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Vibronic coupling with low-frequency vibrations in quasi-forbidden Frenkel exciton states
4 November 1994
ELSEVIER
CHEMICAL PHYSICS LETTERS
Chemical Physics Letters 229 (1994) 439-442
Vibronic coupling with low-frequency vibrations in quasi-forbidden Frenkel exciton states Piotr Petelenz K. GumiAski Department of Theoretical Chemistry, Jagiellonian University, Ingardena 3, 30-060 Cracow, Poland Received 28 July 1994
Abstract A simple dimer model is used to demonstrate that in weakly allowed Frenkel exciton states, where the factor-group splitting is largely due to charge transfer interactions, the exciton-phonon coupling constant may be considerably different for the two Davydov components. The model is applied to the S, transition of the naphthalene crystal and suggests a smaller Davydov splitting than normally assumed (60 cm- ’ rather than 150 cm- ’ ).
1. Introduction
Due to very weak dipolar interactions of quasi-forbidden excited states, the Davydov splitting of the corresponding Frenkel excitons is usually small. Its main contributions come from higher multipole interactions [ 1 ] and, as shown recently [ 2-61, charge transfer (CT) effects. Both types of terms mentioned above strongly depend on the distance between the interacting molecules. Hence, it is reasonable to expect that in the resulting eigenstates the crystal lattice may be considerably distorted with respect to the ground state. This should result in strong exciton-phonon coupling and affect the optical spectrum. As the coupling is due to the dependence of the off-diagonal (in local basis) matrix elements of the Hamiltonian on nuclear coordinates, it is likely to affect differently each of the Davydov components (as demonstrated by Zgierski [ 7 ] ) . The effect should be observable e.g. in the very weakly allowed SI state of the naphthalene crystal,
whose properties are strongly affected by CT effects [2-61. The spectrum of the crystal in the S, region starts with a narrow line at 3 1476 cm- ’ (a polarized [ 81, referred to as the A band [ 9]), followed by a very broad, almost structureless band at 31623 cm-’ (predominantly b polarized [ 81, referred to as the B band [ 91). Commonly, these bands are interpreted as the two Davydov components of the corresponding Frenkel exciton state. This explanation fails to account for the enormous difference in the width of the two bands. An alternative interpretation was suggested recently [ 9 1. According to the hypothesis proposed by Brovchenko et al., the broad band observed at 3 1623 cm-’ would not correspond to the b polarized Davydov component of the O-O transition, but would represent a strongly depolarized phonon sideband built upon the a component (which would represent the zero-phonon line, ZPL). In fact, the large width and the lack of structure of the 3 1623 cm- ’ band very well agree with the known characteristics of typical phonon sidebands.
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDIOOO9-2614(94)01089-7
P. Petelenz / Chemical Physics Letters 229 (1994) 439-442
440
Although some aspects of the interpretation presented in Ref. [ 91 are not entirely clear, the idea is attractive and very likely correct. However, there is still a problem: where is the other Davydov component? If the corresponding absorption line is as sharp as the A band, it should be discernible from the broad background provided by the phonon sideband. It will be shown that the peculiarities of phonon coupling to quasi-forbidden exciton states might explain the apparent absence of the b polarized counterpart of the A band.
2. Model For the sake of simplicity, we represent the two molecules from the unit cell of the naphthalene crystal as two moieties of a model dimer, and focus our attention on the translational intermolecular vibration which is supposed to mimic the lattice phonon modes of the crystal. Vibronic coupling with intramolecular vibrations is disregarded. The treatment is confined to four electronic states, symbolically denoted as 1A*B), I.4B*), I A+B- ) and )A-B+), where the asterisk denotes electronic excitation. These (‘local’) states describe the exciton and the charges as strictly localized at one of the moieties. Only one electronic excited state is taken into account for each monomer. Correct to terms linear in intermolecular overlap integrals, in this basis the electronic Hamiltonian assumes the form E(q)
M(q)
D,(q)
M(q) De(q)
E(q) D,(q)
&(q) E”(q)
i D,(q)
D,(q)
0
D,(q) D,(q) 0
'
(1)
E"(q) 1
where E(q)=(A*B]H]A*B)=(AB*]H]AB*) and Em(q)=(.4+B-]HlA+B-)=(A-B+]H]A-B+) are the diagonal energies of the local Frenkel and local CT states, M= (A*B ]H] AB*) is the exciton transfer (resonance) integral, while D, = (A*B]H(A+B-) and Dh=(A*B/H]A-B+) stand for the exciton dissociation integrals with electron and hole transfer, respectively. Out of all vibrational coordinates, we include only one intermolecular mode, presumably translational, which is denoted by q.
