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Optical spectrum of the icosahedral C60- “follene-60”
CHEMICAL PHYSICS LETTERS
Volume 137, number 6
OPTICAL SPECTRUM Sven LARSSON,
Andrey
OF THE ICOSAHEDRAL
3 July 1987
Cso - “FOLLENE-60”
VOLOSOV
Department ofPhysical Chemistry, Chalmers University of Technology and Umverslty of Gothenburg, Gothenburg, Sweden
and Ame ROSBN Department of Physics, Chalmers University of Technology and Umverslty of Gothenburg, Gothenburg, Sweden
Received 3 March 1987; in final form 14 April 1987
The electronic structure and spectra of the icosahedral C&, “follene-60”. are examined by use of the CNDO/S method. The calculated ionization energy and electron affinity are in accord with experimental results. The first allowed optical transition is at higher energy than calculated previously.
A molecule Cbo has recently been discovered in vaporization of carbon species from graphite into a high density helium flow using focused laser radiation [ 11. This fascinating discovery has resulted in further experimental studies of positive and negative carbon clusters [ 2-51, C,La complexes [ 6,7] and the reactivity of large carbon clusters [ 81. The existence of CGOwith unusual stability in an icosahedral structure was anticipated from Hilckel calculations as early as 198 1 [ 91. Recent theoretical studies support this prediction that CeO may be stable in an icosahedral football-like configuration [ 10-l 81 although it has been suggested that different arrangements of the 12 pentagons and 20 hexagons would form possible isomers [ 191. According to one of the most accurate calculations [ 131 the bond lengths are alternating, with the pentagonal edge ones 0.07 8, longer than the bonds shared with hexagonal rings. Since the IUPAC name is somewhat unwieldy [ 201, footballene may be a suitable name in spite of the bond length alternation. However, here we will use “follene-60” which is derived from the latin word (follis) for football. In a recent calculation [ 141 using the discrete variational (DVM) Xa method [ 2 11, Hale obtained ionization energies and optical transition energies for 0 009-2614/87/$ (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division)
the equal-bond-length case. Internal configuration interaction, however, usually plays an important role in aromatic molecules, for instance in the case of approximate peripheral quantization [ 221. In the present case approximate quantization on a spherical surface should apply [ 111. It is therefore of interest to compare with results obtained within the CNDO/S method [ 231, where configuration interaction is carried out between singly excited states of a singlet configuration. The configuration interaction calculations included 250 configurations with the lowest energies. In the present application we have used the Mataga-Nishimoto approximation [24] for two-centre repulsion integrals and the method by Volosov et al. [25] for evaluation of intensities of dipole transitions. As a check of the accuracy of this method calculations were done for naphthalene for which we obtain a theoretical value of 8.7 eV for the ionization energy compared to an experimental value of 8.3 eV [ 261. Evaluation of optical transitions for naphthalene within this approximation shows strong contiguration interaction between the four lowest singly excited states with the energies 4.1, 4.6, 5.6 and 6.0 eV above the ground state with the oscillator strengths B.V.
501
Volume 137, number
6
CHEMICAL
PHYSICS
0.0015, 0.012, 0.60 and 0.15, respectively, in good agreement with experiment [ 271. The CNDO/S method has been tested previously on aromatic molecules with up to seven rings [28]. The error in the S,+S, transition was always below 0.5 eV. As a certain type of molecule was extended in size (naphthalene, anthracene, tetracene, etc.) the correct trend was predicted. Calculations for follene-60 were done for the equalbond-length case (1.42 1 A) and for the alternatingbond-length case (1.474 8, for pentagonal edge and 1.400 8, for bonds shared by hexagonal rings). Energy eigenvalues for the highest occupied valence levels and the lowest unoccupied orbitals for the case with unequal bond lengths are given in fig. 1. Qualitatively similar results are obtained for the case with equal bond lengths. The spherical surface quantization [ 111 applies approximately for 7~orbitals. Not
LETTERS
3 July 1987
shown in the figure are the levels for these orbitals with MO energies at - 17.7, - 16.4 and - 14.3 eV for I=O, 1 and 2, respectively. For I= 3 we obtain a split level with MO energies at - 12.0 and - 11.6 eV, respectively. for I= 4 we obtain molecular levels with energies -8.95 (g.J and -8.72 eV (4). The HOMO and LUMO levels are still recognizable as I= 5 orbitals with symmetries h, and flu, respectively. The LUMO+ 1 orbital is oft,, symmetry, however. The orbital order and splittings agree well with the results of Hale [ 141. The HOMO energy is - 7.55 eV, which should be compared with a value of -7.35 eV for the equal bond length case and the calculated ionization energy of 6.4 eV by Hale. In fig. 1 electronic transitions between molecular orbitals i and a are denoted by an arrow. Their energies above the ground state are given by El--E~=E,-c,-J,,+2K,,
Orbital energy (eV)
t 0
-_____--______-_ -----------__--_
-1 -2
--3
r:
L G
-7
%I
-a -9
--
“g --_-----_
Fig. 1. Orbital excitations ordered according to the energy expectation value of singlet projected Slater determinants. For each set of orbital transitions i&a only the highest and lowest in energy are given along the energy axis in the lower part of the figure. The calculated transition energies are given in the lower part of the figure, plotted as number of transitions in an energy interval of 0.01 eV.
