Relative Position and Relative Rotation in Supramolecular Systems through the Analysis of the Principal Axes of Inertia: Ferrocene/Cucurbit[7]uril and Ferrocenyl Azide/β-Cyclodextrin Case Studies

Abstract

Parameters comprising the relative position and relative rotation of molecules can be evaluated when the principal axes of inertia of the entities in a supramolecular association are employed as reference. Such information applies to the characterization and identification of experimental and theoretical nonbonded systems. The parameters are relevant to geometric comparison (for theory and experiment) and, for instance, to monitoring structures by theoretical simulations. This work introduces a software developed to obtain such parameters through the discussion of some intriguing host–guest systems, the ferrocene/cucurbit[7]uril and ferrocenyl azide/β-cyclodextrin. The ideas within this contribution naturally apply to the study of other nonbonded associations beyond host–guest chemistry. A modified version of the software discussed herein serves to obtain user-defined spatial arrangements for two nonbonded entities. Therefore, with a given geometry, for instance, from X-ray data, the parameters can be derived, and with the parameters, from a theoretical perspective, a spatial arrangement can be obtained.

Introduction

There are many ways to describe the geometry of a given supramolecular system. Also, for each experimental or theoretical contribution in this field, the authors can provide the Cartesian coordinates of all arrangements under investigation as the Supporting Information. On the other hand, as will be discussed herein, it may be useful to establish a simple set of geometric descriptors easy to evaluate and reproduce, applicable for identification and discussion of methodology, and to avoid the need to make available all of the Cartesian coordinates within a contribution. In the field of host–guest chemistry, we regularly face issues related to the identification of the systems under study.

According to an impressive contribution of Jeon and co-workers,¹ the crystallographic data of the host–guest system formed between ferrocene (Fc) and a cucurbituril,² specifically CB[7], attest the existence of two independent arrangements for the compound identified as Fc@CB[7] in the solid state. According to X-ray data, the guest molecule resides inside the host cavity with iron atoms close to the geometric center with distinct inclinations.¹ After the publication of the experimental work, the Fc@CB[7] host–guest systems, among others, were theoretically investigated by Pinjari and Gejji.³ In their important contribution, the distinct spatial arrangements were identified as parallel or perpendicular.

The theoretical investigation attested the reliability of Lee, Yang, and Parr’s (B3LYP) correlation functional^4,5 since the predicted geometries of these inclusion compounds agree well with the crystal structures.¹ Inspired by the experimental¹ and theoretical³ contributions, we can additionally suppose the existence of other distinct spatial arrangements for the Fc inside the CB[7] cavity. In this sense, how can we expand the classification (parallel or perpendicular) to account for other spatial arrangements for the Fc@CB[7] host–guest system?

In another interesting host–guest study, Walla and co-workers⁶ investigated the system formed by ferrocenyl azide (FcN₃) and β-cyclodextrin (β-CD).⁷ According to the authors, the FcN₃ acquires two distinct overall arrangements named equatorial or axial, depending on its relative position inside the β-CD cavity. The stability of such spatial arrangements, in solution, varies depending on the solvent and the temperature.⁶ Again, we face a very intriguing system. A theoretical study can be conducted to define an appropriate approach to model such a type of system that will result in the identification of the Cartesian coordinates of the most favorable arrangement (equatorial or axial) in distinct conditions. However, due to the flexibility of a host–guest system, equatorial or axial refers to a set of supramolecular systems. Within this context, how to address the representative geometries compatible with the proposed experimental identifications?

The excellent processor capacity of modern computers allows us to explore the complex potential energy surface (PES) of a particular host–guest system under study. In host–guest chemistry, this can be done through docking,⁸ molecular dynamics (MD),⁹ quantum mechanics (QM),¹⁰ or hybrid methods such as QM/molecular mechanics (MM) simulations¹¹ or ONIOM,¹² among others. The methodologies can be applied in isolation or in conjunction. For instance, the docking approach can provide some starting association to carry out a molecular dynamics simulation. Among such methods, it is well established that quantum mechanics gives accurate geometries and thermodynamic quantities for the molecular systems under investigation.

The choice of a particular methodology depends on several factors and, fundamentally, the computational cost related to the study of a specific type of system. The flexibility of the host–guest compounds implies several starting associations in modeling. Besides, for cyclodextrin-based host–guest systems, the existence of distinct rims (head and tail) implies additionally at least two inclusion modes in the study. The considerable number of starting arrangements to be considered in a theoretical investigation gives rise to a challenging issue related to the identification of each one and the reproducibility of a given contribution. This issue may not depend on the chosen methodology (docking, molecular dynamics, QM/MM simulations, ONIOM, or other molecular approaches).

Within this work, the ferrocene/cucurbit[7]uril and ferrocenyl azide/β-cyclodextrin systems were employed as case studies. The main idea of this paper is to discuss general spatial descriptors to account for the geometry of noncovalently bonded associations. Additionally, this work also introduces a software written in Fortran programing language (available for any researcher) designed to obtain such geometric descriptors for a given supramolecular system. A modified version of the software discussed herein applies to get Cartesian coordinates of a user-defined supramolecular association. The ideas discussed herein improve the discussion in the field of supramolecular chemistry because with a given geometry, for instance, from X-ray data, the parameters can be obtained, and with the parameters, from a theoretical perspective, a spatial arrangement can be obtained.

Computational Methods

Fortran Implementation

The parameters discussed herein take into account the relative position and the relative rotation between two molecular entities in a supramolecular system. The written code in Fortran language reads a simple text file containing the Cartesian coordinates of two molecules: the host, the heavier, and the guest, the lighter. The coordinates of the center of mass of each molecule determine their relative distance (r). The definition of two right-handed Cartesian systems, each one mounted over the center of the mass of each molecular unit, determines the θ polar angle; the φ azimuthal angle; and the α, β, and γ Euler rotation angles for the guest with respect to the host molecule. The first step in the implementation consists of the construction of two tensors of the moment of inertia, one for each molecule of the supramolecular system under investigation. For each unit (host and guest), with an internal coordinate system centered in its center of mass, the diagonal elements of the tensor of the moment of inertia are evaluated by eqs 1–3, for which m_i accounts for the mass of the ith atom.

1

2

3

The nondiagonal elements of the tensors for each molecule are evaluated by eqs 4–6.