In the harmonic gies read E(q)=tq*+b,q E”(q)
approximation,
the diagonal ener-
>
(2a)
=A+ fq2+b2q,
(2b)
where 6, and 6, represent the displacements of the equilibrium position in the intermolecular mode upon creation of a local Frenkel and local CT state, respectively, and can be expressed as the derivatives a W/ tlq of the intermolecular interaction energy in the corresponding local states. .4 is the energy gap between the two kinds of local states. The standard dimensionless units are used for energy and length, and the energy of the local Frenkel state at equilibrium position is taken as zero. In the basis of symmetry adapted (‘delocalized’) electronic states [F&)=2-I’*( ]CT?)=2-I”(
]A*B) + ]AB*))
,
]A+B-)+]A-B+)),
(3a) (3b)
the matrix of Eq. ( 1) can be split into effective Hamiltonians for the Davydov components ;q2+b,qkM(q)
(
D,(q)
Dk (q)
A+ $q2+b2q
>’
(4)
whereD,(q)=D,(q)kD,(q). As in the naphthalene crystal the CT states are located above the lowest intramolecular excited state [ lo], the lower eigenvalue of each of the two Hamiltonian matrices of Eq. (4) represents the approximate potential energy for the corresponding Davydov component of the Frenkel state. The dissociation integrals D, and Dh depend linearly on the intermolecular overlap integral, which in turn should depend exponentially on intermolecular distance. Therefore, it is resonable to approximate the explicit form of D+ (q) as D, (4) =Di
(0) exp( -&)
,
(5)
“’ is the Slater exponent for carwhere S=,~(h/mo) bon 2p, orbitals (3.09 A-’ ), expressed in the harmonic oscillator dimensionless length units. m is the reduced mass for the intermolecular vibration, equal to one half of the molecular mass. With Eq. (5 ) and taking advantage of the fact that D+ (0)/A<< 1, the potentials may be approximated
P. Petelenz /Chemical Physics Letters 229 (1994) 439-442
by the perturbational der in q)
result (correct to the second or-
+ [b, -2D2,(0)d/AtM’(O)]q
+D2,/AfM(O).
(6)
The above expression shows that the coupling with charge transfer states modifies both the effective force constant for the intermolecular mode and its equilibrium position in the states of Frenkel parentage. The equilibrium position is evidently different for each of the two Davydov components, which will affect the observed width of the corresponding transitions.
441
differentiation with respect to R (expressed in the dimensionless units of the 50 cm-’ oscillator) yields b, = -0.95. With the above input data, the displacement parameters corresponding to the two Davydov components are b_ = [b, -202(0)6/A-M’(O)]
= -0.95,
b+=[b,-20:(0)6/A+M’(O)]=-1.3.
(7)
The difference between the displacement parameters in the two Davydov components is evidently substantial. In reality, it is likely to be even larger.
