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J. Phys. Chem. B 2001, 105, 11060-11066
ARTICLES Na+ Complexes with Crown Ethers: Calculation of Coupling Constants
23Na
NMR Shieldings and Quadrupole
J. A. Tossell Department of Chemistry and Biochemistry, UniVersity of Maryland, College Park, Maryland 20742 ReceiVed: April 23, 2001; In Final Form: September 18, 2001
Na+ forms complexes and compounds with both organic ring systems, such as crown ethers, and inorganic ring systems, such as the aluminosilicate “six ring” in the mineral sodalite Na6(SiAlO4)6. 23Na NMR shieldings and quadrupole coupling constants have been calculated at the Hartree-Fock level, using the 6-31G* basis sets for (12-crown-4)2Na+, 15-crown-5Na+, and 18-crown-6Na+ complexes (and with additional H2O for the 15-crown-5 and 18-crown-6 ligands). Geometries have been optimized using the Hartree-Fock method and valence-electron only polarized double-ζ bases, sometimes using starting geometries from crystallographic data. For the 15-crown-5 and 18-crown-6 hydrated compounds, the calculated shieldings agree well with a recent experimental study (Wong, A.; Wu, G. J. Phys. Chem. A 2000, 104, 11844), although the calculated quadrupole coupling constants are sometimes too large. For the (12-crown-4)2Na+ cation, several structures with different symmetry and bond distances are considered, but none matches both the NMR shift and the quadrupole coupling constant observed for solid (12-crown-4)2NaClO4. Our results indicate that the 23Na NMR shift, compared to Na(OH2)6+ (our model for aqueous Na+), can be accurately calculated at the HartreeFock level with small basis sets such as 6-31G*, once the geometry is accurately known. The main problem is to obtain a sufficiently accurate geometry for the large (and sometimes flexible) ring molecules which bond weakly to the Na+ in the complex. The absolute deshieldings (referred to free gas-phase Na+) obtained from calculations on these crown-ether complexes can be approximated well using a simple additivity model based on Na+ interactions with isolated (CH3)2O molecules. The Na deshielding in the (CH3)2ONa+ complex is calculated to show a strong but simple exponential distance dependence. The calculated absolute deshieldings in the complexes are also strongly correlated with the bond strength sum received by the Na+. A Si3O3O18H12Na-2 model for sodalite gives a 23Na NMR shift of about -8 ppm and a quadrupole coupling constant of 4.4 MHz, similar to the Na NMR results for 18-crown-6Na+.
Introduction Although alkali metal cations form rather weak chemical bonds, they are important in determining properties of both biological and mineralogical systems. In biology, the interaction of alkali cations with membrane-spanning ion channels is important in the excitability of nerve and muscle.1 In mineralogy, the interaction of volatiles such as water with aluminosilicate melts (which greatly decreases the melt viscosity) appears to strongly modify only the environment of the alkali cation.2 Both biochemical and mineralogical regimes involve very complicated and sometimes poorly characterized structures, so that it is useful to initially study simpler analogues. Crown ethers have attracted much interest as simple analogues for biological channels or diffusive carriers for alkali metal cations.3,4 A crown ether is a cyclic array of ether O atoms linked by organic spacers, usually -CH2CH2- groups, giving a general formula (C2H4O)n. Complexes of 15-crown-5, i.e., (C2H4O)5, with Na+ alone and with Na+ and H2O (with geometries calculated as described below) are shown in Figure 1 parts a and b. Sodalites are a group of minerals in which Na+ is coordinated to an inorganic aluminosilicate ring containing three Al and three Si atoms.5 A
