predicting the crystal structure of a molecule from its structural formula is known as crystal structure prediction [6, 12, 33]. The method of crystal structure prediction in CONFLEX is essentially the same as those employed by other groups, with the strategy of (1) generating possible crystal structures (Fig. 10.2a), (2) precise calculation of the relative energy of the crystal (Fig. 10.2b), and (3) selection of suitable predicted structures (Fig. 10.2c) [12].

Fig. 10.2

Crystal structure prediction algorithm: a Generate possible crystal structures, b calculate accurate relative crystal energies, and c select the predicted structure based on their energies (Reprinted (adapted) with permission from Ref. [12])

The first step is to generate trial crystal structures with independent parameters describing the crystal structure as variables. The unique crystal polymorph obtained by structural optimization is considered as a possible crystal structure. This procedure is called “crystal polymorphism search.“ As the search space for polymorphs is large, the reliability of crystal structure prediction depends on the ability to search efficiently without missing any possible stable polymorphs.

In the CONFLEX search for polymorphs, the three-dimensional structure is first constructed based on the two-dimensional structural formula of the target molecule, and the structure of the isolated molecule is subsequently optimized. If the target molecule is flexible, conformational search is performed by local deformation of the molecular structure using corner flap, edge flip, and stepwise rotation and by reservoir-filling algorithm to obtain stable conformational isomers [22,23,24]. From the obtained conformational isomers, one or more conformational isomers are selected in principle in the order of their energies, considering the steric structures and energies, and trial crystal structures are generated for each of them as follows. According to the CSD Space Group Frequency Ordering [34], ten different space groups (P2₁/c, P $\overline{1 }$, C2/c, P2₁2₁2₁, P2₁, Pbca, Pna2₁, Cc, Pnma, and P1) were determined to account for approximately 90% of organic crystals registered in the CSD. The choice of the space group must be flexible, and the symmetry of the target molecule and other factors must be considered as it determines the final crystal structure obtained [21]. The trial crystal structures generated are all subjected to crystal energy minimization and the unique optimized structure obtained is the possible crystal structure of the target molecule. We obtained a structure consistent with the experimental crystal structure [35]. We applied the method to ten different space groups, where the rotation angle of the molecular orientation was set to 20º increments for a single conformation. Approximately 100,000 trial crystal structures were generated. Although the crystal force field calculation can evaluate crystal energies faster than first-principles calculations, the computational load increases substantially owing to the expansion of the search space. Thus, it is important to develop more efficient search algorithms in the future.

In the second step, structures optimized by crystal force field calculations during the search are often re-evaluated for the relative energies between crystal polymorphs using ab initio calculations, especially the DFT-D method under periodic boundary conditions, which is considered superior. However, accurate evaluation is computationally expensive. Ideally, if the crystal structure optimized by the crystal force field calculation is sufficiently consistent with that of the DFT-D method, the large consumption of computation time can be avoided.

In the third step, we generally produce a ranking of the crystal polymorphs by crystal energy. It is desirable that the crystal structure observed in the experiment is consistent with the crystal polymorph with the lowest energy. However, even if we use the DFT-D method, we cannot be sure that the relative energies between the crystalline polymorphs are correctly evaluated. Indeed, in the fifth- and sixth-blind tests, the evaluation by the DFT-D method did not accurately predict the ranking of the crystal polymorphs, even though it could find structures consistent with the measured crystal structures [10, 11]. The accuracy of the evaluation by the crystal force field calculation was inferior to that by the DFT-D calculation. CONFLEX solves this problem by obtaining a ranking that considers the similarity of the crystal structures compared with the predicted crystal structure patterns when data of powder X-ray diffraction (PXRD) patterns are available, even if the crystal structures have not been determined. The ranking based on the similarity can be available for re-evaluation to be predicted polymorphic structure to produce a likely experimental structure [21].

3 Studies on Soft Crystals Using Crystal Force Field Calculations

In this section, we present our joint work on the application of crystal force field calculations to isocyanide-gold complexes, lanthanide (Ln) complexes, and disilane-bridged macrocycles belonging to soft crystals. For the gold isocyanide and Ln complex calculations, we have newly developed the necessary crystal force fields, which are also briefly introduced.

