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The polarized absorption spectra of acenaphthylene
Spectrochimica AC&vol.2W, pp.1376to1380. Persamos Press 1969.Printed inNorthern Ireland
The polarized absorptionspectra of acenaphthylene* A. BRIE, C. Y. FAN and R. A. KYDD Chemistry Department, University of British Columbi8, Vancouver 8, B.C., Canada (Received 23 December 1908) Abstr&--The polarized absorption spectra of 8ccenaphthylene in acenaphtheneand in durene host crystals at low temperature are reported from 6000 A to the matrix cut-off. In order of increasing energy, the electronic transitions are polarized along the long, short 8nd long 8xes of the molecule. The llrst system contained many sharp lines which, in the 8cenaphthene matrix, were clearly separ8ted into two groups correspondingto A, + A, and B, + A, vibronic transitions. The second 8nd third electronic transitions were studied in a durene matrix. The lines were broad rend8 complete vibrational andysis ~8s not attempted.
FOURdistinct bands are observed in the solution spectrum of aoenaphthylene. Lowest in energy is a weak band (f = 0.0042) located near 21,500 cm-l. A stronger and more complex region cf = 0.165) is seen with maximum intensity at 31,000 cm-l, and this is followed by a weaker band df = 0.050) near 38,000 cm-i and a strong, structureless system (f = 0.867) centered at 43,500 cm-l. The polarizations of the five electronic transitions below 45,000 cm-l of acenaphthylene relative to the first weak band at 4700 A have been reported by HEILBRONNER et al. [l], who have also quoted the stretched-film studies of EDGERS [2] to show that the low-energy transition is long-axis polarized. The objection could be made that the assignment of the weak band was vibronic rather than electronic. In an attempt to con&m these assignments, we decided to study the spectra of acenaphthylene in suitable matrices at low temperature. EXPERIMENTAL
Acenaphthene and durene were chosen as the host materials and each compound was puri.tIedby chromatography followed by zone refining. Single crystals doped with acenaphthylene were grown in a Bridgman furnace. The concentrations of the acenaphthylene in the mixed crystals were found by weighing the crystalsimmediately after running their spectra, dissolving them in n-pentane and measuring the optical densities of the acenaphthylene bands in solution. Acenaphthene is an especially useful matrix because the molecules in the lattice [3] have site symmetry such that the molecular long-axis lies along the 13crystallographic axis. Transitions directed along the short in-plane molecular axis have the fraction 0.386 of their strength along c in the oriented-gas approximation. The spectra were * This researchwas supported by 8 grant from the N8tion81 Research Council of Canada. [l] E. ~ILBRONNER, J. P. WEBER, J. MICHL8nd R. ZAHRADNIK,!Z%eoret.Chem. Acta 6, 141 (1966). [2] J. H. EDGERSand E. W. THULSTRXJ~, lecture at the 8th European Congr. Mol. Spectry (1965). [3] H. W. EHRLICH,Acta Cq8t. 10, 699 (1957). 1376
A. BREE, C. Y. FAN and R. A. KYDD
1376
measured using the bc cleavage plane which was readily identified from the optical properties of the crystal [4]. The secondary cleavage plane bc was also used with durene as the matrix. This section was distinguished from the ab cleavage plane by the appearance of a single brush at extinction (rather than a cross) under conoscopic examination. The crystal structure of durene [5] shows that for an oriented-gas model a long-axis transition has 0.563 and 0.029 and a short-axis transition 0.0004 and 0.914 of its strength along the b and c axes, respectively. In this work, the measured intensities have been manipulated SO that 3Esoiution= E,, + Ed + E, and fsoilltion = .f,t +fb +-fc; a’ is defined to be normal to both b and c. SYSTEM I IN AN ACENAPHTHENE~TRIX The weak absorption system of acenapthylene in acenaphthene at 6°K is shown in Fig. 1. There were about 200 lines measured and analysed within 4500 cm-l
I
Fig. 1. The low-energy
I
system
of acenaphthylene st 6OK.