simple cluster model for the sodalite Na6(SiAlO4)6 is Si3Al3O18H12Na-2, shown in Figure 2. The crown ethers and sodalites show many structural similarities. For example, Na-O distances are similar in the complexes with 18-crown-6 and Si3Al3O18H12-3, although the crown ether is more distorted from planarity. Similarly, small ions and neutrals such as OH-, Cl-, NO2-, CO3-2, H2O, and others can be readily incorporated into either crown ether or sodalite structures. There are also differences of course: the O donor atoms are in C-O-C groups in the crown ethers and Si-O-Al groups in the sodalites and the crown ethers are generally more flexible than the aluminosilicate rings in sodalites. Although 23Na is a demanding NMR nuclide, because of its large nuclear quadrupole moment coupled with the lack of symmetry in many Na chemical environments, methods have been recently developed to determine both NMR chemical shifts and quadrupole coupling constants for solids containing Na, using dynamic-angle spinning and double-rotation or multiplequantum magic angle spinning techniques.6 Such MAS techniques have been applied to crown ethers,7 sodalites,8 and other Na-containing solids.9,10 Recently we showed that Na NMR shifts could be accurately calculated using conventional coupled
10.1021/jp0115143 CCC: $20.00 © 2001 American Chemical Society Published on Web 10/23/2001
Na+ Complexes with Crown Ethers
J. Phys. Chem. B, Vol. 105, No. 45, 2001 11061 experimental data. A good correlation was also found between the observed Na NMR shifts and the bond strength sum received by the Na+ (calculated from the experimental bond distances, according to the method of Brown and Altermatt in ref 13). We here extend that approach to the calculation of 23Na NMR properties for the crown ether complexes. Our goal is to have a method accurate enough yet simple enough so that we can calculate the changes in Na NMR properties as the Na+ ion passes through a microscopic model for an ion channel or as its environment in an aluminosilicate melt changes upon addition of water. Computational Methods
Figure 1. Structures calculated at the polarized SBK Hartree-Fock level for (a) 15-crown-5Na+ and (b) 15-crown-5NaH2O+.
Figure 2. Structure calculated at the polarized SBK Hartree-Fock level for Si3Al3O18H12Na+.
Hartree-Fock methods with fairly small basis sets (such as 6-31G*) applied to cluster models for the Na environment in the solid,11 using the gauge-including atomic orbital (GIAO, ref 12) approach. Calculated quadrupole coupling constants also agreed well with the experiment. One of the compounds considered in that paper was Si6O6H12Na+, an even simpler model for sodalite (with H’s terminating the Si’s and no Al’s) which nonetheless gave reasonable agreement with the sodalite
To evaluate Na NMR shieldings, electric field gradients at Na and other properties we carried out Hartree-Fock calculations14 on a number of model molecules. To evaluate the equilibrium geometries, we generally employed valence-electron only relativistic effective core potential basis sets16 designated SBK, with polarization functions of d type on each heavy atom. When crystal structures with H positions determined were available, we used them as starting geometries, extracting the coordinates using CrystalMaker 3 software.15 We used all electron 6-31G* basis sets for most of the calculations of NMR shieldings and EFGs. We used the software package GAMESS17 to evaluate energies and equilibrium geometries. The NMR shieldings and electric field gradients