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Charge transfer effects in the pressure dependence of the ultraviolet absorption spectra of polyacene crystals
Chemical Physics 138 (1989) 35-44 North-Holland, Amsterdam
CHARGE TRANSFER EFFECTS IN THE PRESSURE DEPENDENCE OF THE ULTRAVIOLET ABSORPTION SPECTRA OF POLYACENE CRYSTALS Piotr PETELENZ K. Gumiriski Department of Theoretical Chemistry, Jagiellonian University, Karasia 3, 30-060 Cracow, Poland
Received 9 May 1989
The model applied previously for the interpretation of the electroabsorption spectra of anthracene is used to calculate the pressure dependence of the Davydov splitting and of the gas-to-crystal shift in the naphthalene and anthracene crystals. The results suggest that the overlap-dependent contributions due to the coupling with the CT states may dominate the pressure dependence of both quantities. For anthracene, these contributions may be larger than it has been previously supposed. Some suggestions regarding more complete calculations to be done in the future are given.
1. Introduction
temative to that of ref. [ 91. Our results for naphthalene (presented in section 4) essentially confirm the conclusions of ref. [ 91 and demonstrate the equivalence of the two models. In section 5, we apply the same model for the anthracene crystal and show that the overlap-dependent terms may considerably contribute to the energy of Frenkel excitons and to its pressure dependence. The results suggest that the inclusion of the CT interactions in a more comprehensive treatment might improve the description of the pressure dependence of the Davydov splitting, as Gisby and Walmsley [ 9,101 predicted. However, contrary to their tentative conclusion, we will argue that overlap-dependent terms may also contribute significantly to the gas-to-crystal shift. Some contentious points of the present results will be discussed in section 6.
The calculations of the Davydov splitting of singlet Frenkel excitons in organic molecular crystals are relatively common in the literature, mostly in the context of band structure calculations [ l-81. However, a systematic study of the effect of pressure on that splitting has been undertaken only recently. In their recent papers [ 9, lo], Gisby and Walmsley presented an ingenious scaling approach which allowed them to interpret the pressure dependence of the Davydov splitting and of the gas-to-crystal shift for the lowest singlet exciton of the naphthalene and anthracene crystals. For naphthalene [ 91, the terms depending on intermolecular overlap were included and found to be rather important. For anthracene [ lo], the overlapdependent terms were neglected entirely, resulting in a much too weak pressure dependence of the Davydov splitting. The objective of the present work is to probe just that part of the neglected contributions which is linear in nearest neighbour intermolecular overlap integrals and describes the coupling between the Frenkel excitons and the charge transfer (CT) excitons. The treatment, admittedly semiquantitative, is based on a model applied previously for the interpretation of electroabsorption spectra [ 1 l- 15 ] (introduced in section 2 and slightly modified in section 3), and al0301-0104/89/$03.50 0 Elsevier Science Publishers ( North-Holland Physics Publishing Division )
2. Model Hamiltonian for a linear crystal As in our previous papers on polyacenes [ 1l- 15 1, we use a model which describes the electronic excited states of a linear crystal with two molecules in the unit cell. The corresponding model Hamiltonian reads [ 11-151: H=
B.V.
C H(m), m
(1)
P. Petelenz /Absorption spectra ofpolyacene crystals
36
where
Jh = (i2,,-
H(m)=H,+H,+H,+H,+H,+H,,
The Frenkel form
+ HJ + HJ, ,
t2a)
Hx =E 1 B&B,, I7
+M(O) (B:,B,,
M(n)
+ C M(n) tB:,,&,,+.+h.c.) n
H,=EP
1 T,:,Tm: c
(2c)
HL =EL 1 L,:mLwi d
3
Ho = 1 (DeB,:,L,,, o.n
+D,BZmLvn+n,-n
+h.c.) HP=
,
I W;,j2,
=O
I
I W2,
transfer
> .
integrals
are of the
> =kf,
I Wimi2,m+n
>
ifn=-1, otherwise ,
The remaining matrix elements are referred to as exciton dissociation integrals and are defined as
(2d)
C [D:BiATm+Pom) 0
+ Dk(B;,+w H,=
exciton
= (hmjim+n =A4
t2b) > HT=E’
1 P,:,P,,, ”
I WI, > =
M(O) =
+h.c.)
+ 1 Wn)B,:,%,,+,, n.n
I
T,.,,+B&,P,,,)+h.c.]
C (J,P,:,T,,+J,,P;,_,, ”
HJ = C [J:(PS,,,, ” + J;,(pt,+n,L,~,,~
,
T,,+h.c.)
t2e) (2f1
+ T&,&z-,,) + TX-,,,,,-.,)
13
(2g)
with n=&l,
a=1,2,
fJ’#cJ, for (T= 1 ,
n’=&(a)=1 c-1
fora=2.
BZ,,,PA,, TA,, and LA; are the creation operators for Frenkel excitons and for intracell, crosswise and lengthwise CT excitons, respectively, with their Hermitian conjugates standing for the corresponding annihilation operators. According to the model proposed previously [ 14161, we consider only interactions between molecules that are nearest neighbours. The electron and hole transfer integrals are then given by J, = (j&,,- I I Wbm > =
I I W,,, I W;,
In the above formulas, j,, and j& denote the highest occupied orbital (HOMO) and the lowest unoccupied orbital (LUMO), respectively, of the molecule located at site o of the mth unit cell. V stands for the intermolecular interaction operator. The notation strictly follows ref. [ 161, to which the reader is referred for details. (Note some typographical errors in eq. ( 10) of ref. [ 161, corrected in the above formulas. )
> = = (j;,,+
I I W,
> 9 > , ) ,
3. Modification and diagonalization Hamiltonian
of the model
The applicability of the above model Hamiltonian to a three-dimensional crystal was discussed before [ 12,13 1. The argument was based on the analysis of the values of the CT integrals calculated in refs. [ 1618 ] and led to the conclusion that the CT interaction between the molecules belonging to different ab planes of the anthracene crystal is much weaker than the corresponding interaction along the b axis of the crystal. Since the CT interactions along the a axis are equally weak, the anthracene crystal, from the point of view of its lowest CT states, could be split into a
P. Petelenz /Absorption spectra ofpolyacene crystals
set of linear crystals with two molecules in the unit cell and the stacking axis directed along the b axis of the actual three-dimensional crystal [ 1 1- 15 1. However, this is not true from the point of view of Frenkel excitons. In fact, a Frenkel exciton located at moleculeAofIig. 1 iscoupledtotheA’Br andA’CT charge transfer states (which are included in the model linear crystal), but also to A’DT and A’E’, which are not included. The neglected interactions affect the CT states of the linear crystal enclosed between the two lines only in a higher order, but they affect the Frenkel state just as much as the interactions (with A’BT and A’Cr ) included in the model crystal do. As the description of the Frenkel manifold is the primary objective of this paper, we will make an ad hoc supplement to the Hamiltonian of eqs. ( 1) , (2a)- (2g) by replacing the energy E of the intramolecular exciton by EF,the value renormalized by the CT interaction with the two extra molecules (i.e. four extra CT states). Accordingly, EF reads EF=E-2(D;2+D;2)/(EP-E).
(3)
(It should be noted that this modification affects the Davydov splitting only to a very minor extent. However, it is important for the reproduction of the gasto-crystal shift. ) Secondly, the dipole-dipole interaction between molecule A and molecules D and E is also neglected in our model. This may be remedied by replacing the exciton transfer integral M by the corresponding lattice sum. The Hamiltonian of eqs. ( 1) , (2a)- ( 2g) is a purely electronic one. As suggested previously [ 151, the main effect of molecular vibrations consists in the reduction of the matrix elements by the vibrational overlap factors. It results in the following moditication of the matrix elements:
@ 0
0 @
@ 0
0 0
0 0
~=w(~:I%)12,
(da)
fi=J4(~3~o)12,
(4b)
dey=D,y(v+1)-“2
i r=O
(m: Iml')(mOImL--i)
3
(4c)
(4d)
(de) &=Jh*(u+l)-~,~o
l(m:Imo)12,
(W
where I mc) denotes the wavefunction for the vth vibrational state of the molecule, with p= 0 (which is suppressed) for the ground electronic state of the neutral molecule, p= * for the first electronic excited state of the neutral molecule, p= - for the ground electronic state of the positively and negatively ionized molecule, respectively. v stands either for a blank (for charge transfer between molecules in the same crystallographic position) or for a prime (for charge transfer between molecules in different crystallographic positions). As before [ 15,19 ] we consider only one intramolecular stretching mode of frequency 0.174 eV which, as is known from the spectra of isolated polyacene molecules, couples most strongly with the electronic degrees of freedom. When the change of vibrational frequency upon electronic excitation or ionization of a molecule is neglected, the vibrational overlap integrals which appear in eqs. (4a)-( 4f) involve displaced harmonic oscillator wavefunctions and may be easily calculated based on standard formulas. Due to its translational symmetry, the Hamiltonian of eqs. (l), (2a)-(2g) with or without the modifications of eqs. (4a)- (4f) may be block-diagonalized by means of the Fourier transformation +
,
0 Xk=N-‘12 C exp(ikm)X,,, m
0 0
37
0
Fig. 1. The model linear crystal (between the solid lines) and the supplementary CT interactions.