502
Oscillator strength
1.5
3 July 1987
CHEMICAL PHYSICS LETTERS
Volume 137, number 6
F
5
Energy (eV)
Fig. 2. Cakulated oscillator strengths of aliowed transitions.
and represented along the horizontal axis. Arrows are drawn in fig. 1 only for the highest and lowest transitions along this axis. The excited states resulting from the CI calculation are given along the same energy axis in fig. 1 disregarding the oscillator strengths. In fig. 2 the oscillator strengths have been included. There are 15 symmetry-forbidden HOMO+LUMO transitions in the range 2.6-2.8 eV. The allowed tmnsitions from HOMO~LUMO + 1, HOMO-+ LUMO + 2, HOMO - 1+LUMO interact considerably by CI which in general shifts the intensity from 5.4 and 5.6 eV. The first (triply degenerate) allowed transition is at 3.6 eV with a total oscillator strength of 0.8 which is very small compared to the oscillator strenghts in the range of 0.25-1.3 for the transitions at higher energies. For the equal-bond-length case we obtain a value of 3.2 eV for the first allowed transitions from HOMO -+LUMO + 1 whereas Hale [ 141 obtained 2.5 eV. As seen in fig. 1 the co~esponding transition in the different-bond-length case at 3.6 eV is not much affected by CI. There is thus a difference of 0.7 eV due to the two completely different calculational approaches and a further difference of 0.4 eV due to unequal bond lengths. The intensity is shifted by CI and this shift cannot be accounted for in an orbital excitation model. We conclude that the dipoleallowed transitions should start at a 1 eV higher energy than calculated by Hale and that much stronger transitions appear higher up in the spectrum. Experimentally it has been found that the C, clusters with n = 40-l 00 have ionization th~shol~ above the ArF photon energy of 6.42 eV and below the Fz
energy of 7.87 eV [ 7,293. Further, it was found that the C, ion signal was two orders of magnitude larger when ionizing with the 7.87 eV photons than with the 6.42 eV photons [ 281. This may indicate a nearresonant ionization with the 7.87 eV photons while multiphoton ionization may take place with the 6.42 eV photons. Our calculation of the ionization energy seems therefore to be in good agreement with the present experimental data although we have not included relaxation and correlation effects which may cause some shifts. Some indication of the accuracy of our calculations can be obtained from the earlier mentioned results for naphthalene for which the difference between the experimental and theoretical ionization energy was only 0.4 eV. To get a further test of the calculations the optical spectrum should be checked with, for example, two-photon ionization or two-colour expe~ments. Recently there has been some controversey in the literature regarding the formation mechanism for Cc0 [ 31. It now appears that C, is formed from CGO [ 51. This may be an inefficient process, however, as may be understood from the very high electron affinity of CeO(2.4 eV). At the same time the bond-length changes from CeOto C& are probably very small since the LUMO, which hosts the new electron, is delocalized over the whole molecule. The energy surfaces for C60 + e- and Cc0 will run almost parallel, preventing crossing between their energy surfaces. However, if there is clustering the excessive energy may be picked up by other molecules and electron transfer be facile. We are grateful for support from NFR, the Swedish Natural Science Research Council and STUF, Engineering Research Council of STU. Special thanks should be given to Professor Thure Jansson and M.Sc. Sten Ljungstram for help with interpretation of the latin word follis.