4. Effective mode approach and scaling 3. Estimates for the translational mode The order of magnitude of the effect may be estimated as follows: Let us assume the intermolecular mode to have a frequency corresponding to low frequency lattice lattice phonons, e.g. 50 cm-‘. This vibrational quantum sets the energy unit for Eq. ( 6 ), and the value of 6=0.32. Based on the experimental energy of the S, state ( = 3 1000 cm- ’ ) and the calculated energy of the lowest CT state of the naphthalene crystal (4.38 eV [lo] ), A=75. In a different context [6], the dissociation integrals ( - 0.02 12 and - 0.0 19 1 eV for the electron and the hole, respectively) were estimated from charge transfer integrals calculated by Tiberghien and Delacote [ 111, yielding D, (0) = - 6.5, D_ (0) = -0.34. The corresponding estimate [ 61 of the exciton resonance integral ( = 0.0005 eV) yields M(O)=O.OS. For a dipole allowed transition, M should be proportional to R -n, with n= 3. For the S, naphthalene state, a stronger distance dependence is expected. However, with the above value of M(0) even with n = 5 M’( 0) = 0.0 1 and is numerically irrelevant. The energy W of the interaction between the molecules is due primarily to the dispersion term, proportional to R -6 [2,3 1. Its change between the ground and excited electronic state determines the gas-tocrystal shift. Following the approach of Gisby and Walmsley [ 2,3], the proportionality constant may be evaluated from the experimental shift of about 400 cm-’ (after deducting the CT contribution). Then,
For the lower component, the predicted Stokes shift is S_ = bZ hw/2 z 25 cm-‘. In the crystal spectrum, it should roughly correspond to the maximum of the phonon sideband. Then, if the band peaking at 3 1623 cm-’ really has a considerable contribution from the phonon sideband of the 31476 cm-’ transition (which, after Brovchenko et al. [ 91, we accept as a working hypothesis), b_ seems to be seriously underestimated. This is not surprizing since we have included only one intermolecular mode, completely ignoring the librations, and having no way to include in the dimer model the effect of phonon dispersion, coupling with acoustic phonons, etc. On that view, it seems reasonable to treat the q mode as an effective one and scale its displacement parameters in such a way that the Stokes shift for the lower Davydov component in the crystal be approximately reproduced. This increases b by a factor of 2, so that beff_ = 1.9: beff+ = 2.6. With these values, the expected intensity ratio I0 of the zero phonon line to the phonon sideband is exp(-+b&)=0.16 and exp(-tb&+)=0.03 for the lower and the upper Davydov component, respectively. Therefore, the relative intensity of the ZPL for the upper component is smaller by an order of magnitude, and the line is very likely to fuse with the background and with its own sideband (which carries most of the intensity). On that view, it is not surprizing that no sharp b polarized line is experimentally observed. The above conclusion rests on the tenuous scaling
442
P. Perelenz /Chemical Physics Letters 229 (1994) 439-442
argument, which assumes that the sensitivity of the dissociation integrals to changes of other intermolecular coordinates is as strong as for the translational mode. At least for some librations, this does not seem unlikely [ 121. Even if the argument is not quantitatively correct, it definiteiy shows a substantial difference in the Z, values for the two Davydov components.
5. Davydov splitting in naphthalene In view of Section 4, it is reasonable to suppose that the B band of the naphthalene crystal is in fact composite, consisting of the (depolarized) phonon sideband of the lower (A) Davydov component, and of the genuine transition to the upper Davydov component. The latter should have contributions both from the (probably dominant) phonon sideband and from the zero-phonon line which may be difficult to resolve (due to its low intensity and di~useness}. However, the zero phonon line is expected to be asymmetric: diffuse and likely to amalgamate with the sideband at higher energies, but sharper on the low-energy side. This makes one expect a relatively steep absorption edge at that energy. Such an edge is indeed observed in the spectra of Brovchenko et al. [ 91. There is a shoulder due to the steep incline on the low energy side of the B band (at about 31535 cm-‘). It is observed in b polarization, but absent in a polarization -just as expected for the ZPL. Very likely, that energy represents the actual position of the zero phonon line of the upper Davydov component. If this interpretation is accepted, the Davydov splitting between the a and b polarized components is about 60 cm-‘. This contradicts the normally accepted value of 150 cm-’ [ 2,8], and suggests that the arguments leading to that value should be reexamined. The Davydov splitting in the Sr state of the naphthalene crystal was normally identified with the gap between the A band and the maximum of the B band in the crystal spectrum. Even if the B band were due
exclusively to the upper Davydov component (with the zero phonon line unresolved from the sideband), the arguments of Section 4 suggest that the zerophonon line (being the b polarized counterpart of the A band) should not correspond to the band maximum but should be Iocated at lower energy. Hence, even in that case the Davydov splitting would be less than the 150 cm-’ estimated from the position of the peak. The difference might be considerable and the steep edge-like shoulder at 31535 cm-i could very well correspond to the zero-phonon line of the b palarized exciton state. This value also agrees very well with the theoretical results obtained from the cluster model [ 61 (which, for a neat crystal, are equivalent to those of standard exciton band structure calculations [ 5 ] ) . In fact, with the most accurate parametrization available to date [ 61, the calculated Davydov splitting of the S, state is about 70 cm-‘.