3.1 Force Field Parameters for Isocyanide Gold Complexes

The soft crystal phenomenon of phenyl(phenyl isocyanide)gold(I) (Fig. 10.3, R¹ = R² = H) synthesized by Ito et al. is very interesting that the two crystal structures of the isocyanide gold complex, namely Ib and IIy, induce a single crystal to undergo a single crystal phase transition from Ib to IIy simply by bringing them into contact with each other (Fig. 10.4) [36]. The blue Ib phase gradually changes to the yellow IIy phase upon ultraviolet (UV) irradiation. In addition, although the formed IIy does not revert to Ib by cooling, Ib can be obtained again by recrystallization in a solvent. Upon closer inspection of the difference between the two crystal structures, we notice that during the transition from Ib to IIy, a loose Au–Au bond is formed and the dihedral angle between the two Ph rings is changed. More interestingly, by introducing substituents at two ortho positions of this molecule, various combinations of luminescent colors can be observed before and after crystal rubbing [37].

Fig. 10.3

Pheny(phenylisocyanide)gold(I) and its derivatives: The bond between the phenyl group and Au is assumed to be covalent-like, while the Au···CN bond is a coordination bond

Fig. 10.4

Polymorphs of phenyl(phenylisocynide)gold(I) exhibiting Ib (upper) and IIy (lower) (Reprinted (adapted) with permission in Ref. [38])

To deal with this phenomenon using crystal force field calculations, the atomic type of Au was newly defined, the Au–C covalent and Au…CN coordination bonds between Au and C were considered (Fig. 10.3), and the force field parameters for Au, including bond stretching, angle bending, stretch-bend interactions, electrostatics, and vdW, for Au were developed [38]. The crystal structures Ib and IIy were optimized using the newly constructed crystal force field and the obtained lattice constants were compared with the measured values. These results are shown in Table 10.1. The difference in RMSD₂₀ [39, 40] from the experimental structure by superposition of 20 molecules in the crystal is 0.468 Å for Ib and 0.595 Å for IIy. The Au…Au distance for IIy (Fig. 10.5) is also reproduced relatively well.

Table 10.1 Experimental (Exptl.) and computational (Calcd.) lattice constants for phenyl(phenylisocynide)gold(I) complexes (Units are Å for a, b, c, and degrees for α, β, γ)

Fig. 10.5

Au···Au distances (Å) in experimental (upper) and computational (lower) crystal structures of IIy (Reprinted (adapted) with permission from Ref. [38])

The RMSD of the lattice parameters from the experimental data for the 27 structures (including Ib and IIy) with substituents at the ortho-position were determined to be 0.70 Å for the lattice lengths a, b, and c and 3.3° for the lattice angle. Therefore, the newly developed crystal force field reproduced the crystal structure of gold isocyanide complexes.

3.2 Force Field Parameters for Helicate Lanthanide Complexes

Among the rare earth elements, the Ln series elements exhibit a unique phenomenon called Ln contraction, in which the ion size decreases as the atomic number increases. The six-coordinated Ln complexes with luminescent properties synthesized by Hasegawa et al. (Fig. 10.6, hereafter referred to as LnL complexes) have a crystal structure in which ten types of Lns, namely Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, and Tm, adopt similar coordination structures [41, 42]. In these crystal structures, the distance between the N atom of the ligand ethylenediamine derivative and the central metal Ln changes regularly, which is one of the cases where the effect of Ln contraction is manifested. It is also interesting to note that the chiral helical structure formed by the coordinating ligand changes with Ln, which is a very interesting aspect of the applicability of this material.

