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An experimental analysis of linewidth variations in electron spin resonance spectra incorporating the effects of anisotropic rotational diffusion
JOURNAL
OF MAGNETIC
RESONANCE
16,165-171
(1974)
An Experimental Analysis of Linewidth Variations in Electron Spin ResonanceSpectra Incorporating the Effects of Anisotropic Rotational Diffusion T.E. GOUGHAND Department
of Chemistry,
University
R.G. HACKER
of Waterloo,
Waterloo,
Ontario,
Canada
Received February 25,1974 Linewidth variations within the electron spin responance spectra of the species [M,-pyrazine]+ (M = 7Li, 23Na) are measured and analyzed in terms of a model in which the rotational diffusion about the N-N axis is considered to be more rapid than that about the remaining mutually perpendicular axes. Although the analysis is successful for effects generated by the nitrogen and alkali metal nuclei, it fails when applied to the ring protons. INTRODUCTION
In a previous paper (I) we described a detailed investigation of linewidth variations in the electron spin resonance spectrum of [Na,-pyrazine]+. An attempt was made to interpret the measured variations within the framework of the relaxation-matrix theory of Freed and Fraenkel (2, 3), which assumes a single correlation time, $, is sufficient to describe the rotational diffusion of the solute. Such an analysis was perfectly feasible, but the data emerging from the analysis were physically somewhat unrealistic. Thus the correlation time $ at -30°C was found to be 2.74 x lo-l1 set, while the N-**Na interionic distance was estimated at 1.91 A. The former value seems too short, while the latter is obviously too small. The situation is reminiscent of other investigations (4-6) in which it was found that Freed and Fraenkel theory did not produce numerically correct results. It is customary to point out that these numerical anomalies are probably the results of neglecting anisotropy in the rotational diffusion; however, when an attempt was made to extend the theory to include anisotropic rotational diffusion, a negative diffusion coefficient was obtained (7). By the introduction of postulated fluctuations in isotropic hyperfine splitting constants, it was found possible to obtain more acceptable results and the general conclusion was reached that anisotropic rotational diffusion is not the source of the often observed anomalous lack of L@; broadening from ring protons in aromatic radical anions. Nevertheless, it is obvious that the spectrum must be sensitive to anisotropy in the rotational diffusion. Consider a radical containing a nucleus having an axially symmetric hyperfine tensor (e.g., a nitroxide). Rotation about the axis of symmetry, say the Z axis, does not modulate the hyperfine splitting constant and, therefore, the spectrum is influenced only by the Xand Y tumbling modes. Should the radical contain a second nucleus with a hyperfine tensor axially symmetric along the X axis, then this nucleus affects the spectrum via the Y and Z tumbling modes. Thus the total appearance Copyright 0 1974 by Academic Press, Inc. All rights of reproduction in any form reserved. Printec. in Great Britain
165
166
GOUGH
AND
HACKER
of the spectrum is affected by the relative rates of X, Y, and 2 tumbling, i.e., by the anisotropy of rotational diffusion. In this work, therefore, we attempt to use Freed’s extension (7) of Freed and Fraenkel theory (2,3) so that we may investigate the effects of anisotropic rotational diffusion upon an electron spin resonance spectrum. However, because of the difficulties experienced by previous workers in interpreting information concerning ring protons, we omit such information from our analysis. The systems chosen for study are [23Na,-pyrazine]+ and [‘Li,-pyrazine]+ : for both systems previous investigations (I, 8,9) have determined that the alkali metal cations are symmetrically located on the N-N axis of the anion. FORMULATION
OF
THE
LINEWIDTH
EXPRESSION
The following formulation uses Freed’s extension (7) of standard Freed and Fraenkel theory (2,3) and wherever possible employs the associated symbols and terminology. The species [M,-pyrazine]+ contains two completely equivalent nitrogen and two completely equivalent alkali metal nuclei; however, the four equivalent hydrogen nuclei must be subdivided into two equivalent pairs (2, 5 and 3, 6) thereby generating off-diagonal elements in the relaxation matrix. The linewidths of the spectrum are determined by the eigenvalues of the relaxation matrix, and formulae have been derived relating the width of the ith line in the spectrum to the various spectral densities (3). For [M,-pyrazine]+ the appropriate formula is shown in Eq. [I], where, because of the size of the nitrogen hyperfine splitting, its effects have been taken to second order. (~;‘(~N,JNM”,n;i,>>*
=AN(JN(JN+ 1) +5/3&g* +4l&$,* + AM(JM(& + 1) + 5/3A?i) A&, {
+j=Nsp@j, i + j +k=N,H,M c >,
CJk~~,*fik, I + x VI
The terms in A, are the purely dipolar contributions to the linewidth from the jth group of nuclei, the terms in B., are the dipolar g-tensor broadening contributions, while the C,, terms are the dipolar cross-term contributions from the three groups of nuclei taken in pairs. Xis a residual linewidth term independent of the various spectral indices. The A,, B,, and C,, are related to the spectral densities by the formulae [2]: AN =j$$(O) AM =j&(O) B, = 16/3&j;“” (0) s, C,, = 16/3j$:’ (0) eJsk. The required spectral densities may be deduced from Eq. 2.13 of (7) as in and [4]:
PI Eqs.
