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A dynamical model of excimer instability in molecular crystals
141
Chemical Physics 116 (1987) 141-149 North-Holland, Amsterdam
A DYNAMICAL MODEL OR EXCIMER INSTABILITY
IN MOLECULAR
CRYSTALS
Raffaele Guido DELLA VALLE and Aldo BRILLANTE Dipartimento di Chieica Fisica ed Inorganica, Universid Viale de1 Riso&imento 4, 40136 Bologna, Italy
di Bologna,
Received 23 January 1987; in final form 18 May 1987
Catastrophe theory is applied to the analysis of compb bandshapes observed in the excimer emission of molecular crystals. Excited state potential surfaces in one- and two-well regimes are computed by means of a simple anharmonic quartic potential, modulated by the macroscopic parameters. Model calculations are reported for the pressure dependence of the excimer emission of 9-cyano- and of P-9,10dichloroanthracene, and a satisfactory fit is found.
1. Introduction
Excimer emission is observed when a pair of molecules tightly bound in an electronically excited state falls to a weakly bound or unbound ground state [l]. In an absorption process the system starts near its equilibrium ground state position, jumps vertically (in Condon’s sense) to the excited state potential surface and then relaxes towards a minimum. If the minimum is not too deep (not a stable dimer) the system may fall again to the ground surface through fluorescence emission. Excimer spectra are thus determined by the structure of the ground and excited state Born-Oppenheimer potential surfaces and by their deformations with changing pressure and temperature. These deformations play a most important role in determining the position, shape and width of the excimer emission. In particular, an increased Stokes shift is an indication of a larger stability of the molecular pair in the lattice, due to a more complete overlap with higher binding energy. The existence of complicated patterns of the potential curves of the excited states can be argued on the basis of the excimer profiles experimentally determined by varying temperature [1,2] and pressure [3,4]. The excimeric emission can give rise to more than one peak, or the peaks may become very
asymmetric, reflecting the instability of the molecular interactions when the intermolecular distances are drastically altered under the effect of external parameters. Under these particular conditions one has to explain the experimental spectra by searching for potentials that may exhibit multiple minima. Before trying to understand from the basic physical principles the shape and behaviour of these potential surfaces, we will start using Thorn’s catastrophe theory [5] to classify those features that might have relevant effects on excimer bandshape. Although this classification cannot substitute for a fuller causal analysis, explaining the physical origin of the potential, it does produce useful results by effectively reducing the arbitrariness of a potential model. The theory closely resembles Landau’s description of instabilities in phase transitions, and the final results may be easily obtained adapting Landau’s treatment. Catastrophe theory supplements that treatment providing systematic methods to remove inessential variables, to identify the simplest potential function compatible with the physics of the problem, and to guarantee the stability of the results against small perturbation.’ Catastrophe theory is described in several standard texts [6,7]. A simple introduction is given in section 2 in order to make the paper self-contained. The general properties of ground and ex-
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R.G. Della Valle, A. Brillante / Excimer instability
cited state potential surfaces are discussed in section 3, showing that the simplest non-trivial (not always a single-minimum) surface is represented by a perturbed quartic well. Complex excimer bandshapes result from the quantum-mechanical analysis of such wells (section 4), and the dependence of the shape on the macroscopic parameters is found to be consistent with the experimentally observed pressure dependence of the excimer emission in some anthracene derivatives (section 5).
2. Catastrophe theory In the investigation on how the shape of the potential surfaces is affected by the macroscopic parameters, we will mainly be interested in their “critical structure”, i.e. the number and disposition of maxima and minima. In fact it will be shown that extrema of the potential are related to extrema in the lowest vibrational wavefunctions and in the excimer spectra, irrespectively of the finer details of the potential. Two potentials are “structurally equivalent” if they have the same critical structure, i.e. roughly the same shape. All single-minimum potentials are thus structurally equivalent. A potential function is “structurally stable” if its critical structure does not change by addition of, a small perturbation. It must be stressed that it is the shape of the potential that must be stable, not the system itself. Potential functions are expected to be almost always structurally stable, but may lose stability for exceptional values of the macroscopic parameters. Physical considerations restrict the kind of structural instability, or “catastrophe”, that may develop. By changing the macroscopic parameters along some trajectory we may cross one of these exceptional points where the potential changes its shape. This is somehow analogous to a phase transition while a thermodynamical critical surface is crossed. More formally we may say that the potential is perturbed by moving away from a point where it is structurally unstable. Its critical structure and stability properties may thus be changed.
Once just sufficient perturbations have been added to make the potential structurally stable, further perturbations have no effect. Thus it suffices to study the behaviour of this minimal potential, the “universal unfolding” of the catastrophe, to understand the effect of any possible weak perturbation on the critical structure.