4

5

6

The diagonalization of the tensor gives the principal inertial axes for each molecule.¹³ In this procedure, the LAPACK¹⁴ diagonalization subroutine DSYEV was applied to obtain the eigenvectors of the tensor of the moment of inertia. The eigenvectors define the principal A, B, and C inertial axes. The related eigenvalues correspond to the inertial moments I_A, I_B, and I_C employed, for instance, in the evaluation of the rotational constants for each molecule, with I_C > I_B > I_A. The parameters related to the angular characterization of the supramolecular system depend on the coordinate systems mounted in the center of the mass of each molecule.

Within this work, the Fortran implementation evaluates the parameters assuming the existence of two coordinate systems, X, Y, Z, and x, y, z, named herein the reference system and derived system, respectively. The reference system (X, Y, and Z) corresponds to the coordinate system mounted over the center of the mass of the heavier molecule (the host). The derived system (x, y, and z) corresponds to the coordinate system mounted over the center of the mass of the lighter molecule (the guest). For the host molecule, the Z-axis coincides with the eigenvectors associated with the higher inertial moment (I_C). For the guest molecule, the z-axis corresponds to the eigenvectors associated with the smaller inertial moment (I_A). Both Z- and z-axes and the corresponding right-handed coordination systems were mounted over each center of mass.

At this point, a discussion concerning the definition of the positive direction of both reference and derived systems is required. The diagonalization procedure of the tensor of the moment of inertia gives the eigenvectors and the corresponding eigenvalues (moments of inertia for each principal axis). Three eigenvectors are promptly obtained through the application of the diagonalization procedure.¹⁴ It is noticeable that the opposite eigenvectors are also solutions to the eigenvalue problem related to the diagonalization of the inertial tensor. Therefore, six eigenvectors are evaluated through the diagonalization procedure. The principal axes of inertia do not determine any preferable direction. Consequently, it is not possible at first glance to determine, in a positive mathematical sense, instructions to construct the X, Y, and Z right-handed reference system. The same is valid for the derived system (x, y, and z).

Some molecules such as CB[7] are highly symmetric, and there is no preference for the evaluation of a particular positive direction coherent with a right-handed X, Y, and Z reference system. Other molecules, such as cyclodextrins (CDs), are not so symmetric. Such class of carbohydrates presents two rims, one comprising the primary hydroxyl groups (tail) and another containing the secondary hydroxyl groups (head). There is a distinction between the inclusion through the head or tail portion. How to differentiate such an association from a geometric perspective?

Heteroatomic molecules possess a center of mass for a given group of atoms (atoms with the same atomic number). Native CDs are heteroatomic molecules constructed with hydrogen, carbon, and oxygen atoms. The center of mass of the group of atoms with the higher atomic number, the oxygen atoms, can be employed as a reference to determine some preferable directions to mount the reference system. The convention adopted and implemented within the Fortran software APARM¹⁵ introduced in this work assumes a positive direction in a mathematical sense for the one compatible with the small angle evaluated between a particular eigenvector along an inertial axis and the center of mass for the set of atoms with the higher atomic number.

Within this work, the following colors were used: blue for the I_A axis (or eigenvector direction), green for the I_B axis (or eigenvector direction), and red for the I_C axis (or eigenvector direction), with I_A, I_B, and I_C being the moments of inertia of a molecule. Within this convention, for an asymmetric top, I_C > I_B > I_A. For the hosts CB[7] and β-CD and the guests Fc and FcN₃, Figure 1 shows the corresponding eigenvectors (with their proper colors) for each compound. Due to their symmetry, the eigenvector directions for CB[7] and Fc may vary, assuming supplementary angles. It is noticeable that the eigenvector point toward the smaller moment of inertia (I_A, blue vector) for FcN₃ is not precisely passing through the geometric center of the two rings of the molecule such as for the Fc (Figure 1).

Figure 1.

Eigenvectors employed in the construction of the reference system (XYZ) for the hosts CB[7] and β-CD and the derived system (xyz) for the guests, Fc and FcN₃. Cartesian coordinates and arrows.bild (file with axes) obtained from the Fortran software UD-APARM.¹⁵ Colors adopted herein: blue for I_A, green for I_B, and red for I_C eigenvectors assuming that I_C > I_B > I_A for an asymmetric top. The three-dimensional (3D) molecular models were obtained from UCSF Chimera.¹⁶

All possible combinations of the eigenvectors of the tensor of the moment of inertia with the corresponding vector pointing from the origin of the reference system (X, Y, and Z) toward the center of mass of the set of atoms with a higher atomic number (higher-Z-CM) are considered in the Fortran implementation. A similar analysis is also implemented for the derived system (x, y, and z). This procedure is not quite simple and is responsible for roughly 2700 lines of code in the Fortran implementation. Such a distinction is attributed to a comparison between the angle formed by one eigenvector and the higher-Z-CM direction and the opposite eigenvector and the higher-Z-CM direction. When a difference between those angles is greater than 10°, the flag “Y” is attributed to that particular axis of inertia. For the three Cartesian axes, eight possibilities can be addressed: NNN, YNN, NYN, NNY, YYN, YNY, NYY, and YYY. For an axis with “Y” attribution, the smaller difference in degrees between one eigenvector and the higher-Z-CM direction will determine the positive Z- or z-axis for both the reference and the derived systems, respectively, which guarantees the evaluation of the Z or z positive direction independently of the spatial orientation of the reference molecule (the host), for instance.

This analysis is crucial to establish universal parameters to characterize the association of a particular class of molecules. For instance, such evaluation will determine the positive Z-axis for CDs essential to distinguish the inclusion by the head or by the tail cavity portion of such a type of molecule. No matter the orientation of the CD, the positive Z direction will always point toward the secondary hydroxyl groups, the head portion of the cavity of the standard CDs (Figure 1). Attribution of the “Y” flag is also printed in the output “LOG-PARAM-ONLY.txt” after the execution of the APARM Fortran software.¹⁵ The output contains much other information, such as the determination of the higher-Z-CM, evaluation of the moments of inertia, and the rotational constants for each of two molecules in a supramolecular system.

As discussed, a similar procedure is carried out for the construction of the derived system. Once the reference (X, Y, and Z) and the derived system (x, y, and z) are defined, the parameters are evaluated. When not applicable, the right-handed coordinate system is mounted through the use of the eigenvectors obtained along with the LAPACK implementation without the mass distribution analysis, which takes place for highly symmetric molecules such as CB[7]. To allow reproducibility, for highly symmetric molecules such as CB[7], the definitive evaluation of the preferable axes to mount the reference system takes place after the alignment of the principal axes of inertia of the molecule under study and the Cartesian axes. The center of mass of each molecule is used to determine the relative position parameters related to distance (r) and polar (θ) and azimuth angles (φ). Within this work, the positive sign for the azimuth angle (φ) is measured in the counterclockwise sense from the reference direction X on the reference plane XY mounted over the center of mass of the host molecule (the origin). The modification of such parameters, from a supramolecular perspective, is shown in Figure 2.