Fig. 10.6

Molecular structure of the helicate complex, LnL

In the case of the isocyanide gold complex mentioned above, it was necessary to define both covalent and linear anisotropic coordination bonds between Au and C. However, in this LnL complex, we decided to model it merely as a free Ln ion [42]. In other words, only isotropic non-bonded interactions need to be considered to determine the interaction between the Ln ion and ligand. In addition, if the Ln has a formal charge of +3, the electrostatic interaction can be calculated conventionally, so the only new parameter required is in terms of the vdW interaction. Then, among the force field parameters for the vdW interaction of Ln metal ions, N and G in Eq. (10.7) and A in Eq. (10.8) should be the same values as for the iodine atom common to the Ln series. Thus, 1.404, 6.95, and 3.08 must be adopted for N, G, and A, respectively, according to the outermost electron configuration rules described in Sect. 10.2.3. Thus, the only force field parameter we need to determine is the atomic polarization factor α. The optimized α to reproduce the X-ray crystal structures formed by the ten Ln complexes are listed in Table 10.2 [42]. The vdW radii ${R}_{II}^{*}$ from Eq. (10.8) with the α of each Ln are smaller in order of atomic number and Ln contraction occurs.

Table 10.2 Force field parameters and van der Waals radii of Lanthanide ions

Comparing the structure-optimized crystal structures with the experimental structures using the new force-field parameters, all the optimized structures are in agreement with the experimental structures, ranging from 0.32–0.46 Å in the RMSD₂₀ and from 0.031–0.064 Å in the absolute error of the distance between Ln···L (Fig. 10.7) [42]. To further verify the robustness of the optimized force-field parameters, we compared the force-field-optimized crystal structures with the experimental structures for the four Eu derivatives that were not used as reference structures for optimization. The results show that the absolute errors of the RMSD₂₀ and Ln···L distances are 0.40–0.61 Å and 0.018–0.041 Å, respectively.

Fig. 10.7

Superimposed representation of observed (green) and optimized structures (yellow) of a series of Ln complexes with the helicate ligand L (Reprinted (adapted) with permission from Ref. [42])

This study is the first direct confirmation of Ln contraction by deriving a scientifically significant value of the vdW radius from the experimental structure. Although it is a recent trend to refer to high-precision first-principles calculations for determining force-field parameters, a large-scale computer is required to deal with this family of crystal structures by first-principles calculations. At present, the combination of experimental structure and crystal force field calculations is very useful for such a large system. It is also known that the Buffered 14–7 type function employed here to reproduce the vdW interaction is more accurate than the Lennard–Jones type function [26, 32]. As it can also be used in molecular dynamics simulations, it is expected to have applications in future developmental studies.

3.3 Prediction of the Vapochromic Crystal Structures of Ni(II)-Quinonoid Complexes

The Ni quinonoid complex crystal with a methyl group discovered by Kar et al. selectively adsorbs methanol vapor and changes its spin state to paramagnetic, and the solid exhibits vapochromism, changing its color from purple (MeP) to orange (MeO) (Fig. 10.8) [43]. Furthermore, the crystal with an ethyl group (EtP) is a purple solid as MeP and its structure and electronic state are similar to those of MeP. However, it does not adsorb methanol, as suggested by the experiment. The crystal structure of MeP is key to analyzing the methanol-selective adsorption mechanism but it has not yet been identified. Therefore, we attempted to construct crystal structure models of MeP based on the known crystal structures of MeO and EtP but we have not been able to find any crystal structure that is sufficiently consistent with the experimentally obtained PXRD patterns.

Fig. 10.8

Methanol vapor uptake of a Ni-quinonoid complex

Therefore, we created new force field parameters for Ni to reproduce the three-dimensional structures of these complexes and modified the parameters of the ligands. Using the crystal force field with these updated parameters, we searched for crystal polymorphs of MeP, assuming the same space group P1 as the crystal structures of MeO and EtP, and obtained 18 crystal structures (Fig. 10.9). These structures were further optimized using the DFT-D method (PBE-D3, PAW), and structures that are lower in crystal energy than those modeled from known crystal structures (MeO and EtP) were determined to be in good agreement with the PXRD patterns [44].