(5)
(where X,,, denotes B,, PO,,,, T,, and L,,, and N stands for the number of unit cells) to yield H= C k
H(k)
.
(6)
38
P. Petelenz /Absorption spectra ofpolyacene crysials
Due to the normal selection rule for optical transitions (Ak= 0), we are interested only in the k= 0 term, which reads H(O)=HO,+HO,+H~+HO,+HO,+H~,
(7a)
+H:+H$, with HO, = (E+2@) + 2&7x
”
HO,=EPx
C B&B,, Y.0
(B,:,Bv.2+h.c.) P&P,,,
1 L,+,,L,, Y.O..n
H;=(&++,,)
(7b)
HO,=ETC
“3 H:=EL
,
Y.0
T&T,,
,
(7c)
C B,+,L,,+h.c., Y.LT.ll
(7d)
+ 6;, 1 B &( P,,, + T,,) + h.c. , V.0 H: = (Je +&)
CP,+,T,,+h.c. Y.0
H$, = C [~U’,+,L,n, Y.0 + &(P,‘,L”,.,.
,
(7e) ( 7f)
+ T,+,Lo.-n, 1
+T:,L,,,,_,,)]
+h.c.
(7g)
The Hamiltonian matrix defined by the above set of equations may be diagonalized analytically [ 121 or by standard numerical methods. In the present paper we will adopt the latter approach. (Note some errors in the analytical formulas of ref. [ 12 1. In eq. ( 2 1) of that paper, J; and J’ should be replaced by J; and J’f , respectively. )
4. Naphthalene The Frenkel excitons deriving from the lowest ‘Lb singlet state of the naphthalene molecule are our first target. To facilitate the comparison with ref. [ 9 1, the treatment is limited to a purely electronic (adiabatic) model, without the modifications of eqs. (4a)-(4f). Most of the matrix elements of the Hamiltonian matrix of eqs. (7a)-( 7g) can be found in the litera-
ture. The energy E of the first electronic excited state of the naphthalene molecule is 3 1000 cm- ’ [ 8 1. The nearest neighbour excitation resonance integrals were calculated in ref. [ 91 and they are: W= -2.4 cm-’ and M= + 10.0 cm-’ (at atmospheric pressure). The sign of A4 depends on the choice of the relative phase of the molecular orbitals of translationally inequivalent molecules. It should be chosen in accordance with the phase convention used in the evaluation of charge transfer integrals and Frenkel exciton dissociation integrals (vide infra). If all the integrals were calculated within the same scheme from the same set of molecular orbitals, consistency would be automatically guaranteed. This is not the case in the present treatment, where A4 is taken from a different paper [ 91 than the CT integrals [ 201. In the lack of an independent argument, this compels us to consider the sign of M as adjustable. The results presented in this paper were obtained with the “plus” sign. In the lack of a properly calculated lattice sum for naphthalene we account for the interaction with the two nearest molecules which are not included in the model crystal (cf. section 3) by doubling the value of M. The energy EP of the lowest ( f, 4, 0) CT state of the naphthalene crystal is about 36 100 cm- ’ [ 2 11, and the energy EL of the (0, 1,0) state is about 650 cm - ’ higher [ 2 11. The electron and hole transfer integrals J are known from several independent studies [ 16-18,201. We will use the values from ref. [ 201, where the pressure dependence of these integrals is also given. For the atmospheric pressure, these values are: J,=74 cm-‘, J,=-350 cm-‘, J:= -150cm-‘,&=120cm-‘. The Frenkel exciton dissociation integrals D normally exceed the corresponding CT integrals Jby 1520% [ 12,221. Therefore, we take the D integrals equal to 1.15 times the corresponding J integrals; this yields D,=SS cm-‘, D,=400 cm-‘, D:= -170 cm-’ and Db = 140 cm - ‘. These estimates are certainly very approximative: Firstly, the observation of the relationship between the J and the D integrals was based on rather crude and incomplete numerical results [ 221. Secondly, this observation pertained originally to the Frenkel state deriving from the same molecular orbitals as the CT states (presumably, HOMO and LUMO). This is not the case for the ‘Lb state. Nevertheless, we believe that these estimates are not grossly in error. In fact, the charge transfer integrals
P. Petelenz /Absorption spectra of polyacene crystals
as well as the exciton dissociation integrals are limited by intermolecular overlap, both being roughly linear in intermolecular overlap integrals. For this reason, one intuitively expects them to be of the same order of magnitude and the results of ref. [ 221 only confirm this expectation. On similar physical grounds, one tends to expect the dissociation integrals for different low-energy Frenkel states not to differ too much. In order to describe the effect of pressure, some explicit functional representation of the dependence of the Hamiltonian matrix elements on lattice parameters has to be adopted. We neglect the effect of pressure on the angles and assume that the intermolecular transfer integrals depend only on the distances between the centres of the molecules. For Frenkel exciton transfer integrals (M and W) we assume this dependence to be of dipole-dipole type, i.e. R -‘. As the charge transfer integrals are approximately linear in intermolecular overlap integrals, we assume their dependence on the distance to be exponential, i.e. exp( --ML Atomic orbitals are normally represented as superpositions of Slater type functions. At different distances and in different directions different terms of such a superposition will give the leading contribution to the wavefunction. Consequently, the actual value of the exponent p determining the distance dependence of the CT integrals will depend on the distance and on the relative orientation of the molecules involved. Obviously, p will also depend on the kind of molecular orbital involved, i.e. it will generally be different for electron and hole transfer integrals. In ref. [ 201, the transfer integrals were calculated for several values of the pressure. At low pressures, the contraction of the lattice is very small so that the exponential function in the distance dependence of the transfer integrals may be approximated by the linear function, which allows one to estimate the exponents p from the values of the transfer integrals at two different pressures. With the pressure dependence of the lattice parameters as listed in ref. [ 201, this produces p equal to (in A-‘) 2.2330, 1.7683, 0.9429 and 3.0882 for J,, J,,, J: and J;, , respectively. Based on the argument presented above, we adopt the same values of the exponents for the distance dependence of the exciton dissociation integrals D. The results of our calculations are presented in figs.
39
2 and 3, and compared with those of ref. [ 91 represented by dotted lines. (The latter were very close to experimental data.) As the long-range interactions are absent from our model and so are all dispersive interactions, the total values of the Davydov splitting A and of the gas-to-crystal shift S (solid lines) are underestimated (the latter seriously). In fact, out of all the interactions which are of the zeroth order in intermolecular transfer integrals, only the nearest
&cm-’
2801
Fig. 2. Davydov splitting Aof the S, state of the naphthalene crystal as a function of crystal density (expressed in the units of the density at atmospheric pressure). Solid line - total value obtained in this paper (with M and W as described in the text), broken line - CT contribution (M= W=O), dotted line - total value from ref. [ 91.
tS,cm-’ -700.
.._-”
:
.-.
-500.
-300J
-100
I
_____--
______----________.-----
______--
I
I. 00
,
I. 04
1.08
I. 12
P/g.
Fig. 3. Gas-to-crystal shift S of the S, state of the naphthalene crystal as a function of crystal density. Broken line - CT contribution obtained in this paper (M= W=O), dotted line - total value from ref. [ 91.