References [ 1] H.W. Kroto, J.R. Heath, S.C. O’Brien, R.F. Curl and R.E. Smalley,Nature318
(1985) 162.
[ 21 Y. Liu, S.C. O’Brien, Q. Zhan& J.R. Heath, F.K. Tittel, R.F. Curl, H.W. Kroto and R.E. Smaliey, Chem. Phys. Letters 126 (1986) 215.
503
Volume 137, number 6
CHEMICAL PHYSICS LETTERS
[3] M.Y. Hahn, E.C. Honea, A.J. Paguia, K.E. Schriver, A.M. Camarena and R.L. Whetten, Chem. Phys. Letters 130 (1986) 12. [4] A. O’Keefe, M.M. Ross and A.P. Baronavski, Chem. Phys. Letters 130 (1986) 17. [5] S.C. O’Brien, J.R. Heath, H.W. Kroto, R.F. Curl and R.E. Smalley, Chem. Phys. Letters 132 (1986) 99. [6] J.R. Heath, S.C. O’Brien, Q. Zhang, Y. Liu, R.F. Curl, H.W. Kroto, F.K. Tittel and R.E. Smalley, J. Am. Chem. Sot. 107 (1985) 7779. [7] D.M. Cox, D.J. Trevor K.C. Reichmann and A. Kaldor, J. Am. Chem. Sot. 108 (I 986) 2457. [8] Q. Zang, S.C. O’Brien, J.R. Heath, Y. Liu, R.F. Curl, H.W. Kroto and R.E. Smalley, J. Phys. Chem. 90 (1986) 525. [9] R. Davidson, Theoret. Chim. Acta 58 (1981) 193. [ IO] A.D.J. Haymet, Chem. Phys. Letters 122 (1985) 421; J. Am. Chem. Sot. 108 (1986) 319. [ 111 R.C. Haddon, L.E. Brus and K. Raghavachari, Chem. Phys. Letters 125 (1986) 459. [ 121D.J. Klein, T.G. Schmalz, G.E. Hite and W.A. Seitz, J. Am. Chem. Sot. 108 (1986) 1301. [ 131 M.D. Newton and R.E. Stanton, J. Am. Chem. Sot. 108 (1986) 2469. [ 141 P.D. Hale, J. Am. Chem. Sot. 108 (1986) 6087. [ 151 R.L. Disch and J.M. Schulman, Chem. Phys. Letters 125 (1986) 465.
504
3 July 1987
[ 161 T.G. Schmalz, W.A. Seitz, D.J. Klein and G.E. Hite, Chem. Phys. Letters 130 (1986) 203. [ 171 S. Satpathy, Chem. Phys. Letters 130 (1986) 545. [ 181 D.S. Marynick and S. Estreicher, Chem. Phys. Letters 132 (1986) 383. [ 191 A.J. Stoneand D.J. Wales, Chem. Phys. Letters 128 (1986) 501. [ 201 J. Castells and F. Serratosa, J. Chem. Educ. 60 (1983) 941; 63 (1986) 630; J. Am. Chem. Sot. 108 (1986) 319. [21] B. Delley and D.E. Ellis, J. Chem. Phys. 76 (1982) 1949, and references therein. [22] J.R. Platt, J. Chem. Phys. 17 (1949) 484; W. Mofftt, J. Chem. Phys. 22 (1954) 1920. [ 231 J. de1 Bene and H.H. Jaffe, J. Chem. Phys. 48 (1968) 1807, 4050. [ 241 N. Mataga and K. Nishimoto, Z. Physik. Chem. (Frankfurt) 12 (1957) 335; 13 (1957) 140. [25] A.P. Volosov, V.A. Zubkov and T.M Birshtein, Tetrahedron 31 (1975) 1259. [ 261 E. Lindholm, C. Fridh and L. Asbrink, Faraday Discussions Chem. Sot. 54 (1972) 127. [ 271 H.H. Perkampus, I. Sandeman and C.J. Timmons, eds, UV atlas of organic compounds (Butterworths, London). [28] B. Dickand B. Nickel, Chem. Phys. 78 (1983) 1. [29] D.M. Cox, D.J. Trevor, K.C. Reichmann and A. Kaldor, quotes as ref [ 41 in ref. [ 7 1.