Fig. 10.9

Density-energy landscape of the polymorph search for MeP: A few crystal structures found in the crystal polymorph search are more common in energy and density than those derived from MeO (red) and EtP (green)

In this study, we show that if appropriate force field parameters can be obtained to reproduce the molecular structure of a complex and estimate the space groups that the crystal structure of the complex can assume, it is possible to identify the crystal structure of a complex that is difficult to analyze by single crystal X-ray analysis through a crystal polymorph search. Furthermore, by performing electronic structure calculations on the obtained structures, we succeeded in evaluating the energies of adsorption processes and analyzing absorption spectra [44]. We believe that this study may be an opportunity to establish a new scheme for vapochromic complexes, for which it is sometimes difficult to identify the crystal structure before and after adsorption, by searching for the crystal structure using the classical force field and analyzing the luminescence mechanism by electronic structure calculation.

3.4 Structural and Energetic Evaluation of the Thermosalience in Disilanyl Macrocycles

The thermosalient effect is a phenomenon in which a crystal piece changes its shape remarkably when the temperature changes. The analysis of the states before and after this phenomenon, which is caused by the instantaneous structural change of molecules in a crystal, is important for clarifying the mechanism of phase transitions. Omoto et al. synthesized a cyclic molecule C4 (Fig. 10.10) consisting of four p-phenylenes bridged by disilane and concluded that its crystal undergoes a thermosalient phenomenon because of a phase transition under low temperature conditions [45].

Fig. 10.10

Disilanyl macrocycle composed of four disilane (Si–Si) bonds and four p-phenylenes

In this system, the crystal structures of the C4α and C4β phases before and after the transition are known. The crystal structures of the C4α and C4β phases were optimized by CONFLEX crystal force field calculations while the respective lattice constants were maintained and the three-dimensional structures of the molecules in the crystals were reproduced. The energy difference between the two crystal structures is attributed to the intermolecular interactions. Furthermore, we optimized the crystal structure by stretching the lattice lengths a, b, and c separately and compared the crystal energies. This corresponds to the observation that the thermosalient phenomena are induced by crystal splitting along the (001) or (010) planes (Fig. 10.11) [45].

Fig. 10.11

a Changes in the intermolecular energy of the virtual crystals of C4α with an extension of the intermolecular distance. b Crystal packing with cell axes and c face-indexing graphics of C4α (Reprinted (adapted) with permission from Ref. [50]. Copyright (2020) American Chemical Society.)

In this system, two crystal structures converge to the same structure when the crystal force field is used for structural optimization, including the lattice parameter. This result differs from experimental observations. To reproduce their crystal structures more accurately, we need to further improve the force field parameters. In particular, the refinement of the crystal force field around Si atoms will be important because of the wide range of applications of compounds with carbon-silicon (C–Si) bonds [46].

3.5 Intermediate Formation by Pyridine Coordination for Lanthanide Complexes

The seven-coordinated Tb dinuclear complex [Tb₂(tmh)₆(4,4’-bpy)] (Tb-dinuclear), in which the green luminescent Tb ion is linked by a bipyridine (bpy), is converted under pyridine (py) vapor into an eight-coordinated Tb complex [Tb(tmh)₃(py)₂] (Tb-pyr2) via an intermediate of the eight-coordinated Tb complex [Tb(tmh)₃(4,4’-bpy)] (Tb-polymer), as revealed by X-ray crystal structure analysis (Fig. 10.12). Hasegawa et al. discovered this three-step deformation and succeeded in linking it with a similarly generated yellow-luminescent Dy complex via a mononuclear intermediate, Tb-pyr2 [47]. The discovery of the rare properties of this rare-earth complex, such as its ability to link and combine two (or more) molecular crystals and transmit optical information in one direction, is extremely important for new developments, such as organic transistors and organic electro-luminescences (ELs).

Fig. 10.12

Three-step deformation of the connected train-type Tb complex (Reprinted (adapted) with permission from Ref. [50])

This three-step transformation process contains several “obscurities”. For example, the pathway py uses to approach Tb in the crystal of Tb-dinuclear, how bpy leaves Tb at that time and where it goes, when py leaves after Tb-pyr2 formation, and what pathway bpy uses to re-approach Tb to form a Tb-polymer. We investigated the dynamic pathways of py and bpy in the crystal by kinetic simulations based on crystal force field calculations using the rare-earth force field parameters developed above (Sect. 10.3.3). However, it was determined that the periodic boundary condition cannot be used for the migration of solvent molecules in the crystal because of symmetry breaking. Thus, it is necessary to construct a sufficiently large virtual lattice with some computational restrictions. Unfortunately, such a large-scale dynamics simulation could not be performed in a limited time.