at the Na were calculated with Gaussian 94.18 For some of the molecules, we also employed 6-3112d,p basis sets in the NMR calculations to test the basis set dependence of our results. Diffuse functions were also added to the basis set for the geometry optimizations, as Feller and co-workers19 have shown they can have a significant effect on Na-O distances in some cases. The reference molecule chosen to represent Na+(aqueous), the species present in the usual 1 M NaCl(aq) standard, was Na(OH2)6+, as in our previous study.11 The value of the quadrupole moment of 23Na, needed to convert the largest magnitude principal component of the EFGs (q) in atomic units into quadrupole coupling constants (NQCC) in MHz, was taken from a recent study by Pyykko and co-workers.20 Theoretical studies on the ground-state geometries and electron densities of some crown-ether Na+ complexes have been previously combined with experimental studies of 1H and 13C NMR, but no calculations of NMR parameters were given.21 In addition, the effect of complexation on the 13C shifts was found to be very small (considerably less than 1 ppm) for several 18-crown-6 complexes of alkali metals. 23Na NMR spectra of the 18-crown-6 complex in solution have also been previously reported.22 This study also reported 23Na NMR parameters for solid crown-ether complexes, but approximations in the analysis of the spectrum prevented the determination of coupling constants and gave inaccurate values for the chemical shifts (see discussion in ref 7). Results Comparison of Cluster MO Results with Experiment. Calculated NMR properties are presented for the crown etherNa+ complexes in Table 1. All of the principal components of the EFG tensor, q, are given, in the normal convention, with the magnitudes of qxx, qyy, and qzz increasing (and qxx + qyy + qzz ) 0). The NQCC is then determined by the principal component of q with the largest magnitude. Experimental NMR shifts and nuclear quadrupole coupling constants in the solid state are available for comparison only for 15-crown-5NaH2O+ (actually its benzo- form), 18-crown-6NaH2O+, and (12-
11062 J. Phys. Chem. B, Vol. 105, No. 45, 2001
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TABLE 1: Calculated 23Na NMR Properties for Na(OH2)6+ and for Crown Ether-Na+ Complexes, with Available Experimental Values in Bolda molecule Na(OH2)6+ 15-crown-5Na+ 15-crown-5NaH2O+ pol. SBK pol. SBK + diffuse exp. 18-crown-6Na+ planar,D6h folded 18-crown-6NaH2O+ exp.geom. opt. geom.: pol. SBK pol. SBK + diffuse 12-crown-4Na+ opt. geom.: pol. SBK pol. SBK + diffuse (12-crown-4)2Na+ S8 pol. SBK pol. SBK + diffuse C4h pol SBK pol. SBK + diffuse exp. a
σ33 - σ11 (ppm)
σNa (ppm)
δNa (ppm)
2.49-2.52 2.20 × 5
588.9 570.5
defined as 0 +18.4
4.0 28.4
-0.015, -0.010, 0.0250 0.149, 0.149, -0.298
2.29 × 5, 2.39 2.36-2.43,2.36 2.28-2.43
576.6 582.9
+12.3 (13) + 6.0
7.5 5.1
0.051, 0.057, -0.108 0.019, 0.034, -0.053
3.5 1.7 (1.45)
2.66-2.69 2.41-2.60
607.3 596.9
-18.4 -8.0
15.3 16.4
0.051, 0.056, -0.107 0.039, 0.058, -0.098
3.5 3.2
2.45-2.62,2.32 2.44-2.64,2.43 2.50-2.66,2.37
587.5 590.6 591.7
+1.4 (+1) -1.7 -2.8
13.5 10.9 12.8
0.002, 0.033, -0.035 0.003, 0.030, -0.033 0.014, 0.037, -0.051
1.2 (0.95) 1.1 1.7
2.32 × 4 2.26 × 4
593.6 588.4
-4.7 +0.5
2.40 × 8 2.54 × 8 2.51 × 8 2.50 × 8 2.47-2.54
579.6 594.1 590.0 589.4
+9.3 -5.2 (-6) -1.1 -0.5
R(Na-O) (Å)
6.3 4.8 20.4 22.8 (14) 22.2 22.3
qxx, qyy, qzz at Na (in au)
NQCC at Na 0.8 9.8
-0.0006, -0.0006, 0.0012 0.010, 0.010, -0.021
0.03 0.7
-0.045, -0.045, 0.091 -0.076, -0.076, 0.151 -0.060, -0.060, 0.120 -0.060, -0.060, 0.119
3.0 (1.1) 5.0 3.9 3.9
Experimental Na-O distances from ref 23.