in aoenaphthene
(bc section)
From the appearance of the spectrum, of the lowest energy line at 21,090 cm-l. lines in the b polarized spectrum were assumed to involve totally symmetric vibrations (with the pure electronic transition at 21,090 cm-l) and those in the c polarization, non-totally symmetric vibrations. This assumption is justifled in the durene matrix work. Thus, A, fundamentals of the excited electronic state are marked by the intervals 414, 526, 654, 780, 986, 1036, 1077, 1234, 1319, 1329, 1378, 1502 and 1571 cm-l with B, fundamentals [6], located through the “false origins”, as 456,474, [4] A. BREE,Mol. Playa. 6, 563 (1963). [5] J. M. ROBERTSON, Proc. Roy. Sot. A14l, 594 (1933). [6] The conventions adopted in this work follow the recommendations of the Joint Commission for Spectroscopy as reported by R. S. MTJLLIKEN, J. Chem. Phya. 23, 1997 (1955). Acoordingly, the long axis (y) of the molecule transforms like B,.
The polarized absorption spectra of acenaphthylene
1377
1009, 1148, 1193, 1351 and 1566 cm-l. The last two A, fundamentals of energy less than 2000 cm-l may be identiGed with the very weak lines 6’78 and 1362 cm-l to the blue of the origin. Other possible B, fundamentals of the excited electronic state occur at 1308, 1420 and 1578 cm- 1. The intervals 41, 61 and 69 cm-l represented phonons of the locally disturbed lattice and were built on the origin band and other strong lines; these intervals have been omitted from Table 1 for brevity although they may be seen in Fig. 1. The 109 cm-l interval was taken to mark a B, fundamental allowed to appear through mixing with the lattice modes. Similarly the 297 cm-l interval may reasonably be taken as an A, molecular fundamental since it appears to be too low for a B, fundamental. These assignments are indicated in Table 1.
Fig. 2. Acenaphthylene absorption spectra. System 1 (left hand scale): full line, libin aoenaphthenematrix at 77’K; dotted line, solution in n-pentane at 300°K. System 2 (right hand scale): full line, 116;broken line, IIcin durenematrix at 77’K; dotted line, solution in methanol at 300’K.
Since the shape of the origin band is the same in the matrix and solution (see Fig. 2), it is interesting to note that q, (-95 1 mole-l cm-l) is three time asolution (~31 1 mole-l cm-‘). Evidently the oriented-gas prediction is correct in this case and mixing with other states has not occurred. The complete polarization of the absorption lines shows that the guest molecules have taken up positions in the lattice with their axes exactly coinciding with those of the host molecules and, because of the close similarity in molecular geometries and optical properties of the crystals [7], there seems to be no reason to suppose that any permutation of the sets of axes has occurred. However, even at the lowest temperatures, no fluorescence was detected so that the electronic origin could not be located [7] A. N. WINCHELL,The Optical Properties (1954).
of OrganicCompounds,pp. 80, 81. Academic Press
A. BREE, C. Y. FAN and R. A.
1378
KYDD
Table 1. The absorption spectra of acenaphthylenein acenaphtheneat -6OK*
Ilb
IIC
21,090 109 297 414 466 474 620 636 664 678 780 890 938 981 986 1000 1009 1036 1063 1066 1077 1088 1144 1148 1179 1186 1193 1234 1237 1307 1308 1319 1329 1334 1344 1361 1362 1378 1420 1436 1441 1486 1602 1609 1613 1634 1661 1666 1671 1677 1678 1606 1674
0.D.t 0.68 0.22 0.02 0.08 0.09 0.03 0.63 0.07 0.20 0.04 0.42 0.10 0.04 0.07 0.33 0.04 0.04 0.21 0.16 0.03 0.41 0.04 0.03 0.09 0.06 0.11 0.04 0.13 0.04 0.32 0.02 0.32 0.21 0.03 0.18 0.20 0.13 0.39 0.03 0.16 0.04 0.19 0.23 0.02 0.23 0.07 0.13 0.13 0.13 0.10 0.04 0.23 0.08
Assignment Origin 109, B,? 297. A,? 414, A, 466, B, 474, B, 626, A, 109 + 626 + 664. A, 678, A,? 780, A, 109 + 780 + 414+ 626-2 466 + 626 986. A, 474 + 626 1009, B, 1036, A, 2 X 626 + 1 414 + 664 1077, A, 109 + 986 109 + 1036 1148, B, 626 + 654 109 + 1077 1193, B, 1234, A, 466 + 780 + 626 + 780+ 1308, B,? 1319, A, 1329, A, 297+ 1036+ 109+ 1234+ 1351, B, 1362, A, 1378, A, 1420, B,? 654 + 780+ 466 + 986 109 + 1378 1602, A, 466+ 1063 626 + 986 + 466+ 1077+ 2 x 780+ 1 1666, B, 1671, A, 3 x 626 - 1 1678. B,? 626+ 1077+ 626 + 1148
1
1 1
2 7 1 1 1
1 1
1. 1
2 1 1
1 1
2
* The position of the origin is in cm-’ and all other entries are differences from the origin in OIU-~. t Optical density defined aa log,, (IO/I).