Volume 137, number 6
OPTICAL SPECTRUM Sven LARSSON,
Andrey
OF THE ICOSAHEDRAL
3 July 1987
Cso - “FOLLENE-60”
VOLOSOV
Department ofPhysical Chemistry, Chalmers University of Technology and Umverslty of Gothenburg, Gothenburg, Sweden
and Ame ROSBN Department of Physics, Chalmers University of Technology and Umverslty of Gothenburg, Gothenburg, Sweden
Received 3 March 1987; in final form 14 April 1987
The electronic structure and spectra of the icosahedral C&, “follene-60”. are examined by use of the CNDO/S method. The calculated ionization energy and electron affinity are in accord with experimental results. The first allowed optical transition is at higher energy than calculated previously.
A molecule Cbo has recently been discovered in vaporization of carbon species from graphite into a high density helium flow using focused laser radiation [ 11. This fascinating discovery has resulted in further experimental studies of positive and negative carbon clusters [ 2-51, C,La complexes [ 6,7] and the reactivity of large carbon clusters [ 81. The existence of CGOwith unusual stability in an icosahedral structure was anticipated from Hilckel calculations as early as 198 1 [ 91. Recent theoretical studies support this prediction that CeO may be stable in an icosahedral football-like configuration [ 10-l 81 although it has been suggested that different arrangements of the 12 pentagons and 20 hexagons would form possible isomers [ 191. According to one of the most accurate calculations [ 131 the bond lengths are alternating, with the pentagonal edge ones 0.07 8, longer than the bonds shared with hexagonal rings. Since the IUPAC name is somewhat unwieldy [ 201, footballene may be a suitable name in spite of the bond length alternation. However, here we will use “follene-60” which is derived from the latin word (follis) for football. In a recent calculation [ 141 using the discrete variational (DVM) Xa method [ 2 11, Hale obtained ionization energies and optical transition energies for 0 009-2614/87/$ (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division)
the equal-bond-length case. Internal configuration interaction, however, usually plays an important role in aromatic molecules, for instance in the case of approximate peripheral quantization [ 221. In the present case approximate quantization on a spherical surface should apply [ 111. It is therefore of interest to compare with results obtained within the CNDO/S method [ 231, where configuration interaction is carried out between singly excited states of a singlet configuration. The configuration interaction calculations included 250 configurations with the lowest energies. In the present application we have used the Mataga-Nishimoto approximation [24] for two-centre repulsion integrals and the method by Volosov et al. [25] for evaluation of intensities of dipole transitions. As a check of the accuracy of this method calculations were done for naphthalene for which we obtain a theoretical value of 8.7 eV for the ionization energy compared to an experimental value of 8.3 eV [ 261. Evaluation of optical transitions for naphthalene within this approximation shows strong contiguration interaction between the four lowest singly excited states with the energies 4.1, 4.6, 5.6 and 6.0 eV above the ground state with the oscillator strengths B.V.
501
Volume 137, number
6
CHEMICAL
PHYSICS
0.0015, 0.012, 0.60 and 0.15, respectively, in good agreement with experiment [ 271. The CNDO/S method has been tested previously on aromatic molecules with up to seven rings [28]. The error in the S,+S, transition was always below 0.5 eV. As a certain type of molecule was extended in size (naphthalene, anthracene, tetracene, etc.) the correct trend was predicted. Calculations for follene-60 were done for the equalbond-length case (1.42 1 A) and for the alternatingbond-length case (1.474 8, for pentagonal edge and 1.400 8, for bonds shared by hexagonal rings). Energy eigenvalues for the highest occupied valence levels and the lowest unoccupied orbitals for the case with unequal bond lengths are given in fig. 1. Qualitatively similar results are obtained for the case with equal bond lengths. The spherical surface quantization [ 111 applies approximately for 7~orbitals. Not
LETTERS
3 July 1987
shown in the figure are the levels for these orbitals with MO energies at - 17.7, - 16.4 and - 14.3 eV for I=O, 1 and 2, respectively. For I= 3 we obtain a split level with MO energies at - 12.0 and - 11.6 eV, respectively. for I= 4 we obtain molecular levels with energies -8.95 (g.J and -8.72 eV (4). The HOMO and LUMO levels are still recognizable as I= 5 orbitals with symmetries h, and flu, respectively. The LUMO+ 1 orbital is oft,, symmetry, however. The orbital order and splittings agree well with the results of Hale [ 141. The HOMO energy is - 7.55 eV, which should be compared with a value of -7.35 eV for the equal bond length case and the calculated ionization energy of 6.4 eV by Hale. In fig. 1 electronic transitions between molecular orbitals i and a are denoted by an arrow. Their energies above the ground state are given by El--E~=E,-c,-J,,+2K,,
Orbital energy (eV)
t 0
-_____--______-_ -----------__--_
-1 -2
--3
r:
L G
-7
%I
-a -9
--
“g --_-----_
Fig. 1. Orbital excitations ordered according to the energy expectation value of singlet projected Slater determinants. For each set of orbital transitions i&a only the highest and lowest in energy are given along the energy axis in the lower part of the figure. The calculated transition energies are given in the lower part of the figure, plotted as number of transitions in an energy interval of 0.01 eV.