crown-4)2Na+. For all of the complexes except 18-crown-6Na+ we calculated optimized geometries using both polarized SBK and polarized SBK + diffuse function basis sets (for 18-crown6Na+ we could not get SCF convergence with diffuse functions in the basis set). For 15-crown-5Na+ the addition of diffuse functions changed the Na-O distance by less than 0.01 Å, but for the other complexes, the changes in optimized geometry were larger, although they showed no consistent trend. The structures calculated at the polarized SBK Hartree-Fock level for 18-crown-6Na+, 18-crown-6NaH2O+, 12-crown-4Na+, and (12-crown-4)2Na+ are shown in Figure 3a-d. We also performed a calculation using the single experimental geometry available, 18-crown-6NaH2O+ (ref 23 b), which was fully defined, with H as well as heavy atom positions tabulated (although based upon assumed C-H distances). For that geometry the agreement of experiment and calculation is very good for both the chemical shift, δ, and the NQCC. The optimized geometry for 18-crown-6NaH2O+ obtained using the experimental geometry as a starting point also gives geometry, shift, and NQCC which agree well with experiment. Similarly, for 15-crown-5NaH2O+, for which we optimized the geometry with the X-ray structure for the heavy atoms as starting point, the agreement with the experimental δ is also very good for the polarized SBK geometry, although the calculated NQCC is somewhat too high. The geometry optimized using diffuse functions in the basis set gives a worse shift value but a better value for the NQCC compared to experiment. The worst discrepancies with experiment occur for the (12-crown-4)2Na+ complex, for which the choice of point group symmetry (S8 or C4h) and the neglect or inclusion of diffuse functions has a strong influence upon the optimized Na-O distance and upon the calculated shift (the S8 structure is found to be of lowest energy, consistent with the experimental symmetry in the solid). In Figure 4, we plot the calculated 23Na NMR shifts vs the calculated Na-O distance for the different calculated geometries of (12-crown-4)2Na+. The good correlation shows that the Na-O distance is the main determinant of the shift. Even a complex with a counterion added, (12-crown-4)2Na+..ClO4-, with optimized Na-O distances of 2.42 Å × 4 and 2.60 Å ×
4 and a calculated σNa of 588.1 ppm, fits this same shift vs distance correlation. The best agreement with the experimental shielding occurs for the S8 optimized geometry using the polarized SBK + diffuse basis, but this geometry gives a NQCC value much larger than experiment. It should be noted that IR studies24 of crown ether-Na+ complexes in solution indicate that the Na+ cation can in some cases be tunneling between multiple energy minima, so that any approach based on a single static geometry may be somewhat problematic. It is not clear whether such dynamic effects could be the source of the discrepancy for (12-crown-4)2Na+. Notice that the effect on the 23Na NMR shielding of adding water to a crown ether-Na+ complex is not a simple one. Addition of H2O increases the shielding for the 15-crown-5 case and reduces it for the 18-crown-6 case. In 15-crown-5Na+, the Na is essentially coplanar with the five ether O atoms, whereas in 15-crown-5NaH2O+, the Na has moved above the ring, increasing its bond distances to the ether O atoms and consequently reducing the magnitude of their deshielding. The added deshielding contribution from the H2O is not large enough to overcome this effect, so the overall deshielding drops when the water is added. For 18-crown-6Na+, addition of water does not significantly move the Na from its central position in the puckered ring, but the H2O does contribute some additional deshielding. Thus, the total deshielding is increased when H2O is added in the 18-crown-6Na+ case. The interaction energy of the crown ether-Na+ complex with H2O is also much more favorable in the 18-crown-6 case, with hydration energies (gas-phase, polarized SBK Hartree-Fock, uncorrected for basis set superposition error) of about -440 kJ/mol for 15-crown-5Na+ and -630 kJ/mol for 18-crown6Na+. This is consistent with the difference between H2O effects upon the Na shielding in 15-crown-5Na+ and 18-crown-6Na+; in the 18-crown-6 case, H2O can bond to the Na+ without significantly weakening its bonding to the O atoms of the cyclic ether, so the overall interaction is more favorable. We have also carried out some NMR shielding calculations with larger basis sets and have optimized some geometries using hybrid Hartree-Fock density functional methods but have not
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J. Phys. Chem. B, Vol. 105, No. 45, 2001 11063
Figure 3. Structures calculated at the polarized SBK Hartree-Fock level for (a) 18-crown-6Na+, (b) 18-crown-6NaH2O+, (c) 12-crown-4Na+, and (d) (12-crown-4)2Na+.