[3]
LINEWIDTHS
AND
ANISOTROPIC
ROTATIONAL
167
DIFFUSION
where ijim) and g@‘) are the spherical components of the various hyperfine tensors and g tensor, respectively, and A,, ,,,, denotes the spherical components of the rotational diffllsion tensor. The position of the ith line wi is given by Eq. [5] in which IV,,defines the center of the spectrum, and once again the nitrogen hyperfine splitting is included to second order: Wi=Wo+
1
- 4
uj@j,*
(JN(JN
+
1) -
&3,/2%,
j=N, H,M
[51
EXPERIMENTAL
Procedures for the production of the species [M,-pyrazine]+ have already been described (8). The only modification to these procedures was the use of isotopically pure (99.5 %) ‘Li (Oak Ridge) both as the reducing agent and as a source of lithium tetraphenylboride which was synthesized via the chloride (IO). The interface used to record the ielectron spin resonance spectra in digital form has been described previously (II), as has the least squares fitting program used to extract the various hyperfine and linewidth parameters which characterize the spectrum (I). RESULTS
AND
DISCUSSION
The electron spin resonance spectrum of [7Li,-pyrazine]+ in tetrahydrofuran was recorded in five replicate experiments, under constant spectrometer settings at a sample temperature of -36.3 + 0.5”C. Each spectrum was analyzed separately, and Table 1 TABLE
EXPERIMENTALLY DETERMINED TERMSFOR [‘Li&‘yrazine]+
HYPERFINE SPLWIINGS IN TETRAHYDROFVRAN
WIDTH
Hyperfme
1
splittings
AND LINEAT -36.3”C
(MHz) --~--
uN = 18.767
+ 0.012
UN = -7.591
f 0.007
UL‘ = -2.182
* 0.002
~-
Linewidth AN = 37.42
terms (kHz)
f 0.45
&=37.21+1.80
CNHa=-8.10k1.20
AHa = 1.81 k 0.74
&“=-1.10+0.39
Cp,,, = 23.26
+ 0.69
ALI = 0.789
&I
C”,,
f 0.57
+ 0.067
= 3.72 + 0.24 X = 65.93
a Calculated
values AH = 15.0,
= -1.34
+ 0.61 BH = 0.37,
CNH = -10.6,
CHLl =
-6.90.
contains the mean experimental values for the hyperfine splittings and linewidth parameters together with the associated standard deviations. Because of the uncertainties discussed in the introduction concerning differential line-broadening effects generated by ‘H nuclei, the analysis which follows restricts
168
GOUGH
AND
HACKER
itself to the use of the six observables AN, &, A,, &, CNM, and gisO.However, seven unknowns are involved in the complete description of line broadening generated by anisotropic rotational diffusion, viz., R 1,2, 3, the tumbling rates about the three principal axes of the molecule, g,,,, 3, the three principal components of the g-tensor, and r,, the interionic N-M+ distance. Accordingly, we simplify the problem by making the physically reasonable approximation that the two “end over end” tumbling modes of the radical have identical rates, while the tumbling rate about the N-N axis is different, because it does not involve motion of the alkali-metal cations and their associated solvation shells. We define axis 1 to be the axis through the nitrogen nuclei, axis 2 to be the remaining in-plane axis, and axis 3 to be the axis perpendicular to the ring. Thus we approximate R, + R, = R3. With this approximation we may then adapt Eq. [3.6] of (7) to give the relationship between the spherical and Cartesian components of the rotational diffusion tensor: &,o = 4X/[IOXY I”,, = d/‘(X-
- 3(X2 + Y”)], Y)/[lOXY-
A**,* = f ;x+2Y/[loxY-
3(X2 + Y2)], 3(X2 + Y2)],
Fl
where X= RI + 5Rz, Y = 3(R, + R,), and the A,,,,,. = A,9 ‘,,, = L,,,,+,,~. Analysis of [‘Liz-Pyrazine]+
Results The experimental value of A, in Table 1 together with a value of @n = -664 MHz
may be used in McConnell’s relationship to calculate the spin density on carbon as 0.1142. Normalization of spin density then shows pN = 0.2716. Assuming that the geometry of the pyrazine anion is unchanged from that of pyrazine (12) and that the molecular orbital occupied by the unpaired electron is a linear combination of Slater 2pz orbitals of carbon and nitrogen with effective nuclear charges, 2, of 3.25 and 3.09, respectively, SF’ is calculated, including nonlocal contributions, as 12.50 MHz. fig*) terms are calculated to be negligible in comparison with &‘), and since the two lithium nuclei are a completely equivalent pair, 6g2) = &*). Using Eqs. [2] and [3] and the experimental values of AN, ALi, CNLi from Table 7, we may write : j$
= 4n2/5@)’
1 o, o =
37,420 see-l,
[71
jif& = 4r~*/5(Bfp,)~& I o + 2 .0 B~<)‘(l, ,2 + i-, , *) + 4.0 &“)&“,) A,,,*) = 789 set-l,
[8]
jLji = 4rcz/5(bg) B;ty)1 o,. + 2.Ob$‘)&)J,,,)
PI
= -4.360sec-‘.
