- Email: [email protected]
Comparative effects of the intermolecular and intramolecular electron transfer on ESR spectra
Volume
23, number
3
CHEMICAL
COMPARATIVE
EFFECTS
PHYSICS
OF INTERMOLECULAR
ELECTRON
TRANSFER
1 December
LETTERS
1973
AND iNTRAMOLECULAR
ON ESR SPECI-RA
N.M. ATHERTON Department
of Chemistry. Received
The University. Sheffield S3 7HF, UK 31 August
1973
The lines in the ESR specrrum of a radical-anion consistins of two identical but magnetically independent halves, and in which rhe unpaired electron is undergoing slew intramolecular transfer between the two halves, aIe superpositions of Iorentzians of different widths. This is in contrast to the situation for slow intermolecular electron transfer, when all the components of a degenerate transition have the same width. The effects of the two processes on the spectra are only indistinguishable when the exchange rate is considerably less than the linewidth in the absence of exchange.
Szwarc and his collaborators have recently described [I, 21 a novel ESR method for studying the mobility and flexibility of polymer chains as characterised by the rate of collisions between the ends of the chain in a molecule. The method was applied to (Y,w-di-l-naphthyln&canes: the radical-anions were prepared and the rate of intramolecular transfer of the unpaired electron between the naphthyl groups measured from the broadening of the lines in the ESR spectra. It was suggested that quantitative kinetic data could be obtained by calibrating the spectra against those obtained on intermolectdar etectron transfer between the radical-anion of a a-(mono)-1-naphthyl-n-alkane and the corresponding neutral molecule, the assumption being that at slow exchange rates the effects of inter- and intra-molecular exchange on the spectra are the same. The purpose of this note is to discuss this assumption_ The difference between the intra- and the intermo!ecular processes is that for the intra- the outcome is completely
determined
before the exchange
occurs
whereas for the inter- it is not. The consequences are that for slow intramolecular exchange the lines are superpositions of lines of different widths while for slow intermolecuIar exchange all the components of a degenerate line have the same width. The formal derivation of these results is readily achieved using the Kaplan-Alexander method [3,4] _ The application of this method to ESR problems has been described several times for both intra- and inter454
molecular exchange [5], and the analysis wil! not be repeated here. The results for the slow exchange limit are obtained by assuming that non-degenerate transitions are well-separated. For intramolecular exchange of the special type under consideration one finds that a line of degeneracy Dj consists of {(EDi)D,J lorentzians with an exchange contribution of (l/1) to their width and Dj components which are unbroadened. Here (7) is the mean lifetime between exchanges and D,- refers to the degeneracy with regard only to the nuclei which have non-zero hyperfine coupling at infmitely slow exchange. A normalising factor cDi/ZZDi) must be included in calculations to keep the correct relative intensities between transitions at different frequencies. For intermolecular exchange one obtains the long-familiar results [6] that a line of degeneracy Di acquires an exchange contribution of {l -(~j/CD~)} X (1 /r) to its width. If the widths of the components of a transition in a system undergoing intramolecular exchange are averaged one obtains the intermolecular exchange width. It has long been recognised [7] that the average lineshape only gives a good description of the actual shape of a superposition of lorentzians if the differences in width of the components are small: in the case being considered here the exchange rate must be really slow. Use of the Kaplan-Alexander method without any approximation enabIes spectra to be calculated over the whole range of rates. The simplest system which exhibits all the interesting
Volume
23. number
3
CHEMICAL
PHYSICS
LETTERS
1
December 1973
broadened spectra is discernible. It seems that analysis of the intramolecular exchange broadening as though it were due to intermolecular exchange would probably be the major source of error in the measurement of the. rate constant. However, at the faster rate the differnot
INTERMOLECULAR
Fig. 1. Effects of intermolecular and intramolecular electron transfer for a system with hyperfme coupling :o two equivalent protons. The complete spectra are always symmetrical and
SJ only one line of relative intensity unity is drawn for each
ence between the two cases is striking. If the peak to peak linewidths of the intra-broadened spectrum are analysed as though they contained a contribution from intermolecular exchange then one only obtains a littte over 50% of the true value of (l/r) from the wing lines, and a little over 25% of the true value from the centre line. It is indeed clear that the equivalence of intermolecular and intramolecular exchange must only be assumed when the rate is well below the natural linewidth. Of course, the slow exchange limit is no longer valid anyway when the rate is sufficient to cause marked overlap of the lines. As Szwarc points out [?I, it is generally necessary to solve the complete lineshape equations. In conclusion, it is worth stressing that the pattern of line-broadening obtained in Szwarc’s type of experiment is different from that found in many other cases where exchange between different equivalent conformations of a radical gives rise to linewidth alternation [B] . The difference arises because couplings are switched between zero and non-zero values, rather than between different non-zero values. The zero coupling constants make the transitions for a non-degenerate spin state of the finitely coupled nuclei actually degenerate, so that at appropriate exchange rates they are superpositions of lines of different widths. It would be interesting to study this type of situation for other sorts of exchange
USC.