1379
The polarized absorption spectra of ammaphthylene
or assigned with certainty. Further, while MIKHAILENKO and TEPLYAKOV [8] have reported that the lowest-energy line in absorption coincided with the highest-energy fluorescence line (strongest component of the multiplet at 21,960 cm-l) when acenaphthylene was held in a polycrystalline n-hexane matrix at 77”K, we have been unable to reproduce their results. We have found that any fluorescence was too weak to be photographed and the origin in absorption occurred at 21,382 cm-i at 77’K; the reason for this discrepancy is not understood. Therefore, because the position of the origin remains in doubt and because the second and third transitions were masked by the acenaphthene absorption, the spectra were remeasured in a durene matrix. RESULTS WITH DURENE AS MATRIX While acenaphthylene readily substituted in the acenaphthene crystal lattice, it was more difilcult to prepare single crystals of durene having a sufficiently high concentration of acenaphthylene to study the weak system in absorption. However, a rather low contrast spectrum was photographed at 6°K. The origin suffered a considerable blue shift (486 cm-l) due to the change of matrix, appearing at 21,576 cm-l. Other absorption lines appeared 523, 788, 1039, 1085, 1323, 1377 and 1620 cm-r to the blue of the origin with about the same relative intensity as in the acenaphthene matrix ; the perturbed durene lattice frequencies 23 and 43 cm-l were present and had the effect of blurring the spectrum at higher energies. A very weak fluorescence was recorded together with phosphorescence from some unknown impurity of durene. The only resonance line appeared at 21,576 cm-l. Thus the pure electronic transition may be assigned B, c A, since the origin was stronger in the b spectrum. This result is in good agreement with the acenaphthene matrix work. The second region of absorption at about 3200 A is shown in Fig. 2. The host molecule in this case is durene, and the relatively broad bands shown here at 77’K did not sharpen appreciably on cooling to 6°K. The system is composed to two transitions, one short-axis polarized with an origin in the c spectrum at 29,050 cm-l and the other long-axis polarized with its origin in the b spectrum at 30,000 cm-l. The coarse vibrational structure in c polarization was interpreted in terms of totally symmetric fundamentals at 520,1035 and 1435 cm-l. The origin at 30,000 cm-l showed evidence of shoulders on its high and low energy edge (see Fig. 2). At 6”K, these features were seen as weak peaks which marked the presence of B, fundamentals at 505, 770 and 1180 cm-l belonging to the short-axis transition. The sum of the oscillator strengths for the two electronic transitions in the 3200 A system in solution was 0.167. The error in the molar extinction coefficients for the crystal spectra shown in Fig. 2 was about 15 ‘A and arose primarily in the determination of the guest concentration. The oscillator strength for the c spectrum was 0.105 and for the b spectrum was 0.088. In the oriented-gas approximation these intensities may be apportioned as shown below:
Ila’ short-axis long-axis
absorption : absorption:
[S] v. I. MIEHAILE NKO and P. A.
TEPLYAKOV,
lib
IIC
0.009
0.000
0.100
0.064
0.088
0.005
Opt. Spektroskopiya 22, 48 (1967).
1380
A. BREE, C. Y. FAN and R. A. KYDD
In this context, long-axis absorption includes contributions from both electronic The sum of transitions and is assumed to have a small component in c polarization. the intensities for the spectra in all crystal axes is expected to be 0.266 (or 0.27) and the difference from the solution value is large enough to be significant. If the crystal spectra are added, the peak at 29,050 cm-l appears to have intensified relative to the solution spectrum. This suggests that intensity has been transferred into the shortaxis transition and this has probably been induced by mixing with states of the durene crystal. Thus we have shown that the low-energy transition is polarized along the longaxis of the molecule and that this is followed at higher energy by a short and a longaxis polarized transition. This conl%ms the results found by HEILBRONNER etal.[11.
