J. Phys. Chem. 1994,98,1289-7299
7289
Energy Gap Law for Nonradiative and Radiative Charge Transfer in Isolated and in Solvated Supermolecules M. Bixon and Joshua Jortner' School of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel
J. Codes, H. Heitele, and M. E. Michel-Beyerle Institut fiir Physikalische und Theoretische Chemie der Technische Universitht Miinchen, Lichtenbergstrasse 4, 85 747 Carching, Germany Received: January 12, 1994; In Final Form: May 23, 1994'
In this paper we explore the foundations and some applications of the energy gap law (EGL) for nonradiative and radiative charge recombination from an ion pair state to the ground electronic state of isolated (solvent-free) and solvated donor (D)-acceptor (A) complexes and DBA bridged (B) supermolecules. The energy gap dependence of the averaged Franck-Condon density AFD(E), which is proportional to the microscopic electrontransfer (ET) rate, &(E),at the excess energy E, was calculated numerically (for a range of E) and by saddle point integration (for E = 0) for a displaced harmonic potential system. The intramolecular electron vibration coupling parameters were inferred from resonance Raman data and from ET emission line shapes. For isolated supermolecules an energy gap (AE)dependence of AFD(E) was derived, which for the electronic origin (E = 0) is a multi-Poissonian, with a Gaussian dependence over a narrow, low AE domain and a superexponential decrease with increasing AE for large AE. The EGL, AFD(0) = A exp(qA,??),holds for large values of A,?? over physically relevant AE domains (of -5000 cm-I), where the theoretical parameters y and A have to be extracted from numerical calculations using a complete set of nuclear frequencies and their coupling parameters. Approximate coarse graining of the coupling parameters over a small number of frequencies reveals that within a few-mode approximation it is important to segregate between medium- and high-frequency modes; the averaged single-modeapproximation is inadequate, while the maximal mode representation (which is valid in the asymptotic limit of huge LE) does not hold in the relevant A,?? domain. The failure of the single-mode approximation forces us to utilize the exponential EGL as a useful empirical relation for the representation of "exact" theoretical results or of experimental data for isolated systems. Focusing on the EGL fo_rsolvated supermolecules, w,e have shown that the first-order solvent correction to the EGL is AFD(0) N- A exp[-.y(AE - As)] with A = A exp(y*X&T) where XS is the solvent reorganization energy, with the y parameter being solvent invariant and determined by the intramolecular dynamics. The EGL for solvated DBA was successfully applied for the analysis of the nonradiative ET rates in the pyrene-substituted barrelene-based donor-acceptor supermolecule in a series of solvents, with the solvent-dependent energy gaps being varied in the range of 0.45 eV, while the As vary in the range XS = 0.16 eV (for n-hexane) to XS = 0.36 eV (for acetonitrile). Finally, we have explored the isomorphism between the description of the nuclear Franck-Condon vibrational overlap for nonradiative and radiative ET processes. We predict an exponential EGL for the low-energy tails in the charge-transfer fluorescence spectra of isolated and solvated supermolecules.
I. Introduction A ubiquity of nonradiativeelectronic processes in isolated large molecules, in condensed phases and in proteins, which occur without the breaking of chemical bonds, involve the conversion of electronic to vibrational energy.1 In condensed phases, these nonradiative relaxation processes span a broad spectrum of intermolecular and intramolecularbehavior, e.g., electron transfer (ET) in solutions and in glasses,2J small polaron motion in solids? electron-hole recombination in semiconductors5and in amorphous solids: electronic energy transfer between molecules and ions in solids? liquids? and glasses? and spin interconversion in transition metal compounds.10 Nonradiative processes in biophysical systems pertain to the primary electron transfer in the photosynthetic reaction center," electron transfer in giubular proteins,12 and group transfer in the low-temperature recombination of dioxygen and carbon monoxide with hemoglobin.13 Radiationless processes in isolated molecules correspond to internal conversion and intersystemcrossing in large isolated m o l e c ~ l e s ~and ~ Jlong~ range photoinduced intramolecular ET within solvent-free suAtmtract published in Adounce ACS Absrrucrs, July 1, 1994.