3. Potential surfaces
The nature of the electronic ground state in a molecular crystal is fairly well understood [8,9]. The multidimensional potential surface has a single minimum, corresponding to the equilibrium structure of the crystal. The ground state potential energy VPmay be expanded in power series of the displacements from the minimum xi (i = 1,2,...,N):
A unitary transformation of coordinates {xi } + { qi} may be found that reduces the quadratic part of the hamiltonian H = T + Vg to diagonal form
(2) (mass weighted coordinates qi are used). The ground state is approximately described in terms of N harmonic oscillators (phonons) of frequency wi. To yield a single minimum, and thus real positive frequencies wi, the hessian matrix (a*V/ i3x,axj) has to be definite positive of rank N (non-zero determinant). The quadratic potential implied by (2) is structurally stable [7]. If the frequencies stay positive and the higher-order terms remain small enough, no new extrema appear. The structure of the excited state has been investigated by several authors in terms of one[lo] and two-dimensional [ll] potential wells, semi-localized travelling excitons [12] and doubleminima wells [13], but a general picture is still
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missing. The excited state potential surface might be extremely complex, with many minima, some of which could be inaccessible to the system, due to high barriers or dynamical effects, like momentum conservation in absorption or decay processes to the ground state faster than the relaxation to the minimum in fluorescence. Changes of macroscopic parameters may strongly affect the excited state surface, shifting minima [2], creating new trapped states [14] or lowering potential barriers PI. It has in fact been experimentally shown that more than one excimeric form may originate from a given compound [3,15,16]. Temperature, pressure, defect and impurity concentration seem to govern the occurrence of different emissions. The energy transfer processes following monomer exciton absorption consequently change according to those parameters. The travelling excitation may then be trapped in one or another of the multiple wells of the excimer potential, giving rise to shifted emissions. We will investigate the effect of the simplest potential shape on the excimer spectra, without giving for the moment a full account of the physical origin of such behaviour. Model computations at this purpose are in progress. If the excited state surface has a single minimum, i.e. all real positive frequencies We,it may be described in terms of independent oscillators, possibly shifted in origin, changed in frequency and Duschinsky-rotated in orientation with respect to those of the ground state [17]. The spectra may be computed through multidimensional harmonic Franck-Condon integrals by standard methods [l&19]. Only few coordinates experience large changes during excimer formation or decay (i.e. those of the molecules involved and of the immediately surrounding environment). Only the corresponding modes are thus vibrationally excited during the transition, while the other ones may be considered as part of the broadening heat bath and otherwise ignored [l]. By changing the macroscopic parameters we may reach a point where one of the frequencies (say wr) vanishes and the hessian has rank N - 1. We may still find (“splitting lemma” [6]) a unitary transformation of the coordinates reducing the
excited state potential to a form where the leading term for all coordinates except q1 is harmonic: izl
+ higher-order terms in q1 + . * . .
(3)
The coordinates with non-zero frequencies change very little with respect to the change of q1 (there is no recall force for small variations of ql) and may thus be ignored. Excluding the unlikely case where two or more harmonic frequencies simultaneously vanish, only a single mono-dimensional potential V,(q), q = q,, needs to be considered. In the vicinity of the thermodynamic point where the frequency vanishes the harmonic approximation breaks down and more terms have to be considered in the Taylor expansion about the equilibrium position:
K(q) = K(O)+ %q+
I a2v --2q2
2! a42
+ 1 a’v, q+4?aq 3 1 ea4v 4+ .... 3! a43 4q
(4)
According to catastrophe theory [7], and in strict analogy to Landau’s theory of second-order phase transitions [20], the nature of the critical point is determined by the sign of the first non-zero coefficient in the Taylor expansion (4). When w is zero all terms before the quartic must vanish. Indeed (aV,/aq), = 0 is the equilibrium condition, (a 2 4, cannot change the local critical structure of the potential [7] and there are no lower-order terms left to add. The expansion (4), truncated to the fourth order, is thus the more general potential ever required. Through a suitable shift and scaling of energy
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R.G. Della Valle, A. Brillante / Excimer instability
and coordinate the potential (4) may be simplified, removing zeroth- and third-order terms and transforming the potential into
v,(q) = q4 +
uq*+uq.
:
(5)
The two “control parameters” u and u are combinations of potential derivatives, determined by the transformation from eq. (4) to eq. (5), and thus must be (hopefully smooth) functions of the macroscopic parameters. Although in general the meaning of u and u may be rather speculative, one will nevertheless see that their effect on the potential is remarkable different. The influence of ZJ has a more abrupt character, therefore u might be identified with a “discontinuous” parameter affecting the bandshape but rather ineffective on the position of the minima. On the contrary, u has a continuous character, like that, for instance, of temperature or pressure. The last term of eq. (5) has the correct form for the work done by the external pressure against an effective area [21], hinting a possible identification of the parameter u with the pressure. This suggestive analogy is by no means a proof, and a derivation from the basic principles would be required. For our purposes we will accept the identification on a phenomenological ground. Indeed, as shown later, a very good fit of the experimental data is obtained. The excited state potential ve(q), eq. (5), is structurally stable and has the minimum number of terms. It is the universal unfolding of the catastrophe q4 [5]. In Thorn’s nomenclature q4 is called the “cusp” catastrophe. Eq. (5) may be conceived as an expansion around the critical point in terms of some parameters u and u, in the frame of Landau’s theory. According to the catastrophe theory that expansion will remain valid in the neighborhood of the critical point, as no small perturbation, involving additional powers of q, or extra variables q2, q3,. . . , can change the local critical structure of the potential. It must be stressed that the shift and scaling required to reduce the potential to the canonical form (5) depend on the macroscopic parameters. Only terms that may cause discontinuities are kept in the transformation, discarding any term not
0
-4 /’ \
/’
A<0 -4
I
0
4
Y
I
Fig. 1. Potential wells and excimer bandshapes for various values of the control parameters. Potential wells and related bandshapes are drawn at or near the corresponding (u, u) points in the control space. In the terms of the scaled units described in the text, the selected points are (0,4), (+5,2), (0, 0), ( f 5, -4), (kO.2, -4), (0, - 5). The normalized bandshapes, computed for a ground state potential 0.53 * (q - 5)* at temperature kT/h = 1, have been gaussian-smoothed in order to reduce sampling artifacts and oscillations of the Franck-Condon coefficients. The broken curve divides the (u, u) control space into single (A > 0) and double (A < 0) minima regions.
affecting the critical structure. Such “continuous” terms have to be considered when comparing the model with the experimental results. Several shapes of the potential ve(q) = q4 + uq2 + uq for different values of u and u are shown in fig. 1. The critical structure of the potential is easily determined. The first derivative of V,(q), ve’(q) = 4q3 + 2uq + u, is a cubic with a single zero (a minimum ) or three zeros (two minima with a maximum in between) according to the positive or negative sign of the discriminant A = 8u3 + 2%~~.