Since it is known that the lithium ions lie along the N-N axis, all &“’ values may be expressed as functions of rLi : therefore the seven equations [6] through [9] are simultaneous equations of the seven variables AO,O,3L0,*, 1+2,2, RI, R,, rLi and as such have the unique solution, for the variables of physical interest, RI = (1.66 f 0.32) x lOlo see-‘, R2 = (2.11 f 0.21) x log set-l, rL1 = 2.26 + 0.09 A, where the errors are estimated from the standard deviation of the experimental parameters. The above procedure was repeated for five sample temperatures, the results being summarized in Table 2. It can be seen that within our estimated experimental error,
LINEWIDTHS
AND
ANISOTROPIC TABLE
ROTATIONAL
169
DIFFUSION
2
THE TEMPERATURE DEPENDENCE OF THE EXPERIMENTALLY DETERMINED HYPERIWE AND LINEWIDTH TERMS OF [‘Li2-Pyrazine]+ IN TETRAHYDROFIJRAN TOGETHER WITH THE CORRESPONDING CALCULATED VALUES OF THE INTERIONIC DISTANCE AND RATES OF ROTATIONAL DIFFUSION Temperature Parameter UN MHz a,, MHz uL, MHz AN kHz AH kHz ALi kHz BN kHz BH kHz BL, kHz C,, kHz CNLI kHz C,,i kHz hi ‘4 R, x 1O-9 sec.’ Rz x 10e9 sec.-’
“C
-53.7
-46.0
-36.3
-26.6
-15.1
18.747 -7.624 -2.278 62.26 4.50 1.224 60.98 -2.0 5.66 -13.25 38.48 -0.32 2.32 11.2 1.20
18.745 -7.603 -2.230 48.82 2.93 1.060 45.30 -1.8 5.06 -10.10 32.60 -1.57 2.30 16.5 1.45
18.767 -7.591 -2.182 37.42 1.81 0.789 37.21 -1.1 3.72 -8.10 23.26 -1.34 2.26 16.6 2.11
18.716 -7.553 -2.128 29.67 0.53 0.545 26.37 -0.78 2.66 -2.23 18.31 -0.18 2.39 28.0 2.36
18.754 -7.545 -2.090 24.40 0.18 0.488 24.04 -0.81 2.51 -2.10 15.95 -0.51 2.36 37.1 2.79
the value of TLi remains constant. However, as expected, the values of RI and Rz are temperature dependent, and Fig. 1 shows the result of expressing this dependence in terms of the Arrhenius equation. Reasonably linear plots are obtained, and from them it may be deduced that R, = 2.7 x 1Ol3exp (-3,40O/RT) set-l, Rz = 3.3 x 1O1lexp (-2,4OO/RT) set-I.
WI
As anticipated, the “end over end” tumbling rate, expressed by Rz, is the slower: fro:m Eq. [lo] it can be seen that the pre-exponential factor for RI is about what would be predicted from collision theory whereas that for Rz is lo2 times smaller, presumably reflecting the more drastic changes in solvation which must accompany the “end over end” tumbling. Once the values of I,,,, are known, the experimental values of & and BLi may be used in conjunction with Eqs. [2] and [4] to calculate 0.0029, g“‘0’=2g,-g,-g,=gt2) = g, -g, = -0.00025. Me:asurement of the isotropic g-value provides a third relationship between the components of the g-tensor: giso
= +(g, + g, + g3) = 2.0034.