Figure 2.

Eigenvectors employed in the construction of the reference and derived systems for the association of β-CD and phenol showing the variation in the supramolecular parameters for relative positions (r, θ, and φ). Cartesian coordinates and arrows.bild (file with axes) obtained from the Fortran software UD-APARM.¹⁵ The 3D molecular models were obtained from UCSF Chimera.¹⁶

The relative Euler angles (α, β, and γ) are evaluated after the definition of the derived system. The definition of the relative rotation though Euler angles is not unique. Within this work, as implemented in APARM and UD-APARM,¹⁵ the rotation matrix defined in the Arfken textbook¹⁷ was used, assuming, as explained, two right-handed coordinate systems mounted in each center of mass of the molecules in a given supramolecular association. The Euler angles (α, β, and γ) employed within this work are shown in Figure 3.

Figure 3.

Eigenvectors employed in the construction of the reference and derived systems for the association of β-CD and phenol showing the variation in the supramolecular parameters for Euler angles (α, β, and γ). Cartesian coordinates and arrows.bild (file with axes) obtained from the Fortran software UD-APARM.¹⁵ The 3D molecular models were obtained from UCSF Chimera.¹⁶

The flowchart for the evaluation of the supramolecular parameters for relative positions (r, θ, and φ) and relative rotations (α, β, and γ) implemented in APARM is illustrated in Figure 4.

Figure 4.

Flowchart for the evaluation of the supramolecular parameters for relative positions and rotations implemented in the APARM¹⁵ Fortran software.

To evaluate the Euler angles, the eigenvectors of the reference system (X, Y, Z), mounted over the center of mass of the host (the heavier molecule), rotate to coincide with the eigenvectors of the derived system (x, y, z) mounted over the center of mass of the guest (the lighter molecule). The Z–Y–Z order of rotation was adopted. The same rotation matrix implemented in the WSolid 1 (solid-state NMR simulations) software¹⁷ and described in the Arfken textbook¹⁸ was used within such procedure.

As also pointed out by Prof. Klaus Eichele, the developer of the WSolid 1, the Z–Y–Z rotation order is adopted in many essential references such as Arfken,¹⁸ Spiess,¹⁹ Mehring,²⁰ Narita et al.,²¹ Freude and Haase,²² and Dye et al.²³ The data concerning the supramolecular parameters were obtained through the APARM Fortran software¹⁵ introduced in this work. Other Euler angle conventions may be adopted in future implementations. One of the critical ideas of this work consists of the use of the axes of inertia of the molecules to construct the reference and derived system in a supramolecular system. The implementation may vary. There are many variations to treat Euler angles. Therefore, the present work is not exhaustive.

Quantum Mechanical Calculations

All host–guest systems investigated herein were fully optimized without any constraint with Lee, Yang, and Parr’s (B3LYP) correlation functional.^4,5 In all quantum mechanical calculations, the 6-31G(d) basis set²⁴ was employed. All calculations were carried out with the Gaussian-09 computational chemistry software package.²⁵ All studied associations were subjected to high-computational-cost harmonic frequency calculations at the B3LYP/6-31G(d) level of theory. The starting geometries for Fc@CB[7] were constructed from the crystallographic data previously reported.¹ The associations for FcN₃@β-CD were constructed with the crystallographic information for FcN₃⁶ and β-CD.²⁶ Within this work, the two modes of inclusion were investigated as starting associations for the inclusion compound formed with β-CD. Therefore, four supramolecular FcN₃@β-CD starting associations were investigated. All quantum mechanical calculations were performed in vacuum conditions.

Within this work, the two crystallographic structures were initially subjected to unconstrained optimization with the LOOSE keyword. According to the Gaussian-09 manual, the use of such a keyword is appropriate for initial optimizations of large molecules using density functional theory (DFT) methods. Due to some difficulty in the optimization with systems comprising CB[7] molecules, the LOOSE keyword was employed initially. In a subsequent step, the optimized spatial arrangements constructed from the crystallographic information were resubjected to another optimization without the use of the LOOSE keyword. At this stage, two modifications of the crystal arrangements were also subjected to unconstrained optimizations. The starting Fc@CB[7] associations were identified herein as 1, 2, 3, and 4, and the corresponding optimized geometries (the derived geometries) were identified as 1′, 2′, 3′, and 4′. For the supramolecular system formed with FcN₃ and β-CD, four inclusion compounds identified herein as 5, 6, 7, and 8 were subjected to unconstrained optimization. For such structures, the LOOSE keyword was not applied. Axial and equatorial arrangements compatible with tail and head inclusion modes were investigated. The optimized associations (the derived structures) were identified herein as 5′, 6′, 7′, and 8′. After the optimization, the 5′ association was not addressed as an inclusion compound due to the distance between the host and guest.

Results and Discussion

Relative Energy Analysis

According to the experimental finding, Fc@CB[7] was identified only with two spatial arrangements in the solid state.¹ The theoretical investigations of the systems formed with Cucurbit[n]urils (n = 5–8), ferrocene (Fc), and their complexes,³ specifically for Fc@CB[7], focused their discussion only on two such possibilities named parallel and perpendicular. For the CB[5] and CB[6] associations with Fc, other spatial arrangements beyond such arrangements were investigated.³ Beyond the parallel or perpendicular identifications, other associations were found as stationary points on the potential surface at the B3LYP/6-31G(d) level of theory, as attested by the data reported herein. The absolute (E), the relative energies (ΔE), and the corresponding first three harmonic frequencies for Fc@CB[7] associations are shown in Table 1.