We reserved the intracrystal dynamics of py to examine how py approaches Tb-dinuclear (Fig. 10.13). We decided to examine the unique four-way pattern in which the first py approaches. We applied the DFT method to the py…Tb-dinuclear in each pattern and evaluated the stability of the py…Tb-dinuclear by structure optimization in pyridine solvent with Polarizable Continuum Model (PCM). The results showed that the eight-coordinate Type 1 structure, such as Tb-pyr2 and the Tb-polymer, is stable and py predominantly approaches from the opposite side of bpy to Tb (Table 10.3). The Type 1 + 4 structure, in which the second py is coordinated from the top of the bpy, is the most stable (Fig. 10.14). From these results, the coordination process of py to Tb-nuclear can be determined, as shown in Fig. 10.15, even though the dynamics of py in the crystal is unknown.

Table 10.3 Energy differences ($\Delta E$) of the first and second pyridine coordination for Tb-dinuclear

Fig. 10.13

Structures of one pyridine coordination approach to a Tb-dinuclear complex: a Type 1 is a pyridine approaching from the opposite side upward for Tb-ion, b Type 2 is from the opposite side downward, c Type 3 is from the same side downward, and d Type 4 is from the same side upward

Fig. 10.14

Structures of another pyridine coordination for a py-coordinated Tb-dinuclear complex: a Type 1 is a second pyridine approaching from the opposite side downward to Tb-dinuclear, b Type 2 is from the same side downward, and c Type 3 is from the same side upward

Fig. 10.15

Scheme of pyridine addition to the Tb-dinuclear complex

The computational cost of analyzing the migration processes and pathways of solvent molecules in molecular crystals in a solvent vapor atmosphere, such as vapochromism, is very high, even if the force field method is employed. It is important to devise a simpler computational model, as in the case of the Tb complex, which can be calculated without losing the essence of the problem through the ingenuity of the researcher. A well-conceived computational model will not only help us to solve the problem precisely but also amplify the interest of the research.

4 Perspectives on Computational Chemistry Methods for Molecular Crystals, Especially Soft Crystals

When soft crystals are dealt with computational chemistry, it is necessary to construct an appropriate computational model that describes how the external weak macroscopic stimuli that trigger the property change propagate into the crystal and what effects they have on the molecules. In the cases of the Ni(II)-quinonoid (Sect. 10.3.3) and Ln (Sect. 10.3.5) complexes, the existing crystal calculations could be applied to the macroscopic stimuli of “exposure to solvent vapor” by neglecting the dynamics of solvent molecules penetrating the crystals. In the case of the disilanyl macrocycle (Sect. 10.3.4), which exhibits thermosalient effects, the tolerance of the crystal structure to sudden volume expansion or contraction upon heating or cooling can be discussed in terms of the lattice length and crystal energy. The model was constructed within the range of available computational methods, and if the obtained results agreed with the experimental results within the error range, it implies that the validity of the calculation is verified. However, verification of the calculation requires some discussion on the validity of the assumptions, hypotheses, and approximations that are the premises that the calculation model is based on, such as neglecting the dynamics of solvent molecules in the crystal or the deformation of the unit lattice. In particular, as traditional hypotheses and approximations are sometimes dependent on the computational capabilities of time, a bold update of these assumptions and approximations to match the current computational capabilities of current, large-scale, high-speed computers may lead to a major breakthrough.

4.1 Domino-Transformation of Isocyanide Gold Complexes

If the transition process between two single crystal polymorphs by structural phase transition is known, it will contribute to the construction of a new molecular design theory, such as control of the phase transition, improvement of the mutating properties, etc. In particular, if two crystal polymorphs are determined, the transition process may be inferred from their comparison. However, in the case of the mechanochromic isocyanide gold complexes described in Sect. 10.3.1, the two crystal structures are quite different and the changes and relationships between the polymorphs require sophisticated analysis techniques to be observed and interpreted accurately.