40
P. Petelenz /Absorption spectra ofpolyacene crystals
neighbour terms M and W corresponding to the ( 4, f , 0) and (0, 1, 0) directions are included. Consequently, one could not really expect the total values to be correctly reproduced. The CT contribution to the Davydov splitting obtained from our model (dashed line in fig. 2a, obtained with M= W=O) agrees very well with the conclusion of ref. [9] that the relative contribution of the overlap-dependent terms to the splitting is about 30% at atmospheric pressure and increases to about 500/b at 1.5 GPa. This demonstrates that the charge transfer terms, linear in intermolecular overlap, are the leading overlap-dependent contributions. This is not surprising, since the higher-order terms are expected to be numerically close to the triplet exciton transfer integrals, and those are known to amount only to a few wavenumbers [ 18 1. It is readily seen that the form of the density dependence of the Davydov splitting is reasonably well reproduced even by the CT contribution alone. The nearest neighbour exciton resonance interactions M and W change the total value of the splitting (bringing it closer to experiment) but hardly affect the form of the density dependence. It seems to indicate that the pressure dependence of A is in naphthalene due primarily to the involvement of the CT terms. The slightly steeper dependence obtained from the more complete calculations of ref. [ 91 is probably due to dispersive interactions, while the discrepancy in the total value of A follows primarily from the neglect of the (considerable) exciton resonance interaction in the ( A, 4, 1) direction. Fig. 3 shows that the CT terms contribute only marginally to the gas-to-crystal shift in naphthalene, but prominently to the density/pressure dependence of the latter. (The total value of S obtained from our model with M# 0, W# 0 differs negligibly from the CT contribution to S, and hence is not displayed.) This agrees very well with the expectation of Gisby and Walmsley [ 9 1, and indicates that the inclusion of overlap dependent terms may be important in a future attempt to ameliorate their approach.
5. Anthracene The case of anthracene is in a way more obvious, since the Frenkel state at hand derives from the same
molecular orbitals as the CT states under consideration, so that the argument of ref. [ 221 is directly applicable for the estimation of the exciton dissociation integrals. Accordingly, from the transfer integrals [ 181 (all quantities in cm-‘): J,= 197, Jh= -357, J:=424, Jb=-337 we get: D,=226.5, D,,=-411, 0: =488, D;, = - 388. The pressure dependence of the transfer integrals is not directly available from the literature. A transfer integral is the difference of the intermolecular resonance integral and the intermolecular exchange integral. While the former were calculated for anthracene at different pressures [ 251, we are not aware of any calculations of the pressure dependence of the latter for this crystal. One would intuitively expect that, account taken of the structural similarity, this dependence should not be very different from that in naphthalene. This expectation may be roughly tested on the resonance integrals, which are known for both crystals [ 20,23 1. (The pressure dependence of the lattice parameters of the anthracene is listed in ref. [ 231. ) In fact, if the distance dependence of these integrals is modelled by the exponential function, the exponents ,u governing the changes of the resonance contribution to J,, J,, and J: do not differ drastically between naphthalene and anthracene. This makes us adopt for J,, J,, and J: (and consequently for De, D,, and Dk) the same ,u values as for the corresponding integrals in naphthalene. There is an unexpected anomaly for Jh , where the corresponding exponent for the resonance integral is 4.0075 8, - ’ for naphthalene but only 0.99 14 A- ’ for anthracene. Nevertheless, we may still hope that the distance dependence of the exchange contribution to the transfer integral is more or less the same for both crystals, since the intermolecular exchange integral (where the leading part of the distance dependence is due to orbital overlap) is expected to be less sensitive to geometry than the intermolecular resonance integral where the Goeppert-Mayer potential (depending exponentially on the distance) is also involved. Consequently, we approximate p for J;, as an average of the value determined for the resonance integral in anthracene (0.99 14 8, - ’ ) and of the value estimated for the distance dependence of the exchange integral in naphthalene (2.842 1 A-’ [ 20]), the two values being weighted by the contributions of the corresponding integrals to Ji (92.5 and 244.5
P. Petelenz /Absorption spectra of polyacene crystals
cm-‘, respectively). This yieldsp=2.3341 A-‘. The interpretation [ 15 ] of the electroabsorption spectrum of the anthracene crystal [ 241 led us to the value EG= 34255 cm - ’ for the optical band gap, and this value is being used in the present paper. With the electrostatic stabilization energies of the CT states known from ref. [ 251 it yields EP=27340 cm-’ and EL=28331 cm-‘. In order to calculate the vibrational overlap integrals we assume that the frequency of the progression forming mode is the same in all electronic states of interest. Then, the overlap integrals are completely defined by the displacement parameters lo, 1+ and 1_ representing the change of the equilibrium position upon electronic excitation and positive or negative ionization, respectively. &,= 1.50 and I + = 1.OO are readily found from the absorption spectrum [ 68 ] and photoemission spectrum [ 26,27 ] of the anthracene molecule. Based on the pairing theorem, we approximate ;i _ as I _ =I + = 1.OO. Supplemented by the value of the free molecule excitation energy to S, (26000 cm-’ [ 6-S]), this set of data allows us to estimate the pressure dependence of the CT contribution to the Davydov splitting. It is represented by the solid line in fig. 4a. It is readily seen that the CT contribution is substantial. As the numerical value EG quoted above is based on an interpretation of the electroabsorption spectrum [ 241 which may at the present moment be considered only hypothetical (due to the invoked involvement of an intramolecular phonon of energy 0.174 eV [ 15]), we repeated the calculations with a more conservative choice of EG= 35657 cm-‘. This led us to the zero-pressure values of the CT contribution to A and S reduced by a factor of 1.5-2, but with the pressure dependence still stronger than for the original choice of EG. As even with the original choice of EG the pressure dependence of the total values of A and S comes out slightly too strong compared to experiment, we believe that the pressure dependence is consistent with the gap of 34255 cm-‘, as proposed in ref. [ 15 ] , rather than with the “conservative” value of 35657 cm-‘. Of course, in view ofthe limited overall accuracy of the present model approach, this argument cannot be considered as a conclusive proof in favour of the interpretation of ref. 1151. However,
even with the “conservative”
choice of
A,cm-’ 800
a)
I
400.
I
l
1.08
I. 00
I. 16
3190
A,cm-' T
b)
400.
^. ,
1.00
1.08
1.16
3+
Fig. 4. Davydav splitting of the S, state of the anthracene crystal as a function of crystal density. Broken lines - experimental, dotted lines - results of ref. [ lo], solid lines - calculated in this paper: (a) CT contribution, (b) total value. The subsequent lines ofeach kind represent (from top to bottom) the O-O, O-l and O2 vibronic transitions.
EG, the resulting CT contribution to A and S is still considerable, which highlights the necessity of its inclusion in the description of the pressure dependence, suggested by Gisby and Walmsley [ lo]. Fig. 4a shows that, like in naphthalene, the slopes of the curves describing the density dependence of the CT contribution to the Davydov splitting are close to the slopes of the corresponding experimental A(p) dependences, in particular for the O-O transition, On the other hand, the curves calculated by Gisby and Walmsley [ lo] with the complete neglect of the overlap-dependent contributions (dotted lines) are much too flat. This seems to indicate that no matter how large the CT contribution really is, its inclusion is of crucial importance for the description of the pressure dependence of the Davydov splitting.
