The Journal of Physical Chemistry, Vol. 82, No. 10, 1978 1105
Crystal Structure of KBpa2Ttg
Soc., in press. (25) A. Rembaum and A. Eisenberg, Macromoi. Rev., 1, 57 (1966). (26) K. S. V. Santhanam, N. Jespersen, and A. J. Bard, J. Am. Chem. Soc., 99, 274 (1977). (27) M. T. Stankovich and A. J. Bard, J. Nectroanal. Chem., 85, 173 (1977).
(21) F. Ammar and J. M. Savgant, J. Nectroanal. Chem., 47, 115 (1973). (22) T. Teherani, L. A. Tinker, and A. J. Bard, J . Nectroanal. Chem., in press. (23) J. B. Flanagan, Ph.D. Dissertation, California Institute of Technology, 1978. (24) J. B. Fbnagan, S. Margel, A. J. Bard, and F. C. Anson, J. Am. Chem.
Molecular and Magnetic Structure of the Paramagnetic Ion Pair Bis(tetrag1yme)potassium Biphenyl J. H. Noordik, J. Schreurs, R. 0. Gould, J. J. Mooij, and E. de Boer* Research Institute for Materials (RIM), University of N#megen, Toernooiveid, Nijmegen, The Netherlands (Received July 5, 1977) Publication costs assisted by the University of Nijmegen
The crystal structure of K+ [CH30(CHzCHz0)4CH3]2ClzH~~ has been determined by x-ray diffraction at 120 K. The crystal data are as follows: space group Cc or C2/c, a = 9.654(3), b = 16.803(9),c = 21.845(7) 8, fl = 96.03(2)", 2 = 4, and d, = 1.03 g ~ m - The ~ . ten oxygen atoms of the two tetraglyme molecules spherically surround each K+ ion, such as to give a solvent-separated ion pair structure. Susceptibility measurements point to a ferromagnetic coupling between the spins. The ESR spectrum consists of d single exchange narrowed line with a line width depending on the orientation of the magnetic field with respwt to the crystal. The molecular g tensor of the biphenyl anion has been determined. The g tensor calculated using Stone's theory agreed well with the experimental g tensor. A satisfactory explanation for the angular dependent line width could not be given.
Introduction The concept of an ion pair was introduced in 1926 by Bjerrum-l That different types of ion pairs could exist in solution was suggested in 1954 by Fuoss2 and W i n ~ t e i n . ~ Szwarc4 referred to these distinct ion pairs as loose and tight ion pairs. In loose ion pairs the partners of the pair are separated by solvent molecules (solvent-separated ion pairs), in tight ion pairs the partners are in close contact and solvated only on the outside (contact ion pairs). In numerous papers, Szwarc and his c o - ~ o r k e r s using ,~ various techniques, shed much light on the structure of these ion pairs in solution, so that by now the concept of an ion pair is well established. Recently we reported on the structure and magnetic properties of ion pairs in the solid state.5 Examples were found both of contactSbpcand of solvent-separated ion pairs.5a In this paper we report the crystal structure and the magnetic properties of bis(tetrag1yme)potassium biphenyl (KBp2Ttg). As in the rubidium analogue5"the ions occur in the crystals as solvent-separated ion pairs. Despite the correspondence in chemical composition the compounds are not isomorphous. The unit cell of KBp2Ttg is half that of RbBp-2Ttg. The large concentration of electron spins in the crystals causes a strong exchange interaction between these spins. Under these circumstances the spin Hamiltonian for these crystals in a magnetic field is N
observed, its g value corresponds to the average value of the g values of the two sites. The exchange interaction is isotropic since no orbital angular momentum is associated with the ground states of the interacting molecules. The magnetic dipole-dipole interaction has to be included in the Hamiltonian when the electron spins are not largely separated. The components of the traceless symmetric dipolar tensor Djk are given by jk
P, 4
= x, Y ,
where r,kis the vector between the two relevant electrons, 1c/ is the electron wave function. The average g values have been measured in three planes normal to the crystallographic axes. Utilizing the crystal structure, the molecular g tensor has been derived. These values have been compared with the theoretical g values, calculated with the theory of Stone.6 The ESR line width depends on the orientation of the magnetic field with respect to the crystal axes. Using the theory of Anderson and Weiss7 and van Vleck,8 an explanation for this was sought. Susceptibility measurements revealed a ferromagnetic coupling in the crystals, contrary to antiferromagnetic coupling observed for the crystals of RbB~e2Ttg.'~
j and k , and
Experimental Section
N