permolecules.16J7 An important generalization of a variety of nonradiativeelectronic processes (both in condensed medium and in isolated molecules) pertains to the energy gap dependence of the nonradiative rate. A celebrated example is the Marcus parabolic (free) energy gap dependence (AE)of ET rates in a dense medium,l*J9 k a exp[-(AE - &)2/4ksm], where & is the medium reorganization energy and AEis theenergy difference between the electronic origins of the initial and the final states (note that this definition of AE has the opposite sign to the conventional one in the field of ET). An isomorphous parabolic expression was derived by Kubo and Toyozawa20Jl for thermal ionization and electron-hole recombination in semiconductors. Another notable generalization involves the "energy gap law" (EGL)22.23 for the exponential decrease of the nonradiative rate, i.e., k a exp(-yAE) for large energy gaps (AE)between the initial and final states with relatively small changes in their equilibrium geometries (i.e., the weak coupling limit22). The EGL22-45 was originally advanced for radiationless transitions in isolated large molecules,- e.g., intramolecular internal conversion and intersystem crossing.2429 The EGL was examined and applied for a variety of radiationless processes, Le., f-f relaxation
0 1994 American Chemical Society Q022-3654/94/2098-1289~Q4.5Q~Q
7290 The Journal of Physical Chemistry, Vol. 98, No. 30, 1994
5=3
Bixon et al. Following our recent analysis of ET in isolated, solvent-free supermolecules,61we focus on the population decay of initially excited vibronic state(s) at the initial energy E in the DBA manifold (Figure 1). The initial state(s) is (are) prepared by photoselection. The energy-dependent rate constant k(E) at the energy E (Figure 1) is given by
k ( E ) = (2?rV2/h)AFD(E)
(11.2)
where Vis the electroniccoupling. The averaged Franck-Condon density (AFD) was calculated for a simpleharmonic model system with two displaced nuclear potential surfaces Uj(q)and Uhq), which are characterized by the same frequencies ( 0 1 ~ 0 2 , ..., wn) and by the (dimensionless) displacements of the equilibrium positions of the minima of the potential surfaces (AI, A2, ..., An}. The intramolecular reorganization energy is n
x =p o p , * 9,
_ I
Aqa
AE is the energy gap between the minima of the potential surfaces
Figure 1. A schematic representation of the nuclear potential surfaces of DBA and D+BA-. The correspondingladder diagram of thevibrational states is drawn on the right-hand side of the figure.
of lanthanide c0mplexes3G3~and electronic energy transfer4’ as well as electron transfer in ion pairs38 and in solvated metalpolypiridine complexes.3g-44 In this paper we explore the foundations and the applications of the EGL for ET dynamics in isolated and in solvated organic supermolecules. In view of the close analogy between the radiative and radiationless processes in condensed media or in large mole~ules,2~~2~.5* we shall also investigate the exponential EGL for low-energy tails of chargetransfer optical fluorescence and the high-energy tails of the charge-transfer absorption spectra.4657 11. Energy Gap Law for Nonradiative Charge Recombination
in an Isolated Supermolecule We consider first long-range ET in a structurally rigid, solventfree DBA molecule, which consists of a bridged (B) electron donor (D) and electron acceptor (A).6J73941 A theoretical analysis of direct16961 and sequential17 intramolecular ET was presented, and an experimental demonstration of long-range ET in a semirigid59 and in a rigid60 jet cooled supermolecule was provided by Verhoeven et al. In the exploration of the energy gap law in isolated supermolecules, we shall consider ET dynamics for large energy gaps. Accordingly, we shall focus on the radiationless charge recombination reaction D’BA-
-
(11.3)
=
DBA
(11.1)
where the charge-transfer state D+BA- constitutes the lowest spin-allowed excited state of the supermolecule. The intramolecular radiationless transition in the isolated supermolecule (Figure 1) does not take place at the crossing point (region) of the two nuclear potential surfaces. This is in contrast to the conventional description of ET in solution as occurring at the crossing region, which becomes accessible by solvent-induced vibrational excitation. Rather, for the large energy gap AE,as appropriate for reaction 11.1, the lowest crossing point of the nuclear potential surfaces (Figure 1) is high in energy, whereupon the intramolecular ET in the solvent-free supermolecule occurs via nuclear tunneling from low-lying, optically excited vibronic states of D+BA- to the final DBA manifold. Adopting the terminology of the ET theory3J8J9 and of intramolecular dynamics,1J5s22 we consider nuclear tunneling in the inverted region (for ET), which is isomorphous to an intramolecular radiationless transition in the weak coupling limit.
(Figure 1) and S, = A?/2 is the mode-specific electron-nuclear coupling strength for a distinct vibration mode W I . The initial and final vibronic states will be specified in terms of the occupation numbers of the vibrational modes j (il, j 2 , ...,in) and f VI&, ...,fn), respectively. The averaged Franck-Condon density (AFD) is62 AFD(E) = [N(E)]-’ (6E)-’
rrOF(ji&)~5~~, (11.4)
VI
v)
-1
FU1;fr)are the Franck-Condon factors between the initial j 1 and the final fr states of mode I . minu8
[5
FG j&) = exp(-S)j! f!