Solving these three simultaneous equations yields gl = 2.0037, g, = 2.0040, and g, := 2.0024. These results provide further confirmation that the procedures used have
170
GOUGH
AND
HACKER
quantitative validity since it is expected from theoretical arguments (23) that for an aromatic radical anion g, equals the free-spin value while the remaining components are larger. Back calculation of the experimental parameters AH, BH, CH,i is also possible once the various A,,,, are defined. The results have been included in Table 1: as expected, the experimental result for A, is indeed anomalously small which leads one to suspect
24-
-‘lo 10
22--O---Y O--IO R2 O---l.
I 4.0
O-
4.5 l/T
X IO3
OK-’
FIG. 1, Arrhenius plots for the rotational diffusion rate constantsof [‘L&pyrazine]+ in tetrahydrofuran.
that the partitioning of AH into contributions from j&(O) and&f&(O), which was performed for isotropic rotational diffusion, may not be carried over in identical form to the anisotropic case. In the calculation of CNH, this problem does not arise because the large value of fig’ means that only terms in &” &‘) are important ; this in turn means that for the calculation of CNH, all the hydrogens may be regarded as “pseudototally equivalent”. It is worth noting, in passing, that the isotropic model necessarily predicts a positive value for C,,. Summarizing the above analysis, we conclude that the incorporation of anisotropic rotational diffusion into the theory of the linewidths of electron spin resonance spectra does lead to physically reasonable results whenever totally equivalent nuclei are considered. However, inclusion of rotational diffusion does not improve the situation concerning equivalent, but not totally equivalent, nuclei. It would seem that the initial approximation made by Freed and Fraenkel (2) to handle such situations should be re-examined. Re-examination of [Na,-Pyrazine]+
Spectra
Because the above analysis of [Li,-pyrazine]+ gave results which are physically more realistic than those previously obtained from [Na,-pyrazine]+, it was decided to re-examine the spectrum of the latter species, at -4O”C, and to analyze the results
LINEWIDTHS
AND
ANISOTROPIC
ROTATIONAL
171
DIFFUSION
including anisotropic rotational diffusion. Best fit results for the 6 observables were r Na -- 1.69 A, R1 = 3.22 x 10’ see-l, R2 = 7.89 x log see-l, g, = 2.0038, g, = 2.0039, and g3 = 2.0026. Inclusion of anisotropic rotational diffusion, rather than improving the situation, makes r,, even smaller and predicts the reverse of the expected relationship between RI and R,. In order to understand these results, we postulate that the isotropic sodium hyperfine coupling constant is modulated by motion of the sodium ion with respect to the pyrazine anion. The source of this motion is presumably an equilibrium existing between different solvates of the ionic associate; such effects have been previously postulated by Hirota (14) to account for @r$ broadening of sodium naphthalenide ion pairs, and by Al-Baldawi and Gough (8) to account for the aUOItXilOUS temperature dependence Of aNa for [Na,-pyrazine]+. This additional broa.dening can only be incorporated into the present analysis by spurious shortening of ha, which will then lead to greater calculated broadening effects for the sodium nucleus. REFERENCES GOUGH AND R. G. HACKER, J. Magn. Resonance 6,129 (1972). FREED AND G. K. FRAENKEL, J. Chem. Phys. 39,326 (1963). FRAENKEL, J. Phys. Chem. 71, 139 (1967). FREED AND G. K. FRAENKEL, J. Chem. Phys. 40,1815 (1964). BARTON AND G. K. FRAENKEL, J. Chem. Phys. 41,695 (1964). SEGAL, A. RAYMOND, AND G. K. FRAENKEL, J. Chem. Phys. S&l336 FREED, J. Chem. Phys. 41,2077 (1964). AL BALDAWI AND T. E. GOUGH, Can. J. Chem. 48,2798 (1970). AL BALDAWI AND T. E. GOUGH, Can. J. Chem. 49,2059 (1971). IO. M. Szwmc, J. Phys. Chem. 69,608 (1965). 11. T. E. GOUGH AND F. W. GROSSMAN, J. Magn. Resonance 7,24 (1972). 12. T. A. CLAXTON AND D. MCWILLIAMS, Trans. Faraday Sot. 65,2129 (1969). 13. A. J. STONE, Mol. Phys. 6, 509 (1963). 14. N. HIROTA, J. Whys. Chem. 71,127 (1967). 1. 2. 3. 4. 5. 6. 7. 8. 9.
‘I”. J. G. J. B. I#. J. S. S.