The metastable Ib of gold isocyanide complexes is triclinic $P\stackrel{-}{1,}$ while the most stable IIy is tetragonal $I\overline{4 }2d$ (Fig. 10.4) [36]. Therefore, the structural transformation from Ib to IIy cannot be explained by a simple conformational change or molecular reorientation. Ito et al. suggested that the thermal phase transition may proceed via a mechanism similar to the proposed epitaxial mechanism [48]. This phase transition mechanism induced by mechanical stimuli is known as a “domino-like transformation” (Fig. 10.4 in ref. 37). The mechanism is based on the assumption that mechanical and thermal stimuli induce microcracks. The nucleation of the daughter phase (IIy) occurs in the microcracks parallel to the cleavage plane of the parent phase (Ib), and the molecules continuously move (like dominoes) from the opposite side of the microcracks to form the oriented nuclei of the daughter phase. Then, the more stable daughter phase grows inside the metastable parent phase.

In general crystal calculation models, imposing three-dimensional periodic boundary conditions can reasonably reduce the computational complexity, thereby allowing calculations to be performed within the available time. However, to construct a computational model according to this hypothesis, it is necessary to create a virtual lattice that is large enough to allow cracks to form inside the crystal and perform the calculation under a two-dimensional periodic boundary condition. This can be accomplished using the atomic and molecular level dynamics simulations employed in the field of tribology for material surface phenomena. However, it would be computationally more expensive and large-scale than the three-dimensional periodic boundary conditions.

We departed from the hypothesis of a crack space in the crystal and focused on the thermal phase transition from Ib to IIy that occurs at 64.5–74.9 °C, assuming the repetition of many polymorphic interphase transitions, which have not been observed but are expected to be present. Forming period-free regions in the crystal with our available crystal calculation methods is challenging. Therefore, we developed a crystal dynamics method that can apply vibration vectors as heat by modifying the developed crystal structure optimization method.

We performed lattice-variable crystal dynamics simulations to follow the transformation process from Ib to IIy (Fig. 10.16) and generated a virtual lattice model in which the unit lattice of Ib is extended by a factor of 8 (2 × 2 × 2), 16 molecules are set to fit in the unit lattice as in IIy, and the space group is set to P1 to allow all degrees of freedom. The crystal dynamics simulation results show that following the deformation of the crystal structure along the lattice vibration modes of Ib, the crystal energy $\Delta E$, the unit cell volume %CELL, and the rate of change (%) of the lattice constants a, b, c, $\mathrm{\alpha },\upbeta ,\mathrm{ and ~\gamma }$ lead to a new polymorph III to appear in the region beyond the initial energy barrier and eventually return to Ib. By repeating the same procedure for other vibrational modes and new polymorphs, we may find stepwise pathways from Ib to IIy via multiple polymorphs (Fig. 10.16). Unfortunately, we have not been able to reach IIy from Ib presently but will publish the details and results of this methodology.

Fig. 10.16

Molecular crystal dynamics for the Ib form of an isocyanide gold complex: Following the changes in the crystal structure along the lattice vibrational modes of Ib, the potential energy (ΔE, bottom blue line), unit cell volume (%CELL, top line), and lattice constant also change, and new polymorphs of III appear in the intermediate region. Repeating this vibrational mode may help to find a transformation pathway to IIy

4.2 Approach to Organic Superelasticity of Terephthalamide

The organic superelasticity of terephthalamide, which was discovered by Takamizawa et al., is considered a soft crystal because it responds to macroscopic external stimuli. A single crystal phase fixed at one end is subjected to shear forces by pushing in the opposite end. When a shearing force is applied to an $\alpha$ phase with one end fixed by pushing in the opposite end, the intermediate region of the single crystal deforms and another single crystal $\beta$ phase appears (Fig. 10.17). When the force is unloaded, the single crystal returns to its original $\alpha$ phase. From close observation of the shear surface of the single crystal, it was determined that the $\beta$ phase gradually appears along the shear plane (Fig. 10.18) [4, 5, 49].