42
P. Petelenz /Absorption spectra of polyacene crystals
In order to test the relevance of the interaction between different CT states, mediated by the charge transfer integrals J, we have calculated d and S for J, = J,, = J: = J; = 0. It turned out that the energies of the Frenkel states are only very weakly affected by the changes of the J integrals (the shifts being of the order of a few wavenumbers). This was only to be expected, since the effect of these integrals on the Frenkel manifold appears in a higher order of a rapidly converging perturbation series. The estimates of the total values of d and S based on the present approach are again bound to be very poor due to inherent limitations of the nearest neighbour model. They need additional parameters M and W, describing the interactions within the Frenkel manifold. In our model, these parameters represent the nearest neighbour exciton transfer integrals. As has already been mentioned, in order to reasonably approximate the physics of the actual crystal, they have to be considered as effective values, including implicitly the interactions that are not explicitly included. Accordingly, we estimate them from the direction-independent part t,X.ar( 0) [ 61 of the corresponding lattice sums evaluated at the centre of the Brillouin zone. For the S, transition dipole moment of0.61 eA,tableVofref. [6] yields W=-386cm-’ and M= + 123 cm-‘. As mentioned in section 4, the sign of M depends again on the choice of the phase and, in the lack of an independent argument, has to be considered as adjustable. The results are presented for the “minus” sign, which yields a much better agreement with experiment (for the “plus” sign the Davydov splitting is seriously very overestimated). The results are represented in fig. 4b as solid lines. As is readily seen, the overall values of d are seriously overestimated (except for the O-l line, where the agreement seems to be accidental). The discrepancies are not surprising in view of the simplifications of the model mentioned above. It should be emphasized that the model is essentially non-empirical: there are no parameters (apart from the sign of M) adjusted specifically to fit S, A or their pressure dependence. On this view, the agreement correct to a factor not exceeding 2 seems rather reasonable. The present model tends to overestimate the pressure-induced changes while the scaling treatment of ref. [ lo] underestimated them, with the difference
that no scaling is invoked here as all the input data are known from independent calculations or experiments. This seems to suggest that an approach combining both models might be successful. With the overlap-dependent terms estimated in a non-empirical way, the long-range and dispersive contribution could be calculated exactly in the same way as done by Gisby and Walmsley [ 10 1, but with re-optimized values of the adjustable scaling parameters. The negligible importance of the coupling between different CT states (vide supra) would allow one to disregard this coupling completely and include only the direct interaction of the Frenkel state with the nearest neighbour CT states, thereby confining the treatment to a semi-local model similar to that used in ref. [ 9 1. The density dependence of the gas-to-crystal shift S obtained from our calculations is displayed in fig. 5. Following refs. [9,10], we present the value of S with respect to its value at atmospheric pressure, which is taken as zero. The figure demonstrates that within our model the increase of S with increasing density is dominated by the CT contribution. Due to exciton resonance interactions (M and W terms) the total value of the shift exceeds the CT contribution to the shift by about 200 cm-’ ( - 780 versus - 550 cm- ’ for the O-O transition). In contrast to the conclusions of ref. [ lo] the present results seem to suggest that the overlap-dependent terms may contribute significantly not only to
Fig. 5. Gas-to-crystal shift (with respect to its value at atmospheric pressure) of the S, state of the anthracene crystal as a function of crystal density. Broken lines-experimental, solid lines - total values calculated in this paper, dotted lines - CT contributions calculated in this paper.
P. Petelenr /Absorption spectra of polyacene crystals
A, but also to S. This tentative conclusion agrees well with the recent experimental work suggesting the interaction between Frenkel excitons and CT excitons as an important contribution to the energy of local Frenkel states [ 281. It has to be confirmed by a more sophisticated calculation, correctly including the longrange and the dispersive interactions.
6. Discussion In the present paper the naphthalene and anthracene crystals are represented by a model linear crystal of refs. [ 1l-l 51, supplemented by some local contributions which are relevant only for the Frenkel manifold. The model is designed specifically to deal with the CT contribution to the energy of the Frenkel states, linear in intermolecular overlap integrals. It is essentially non-empirical, but is oversimplified as it ignores most of the long-range interactions and all dispersive interactions. Therefore, its conclusions must be analyzed with some care. The validity of the model was tested in this paper for the case of naphthalene. It roughly reproduced the estimate [ 91 of the overlap-dependent contribution to the Davydov splitting of that crystal and of its pressure dependence, thereby confirming the conclusion [ 91 about the importance of overlap-dependent terms. The novel conclusion of this paper is that the pressure dependence not only of the Davydov splitting but also of the gas-to-crystal shift of the S, state of naphthalene is probably dominated by CT terms, with a smaller contribution from dispersive terms. The same model was subsequently applied for the S, state of anthracene. It led to the unexpected conclusion that the CT contribution both to the Davydov splitting and to the gas-to-crystal shift may be very substantial there. In order to relate this conclusion to previous results obtained within overlap-independent models, more complete calculations are necessary, possibly of the scaling type [ 9, lo]. A large contribution of the CT interaction to the Davydov splitting of a singlet Frenkel state has already been suggested for tetracene [ 141. Its relatively large value in anthracene (of the order of 400 cm-’ ) compares reasonably with the naphthalene value of about 50 cm- ’ [ 91, in view of the difference
43
between the corresponding CT integrals by a factor of 1.5-3 [ 18 ] and of the reduced energy gap between the Frenkel and the CT states. This conclusion has a somewhat unexpected aspect. As the CT contribution to the Davydov splitting in anthracene exceeds the experimental value of the splitting, it seems that the sign of other leading contributions must be opposite to that of the CT contribution. In that case, the actual value of the splitting would probably be determined mostly by the CT terms, reduced by a smaller but also considerable contribution from other interactions. Such an interpretation seems to be in conflict with the common belief that the Davydov splitting in anthracene is due mostly to overlap-independent terms, primarily of dipole-dipole origin. In order to eliminate the possibility that this is an artifact of the model used in this paper (which is unlikely, but possible), also the part of the present results concerning the Davydov splitting has to be confirmed by more complete calculations.
References [I ] V.M. Agranovich, Theory of Excitons (Nauka, Moscow, 1968) pp. 91-100, in Russian. [ 21 AS. Davydov, Theory of Molecular Excitons (McGraw-Hill, New York, 1962) p. 92. [ 31 D.P. Craig and S.H. Walmsley, Mol. Phys. 4 ( 1961) 113. [4] D.P. Craig and T. Thirunamachandran, Proc. Phys. Sot. (London) 84 (1964) 781. [5] G.D. Mahan, J. Chem. Phys. 41 (1964) 2930. [6] L.B. Clark and M.R. Philpott, J. Chem. Phys. 53 (1970) 3790. [7] M.R. Philpott, J. Chem. Phys. 50 (1969) 5117; 54 (1971) 111; 59 (1973) 4406; Chem. Phys. Letters 17 (1972) 54. [ 81 D.W. Schlosser and M.R. Philpott, Chem. Phys. 49 (1980) 181. [9] J.A. Gisby and S.H. Walmsley, Chem. Phys. Letters 135 (1987) 275. [IO] J.A. Gisby and S.H. Walmsley, Chem. Phys. 122 (1988) 271. [ 111 P. Petelenz, Proceedings of the XIth Molecular Crystal Symposium, Lugano 1985, p. 230. [ 121 R.W. Munn, P. Petelenz and W. Siebrand, Chem. Phys. 111 (1987) 207. [ 131 P. Petelenz and V.H. Smith Jr., Acta Phys. Polonica A 74 (1988) 435. [ 141 B. Petelenz, P. Petelenz, H.F. Shurvell and V.H. Smith Jr., Chem. Phys. 119 (1988) 25.
44
P. Petelenz /Absorption spectra ofpolyacene crystals
[ 151 P. Petelenz and V.H. Smith Jr., Chem. Phys. 131 ( 1989) 409. [ 161 J.I. Katz, S.A. Rice, S.I. Choi and J. Jortner, J. Chem. Phys. 39 (1963) 1683. [ 17 ] R. Silbey, J. Jortner, S.A. Rice and M.T. Vala, J. Chem. Phys. 42 (1965) 733. [ I8 ] A. Tiberghien and G. Delacote, J. Phys. (Paris) 3 1 ( 1970) 637. [ 191 P. Petelenz, Chem Phys. 133 (1989) 199. [20] M. El Hamamsy, A.C. Damask and S.A. Elnahwy, Mol. Cryst. Liquid Cryst. 95 (1983) 209. [ 211 E.A. Silinsh, Organic Molecular Crystals - Their Electronic States (Springer, Berlin, 1980) p. 108. [ 221 P. Petelenz and M.Z. Zgierski, Phys. Stat. Sol. (b) 69 ( 1975) 133.
[ 231 S.A. Elnahwy, M. El Hamamsy and A.C. Damask, Phys. Rev. B 19 (1979) 1108. [24] L. Sebastian, G. Weiser and H. Blssler, Chem. Phys. 75 (1983) 103. [25] P.J. Bounds, W. Siebrand, I. Eisenstein, R.W. Munn and P. Petelenz, Chem. Phys. 95 ( 1985) 197. [ 261 P.A. Clark, F. Brogli and E. Heilbronner, Helv. Chim. Acta 55 (1972) 1415. [27] R. Boschi, J.N. Murrell and W. Schmidt, Discussions Faraday Sot. 54 (1972) 116. [28] I.V. Brovchenko, N.I. Ostapenko, V.I. Sugakov and M.T. Shpak, Izv. Akad. Nauk SSSR Ser. Fiz. 52 (1988) 659.