This research was supported by grant No. 206699101 (1723/2/91) from the State Committee for Scientific Research (Poland).
References [ I ] D.P. Craig and S.H. Walmsley, Mol. Phys. 4 ( 1961) I 13. [2] J.A. Gisby and S.H. Walmsley, Chem. Phys. Letters 135 (1987) 275. [3] J.A. Gisby and S.H. Wafmsley, Chem. Phys. 122 (1988) 271. f4] P. Petelent, Chem. Phys. 138 (1989) 35. [ 5 ] P. Petelenz and M. Slawik, Chem. Phys. 157 ( 1991) 169. [ 6lI.V. Brovchenko, A. Eilmes and P. Petelenz, J. Chem. Phys. 98 (1993) 3737. [7] M.Z. Zgierski, Chem. Phys. Letters 21 (1973) 525. [8]D.P.Craig,L.E.LyonsandJ.R. Walsh,Mol.Phys.4(1961) 97. [Q] I.V. Brovchenko, V.I. Tovstenko and M.T. Shpak, Fiz. Tverd. Tela. 3 1 ( 1989) 1. [lo] E.A. Silinsh, Organic molecular crystals - their electronic states (Springer, Berlin, 1980) p. 108. [ 111A. Tiberghien and G. Delacote, J. Phys. (Paris) 31 ( 1970) 637. [ 121 H. Sumi, J. Chem. Phys. 70 (1979) 3775.
ELSEVIER
CHEMICAL PHYSICS LETTERS
Chemical Physics Letters 229 (1994) 439-442
Vibronic coupling with low-frequency vibrations in quasi-forbidden Frenkel exciton states Piotr Petelenz K. GumiAski Department of Theoretical Chemistry, Jagiellonian University, Ingardena 3, 30-060 Cracow, Poland Received 28 July 1994
Abstract A simple dimer model is used to demonstrate that in weakly allowed Frenkel exciton states, where the factor-group splitting is largely due to charge transfer interactions, the exciton-phonon coupling constant may be considerably different for the two Davydov components. The model is applied to the S, transition of the naphthalene crystal and suggests a smaller Davydov splitting than normally assumed (60 cm- ’ rather than 150 cm- ’ ).
1. Introduction
Due to very weak dipolar interactions of quasi-forbidden excited states, the Davydov splitting of the corresponding Frenkel excitons is usually small. Its main contributions come from higher multipole interactions [ 1 ] and, as shown recently [ 2-61, charge transfer (CT) effects. Both types of terms mentioned above strongly depend on the distance between the interacting molecules. Hence, it is reasonable to expect that in the resulting eigenstates the crystal lattice may be considerably distorted with respect to the ground state. This should result in strong exciton-phonon coupling and affect the optical spectrum. As the coupling is due to the dependence of the off-diagonal (in local basis) matrix elements of the Hamiltonian on nuclear coordinates, it is likely to affect differently each of the Davydov components (as demonstrated by Zgierski [ 7 ] ) . The effect should be observable e.g. in the very weakly allowed SI state of the naphthalene crystal,
whose properties are strongly affected by CT effects [2-61. The spectrum of the crystal in the S, region starts with a narrow line at 3 1476 cm- ’ (a polarized [ 81, referred to as the A band [ 9]), followed by a very broad, almost structureless band at 31623 cm-’ (predominantly b polarized [ 81, referred to as the B band [ 91). Commonly, these bands are interpreted as the two Davydov components of the corresponding Frenkel exciton state. This explanation fails to account for the enormous difference in the width of the two bands. An alternative interpretation was suggested recently [ 9 1. According to the hypothesis proposed by Brovchenko et al., the broad band observed at 3 1623 cm-’ would not correspond to the b polarized Davydov component of the O-O transition, but would represent a strongly depolarized phonon sideband built upon the a component (which would represent the zero-phonon line, ZPL). In fact, the large width and the lack of structure of the 3 1623 cm- ’ band very well agree with the known characteristics of typical phonon sidebands.