been able to find any method which gives a more accurate fit to the available experimental data. Use of larger basis sets in the NMR calculations, e.g., 6-3112d,p bases, invariably produces larger shifts. For example,18-crown-6NaH2O+ at its experimental geometry has a calculated shift of +6.3 ppm using the larger basis (with respect to Na(OH2)6+ calculated at the same basis set level), compared to shifts of +1.4 ppm calculated at the 6-31G* level and +1 from experiment. The use of B3LYP geometries on the other hand seems to reduce the shift range, e.g., giving a shift of +1.7 for 15-crown-5H2O+ (using the 6-31G* basis), compared with +12.3 from the polarized SBK Hartree-Fock geometry and +13 from experiment. Calculation of Distance Dependence of Deshielding and EFG for Smaller Molecules. We also carried out a number of calculations in which the distance between the Na+ and the O atoms of an ether or crown ether were systematically varied, to determine the distance dependence of the properties. First, we varied the distance between Na+ and the O atom of dimethyl ether, (CH3)2O (the simplest possible model for an ether linkages), holding all other geometric parameters fixed. The calculated 23Na shielding at each distance was subtracted from the shielding of free gas-phase Na+ (623.2 ppm, calculated at
Figure 4. Calculated 23Na NMR shift, δ (ppm), vs calculated Na-O distance from various calculations on (12-crown-4)2Na+ in Table 1. Least-squares fitted line is shown, with r2 value.
11064 J. Phys. Chem. B, Vol. 105, No. 45, 2001
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Figure 6. Absolute deshieldings from cluster MO calculations (for compounds from Table 1) vs absolute deshieldings estimated using additivity approximation and correlation of deshielding vs distance discussed in text and shown in Figure 5a. Least-squares fitted line is shown, with r2 value.
Figure 5. (a) Calculated absolute deshielding of Na vs Na-O distance in (CH3)2ONa+ (b) Calculated EFG at Na, q, vs Na-O distance in (CH3)2ONa+. Best exponential fit is shown, with r2 value.
the same 6-31G* GIAO level), to generate a quantity which we call the absolute deshielding. This quantity is plotted against the Na-O distance in Figure 5a and is well fit by an exponential dependence, ∆σ ) 1862 × 10-1.064R. This result is like that previously reported11 for the interaction of Na+ with H2O, (SiH3)2O, and SiH3ONa+. However, the magnitude of the absolute deshielding at any given Na-O distance is a good deal smaller for (CH3)2O than it was for H2O or any of the siloxane or silicate species in ref 11. The calculated EFG at Na in the complex with (CH3)2O is plotted as a function of Na-O distance in Figure 5b and shows an exponential dependence similar to that for the deshielding. If we now estimate the total deshielding of the Na+ in each of the different crown ether complex geometries listed in Table 1 by simply adding the contributions from the different Na-O bonds in the complex, weighted by the Na-O distance according to the formula above, we can obtain estimated total absolute deshieldings. These are plotted against the absolute deshieldings obtained directly from the cluster MO calculations in Figure 6. It is clear that the absolute deshieldings estimated using this additivity approach correlate well with the absolute deshieldings