CHARGE TRANSFER EFFECTS IN THE PRESSURE DEPENDENCE OF THE ULTRAVIOLET ABSORPTION SPECTRA OF POLYACENE CRYSTALS Piotr PETELENZ K. Gumiriski Department of Theoretical Chemistry, Jagiellonian University, Karasia 3, 30-060 Cracow, Poland
Received 9 May 1989
The model applied previously for the interpretation of the electroabsorption spectra of anthracene is used to calculate the pressure dependence of the Davydov splitting and of the gas-to-crystal shift in the naphthalene and anthracene crystals. The results suggest that the overlap-dependent contributions due to the coupling with the CT states may dominate the pressure dependence of both quantities. For anthracene, these contributions may be larger than it has been previously supposed. Some suggestions regarding more complete calculations to be done in the future are given.
1. Introduction
temative to that of ref. [ 91. Our results for naphthalene (presented in section 4) essentially confirm the conclusions of ref. [ 91 and demonstrate the equivalence of the two models. In section 5, we apply the same model for the anthracene crystal and show that the overlap-dependent terms may considerably contribute to the energy of Frenkel excitons and to its pressure dependence. The results suggest that the inclusion of the CT interactions in a more comprehensive treatment might improve the description of the pressure dependence of the Davydov splitting, as Gisby and Walmsley [ 9,101 predicted. However, contrary to their tentative conclusion, we will argue that overlap-dependent terms may also contribute significantly to the gas-to-crystal shift. Some contentious points of the present results will be discussed in section 6.
The calculations of the Davydov splitting of singlet Frenkel excitons in organic molecular crystals are relatively common in the literature, mostly in the context of band structure calculations [ l-81. However, a systematic study of the effect of pressure on that splitting has been undertaken only recently. In their recent papers [ 9, lo], Gisby and Walmsley presented an ingenious scaling approach which allowed them to interpret the pressure dependence of the Davydov splitting and of the gas-to-crystal shift for the lowest singlet exciton of the naphthalene and anthracene crystals. For naphthalene [ 91, the terms depending on intermolecular overlap were included and found to be rather important. For anthracene [ lo], the overlapdependent terms were neglected entirely, resulting in a much too weak pressure dependence of the Davydov splitting. The objective of the present work is to probe just that part of the neglected contributions which is linear in nearest neighbour intermolecular overlap integrals and describes the coupling between the Frenkel excitons and the charge transfer (CT) excitons. The treatment, admittedly semiquantitative, is based on a model applied previously for the interpretation of electroabsorption spectra [ 1 l- 15 ] (introduced in section 2 and slightly modified in section 3), and al0301-0104/89/$03.50 0 Elsevier Science Publishers ( North-Holland Physics Publishing Division )
2. Model Hamiltonian for a linear crystal As in our previous papers on polyacenes [ 1l- 15 1, we use a model which describes the electronic excited states of a linear crystal with two molecules in the unit cell. The corresponding model Hamiltonian reads [ 11-151: H=
B.V.
C H(m), m
(1)
P. Petelenz /Absorption spectra ofpolyacene crystals
36
where
Jh = (i2,,-
H(m)=H,+H,+H,+H,+H,+H,,
The Frenkel form
+ HJ + HJ, ,
t2a)
Hx =E 1 B&B,, I7
+M(O) (B:,B,,
M(n)
+ C M(n) tB:,,&,,+.+h.c.) n
H,=EP
1 T,:,Tm: c
(2c)
HL =EL 1 L,:mLwi d
3
Ho = 1 (DeB,:,L,,, o.n
+D,BZmLvn+n,-n
+h.c.) HP=
,
I W;,j2,
=O
I
I W2,
transfer
> .
integrals
are of the
> =kf,
I Wimi2,m+n
>
ifn=-1, otherwise ,
The remaining matrix elements are referred to as exciton dissociation integrals and are defined as
(2d)
C [D:BiATm+Pom) 0
+ Dk(B;,+w H,=
exciton
= (hmjim+n =A4
t2b) > HT=E’
1 P,:,P,,, ”
I WI, > =
M(O) =
+h.c.)
+ 1 Wn)B,:,%,,+,, n.n
I
T,.,,+B&,P,,,)+h.c.]
C (J,P,:,T,,+J,,P;,_,, ”
HJ = C [J:(PS,,,, ” + J;,(pt,+n,L,~,,~
,
T,,+h.c.)
t2e) (2f1
+ T&,&z-,,) + TX-,,,,,-.,)
13
(2g)
with n=&l,
a=1,2,
fJ’#cJ, for (T= 1 ,
n’=&(a)=1 c-1
fora=2.
BZ,,,PA,, TA,, and LA; are the creation operators for Frenkel excitons and for intracell, crosswise and lengthwise CT excitons, respectively, with their Hermitian conjugates standing for the corresponding annihilation operators. According to the model proposed previously [ 14161, we consider only interactions between molecules that are nearest neighbours. The electron and hole transfer integrals are then given by J, = (j&,,- I I Wbm > =
I I W,,, I W;,
In the above formulas, j,, and j& denote the highest occupied orbital (HOMO) and the lowest unoccupied orbital (LUMO), respectively, of the molecule located at site o of the mth unit cell. V stands for the intermolecular interaction operator. The notation strictly follows ref. [ 161, to which the reader is referred for details. (Note some typographical errors in eq. ( 10) of ref. [ 161, corrected in the above formulas. )
> =
I I W,
> 9 > , ) ,
3. Modification and diagonalization Hamiltonian
of the model
The applicability of the above model Hamiltonian to a three-dimensional crystal was discussed before [ 12,13 1. The argument was based on the analysis of the values of the CT integrals calculated in refs. [ 1618 ] and led to the conclusion that the CT interaction between the molecules belonging to different ab planes of the anthracene crystal is much weaker than the corresponding interaction along the b axis of the crystal. Since the CT interactions along the a axis are equally weak, the anthracene crystal, from the point of view of its lowest CT states, could be split into a
P. Petelenz /Absorption spectra ofpolyacene crystals
set of linear crystals with two molecules in the unit cell and the stacking axis directed along the b axis of the actual three-dimensional crystal [ 1 1- 15 1. However, this is not true from the point of view of Frenkel excitons. In fact, a Frenkel exciton located at moleculeAofIig. 1 iscoupledtotheA’Br andA’CT charge transfer states (which are included in the model linear crystal), but also to A’DT and A’E’, which are not included. The neglected interactions affect the CT states of the linear crystal enclosed between the two lines only in a higher order, but they affect the Frenkel state just as much as the interactions (with A’BT and A’Cr ) included in the model crystal do. As the description of the Frenkel manifold is the primary objective of this paper, we will make an ad hoc supplement to the Hamiltonian of eqs. ( 1) , (2a)- (2g) by replacing the energy E of the intramolecular exciton by EF,the value renormalized by the CT interaction with the two extra molecules (i.e. four extra CT states). Accordingly, EF reads EF=E-2(D;2+D;2)/(EP-E).
(3)
(It should be noted that this modification affects the Davydov splitting only to a very minor extent. However, it is important for the reproduction of the gasto-crystal shift. ) Secondly, the dipole-dipole interaction between molecule A and molecules D and E is also neglected in our model. This may be remedied by replacing the exciton transfer integral M by the corresponding lattice sum. The Hamiltonian of eqs. ( 1) , (2a)- ( 2g) is a purely electronic one. As suggested previously [ 151, the main effect of molecular vibrations consists in the reduction of the matrix elements by the vibrational overlap factors. It results in the following moditication of the matrix elements:
@ 0
0 @
@ 0
0 0
0 0
~=w(~:I%)12,
(da)
fi=J4(~3~o)12,
(4b)
dey=D,y(v+1)-“2
i r=O
(m: Iml')(mOImL--i)
3
(4c)
(4d)
(de) &=Jh*(u+l)-~,~o
l(m:Imo)12,
(W
where I mc) denotes the wavefunction for the vth vibrational state of the molecule, with p= 0 (which is suppressed) for the ground electronic state of the neutral molecule, p= * for the first electronic excited state of the neutral molecule, p= - for the ground electronic state of the positively and negatively ionized molecule, respectively. v stands either for a blank (for charge transfer between molecules in the same crystallographic position) or for a prime (for charge transfer between molecules in different crystallographic positions). As before [ 15,19 ] we consider only one intramolecular stretching mode of frequency 0.174 eV which, as is known from the spectra of isolated polyacene molecules, couples most strongly with the electronic degrees of freedom. When the change of vibrational frequency upon electronic excitation or ionization of a molecule is neglected, the vibrational overlap integrals which appear in eqs. (4a)-( 4f) involve displaced harmonic oscillator wavefunctions and may be easily calculated based on standard formulas. Due to its translational symmetry, the Hamiltonian of eqs. (l), (2a)-(2g) with or without the modifications of eqs. (4a)- (4f) may be block-diagonalized by means of the Fourier transformation +
,
0 Xk=N-‘12 C exp(ikm)X,,, m
0 0
37
0
Fig. 1. The model linear crystal (between the solid lines) and the supplementary CT interactions.