Table 1. Electronic Energies (E), Relative Energies (ΔE), and Three Smaller Harmonic Frequencies (ω1, ω2, and ω3) Evaluated at the B3LYP/6-31G(d) Level of Theory for the Fully Unconstrained Optimized Fc/CB[7] and FcN₃/β-CD Associations.

system	ID starting identification	E (hartrees)	ΔE (kcal mol–1)	ω1 (cm–1)	ω2 (cm–1)	ω3 (cm–1)
Fc/CB[7]	1′ crystal (parallel)	–5862.9562283	6.40	15.8145	16.1764	21.5964
	2′ other parallel	–5862.9664331	0.00	11.9238	16.0264	18.6967
	3′ crystal (perpendicular)	–5862.9613383	3.20	17.8381	20.0143	26.1076
	4′ other perpendicular	–5862.9612184	3.27	22.9483	26.8588	32.2308
FcN3/β-CD	5′ axial (head)	–6089.4803391	0.13	9.4693	12.8956	14.6314
	6′ axial (tail)	–6089.4756953	3.04	8.5941	12.3381	16.7330
	7′ equatorial (head)	–6089.4805410	0.00	8.6073	14.1233	15.0274
	8′ equatorial (tail)	–6089.4791489	0.87	5.9839	11.0637	13.1312

The data in Table 1 for Fc/CB[7] attests that all optimized associations correspond to true minima in the potential energy surface at the B3LYP/6-31G(d) level of theory (all frequencies are positive real numbers). Furthermore, the absolute values in hartrees (E) attest that all Fc/CB[7] host–guest compounds are distinct, which is also supported by a set of frequencies in Table 1. It is noticeable that the most stable arrangement in vacuum conditions, the association 2′, was not derived from the optimization of some experimental X-ray structure. Interestingly, both structures derived from the solid-state arrangement present relatively higher energies of 3.20 and 6.40 kcal mol^–1 greater than that of the most stable association 2′. As will be discussed in the Geometrical Analysis section, such a stable arrangement was not previously reported for Fc/CB[7] and surprisingly corresponded to a type of parallel association. The comparison between the stability of a particular arrangement in the solid state and gas phase can be debatable, and such analysis is beyond the scope of this work. For the FcN₃/β-CD optimized supramolecular systems, the harmonic frequencies in Table 1 also attest that all optimized associations correspond to true minima in the potential energy surface at the B3LYP/6-31G(d) level of theory. Frequency and energy values (E) indicate that all optimized associations are distinct spatial arrangements, as will also be discussed in the Geometrical Analysis section. According to Table 1, the most favorable arrangement in the vacuum condition corresponds to the association 7′, which is an arrangement slightly more favorable than association 5′. We conclude that the equatorial starting mode gives the corresponding FcN₃/β-CD the most favorable association in vacuum conditions at the B3LYP/6-31G(d) level of theory. The comparison between the energy data for both distinct associations, Fc/CB[7] and FcN₃/β-CD, indicates smaller differences in relative energies for FcN₃/β-CD associations. Finally, the energy values (E) in Table 1, in conjunction with the overall geometries (as will be discussed), attest that 1′ and 4′ correspond to the inclusion compounds previously reported by Pinjari and Gejji.³

Weak Interaction Analysis

To analyze the weak interactions that stabilize the supramolecular compounds studied herein, the most stable associations named 2′ Fc@CB[7] and 7′ FcN₃@β-CD (see Table 1) were investigated through the calculation of quantum theory of atoms in molecules (QTAIM),²⁷ noncovalent interaction (NCI),²⁸ and natural bond orbital (NBO).²⁹⁻³¹ The data were obtained through the Gaussian-09 computational chemistry software package²⁴ in conjunction with Multiwfn, a multifunctional wave function analyzer.³²

The relevant interactions that stabilize the most stable systems under investigation can be visualized in terms of the molecular graphs that include the (3,–1) critical points (CPs) and bond paths, as obtained from QTAIM analysis (Figures S1 and S2, Supporting Information). As illustrated in Figure S1, for 7′ FcN₃@β-CD, intramolecular and intermolecular hydrogen bonds are promptly identified due to the atoms involved along the bond paths. Other intermolecular interactions can also be identified in Figure S1. For 2′ Fc@CB[7], the (3,–1) critical points and bond paths also suggest the existence of many intermolecular interactions between the host and guest. Due to the type of atoms, such interactions cannot be regarded as hydrogen bonds. Topological parameters for both inclusion compounds 2′ Fc@CB[7] and 7′ FcN₃@β-CD were also tabulated (Tables S1 and S2, Supporting Information). The values in Tables S1 and S2 correspond to electron density, ρ(r); Laplacian, ∇²ρ(r); and total electron energy densities, H(r) at some selected bond critical points (BCPs).

According to Nakanishi and co-workers,³³ the stronger the interactions, the larger the ρ(r) and more negative will be the ∇²ρ(r) (Laplacian) and H(r). The data in Table S1 identified for covalent bonds (for β-CD and FcN₃) is coherent with such criteria (exception for iron–carbon BCPs of FcN₃, with positive Laplacian). Weak interactions such as van der Waals present positive values for ∇²ρ(r) (Laplacian) and H(r) at critical points. Furthermore, ρ(r) lies between 0.00 and 0.01 au,³³ which is coherent with some weak interactions in Table S1. For hydrogen bonds, the typical ranges were addressed at CPs:³³ 0.01 < ρ(r) < 0.04; 0.04 < ∇²ρ(r) < 0.12, and −0.004 < H(r) < 0.002. All β-CD intramolecular hydrogen bondings fulfill such criteria. The intermolecular hydrogen bond formed between the N2 atom of the guest (FcN₃) and the H68 of the host (β-CD) presents a slightly smaller value for H(r). The other values are coherent with the ranges. The data for the inclusion compound 2′ Fc@CB[7] (Table S2, Supporting Information) are coherent with the ranges defined for weak interaction and were identified between oxygen and nitrogen atoms from the host CB[7] and hydrogen atoms from the guest, the Fc, as illustrated by the paths and CPs in Figure S2.

CDs and CBs have a fundamental difference in host–guest interactions, with the OH groups in CDs usually responsible for the formation of host–guest intermolecular hydrogen bonds that contribute to the guest binding. For CBs, the carbonyl groups in the cavity allow charge–dipole interactions as well as, depending on the guest, hydrogen bonding.² The regions around carbonyl oxygens in CB[7] are found to be significantly negative, which accounts for the stability in the formation of host–guest systems with positively charged guests.² However, in this study, the Fc guest is neutral and, according to the type of atoms and data in Table S2, does not form any hydrogen bond with the CB[7] host. The formation of an intermolecular hydrogen bond in 7′ FcN₃@β-CD explains its estimated more favorable interaction energy when compared to 2′ Fc@CB[7], in the gas phase, at the B3LYP/6-31G(d) level of theory (data not shown). The comparison of the NCI plots also attests the existence of the hydrogen bond in 7′ FcN₃@β-CD (Figure S3, Supporting Information, marked according to the Multiwfn manual).³²

