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Breakdown of the anisotropic lorentz approximation in p-terphenyl
CHEMICAL PHYSICS LETTERS
Volume 51, number 2
BREAKDOWN
OF THE ANISOTROPIC
LORENTZ
APPROXIMATION
15 October
1977
IN p-TERPHENYL
J.H: MEYLING Laboratory for Physical Chemistry. The ihiversity Groningen. 7’7~IVetherlands
of Gromb~gen.
and PJ. BOUNDS and R.W. MUNN Department of Chemistry, University of Manchester Institute of Technology. Manchester hf60 I QD, UK Received 9 May 1977 Revised manuscript received
6 July 1977
Polarizability changes for tetracene and pentacene in a p-terphenyl host crystai calculated from Stark measurements using the anisotropic Lorentz local field disagree with physical intuition. A local field taking better account of the anisotropic p-tcrphenyl structure gives acceptable polarizability changes.
l_ Introduction In a previous paper [l] one of us reported the anisotropv of the quadratic Stark effect for tetracene and pentacene in a p-terphenyl host crystal (space group P21/a, 2 = 2). An anisotropic Lorentz approximation for the local electric field in the host was used to transform the measured spectral shifts into the molecular difference polarizability tensor of the absorbing states. The resulting values of Aa: were found to be (in cm-24): tetracene
pentacene
A~LL
29 * 3
53*4
&MlW
25+4
90F8
AffAW
5+5
-11+9
where L, Mand iV are the long, medium and normal molecular axes. These values disagree with physical intuition, which leads one to expect A@M~ < A@Q and it was suggested [l] that possibly the approximation used for the local field was responsible for this discrepancy. 234
The anisotropic Lorentz approximation is simple to apply but is hard to justify in very anisotropic crystals such as p-terphenyl. Recent work has therefore sought to calculate the local field rigorously within the approximation of treating molecules as linearly polarizable points [24]. One problem with these calculations is that the results are not unique, although extra experimental information should eliminate the arbitrariness [5,6] _ More seriously, the local fields prove to be unreasonably anisotropic, to the extent that in p-terphenyl the local field along the c’ axis is calculated to be opposite in direction to the applied field. This problem can be traced to the representation of a molecule as a single point when molecules are separated by distances comparable with their dimensions, i.e. to the breakdown of the dipole approximation. Realisation of this fact has prompted caIcuIations treating molecules as sets of pojarizable points [6-8 J , thereby taking account of molecular size, shape, and orientation_ Since these calculations give intuitively reasonable results, we have used the method of ref. [6] to caIcuIate the local field in p-terphenyl in order to reanalyse the Stark effect measurements.
Volume 5 1, number 2
CHEMICAL PHYSICS LETTERS
2. Caicuiations The method entails calcuhting the Lorentz-factor tensor Lkfis between sublattices k and k’ as the average of the tensors describing the interactions between point dipoles at the centre of each aromatic ring; interactions within the same molecule are excluded. This averaging allows us to retain the algebra of the singlepoint treatment for two molecules per unit cell related by a twofold axis. The averaged Lkke for p-terphenyl are given in ref. [6] _ The local field depends on a matrix p which is not uniquely defined in the absence of further experimental data. Here we make the arbitrary but convenient choice p = 1 which allows the local-field tensor for sublattice 1 to be calculated as [9] d, = 1 + %I,
+ L,,)=
x t
(1)
where x is the electric susceptibility tensor. In this approximation the local-field tensor is the simple average of those at the centres of the aromatic rings. it may be noted that although p has been chosen arbitrarily, the LI~Z, LIC’,bb, C’Q,and c’c’ elements of the local-field tensor are independent of this choice [3,4,9] ; how the other elements depend on p is discussed in ref. [s] _ The local-field tensor calculated from eq. (1) is
d~=~~
~~~~
~~~~.
The anisotropic Lorentz local-field tensor dL can be written as [9] dL -1+$x,
(2)
which yields
dL=
-0.134
1.542
0
0
1.615
0
0
I.967 I
( -0.134
.
(This‘ differs slightly from that in ref. [I] which used the principal components of x .) As can be seen, the ordering of the diagonal components of dL is the reverse of that in d, , the change in the c’c’ component being particularly large.