(5)
(where X,,, denotes B,, PO,,,, T,, and L,,, and N stands for the number of unit cells) to yield H= C k
H(k)
.
(6)
38
P. Petelenz /Absorption spectra ofpolyacene crysials
Due to the normal selection rule for optical transitions (Ak= 0), we are interested only in the k= 0 term, which reads H(O)=HO,+HO,+H~+HO,+HO,+H~,
(7a)
+H:+H$, with HO, = (E+2@) + 2&7x
”
HO,=EPx
C B&B,, Y.0
(B,:,Bv.2+h.c.) P&P,,,
1 L,+,,L,, Y.O..n
H;=(&++,,)
(7b)
HO,=ETC
“3 H:=EL
,
Y.0
T&T,,
,
(7c)
C B,+,L,,+h.c., Y.LT.ll
(7d)
+ 6;, 1 B &( P,,, + T,,) + h.c. , V.0 H: = (Je +&)
CP,+,T,,+h.c. Y.0
H$, = C [~U’,+,L,n, Y.0 + &(P,‘,L”,.,.
,
(7e) ( 7f)
+ T,+,Lo.-n, 1
+T:,L,,,,_,,)]
+h.c.
(7g)
The Hamiltonian matrix defined by the above set of equations may be diagonalized analytically [ 121 or by standard numerical methods. In the present paper we will adopt the latter approach. (Note some errors in the analytical formulas of ref. [ 12 1. In eq. ( 2 1) of that paper, J; and J’ should be replaced by J; and J’f , respectively. )
4. Naphthalene The Frenkel excitons deriving from the lowest ‘Lb singlet state of the naphthalene molecule are our first target. To facilitate the comparison with ref. [ 9 1, the treatment is limited to a purely electronic (adiabatic) model, without the modifications of eqs. (4a)-(4f). Most of the matrix elements of the Hamiltonian matrix of eqs. (7a)-( 7g) can be found in the litera-
ture. The energy E of the first electronic excited state of the naphthalene molecule is 3 1000 cm- ’ [ 8 1. The nearest neighbour excitation resonance integrals were calculated in ref. [ 91 and they are: W= -2.4 cm-’ and M= + 10.0 cm-’ (at atmospheric pressure). The sign of A4 depends on the choice of the relative phase of the molecular orbitals of translationally inequivalent molecules. It should be chosen in accordance with the phase convention used in the evaluation of charge transfer integrals and Frenkel exciton dissociation integrals (vide infra). If all the integrals were calculated within the same scheme from the same set of molecular orbitals, consistency would be automatically guaranteed. This is not the case in the present treatment, where A4 is taken from a different paper [ 91 than the CT integrals [ 201. In the lack of an independent argument, this compels us to consider the sign of M as adjustable. The results presented in this paper were obtained with the “plus” sign. In the lack of a properly calculated lattice sum for naphthalene we account for the interaction with the two nearest molecules which are not included in the model crystal (cf. section 3) by doubling the value of M. The energy EP of the lowest ( f, 4, 0) CT state of the naphthalene crystal is about 36 100 cm- ’ [ 2 11, and the energy EL of the (0, 1,0) state is about 650 cm - ’ higher [ 2 11. The electron and hole transfer integrals J are known from several independent studies [ 16-18,201. We will use the values from ref. [ 201, where the pressure dependence of these integrals is also given. For the atmospheric pressure, these values are: J,=74 cm-‘, J,=-350 cm-‘, J:= -150cm-‘,&=120cm-‘. The Frenkel exciton dissociation integrals D normally exceed the corresponding CT integrals Jby 1520% [ 12,221. Therefore, we take the D integrals equal to 1.15 times the corresponding J integrals; this yields D,=SS cm-‘, D,=400 cm-‘, D:= -170 cm-’ and Db = 140 cm - ‘. These estimates are certainly very approximative: Firstly, the observation of the relationship between the J and the D integrals was based on rather crude and incomplete numerical results [ 221. Secondly, this observation pertained originally to the Frenkel state deriving from the same molecular orbitals as the CT states (presumably, HOMO and LUMO). This is not the case for the ‘Lb state. Nevertheless, we believe that these estimates are not grossly in error. In fact, the charge transfer integrals
P. Petelenz /Absorption spectra of polyacene crystals
as well as the exciton dissociation integrals are limited by intermolecular overlap, both being roughly linear in intermolecular overlap integrals. For this reason, one intuitively expects them to be of the same order of magnitude and the results of ref. [ 221 only confirm this expectation. On similar physical grounds, one tends to expect the dissociation integrals for different low-energy Frenkel states not to differ too much. In order to describe the effect of pressure, some explicit functional representation of the dependence of the Hamiltonian matrix elements on lattice parameters has to be adopted. We neglect the effect of pressure on the angles and assume that the intermolecular transfer integrals depend only on the distances between the centres of the molecules. For Frenkel exciton transfer integrals (M and W) we assume this dependence to be of dipole-dipole type, i.e. R -‘. As the charge transfer integrals are approximately linear in intermolecular overlap integrals, we assume their dependence on the distance to be exponential, i.e. exp( --ML Atomic orbitals are normally represented as superpositions of Slater type functions. At different distances and in different directions different terms of such a superposition will give the leading contribution to the wavefunction. Consequently, the actual value of the exponent p determining the distance dependence of the CT integrals will depend on the distance and on the relative orientation of the molecules involved. Obviously, p will also depend on the kind of molecular orbital involved, i.e. it will generally be different for electron and hole transfer integrals. In ref. [ 201, the transfer integrals were calculated for several values of the pressure. At low pressures, the contraction of the lattice is very small so that the exponential function in the distance dependence of the transfer integrals may be approximated by the linear function, which allows one to estimate the exponents p from the values of the transfer integrals at two different pressures. With the pressure dependence of the lattice parameters as listed in ref. [ 201, this produces p equal to (in A-‘) 2.2330, 1.7683, 0.9429 and 3.0882 for J,, J,,, J: and J;, , respectively. Based on the argument presented above, we adopt the same values of the exponents for the distance dependence of the exciton dissociation integrals D. The results of our calculations are presented in figs.
39
2 and 3, and compared with those of ref. [ 91 represented by dotted lines. (The latter were very close to experimental data.) As the long-range interactions are absent from our model and so are all dispersive interactions, the total values of the Davydov splitting A and of the gas-to-crystal shift S (solid lines) are underestimated (the latter seriously). In fact, out of all the interactions which are of the zeroth order in intermolecular transfer integrals, only the nearest
&cm-’
2801
Fig. 2. Davydov splitting Aof the S, state of the naphthalene crystal as a function of crystal density (expressed in the units of the density at atmospheric pressure). Solid line - total value obtained in this paper (with M and W as described in the text), broken line - CT contribution (M= W=O), dotted line - total value from ref. [ 91.
tS,cm-’ -700.
.._-”
:
.-.
-500.
-300J
-100
I
_____--
______----________.-----
______--
I
I. 00
,
I. 04
1.08
I. 12
P/g.
Fig. 3. Gas-to-crystal shift S of the S, state of the naphthalene crystal as a function of crystal density. Broken line - CT contribution obtained in this paper (M= W=O), dotted line - total value from ref. [ 91.