The polarized absorptionspectra of acenaphthylene* A. BRIE, C. Y. FAN and R. A. KYDD Chemistry Department, University of British Columbi8, Vancouver 8, B.C., Canada (Received 23 December 1908) Abstr&--The polarized absorption spectra of 8ccenaphthylene in acenaphtheneand in durene host crystals at low temperature are reported from 6000 A to the matrix cut-off. In order of increasing energy, the electronic transitions are polarized along the long, short 8nd long 8xes of the molecule. The llrst system contained many sharp lines which, in the 8cenaphthene matrix, were clearly separ8ted into two groups correspondingto A, + A, and B, + A, vibronic transitions. The second 8nd third electronic transitions were studied in a durene matrix. The lines were broad rend8 complete vibrational andysis ~8s not attempted.
FOURdistinct bands are observed in the solution spectrum of aoenaphthylene. Lowest in energy is a weak band (f = 0.0042) located near 21,500 cm-l. A stronger and more complex region cf = 0.165) is seen with maximum intensity at 31,000 cm-l, and this is followed by a weaker band df = 0.050) near 38,000 cm-i and a strong, structureless system (f = 0.867) centered at 43,500 cm-l. The polarizations of the five electronic transitions below 45,000 cm-l of acenaphthylene relative to the first weak band at 4700 A have been reported by HEILBRONNER et al. [l], who have also quoted the stretched-film studies of EDGERS [2] to show that the low-energy transition is long-axis polarized. The objection could be made that the assignment of the weak band was vibronic rather than electronic. In an attempt to con&m these assignments, we decided to study the spectra of acenaphthylene in suitable matrices at low temperature. EXPERIMENTAL
Acenaphthene and durene were chosen as the host materials and each compound was puri.tIedby chromatography followed by zone refining. Single crystals doped with acenaphthylene were grown in a Bridgman furnace. The concentrations of the acenaphthylene in the mixed crystals were found by weighing the crystalsimmediately after running their spectra, dissolving them in n-pentane and measuring the optical densities of the acenaphthylene bands in solution. Acenaphthene is an especially useful matrix because the molecules in the lattice [3] have site symmetry such that the molecular long-axis lies along the 13crystallographic axis. Transitions directed along the short in-plane molecular axis have the fraction 0.386 of their strength along c in the oriented-gas approximation. The spectra were * This researchwas supported by 8 grant from the N8tion81 Research Council of Canada. [l] E. ~ILBRONNER, J. P. WEBER, J. MICHL8nd R. ZAHRADNIK,!Z%eoret.Chem. Acta 6, 141 (1966). [2] J. H. EDGERSand E. W. THULSTRXJ~, lecture at the 8th European Congr. Mol. Spectry (1965). [3] H. W. EHRLICH,Acta Cq8t. 10, 699 (1957). 1376
A. BREE, C. Y. FAN and R. A. KYDD
1376
measured using the bc cleavage plane which was readily identified from the optical properties of the crystal [4]. The secondary cleavage plane bc was also used with durene as the matrix. This section was distinguished from the ab cleavage plane by the appearance of a single brush at extinction (rather than a cross) under conoscopic examination. The crystal structure of durene [5] shows that for an oriented-gas model a long-axis transition has 0.563 and 0.029 and a short-axis transition 0.0004 and 0.914 of its strength along the b and c axes, respectively. In this work, the measured intensities have been manipulated SO that 3Esoiution= E,, + Ed + E, and fsoilltion = .f,t +fb +-fc; a’ is defined to be normal to both b and c. SYSTEM I IN AN ACENAPHTHENE~TRIX The weak absorption system of acenapthylene in acenaphthene at 6°K is shown in Fig. 1. There were about 200 lines measured and analysed within 4500 cm-l
I
Fig. 1. The low-energy
I
system
of acenaphthylene st 6OK.
in aoenaphthene
(bc section)
From the appearance of the spectrum, of the lowest energy line at 21,090 cm-l. lines in the b polarized spectrum were assumed to involve totally symmetric vibrations (with the pure electronic transition at 21,090 cm-l) and those in the c polarization, non-totally symmetric vibrations. This assumption is justifled in the durene matrix work. Thus, A, fundamentals of the excited electronic state are marked by the intervals 414, 526, 654, 780, 986, 1036, 1077, 1234, 1319, 1329, 1378, 1502 and 1571 cm-l with B, fundamentals [6], located through the “false origins”, as 456,474, [4] A. BREE,Mol. Playa. 6, 563 (1963). [5] J. M. ROBERTSON, Proc. Roy. Sot. A14l, 594 (1933). [6] The conventions adopted in this work follow the recommendations of the Joint Commission for Spectroscopy as reported by R. S. MTJLLIKEN, J. Chem. Phya. 23, 1997 (1955). Acoordingly, the long axis (y) of the molecule transforms like B,.