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDIOOO9-2614(94)01089-7
P. Petelenz / Chemical Physics Letters 229 (1994) 439-442
440
Although some aspects of the interpretation presented in Ref. [ 91 are not entirely clear, the idea is attractive and very likely correct. However, there is still a problem: where is the other Davydov component? If the corresponding absorption line is as sharp as the A band, it should be discernible from the broad background provided by the phonon sideband. It will be shown that the peculiarities of phonon coupling to quasi-forbidden exciton states might explain the apparent absence of the b polarized counterpart of the A band.
2. Model For the sake of simplicity, we represent the two molecules from the unit cell of the naphthalene crystal as two moieties of a model dimer, and focus our attention on the translational intermolecular vibration which is supposed to mimic the lattice phonon modes of the crystal. Vibronic coupling with intramolecular vibrations is disregarded. The treatment is confined to four electronic states, symbolically denoted as 1A*B), I.4B*), I A+B- ) and )A-B+), where the asterisk denotes electronic excitation. These (‘local’) states describe the exciton and the charges as strictly localized at one of the moieties. Only one electronic excited state is taken into account for each monomer. Correct to terms linear in intermolecular overlap integrals, in this basis the electronic Hamiltonian assumes the form E(q)
M(q)
D,(q)
M(q) De(q)
E(q) D,(q)
&(q) E”(q)
i D,(q)
D,(q)
0
D,(q) D,(q) 0
'
(1)
E"(q) 1
where E(q)=(A*B]H]A*B)=(AB*]H]AB*) and Em(q)=(.4+B-]HlA+B-)=(A-B+]H]A-B+) are the diagonal energies of the local Frenkel and local CT states, M= (A*B ]H] AB*) is the exciton transfer (resonance) integral, while D, = (A*B]H(A+B-) and Dh=(A*B/H]A-B+) stand for the exciton dissociation integrals with electron and hole transfer, respectively. Out of all vibrational coordinates, we include only one intermolecular mode, presumably translational, which is denoted by q.
In the harmonic gies read E(q)=tq*+b,q E”(q)
approximation,
the diagonal ener-
>
(2a)
=A+ fq2+b2q,
(2b)
where 6, and 6, represent the displacements of the equilibrium position in the intermolecular mode upon creation of a local Frenkel and local CT state, respectively, and can be expressed as the derivatives a W/ tlq of the intermolecular interaction energy in the corresponding local states. .4 is the energy gap between the two kinds of local states. The standard dimensionless units are used for energy and length, and the energy of the local Frenkel state at equilibrium position is taken as zero. In the basis of symmetry adapted (‘delocalized’) electronic states [F&)=2-I’*( ]CT?)=2-I”(
]A*B) + ]AB*))
,
]A+B-)+]A-B+)),
(3a) (3b)
the matrix of Eq. ( 1) can be split into effective Hamiltonians for the Davydov components ;q2+b,qkM(q)
(
D,(q)
Dk (q)
A+ $q2+b2q
>’
(4)
whereD,(q)=D,(q)kD,(q). As in the naphthalene crystal the CT states are located above the lowest intramolecular excited state [ lo], the lower eigenvalue of each of the two Hamiltonian matrices of Eq. (4) represents the approximate potential energy for the corresponding Davydov component of the Frenkel state. The dissociation integrals D, and Dh depend linearly on the intermolecular overlap integral, which in turn should depend exponentially on intermolecular distance. Therefore, it is resonable to approximate the explicit form of D+ (q) as D, (4) =Di
(0) exp( -&)
,
(5)
“’ is the Slater exponent for carwhere S=,~(h/mo) bon 2p, orbitals (3.09 A-’ ), expressed in the harmonic oscillator dimensionless length units. m is the reduced mass for the intermolecular vibration, equal to one half of the molecular mass. With Eq. (5 ) and taking advantage of the fact that D+ (0)/A<< 1, the potentials may be approximated
P. Petelenz /Chemical Physics Letters 229 (1994) 439-442
by the perturbational der in q)
result (correct to the second or-
+ [b, -2D2,(0)d/AtM’(O)]q
+D2,/AfM(O).