from the full cluster MO calculations, with a correlation coefficient of 0.954, even better than in our previous study11 of silicates and aluminosilicates. Note though that the directly calculated deshieldings are about 20% larger than those obtained from the additivity approach, possibly because (CH3)2O is a fairly crude model for the ether O in the crown ether. We have performed a similar analysis for the 12-crown-4 ligand interacting with Na+, varying the Na-O distance in a symmetric (C4V) approach geometry. For the complexes of the other crown ethers, such a restricted geometry would not be realistic because the larger crowns “wrap around” the cation, but for 12-crown-4, the crown remains fairly rigid upon approach of the cation. The dependence of the absolute deshielding on Na-O distance is given in Figure 7 and is qualitatively of the same form as that seen for the interaction of Na+ with (CH3)2O. The deshielding calculated for the double12-crown-4 complex, at a given Na-O distance, is about 1.8 times as large as that for the single 12-crown-4. Thus, the deshielding apparently becomes “saturated” so that the effect of the second ligand is somewhat smaller than that of the first, as seen in our previous study.11 It is also interesting to examine the variation in calculated EFG with Na-O distance in the 12crown-4Na+ case. Rather than the continuous increase in q with reduction in distance observed in the (CH3)2ONa+ case, we see that the q values actually decrease as the Na-O distance decreases, approaching zero for a distance around 2.3 Å. Thus, the small value observed experimentally for the NQCC in (12-crown-4)2Na+ could be explained by a reduced value for the Na-O distance. In our previous study of 23Na NMR of solids, we observed a correlation between the bond strength sum received by the Na and the absolute deshielding. Bond strength sums were calculated using the approach of Brown and Altermatt which relates the bond strength to the bond distance.13 For Na-O bonds, they give the functional dependence of the bond strength, si, as
si ) exp[(1.803 - R)/0.37] where R is the Na-O bond length in Å. The bond strength sum is simply the sum of the si for the different bonds to Na+. Applying this method to the complexes in Table 1, we obtain bond strength sums whose correlation with our calculated
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J. Phys. Chem. B, Vol. 105, No. 45, 2001 11065
Figure 8. Calculated absolute deshielding vs bond strength sum (using method of ref 13 and calculated geometries). Least-squares fitted line is shown, with r2.
TABLE 2: Calculated 17O and 13C NMR Properties for Free Ligands and Complexes of 12-crown-4 and 15-crown-5, Using Polarized SBK Geometries, with Shieldings in ppm and EFGs in Atomic Units
Figure 7. (a) Absolute deshieldings calculated for various Na-O distances in 12-crown-4Na+. (b) EFG at Na, q, calculated for various Na-O distances in 12-crown-4Na+. Exponential fit is shown for deshieldings and broken line is given as guide to eye for the EFGs.
absolute deshielding is shown in Figure 8. The correlation is quite strong, with an r2 of 0.956. Therefore, we can predict the deshielding rather accurately from the bond strength sum alone, if we know the structure. An even more detailed analysis of the shielding is possible, focusing upon the correlation between absolute deshielding and the valence orbital population on Na, as shown in our previous work. However, this relationship is exponential as well as being basis set dependent, so it has less general applicability than does the deshielding vs bond strength sum relationship shown in Figure 8. Calculation of 17O and 13C Shieldings in Crown Ether Complexes and 23Na Shieldings for Crown Ether Complexes in Solution and for Sodalites. We have also calculated NMR parameters for 17O and 13C for the free crown ethers and their Na+ complexes. Representative results for the more symmetric complexes are given in Table 2. We find that changes in the C shielding are small and apparently unsystematic. By contrast, the O atoms in the complexes are systematically shielded with respect to the free ligand and their quadrupole coupling constants
molecule
σO(ppm)
qO(au)
σC(ppm)
12-crown-4 12-crown-4Na+ (12-crown-4)2Na+ 15-crown-5 15-crown-5Na+ 15-crown-5NaH2O+
349.7 357.6 354.6 344.3 355.1 352.9-353.8
2.085 1.947 1.971 2.009 1.902 1.916-1.930
138.8 137.7 136.7 140.6 141.3 140.6-141.3