The polarized absorption spectra of acenaphthylene
1377
1009, 1148, 1193, 1351 and 1566 cm-l. The last two A, fundamentals of energy less than 2000 cm-l may be identiGed with the very weak lines 6’78 and 1362 cm-l to the blue of the origin. Other possible B, fundamentals of the excited electronic state occur at 1308, 1420 and 1578 cm- 1. The intervals 41, 61 and 69 cm-l represented phonons of the locally disturbed lattice and were built on the origin band and other strong lines; these intervals have been omitted from Table 1 for brevity although they may be seen in Fig. 1. The 109 cm-l interval was taken to mark a B, fundamental allowed to appear through mixing with the lattice modes. Similarly the 297 cm-l interval may reasonably be taken as an A, molecular fundamental since it appears to be too low for a B, fundamental. These assignments are indicated in Table 1.
Fig. 2. Acenaphthylene absorption spectra. System 1 (left hand scale): full line, libin aoenaphthenematrix at 77’K; dotted line, solution in n-pentane at 300°K. System 2 (right hand scale): full line, 116;broken line, IIcin durenematrix at 77’K; dotted line, solution in methanol at 300’K.
Since the shape of the origin band is the same in the matrix and solution (see Fig. 2), it is interesting to note that q, (-95 1 mole-l cm-l) is three time asolution (~31 1 mole-l cm-‘). Evidently the oriented-gas prediction is correct in this case and mixing with other states has not occurred. The complete polarization of the absorption lines shows that the guest molecules have taken up positions in the lattice with their axes exactly coinciding with those of the host molecules and, because of the close similarity in molecular geometries and optical properties of the crystals [7], there seems to be no reason to suppose that any permutation of the sets of axes has occurred. However, even at the lowest temperatures, no fluorescence was detected so that the electronic origin could not be located [7] A. N. WINCHELL,The Optical Properties (1954).
of OrganicCompounds,pp. 80, 81. Academic Press
A. BREE, C. Y. FAN and R. A.
1378
KYDD
Table 1. The absorption spectra of acenaphthylenein acenaphtheneat -6OK*
Ilb
IIC
21,090 109 297 414 466 474 620 636 664 678 780 890 938 981 986 1000 1009 1036 1063 1066 1077 1088 1144 1148 1179 1186 1193 1234 1237 1307 1308 1319 1329 1334 1344 1361 1362 1378 1420 1436 1441 1486 1602 1609 1613 1634 1661 1666 1671 1677 1678 1606 1674
0.D.t 0.68 0.22 0.02 0.08 0.09 0.03 0.63 0.07 0.20 0.04 0.42 0.10 0.04 0.07 0.33 0.04 0.04 0.21 0.16 0.03 0.41 0.04 0.03 0.09 0.06 0.11 0.04 0.13 0.04 0.32 0.02 0.32 0.21 0.03 0.18 0.20 0.13 0.39 0.03 0.16 0.04 0.19 0.23 0.02 0.23 0.07 0.13 0.13 0.13 0.10 0.04 0.23 0.08
Assignment Origin 109, B,? 297. A,? 414, A, 466, B, 474, B, 626, A, 109 + 626 + 664. A, 678, A,? 780, A, 109 + 780 + 414+ 626-2 466 + 626 986. A, 474 + 626 1009, B, 1036, A, 2 X 626 + 1 414 + 664 1077, A, 109 + 986 109 + 1036 1148, B, 626 + 654 109 + 1077 1193, B, 1234, A, 466 + 780 + 626 + 780+ 1308, B,? 1319, A, 1329, A, 297+ 1036+ 109+ 1234+ 1351, B, 1362, A, 1378, A, 1420, B,? 654 + 780+ 466 + 986 109 + 1378 1602, A, 466+ 1063 626 + 986 + 466+ 1077+ 2 x 780+ 1 1666, B, 1671, A, 3 x 626 - 1 1678. B,? 626+ 1077+ 626 + 1148
1
1 1
2 7 1 1 1
1 1
1. 1
2 1 1
1 1
2
* The position of the origin is in cm-’ and all other entries are differences from the origin in OIU-~. t Optical density defined aa log,, (IO/I).