(6)
The above expression shows that the coupling with charge transfer states modifies both the effective force constant for the intermolecular mode and its equilibrium position in the states of Frenkel parentage. The equilibrium position is evidently different for each of the two Davydov components, which will affect the observed width of the corresponding transitions.
441
differentiation with respect to R (expressed in the dimensionless units of the 50 cm-’ oscillator) yields b, = -0.95. With the above input data, the displacement parameters corresponding to the two Davydov components are b_ = [b, -202(0)6/A-M’(O)]
= -0.95,
b+=[b,-20:(0)6/A+M’(O)]=-1.3.
(7)
The difference between the displacement parameters in the two Davydov components is evidently substantial. In reality, it is likely to be even larger.
4. Effective mode approach and scaling 3. Estimates for the translational mode The order of magnitude of the effect may be estimated as follows: Let us assume the intermolecular mode to have a frequency corresponding to low frequency lattice lattice phonons, e.g. 50 cm-‘. This vibrational quantum sets the energy unit for Eq. ( 6 ), and the value of 6=0.32. Based on the experimental energy of the S, state ( = 3 1000 cm- ’ ) and the calculated energy of the lowest CT state of the naphthalene crystal (4.38 eV [lo] ), A=75. In a different context [6], the dissociation integrals ( - 0.02 12 and - 0.0 19 1 eV for the electron and the hole, respectively) were estimated from charge transfer integrals calculated by Tiberghien and Delacote [ 111, yielding D, (0) = - 6.5, D_ (0) = -0.34. The corresponding estimate [ 61 of the exciton resonance integral ( = 0.0005 eV) yields M(O)=O.OS. For a dipole allowed transition, M should be proportional to R -n, with n= 3. For the S, naphthalene state, a stronger distance dependence is expected. However, with the above value of M(0) even with n = 5 M’( 0) = 0.0 1 and is numerically irrelevant. The energy W of the interaction between the molecules is due primarily to the dispersion term, proportional to R -6 [2,3 1. Its change between the ground and excited electronic state determines the gas-tocrystal shift. Following the approach of Gisby and Walmsley [ 2,3], the proportionality constant may be evaluated from the experimental shift of about 400 cm-’ (after deducting the CT contribution). Then,
For the lower component, the predicted Stokes shift is S_ = bZ hw/2 z 25 cm-‘. In the crystal spectrum, it should roughly correspond to the maximum of the phonon sideband. Then, if the band peaking at 3 1623 cm-’ really has a considerable contribution from the phonon sideband of the 31476 cm-’ transition (which, after Brovchenko et al. [ 91, we accept as a working hypothesis), b_ seems to be seriously underestimated. This is not surprizing since we have included only one intermolecular mode, completely ignoring the librations, and having no way to include in the dimer model the effect of phonon dispersion, coupling with acoustic phonons, etc. On that view, it seems reasonable to treat the q mode as an effective one and scale its displacement parameters in such a way that the Stokes shift for the lower Davydov component in the crystal be approximately reproduced. This increases b by a factor of 2, so that beff_ = 1.9: beff+ = 2.6. With these values, the expected intensity ratio I0 of the zero phonon line to the phonon sideband is exp(-+b&)=0.16 and exp(-tb&+)=0.03 for the lower and the upper Davydov component, respectively. Therefore, the relative intensity of the ZPL for the upper component is smaller by an order of magnitude, and the line is very likely to fuse with the background and with its own sideband (which carries most of the intensity). On that view, it is not surprizing that no sharp b polarized line is experimentally observed. The above conclusion rests on the tenuous scaling
442
P. Perelenz /Chemical Physics Letters 229 (1994) 439-442
argument, which assumes that the sensitivity of the dissociation integrals to changes of other intermolecular coordinates is as strong as for the translational mode. At least for some librations, this does not seem unlikely [ 121. Even if the argument is not quantitatively correct, it definiteiy shows a substantial difference in the Z, values for the two Davydov components.
