Host–guest complexes of mixed glycol-bipyridine cryptands: prediction of ion selectivity by quantum chemical calculations, part V

The selectivity of the cryptands [2.2.bpy] and [2.bpy.bpy] for the endohedral complexation of alkali, alkaline-earth and earth metal ions was predicted on the basis of the DFT (B3LYP/LANL2DZp) calculated structures and complex-formation energies. The cavity size in both cryptands lay between that for [2.2.2] and [bpy.bpy.bpy], such that the complexation of K+, Sr2+ and Tl3+ is most favorable. While the [2.2.bpy] is moderately larger, preferring Rb+ complexation and demonstrating equal priority for Sr2+ and Ba2+, the slightly smaller [2.bpy.bpy] yields more stable cryptates with Na+ and Ca2+. Although the CH2-units containing molecular bars fixed at the bridgehead nitrogen atoms determine the flexibility of the cryptands, the twist angles associated with the bipyridine and glycol building blocks also contribute considerably.


Introduction
The present report continues a series of contributions from our group dealing with quantum chemical investigations of the selective complexation of alkali and alkaline-earth metal cations by supramolecular species, predominantly cryptands and their derivatives [1][2][3][4]. The current state of research relevant for our studies was carefully explored and illustrated in the mentioned publications and will therefore not be repeated in detail in this work.
The current work extends our explorations into the outlined topic. Here we discuss two hybrid cryptands between [ Figure 2.

Results and Discussion
Although as of spring 2013 more than 650 X-ray structures are listed in the Cambridge Structural Database for [2.2.2] and mostly its alkali and alkaline-earth metal cryptate complexes, only the structure of [Na 2.2.bpy]Br has been published [61], and to the best of our knowledge there are no further experimental structures for [2.2.bpy] and [2.bpy.bpy] cryptands and cryptate complexes. In earlier investigations on related supramolecular systems the applied method (RB3LYP/LANL2DZp) provided satisfactory results [1][2][3][4]. In all these cases the calculated bond length between the guest ions and the donor atoms was elongated compared to the analogous bonds in the X-ray structures. The same behavior is found when comparing [Na 2.2.bpy]Br and the (B3LYP/LANL2DZp) calculated C 2 symmetric [Na 2.2.bpy] + -ion. The bonds between the sodium cation and the donor atoms are around 5.5% longer than in the averaged solid-state structure (Table 1). Whereas Table 1 and  Table 2 present significant data for all discussed structures, Figure 3 and Figure 4 [61].  [1][2][3][4]. A comparison of the bond distances between the donor atoms and the metal cation complexed endohedrally by the cryptand or by the solvent molecules, e.g., pyridine or water, can be drawn. This method only provides reliable results if the donor atoms that coordinate to the metal center are the same and in an equal         Table 5, and against [M(H 2 O) n ] m+ (n = 4 and 6 for Li + and Be 2+ and 6 for all others), see Table 6.
A direct comparison of the calculated data for the metal donor atom bonds in cryptates and for solvated metal ions is given in Table 7 for [2.2.bpy] and Table 8 for [2.bpy.bpy]. To illustrate the situation more clearly, the results are presented in Figure 6 and Figure 7 for [2.2.bpy] and in Figure 8 and Figure 9 for [2.bpy.bpy], where bisecting lines point to the cases in which coordination is most likely to occur. The ions above the line are somewhat too small, whereas the ions below the line are too large for the studied cryptand.
As depicted in Figure 6, the interaction of both N sp2 atoms of the bipyridine side of the [2.2.bpy] with all presented cations is essential. The bridgehead N sp3 atoms also play an important role for many of the studied ions, such as Ga 3+ , Tl 3+ , In 3+ , Ca 2+ , Sr 2+ , K + , Ba 2+ and Rb + . However, for the larger (Na + , Cs + ) and especially for the smaller ions (Mg 2+ , Al 3+ , Li + ), the N sp3 -M m+ interaction seems to be of lesser importance. The structure of [Be 2.2.bpy] 2+ presents a special case: in the energetically more stable C 1 symmetry (compared with C 2 , see later discussion), the Be 2+ ion seems to be shifted towards one of the N sp3 atoms, as the calculated bond lengths differ significantly. Its fourth coordination site is occupied by one of the oxygen donor atoms.