(6)
V,(q) has two minima for the points inside the “cusp” curve A = 0 (broken curve in fig. 1) and one outside. It is thus easy to see that the potential q4 is not structurally stable. The addition of an infinitesimal perturbation - cq* brings it from the one-well to the two-well regime.
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R.G. Della ValIe, A. Brillnnte / Excimer instability
4. Excimer bandshape
Potential functions siinilar to (5) have been treated by several authors [22,23]. The eigenvalues En and eigenfunctions w,,(q) of the hamiltonian H = T + V, may be obtained to any desired accuracy expanding the eigenfunctions as linear combinations of a suitable number of harmonic oscillator wavefunctions and diagonalizing the secular matrix. We have found a basis of thirty-three wavefunctions to be adequate to reach complete convergence of the eigenvalues. The first few eigenvalues and eigenfunctions for a selected potential well in the two-minima region are shown in fig. 2. As usual for mono-dimensional wells [24], there are no degenerate levels and the n th wavefunction has n nodes. In the two-well regime the lowest vibrational wavefunction exhibits a minimum, or at least a shoulder, provided that the barrier height is larger than Ea. When the two wells are slightly asymmetric, the lowest vibrational state lies mainly in the lower well, while the next state, slightly higher in energy, lies in the other well. In the single-well regime the lowest state has a single maximum and the next level is much higher. According to first-order perturbation theory, by using Born-Oppenheimer wavefunctions and neglecting the dependence of the electric dipole moment on the nuclear coordinates, the fluorescence spectrum for transitions from electronically excited states to lowest states is proportional to 1251:
where P,,f is the population of the state induced by the probe beam. If the relaxation among the excited levels is faster than the decay to the ground state, the population of the excited states thermalizes and a Boltzmann statistics may be used P,t 0: exp( - Ent/kT).
Fig. 2. A typical double-well potential and the wavefunctions in the ground and excited states with the corresponding extimer bandshape. All parameters are those used for the point (0, -5) in fig. 1. One of the more likely transition processes, corresponding to a minimum of the upper well and to a peak of the bandshape, is symbolically pictured.
ground state well [l] (fig. 2). The ground state sublevels are very dense and cannot be individually resolved. Some typical bandshapes (7) for a purely harmonic ground state potential and various control parameters u and u for the upper state potential are shown in fig. 1. The overlap integral (qnt ( 4,) has been expanded in a sum of the proper harmonic overlap integrals [26]. The gross structure of the bandshape turns out to be rather insensitive to both ground state frequency and shift from the equilibrium position. It appears from the figure that the critical structure of the potential and the bandshape are strictly related. These findings may be rationalized in terms of the semiclassical Franck-Condon principle. After an optical transition, the system, subjected to a new potential, will not be in equilibrium. The important final states for the transitions will be highly excited states with high quantum numbers which, according to the Bohr correspondence principle, behave quasi-classically. The final vibrational state q,(q) will oscillate rapidly except near the classical turning point q where Vg(q) = E,,. Thus a good approximation to (7) is [25]:
(8)
Under conditions appropriate to excimer formation, the excited state potential well is deep and strongly shifted with respect to the shallower
x6(E,f
- v,(q)
- hw).
(9)
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R.G. Della Valle, A. Brillante / Excimer instability
variations of the excimer frequency, relative intensity and bandshape of 9-cyanoanthracene (9CNA) [3] and P-9,10-dichloroanthracene (DClA) [4] with pressure. This simple model breaks down when moving too far away from the thermodynamic point where the frequency wi, related to the transition coordinate q = ql, vanishes. Indeed a highly asymmetric double well (AE B kT) results for u -=z0 and u # 0. The highest of the two wells is essentially unpopulated and the system is effectively reduced to the single-minimum case.
The ground state potential VJq) is monotonically decreasing in the region of interest (fig. 2), thus the contribution from the state $,,t to the emission spectra has the same shape of the probability density I b(q) I 2, sligthly deformed by the curvature on Vs(q), shifted in origin by an amount E,,r and weighted by P,,. At low temperatures only the first few levels are appreciably populated. A simple pattern suggested by these general considerations is in qualitative agreement with the experimental findings. When a macroscopic parameter (e.g. the pressure) changes, the corresponding control parameters u and u will move along some trajectory in the control space. The bandshape has a single peak, slowly drifting in frequency, while u and u move in the singleminimum region. A new peak appears at higher (or lower) frequencies upon entering the twominima region from the left (or right). While crossing through the cusp region the intensity is progressively transferred from the former peak to the new one. The peaks are symmetric when well inside the single-minimum region, becoming slightly asymmetric near or inside the two-minima region. Two trajectories, corresponding to u = 3 and U= - 5 in fig. 1 might properly describe the
5. Pressure dependence of excimer frequencies
A quantitative picture may be obtained by a few approximations on eq. (9). At low temperatures only the lowest excited levels (or a few nearly degenerated levels) are appreciably populated. The sum on the excited state n’ may be restricted to these states and the energy E,,t may be considered as a constant. If the upper and lower potential wells are displaced enough, V,(q) may be linearized in q and the integral on Dirac’s delta may be carried out at once.
21.000
/
/
I
JO
20
P Ikbarl Fig. 3. Fit of excimer emission frequency for 9CNA under pressure.
80
R G. Della Valle, A. Brillante / Excimer instability
Recalling the relationship between the probability density and the potential shape, we may assume that the peaks of the excimer spectra coincide with the extrema of the excited state potential (proper units, incorporating all parameters of the ground state potential, are used). With a final scaling of q and u in K’(q) (by respectively) we may and (u(-~‘~ lu(+2 eliminate the dependence of the extrema of I/,(q) on the magnitude (but not on the sign) of U. We are thus left with two universal dimensionless “frequency” and “macroscopic” parameters q and u, related by 4qQ2q+udl,
00)
where the positive sign holds in the structurally stable regime (U > 0) and the negative sign in the unstable regime (U < 0). In order to relate the model to the experimental data, we have to specify the origin and the scale of the macroscopic and frequency parameters. Furthermore it may be necessary to add any “continuous” term discarded in the transformation from eq. (4) to eq. (5). In the case of 9CNA and DClA, using pressure as the macroscopic parame-
147
ter, we have found an additional linear pressuredependent frequency shift to be sufficient for a satisfactory fit: P=Po+Plu,
(11)
w=wo+w,q+ar(p-PO).