where the three terms represent the Zeeman, the exchange, and the dipolar interactions, respectively. The summation indices j and k run over all pairs of spins in the crystal. The crystals under investigation contain two magnetically nonequivalent sites in the unit cell. Because of the exchange, only a single exchange-narrowed ESR line is 0022-3654/78/2082-1105$0 1.OO/O
Preparation of t h e Crystals. A solution of KBp in Ttg was prepared under high vacuum using standard techn i q u e ~ .From ~ this solution, single crystals were obtained by slowly cooling the solution to about 10 "C, 1 "C/h. Since the crystals are very sensitive to air and moisture, they had to be mounted in thin glass capillaries under a He atmosphere in a glove box. These manipulations had to be carried out at a temperature of about -20 "C because of the low melting point of the KBp-2Ttg crystals (about 40 "C). The crystals used for the x-ray data collection at
0 1978 American Chemical Society
1106
The Journal of Physical Chemistry, Vol. 82, No. 10, 1978
120 K were fixed in the glass capillary by melting the crystal surface slightly, then cooling rapidly, thereby attaching the crystals to the glass by a thin film of polycrystalline or amorphous material. Structure Determination. Unit cell dimensions and x-ray intensity data of KBpe2Ttg at 120 K were obtained with a CAD4 diffractometer, equipped with a devicelo to cool the crystal in a stream of cold nitrogen gas. Graphite-monochromated Mo Kcu radiation was used (A 0.71069 A) and the intensity data were measured employing the 6-28 scan technique. The unit cell dimensions have been calculated as the weighted average of several independently determined sets of cell dimensions, each set obtained by least-squares adjustment of the setting angles of 25 reflections with 15' < 0 C 20". Systematic absences are hkl for h + k odd, h01 for 1 odd, and OkO for k odd, consistent with the space groups Cc and C2/c. A satisfactory chemical model could only be obtained by refinement of a partially disordered structure in the centric space group. The crystal data are as follows (at 120 K): a = 9.654(3), b = 16.803(9), c = 21.845(7) A, ,f3 = 96.03(2)', V = 3465 A3, d, = 1.03 g ~ m - 2~ =, 4, linear absorption coefficient p(Mo Ka) = 1.97 cm-l, The unit cell parameters at room temperature are a = 9.763, b = 17.140, c = 22.044 A, /3 = 95.87'. To collect a full set of data, two different crystals had to be used. The data were scaled by means of the intensity control reflections measured every hour. There was no indication of crystal decomposition during the data collection period. Up to 30" in 0, 20 993 possible reflections were measured, one quarter of which are symmetry independent. After averaging, 5142 symmetry-independent reflections remained, 3417 of which were considered observed at the 3a significance level, a(n being obtained from counting statistics. No absorption correction was applied since the shape of the crystal could not be determined accurately: the maximum calculated effect of the neglect of absorption correction is less than 5% in I. The discrepancy index for the averaging (Z,=lNA/L1,=lNOwas 0.028. The intensity data were corrected for Lorentz and polarization effects and reduced to F, values. Only the observed reflections were used in the refinement of the structure. The structure was solved by direct methods ( M U L T A N ) ~ ~ and completed by Fourier methods, using a preliminary data set collected at room temperature. With these data, positions for all atoms could be found, assuming the centric space group. With 2 = 4, this requires two tetraglyme molecules to be related by a twofold axis and the biphenyl ions to lie on a center of symmetry. Least-squares refinement of this model, on the preliminary data set in space group C2/c, converged at R = 0.105 with very high anisotropic thermal parameters and C-C and C-0 distances in the tetraglyme molecule ranging from 1.1to 1.7 A for the atoms between C ( l ) and C(4). Then both a disordered centric model and an ordered acentric model were tried for the Ttg chains, but correlation between parameters was very high, and refinement did not proceed sensibly. Because of these problems, the low temperature data set was collected and the refinement was carried out with these data. However, the centric and acentric refinements still did not proceed sensibly, indicating substantial errors in the coordinates, in particular in those of the atoms C(1) to C(4) in the Ttg chain. A comparison of the coordination of the cation in this structure with that in the Rb structureSamade it possible to improve these coordinates. By fitting the coordinates of the cation Ttg moiety of the Rb structure as a rigid group to the trial
de Boer et al.