process.
feeatures is one with hyperfine coupling to a pair of equivalent spin-l /2 nuclei and calculations have been
References
carried out for this model at the slow exchange limit. For intramolecuiar exchange the wing lines are the superposition of one unbroadened lorentzian and three with an exchange contribution of (l/r) to the width,
the central line is a superposition of equal numbers of broadened and unbroadened lorentzians. For intermolecular exchange the wing lines have an extra width of (3/47) and the central line one of (1/2r). The calculated derivative lineshapes are shown in fig. 1 for r = ST, and T = T2, together with 7 = m for comparison. At the slower exchange rate very little difference between the inter- and the intramolecular excbange-
[II K. Shirnadn.
G. hloshuk. H.D. Connor, P. Cnluwe and h¶. Szwarc. Chem. Phys. Letters 14 (1972) 396. PI ht. Szwsrc. in: Nobel symposium, Vol. 22. eds. P.-O. Kinell, B. Rhnby and V. Runnstrijm-Reio (Almqvist and Wiksell, Stockholm, 1973) p. 291. [31 J. Kaplan, J. Chem. Phys. 28 (1958) 278; 29 (1958) 462. [41 S. Alexander, J. Chem. Phys. 37 (1962) 967,974. Ph.D. Thesis. Universily of Amsterdam 151 A.F. NeivaCorreia, (1967); J.R. Norris. Chem. Phys. Letters 1 (1967) 333; R.F. Adams and N.M. Atherton, Chem. Phys. Letters 1 (1967) 351; R.F. Adams, NM Atherton and A.J. Bhckhuat, Trans. Faraday Sdc. 65 (1969) 2961.
455
Volume 23, number 3
CHEMICAL PHYSICS LETTERS
[6] R.L. Ward and S.I. Weissman, J. em. Chem. Sot. 79 (1957) 2086; P.J:Zandstia aad S.I. Weissman, J. Chem. Phys. 35 (1961)
7.51.
1 December 1973
[8] A. Hudson and G.R. Luckhurst,
Chem.
Rev. 69 (19693
191;
P.D. Sullivan and J.R. Bolton, (1970) 39.
Advan.
hlag.
Reson. 4
[7] J.H. Freed and G.K. Fraenkel, J. Chem. Phys. 39 (1963) 326.
ADDENDUM
ERRATUM
13. Benoit and G. Rabii, Nuclear magnetic resonance of rapidly rotated elastomers, Chem. Phys. Letters 21 (1973)
Phys. Letters
466.
An article of DoskoEilovj. and Schneider
11J came
recently to our attention: they describe the NMR narrowing by magic angle rotation of a polymer in solution, Iike in our experiments; the physical situation appears to be different, their polymer is in the helical form and the slow mechanism corresponds to the reorientation of the helix axes.
il] D. DoskoEilovi and B. Schneider, hlacromolecul:s (1973) 76.
‘.456:. _: .