The NBO data for 7′ FcN₃@β-CD (Table S3, Supporting Information) suggests that the intermolecular interaction formed between the host and guest is important in the stabilization of the host–guest system. The stabilization energy E⁽²⁾ from second-order perturbation theory in natural bond orbital population analysis associated with the intermolecular hydrogen in such an inclusion compound is high when compared to the values for intramolecular hydrogen bonds in β-CD. It is noticeable that the use of NBO data to study hydrogen bonding is debatable. According to Stone,³⁴ the charge-transfer component of the energy of interaction between molecules, a controversial issue, cannot be addressed based on NBO analysis that gives values larger than those obtained by other methods. In this sense, one alternative consists of the use of the QTAIM topological parameters obtained through the Multiwfn.³² In a very interesting contribution of Lu and collaborators,³⁵ a new classification for the intermolecular hydrogen bond was proposed, and on the basis of a correlation analysis, it was discussed that the ρ(r) at the bond critical point (BCP) could be employed for quick and reliable prediction of the bond energy (BE) associated with hydrogen bonds for neutral and charged complexes. The BEs associated with intermolecular and intramolecular hydrogen bonds in 7′ FcN₃@β-CD were evaluated as proposed by Lu and collaborators (Table S4, Supporting Information). According to Table S4, the BE for the intermolecular hydrogen bond formed by N2 and H68 corresponds to −5.43 kcal mol^–1, a hydrogen bond with strength classified in the range “weak to medium” (BE varying from −2.5 to −14.0 kcal mol^–1) with electrostatics as major nature.³⁵ Finally, the application of such an approach to determine the BE for the intramolecular hydrogen bond belt in β-CD gives values varying from −4.50 to −5.62 kcal mol^–1 (Table S4, Supporting information). Such values are coherent with the stability of the intramolecular hydrogen bond in β-CD.

Geometrical Analysis

All starting associations and the corresponding unconstrained optimized geometries at the B3LYP/6-31G(d) level of theory are depicted in Figures 5 and 6 for the Fc/CB[7] and FcN₃/β-CD systems, respectively. The associations in such figures were obtained as an output for the APARM Fortran software.¹⁵

Figure 5.

Starting associations for the Fc/CB[7] system (1–4) and the corresponding fully unconstrained optimized geometries (1′–4′) at the B3LYP/6-31G(d) level of theory. When applicable, some identifications of the arrangements are shown. For the sake of clarity, the hydrogen atoms are not shown.

Figure 6.

Starting associations for the FcN₃/β-CD system (5–8) and the corresponding fully unconstrained optimized geometries (5′–8′) at the B3LYP/6-31G(d) level of theory. When applicable, some identifications of the arrangements are shown. For the sake of clarity, the hydrogen atoms are not shown.

The data comprising the relative position (r, θ, and φ) and relative rotation (α, β, and γ Euler angles) evaluated through the APARM software are collected in Table 2.

Table 2. Relative Position (r, θ, and φ) and Relative Rotation (α, β, and γ Euler Angles) Evaluated before and after Optimization of the Fc/CB[7] and FeN₃/β-CD Supramolecular Systems^a.

system	ID	r (Å)	θ (deg)	φ (deg)	α (deg)	β (deg)	γ (deg)
Fc/CB[7]	1	0.3	20	329	227	24	10
	1′	0.3	17	270	270	32	90
	2	0.3	20	329	214	4	22
	2′	2.2	0	281	180	180	270
	3	0.1	12	345	342	79	92
	3′	0.1	90	0	90	76	90
	4	0.1	14	26	256	91	221
	4′	0.4	15	180	87	90	90
FcN3/β-CD	5	1.5	17	134	101	133	86
	5′	5.6	20	346	23	80	147
	6	0.6	42	95	300	49	65
	6′	2.8	15	244	62	16	313
	7	1.0	14	91	261	141	305
	7′	2.3	13	88	108	134	248
	8	0.5	146	43	78	29	280
	8′	2.6	4	233	103	19	270

^aData evaluated through the APARM Fortran software.¹⁵

The software does not modify any of the geometries. Along with the evaluation of the principal axes of inertia and other data, the system position is modified to put the origin of the Cartesian system in the center of mass of the host molecule, followed by the alignment of the axes. Therefore, the Cartesian coordinate system coincides with the XYZ reference system. This procedure facilitates the discussion related to the visualization and comparison between spatial arrangements. All arrangements in Figures 5 and 6 are depicted keeping the same view concerning the XYZ reference system. Initially, we will focus on the arrangements that correspond to the starting (1–4) and optimized (1′–4′) associations for the Fc/CB[7] system.

According to Figure 5, the optimized associations are quite similar to the corresponding starting associations. The 2′ structure is an exception. We can see from the analysis of the geometries (Figure 5) that the Fc molecule does not keep its original position along with the optimization procedure of the system 2. The data in Table 2 also attest such a modification. For such a system, the distance of the center of mass (the r parameter) is modified from 0.3 to 2.2 Å. Interestingly, such an arrangement (2′) in vacuum conditions at the B3LYP/6-31G(d) level of theory corresponds to the most favorable one (see Table 1).

Using the A@B[r θ φ](α β γ) notation comprising the position and rotation for an inclusion compound formed by the A guest and B host, we can state that the most stable association in vacuum conditions corresponds to the inclusion compound Fc@CB[7][2.2 0 281](180 180 270), an arrangement not previously reported at the B3LYP/6-31G(d) level of theory.

The starting associations 1 and 3 correspond to experimental X-ray structures.¹ According to the experimental report, the inclination of the guest Fc inside the CB[7] host molecule corresponds to 22 and 79° for 1 and 3, respectively.¹ Such data can be interpreted as the angle between some of the principal axes of inertia of the host and the guest. For CB[7], a highly symmetric host, one of the principal axes of inertia associated with the higher moment of inertia (I_C), passes through the center of the CB[7] cavity. For the Fc molecule, a highly symmetric guest, the axis associated with the smaller moment of inertia (I_A), passes through the center of mass and the geometric center of the two cyclopentadienyl rings, as illustrated in Figure 1. Therefore, as I_C > I_B > I_A, experimental guest inclination inside the host cavity for the Fc@CB[7] system corresponds to the angle between the I_A axis from Fc and the I_C axis from CB[7]. The angle formed by such axes of inertia as implemented in APARM corresponds to the β Euler angles in Table 2. Due to symmetry, two supplementary angles can be attributed to such inclination in a mathematical sense. For the sake of comparison, just one of such solutions was collected. The current implementation of the APARM does not search for all possible solutions in the case of highly symmetric molecules.