parameters of the K structure, new starting parameters for least-squares refinement of the K structure were obtained. The final agreement between the two sets of coordinates is shown in Table I, and it may be noted here that the atoms in the two symmetry-independent Ttg chains in the Rb structure from C(17) to C(22) and from C(27) to C(32) have an approximate twofold relationship, while those in the rest of the chain do not. A partially disordered Ttg molecule with site occupancies of 0.5 for the atoms C(1) to C(4) refined sensibly in the centric space group. A similar fitting of the coordinates of the R b structure to the parameters for a biphenyl ion in the K structure was also tried. Both possible orientations of a biphenyl ion, relative to a cation Ttg moiety, were refined in an acentric ordered structure and in a disordered centric structure. However, in all refinements the biphenyl ion converged to a centric ordered group, and the acentric model again gave high correlations and poor convergence. Only the centric model with the disordered Ttg chain and the ordered biphenyl ion could be refined to a chemically acceptable structure. Then hydrogen atoms (invariant, B = 5.0 A2) were entered in calculated positions (biphenyl: H at bisector of C-H-C angle at a C-H distance of 1.1A. Ttg: C-H distance 1.1A, H-C-H angle 109.5', H atoms at end carbon atoms staggered with respect to tetraglyme chains). Refinement of position parameters, anisotropic thermal parameters of all non-hydrogen atoms except the disordered ones which were kept isotropic, and population parameters of the disordered atoms, gave an R value of 0.077. The sum of the population parameters of the disordered atoms stayed close to 1,with individual values not significantly different from 0.5 for both possible chain conformations. With population parameters fixed at 0.5, anisotropic refinement of all nonhydrogen atoms reduced R to a final value of 0.060 and gave the parameters listed in Tables I1 and 111. In the last cycles of refinement, a weighting scheme of the form a2 = a: + 0.0001F,2 was used. The atomic scattering factors used for K', 0, and C were taken from ref 12, and for H from ref 13. Magnetic Measurements. The ESR measurements were carried out at room temperature on a Varian E-12 ESR spectrometer. The microwave frequency was counted with a H P 5245L frequency counter. The magnetic field was measured with the NMR probe of an AEG gaussmeter. For the calibration of the g values we used a sample of perylene in concentrated sulfuric acid, for which the g value is accurately known and is equal to 2.002 564(6).14 This sample was measured in the same position as the KBp. 2Ttg crystal had been before. Single crystals were aligned on a brass goniometer head along one of the crystal axes a, b, or c, by the double oscillation method15 using a Weisenberg camera. The goniometer head with the aligned crystal was moved from the Weisenberg camera to the ESR cavity with an accuracy of better than 1'. Then the angular dependence of the g values and line widths was measured with the magnetic field normal to the a, b, and c axes, respectively. The procedure followed to correct the measured g values for systematic errors is the same as that used in the measurements of single crystals of RbBp-2TtgaSeFor the evaluation of the corrected g values the computer program G A P L S D ~was ~ used. The susceptibility of KBp-2Ttg was measured as a function of the temperature with a foner vibrating sample magnetometer in the low temperature region (3-130 K) and with a Gouy balance in the high temperature region (80-300 K). Results and Discussion Crystal Structure. The molecular structure of KBp.
Crystal Structure of KBp-PTtg
The Journal of Physical Chemistry, Vol. 82, No. IO, 1078