15 October
1977
Comparison of eqs. (I) and (2) shows that the anisotropic Lorentz approximation is equivalent to taking Lll + L12 as isotropic. This is far from true inp-terphenyl, where the c’c’ component of LI1 + t12 is less than a quarter of the ~%tand bb components. Physical arguments also indicate that dL will always tend to have the wrong anisotropy, because it depends only on x, which is a product of the local-field and molecular polarizability tensors. Molecules are farthest apart (implying small local fields) in the direction of their greatest dimension (implying large polarizability), so that in x a small local field tends to be masked by a large polarizability. Thus the anisotropy of dL is less than that of the true local field and may even be reversed, as in p-terphenyl. Recalculating Aa: from the Stark experiments described in ref. [l] using dl we obtain (in cm-24); tetracene
pentacene
6.5 + 3
131 t 10
23-t-4
76 + 6
-St4
-3229
The marked reduction of the c’c’ component of d, compared with the corresponding component of d L leads to an increase in the value of AorLL required to reproduce the experimental results.
3. Discussion The revised components of Ao now agree with the physical expectation that the change along the long molecular axis should exceed that perpendicular to the axis in the molecular plane. The fractional uncertainties are smaller than in the original estimates, permitting the conclusion that Q!probably decreases normal to the molecular plane on excitation. For tetracene, Aor,, and the average z agree with the values reported by Liptay et al. [lo], as before, but 3LsariA%WfiI now also agrees with the vaiue reported by Barnett et al. [l l] . This agreement is particularly encouraging since the various values refer to tetracene in different environments: p-terphenyl crystal host (this work), cyclohexane solution [lo], and polystyrene fiim ill]. 235
Volume 51, number 2
The Iocal field used here is subject to uncertainty from two sources. One is the arbitrary choice of p, but this affects only four of the elements of d and is unlikely to have more than a moderate quantitative effect on the analysis. The other is the choice of three points to represent the p-terphenyl molecule, but experience indicates that the major qualitative features should be thereby included, although some quantitative changes could ensue from a choice of more points [I 21. The anaiysis of the Stark measurements has also been simplified by neglecting the reaction field, which yields contributions depending on the square and higher powers of ACY[4,13] . Nevertheless, we are confident that the results for Aa! presented here are qualitatively correct_ We conclude that the anisotropic Lorentz approximation tends to give local Gelds with-the wrong anisotropy and may lead to qualitatively incorrect conclusions in highly anisotropic crystals. This possibility is exaggerated in analysing the quadratic Stark effect because the square of the local field is required. The revised analysis of the Stark effect measurements gives values for Aa! in tetracene and pentacene which accord with physical intuition and agree with values derived by other methods.
236
15 October 1977
CHEMICAL PHYSICS LETTERS
Acknowledgement We thank the Science Research a Research Studentship (PJB).
Council (U.K.)
for
References [II J.H. Meyling, H.H. Hessetink and D.A. Wiersma, Chem. Phys. 17 (1976)
353.
121 P.G. Cummins, D.A. Dunmur and R-W- Munn, Chem.
Phys. Letters 36 (1975) 199. L31 F-P. Chen, D-M. Hanson and D. Fox, J. Chem. Phys. 63 (1975) 3878. r41 D.A. Dunmur and R.W. Mm-m, Chem. Phys. 11 (1975) 297. 151 D.A. Dunmur, W.H. Miller and R.W. Munn, Chem. Phys. Letters 47 (1977) 592. 161 P.J. Bounds and R.W. Munn, C&em. Phys. 24 (1977) 343. r71 N.J. Bridge and L-P. Gianneschi, J. Chem. Sot. Faraday II 72 (1976) 1622. 181 T. Luty, Chem. Phys. Letters 43 (1976) 335. 191 A.H. Price, J.O. Williams and R.W. Munn, Chem. Phys. 14 (1976) 413. [ 101 W. Liptay, G. Walz, 1%‘.Baumann, H.J. Schlosser, H. Deckers and N. Detzer, 2. Naturforsch. 263 (1971) 2020. [ 1 I] G.P. Bamett, M.A. Kurzmack and M.M. Malley, Chem. Phys. Letters 23 (1973) 237. 1121 NJ. Bridge, private communication; P.J. Bounds and R.W. Murm, unpublished work. [ 131 F-P. Chen, D-M. Hanson and D. Fox, 1. Chem. Phys. 66 (1977) 4954.