1379
The polarized absorption spectra of ammaphthylene
or assigned with certainty. Further, while MIKHAILENKO and TEPLYAKOV [8] have reported that the lowest-energy line in absorption coincided with the highest-energy fluorescence line (strongest component of the multiplet at 21,960 cm-l) when acenaphthylene was held in a polycrystalline n-hexane matrix at 77”K, we have been unable to reproduce their results. We have found that any fluorescence was too weak to be photographed and the origin in absorption occurred at 21,382 cm-i at 77’K; the reason for this discrepancy is not understood. Therefore, because the position of the origin remains in doubt and because the second and third transitions were masked by the acenaphthene absorption, the spectra were remeasured in a durene matrix. RESULTS WITH DURENE AS MATRIX While acenaphthylene readily substituted in the acenaphthene crystal lattice, it was more difilcult to prepare single crystals of durene having a sufficiently high concentration of acenaphthylene to study the weak system in absorption. However, a rather low contrast spectrum was photographed at 6°K. The origin suffered a considerable blue shift (486 cm-l) due to the change of matrix, appearing at 21,576 cm-l. Other absorption lines appeared 523, 788, 1039, 1085, 1323, 1377 and 1620 cm-r to the blue of the origin with about the same relative intensity as in the acenaphthene matrix ; the perturbed durene lattice frequencies 23 and 43 cm-l were present and had the effect of blurring the spectrum at higher energies. A very weak fluorescence was recorded together with phosphorescence from some unknown impurity of durene. The only resonance line appeared at 21,576 cm-l. Thus the pure electronic transition may be assigned B, c A, since the origin was stronger in the b spectrum. This result is in good agreement with the acenaphthene matrix work. The second region of absorption at about 3200 A is shown in Fig. 2. The host molecule in this case is durene, and the relatively broad bands shown here at 77’K did not sharpen appreciably on cooling to 6°K. The system is composed to two transitions, one short-axis polarized with an origin in the c spectrum at 29,050 cm-l and the other long-axis polarized with its origin in the b spectrum at 30,000 cm-l. The coarse vibrational structure in c polarization was interpreted in terms of totally symmetric fundamentals at 520,1035 and 1435 cm-l. The origin at 30,000 cm-l showed evidence of shoulders on its high and low energy edge (see Fig. 2). At 6”K, these features were seen as weak peaks which marked the presence of B, fundamentals at 505, 770 and 1180 cm-l belonging to the short-axis transition. The sum of the oscillator strengths for the two electronic transitions in the 3200 A system in solution was 0.167. The error in the molar extinction coefficients for the crystal spectra shown in Fig. 2 was about 15 ‘A and arose primarily in the determination of the guest concentration. The oscillator strength for the c spectrum was 0.105 and for the b spectrum was 0.088. In the oriented-gas approximation these intensities may be apportioned as shown below:
Ila’ short-axis long-axis
absorption : absorption:
[S] v. I. MIEHAILE NKO and P. A.
TEPLYAKOV,
lib
IIC
0.009
0.000
0.100
0.064
0.088
0.005
Opt. Spektroskopiya 22, 48 (1967).
1380
A. BREE, C. Y. FAN and R. A. KYDD
In this context, long-axis absorption includes contributions from both electronic The sum of transitions and is assumed to have a small component in c polarization. the intensities for the spectra in all crystal axes is expected to be 0.266 (or 0.27) and the difference from the solution value is large enough to be significant. If the crystal spectra are added, the peak at 29,050 cm-l appears to have intensified relative to the solution spectrum. This suggests that intensity has been transferred into the shortaxis transition and this has probably been induced by mixing with states of the durene crystal. Thus we have shown that the low-energy transition is polarized along the longaxis of the molecule and that this is followed at higher energy by a short and a longaxis polarized transition. This conl%ms the results found by HEILBRONNER etal.[11.