1107
TABLE I : Comparison of the Final Refined Coordinates of the Cation Ttg Moiety in I with Those in I1 Placed on a Consistent Cartesian Axial System Atom
Y
z
X
Y
z
D
0.000 -0.704 -1.345 - 2.612 - 2.653 -2.154 -2.180 - 1.644 -0.276 0.410 1.777 1.671 2.911 2.713 1.942 1.608
0.000 3.860 2.739 2.495 1.885 0.560 - 0.191 - 1.567 - 1.432 - 2.658 - 2.475 - 2.216 -2.107 - 2.232 - 1.122 - 1.298
0.073 1.629 0.780 1.373 2.560 2.102 3.186 2.744 2.467 2.526 2.032 0.614 0.052 -1.458 - 1.793 -3.175
-0.151 -0.514 - 1.439 - 2.725 - 2.698 - 2.321 - 2.364 -1.759 -0.368 0.406 1.795 1.716 2.970 2.800 1.912 1.512
0.090 3.512 2.762 2.559 1.625 0.305 -0.365 - 1.929 - 1.701 - 2.926 - 2.629 - 2.395 -2.048 -2.024 -0.936 - 1.004
0.190 0.399 0.107 0.134 0.266 0.305 0.485 0.393 0.329 0.532 0.502 0.432 0.473 0.512 0.411 0.482
- 1.043 - 1.522
- 3.362 - 2.387 - 2.289
- 1.423 - 1.017
- 0.809 -1.553 - 2.672 - 3.396 - 2.636 - 2.431 -1.731 -0.365 0.400 1.826 1.810 3.080 3.019 2.072 1.895
-3.606 -2.436 -2.225 - 0.974 0.221 0.679 2.013 1.790 2.983 2.671 2.487 2.061 2.006
0.283 0.176 0.116 0.251 0.270 0.555 0.483 0.374 0.508 0.499 0.473 0.511 0.617 0.365 0.487
X
K
0.000
Atom
-2.751 -1.908 -1.011 - 3.264 - 1.434 0.073 -2.456 - 1.510 0.239 - 2.264 - 2.863 - 1.644 1.566 - 2.807 1.432 -0.276 - 2.367 0.410 2.658 - 2.457 2.475 1.777 - 2.011 1.671 2.216 - 0.585 2.911 2.107 -0.066 2.713 2.232 1.432 1.942 1,122 1,010 1.835 1.608 1.298 3.253 1.034 a D is the distance between the two corresponding atoms. Root-mean-square value of D = 0.409 A . Mean value of D = 0.385 A . TABLE 11: Final Positional Parameters X 10' for the Non-Hydrogen Atoms with Standard Deviations in Parentheses Atom
X
0 (0)
Y
4997 ( 4 )
z
25000 ( 0 )
Biphenyl Anion 33171 (23) 14897 ( 1 3 ) 45734 (27) 10950 ( 1 5 ) 57412 (25) 14601 (18) 56099 (23) 22358 (17) 43687 (22) 26447 (14) 31483 (20) 22890 (12)
52216 (10) 52340 (11) 50355 (11) 48160 (11) 48034 ( 1 0 ) 50085 ( 9 )
Tetraglyme 13017 (174) 807 ( 8 8 ) 12411 (223) -1210 (121) 5558 (80) -3004 (60) 9660 ( 9 9 ) - 4062 ( 7 3 ) 11178 (177) -10545 (106) 16685 (235) -11377 (128) 24119(127) -10791 ( 8 3 ) 11570 (123) -14426 ( 4 1 ) 21044 (54) -7819 (27) 14309 ( 6 5 ) -9617 (31) 32508 (87) - 7974 (50) 26877 (119) -8477 (101) 29095 ( 3 4 ) - 4784 ( 2 3 ) 25383 (20) 3352 (13) 24295 (33) 7439 ( 2 3 ) 18805 (31) 15573 (19) 4726 (19) 14944 (11) -1997 ( 3 2 ) 22323 (16) - 17439 ( 3 1 ) 21145 ( 1 6 ) -21129 (19) 16555 (12) -35299 (35) 14566 ( 2 5 )
42671 (68) 41625 ( 8 5 ) 37538 ( 4 1 ) 35929 ( 5 1 ) 36420 ( 8 0 ) 35479 (116) 33630 (63) 29628 (52) 27562 ( 2 2 ) 24666 (29) 24124 (50) 23905 (71) 17829 (14) 18443 ( 9 ) 12833 (14) 13671 ( 1 3 ) 14857 ( 8 ) 15356 (12) 14784 ( 1 2 ) 19865 ( 8 ) 19057 (17)
2Ttg (I) (Figures 1 and 2) is essentially the same as that of RbBp-2Ttg (11). Both are solvent-separated ion pair structures, consisting of a biphenyl anion and an alkali
Figure 1.
The
bc
projection of the unit cell of K+ [CH,O(CH,CH,-
0)4CH312C12H10'.
metal cation coordinated by two tetraglyme molecules. However despite the close correspondence in chemical composition and the similarity in the molecular structure, the packing in the two crystals is very different. The unit cell of I is half the size of that of 11, thus in I the potassium and biphenyl anions lie on symmetry elements. The glyme molecules in I show on the average twofold symmetry but their arrangement is very similar to that in 11, where the glyme chains are symmetry independent. Table I shows the agreement between the final coordinates, of the alkali metal Ttg part of the structure, in I and 11.
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The Journal of Physical Chemistry, Vol. 82,No. 70, 1978
/
de Boer et al.
I
I
fl
Figure 2. A stereoscopic view of the unit cell along the a direction.