02)
Figs. 3 and 4 show fits to the experimental excimer fluorescence frequencies, according to eq. (10) with the positive sign for 9CNA and the negative sign for DClA. The scalings (11) and (12) are used, with p,, = 8 kbar, p1 = 0.93 kbar, w0 = 19780 cm-‘, oi = 600 cm-‘, a = - 23 cm-‘/kbar for 9CNA, and p. = 52 kbar, p1 = -45 kbar, w. = 21100 cm-‘, w1 = 2000 cm-l, a = -53 cm-‘/kbar for DClA. The phenomenological meaning of po, p1 and oo, w, is an indication of the gap position and size, in the pressure and energy scales respectively.
6. Conclusions The model developed in this paper shows that a simple anharmonic quartic potential, whose shape
21.000
20.000
19.000
(cm”1 1.3.000
17.000
P Ikbarl Fig. 4. Fit of excimer emission frequency of DClA under pressure. The back-bending part of the theoretical curve in the double-peak region is related to the local maximum of the potential well and has no experimental relevance.
148
R.G. Della Valle, A. Brillan!e / Excimer instabiliry
is modulated by the macroscopic parameters, may explain the variations in frequency and bandshape of the excimer emission in a molecular crystal under pressure. It must be stressed that our resulting bandshapes and frequency dependence are “structurally stable”. This implies that any physically reasonable potential will qualitatively produce the same results as the universal unfolding Ve(q) of eq. (5), provided that the two potentials have the same critical structure. The geometry of the excimer is in most cases preformed in the crystal lattice, originating from adjacent molecules arranged in “pair” or “stack” structures [l]. For monosubstituted anthracene derivatives the additional distinction between the “cis” and “ tram” configurations of the molecular pair can be made [3,27]. One should expect an intimate relationship between type of emission and excimer geometry, as a consequence of the different extent of the intermolecular interaction between adjacent molecules. The present paper suggests that any realistic model explaining the physical origin of the potential modulation by the macroscopic parameters must justify the possible occurrence of two, or possible more, competing excimer configurations. We have identified, to test our model, two different cases where differently bound molecular pairs yield competing emissions in different pressure regimes. In the case of 9CNA, increasing pressure (and decreasing temperature) acts as a drive to induce the “ topochemical” cis-excimer emission. The net result is a stabilization of the total energy with an increased Stokes shift of a thousand wavenumbers. The opposite effect is observed for DClA, where a more weakly bound excited state is favoured by pressure. In both cases the different constraints on the molecular pair, very sensitive to application of external pressure, are clearly responsible for the observed behaviour. Contrary to 9CNA, stabilization of the excimer state in DClA is inhibited at high pressure, when the steric hindrance of the 9,10-substitution becomes efficient (closer intermolecular contacts). These considerations might account for the stable or unstable regime of the pressure dependence, i.e. the positive or negative
sign given in eq. (10). It seems very encouraging that both cases could be described by the model here presented, identifying a promising starting point for further developments of a theoretical treatment of the excited state potential of these systems.
Acknowledgement This work was supported by the Italian Minister0 della Pubblica Istruzione and by CNR. We thank Professor C. Zauli for stimulating discussions and useful suggestions.
References [l] V. Yakhot, M.D. Cohen and Z. Ludmer, Advan. Photothem. 1 (1979) 489. [2] M.D. Cohen, Z. Ludmer and V. Yakhot, Phys. Stat. Sol. B 67 (1975) 57. [3] A. Brillante, M. Hanfland and K. Syassen, Chem. Phys. Letters 119 (1985) 42. [4] A. Brillante, K. Strijssner and K. Syassen, Chem. Phys. Letters 134 (1987) 331. [5] R. Thorn, Structural stability and morphogenesis (Benjamin, New York, 1975). [6] P.T. Saunders, An introduction to catastrophe theory (Cambridge Univ. Press, Cambridge, 1980). [7] E.C. Zeeman, Catastrophe theory (Addison-Wesley, Reading, MA, 1977). [8] M. Born and K. Huang, Dynamical theory of crystal lattices (Oxford Univ. Press, Oxford, 1954). [9] S. Califano, V. Schettino and N. Neto, Lattice dynamics of molecular crystals (Springer, Berlin, 1981). [lo] B.N. Srinivasan, J.V. Russell and S.P. Glynn, J. Chem. Phys. 48 (1968) 1931. [ll] M.D. Cohen, R. Haberkom, E. Huler, Z. Ludmer, M.E. Michel-Beyerle, D. Rabinovich, R. Sharon, A. Warshel and V. Yakhot, Chem. Phys. 27 (1978) 211. [12] L.A. Dissado and S.H. Walmsley, Chem. Phys. Letters 87 (1982) 74. [13] H. Auweter, D. Ramer, B. Kunze and H.C. Wolf, Chem. Phys. Letters 85 (1982) 325. [14] B. Walker, H. Port and H.C. Wolf, Chem. Phys. 92 (1985) 177. [15] Z. Ludmer, Phys. Stat. Sol. 43a (1977) 695. [16] L. Zeiri, Chem. Phys. Letters 121 (1985) 504. [17] F. Duschinsky, Acta Physic&rim. URSS 7 (1937) 551. 1181 A.R. Chigiryov, J. Mol. Spectry. 115 (1986) 419. [19] T.R. Faulkner and F.S. Richardson, J. Chem. Phys. 70 (1979) 1201.