40
P. Petelenz /Absorption spectra ofpolyacene crystals
neighbour terms M and W corresponding to the ( 4, f , 0) and (0, 1, 0) directions are included. Consequently, one could not really expect the total values to be correctly reproduced. The CT contribution to the Davydov splitting obtained from our model (dashed line in fig. 2a, obtained with M= W=O) agrees very well with the conclusion of ref. [9] that the relative contribution of the overlap-dependent terms to the splitting is about 30% at atmospheric pressure and increases to about 500/b at 1.5 GPa. This demonstrates that the charge transfer terms, linear in intermolecular overlap, are the leading overlap-dependent contributions. This is not surprising, since the higher-order terms are expected to be numerically close to the triplet exciton transfer integrals, and those are known to amount only to a few wavenumbers [ 18 1. It is readily seen that the form of the density dependence of the Davydov splitting is reasonably well reproduced even by the CT contribution alone. The nearest neighbour exciton resonance interactions M and W change the total value of the splitting (bringing it closer to experiment) but hardly affect the form of the density dependence. It seems to indicate that the pressure dependence of A is in naphthalene due primarily to the involvement of the CT terms. The slightly steeper dependence obtained from the more complete calculations of ref. [ 91 is probably due to dispersive interactions, while the discrepancy in the total value of A follows primarily from the neglect of the (considerable) exciton resonance interaction in the ( A, 4, 1) direction. Fig. 3 shows that the CT terms contribute only marginally to the gas-to-crystal shift in naphthalene, but prominently to the density/pressure dependence of the latter. (The total value of S obtained from our model with M# 0, W# 0 differs negligibly from the CT contribution to S, and hence is not displayed.) This agrees very well with the expectation of Gisby and Walmsley [ 9 1, and indicates that the inclusion of overlap dependent terms may be important in a future attempt to ameliorate their approach.
5. Anthracene The case of anthracene is in a way more obvious, since the Frenkel state at hand derives from the same
molecular orbitals as the CT states under consideration, so that the argument of ref. [ 221 is directly applicable for the estimation of the exciton dissociation integrals. Accordingly, from the transfer integrals [ 181 (all quantities in cm-‘): J,= 197, Jh= -357, J:=424, Jb=-337 we get: D,=226.5, D,,=-411, 0: =488, D;, = - 388. The pressure dependence of the transfer integrals is not directly available from the literature. A transfer integral is the difference of the intermolecular resonance integral and the intermolecular exchange integral. While the former were calculated for anthracene at different pressures [ 251, we are not aware of any calculations of the pressure dependence of the latter for this crystal. One would intuitively expect that, account taken of the structural similarity, this dependence should not be very different from that in naphthalene. This expectation may be roughly tested on the resonance integrals, which are known for both crystals [ 20,23 1. (The pressure dependence of the lattice parameters of the anthracene is listed in ref. [ 231. ) In fact, if the distance dependence of these integrals is modelled by the exponential function, the exponents ,u governing the changes of the resonance contribution to J,, J,, and J: do not differ drastically between naphthalene and anthracene. This makes us adopt for J,, J,, and J: (and consequently for De, D,, and Dk) the same ,u values as for the corresponding integrals in naphthalene. There is an unexpected anomaly for Jh , where the corresponding exponent for the resonance integral is 4.0075 8, - ’ for naphthalene but only 0.99 14 A- ’ for anthracene. Nevertheless, we may still hope that the distance dependence of the exchange contribution to the transfer integral is more or less the same for both crystals, since the intermolecular exchange integral (where the leading part of the distance dependence is due to orbital overlap) is expected to be less sensitive to geometry than the intermolecular resonance integral where the Goeppert-Mayer potential (depending exponentially on the distance) is also involved. Consequently, we approximate p for J;, as an average of the value determined for the resonance integral in anthracene (0.99 14 8, - ’ ) and of the value estimated for the distance dependence of the exchange integral in naphthalene (2.842 1 A-’ [ 20]), the two values being weighted by the contributions of the corresponding integrals to Ji (92.5 and 244.5
P. Petelenz /Absorption spectra of polyacene crystals
cm-‘, respectively). This yieldsp=2.3341 A-‘. The interpretation [ 15 ] of the electroabsorption spectrum of the anthracene crystal [ 241 led us to the value EG= 34255 cm - ’ for the optical band gap, and this value is being used in the present paper. With the electrostatic stabilization energies of the CT states known from ref. [ 251 it yields EP=27340 cm-’ and EL=28331 cm-‘. In order to calculate the vibrational overlap integrals we assume that the frequency of the progression forming mode is the same in all electronic states of interest. Then, the overlap integrals are completely defined by the displacement parameters lo, 1+ and 1_ representing the change of the equilibrium position upon electronic excitation and positive or negative ionization, respectively. &,= 1.50 and I + = 1.OO are readily found from the absorption spectrum [ 68 ] and photoemission spectrum [ 26,27 ] of the anthracene molecule. Based on the pairing theorem, we approximate ;i _ as I _ =I + = 1.OO. Supplemented by the value of the free molecule excitation energy to S, (26000 cm-’ [ 6-S]), this set of data allows us to estimate the pressure dependence of the CT contribution to the Davydov splitting. It is represented by the solid line in fig. 4a. It is readily seen that the CT contribution is substantial. As the numerical value EG quoted above is based on an interpretation of the electroabsorption spectrum [ 241 which may at the present moment be considered only hypothetical (due to the invoked involvement of an intramolecular phonon of energy 0.174 eV [ 15]), we repeated the calculations with a more conservative choice of EG= 35657 cm-‘. This led us to the zero-pressure values of the CT contribution to A and S reduced by a factor of 1.5-2, but with the pressure dependence still stronger than for the original choice of EG. As even with the original choice of EG the pressure dependence of the total values of A and S comes out slightly too strong compared to experiment, we believe that the pressure dependence is consistent with the gap of 34255 cm-‘, as proposed in ref. [ 15 ] , rather than with the “conservative” value of 35657 cm-‘. Of course, in view ofthe limited overall accuracy of the present model approach, this argument cannot be considered as a conclusive proof in favour of the interpretation of ref. 1151. However,
even with the “conservative”
choice of
A,cm-’ 800
a)
I
400.
I
l
1.08
I. 00
I. 16
3190
A,cm-' T
b)
400.
^. ,
1.00
1.08
1.16
3+
Fig. 4. Davydav splitting of the S, state of the anthracene crystal as a function of crystal density. Broken lines - experimental, dotted lines - results of ref. [ lo], solid lines - calculated in this paper: (a) CT contribution, (b) total value. The subsequent lines ofeach kind represent (from top to bottom) the O-O, O-l and O2 vibronic transitions.
EG, the resulting CT contribution to A and S is still considerable, which highlights the necessity of its inclusion in the description of the pressure dependence, suggested by Gisby and Walmsley [ lo]. Fig. 4a shows that, like in naphthalene, the slopes of the curves describing the density dependence of the CT contribution to the Davydov splitting are close to the slopes of the corresponding experimental A(p) dependences, in particular for the O-O transition, On the other hand, the curves calculated by Gisby and Walmsley [ lo] with the complete neglect of the overlap-dependent contributions (dotted lines) are much too flat. This seems to indicate that no matter how large the CT contribution really is, its inclusion is of crucial importance for the description of the pressure dependence of the Davydov splitting.