TABLE 111: Anisotropic Thermal Parameters X l o 5for the Non-Hydrogen Atomsu Atom K C(11) C(12) C(13) C(14) C(15) C(16)
TABLE IV: Bond Distances (in A ) and Angles (in Degrees) in the KTtg Part of I
u,,
u?.l
'33
2619
2281
3093
0
384
0
3494 5005 3388 2386 2526 2419
Biphenyl Anion 3359 2615 533 4259 3001 1966 7055 3427 2609 942 7470 3171 4743 2735 278 3297 1748 393
209 400 583 519 318 64
-274 -174 56 707 381 -231
u,2
u13
'13
Tetragly me C(l) 6372 C(1') 10343 O(1) 4442 O(1') 6920 C(2) 9400 C(2') 18919 C(3) 8306 C(3') 16752 O(2) 5474 O(2') 9179 C(4) 6045 6723 C(4') C(5) 6330 O(3) 5147 5682 C(6) C(7) 5257 O(4) 4965 6921 C(8) 6526 C(9) O(5) 5051 C(10) 5100
5405 5019 1146 -3525 -387 13493 10010 5657 3720 7997 654 5072 4464 -502 1564 7123 8840 601 -1247 4071 3581 6730 9 -229 2934 4733 13683 3517 -11974 925 5897 8316 4524 1562 2320 3496 13478 1850 -4801 2315 -624 -195 5817 4268 1791 -862 5987 7362 2912 -51 5867 5037 5140 1256 349 -986 -2208 19991 5736 2220 516 -1895 10272 4529 4639 7494 4262 321 692 -1031 1235 -732 9786 3940 -516 1482 7263 4347 -2406 957 350 4887 4320 -1118 344 -737 174 4204 4088 -979 820 -555 1271 4551 3518 7233 3602 1478 165 1319 1185 2634 11249 6230 2044
a Thermal parameters are expressed in the form T = e ~ p ( - 2 n ~ ( U , , h ~t a U,,kzb*z *~ + U,312c*2t 2U,,hka*b* + 2U,,hla*c* t 2UZ3klb*c*))where the V, values are in A .
The approximate twofold relationship between the atoms C(27) to C(32) and C(17) to C(22) in I1 is a real twofold relationship between atoms C(5) to C(10) and C(5') to C(l0') in I. The atoms C(1) to C(4) and C(1') to C(4') in I correspond closely with the atoms C(23) to C(26) and C(13) to C(16) in 11. The mean distance between two corresponding atoms in I and I1 is 0.385 A. In order to refine the two glyme chain conformations at positions around a twofold axis, the disorder in the section C(1) to C(4) has to be introduced. In Table IV some relevant bond distances and angles are given. The average value of the K-0 separation is 0.085 A shorter than the Rb-0 separation in I1 (cf. Table I11 in ref 5a), while the difference in ionic radii for Rb and K is 0.14 A.17 The average C-0 and C-C distances in I are smaller than in 11, the average C-0-C and 0-C-C
K-O(1) K-O(1') K-O(2) K-O(2')
3.048(9) 2.902 (11) 2.973 (5) 2.822 (6)
K-0(3) K-0(4) K-O(5)
2.978 ( 2 ) 2.850 ( 2 ) 2.951 (2)
Tetraglyme molecules Smallest Largest value
value
Av value
0-C-C
1.267 1.417 109.1 104.7
1.427 1.501 117.0 117.6
1.400 1.476 112.6 109.8
1.392
1.425
c-0 c-c c-0-c
Figure 3. Bond distances (in A) and angles (in degrees) in the biphenyl anion. Standard deviations are 0.003 A or 0.2'.
angles are the same in I and I1 within experimental accuracy. (See Table IV in present paper and Table I11 in ref 5a.) In Figure 3 the bond angles and distances in the biphenyl anion are given. They resemble very closely those in I1 (cf. Figure 3 in ref 5a). The changes in the bond distances compared with those in the neutral molecule correspond with the calculated bond order changes on going from the neutral molecule to the anion (cf. Figure 4 of ref sa). The position of the biphenyl anion on a center of symmetry requires this ion to be planar, whereas in I1 the dihedral angle between the planes of the rings is 9.4". The packing of the molecules in I is substantially different from that in 11. In RbBp.2Ttg there exists a pair of biphenyl molecules, related by a twofold axis, having a distance between their centers of 7.58 A, which is much shorter than the distance of 9.79 A to the next nearest neighbors. In KBp2Ttg the shortest distance between two biphenyl anions is 9.65 A (translation along 6) closely followed by one of 9.69 A (C-face centering). Additional Remarks Concerning t h e Disorder. The planarity of the biphenyl ions was not considered significant. The planar structure, resulting from the refinement in the centric space group, has relatively high thermal parameters which suggest that a structure containing a disordered arrangement of nonplanar biphenyls
The Journal of Physical Chemistry, Vol. 82, No. 10, 1978 1109
Crystal Structure of KBpPTtg
paramagnetic susceptibility of X R p . 2 T t j Foner
.