502
Oscillator strength
1.5
3 July 1987
CHEMICAL PHYSICS LETTERS
Volume 137, number 6
F
5
Energy (eV)
Fig. 2. Cakulated oscillator strengths of aliowed transitions.
and represented along the horizontal axis. Arrows are drawn in fig. 1 only for the highest and lowest transitions along this axis. The excited states resulting from the CI calculation are given along the same energy axis in fig. 1 disregarding the oscillator strengths. In fig. 2 the oscillator strengths have been included. There are 15 symmetry-forbidden HOMO+LUMO transitions in the range 2.6-2.8 eV. The allowed tmnsitions from HOMO~LUMO + 1, HOMO-+ LUMO + 2, HOMO - 1+LUMO interact considerably by CI which in general shifts the intensity from 5.4 and 5.6 eV. The first (triply degenerate) allowed transition is at 3.6 eV with a total oscillator strength of 0.8 which is very small compared to the oscillator strenghts in the range of 0.25-1.3 for the transitions at higher energies. For the equal-bond-length case we obtain a value of 3.2 eV for the first allowed transitions from HOMO -+LUMO + 1 whereas Hale [ 141 obtained 2.5 eV. As seen in fig. 1 the co~esponding transition in the different-bond-length case at 3.6 eV is not much affected by CI. There is thus a difference of 0.7 eV due to the two completely different calculational approaches and a further difference of 0.4 eV due to unequal bond lengths. The intensity is shifted by CI and this shift cannot be accounted for in an orbital excitation model. We conclude that the dipoleallowed transitions should start at a 1 eV higher energy than calculated by Hale and that much stronger transitions appear higher up in the spectrum. Experimentally it has been found that the C, clusters with n = 40-l 00 have ionization th~shol~ above the ArF photon energy of 6.42 eV and below the Fz
energy of 7.87 eV [ 7,293. Further, it was found that the C, ion signal was two orders of magnitude larger when ionizing with the 7.87 eV photons than with the 6.42 eV photons [ 281. This may indicate a nearresonant ionization with the 7.87 eV photons while multiphoton ionization may take place with the 6.42 eV photons. Our calculation of the ionization energy seems therefore to be in good agreement with the present experimental data although we have not included relaxation and correlation effects which may cause some shifts. Some indication of the accuracy of our calculations can be obtained from the earlier mentioned results for naphthalene for which the difference between the experimental and theoretical ionization energy was only 0.4 eV. To get a further test of the calculations the optical spectrum should be checked with, for example, two-photon ionization or two-colour expe~ments. Recently there has been some controversey in the literature regarding the formation mechanism for Cc0 [ 31. It now appears that C, is formed from CGO [ 51. This may be an inefficient process, however, as may be understood from the very high electron affinity of CeO(2.4 eV). At the same time the bond-length changes from CeOto C& are probably very small since the LUMO, which hosts the new electron, is delocalized over the whole molecule. The energy surfaces for C60 + e- and Cc0 will run almost parallel, preventing crossing between their energy surfaces. However, if there is clustering the excessive energy may be picked up by other molecules and electron transfer be facile. We are grateful for support from NFR, the Swedish Natural Science Research Council and STUF, Engineering Research Council of STU. Special thanks should be given to Professor Thure Jansson and M.Sc. Sten Ljungstram for help with interpretation of the latin word follis.