The data set describing the interaction of oxygen donor atoms with the studied ions is shown in Figure 7. Only three cations     Table 7).  Table 8).  Table 7).  Table 8).
points for Ga 3+ (C 1 ) and Mg 2+ (C 1 ) also lie close to the bisecting line, pointing to possible coordination. Finally, Li + , In 3+ , Tl 3+ and Na + as well as the residual data points for Be 2+ , Ga 3+ and Mg 2+ are positioned too far away from the line for the potential coordination to be relevant. At a first glance this finding seems to contradict the experimentally obtained results for Na + , but the solid-state structure of [Na 2.2.bpy]Br is somehow distorted (X-ray: C 1 symmetry compared to calculated C 2 ) [61], to allow the cationic sodium center to interact with the available cryptand donor atoms and to get a better stabilization as in the applied weakly coordinating solvent acetonitrile [63][64][65].
It can be concluded that an interaction between Ca 2+ , Sr 2+ , K + , Ba 2+ and Rb + and all present donor atoms (N sp2 , N sp3 and O) plays an important role. This can be explained, since especially the larger K + , Ba 2+ and Rb + ions prefer a higher coordination number than six [3]. They are also energetically favored (see further discussion) and seem to fit well into the cavity. In 3+ and Tl 3+ prefer the interaction with nitrogen donor atoms, completing the vacant positions with O-atoms. Among the earth metal ions Tl 3+ is also energetically favored (see further discussion). Be 2+ has a special position among the cations as described above, but it is still too small for the cryptand, as are also Li + , Mg 2+ , Al 3+ and Ga 3+ . For Na + and Cs + , the interaction with the bridgehead nitrogen appears not to be essential, more important are the connections to nitrogen atoms of the bipyridine side of the ligand and oxygen donor atoms. These cations also do not fit well into the cavity, being too small or too large, respectively.
As can be seen from Figure 8, the interaction with N sp2 atoms of the bipyridine site of [2.bpy.bpy] is significant for all studied ions, though the largest (Rb + , Cs + ) deviate slightly from the general trend. A similar situation occurs for the bridgehead nitrogen atoms: most of the investigated cations are placed along the bisecting line, indicating an important role of the potential coordination. The large (Rb + , Cs + ) and small (Li + , Mg 2+ (C 1 )) ions lie further away from the line, as this kind of interaction is less relevant for them. Be 2+ again presents a special case, as it is shifted towards one of the N sp3 donor atoms and coordinates furthermore to three of the N sp2 atoms.
The interaction between oxygen donor atoms and the studied cations is less important in the case of [2.bpy.bpy], as there are six nitrogen donor atoms present to fill the coordination sphere. Even so, the bridgehead nitrogens are in some cases too far away for an effective interaction. As shown in Figure 9, only K + , Ba 2+ and Rb + lie on the bisecting line, while Mg 2+ , Ca 2+ , Sr 2+ and Cs + are placed near it. The residual cations, especially the small Be 2+ , Li + , Al 3+ and Ga 3+ are too far away for any kind of significant interaction. Summing up, Ca 2+ , Sr 2+ , K + , Ba 2+ and Rb + fit best into the cavity of [2.bpy.bpy], as was the case for [2.2.bpy], though the smaller cations (Ca 2+ and Sr 2+ ) prefer the interaction with the nitrogen donor atoms and the larger cations (K + , Ba 2+ and Rb + ) tend to interact with the oxygen donors. The smaller cations are nested against the nitrogen atoms, though they do not fit well into the cavity. Among the earth metal ions, In 3+ fits best, lying closer to the nitrogen donors and being also one of the energetically favored cations (Table 4).
According to the performed comparison of the bond lengths between metal cations and N/O donor atoms in cryptates with the same bond lengths of the solvated metal centers, both cavities have a size similar to [2.2.2] and prefer cations with larger radii in every one of the studied main groups. However, an additional energy consideration is necessary to provide detailed and more precise information about the favorable coordination layout of the investigated cryptands.
Besides the structural evaluation of the computed systems, the examination of the energy of a model reaction, depicted in Scheme 1, provides valuable results, important for the prediction of the cryptand's selectivity. All cations were calculated in the six-fold coordination environment, to maintain equal conditions for all studied systems. For the lithium [66] and beryllium [67] cations the preferable four-fold coordinated structures were found. They show additional water molecules, which do not interact directly with the metal center, but are held in the second coordination sphere by hydrogen bonds. However, the gas phase [Li(H 2 O) 6 ] + and [Be(H 2 O) 6 ] 2+ exist as local minima. The complexation energies computed in this way are shown in Table 3 and Table 4, and are plotted against the ionic radii, see Figure 10 and Figure 11.   [68,69]. Cryptand [2.bpy.bpy] also prefers to bind K + , this time followed by Na + ; and Sr 2+ , followed by Ca 2+ . Finally the combination with Tl 3+ yields the most stable  Table 3.  Table 4. cryptate. In general, the computed energies confirm our conclusions drawn from the bond lengths between the metal centers and donor atoms of the cryptands and/or solvent molecules.  2+ can be considered as structures allowing appropriate coordination and stabilization in the gas phase, but in solution these structures will surely not be superior to solvated Be 2+ and an empty cryptand. In general, the computed complexation energies allow conclusions about the stability order of the endohedral cryptate complexes and correlated with it about the ion selectivity of the respective cryptands, as will be explained below.