R.G. Della Valle, A. Briliante / Excimer instabdity
[20] L.D. Landau and E. Lifchitz, Statistical physics (Pergamon Press, Oxford, 1958). [21] H.G. Drickamer and C.W. Frank, Electronic transitions and the high pressure chemistry and physics of solids (Chapman and Ha& London, 1973). 1221 R.L. Somorjai and D.F. Hornig, J. Chem. Phys. 36 (1961) 1980. [23] S.I. Chan, D. Stelman and L.E. Thompson, J. Chem. Phys. 41 (1964) 2828.
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[24] L.D. Landau and E. Lifchitz, Quantum mechanics (Pergamon Press, Oxford, 1958). [25] M. Lax, J. Chem. Phys. 20 (1952) 1752. [26] E. Hutchisson, Phys. Rev. 36 (1930) 410. [27] E.-Z.M. Ebeid, S.E. Morsi, M.A. El-Bayoumi and J.O. Williams, J. Chem. Sot. Faraday Trans. I 74 (1978) 1457.
Chemical Physics 116 (1987) 141-149 North-Holland, Amsterdam
A DYNAMICAL MODEL OR EXCIMER INSTABILITY
IN MOLECULAR
CRYSTALS
Raffaele Guido DELLA VALLE and Aldo BRILLANTE Dipartimento di Chieica Fisica ed Inorganica, Universid Viale de1 Riso&imento 4, 40136 Bologna, Italy
di Bologna,
Received 23 January 1987; in final form 18 May 1987
Catastrophe theory is applied to the analysis of compb bandshapes observed in the excimer emission of molecular crystals. Excited state potential surfaces in one- and two-well regimes are computed by means of a simple anharmonic quartic potential, modulated by the macroscopic parameters. Model calculations are reported for the pressure dependence of the excimer emission of 9-cyano- and of P-9,10dichloroanthracene, and a satisfactory fit is found.
1. Introduction
Excimer emission is observed when a pair of molecules tightly bound in an electronically excited state falls to a weakly bound or unbound ground state [l]. In an absorption process the system starts near its equilibrium ground state position, jumps vertically (in Condon’s sense) to the excited state potential surface and then relaxes towards a minimum. If the minimum is not too deep (not a stable dimer) the system may fall again to the ground surface through fluorescence emission. Excimer spectra are thus determined by the structure of the ground and excited state Born-Oppenheimer potential surfaces and by their deformations with changing pressure and temperature. These deformations play a most important role in determining the position, shape and width of the excimer emission. In particular, an increased Stokes shift is an indication of a larger stability of the molecular pair in the lattice, due to a more complete overlap with higher binding energy. The existence of complicated patterns of the potential curves of the excited states can be argued on the basis of the excimer profiles experimentally determined by varying temperature [1,2] and pressure [3,4]. The excimeric emission can give rise to more than one peak, or the peaks may become very
asymmetric, reflecting the instability of the molecular interactions when the intermolecular distances are drastically altered under the effect of external parameters. Under these particular conditions one has to explain the experimental spectra by searching for potentials that may exhibit multiple minima. Before trying to understand from the basic physical principles the shape and behaviour of these potential surfaces, we will start using Thorn’s catastrophe theory [5] to classify those features that might have relevant effects on excimer bandshape. Although this classification cannot substitute for a fuller causal analysis, explaining the physical origin of the potential, it does produce useful results by effectively reducing the arbitrariness of a potential model. The theory closely resembles Landau’s description of instabilities in phase transitions, and the final results may be easily obtained adapting Landau’s treatment. Catastrophe theory supplements that treatment providing systematic methods to remove inessential variables, to identify the simplest potential function compatible with the physics of the problem, and to guarantee the stability of the results against small perturbation.’ Catastrophe theory is described in several standard texts [6,7]. A simple introduction is given in section 2 in order to make the paper self-contained. The general properties of ground and ex-
0301-0104/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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cited state potential surfaces are discussed in section 3, showing that the simplest non-trivial (not always a single-minimum) surface is represented by a perturbed quartic well. Complex excimer bandshapes result from the quantum-mechanical analysis of such wells (section 4), and the dependence of the shape on the macroscopic parameters is found to be consistent with the experimentally observed pressure dependence of the excimer emission in some anthracene derivatives (section 5).
2. Catastrophe theory In the investigation on how the shape of the potential surfaces is affected by the macroscopic parameters, we will mainly be interested in their “critical structure”, i.e. the number and disposition of maxima and minima. In fact it will be shown that extrema of the potential are related to extrema in the lowest vibrational wavefunctions and in the excimer spectra, irrespectively of the finer details of the potential. Two potentials are “structurally equivalent” if they have the same critical structure, i.e. roughly the same shape. All single-minimum potentials are thus structurally equivalent. A potential function is “structurally stable” if its critical structure does not change by addition of, a small perturbation. It must be stressed that it is the shape of the potential that must be stable, not the system itself. Potential functions are expected to be almost always structurally stable, but may lose stability for exceptional values of the macroscopic parameters. Physical considerations restrict the kind of structural instability, or “catastrophe”, that may develop. By changing the macroscopic parameters along some trajectory we may cross one of these exceptional points where the potential changes its shape. This is somehow analogous to a phase transition while a thermodynamical critical surface is crossed. More formally we may say that the potential is perturbed by moving away from a point where it is structurally unstable. Its critical structure and stability properties may thus be changed.
Once just sufficient perturbations have been added to make the potential structurally stable, further perturbations have no effect. Thus it suffices to study the behaviour of this minimal potential, the “universal unfolding” of the catastrophe, to understand the effect of any possible weak perturbation on the critical structure.