E. H. K. H. L. G. H. A. A.
(1969).
OF MAGNETIC
RESONANCE
16,165-171
(1974)
An Experimental Analysis of Linewidth Variations in Electron Spin ResonanceSpectra Incorporating the Effects of Anisotropic Rotational Diffusion T.E. GOUGHAND Department
of Chemistry,
University
R.G. HACKER
of Waterloo,
Waterloo,
Ontario,
Canada
Received February 25,1974 Linewidth variations within the electron spin responance spectra of the species [M,-pyrazine]+ (M = 7Li, 23Na) are measured and analyzed in terms of a model in which the rotational diffusion about the N-N axis is considered to be more rapid than that about the remaining mutually perpendicular axes. Although the analysis is successful for effects generated by the nitrogen and alkali metal nuclei, it fails when applied to the ring protons. INTRODUCTION
In a previous paper (I) we described a detailed investigation of linewidth variations in the electron spin resonance spectrum of [Na,-pyrazine]+. An attempt was made to interpret the measured variations within the framework of the relaxation-matrix theory of Freed and Fraenkel (2, 3), which assumes a single correlation time, $, is sufficient to describe the rotational diffusion of the solute. Such an analysis was perfectly feasible, but the data emerging from the analysis were physically somewhat unrealistic. Thus the correlation time $ at -30°C was found to be 2.74 x lo-l1 set, while the N-**Na interionic distance was estimated at 1.91 A. The former value seems too short, while the latter is obviously too small. The situation is reminiscent of other investigations (4-6) in which it was found that Freed and Fraenkel theory did not produce numerically correct results. It is customary to point out that these numerical anomalies are probably the results of neglecting anisotropy in the rotational diffusion; however, when an attempt was made to extend the theory to include anisotropic rotational diffusion, a negative diffusion coefficient was obtained (7). By the introduction of postulated fluctuations in isotropic hyperfine splitting constants, it was found possible to obtain more acceptable results and the general conclusion was reached that anisotropic rotational diffusion is not the source of the often observed anomalous lack of L@; broadening from ring protons in aromatic radical anions. Nevertheless, it is obvious that the spectrum must be sensitive to anisotropy in the rotational diffusion. Consider a radical containing a nucleus having an axially symmetric hyperfine tensor (e.g., a nitroxide). Rotation about the axis of symmetry, say the Z axis, does not modulate the hyperfine splitting constant and, therefore, the spectrum is influenced only by the Xand Y tumbling modes. Should the radical contain a second nucleus with a hyperfine tensor axially symmetric along the X axis, then this nucleus affects the spectrum via the Y and Z tumbling modes. Thus the total appearance Copyright 0 1974 by Academic Press, Inc. All rights of reproduction in any form reserved. Printec. in Great Britain
165
166
GOUGH
AND
HACKER
of the spectrum is affected by the relative rates of X, Y, and 2 tumbling, i.e., by the anisotropy of rotational diffusion. In this work, therefore, we attempt to use Freed’s extension (7) of Freed and Fraenkel theory (2,3) so that we may investigate the effects of anisotropic rotational diffusion upon an electron spin resonance spectrum. However, because of the difficulties experienced by previous workers in interpreting information concerning ring protons, we omit such information from our analysis. The systems chosen for study are [23Na,-pyrazine]+ and [‘Li,-pyrazine]+ : for both systems previous investigations (I, 8,9) have determined that the alkali metal cations are symmetrically located on the N-N axis of the anion. FORMULATION
OF
THE
LINEWIDTH
EXPRESSION
The following formulation uses Freed’s extension (7) of standard Freed and Fraenkel theory (2,3) and wherever possible employs the associated symbols and terminology. The species [M,-pyrazine]+ contains two completely equivalent nitrogen and two completely equivalent alkali metal nuclei; however, the four equivalent hydrogen nuclei must be subdivided into two equivalent pairs (2, 5 and 3, 6) thereby generating off-diagonal elements in the relaxation matrix. The linewidths of the spectrum are determined by the eigenvalues of the relaxation matrix, and formulae have been derived relating the width of the ith line in the spectrum to the various spectral densities (3). For [M,-pyrazine]+ the appropriate formula is shown in Eq. [I], where, because of the size of the nitrogen hyperfine splitting, its effects have been taken to second order. (~;‘(~N,JNM”,n;i,>>*
=AN(JN(JN+ 1) +5/3&g* +4l&$,* + AM(JM(& + 1) + 5/3A?i) A&, {
+j=Nsp@j, i + j +k=N,H,M c >,
CJk~~,*fik, I + x VI
The terms in A, are the purely dipolar contributions to the linewidth from the jth group of nuclei, the terms in B., are the dipolar g-tensor broadening contributions, while the C,, terms are the dipolar cross-term contributions from the three groups of nuclei taken in pairs. Xis a residual linewidth term independent of the various spectral indices. The A,, B,, and C,, are related to the spectral densities by the formulae [2]: AN =j$$(O) AM =j&(O) B, = 16/3&j;“” (0) s, C,, = 16/3j$:’ (0) eJsk. The required spectral densities may be deduced from Eq. 2.13 of (7) as in and [4]:
PI Eqs.