Earlier reports from our [1][2][3][4] and other [70] groups demonstrated that the cryptand does not remain unaltered throughout the complexation process; it twists in order to adjust for the optimal interaction with the metal cation. The effect of this twisting is observable also in the experimentally achieved solidstate structures. For example, [phen.phen.phen] has a very rigid structure, so the resulting average bond length between the metal center and the aromatic nitrogen donor atoms in [Na phen.phen.phen] + are longer than the equivalent bonds in [Na 2.2.bpy] + (av N sp2 -Na = 2.70 Å [71] and av N sp2 -Na = 2.61 Å [61], respectively). The [2.2.bpy] cryptand has namely two glycol-containing molecular arms, which can wrap flexibly around the cation, allowing closer proximity between the metal center and bipyridine ligand. The increasing size of the guest ions is accompanied by a general enlargement of the metal-donor bond length, as can be concluded from the elucidation of the calculated distances (see Table 1 and Table 2). Apart from various contributions of the guest ions, the observed behavior is evidence for the possible flexibility of the studied cryptands. A proper flexibility requires an adaptable molecular moiety.
While [M 2.2.bpy] m+ has one molecular bar with N-CH 2 (pyridine) motifs adjacent to the N bridgeheads and a conformationally nearly unhampered C-C bond bridging the two pyridine rings and two molecular bars C 2 H 4 -O-C 2 H 4 -O-C 2 H 4 adjacent to the N bridgehead atoms, the [M 2.bpy.bpy] m+ has two of the bipyridine and one of the glycol building blocks. Descriptors for the twist and tilt of these moieties are the torsion angles such as CH 2 -N sp3 -N sp3 -CH 2 , N sp2 -C-C-N sp2 , (CH 2 ) 2 -N sp3 -N sp3 -(CH 2 ) 2 and O-C-C-O. They all show a qualitative linear behavior that mainly depends on the size of the ion, see Figure 12, Figure 13 and Figure 14. The torsion angles CH 2 -N sp3 -N sp3 -CH 2 and (CH 2 ) 2 -N sp3 -N sp3 -(CH 2 ) 2 show very similar behavior, becoming more positive within the studied main groups, with the only exception presented by the earth metals, whose values are slightly scattered but still fit well in the general trend, see Figure 12. These angles cover the widest range in both host-guest systems, see Table 9.
The illustrated occurrence evidences the importance of the possible alteration of the angles between the CH 2 groups and the N sp3 bridgehead for optimal matching between the host and the guest molecules. The stereochemistry of all investigated cryptates is the same and can be described as λ, as the pointed angles are negative [3]. The only exception is the (CH 2 ) 2 -N sp3 -N sp3 -(CH 2 ) 2 angle in [Mg 2.2.bpy] 2+(TS) .
The N sp2 -C-C-N sp2 torsion angle covers a somewhat smaller range than both angles at the bridgehead nitrogen atoms, as shown in Table 9, because of the greater rigidity of the connected pyridine rings compared to the aliphatic CH 2 groups. However, the differences are much larger than in the case of [M 2.2.phen] m+ and [M 2.phen.phen] m+ [1], which is easy to understand since the polycyclic heteroaromatic phenantroline system is inflexible, while the conformationally nearly unhampered C-C bond bridging the two pyridine moieties allows more flexible arrangement of host and guest. The main exceptions of the linear trend are the cryptates with Be 2+ , Ga 3+ , Al 3+ and Li + . As explained above, Be 2+ cation presents a special case for both cryptates. In [Be 2.bpy.bpy] 2+ , the coordination sphere of the cation consists of four nitrogen atoms, viz., three coming from bpy-groups and the fourth being a bridgehead nitrogen. According to this, the cryptate in the C 1 symmetry, which is more stable than the C 2 symmetry, shows two N sp2 -C-C-N sp2 twist angles: one of −0.3°, which lies above and one of −42.9°, which fits well in the general linear trend. In the case of [Be 2.2.bpy] 2+ the cation is again fourfold coordinated and shifted to one of the bridgehead nitrogen atoms, though this time one of the O-atoms is included in the coordination sphere. Because of its small radius, it apparently does not need to twist the chelating group as much as one would expect from the extrapolation of the other values. Similar behavior can be ascribed to other small cations: Al 3+ , Li + and Ga 3+ , while [Ga 2.2.bpy] 3+ in C 2 symmetry is present as a transition state, which can additionally contribute to the observed deviation. Throughout the series, cryptates change their stereochemistry. The twist angles for the smaller cations are negative and therefore the cryptates show a λ configuration; the larger ions cause positive angles and as a result the cryptates have a δ configuration.  [1]. The small values indicate that the twist of the molecules in this region is less important for the mutual adjustment of the host and guest. Apart from the exceptions described above, the coordination of the O atoms is mostly important for the larger cations for which Figure 14 shows a good linear trend, while the smaller ions deviate from it. In both cryptate series the angles are positive and become larger in the studied main groups. Hence, they show δ stereochemistry.