,.;/ 0
I
50
1
I
100
150
4
zoo
I
I
250
30
temperature (‘K
c t0
1
Figure 4. The reciprocal of the relative paramagnetic susceptibility vs. temperature for KBpPTtg.
may be possible, A nonplanar structure similar to that in 11, disordered on apparent centers of symmetry, would not be refinable, as the maximum expected separation between atoms related by the false symmetry would be 0.2 A, much smaller than the resolution of the data (0.7 A). An ordered arrangement of nonplanar biphenyl ions in the acentric space group Cc may be ruled out by the fact that all such models refine to a centric structure. In the portion of the tetraglyme for which separate positions have been refined, there is no difficulty in distinguishing the two chain conformations. In every case, substitution of one or more atoms C(1) to C(4) from one chain in the other (C(1’) to C(4’)) or vice versa, gives at least one bond length well outside the range of normal C-C and C-0 distances. It can also be concluded that it is impossible for one cation to be coordinated by two glyme molecules both containing “primed” atoms C(1’) to C(4’), as this brings O(2’) for one glyme molecule to within 2.8 A of C(3’) for the other. Further, refinement of population parameters showed no great excess of one conformation over the other. In view of the similarity to the Rb structure, it seems most likely that every K+ ion is coordinated by glyme molecules in both conformations, which gives rise to the apparent disorder about the twofold axis. Susceptibility. In Figure 4 the reciprocal of the relative paramagnetic susceptibility of KBp.2Ttg is shown as a function of the temperature. From the high temperature points, for which the Curie-Weiss law is satisfied, a positive Weiss constant can be derived of 7 (lt3) K, indicating that the exchange interaction induces a ferromagnetic coupling. Interestingly for the crystal of RbBp-2Ttg an antiferromagnetic coupling was found. The susceptibility behavior of RbBp-2Ttg could be explained with a singlet-triplet dimer model, because in the crystal two nearest neighbors could be discerned. There, the separation between the singlet and triplet state was found to be 17 cm-l. The crystal structure of KBp2Ttg reveals that “dimers” cannot be discerned in this crystal as was supposed in one of our earlier publications.5” Accordingly the exchange interaction between the spins involves more than one neighbor, so that the interaction has to be described with a three-dimensional model. Apparently, the final result is a positive value of the exchange coupling constant J. g Tensor. The single crystals of KBp2Ttg contain two magnetically nonequivalent sites in the unit cell. Because of the presence of a large number of spins, a strong ex-
I
C* I
30
60
1 90
, 120
1
150 180 rotation angle e
Flgure 5. The variation of the g value and the derivative line width, AH,, (in Gauss), as a function of the magnetic field, rotating in the ac plane. The solid line in the lower part of the figure represents the calculated second moment, ( Aw2).
TABLE V: Experimental and Calculated Values of the g Tensor of the Bp Anion in KBp. 2Ttg and RbBp 2Ttg Exptl gx
g, g, g,, giso
KBp. 2Ttg
RbBp. 2Ttg
Calcd
2,00339 (1) 2.00264 (2) 2.00227 ( 2 ) 2.00276 ( 1 ) 2.002757 ( 6 )
2.00334 ( 2 ) 2.00262 ( 2 ) 2.00230 ( 2 ) 2.00275 ( 1 )
2.00328 2.00256 2.00238 2.00274
change interaction exists in the crystals, which averages out all intramolecular interactions. As a result, a single exchange narrowed ESR line is observed. The resonance position of this line is the average value corresponding to the two nonequivalent sites, and the measured g tensor is the average of two tensors with equal principal values, but with different orientations. In Figure 5 the g value is plotted as a function of the magnetic field in the ac plane. The principal values are found along the crystallographic directions within an accuracy of 2O, and are equal to g, = 2.00320 (l),gb = 2.002 78(1), and g,, = 2.002 31(1) (E* perpendicular to 6). The average g value is equal to 2.002 76. The isotropic g value of the “free” biphenyl anion measured in tetraglyme amounts to 2.002 757(6). The close correspondence between the average g value and this isotropic value points to a negligible contribution of the alkali spin-orbit coupling to the measured g tensor, in agreement with the solvent-separated ion pair structure of the crystal. Utilizing the known crystal structure, we can transform the average g tensor to the molecular g tensor. The method employed has been described in ref 5e for the crystal of RbBp.2Ttg. In Table V the results of these transformations are tabulated. The principal values are found along the symmetry axes of the biphenyl molecule (see for direction of axes Figure 3). As can be seen from Table V, the results of the two independent measurements agree quite well. The calculated values were obtained with the theory of Stone,6using for Stone’s empirical parameter (yl- yz)a value of -20 X 10-5.5“Excellent agreement exists between the experimental and calculated values.