5. Davydov splitting in naphthalene In view of Section 4, it is reasonable to suppose that the B band of the naphthalene crystal is in fact composite, consisting of the (depolarized) phonon sideband of the lower (A) Davydov component, and of the genuine transition to the upper Davydov component. The latter should have contributions both from the (probably dominant) phonon sideband and from the zero-phonon line which may be difficult to resolve (due to its low intensity and di~useness}. However, the zero phonon line is expected to be asymmetric: diffuse and likely to amalgamate with the sideband at higher energies, but sharper on the low-energy side. This makes one expect a relatively steep absorption edge at that energy. Such an edge is indeed observed in the spectra of Brovchenko et al. [ 91. There is a shoulder due to the steep incline on the low energy side of the B band (at about 31535 cm-‘). It is observed in b polarization, but absent in a polarization -just as expected for the ZPL. Very likely, that energy represents the actual position of the zero phonon line of the upper Davydov component. If this interpretation is accepted, the Davydov splitting between the a and b polarized components is about 60 cm-‘. This contradicts the normally accepted value of 150 cm-’ [ 2,8], and suggests that the arguments leading to that value should be reexamined. The Davydov splitting in the Sr state of the naphthalene crystal was normally identified with the gap between the A band and the maximum of the B band in the crystal spectrum. Even if the B band were due
exclusively to the upper Davydov component (with the zero phonon line unresolved from the sideband), the arguments of Section 4 suggest that the zerophonon line (being the b polarized counterpart of the A band) should not correspond to the band maximum but should be Iocated at lower energy. Hence, even in that case the Davydov splitting would be less than the 150 cm-’ estimated from the position of the peak. The difference might be considerable and the steep edge-like shoulder at 31535 cm-i could very well correspond to the zero-phonon line of the b palarized exciton state. This value also agrees very well with the theoretical results obtained from the cluster model [ 61 (which, for a neat crystal, are equivalent to those of standard exciton band structure calculations [ 5 ] ) . In fact, with the most accurate parametrization available to date [ 61, the calculated Davydov splitting of the S, state is about 70 cm-‘.
This research was supported by grant No. 206699101 (1723/2/91) from the State Committee for Scientific Research (Poland).
References [ I ] D.P. Craig and S.H. Walmsley, Mol. Phys. 4 ( 1961) I 13. [2] J.A. Gisby and S.H. Walmsley, Chem. Phys. Letters 135 (1987) 275. [3] J.A. Gisby and S.H. Wafmsley, Chem. Phys. 122 (1988) 271. f4] P. Petelent, Chem. Phys. 138 (1989) 35. [ 5 ] P. Petelenz and M. Slawik, Chem. Phys. 157 ( 1991) 169. [ 6lI.V. Brovchenko, A. Eilmes and P. Petelenz, J. Chem. Phys. 98 (1993) 3737. [7] M.Z. Zgierski, Chem. Phys. Letters 21 (1973) 525. [8]D.P.Craig,L.E.LyonsandJ.R. Walsh,Mol.Phys.4(1961) 97. [Q] I.V. Brovchenko, V.I. Tovstenko and M.T. Shpak, Fiz. Tverd. Tela. 3 1 ( 1989) 1. [lo] E.A. Silinsh, Organic molecular crystals - their electronic states (Springer, Berlin, 1980) p. 108. [ 111A. Tiberghien and G. Delacote, J. Phys. (Paris) 31 ( 1970) 637. [ 121 H. Sumi, J. Chem. Phys. 70 (1979) 3775.