are reduced. The magnitude of the effect increases as the Na-O distance in the complex decreases. Thus, it may be possible to identify these complexes by virtue of changes in their 17O NMR. Finally, it is well established that 23Na NMR shifts are often different in solutions of crown ether salts compared to their solid-state values. For example, Saito and Tabeta22 report δNa ) -10.9 ppm for the shift of the 18-crown-6Na+ complex in CDCl3 solution (compared to the value of +1 ppm in the solid state from Wong and Wu, ref 7). Although we have no direct information on the structure of this complex in solution, our calculated shift for the most stable form of 18-crown-6Na+ (without coordinated water) is -8.0 ppm, suggesting that this structure provides a good description of the 18-crown-6Na+ complex in the weakly complexing CDCl3 solvent. As we noted earlier, the sodalites bear many similarities to the crown ether complexes. For the Si3Al3O18H12Na-2 complex, we calculate an essentially planar geometry about the Na+, with three Na-O distances of 2.58 and three of 2.73 Å, which compares well with experimental values25 of 2.56 and 2.69 Å. The Na shielding calculated is 597.7 ppm, which is similar to our previous result of 593.4 ppm for Si6O6H12Na+, which has the same planar six-coordinate geometry about the Na+. The calculated q value in Si3Al3O18H12Na-2 is 0.134, giving a NQCC of 4.4 MHz, somewhat smaller than the reported value of 5.9 MHz.26d Unfortunately, the experimental values reported for the 23Na NMR shift show a wide range, with values from about -1026a to +10 ppm26c,d reported (with respect to Na+(aq) as reference). Our calculated shielding yields a shift of -8.8 ppm, more consistent with the earlier results.
11066 J. Phys. Chem. B, Vol. 105, No. 45, 2001 Conclusion Calculations at the Hartree-Fock GIAO level with small basis set, such as 6-31G*, yield accurate 23Na NMR shifts and quadrupole coupling constants for crown-ether and sodalite complexes if accurate geometries are available. The problem is to determine geometries for species with weak bonding and large flexible ligands such as the crown ethers. It also appears that 23Na NMR deshieldings can be calculated fairly accurately using an additivity approach, in which contributions from individual Na-O bonds (strongly dependent upon Na-O distance) are summed, at least for complexes in which all of the O atoms are in ether linkages. It remains to be seen if this approach can be extended to large cryptands, which may have chemically inequivalent O atoms, such as ether O’s and carbonyl O’s, coordinated to the Na+. Calculated deshieldings at Na are also shown to correlate well with bond strength sums, calculated using the bond distance based approach of Brown and Altermatt. Acknowledgment. This work was supported by NSF Grant EAR-0001031 and DOE Grant DE-FG02-94ER14467. References and Notes (1) Lauger, P. Angew. Chem., Int. Ed. Engl. 1985, 24, 905-923. (2) Kohn, S. C. Mineralog. Mag. 2000, 64, 389-408. (3) Lehn, J.-M. Angew. Chem., Int. Ed. Engl. 1988, 27, 89-112. (4) Pregel, M. J.; Jullien, L.; Lehn, J.-M. Angew. Chem., Int. Ed. Engl. 1992, 31, 1637-1640. (5) Engelhardt, L.; Felsche, J.; Sieger, P. J. Am. Chem. Soc. 1992, 114, 1173-1182. (6) (a) Mueller, K. T.; Sun, B. Q.; Chingas, G. C.; Zwanziger, J. W.; Terao, T.; Pines, A. J. Magn. Reson. 1990, 86, 470-487. (b) Farnan, I.; Grandinetti, P. J.; Baltisberger, J. H.; Stebbins, J. F.; Werner, U.; Eastman, M. A.; Pines, A. Nature 1992, 358, 31-35. (c) Frydman, L.; Harwood, J. S. J. Am. Chem. Soc. 1995, 117, 5367-5367. (7) Wong, A.; Wu, G. J. Phys. Chem. A 2000, 104, 11844-11852. (8) Engelhardt, G.; Sieger, P.; Felsche, J. Anal. Chim. Acta 1993, 283, 967-985. (9) Koller, H.; Engelhardt, G.; Kentgens, A. P. M.; Sauer, A. J. Phys. Chem. 1994, 98, 1544-1551.
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