3. Potential surfaces
The nature of the electronic ground state in a molecular crystal is fairly well understood [8,9]. The multidimensional potential surface has a single minimum, corresponding to the equilibrium structure of the crystal. The ground state potential energy VPmay be expanded in power series of the displacements from the minimum xi (i = 1,2,...,N):
A unitary transformation of coordinates {xi } + { qi} may be found that reduces the quadratic part of the hamiltonian H = T + Vg to diagonal form
(2) (mass weighted coordinates qi are used). The ground state is approximately described in terms of N harmonic oscillators (phonons) of frequency wi. To yield a single minimum, and thus real positive frequencies wi, the hessian matrix (a*V/ i3x,axj) has to be definite positive of rank N (non-zero determinant). The quadratic potential implied by (2) is structurally stable [7]. If the frequencies stay positive and the higher-order terms remain small enough, no new extrema appear. The structure of the excited state has been investigated by several authors in terms of one[lo] and two-dimensional [ll] potential wells, semi-localized travelling excitons [12] and doubleminima wells [13], but a general picture is still
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R. G. Della Valle, A. Brillanie / Excimer instability
missing. The excited state potential surface might be extremely complex, with many minima, some of which could be inaccessible to the system, due to high barriers or dynamical effects, like momentum conservation in absorption or decay processes to the ground state faster than the relaxation to the minimum in fluorescence. Changes of macroscopic parameters may strongly affect the excited state surface, shifting minima [2], creating new trapped states [14] or lowering potential barriers PI. It has in fact been experimentally shown that more than one excimeric form may originate from a given compound [3,15,16]. Temperature, pressure, defect and impurity concentration seem to govern the occurrence of different emissions. The energy transfer processes following monomer exciton absorption consequently change according to those parameters. The travelling excitation may then be trapped in one or another of the multiple wells of the excimer potential, giving rise to shifted emissions. We will investigate the effect of the simplest potential shape on the excimer spectra, without giving for the moment a full account of the physical origin of such behaviour. Model computations at this purpose are in progress. If the excited state surface has a single minimum, i.e. all real positive frequencies We,it may be described in terms of independent oscillators, possibly shifted in origin, changed in frequency and Duschinsky-rotated in orientation with respect to those of the ground state [17]. The spectra may be computed through multidimensional harmonic Franck-Condon integrals by standard methods [l&19]. Only few coordinates experience large changes during excimer formation or decay (i.e. those of the molecules involved and of the immediately surrounding environment). Only the corresponding modes are thus vibrationally excited during the transition, while the other ones may be considered as part of the broadening heat bath and otherwise ignored [l]. By changing the macroscopic parameters we may reach a point where one of the frequencies (say wr) vanishes and the hessian has rank N - 1. We may still find (“splitting lemma” [6]) a unitary transformation of the coordinates reducing the
excited state potential to a form where the leading term for all coordinates except q1 is harmonic: izl
+ higher-order terms in q1 + . * . .
(3)
The coordinates with non-zero frequencies change very little with respect to the change of q1 (there is no recall force for small variations of ql) and may thus be ignored. Excluding the unlikely case where two or more harmonic frequencies simultaneously vanish, only a single mono-dimensional potential V,(q), q = q,, needs to be considered. In the vicinity of the thermodynamic point where the frequency vanishes the harmonic approximation breaks down and more terms have to be considered in the Taylor expansion about the equilibrium position:
K(q) = K(O)+ %q+
I a2v --2q2
2! a42
+ 1 a’v, q+4?aq 3 1 ea4v 4+ .... 3! a43 4q
(4)
According to catastrophe theory [7], and in strict analogy to Landau’s theory of second-order phase transitions [20], the nature of the critical point is determined by the sign of the first non-zero coefficient in the Taylor expansion (4). When w is zero all terms before the quartic must vanish. Indeed (aV,/aq), = 0 is the equilibrium condition, (a 2 4, cannot change the local critical structure of the potential [7] and there are no lower-order terms left to add. The expansion (4), truncated to the fourth order, is thus the more general potential ever required. Through a suitable shift and scaling of energy
144
R.G. Della Valle, A. Brillante / Excimer instability
and coordinate the potential (4) may be simplified, removing zeroth- and third-order terms and transforming the potential into
v,(q) = q4 +
uq*+uq.
:
(5)
The two “control parameters” u and u are combinations of potential derivatives, determined by the transformation from eq. (4) to eq. (5), and thus must be (hopefully smooth) functions of the macroscopic parameters. Although in general the meaning of u and u may be rather speculative, one will nevertheless see that their effect on the potential is remarkable different. The influence of ZJ has a more abrupt character, therefore u might be identified with a “discontinuous” parameter affecting the bandshape but rather ineffective on the position of the minima. On the contrary, u has a continuous character, like that, for instance, of temperature or pressure. The last term of eq. (5) has the correct form for the work done by the external pressure against an effective area [21], hinting a possible identification of the parameter u with the pressure. This suggestive analogy is by no means a proof, and a derivation from the basic principles would be required. For our purposes we will accept the identification on a phenomenological ground. Indeed, as shown later, a very good fit of the experimental data is obtained. The excited state potential ve(q), eq. (5), is structurally stable and has the minimum number of terms. It is the universal unfolding of the catastrophe q4 [5]. In Thorn’s nomenclature q4 is called the “cusp” catastrophe. Eq. (5) may be conceived as an expansion around the critical point in terms of some parameters u and u, in the frame of Landau’s theory. According to the catastrophe theory that expansion will remain valid in the neighborhood of the critical point, as no small perturbation, involving additional powers of q, or extra variables q2, q3,. . . , can change the local critical structure of the potential. It must be stressed that the shift and scaling required to reduce the potential to the canonical form (5) depend on the macroscopic parameters. Only terms that may cause discontinuities are kept in the transformation, discarding any term not
0
-4 /’ \
/’
A<0 -4
I
0
4
Y
I
Fig. 1. Potential wells and excimer bandshapes for various values of the control parameters. Potential wells and related bandshapes are drawn at or near the corresponding (u, u) points in the control space. In the terms of the scaled units described in the text, the selected points are (0,4), (+5,2), (0, 0), ( f 5, -4), (kO.2, -4), (0, - 5). The normalized bandshapes, computed for a ground state potential 0.53 * (q - 5)* at temperature kT/h = 1, have been gaussian-smoothed in order to reduce sampling artifacts and oscillations of the Franck-Condon coefficients. The broken curve divides the (u, u) control space into single (A > 0) and double (A < 0) minima regions.