[3]
LINEWIDTHS
AND
ANISOTROPIC
ROTATIONAL
167
DIFFUSION
where ijim) and g@‘) are the spherical components of the various hyperfine tensors and g tensor, respectively, and A,, ,,,, denotes the spherical components of the rotational diffllsion tensor. The position of the ith line wi is given by Eq. [5] in which IV,,defines the center of the spectrum, and once again the nitrogen hyperfine splitting is included to second order: Wi=Wo+
1
- 4
uj@j,*
(JN(JN
+
1) -
&3,/2%,
j=N, H,M
[51
EXPERIMENTAL
Procedures for the production of the species [M,-pyrazine]+ have already been described (8). The only modification to these procedures was the use of isotopically pure (99.5 %) ‘Li (Oak Ridge) both as the reducing agent and as a source of lithium tetraphenylboride which was synthesized via the chloride (IO). The interface used to record the ielectron spin resonance spectra in digital form has been described previously (II), as has the least squares fitting program used to extract the various hyperfine and linewidth parameters which characterize the spectrum (I). RESULTS
AND
DISCUSSION
The electron spin resonance spectrum of [7Li,-pyrazine]+ in tetrahydrofuran was recorded in five replicate experiments, under constant spectrometer settings at a sample temperature of -36.3 + 0.5”C. Each spectrum was analyzed separately, and Table 1 TABLE
EXPERIMENTALLY DETERMINED TERMSFOR [‘Li&‘yrazine]+
HYPERFINE SPLWIINGS IN TETRAHYDROFVRAN
WIDTH
Hyperfme
1
splittings
AND LINEAT -36.3”C
(MHz) --~--
uN = 18.767
+ 0.012
UN = -7.591
f 0.007
UL‘ = -2.182
* 0.002
~-
Linewidth AN = 37.42
terms (kHz)
f 0.45
&=37.21+1.80
CNHa=-8.10k1.20
AHa = 1.81 k 0.74
&“=-1.10+0.39
Cp,,, = 23.26
+ 0.69
ALI = 0.789
&I
C”,,
f 0.57
+ 0.067
= 3.72 + 0.24 X = 65.93
a Calculated
values AH = 15.0,
= -1.34
+ 0.61 BH = 0.37,
CNH = -10.6,
CHLl =
-6.90.
contains the mean experimental values for the hyperfine splittings and linewidth parameters together with the associated standard deviations. Because of the uncertainties discussed in the introduction concerning differential line-broadening effects generated by ‘H nuclei, the analysis which follows restricts
168
GOUGH
AND
HACKER
itself to the use of the six observables AN, &, A,, &, CNM, and gisO.However, seven unknowns are involved in the complete description of line broadening generated by anisotropic rotational diffusion, viz., R 1,2, 3, the tumbling rates about the three principal axes of the molecule, g,,,, 3, the three principal components of the g-tensor, and r,, the interionic N-M+ distance. Accordingly, we simplify the problem by making the physically reasonable approximation that the two “end over end” tumbling modes of the radical have identical rates, while the tumbling rate about the N-N axis is different, because it does not involve motion of the alkali-metal cations and their associated solvation shells. We define axis 1 to be the axis through the nitrogen nuclei, axis 2 to be the remaining in-plane axis, and axis 3 to be the axis perpendicular to the ring. Thus we approximate R, + R, = R3. With this approximation we may then adapt Eq. [3.6] of (7) to give the relationship between the spherical and Cartesian components of the rotational diffusion tensor: &,o = 4X/[IOXY I”,, = d/‘(X-
- 3(X2 + Y”)], Y)/[lOXY-
A**,* = f ;x+2Y/[loxY-
3(X2 + Y2)], 3(X2 + Y2)],
Fl
where X= RI + 5Rz, Y = 3(R, + R,), and the A,,,,,. = A,9 ‘,,, = L,,,,+,,~. Analysis of [‘Liz-Pyrazine]+
Results The experimental value of A, in Table 1 together with a value of @n = -664 MHz
may be used in McConnell’s relationship to calculate the spin density on carbon as 0.1142. Normalization of spin density then shows pN = 0.2716. Assuming that the geometry of the pyrazine anion is unchanged from that of pyrazine (12) and that the molecular orbital occupied by the unpaired electron is a linear combination of Slater 2pz orbitals of carbon and nitrogen with effective nuclear charges, 2, of 3.25 and 3.09, respectively, SF’ is calculated, including nonlocal contributions, as 12.50 MHz. fig*) terms are calculated to be negligible in comparison with &‘), and since the two lithium nuclei are a completely equivalent pair, 6g2) = &*). Using Eqs. [2] and [3] and the experimental values of AN, ALi, CNLi from Table 7, we may write : j$
= 4n2/5@)’
1 o, o =
37,420 see-l,
[71
jif& = 4r~*/5(Bfp,)~& I o + 2 .0 B~<)‘(l, ,2 + i-, , *) + 4.0 &“)&“,) A,,,*) = 789 set-l,
[8]
jLji = 4rcz/5(bg) B;ty)1 o,. + 2.Ob$‘)&)J,,,)
PI
= -4.360sec-‘.