The four presented torsion angles describe the twist of the cryptand around the cations. The CH 2 -N sp3 -N sp3 -CH 2 and (CH 2 ) 2 -N sp3 -N sp3 -(CH 2 ) 2 angles lie in the middle of the molecule, pointing to the bipyridine and glycole building blocks, respectively. The N sp2 -C-C-N sp2 and O-C-C-O angles lie above and under the middle of the molecule and the hosted cation at these molecular parts.
Comparison of the data given in Table 1 and Table 2 shows a mutual shift in the calculated values of the twist angles at the respective molecule halves. While the magnitude of the CH 2 -N sp3 -N sp3 -CH 2 or (CH 2 ) 2 -N sp3 -N sp3 -(CH 2 ) 2 angle becomes smaller, the N sp2 -C-C-N sp2 or O-C-C-O angle is getting larger, for both cryptates alike, with the main exception presented by earth metals bound by [2.2.bpy], see Figure 15 and Figure 16. Hence, the cryptands coil around the hosted cations in order to get more tilted and closer to them.
As shown above, the analysis of the computed complexation energies plotted against the ionic radii allows conclusions to be made about the stability order of the studied endohedral cryptate complexes. Thereby the preferred selectivity of the alkaline earth ions plays the more important role, since the alkali cations are not very sensitive in this range, as the ion radius difference for Li + , Na + and K + is too large. The observed preference for the cation size can be correlated with the cavity size and, hence, with the selectivity of the cryptand. An overview of the preferred ion selectivity of several cryptand families recently studied in our group ( [1][2][3][4] and the present contribution) is given in Table 10.   . In sarcophagine, the bridgehead nitrogen atoms are replaced by carbon atoms and in sepulchrate every aliphatic chain connecting the bridgehead nitrogen atoms is shortened by two CH 2 units. These structural changes lead to more constitutional rigidity of the ligands and smaller cavity sizes, such that these two prefer the smallest cations of all cryptands examined throughout our study.
To demonstrate our results more clearly, we arranged the cryptands schematically in a spiral shape shown in Figure 17, which not only includes already investigated ligands, but also provides space for further macromolecules that will be studied in future. Thereby, molecules of different size, viz., larger, smaller and those in between, will fit well into the given layout. The flexibility of the cryptands, which is important for selective host binding, is dominated by the flexibility of the CH 2units adjacent to the bridgehead nitrogen atoms. However, the contribution of the torsion angles of the bipyridine and glycole building blocks also plays an important role, the synchronous movement of the opposite molecule sides allows the cryptand to coil around the complexed cation.

Conclusion
The algebraic sign of the calculated angles allows the assignment of their stereochemistry. Throughout the series of all investigated cryptates, the molecular bars adjacent to the bridgehead nitrogen atoms show λ and the aliphatic di-ether chains δ stereochemistry, corresponding to negative and positive algebraic signs. Conversely, the steric configuration of the coordinating bipyridine moiety alternates depending on the size of the complexed metal ion.

Experimental Quantum chemical methods
We performed B3LYP/LANL2DZp hybrid density functional calculations, i.e., with pseudo-potentials on the heavy elements and the valence basis set augmented with polarization functions [75][76][77][78][79][80][81][82][83][84][85][86]. During the optimization of the structures no other constraints than symmetry were applied. In addition, the resulting structures were characterized as minima, transition structures, etc., by computation of vibrational frequencies. The relative energies were corrected for zero-point vibrational energies (ZPE). We deliberately did not include any solvent model for the sake of comparability with earlier studies [1][2][3][4] and to exclude further approximations. The GAUSSIAN suite of programs was used [87].