(-1 )i+f-r(,9)(l+l-W2
3*
(11.5) r!(j-r)!u-r)! where on the right-hand side of eq 11.5 we abbreviate S = SI, j = j r , and f =fr. The Kroneker delta function in eq 11.4 restricts the sums to degenerate initial and final states. The normalization factors for the AFD (eq 11.4) involve the number N(E) of initial vibronic states in the energy range 6E around E. 6E is chosen as the common divider of the vibrational frequencies and of the energy gap. Equations 11.2-11.5 provide microscopic ET rates, when the intramolecular vibrational energy redistribution (IVR) in the initial DBA manifold is slow on the time scaleof microscopic ET. Our computational method involves a coarse graining procedure over the initial “active” states in a “small” energy domain 6E. This scheme precludes the Occurrence of efficient IVR with the “nonactive”vibrational modes, but is rather implied by the discretization procedure for the frequencies of the active modes and for the energy gap. We have shown6’ that this coarse graining over the initial states in the energy range 6E is valid and practical, provided that the variations of the individual FranckCondon densities are small, a condition well satisfied61.62 for 6E = 100 cm-1 in conjunction with the frequencies to be used herein. This procedure results in mode-specific ET rates for low values of E, while for higher values of E averaged k(E) values are obtained. This averaging corresponds only to the subset of the “active” modes in the energy domain 6E, for which F(il;fr) # 0, and does not involve IVR over all intramolecular vibrational modes. In particular, the AFD for the electronic origin, Le., AFD(O), is given by eq 11.4 with N(0) = 1 and the sum over Q} being eliminated, representing an initial state-selective rate. We shall present model calculations of the energy dependence of the microscopic AFD(E), which determine k(E), eq 11.2, for ET in isolated, solvent-free, supermolecules, where optical selection of initial states at a given energy can be accomplished.
The Journal of Physical Chemistry, Vol. 98, No. 30, 1994 7291
Energy Gap Law for Charge Transfer in Supermolecules We shall limit ourselves to the electronic origin (E = 0) and to moderately low values of the excess energy, i.e., E d X/2 (i.e., E = 2000 cm-1). Our model calculations attempt to mimic the dependence of AFD(E) (and k(E)) on the relevant energetic parameters, e.g., the energy gap AE and the intramolecular reorganization energy (eq 11.3), A = XIXI, with XI = Slhw~. Information on the mode-specific XI and S~parameters(with hwl = XI/SI) was taken from resonance Raman spectroscopy of DA complexes in ~olution,6~ while less detailed information emerges from the analysis of the line shapes for radiative D+A- + hv DA52 and D+BA- + hv DBA5I-57 recombination in solution. Guided by these data, the following mode-specific SI and 01 parameters were chosen for our model calculations. (i) Typical DA Complexes. On the basis of resonance Raman data for the hexamethylbenzene-tetracyanoethylene (HMB/ TCE) charge-transfer complex,63we have used the following sets of parameters, which are specified by the frequencies o = (WI, w2, ..., on)and coupling strengths S (SI, SZ,...,S,,). (iA) The full set of coupled modes consisting of 11 vibrations in the range 150-2200 cm-I. It is reduced to a set of nine modes that have a common multiplier of Sw = 50 cm-'.
-
-
(w/cm-') = (150,450,550,600,1000,1300,1400,1550,2200)
S = (8,
0.9, 0.2, 0.2, 0.1, 0.3, 0.1, 0.6, 0.1) (11.6)
The intramolecular reorganization energy is X = 3600 cm-l. The analysis of the optical absorption line shape in conjunction with the Raman (SI)data63 of the HMB/TCE complex in CCL results in a very high value of AS = 3500 cm-' for the medium reorganization energy. In order to reduce the value of Xs to the Yreasonable"values of 1000 cm-1, we construct a second set of (SI)by scaling the Raman (SI)data by a numerical factor of 1.5 to give X = 5400 cm-1. (iB) Grouped modes. We have subdivided the full set of modes into groups each containing severalmodes with couplingstrengths (S,) and frequencies (w,). Each group is represented by the single frequency uj= cS#,/cSd and the coupling Sj = E&. We have partitioned the molecularmodes (11.6) into four averaged modes (w/cm-') = (200,500,1200,1500)
S = (6, 3,
1,
1)
(11.7)
The implications of this approximate grouping procedure on the calculations of the ET rates will be examined in section IV. We note in passing that the W I mode in schemes 11.6 and 11.7 presumably corresponds to the intermolecular motion of the two components and may be absent in rigid supermolecules. (ii) Typical DBA Supermolecules. Guided by an optical line shape analysis for CT emission in barrelene-based DBA supermolecules,57 which for a two-mode fit resulted in OL = 400 cm-l with SL= 5 and WH = 1400 cm-1 with SH = 1.8, we have used four vibrations with the following sets of coupling parameters (w/cm-') = (400,600,1300,1500) S = (1,
1,
0.8,
1)
(11.8)
which gives A = 3540 cm-I. Taking S = (3, 2,0.8, 1) with the same frequencies yields A = 4900 cm-l. The total intramolecular reorganization energies X = 36005400 cm-1 for case i and X = 3500-4900 cm-1 for case ii are in accord with recent spectroscopic data for barrelene-based DBA molecules in hexane, which yield X = 4000 ~m-1.5~A typical energy gap for charge recombination in DA complexes is AE = 10 000-15 OOO cm-I. The energy gap for charge recombination