Since it is known that the lithium ions lie along the N-N axis, all &“’ values may be expressed as functions of rLi : therefore the seven equations [6] through [9] are simultaneous equations of the seven variables AO,O,3L0,*, 1+2,2, RI, R,, rLi and as such have the unique solution, for the variables of physical interest, RI = (1.66 f 0.32) x lOlo see-‘, R2 = (2.11 f 0.21) x log set-l, rL1 = 2.26 + 0.09 A, where the errors are estimated from the standard deviation of the experimental parameters. The above procedure was repeated for five sample temperatures, the results being summarized in Table 2. It can be seen that within our estimated experimental error,
LINEWIDTHS
AND
ANISOTROPIC TABLE
ROTATIONAL
169
DIFFUSION
2
THE TEMPERATURE DEPENDENCE OF THE EXPERIMENTALLY DETERMINED HYPERIWE AND LINEWIDTH TERMS OF [‘Li2-Pyrazine]+ IN TETRAHYDROFIJRAN TOGETHER WITH THE CORRESPONDING CALCULATED VALUES OF THE INTERIONIC DISTANCE AND RATES OF ROTATIONAL DIFFUSION Temperature Parameter UN MHz a,, MHz uL, MHz AN kHz AH kHz ALi kHz BN kHz BH kHz BL, kHz C,, kHz CNLI kHz C,,i kHz hi ‘4 R, x 1O-9 sec.’ Rz x 10e9 sec.-’
“C
-53.7
-46.0
-36.3
-26.6
-15.1
18.747 -7.624 -2.278 62.26 4.50 1.224 60.98 -2.0 5.66 -13.25 38.48 -0.32 2.32 11.2 1.20
18.745 -7.603 -2.230 48.82 2.93 1.060 45.30 -1.8 5.06 -10.10 32.60 -1.57 2.30 16.5 1.45
18.767 -7.591 -2.182 37.42 1.81 0.789 37.21 -1.1 3.72 -8.10 23.26 -1.34 2.26 16.6 2.11
18.716 -7.553 -2.128 29.67 0.53 0.545 26.37 -0.78 2.66 -2.23 18.31 -0.18 2.39 28.0 2.36
18.754 -7.545 -2.090 24.40 0.18 0.488 24.04 -0.81 2.51 -2.10 15.95 -0.51 2.36 37.1 2.79
the value of TLi remains constant. However, as expected, the values of RI and Rz are temperature dependent, and Fig. 1 shows the result of expressing this dependence in terms of the Arrhenius equation. Reasonably linear plots are obtained, and from them it may be deduced that R, = 2.7 x 1Ol3exp (-3,40O/RT) set-l, Rz = 3.3 x 1O1lexp (-2,4OO/RT) set-I.
WI
As anticipated, the “end over end” tumbling rate, expressed by Rz, is the slower: fro:m Eq. [lo] it can be seen that the pre-exponential factor for RI is about what would be predicted from collision theory whereas that for Rz is lo2 times smaller, presumably reflecting the more drastic changes in solvation which must accompany the “end over end” tumbling. Once the values of I,,,, are known, the experimental values of & and BLi may be used in conjunction with Eqs. [2] and [4] to calculate 0.0029, g“‘0’=2g,-g,-g,=gt2) = g, -g, = -0.00025. Me:asurement of the isotropic g-value provides a third relationship between the components of the g-tensor: giso
= +(g, + g, + g3) = 2.0034.