According to the data in Table 2, the APARM found 24 and 79° for the experimental X-ray crystallographic inclinations for the Fc@CB[7] host–guest compounds 1 and 3, respectively. The method employed in the evaluation of such data in the experimental work was not clearly described.¹ What is interesting is that when the distance of the center of mass between the host and guest is almost negligible (0.1 Å), we found an agreement between the inclination angles (79°). When such a difference corresponds to 0.3 Å, the inclination angles differ by 2°. Due to the accuracy in the implementation of the Fortran software discussed herein (and other geometrical assumptions), 24° will be assumed as the reference for the experimental inclination. In this sense and based on other data in Table 2, we can state that the geometry of the Fc@CB[7] compound in the parallel form (structure 1 in Figure 5) is well predicted by the B3lYP\6-31G(d) level of theory, as also pointed out by Pinjari and Gejji³ who obtained the same spatial arrangement at this level, the structure 1′ in Figure 3.

For the perpendicular X-ray structure, with an inclination of 79°, Pinjari and Gejji³ found the very similar structure 4′ (undoubtedly addressed based on the data in Table 1). The association parameter found an inclination corresponding to 90°. It is noticeable that the distance of the center of mass between the host and guest (r) corresponds to 0.4 Å for the optimized association 4′. Probably, this accounts for some difference in the inclination reported previously and evaluated by the APARM Fortran software. Interestingly, compound 3′ was found by the optimization procedure. The distance of the center of mass (r) corresponding to 0.1 Å and the inclination of 76° suggests that such a supramolecular association corresponds to the better theoretical geometry predicted at the B3LYP/6-31G(d) level of theory for the experimental perpendicular Fc@CB[7] compound (compound 3). Furthermore, such an association is slightly favorable than 4′ (Table 1). Based on Figure 5 and the data in Table 2, we can state that 3′ association, the compound identified as Fc@CB[7][0.1 90 0](90 76 90), an association not previously reported, is very close in geometry to the X-ray compound 3, identified as Fc@CB[7][0.1 12 345](342 79 92).

Now, we focus on the FcN₃/β-CD system. The starting associations (5–8) and the corresponding optimized associations (5′–8′) are depicted in Figure 6. Within this study, some representative starting FcN₃@β-CD associations with the axial-head (5), axial-tail (6), equatorial-head (7), and equatorial-tail (8) general identification were subjected to unconstrained optimization at the B3LYP/6-31G(d) level of theory.

According to Figure 6, the FcN₃ inside the β-CD cavity does not keep its original position along with the optimization procedure. The association 5′ probably cannot be considered a real host–guest system. The analysis of the r parameter in Table 2 also attests the changes in the position of the FcN₃ guest inside the β-CD cavity. According to Table 2, the differences in r for starting and optimized associations correspond to 4.1 Å (5–5′), 2.2 Å (6–6′), 1.3 Å (7–7′), and 2.1 Å (8–8′). For the 5′, the FcN₃@β-CD overall notation should be replaced by FcN₃···β-CD. The distance of the center of mass between the host and guest for such an optimized system corresponds to 5.6 Å. Therefore, the 5′ association probably cannot be addressed as an inclusion compound. Usually, the notation for the starting association cannot be transferred for the optimized one. Cyclodextrin-based host–guest systems, in general, are flexible, which implies that a well-defined starting association with its general or specific identification probably cannot be identified after the optimization with a similar notation. The data evaluated from the APARM software can aid in the identification of complex geometries, as will be discussed. Only based on the visual comparison (Figure 6), the 6′ association resembles 8′. Finally, based on the overall arrangement depicted in Figure 6, we can suppose that the optimized association 7′ is quite similar to the starting association 5. A more detailed analysis can be achieved with the data in Table 2.

For associations 5 and 5′, Table 2 indicates, beyond the increase in the distance between the host and guest (growth in r) with the maintenance of the relative polar angle (slight difference in θ from 17 to 20°) and a difference in the azimuth angle (φ from 134 to 345°), the change in the rotation along the I_A axis (α Euler angle changes from 101 to 23°) and the difference in the inclination of FcN₃ inside the β-CD cavity (β Euler angle changes from 33 to 80°), followed by another rotation along the Z axis (γ Euler angle changes from 86 to 147°). The rotation along the Z axis for the FcN₃ molecule may be challenging to visualize due to the asymmetric nature of such a molecule (as illustrated in Figure 1). It is noticeable that the 5′ arrangement is one of the most stable FcN₃/β-CD associations in vacuum conditions at the B3LYP/6-31G(d) level of theory (see Table 1).

Except for the difference in energy (Table 1), associations 6′ and 8′ seem to be similar not only visually but also compared to the data in Table 2. We see the close distance between the host and guest (2.8 Å for 6′ and 2.6 Å for 8′) and close values for the polar (θ) and azimuth (φ) angles with a difference of approximately 10°. The inclinations of the guest inside the cavity are also similar (β Euler angles 16° for 6′ and 19° for 8′). Here, we attest the lack of the use of general identification for such a flexible system. Visually, 6′ and 8′ could be identified by the same general term, for instance (for the analogy), an “expanded equatorial tail” association. However, their apparent similarity does not reflect in their difference of ΔE of 2.17 kcal mol^–1, according to Table 1. Therefore, instead of using a general identification, we can identify most adequately 6′ with the undoubted notation FcN₃@β-CD[2.8 15 244](62 16 313) and the corresponding 8′ with its similar notation FcN₃@β-CD[2.6 4 233](103 19 270). 7′ is the most stable association for FcN₃@β-CD identified in vacuum conditions at the B3LYP/6-31G(d) level of theory (see Table 1). In general terms, it can be classified as the axial-head association due to the general orientation of the guest inside the β-CD cavity. The host–guest distance is close (1.5 Å for 5 and 2.3 Å for 7′), similar to the inclination of the guest inside the β-CD cavity (β Euler angle 133° for 5 and 134° for 7′). Finally, the rotation along the I_A axis as illustrated by the difference in the rotation of the guest inside the CD cavity (see Figure 6) seems not to be defined by the α Euler angle (101° for 5 and 108° for 7′) but by the γ Euler angle (86° for 5 and 248° for 7′). As recommended and discussed herein, the 7′ host–guest system can be identified more precisely as FcN₃@β-CD[2.3 13 88](108 134 248).