42
P. Petelenz /Absorption spectra of polyacene crystals
In order to test the relevance of the interaction between different CT states, mediated by the charge transfer integrals J, we have calculated d and S for J, = J,, = J: = J; = 0. It turned out that the energies of the Frenkel states are only very weakly affected by the changes of the J integrals (the shifts being of the order of a few wavenumbers). This was only to be expected, since the effect of these integrals on the Frenkel manifold appears in a higher order of a rapidly converging perturbation series. The estimates of the total values of d and S based on the present approach are again bound to be very poor due to inherent limitations of the nearest neighbour model. They need additional parameters M and W, describing the interactions within the Frenkel manifold. In our model, these parameters represent the nearest neighbour exciton transfer integrals. As has already been mentioned, in order to reasonably approximate the physics of the actual crystal, they have to be considered as effective values, including implicitly the interactions that are not explicitly included. Accordingly, we estimate them from the direction-independent part t,X.ar( 0) [ 61 of the corresponding lattice sums evaluated at the centre of the Brillouin zone. For the S, transition dipole moment of0.61 eA,tableVofref. [6] yields W=-386cm-’ and M= + 123 cm-‘. As mentioned in section 4, the sign of M depends again on the choice of the phase and, in the lack of an independent argument, has to be considered as adjustable. The results are presented for the “minus” sign, which yields a much better agreement with experiment (for the “plus” sign the Davydov splitting is seriously very overestimated). The results are represented in fig. 4b as solid lines. As is readily seen, the overall values of d are seriously overestimated (except for the O-l line, where the agreement seems to be accidental). The discrepancies are not surprising in view of the simplifications of the model mentioned above. It should be emphasized that the model is essentially non-empirical: there are no parameters (apart from the sign of M) adjusted specifically to fit S, A or their pressure dependence. On this view, the agreement correct to a factor not exceeding 2 seems rather reasonable. The present model tends to overestimate the pressure-induced changes while the scaling treatment of ref. [ lo] underestimated them, with the difference
that no scaling is invoked here as all the input data are known from independent calculations or experiments. This seems to suggest that an approach combining both models might be successful. With the overlap-dependent terms estimated in a non-empirical way, the long-range and dispersive contribution could be calculated exactly in the same way as done by Gisby and Walmsley [ 10 1, but with re-optimized values of the adjustable scaling parameters. The negligible importance of the coupling between different CT states (vide supra) would allow one to disregard this coupling completely and include only the direct interaction of the Frenkel state with the nearest neighbour CT states, thereby confining the treatment to a semi-local model similar to that used in ref. [ 9 1. The density dependence of the gas-to-crystal shift S obtained from our calculations is displayed in fig. 5. Following refs. [9,10], we present the value of S with respect to its value at atmospheric pressure, which is taken as zero. The figure demonstrates that within our model the increase of S with increasing density is dominated by the CT contribution. Due to exciton resonance interactions (M and W terms) the total value of the shift exceeds the CT contribution to the shift by about 200 cm-’ ( - 780 versus - 550 cm- ’ for the O-O transition). In contrast to the conclusions of ref. [ lo] the present results seem to suggest that the overlap-dependent terms may contribute significantly not only to
Fig. 5. Gas-to-crystal shift (with respect to its value at atmospheric pressure) of the S, state of the anthracene crystal as a function of crystal density. Broken lines-experimental, solid lines - total values calculated in this paper, dotted lines - CT contributions calculated in this paper.
P. Petelenr /Absorption spectra of polyacene crystals
A, but also to S. This tentative conclusion agrees well with the recent experimental work suggesting the interaction between Frenkel excitons and CT excitons as an important contribution to the energy of local Frenkel states [ 281. It has to be confirmed by a more sophisticated calculation, correctly including the longrange and the dispersive interactions.
6. Discussion In the present paper the naphthalene and anthracene crystals are represented by a model linear crystal of refs. [ 1l-l 51, supplemented by some local contributions which are relevant only for the Frenkel manifold. The model is designed specifically to deal with the CT contribution to the energy of the Frenkel states, linear in intermolecular overlap integrals. It is essentially non-empirical, but is oversimplified as it ignores most of the long-range interactions and all dispersive interactions. Therefore, its conclusions must be analyzed with some care. The validity of the model was tested in this paper for the case of naphthalene. It roughly reproduced the estimate [ 91 of the overlap-dependent contribution to the Davydov splitting of that crystal and of its pressure dependence, thereby confirming the conclusion [ 91 about the importance of overlap-dependent terms. The novel conclusion of this paper is that the pressure dependence not only of the Davydov splitting but also of the gas-to-crystal shift of the S, state of naphthalene is probably dominated by CT terms, with a smaller contribution from dispersive terms. The same model was subsequently applied for the S, state of anthracene. It led to the unexpected conclusion that the CT contribution both to the Davydov splitting and to the gas-to-crystal shift may be very substantial there. In order to relate this conclusion to previous results obtained within overlap-independent models, more complete calculations are necessary, possibly of the scaling type [ 9, lo]. A large contribution of the CT interaction to the Davydov splitting of a singlet Frenkel state has already been suggested for tetracene [ 141. Its relatively large value in anthracene (of the order of 400 cm-’ ) compares reasonably with the naphthalene value of about 50 cm- ’ [ 91, in view of the difference
43
between the corresponding CT integrals by a factor of 1.5-3 [ 18 ] and of the reduced energy gap between the Frenkel and the CT states. This conclusion has a somewhat unexpected aspect. As the CT contribution to the Davydov splitting in anthracene exceeds the experimental value of the splitting, it seems that the sign of other leading contributions must be opposite to that of the CT contribution. In that case, the actual value of the splitting would probably be determined mostly by the CT terms, reduced by a smaller but also considerable contribution from other interactions. Such an interpretation seems to be in conflict with the common belief that the Davydov splitting in anthracene is due mostly to overlap-independent terms, primarily of dipole-dipole origin. In order to eliminate the possibility that this is an artifact of the model used in this paper (which is unlikely, but possible), also the part of the present results concerning the Davydov splitting has to be confirmed by more complete calculations.
References [I ] V.M. Agranovich, Theory of Excitons (Nauka, Moscow, 1968) pp. 91-100, in Russian. [ 21 AS. Davydov, Theory of Molecular Excitons (McGraw-Hill, New York, 1962) p. 92. [ 31 D.P. Craig and S.H. Walmsley, Mol. Phys. 4 ( 1961) 113. [4] D.P. Craig and T. Thirunamachandran, Proc. Phys. Sot. (London) 84 (1964) 781. [5] G.D. Mahan, J. Chem. Phys. 41 (1964) 2930. [6] L.B. Clark and M.R. Philpott, J. Chem. Phys. 53 (1970) 3790. [7] M.R. Philpott, J. Chem. Phys. 50 (1969) 5117; 54 (1971) 111; 59 (1973) 4406; Chem. Phys. Letters 17 (1972) 54. [ 81 D.W. Schlosser and M.R. Philpott, Chem. Phys. 49 (1980) 181. [9] J.A. Gisby and S.H. Walmsley, Chem. Phys. Letters 135 (1987) 275. [IO] J.A. Gisby and S.H. Walmsley, Chem. Phys. 122 (1988) 271. [ 111 P. Petelenz, Proceedings of the XIth Molecular Crystal Symposium, Lugano 1985, p. 230. [ 121 R.W. Munn, P. Petelenz and W. Siebrand, Chem. Phys. 111 (1987) 207. [ 131 P. Petelenz and V.H. Smith Jr., Acta Phys. Polonica A 74 (1988) 435. [ 141 B. Petelenz, P. Petelenz, H.F. Shurvell and V.H. Smith Jr., Chem. Phys. 119 (1988) 25.
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P. Petelenz /Absorption spectra ofpolyacene crystals
[ 151 P. Petelenz and V.H. Smith Jr., Chem. Phys. 131 ( 1989) 409. [ 161 J.I. Katz, S.A. Rice, S.I. Choi and J. Jortner, J. Chem. Phys. 39 (1963) 1683. [ 17 ] R. Silbey, J. Jortner, S.A. Rice and M.T. Vala, J. Chem. Phys. 42 (1965) 733. [ I8 ] A. Tiberghien and G. Delacote, J. Phys. (Paris) 3 1 ( 1970) 637. [ 191 P. Petelenz, Chem Phys. 133 (1989) 199. [20] M. El Hamamsy, A.C. Damask and S.A. Elnahwy, Mol. Cryst. Liquid Cryst. 95 (1983) 209. [ 211 E.A. Silinsh, Organic Molecular Crystals - Their Electronic States (Springer, Berlin, 1980) p. 108. [ 221 P. Petelenz and M.Z. Zgierski, Phys. Stat. Sol. (b) 69 ( 1975) 133.
[ 231 S.A. Elnahwy, M. El Hamamsy and A.C. Damask, Phys. Rev. B 19 (1979) 1108. [24] L. Sebastian, G. Weiser and H. Blssler, Chem. Phys. 75 (1983) 103. [25] P.J. Bounds, W. Siebrand, I. Eisenstein, R.W. Munn and P. Petelenz, Chem. Phys. 95 ( 1985) 197. [ 261 P.A. Clark, F. Brogli and E. Heilbronner, Helv. Chim. Acta 55 (1972) 1415. [27] R. Boschi, J.N. Murrell and W. Schmidt, Discussions Faraday Sot. 54 (1972) 116. [28] I.V. Brovchenko, N.I. Ostapenko, V.I. Sugakov and M.T. Shpak, Izv. Akad. Nauk SSSR Ser. Fiz. 52 (1988) 659.