Fig. 10.17

Crystal structures of terephtalamide: α-form (left) and β-form (right) (Reprinted (adapted) with permission from Ref. [50])

Fig. 10.18

Shear deformation of a terephthalamide crystal (Reprinted (adapted) with permission from Ref. [50])

This phenomenon is different from any of the aforementioned soft crystal phenomena. The reason is that the $\beta$ phase appears stably at room temperature in a shear deformation state, namely a state of continuously applied stress, and the stress inside the crystal in the $\beta$ phase should have the same magnitude of shear stress applied externally and in the opposite direction [50]. To obtain the crystal structure under high pressure, the crystal enthalpy is defined in the CONFLEX crystal calculations (Eq. (10.9)).

$$H_{{{\text{crystal}}}} = E_{{{\text{crystal}}}} + pV$$

(10.9)

where p is the pressure and V is the unit lattice volume. The crystal structure at a given pressure p can be obtained by performing structural optimization so that ${H}_{\mathrm{crystal}}$ (Eq. (10.9)) is minimized. Thus, a crystal structure can be obtained at a specified pressure p. Generally, under atmospheric pressure, the pV term can be neglected. However, under very high pressures, the effect of this term on the crystal structure is large.

Pressure, including atmospheric pressure, applies isotropic compressive (expansive) stress to the unit lattice and other stresses are anisotropic, which include tensile, compressive, bending, and torsional besides shear. From the macroscopic mechanical property tests of solid materials, we can deduce the type of deformations. To describe the state under general stresses, including pressure, the microscopic work per unit lattice associated with the microscopic deformation of the crystal structure $\Delta W$ is defined by Eq. (10.10):

$$\Delta W\left( {\sigma , \varepsilon V} \right) = \mathop \sum \limits_{ij} \sigma_{ij} \varepsilon_{ij} V$$

(10.10)

where i and j denote the spatial coordinate axes x, y, and z, ${\sigma }_{ij}$ is the stress tensor, and ${\varepsilon }_{ij}$ is the strain tensor. The stress tensor ${\sigma }_{ij}$ is decomposed into an isotropic stress term and a traceless (TL, sum of diagonal terms is zero) term, which is a shear stress, as in Eq. (10.11).

$$\sigma_{ij} = {\text{p}}\delta_{ij} + {\text{q}}\sigma_{ij}^{{{\text{TL}}}}$$

(10.11)

The coefficient p corresponds to the pressure and q indicates the magnitude of the shear stress. In addition to shear stress, the traceless term can include volume-invariant vertical and rotational deformations, which can be combined to add various deformation stresses to the crystal structure. Thus, it considers the work performed by the externally applied stresses $\Delta W$. The enthalpy of a crystal, considering the work done by such externally applied stresses, can be defined as

$$H_{{{\text{crystal}}}} = E_{{{\text{crystal}}}} + \Delta W\left( {\sigma , \varepsilon V} \right)$$

(10.12)

If this equation holds, the crystal structure under external stress can be obtained by optimizing the crystal lattice so that ${H}_{\mathrm{crystal}}$ is minimized.

This methodology was applied to terephthalamide superelastic crystals. To define the difference in the number of molecules between the unit lattice and the disordered phase, the α phase was constructed as a virtual unit lattice (supercell) containing four molecules (2 × 2 × 1) and the β phase as 1 × 1 × 1. The disordered phase is defined as the phase in which all four molecules have different conformations. Figure 10.19 shows the phase diagram when the crystal structure is optimized for different pressures (isotropic compressive stress) p and anisotropic shear stress q. Although the calculation is not refined, the phase transition from the α to β phase occurs simply by optimizing the crystal structure with a stress q of approximately 0.6. Figure 10.20 also shows the stress–strain diagram. The abscissa is the rate of volume change of the supercell (strain tensor norm) and the ordinate is the applied shear stress q. Similar to the pressure–stress phase diagram described above, the β phase appears at a q of approximately 0.6, and it can be seen that the crystal order is destroyed by applying only a small amount of stress above that value.