T.A. Cool and J.R. Airey, Vibrational
6
deactivation
of CO,(OO” 1) molecules by ONF, COF, and 03, Chem. 20 (1973)
The cross-section
67.
values in the last column of table
1 should be reduced by a factor of 10. That is, the entries in this column should read 0.333,0.190,0.224, and
0.026 A2, respectively.
23, number
3
CHEMICAL
COMPARATIVE
EFFECTS
PHYSICS
OF INTERMOLECULAR
ELECTRON
TRANSFER
1 December
LETTERS
1973
AND iNTRAMOLECULAR
ON ESR SPECI-RA
N.M. ATHERTON Department
of Chemistry. Received
The University. Sheffield S3 7HF, UK 31 August
1973
The lines in the ESR specrrum of a radical-anion consistins of two identical but magnetically independent halves, and in which rhe unpaired electron is undergoing slew intramolecular transfer between the two halves, aIe superpositions of Iorentzians of different widths. This is in contrast to the situation for slow intermolecular electron transfer, when all the components of a degenerate transition have the same width. The effects of the two processes on the spectra are only indistinguishable when the exchange rate is considerably less than the linewidth in the absence of exchange.
Szwarc and his collaborators have recently described [I, 21 a novel ESR method for studying the mobility and flexibility of polymer chains as characterised by the rate of collisions between the ends of the chain in a molecule. The method was applied to (Y,w-di-l-naphthyln&canes: the radical-anions were prepared and the rate of intramolecular transfer of the unpaired electron between the naphthyl groups measured from the broadening of the lines in the ESR spectra. It was suggested that quantitative kinetic data could be obtained by calibrating the spectra against those obtained on intermolectdar etectron transfer between the radical-anion of a a-(mono)-1-naphthyl-n-alkane and the corresponding neutral molecule, the assumption being that at slow exchange rates the effects of inter- and intra-molecular exchange on the spectra are the same. The purpose of this note is to discuss this assumption_ The difference between the intra- and the intermo!ecular processes is that for the intra- the outcome is completely
determined
before the exchange
occurs
whereas for the inter- it is not. The consequences are that for slow intramolecular exchange the lines are superpositions of lines of different widths while for slow intermolecuIar exchange all the components of a degenerate line have the same width. The formal derivation of these results is readily achieved using the Kaplan-Alexander method [3,4] _ The application of this method to ESR problems has been described several times for both intra- and inter454
molecular exchange [5], and the analysis wil! not be repeated here. The results for the slow exchange limit are obtained by assuming that non-degenerate transitions are well-separated. For intramolecular exchange of the special type under consideration one finds that a line of degeneracy Dj consists of {(EDi)D,J lorentzians with an exchange contribution of (l/1) to their width and Dj components which are unbroadened. Here (7) is the mean lifetime between exchanges and D,- refers to the degeneracy with regard only to the nuclei which have non-zero hyperfine coupling at infmitely slow exchange. A normalising factor cDi/ZZDi) must be included in calculations to keep the correct relative intensities between transitions at different frequencies. For intermolecular exchange one obtains the long-familiar results [6] that a line of degeneracy Di acquires an exchange contribution of {l -(~j/CD~)} X (1 /r) to its width. If the widths of the components of a transition in a system undergoing intramolecular exchange are averaged one obtains the intermolecular exchange width. It has long been recognised [7] that the average lineshape only gives a good description of the actual shape of a superposition of lorentzians if the differences in width of the components are small: in the case being considered here the exchange rate must be really slow. Use of the Kaplan-Alexander method without any approximation enabIes spectra to be calculated over the whole range of rates. The simplest system which exhibits all the interesting
Volume
23. number
3
CHEMICAL
PHYSICS
LETTERS
1
December 1973
broadened spectra is discernible. It seems that analysis of the intramolecular exchange broadening as though it were due to intermolecular exchange would probably be the major source of error in the measurement of the. rate constant. However, at the faster rate the differnot
INTERMOLECULAR
Fig. 1. Effects of intermolecular and intramolecular electron transfer for a system with hyperfme coupling :o two equivalent protons. The complete spectra are always symmetrical and
SJ only one line of relative intensity unity is drawn for each
ence between the two cases is striking. If the peak to peak linewidths of the intra-broadened spectrum are analysed as though they contained a contribution from intermolecular exchange then one only obtains a littte over 50% of the true value of (l/r) from the wing lines, and a little over 25% of the true value from the centre line. It is indeed clear that the equivalence of intermolecular and intramolecular exchange must only be assumed when the rate is well below the natural linewidth. Of course, the slow exchange limit is no longer valid anyway when the rate is sufficient to cause marked overlap of the lines. As Szwarc points out [?I, it is generally necessary to solve the complete lineshape equations. In conclusion, it is worth stressing that the pattern of line-broadening obtained in Szwarc’s type of experiment is different from that found in many other cases where exchange between different equivalent conformations of a radical gives rise to linewidth alternation [B] . The difference arises because couplings are switched between zero and non-zero values, rather than between different non-zero values. The zero coupling constants make the transitions for a non-degenerate spin state of the finitely coupled nuclei actually degenerate, so that at appropriate exchange rates they are superpositions of lines of different widths. It would be interesting to study this type of situation for other sorts of exchange
USC.