Finally, bearing in mind the structures for which experimental X-ray crystallographic data was reported, empty CB[7]³⁶ and the Fc@CB[7] compound,¹ it is worth noting that the geometries of the CB[7] host when comprising in its cavity the Fc guest are not significantly distorted (Figure S4, Supporting Information). Therefore, the inclusion of the Fc guest does not change the CB[7] geometry considerably.

Related Works

The use of information related to the molecular axes of inertia and Euler rotation angles can be found in the literature in several relevant contributions. Without being exhaustive, some papers can be addressed here and have some relation to this work or had inspired the conducted research.

Foot and Raman³⁷ discussed the relation between the principal axes of inertia and ligand binding. Based on crystallographic data, they explain that with some frequency, at least one of the principal axes of inertia of a protein penetrates the region used for ligand binding, an interesting observation. Another creative contribution related to the study of principal axes for molecules corresponds to the Galek use of moments of inertia to study crystal packing.³⁸ Within the paper, a new method to compare the crystal pack based on moments of inertia tensor is presented. According to the author, the developed method allows the comparison of any two crystal structures irrespective of chemical connectivity.

Concerning the Euler angles and their application to the study of molecular systems, many relevant and inspiring contributions can be addressed. For instance, we have the work of Zannoni³⁹ concerning simulations of novel mesophases for which attractive–repulsive models of the Gay–Berne⁴⁰ potential type, relevant to the study of liquid crystals, are discussed. This potential comprises a term (a strength parameter) that is a function of the orientation vectors of two particles and their separation vector. As discussed by Zannoni and co-workers previously,⁴¹ the biaxial order parameter, a function of the set of Euler angles defining the orientation of a molecule, is the most convenient one for the monitoring phase biaxiality. As mentioned, Euler angles are essential in the study of liquid crystals. Within this context, experimental techniques such as polarized Raman spectroscopy and X-ray diffraction allow the evaluation of orientation order parameters.⁴²

Another exciting application of Euler angles can be understood in terms of the work of van Gurp⁴³ related to the use of rotation matrices for the description of molecular orientation. Within his contribution, orientational distributions are discussed after the definition of a direction of a molecule element (molecule, chain segment, or crystal) as being the spatial position of such element relative to a right-handed macroscopy system of axes, unambiguously defined by three Euler angles. According to van Gurp,⁴³ the discussed mathematical formalism applies to several experimental techniques used to measure the degree of orientation in polymers.

The use of Euler angles to describe the crystal orientation and the issues related to the lack of standard definitions for the unit cell reference settings and specimen axes are well discussed in a recent contribution of Nolze.⁴⁴ According to the author, the application of Euler angles is still widespread nowadays for crystal orientation descriptions from an experimental perspective. However, the use of Euler angles instead of quaternions requires standardization, at least in fields such as electron backscatter diffraction (EBSD). Within Nolze contribution,⁴⁴ Euler rotation angles related to the crystal coordinate system (crystal basis vectors) and the sample coordinate system used in EBSD orientation mapping were discussed.

In another interesting and experimental applicable contribution, the quantitative determination of the spatial orientation of graphene (G) sheets by polarized Raman spectroscopy is discussed.⁴⁵ Within such a work, the local graphene flake orientation inside a specimen is characterized by the surface normal vectors. Such an orientation is related to the coordinate system of the specimen by Euler angles. The orientation distribution function (ODF) is defined in terms of the local orientation of the graphene sheet within the specimens under analysis and the specimen relative to experimental polarized Raman spectroscopy measurement parameters. Polarized Raman spectroscopy is a powerful technique. Another interesting example of the application of such a method is related to the quantification of the alignment of single-wall nanotubes (SWNTs) in a polymer matrix.⁴⁶

Focusing on specific contributions to the field of host–guest chemistry, one of the principal discussions of this work is related to the identification of starting associations and the characterization of resulting arrangements evaluated after the optimization procedure. Many interesting contributions, in this sense, can be discussed and addressed. One effective strategy, possible today due to the enormous processor capacity, consists of the analysis of several optimized starting arrangements through the use of a low-cost computational method before subjecting a representative spatial arrangement to the optimization through the application of a more robust and high-cost computation approach, as reported by Yahia and co-workers.⁴⁷

Within the Yahia and co-workers’ contribution,⁴⁷ the modification of the distance of the center of the mass between the host and guest followed by a rotation along the axis of inclusion was responsible for the generation of several starting host–guest arrangements. Such associations were subjecting to unconstrained optimization in a low-cost computational method, the PM6.⁴⁸ In a subsequent step, a high-cost computational methodology was applied, in which the conventional exchange–correlation B3LYP functional, the Coulomb-attenuating method (CAM-B3LYP),⁴⁹ and Minnesota functionals that take into account the dispersion (M05-2X and M06-2X)^50,51 were used. The methodology applied, as also pointed out by the authors, followed the Liu and Guo procedure.⁵²

Within such a method,⁵² glycosidic oxygen atoms of CDs are used to define an XY plane. The center of such a plane is used as the center of the coordination system. The guest molecule is then placed along the Z-axis of the coordination system, and a different orientation of the guest relative to CD must be taken. Finally, the guest is allowed to enter and pass through the CD molecule in steps. The optimization of each step corresponds to a search method to find out a global minimum host–guest arrangement.⁵²

The Liu and Guo Z-axis is geometrically analogous to the principal axis of inertia related to the CD higher moment of inertia (I_C), the Z-axis, defined in this work (see Figure 1). Such an axis passes through the cavity of the CD. What is fundamental at this point is to bear in mind that to each step of the widespread applied Liu and Guo procedure,⁵² we can attribute a set of the association parameters discussed herein. Therefore, we will have for each of Guo and Liu steps an arrangement that can be identified by the [r θ φ](α β γ) notation. The coordination system defined herein, which uses the principal axes of inertia of CD, through the aid of a user-defined version of APARM, the UD-APARM software,¹⁵ can be applied to generate steps along the I_C axis (Z-axis of the described XYZ reference system), analogous to the Liu and Guo procedure. We now cannot address whether the optimization of steps defined along the principal axes of inertia I_C produces the same global minimum arrangement that can be found with the original Liu and Gue procedure.⁵² Such a discussion is beyond the scope of this work.