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The Journal of Physical Chemistry, Vol. 82, No. 70, 7978
Line Width. The line width of an exchange narrowed ESR line depends on the orientation of the single crystal with respect to the external magnetic field. Expressions for the line width of an exchange narrowed line were derived by Anderson and Weiss7 and by van Vleck.8 Anderson and Weiss showed that, if the electron Zeeman frequency, wo, is larger than the exchange frequency, we, the half-width at half-height, w1/2, is given by ~ 3 1 1 2=
(AW')/w,
(3)
where ( A d ) is the second moment in the absence of exchange. Under this condition (wo > w e ) , van Vleck derived an expression for the second moment: ( A w 2 ) = 3/4S(S f 1)y 4 h Z z 5 k " ( 3 cos2 8, - 1)2 (4)
rlk is the distance vector between the spins j and k and the angle between r,k and the magnetic field. Equations 3 and 4 show that the line width depends on the orientation of the crystal with respect to the magnetic field. The dependence is caused by the dipolar interactions between the magnetic dipoles. In Figure 5 the behavior of the line width is shown for a rotation of the magnetic field in the ac plane. In this plane all biphenyl molecules are magnetically equivalent, so that there is no line broadening due to different resonance positions. The line width given, AHpp,is the peak-to-peak separation in the first derivative of the ESR absorption line, which is Lorentzian a t all observed orientations. In Figure 5, ( Aw2) is plotted on an arbitrary scale. In the calculation, the point dipoles are placed at the centers of the biphenyl anions and contributions from surrounding dipoles k as far as 60 A away from an arbitrary central dipole j are taken into account. A comparison between the experimental curve (dashed curve) and the calculated one (solid line) shows that the positions of maxima are predicted correctly, but that the calculated widths and the positions of minima deviate from the experimental data. For instance at 0 = 0 where the magnetic field coincides with the direction of the shortest distance between the centers of the two nearest biphenyl anions (9.763 8, at room temperature), the calculated width is at its maximum, whereas the experimental width is much lower. We have tried to improve the correspondence by taking into account the delocalized character of the electrons. Magnetic dipoles are placed at each carbon atom weighted by spin densities as calculated by the method of McLachlan.18 Only a slight improvement was achieved. One may conclude from this that the discrepancies between calculated and measured line widths are not due to the use of a dipole representation of unpaired electrons, instead of using wave functions. Also the condition wo > we necessary for the validity of eq 3 and 4 is fulfilled. From the calculated second moment and the line width at 0 ' (magnetic field along the a axis), we calculate w e = 1.8 X 1O1O s-l, which renders w: >> we2 (coo = 6 X 1O1O s-l at X-band). The discrepancy may be caused by taking into account incorrectly the contributions of the nearest neighbors, because their spins are strongly coupled by the exchange interaction. However, omitting these contributions or accounting for them by formulas holding for the case we > mol9 does not lead to a significantly better correspondence.22 Comparison between KBp in the Crystal and in Solution. The crystal structure of KBpq2Ttg shows that this ion pair can be classified as a solvent-separated ion pair. 01,
de Boer et al.
It is of interest to compare this with structural results obtained from experiments on KBp in solution. Canters and de Boerz0have carried out alkali NMR experiments on alkali biphenyl ion pairs in a range of solvents of varying polarity and solvating power and as a function of the temperature. It was observed that in tetraglyme the potassium hyperfine splitting constant, amounting only about 20 mG, did not vary with the temperature. The line width of the potassium resonance line proved to be linearly related to the viscosity of the solvent divided by the temperature. From these facts it was concluded that in solution the ion pairs occur as solvent-separated ion pairs, as is now also borne out in the solid state. Ishizu et a1.21 performed ENDOR experiments on anions of biphenyl. For a solution of KBp in 1,2-dimethoxyethane, they deduced the existence of solvated-separated ion pairs with a separation between the radical anion and the counterion of more than 8.3 8,. The closest distance between the cation and the center of the anion in our crystal amounts to 6.65 A, the next shortest distance being 7.99 A (both values calculated from the 120 K data). Supplementary Material Available: A complete listing of structure factors for KBp.2Ttg (17 pages). Ordering information is available on any current masthead page.