process.
feeatures is one with hyperfine coupling to a pair of equivalent spin-l /2 nuclei and calculations have been
References
carried out for this model at the slow exchange limit. For intramolecuiar exchange the wing lines are the superposition of one unbroadened lorentzian and three with an exchange contribution of (l/r) to the width,
the central line is a superposition of equal numbers of broadened and unbroadened lorentzians. For intermolecular exchange the wing lines have an extra width of (3/47) and the central line one of (1/2r). The calculated derivative lineshapes are shown in fig. 1 for r = ST, and T = T2, together with 7 = m for comparison. At the slower exchange rate very little difference between the inter- and the intramolecular excbange-
[II K. Shirnadn.
G. hloshuk. H.D. Connor, P. Cnluwe and h¶. Szwarc. Chem. Phys. Letters 14 (1972) 396. PI ht. Szwsrc. in: Nobel symposium, Vol. 22. eds. P.-O. Kinell, B. Rhnby and V. Runnstrijm-Reio (Almqvist and Wiksell, Stockholm, 1973) p. 291. [31 J. Kaplan, J. Chem. Phys. 28 (1958) 278; 29 (1958) 462. [41 S. Alexander, J. Chem. Phys. 37 (1962) 967,974. Ph.D. Thesis. Universily of Amsterdam 151 A.F. NeivaCorreia, (1967); J.R. Norris. Chem. Phys. Letters 1 (1967) 333; R.F. Adams and N.M. Atherton, Chem. Phys. Letters 1 (1967) 351; R.F. Adams, NM Atherton and A.J. Bhckhuat, Trans. Faraday Sdc. 65 (1969) 2961.
455
Volume 23, number 3
CHEMICAL PHYSICS LETTERS
[6] R.L. Ward and S.I. Weissman, J. em. Chem. Sot. 79 (1957) 2086; P.J:Zandstia aad S.I. Weissman, J. Chem. Phys. 35 (1961)
7.51.
1 December 1973
[8] A. Hudson and G.R. Luckhurst,
Chem.
Rev. 69 (19693
191;
P.D. Sullivan and J.R. Bolton, (1970) 39.
Advan.
hlag.
Reson. 4
[7] J.H. Freed and G.K. Fraenkel, J. Chem. Phys. 39 (1963) 326.
ADDENDUM
ERRATUM
13. Benoit and G. Rabii, Nuclear magnetic resonance of rapidly rotated elastomers, Chem. Phys. Letters 21 (1973)
Phys. Letters
466.
An article of DoskoEilovj. and Schneider
11J came
recently to our attention: they describe the NMR narrowing by magic angle rotation of a polymer in solution, Iike in our experiments; the physical situation appears to be different, their polymer is in the helical form and the slow mechanism corresponds to the reorientation of the helix axes.
il] D. DoskoEilovi and B. Schneider, hlacromolecul:s (1973) 76.
‘.456:. _: .
T.A. Cool and J.R. Airey, Vibrational
6
deactivation
of CO,(OO” 1) molecules by ONF, COF, and 03, Chem. 20 (1973)
The cross-section
67.
values in the last column of table
1 should be reduced by a factor of 10. That is, the entries in this column should read 0.333,0.190,0.224, and
0.026 A2, respectively.