affecting the critical structure. Such “continuous” terms have to be considered when comparing the model with the experimental results. Several shapes of the potential ve(q) = q4 + uq2 + uq for different values of u and u are shown in fig. 1. The critical structure of the potential is easily determined. The first derivative of V,(q), ve’(q) = 4q3 + 2uq + u, is a cubic with a single zero (a minimum ) or three zeros (two minima with a maximum in between) according to the positive or negative sign of the discriminant A = 8u3 + 2%~~.
(6)
V,(q) has two minima for the points inside the “cusp” curve A = 0 (broken curve in fig. 1) and one outside. It is thus easy to see that the potential q4 is not structurally stable. The addition of an infinitesimal perturbation - cq* brings it from the one-well to the two-well regime.
145
R.G. Della ValIe, A. Brillnnte / Excimer instability
4. Excimer bandshape
Potential functions siinilar to (5) have been treated by several authors [22,23]. The eigenvalues En and eigenfunctions w,,(q) of the hamiltonian H = T + V, may be obtained to any desired accuracy expanding the eigenfunctions as linear combinations of a suitable number of harmonic oscillator wavefunctions and diagonalizing the secular matrix. We have found a basis of thirty-three wavefunctions to be adequate to reach complete convergence of the eigenvalues. The first few eigenvalues and eigenfunctions for a selected potential well in the two-minima region are shown in fig. 2. As usual for mono-dimensional wells [24], there are no degenerate levels and the n th wavefunction has n nodes. In the two-well regime the lowest vibrational wavefunction exhibits a minimum, or at least a shoulder, provided that the barrier height is larger than Ea. When the two wells are slightly asymmetric, the lowest vibrational state lies mainly in the lower well, while the next state, slightly higher in energy, lies in the other well. In the single-well regime the lowest state has a single maximum and the next level is much higher. According to first-order perturbation theory, by using Born-Oppenheimer wavefunctions and neglecting the dependence of the electric dipole moment on the nuclear coordinates, the fluorescence spectrum for transitions from electronically excited states to lowest states is proportional to 1251:
where P,,f is the population of the state induced by the probe beam. If the relaxation among the excited levels is faster than the decay to the ground state, the population of the excited states thermalizes and a Boltzmann statistics may be used P,t 0: exp( - Ent/kT).
Fig. 2. A typical double-well potential and the wavefunctions in the ground and excited states with the corresponding extimer bandshape. All parameters are those used for the point (0, -5) in fig. 1. One of the more likely transition processes, corresponding to a minimum of the upper well and to a peak of the bandshape, is symbolically pictured.
ground state well [l] (fig. 2). The ground state sublevels are very dense and cannot be individually resolved. Some typical bandshapes (7) for a purely harmonic ground state potential and various control parameters u and u for the upper state potential are shown in fig. 1. The overlap integral (qnt ( 4,) has been expanded in a sum of the proper harmonic overlap integrals [26]. The gross structure of the bandshape turns out to be rather insensitive to both ground state frequency and shift from the equilibrium position. It appears from the figure that the critical structure of the potential and the bandshape are strictly related. These findings may be rationalized in terms of the semiclassical Franck-Condon principle. After an optical transition, the system, subjected to a new potential, will not be in equilibrium. The important final states for the transitions will be highly excited states with high quantum numbers which, according to the Bohr correspondence principle, behave quasi-classically. The final vibrational state q,(q) will oscillate rapidly except near the classical turning point q where Vg(q) = E,,. Thus a good approximation to (7) is [25]:
(8)
Under conditions appropriate to excimer formation, the excited state potential well is deep and strongly shifted with respect to the shallower
x6(E,f
- v,(q)
- hw).
(9)
146
R.G. Della Valle, A. Brillante / Excimer instability
variations of the excimer frequency, relative intensity and bandshape of 9-cyanoanthracene (9CNA) [3] and P-9,10-dichloroanthracene (DClA) [4] with pressure. This simple model breaks down when moving too far away from the thermodynamic point where the frequency wi, related to the transition coordinate q = ql, vanishes. Indeed a highly asymmetric double well (AE B kT) results for u -=z0 and u # 0. The highest of the two wells is essentially unpopulated and the system is effectively reduced to the single-minimum case.
The ground state potential VJq) is monotonically decreasing in the region of interest (fig. 2), thus the contribution from the state $,,t to the emission spectra has the same shape of the probability density I b(q) I 2, sligthly deformed by the curvature on Vs(q), shifted in origin by an amount E,,r and weighted by P,,. At low temperatures only the first few levels are appreciably populated. A simple pattern suggested by these general considerations is in qualitative agreement with the experimental findings. When a macroscopic parameter (e.g. the pressure) changes, the corresponding control parameters u and u will move along some trajectory in the control space. The bandshape has a single peak, slowly drifting in frequency, while u and u move in the singleminimum region. A new peak appears at higher (or lower) frequencies upon entering the twominima region from the left (or right). While crossing through the cusp region the intensity is progressively transferred from the former peak to the new one. The peaks are symmetric when well inside the single-minimum region, becoming slightly asymmetric near or inside the two-minima region. Two trajectories, corresponding to u = 3 and U= - 5 in fig. 1 might properly describe the
5. Pressure dependence of excimer frequencies
A quantitative picture may be obtained by a few approximations on eq. (9). At low temperatures only the lowest excited levels (or a few nearly degenerated levels) are appreciably populated. The sum on the excited state n’ may be restricted to these states and the energy E,,t may be considered as a constant. If the upper and lower potential wells are displaced enough, V,(q) may be linearized in q and the integral on Dirac’s delta may be carried out at once.