Solving these three simultaneous equations yields gl = 2.0037, g, = 2.0040, and g, := 2.0024. These results provide further confirmation that the procedures used have
170
GOUGH
AND
HACKER
quantitative validity since it is expected from theoretical arguments (23) that for an aromatic radical anion g, equals the free-spin value while the remaining components are larger. Back calculation of the experimental parameters AH, BH, CH,i is also possible once the various A,,,, are defined. The results have been included in Table 1: as expected, the experimental result for A, is indeed anomalously small which leads one to suspect
24-
-‘lo 10
22--O---Y O--IO R2 O---l.
I 4.0
O-
4.5 l/T
X IO3
OK-’
FIG. 1, Arrhenius plots for the rotational diffusion rate constantsof [‘L&pyrazine]+ in tetrahydrofuran.
that the partitioning of AH into contributions from j&(O) and&f&(O), which was performed for isotropic rotational diffusion, may not be carried over in identical form to the anisotropic case. In the calculation of CNH, this problem does not arise because the large value of fig’ means that only terms in &” &‘) are important ; this in turn means that for the calculation of CNH, all the hydrogens may be regarded as “pseudototally equivalent”. It is worth noting, in passing, that the isotropic model necessarily predicts a positive value for C,,. Summarizing the above analysis, we conclude that the incorporation of anisotropic rotational diffusion into the theory of the linewidths of electron spin resonance spectra does lead to physically reasonable results whenever totally equivalent nuclei are considered. However, inclusion of rotational diffusion does not improve the situation concerning equivalent, but not totally equivalent, nuclei. It would seem that the initial approximation made by Freed and Fraenkel (2) to handle such situations should be re-examined. Re-examination of [Na,-Pyrazine]+
Spectra
Because the above analysis of [Li,-pyrazine]+ gave results which are physically more realistic than those previously obtained from [Na,-pyrazine]+, it was decided to re-examine the spectrum of the latter species, at -4O”C, and to analyze the results
LINEWIDTHS
AND
ANISOTROPIC
ROTATIONAL
171
DIFFUSION
including anisotropic rotational diffusion. Best fit results for the 6 observables were r Na -- 1.69 A, R1 = 3.22 x 10’ see-l, R2 = 7.89 x log see-l, g, = 2.0038, g, = 2.0039, and g3 = 2.0026. Inclusion of anisotropic rotational diffusion, rather than improving the situation, makes r,, even smaller and predicts the reverse of the expected relationship between RI and R,. In order to understand these results, we postulate that the isotropic sodium hyperfine coupling constant is modulated by motion of the sodium ion with respect to the pyrazine anion. The source of this motion is presumably an equilibrium existing between different solvates of the ionic associate; such effects have been previously postulated by Hirota (14) to account for @r$ broadening of sodium naphthalenide ion pairs, and by Al-Baldawi and Gough (8) to account for the aUOItXilOUS temperature dependence Of aNa for [Na,-pyrazine]+. This additional broa.dening can only be incorporated into the present analysis by spurious shortening of ha, which will then lead to greater calculated broadening effects for the sodium nucleus. REFERENCES GOUGH AND R. G. HACKER, J. Magn. Resonance 6,129 (1972). FREED AND G. K. FRAENKEL, J. Chem. Phys. 39,326 (1963). FRAENKEL, J. Phys. Chem. 71, 139 (1967). FREED AND G. K. FRAENKEL, J. Chem. Phys. 40,1815 (1964). BARTON AND G. K. FRAENKEL, J. Chem. Phys. 41,695 (1964). SEGAL, A. RAYMOND, AND G. K. FRAENKEL, J. Chem. Phys. S&l336 FREED, J. Chem. Phys. 41,2077 (1964). AL BALDAWI AND T. E. GOUGH, Can. J. Chem. 48,2798 (1970). AL BALDAWI AND T. E. GOUGH, Can. J. Chem. 49,2059 (1971). IO. M. Szwmc, J. Phys. Chem. 69,608 (1965). 11. T. E. GOUGH AND F. W. GROSSMAN, J. Magn. Resonance 7,24 (1972). 12. T. A. CLAXTON AND D. MCWILLIAMS, Trans. Faraday Sot. 65,2129 (1969). 13. A. J. STONE, Mol. Phys. 6, 509 (1963). 14. N. HIROTA, J. Whys. Chem. 71,127 (1967). 1. 2. 3. 4. 5. 6. 7. 8. 9.
‘I”. J. G. J. B. I#. J. S. S.
E. H. K. H. L. G. H. A. A.
(1969).