Volume 51, number 2
BREAKDOWN
OF THE ANISOTROPIC
LORENTZ
APPROXIMATION
15 October
1977
IN p-TERPHENYL
J.H: MEYLING Laboratory for Physical Chemistry. The ihiversity Groningen. 7’7~IVetherlands
of Gromb~gen.
and PJ. BOUNDS and R.W. MUNN Department of Chemistry, University of Manchester Institute of Technology. Manchester hf60 I QD, UK Received 9 May 1977 Revised manuscript received
6 July 1977
Polarizability changes for tetracene and pentacene in a p-terphenyl host crystai calculated from Stark measurements using the anisotropic Lorentz local field disagree with physical intuition. A local field taking better account of the anisotropic p-tcrphenyl structure gives acceptable polarizability changes.
l_ Introduction In a previous paper [l] one of us reported the anisotropv of the quadratic Stark effect for tetracene and pentacene in a p-terphenyl host crystal (space group P21/a, 2 = 2). An anisotropic Lorentz approximation for the local electric field in the host was used to transform the measured spectral shifts into the molecular difference polarizability tensor of the absorbing states. The resulting values of Aa: were found to be (in cm-24): tetracene
pentacene
A~LL
29 * 3
53*4
&MlW
25+4
90F8
AffAW
5+5
-11+9
where L, Mand iV are the long, medium and normal molecular axes. These values disagree with physical intuition, which leads one to expect A@M~ < A@Q and it was suggested [l] that possibly the approximation used for the local field was responsible for this discrepancy. 234
The anisotropic Lorentz approximation is simple to apply but is hard to justify in very anisotropic crystals such as p-terphenyl. Recent work has therefore sought to calculate the local field rigorously within the approximation of treating molecules as linearly polarizable points [24]. One problem with these calculations is that the results are not unique, although extra experimental information should eliminate the arbitrariness [5,6] _ More seriously, the local fields prove to be unreasonably anisotropic, to the extent that in p-terphenyl the local field along the c’ axis is calculated to be opposite in direction to the applied field. This problem can be traced to the representation of a molecule as a single point when molecules are separated by distances comparable with their dimensions, i.e. to the breakdown of the dipole approximation. Realisation of this fact has prompted caIcuIations treating molecules as sets of pojarizable points [6-8 J , thereby taking account of molecular size, shape, and orientation_ Since these calculations give intuitively reasonable results, we have used the method of ref. [6] to caIcuIate the local field in p-terphenyl in order to reanalyse the Stark effect measurements.
Volume 5 1, number 2
CHEMICAL PHYSICS LETTERS
2. Caicuiations The method entails calcuhting the Lorentz-factor tensor Lkfis between sublattices k and k’ as the average of the tensors describing the interactions between point dipoles at the centre of each aromatic ring; interactions within the same molecule are excluded. This averaging allows us to retain the algebra of the singlepoint treatment for two molecules per unit cell related by a twofold axis. The averaged Lkke for p-terphenyl are given in ref. [6] _ The local field depends on a matrix p which is not uniquely defined in the absence of further experimental data. Here we make the arbitrary but convenient choice p = 1 which allows the local-field tensor for sublattice 1 to be calculated as [9] d, = 1 + %I,
+ L,,)=
x t
(1)
where x is the electric susceptibility tensor. In this approximation the local-field tensor is the simple average of those at the centres of the aromatic rings. it may be noted that although p has been chosen arbitrarily, the LI~Z, LIC’,bb, C’Q,and c’c’ elements of the local-field tensor are independent of this choice [3,4,9] ; how the other elements depend on p is discussed in ref. [s] _ The local-field tensor calculated from eq. (1) is
d~=~~
~~~~
~~~~.
The anisotropic Lorentz local-field tensor dL can be written as [9] dL -1+$x,
(2)
which yields
dL=
-0.134
1.542
0
0
1.615
0
0
I.967 I
( -0.134
.
(This‘ differs slightly from that in ref. [I] which used the principal components of x .) As can be seen, the ordering of the diagonal components of dL is the reverse of that in d, , the change in the c’c’ component being particularly large.