References and Notes N. Bjerrum, Kgt. Dan. Vidensk. Selsk., 7, 9 (1926). H. Sadek and R. M. Fuoss, J. Am. Chem. SOC., 76, 5897, 5905 (1954). S. Winstein, E. Clippinger, A. H. Fainberg, and G. C. Robinson, J. Am. Chem. SOC., 76, 2597 (1954). M. Szwarc, "Ion and Ion Pairs in Organic Reactions", Vol. 1, Wiley-Interscience, New York, N.Y., 1972. (a) J. J. Mooij, A. A. K. Klaassen, E. de Boer, H. M. L. Degens, Th. IE. M. van den Hark, and J. H. Noordik, J. Am. Chem. Soc., 96, 680 (1976); (b) J. H. Noordik, Th. E. M. van den Hark, J. J. Mooij, and A. A. K. Klaassen, Acta Crystallogr., Sect. 8 ,30, 833 (1974); (c) J. H. Noordik, H. M. L. Degens, and J. J. Mooij, bid., 31, 2144 (1975); (d) J. J. Mooij and E. de Boer, Mol. Phys., 32, 113 (1976); (e) J. J. Mooij, A. A. K. Klaassen, and E. de Boer, ibid., 32, 879 (1976). (a) A. J. Stone, Mol. Phys., 6,509 (1963); (b) ibid., 7, 311 (1964). P. W. Anderson and P. R. Weiss, Rev. Mod. Phys., 25, 269 (1953). J. H. van Vleck, Phys. Rev., 74, 1168 (1946). G. W. Canters, A. A. K. Klaassen, and E. de Boer, J . Phys. Chem., 74, 3299 (1970). F. van Bolhuis, J . Appl. Crystaliogr., 4, 263 (1971). MULTAN, a system of computer programs for the automatic solution of crystal structures from x-ray diffraction data. P. Main, M. 'M. Wwlfson, and L. Lessinger, Department of physics, Unlversity of York, England. G. Germin and J. P. Declerq, Laboratoire de chimie physique et de crystallographic, bhtiment Lavoisier, Place de Louis Pasteur, 1348 Louvain-la-Neuve, Belgique, Dec 1974. D. Cromer and J. Mann, Acta Ctystaiiogr., Sect. A , 24, 321 (1968). R. F. Stewart, E. R. Davidson, and W. T. Simpson, J. Chem. Phys., 42, 3175 (1965). (a) B. G. Segal, M. Kaplan, and G. K. Fraenkel, J. Chem. Phys., 43, 4191 (1965); (b) R. D. Allendoerfer, ibid., 55, 3615 (1971). G. H. Stout and L. H. Jensen, "X-ray Structure Determination", MacMillan, New York, N.Y., 1965. C. P. Keijzers, G. F. M. Paulussen, and E. de Boer, Mol. Phys., 29, 973 (1975). 8. S. Gourary and F. J. Adrian, Solid State Phys., I k , 127 (1960). A. D. McLachlan, Mol. Phys., 3, 233 (1960). J. E. Gulley, D. Hone, D. J. Scalapino, and B. G. Silbernagel, Phys. Rev. B , 1, 1020 (1970). (a) G. W. Canters and E. de Boer, Mol. Phys., 26, 1185 (1973); (b) ibid., 27, 665 (1974). K. Ishizu, M. Ohnishi, and H. Shikata, Bull. Chem. SOC. Jpn., 50, 76 (1977). Figure 5 in ref 5a shows that the ESR line width of RbBp-2Ttg measured in the ac plane as a function of the angle 0 behaves as (3 cos' 0 - 1)'. After this publication we were able to align the crystal along a crystallographic axis, in such a way that the direction of the magnetic field with respect to the other axes was also known. We came then to the conclusion that the maximum line width was not measured when the magnetic field is parallel to the vector r, joining nearest neighbor spins, but in a direction just perpendlcular to r. Thls means that also in the case of RbBp-2Ttg there is a dlscrepancy between the experimental and calculated line width behavlor. I