21.000
/
/
I
JO
20
P Ikbarl Fig. 3. Fit of excimer emission frequency for 9CNA under pressure.
80
R G. Della Valle, A. Brillante / Excimer instability
Recalling the relationship between the probability density and the potential shape, we may assume that the peaks of the excimer spectra coincide with the extrema of the excited state potential (proper units, incorporating all parameters of the ground state potential, are used). With a final scaling of q and u in K’(q) (by respectively) we may and (u(-~‘~ lu(+2 eliminate the dependence of the extrema of I/,(q) on the magnitude (but not on the sign) of U. We are thus left with two universal dimensionless “frequency” and “macroscopic” parameters q and u, related by 4qQ2q+udl,
00)
where the positive sign holds in the structurally stable regime (U > 0) and the negative sign in the unstable regime (U < 0). In order to relate the model to the experimental data, we have to specify the origin and the scale of the macroscopic and frequency parameters. Furthermore it may be necessary to add any “continuous” term discarded in the transformation from eq. (4) to eq. (5). In the case of 9CNA and DClA, using pressure as the macroscopic parame-
147
ter, we have found an additional linear pressuredependent frequency shift to be sufficient for a satisfactory fit: P=Po+Plu,
(11)
w=wo+w,q+ar(p-PO).
02)
Figs. 3 and 4 show fits to the experimental excimer fluorescence frequencies, according to eq. (10) with the positive sign for 9CNA and the negative sign for DClA. The scalings (11) and (12) are used, with p,, = 8 kbar, p1 = 0.93 kbar, w0 = 19780 cm-‘, oi = 600 cm-‘, a = - 23 cm-‘/kbar for 9CNA, and p. = 52 kbar, p1 = -45 kbar, w. = 21100 cm-‘, w1 = 2000 cm-l, a = -53 cm-‘/kbar for DClA. The phenomenological meaning of po, p1 and oo, w, is an indication of the gap position and size, in the pressure and energy scales respectively.
6. Conclusions The model developed in this paper shows that a simple anharmonic quartic potential, whose shape
21.000
20.000
19.000
(cm”1 1.3.000
17.000
P Ikbarl Fig. 4. Fit of excimer emission frequency of DClA under pressure. The back-bending part of the theoretical curve in the double-peak region is related to the local maximum of the potential well and has no experimental relevance.
148
R.G. Della Valle, A. Brillan!e / Excimer instabiliry
is modulated by the macroscopic parameters, may explain the variations in frequency and bandshape of the excimer emission in a molecular crystal under pressure. It must be stressed that our resulting bandshapes and frequency dependence are “structurally stable”. This implies that any physically reasonable potential will qualitatively produce the same results as the universal unfolding Ve(q) of eq. (5), provided that the two potentials have the same critical structure. The geometry of the excimer is in most cases preformed in the crystal lattice, originating from adjacent molecules arranged in “pair” or “stack” structures [l]. For monosubstituted anthracene derivatives the additional distinction between the “cis” and “ tram” configurations of the molecular pair can be made [3,27]. One should expect an intimate relationship between type of emission and excimer geometry, as a consequence of the different extent of the intermolecular interaction between adjacent molecules. The present paper suggests that any realistic model explaining the physical origin of the potential modulation by the macroscopic parameters must justify the possible occurrence of two, or possible more, competing excimer configurations. We have identified, to test our model, two different cases where differently bound molecular pairs yield competing emissions in different pressure regimes. In the case of 9CNA, increasing pressure (and decreasing temperature) acts as a drive to induce the “ topochemical” cis-excimer emission. The net result is a stabilization of the total energy with an increased Stokes shift of a thousand wavenumbers. The opposite effect is observed for DClA, where a more weakly bound excited state is favoured by pressure. In both cases the different constraints on the molecular pair, very sensitive to application of external pressure, are clearly responsible for the observed behaviour. Contrary to 9CNA, stabilization of the excimer state in DClA is inhibited at high pressure, when the steric hindrance of the 9,10-substitution becomes efficient (closer intermolecular contacts). These considerations might account for the stable or unstable regime of the pressure dependence, i.e. the positive or negative
sign given in eq. (10). It seems very encouraging that both cases could be described by the model here presented, identifying a promising starting point for further developments of a theoretical treatment of the excited state potential of these systems.
Acknowledgement This work was supported by the Italian Minister0 della Pubblica Istruzione and by CNR. We thank Professor C. Zauli for stimulating discussions and useful suggestions.
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R.G. Della Valle, A. Briliante / Excimer instabdity
[20] L.D. Landau and E. Lifchitz, Statistical physics (Pergamon Press, Oxford, 1958). [21] H.G. Drickamer and C.W. Frank, Electronic transitions and the high pressure chemistry and physics of solids (Chapman and Ha& London, 1973). 1221 R.L. Somorjai and D.F. Hornig, J. Chem. Phys. 36 (1961) 1980. [23] S.I. Chan, D. Stelman and L.E. Thompson, J. Chem. Phys. 41 (1964) 2828.
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[24] L.D. Landau and E. Lifchitz, Quantum mechanics (Pergamon Press, Oxford, 1958). [25] M. Lax, J. Chem. Phys. 20 (1952) 1752. [26] E. Hutchisson, Phys. Rev. 36 (1930) 410. [27] E.-Z.M. Ebeid, S.E. Morsi, M.A. El-Bayoumi and J.O. Williams, J. Chem. Sot. Faraday Trans. I 74 (1978) 1457.