Focusing again on the Yahia and co-workers’ contribution,⁴⁷ the visual inspection of the global minimum is made through the identification of some relevant distances, a traditional identification usually present in host–guest theoretical studies.⁵³⁻⁵⁵ The use of a set of parameters discussed herein can serve as an additional tool in the discussion of the principal arrangements focused on a specific study. Beyond any identification, as proposed and discussed herein, the authors always have the option to address the complete coordinates of each arrangement investigated as the Supporting Information.⁵⁶

In summary, the critical idea of this work is the construction of a reference system (X, Y, and Z) over the principal axes of the host molecule. The additional development of the derived system (x, y, and z) over the principal axes of the guest allows us to evaluate a set of parameters accounting for the relative position and rotation between a pair of molecules. The central idea is simple. The implementation comprises some tricky steps, such as the evaluation among some possibilities of adequate vectors to represent the right-handed coordination systems mounted over each molecule. The notation and ideas do not have the intent to replace the traditional identification and visual analysis related to host–guest chemistry but improve the discussion related to such systems. Finally, the principle discussed herein naturally applies to other supramolecular systems beyond host–guest chemistry, as will be discussed in the next section.

Beyond Host–Guest Chemistry

The applicability of the developed approach to other supramolecular systems beyond host–guest chemistry will be illustrated through the analysis of a preliminary study concerning the interaction of the indigo carmine (IC) dye, one of the oldest and most importantly used dyes,⁵⁷ and a graphene (G)⁵⁸ model comprising 708 atoms. Such a supramolecular system has been studied recently in my research group through molecular dynamics (MD) simulations.

After an initial minimization, the simulations were carried out through the use of the Amber 18 package⁵⁹ with the general AMBER force field (GAFF)⁶⁰ in implicit solvent (water) at 300 K (100 ps of length). Structures of the systems with energy close to the average were collected at 4, 59, and 97 ps and were analyzed by the APARM software discussed herein.¹⁵ The supramolecular associations collected by the MD simulations with the corresponding parameters defined and discussed herein are depicted in Figure 7.

Figure 7.

Geometries of the supramolecular system formed with the indigo carmine (IC) dye and a 708-atom graphene (G) model collected by MD simulations carried out in water (implicit model) at 300 K. Force field: GAFF, length of simulation: 100 ps. The parameters discussed in this work were shown (APARM¹⁵ data). For the sake of clarity, hydrogen atoms are not shown.

According to the geometries depicted in Figure 7, we can see that the dye molecule, based on the right-handed XYZ reference system mounted over the graphene (G), was placed in a negative Z position (below the surface in the figure), which justifies the θ parameter (polar angle) greater than 90°. The position of the IC dye changes throughout the MD simulation, which can be understood in terms of the difference in the r values (4.9, 18.3, and 11.1 Å at 4, 59, and 97 ps, respectively).

According to the geometries in Figure 7, the distance between the plane of carbon atoms and the plane of the IC dye molecule (a plane that passes through the I_A axis for IC) does not change considerably during the MD simulation. The distance between those planes can be evaluated for this system as r cos(180 – θ). Such values correspond to 3.6, 3.0, and 4.0 Å for geometries at 4, 59, and 97 ps, respectively. We also see the rotation of the IC molecule in the surface of the graphene (variation of the φ parameter). The dye molecule keeps its inclination concerning the plane of carbon atoms, as indicated by the small change in the β parameter along the trajectory (90–94° from 4 to 97 ps). The rotation of the dye in the surface of the graphene can also be interpreted by the variation of the γ parameter because the α Euler angle does not change a lot. The very distinct value for γ at 59 ps (163°) accounts for the relative rotation of the dye molecule in such collected geometry. The γ values for 4 and 97 ps are the same (253°), which can be interpreted through the visual analysis of the geometries in Figure 7. Without being exhaustive, the discussion concerning the supramolecular system formed by a dye molecule and a graphene model attests that the developed software and the parameters evaluated apply naturally for other systems beyond host–guest chemistry.

Conclusions

In this work, the Fc/CB[7] and the FeN₃/β-CD intermolecular associations were investigated at the B3LYP/6-31G(d) level of theory. QTAIM analysis for the most stable arrangements obtained theoretically suggests the existence of many intermolecular interactions responsible for the stabilization of the inclusion compounds. NCI and NBO calculations attest the existence of an intermolecular hydrogen bond in the most favorable inclusion compound formed between FeN₃ and β-CD. For such a host–guest system, the hydrogen bond energy evaluated through the use of QTAIM data corresponds to −5.43 kcal mol^–1. According to the investigation of the B3LYP/6-31G(d) potential energy surface (PES) for the Fc@CB[7] and the FeN₃@β-CD host–guest starting arrangements, it can be stated that overall identifications such as parallel, perpendicular, axial, or equatorial correspond to a set of supramolecular structures. Within this context, a collection of parameters to improve the spatial arrangement description was introduced. Through the analysis of the principal axes of inertia of each molecule, the Fortran software discussed in this contribution, named APARM, systematically generates the six parameters accounting for the relative position (r, θ, and φ) and relative rotation (α, β, and, γ Euler angles) of a given supramolecular system. The immediate gain in the discussion of a given experimental or theoretical supramolecular arrangement corresponds to the substitution of the general identification guest@host for the more complete one guest@host[r θ φ](α β γ). Additionally, the parameters discussed herein apply to other nonbonded associations beyond host–guest chemistry for which the notation A···B[r θ φ](α β γ) comprises a more complete description of the supramolecular system formed by the molecules A and B. It is worth noting that a modified version of the APARM, the UD-APARM software, serves to obtain user-defined spatial arrangements for two nonbonded entities. Therefore, with a given geometry, for instance, from X-ray data, the parameters can be derived, and with the parameters, from a theoretical perspective, a geometry can be obtained. Specifically, in theoretical works, the descriptors apply to monitoring of the spatial association of a supramolecular system during an optimization procedure or a molecular dynamics simulation and also apply in the specification of starting spatial arrangements, the additional information that contributes to reproducibility. The last versions of the software discussed herein, along with instructions and other implementations (in a distinct programming language), will be available for download at http://www.dqi.ufla.br/portal/anconi-group-homepage.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.9b03914.

• QTAIM molecular graphs for 7′ FcN₃@β-CD′ and 2′ Fc@CB[7]; QTAIM topological parameters for 7′ FcN₃@β-CD and 2′ Fc@CB[7]; NCI plots for 2′ Fc@CB[7] and 7′ FcN₃@β-CD; NBO relevant data for 7′ FcN₃@β-CD; electron density at BCP and BE for HB in 7′ FcN₃@β-CD; and X-ray crystallographic geometries of CB[7] for comparison (PDF)

The author declares no competing financial interest.