Fig. 10.19

Pressure (p)–stress (q) phase diagram of terephthalamide crystals: α, β, and disordered phases are represented by the red, blue, and yellow areas, respectively (Reprinted (adapted) with permission from Ref. [50])

Fig. 10.20

Stress–strain diagram of terephthalamide crystals, where the pressure (p) was fixed at 0.3. α, β, fracture phases are represented by the red, blue, and yellow curves, respectively (Reprinted (adapted) with permission from Ref. [50])

Such mechanical properties are used for quality testing performed when solid materials are made into market products and are well established in the field of mechanical engineering as macroscopic simulations, such as continuum approximations and finite element methods. The properties of soft crystals as solid materials are expected to be applied to Micro Electro Mechanical Systems (MEMS) and nanomachines. In addition, computational methods for deriving atomistic mechanical properties, such as those presented in this chapter, could become essential in the future.

5 Conclusion

In this chapter, we described studies on computational chemistry that are useful for establishing the scientific theory of soft crystals. The contribution of the CCDC crystal structure prediction blind test has led to the improvement of the accuracy of first-principles calculations for molecular crystals. In particular, crystal structures optimized by the DFT-D method under periodic boundary conditions are more accurate than results obtained experimentally. However, the accuracy of the relative crystal energies among many crystal polymorphs, which is characteristic of molecular crystals, is not sufficient even with the DFT-D method, and further improvements are required to make it practically available. It is expected that the seventh blind test, which is now in progress, can predict crystal structures and energies with even higher accuracy. However, the computational cost of first-principles calculations is still high. To determine crystal polymorphs and crystal calculations of large molecules, we can rely on crystal force field calculations based on classical mechanics to accurately evaluate the crystal structure within the available computational time.

With regard to molecular crystals studies using crystal force field calculations, we introduced several collaborative works in which we applied our CONFLEX crystal force field calculations to various soft crystals. For isocyanide gold, Ln, and nickel complexes, we developed new force field parameters around metal ions, which are missing in the crystal force field, to enable crystal structure predictions and crystal dynamics simulations. In the application of the disilane-bridged macrocycle, the triggering of the thermosalient effect was analyzed in terms of the energy increment associated with the lattice length elongation. For the three-step deformation of luminescent Ln complexes triggered by pyridine vapor spraying, the mechanism of the two-step pyridine-added coordination of pyridine was clarified by devising a computational model.

A common property of many soft crystals is that structural phase transitions, such as single-crystal-to-single-crystal phase transitions, are induced by weak external stimuli, resulting in significant changes in their characteristic properties. It is difficult to analyze all these phenomena by existing computational chemistry methods. Thus, it is necessary to develop new methodologies. In this chapter, we reviewed the crystal dynamics simulations of mechanochromic structural phase transitions in gold isocyanide complexes. Although it has not been clarified whether this phenomenon is epitaxial or multi-step structural phase transition, we believe that crystal dynamics simulation will be an important method to expand its application in the future. A stress-applied crystal structure optimization method for organic superelasticity of terephthalamide, in which the structural phase transition is induced by a continuous external force rather than an instantaneous external stimulus, is presented. This method extends the typical methodology for isotropic stresses, such as pressure, to handle anisotropic stresses. Although quantitative results have not been obtained, the method is expected to be applied to mechanical deformation tests of molecular crystals as solid materials.

In addition to the examples discussed here, various other soft crystals can be analyzed using computational chemistry applications. In particular, computational chemistry is expected to establish the science of soft crystals with many interesting properties, such as the internalization process of small molecules into porous crystals, the deformation mechanism of molecular machines, the properties of organic electronic devices, and anisotropic phonon dispersion analysis. For this, it is necessary to improve the existing methodologies and to further develop new methodologies, as well as to expect the availability of faster and cheaper computers.

This work was supported by Grant-in-Aid for Scientific Research (17H06373, 21K05105) from the Japanese Society for the Promotion of Science.