References [ 1] H.W. Kroto, J.R. Heath, S.C. O’Brien, R.F. Curl and R.E. Smalley,Nature318
(1985) 162.
[ 21 Y. Liu, S.C. O’Brien, Q. Zhan& J.R. Heath, F.K. Tittel, R.F. Curl, H.W. Kroto and R.E. Smaliey, Chem. Phys. Letters 126 (1986) 215.
503
Volume 137, number 6
CHEMICAL PHYSICS LETTERS
[3] M.Y. Hahn, E.C. Honea, A.J. Paguia, K.E. Schriver, A.M. Camarena and R.L. Whetten, Chem. Phys. Letters 130 (1986) 12. [4] A. O’Keefe, M.M. Ross and A.P. Baronavski, Chem. Phys. Letters 130 (1986) 17. [5] S.C. O’Brien, J.R. Heath, H.W. Kroto, R.F. Curl and R.E. Smalley, Chem. Phys. Letters 132 (1986) 99. [6] J.R. Heath, S.C. O’Brien, Q. Zhang, Y. Liu, R.F. Curl, H.W. Kroto, F.K. Tittel and R.E. Smalley, J. Am. Chem. Sot. 107 (1985) 7779. [7] D.M. Cox, D.J. Trevor K.C. Reichmann and A. Kaldor, J. Am. Chem. Sot. 108 (I 986) 2457. [8] Q. Zang, S.C. O’Brien, J.R. Heath, Y. Liu, R.F. Curl, H.W. Kroto and R.E. Smalley, J. Phys. Chem. 90 (1986) 525. [9] R. Davidson, Theoret. Chim. Acta 58 (1981) 193. [ IO] A.D.J. Haymet, Chem. Phys. Letters 122 (1985) 421; J. Am. Chem. Sot. 108 (1986) 319. [ 111 R.C. Haddon, L.E. Brus and K. Raghavachari, Chem. Phys. Letters 125 (1986) 459. [ 121D.J. Klein, T.G. Schmalz, G.E. Hite and W.A. Seitz, J. Am. Chem. Sot. 108 (1986) 1301. [ 131 M.D. Newton and R.E. Stanton, J. Am. Chem. Sot. 108 (1986) 2469. [ 141 P.D. Hale, J. Am. Chem. Sot. 108 (1986) 6087. [ 151 R.L. Disch and J.M. Schulman, Chem. Phys. Letters 125 (1986) 465.
504
3 July 1987
[ 161 T.G. Schmalz, W.A. Seitz, D.J. Klein and G.E. Hite, Chem. Phys. Letters 130 (1986) 203. [ 171 S. Satpathy, Chem. Phys. Letters 130 (1986) 545. [ 181 D.S. Marynick and S. Estreicher, Chem. Phys. Letters 132 (1986) 383. [ 191 A.J. Stoneand D.J. Wales, Chem. Phys. Letters 128 (1986) 501. [ 201 J. Castells and F. Serratosa, J. Chem. Educ. 60 (1983) 941; 63 (1986) 630; J. Am. Chem. Sot. 108 (1986) 319. [21] B. Delley and D.E. Ellis, J. Chem. Phys. 76 (1982) 1949, and references therein. [22] J.R. Platt, J. Chem. Phys. 17 (1949) 484; W. Mofftt, J. Chem. Phys. 22 (1954) 1920. [ 231 J. de1 Bene and H.H. Jaffe, J. Chem. Phys. 48 (1968) 1807, 4050. [ 241 N. Mataga and K. Nishimoto, Z. Physik. Chem. (Frankfurt) 12 (1957) 335; 13 (1957) 140. [25] A.P. Volosov, V.A. Zubkov and T.M Birshtein, Tetrahedron 31 (1975) 1259. [ 261 E. Lindholm, C. Fridh and L. Asbrink, Faraday Discussions Chem. Sot. 54 (1972) 127. [ 271 H.H. Perkampus, I. Sandeman and C.J. Timmons, eds, UV atlas of organic compounds (Butterworths, London). [28] B. Dickand B. Nickel, Chem. Phys. 78 (1983) 1. [29] D.M. Cox, D.J. Trevor, K.C. Reichmann and A. Kaldor, quotes as ref [ 41 in ref. [ 7 1.