15 October
1977
Comparison of eqs. (I) and (2) shows that the anisotropic Lorentz approximation is equivalent to taking Lll + L12 as isotropic. This is far from true inp-terphenyl, where the c’c’ component of LI1 + t12 is less than a quarter of the ~%tand bb components. Physical arguments also indicate that dL will always tend to have the wrong anisotropy, because it depends only on x, which is a product of the local-field and molecular polarizability tensors. Molecules are farthest apart (implying small local fields) in the direction of their greatest dimension (implying large polarizability), so that in x a small local field tends to be masked by a large polarizability. Thus the anisotropy of dL is less than that of the true local field and may even be reversed, as in p-terphenyl. Recalculating Aa: from the Stark experiments described in ref. [l] using dl we obtain (in cm-24); tetracene
pentacene
6.5 + 3
131 t 10
23-t-4
76 + 6
-St4
-3229
The marked reduction of the c’c’ component of d, compared with the corresponding component of d L leads to an increase in the value of AorLL required to reproduce the experimental results.
3. Discussion The revised components of Ao now agree with the physical expectation that the change along the long molecular axis should exceed that perpendicular to the axis in the molecular plane. The fractional uncertainties are smaller than in the original estimates, permitting the conclusion that Q!probably decreases normal to the molecular plane on excitation. For tetracene, Aor,, and the average z agree with the values reported by Liptay et al. [lo], as before, but 3LsariA%WfiI now also agrees with the vaiue reported by Barnett et al. [l l] . This agreement is particularly encouraging since the various values refer to tetracene in different environments: p-terphenyl crystal host (this work), cyclohexane solution [lo], and polystyrene fiim ill]. 235
Volume 51, number 2
The Iocal field used here is subject to uncertainty from two sources. One is the arbitrary choice of p, but this affects only four of the elements of d and is unlikely to have more than a moderate quantitative effect on the analysis. The other is the choice of three points to represent the p-terphenyl molecule, but experience indicates that the major qualitative features should be thereby included, although some quantitative changes could ensue from a choice of more points [I 21. The anaiysis of the Stark measurements has also been simplified by neglecting the reaction field, which yields contributions depending on the square and higher powers of ACY[4,13] . Nevertheless, we are confident that the results for Aa! presented here are qualitatively correct_ We conclude that the anisotropic Lorentz approximation tends to give local Gelds with-the wrong anisotropy and may lead to qualitatively incorrect conclusions in highly anisotropic crystals. This possibility is exaggerated in analysing the quadratic Stark effect because the square of the local field is required. The revised analysis of the Stark effect measurements gives values for Aa! in tetracene and pentacene which accord with physical intuition and agree with values derived by other methods.
236
15 October 1977
CHEMICAL PHYSICS LETTERS
Acknowledgement We thank the Science Research a Research Studentship (PJB).
Council (U.K.)
for
References [II J.H. Meyling, H.H. Hessetink and D.A. Wiersma, Chem. Phys. 17 (1976)
353.
121 P.G. Cummins, D.A. Dunmur and R-W- Munn, Chem.
Phys. Letters 36 (1975) 199. L31 F-P. Chen, D-M. Hanson and D. Fox, J. Chem. Phys. 63 (1975) 3878. r41 D.A. Dunmur and R.W. Mm-m, Chem. Phys. 11 (1975) 297. 151 D.A. Dunmur, W.H. Miller and R.W. Munn, Chem. Phys. Letters 47 (1977) 592. 161 P.J. Bounds and R.W. Munn, C&em. Phys. 24 (1977) 343. r71 N.J. Bridge and L-P. Gianneschi, J. Chem. Sot. Faraday II 72 (1976) 1622. 181 T. Luty, Chem. Phys. Letters 43 (1976) 335. 191 A.H. Price, J.O. Williams and R.W. Munn, Chem. Phys. 14 (1976) 413. [ 101 W. Liptay, G. Walz, 1%‘.Baumann, H.J. Schlosser, H. Deckers and N. Detzer, 2. Naturforsch. 263 (1971) 2020. [ 1 I] G.P. Bamett, M.A. Kurzmack and M.M. Malley, Chem. Phys. Letters 23 (1973) 237. 1121 NJ. Bridge, private communication; P.J. Bounds and R.W. Murm, unpublished work. [ 131 F-P. Chen, D-M. Hanson and D. Fox, 1. Chem. Phys. 66 (1977) 4954.