Electron transfer and intramolecular radiationless transitions

_I. Photochem. Photobiol. A: Chem., 82 (1994) 5-10

Electron Joshua

transfer

Jortner

5

and intramolecular

radiationless

transitions

and M. Bixon

School of Chemistry, Tel Aviv University Tel Aviv 69978 (Israel)

Abstract We outline solvent-free

the conceptual

framework

for the description

of long-range

photoinduced

electron

transfer

in isolated

supermolecules and explore the dependence of the intramolecular electron transfer rates from photoselected states on the molecular parameters, i.e. energy gaps and mode-specific intramolecular reorganization

energies. An energy gap law for the microscopic electron transfer rate was derived which, for relaxation from the electronic origins, exhibits a poissonian energy gap dependence, with an exponential energy gap law for large gaps. The exponential energy gap law manifests the dominant role of nuclear tunnelling in the weak electron-nuclear coupling limit, being quantitatively distinct from the thermal gaussian energy gap dependence.

1. Introduction Electron transfer (ET) reactions in chemistry, physics and biology have been ahnost exclusively explored in donor-acceptor (DA) systems embedded in a medium, e.g. polar solvent, non-polar molecular solvent, glass or protein [l-5]. From the point of view of general methodology, such ET processes correspond to a broad class of nonradiative electronic processes in condensed media, which involve the conversion of electronic to vibrational energy [6]. Another class of non-radiative electronic processes pertains to intramolecular dynamics, i.e. internal conversion and intersystem crossing in large isolated molecules [6-111, where a vibrational quasi-continuum constitutes an intramolecular dissipative channel [6, 9, lo]. Can non-radiative ET be realized as an intramolecular radiationless transition in a solvent-free isolated supermolecule? Recent theoretical [12-141 and experimental [ 15-191 studies have challenged the conventional wisdom regarding the dominating role of medium coupling in ET. We have recently proposed [12] that long-range ET can occur in isolated, solvent-free supermolecules and have analysed the structural constraints for the occurrence of this intramolecular radiationless transition. On the experimental front [15-171, in a single case, conclusive experimental evidence has been reported by Verhoeven and coworkers [15-191 for photoinduced long-range ET in the isolated, jetcooled I-phenyl-C((4-cyano-l-naphthyl)methyl)piperidine semirigid supermolecule and some of 1010.6030/94/$07.00 0 1994 Elsevier Sdence SSDI lOlO-6030(94)02004-7

S.A. All rights resewed

its derivatives under solvent-free conditions. In this paper, we outline the conceptual framework for the description of long-range photoinduced ET in isolated supermolecules and explore the dependence of the intramolecular ET dynamics on the molecular parameters, i.e. energy gaps and mode-specific intramolecular reorganization energies, and on the optical excitation selectivity, i.e. the excess vibrational energy dependence of intramolecular ET.

2. Electron

transfer

in isolated

supermolecules

We consider long-range ET in a structurally rigid, solvent-free DBA supertnolecule (where B represents a molecular bridge). Intramolecular ET will be realized when an “initially excited” vibronic state of DBA is quasi-degenerate with a dense vibronic manifold of D+BAand effectively coupled to it [12], or under a somewhat more complex physical situation, when the D+BAmanifold mediates the relaxation of the initial DA state into another final intramolecular quasi-continuum

P31* The charge transfer state D+BA-

constitutes the lowest spin-allowed electronic excitation, provided that sufficient electrostatic stabilization of D*BAprevails, which implies a practical limit for the D-A (centre-to-centre) separation of R,,<6-7A [12]. Th e order of the electronic states with increasing energy is then &,(DBA), &(D+BA-) and S,((DBA)*) (Fig. l), where

1. Jortner, M. Bixon / Electron tramferand intramolecular

6 E

E

radiationless transitions

&(D+BA-) for (A) and the vibronic manifold of S,(DBA) for (B). Effective interstate coupling prevails provided that the density of final states in the vicinity of Ej PCi)=~6(Ej-E3

0)

is sufficiently large relative to the sequential decay widths rj of the final states, i.e. rj~ X- 1, and the couplingisstrong,sothat2~~Cj)&l(j$jf)126(Ej -E,) x=-1, where A is the system’s hamiltonian. These conditions ensure the occurrence of lj) --+{If>} relaxation in the statistical limit [9, lo]. The intramolecular ET rate (of the initial state b)) in the isolated molecule is [9, lo]

Fig. 1. Molecular level structure for ET processes in an isolated supermolecule. Optical S,(DBA) % S,((DBA)*) excitation selects the vibronic S,((DBA)*) level(s), which undergoes intramolecular charge separation (denoted by horizontal arrow) to manifold. vibronic the S,(D+BA-) &,(DBA) 5 &(D+BA-) excitation selects - the %zz S,(D+BA-) level(s), which undergoes intramolecular charge recombination (denoted by horizontal arrow) to the S,,(DBA) vibronic manifold. This ladder diagram establishes the interrelationship between ET in isolated molecules and intramolecular radiationless transitions. It is gratifying that resonance Raman [28] and optical lineshape 1291 data allow for the quantification of these ladder diagrams.

(DBA)* is a localized spin-allowed electronic excitation of DBA. Two intramolecular ET processes can be realized. (A) Charge separation S,((DBA)*)+S,(D’BA-) with a vibronic level of the electronically excited localized singlet state decaying into the charge transfer vibronic manifold. The initial (doorway) state of S,((DBA)*) can be photoselectively excited by a photon h, (Fig. 1). (B) Charge recombination &(D+BA-), with a vibronic level of the charge transfer state decaying into the vibrationally excited ground state manifold. The initial (doorway) state within S,(D+BA-) can be photoselectively excited by a photon h(Fig. 1). We can characterize the intramolecular ET processes of type (A) or (B) in terms of lj> -{If>} effective coupling and relaxation, where the state b> (with energy Ej) corresponds to a vibronic state of S,((DBA)*) for (A) and a vibronic state of &(D+BA-) for (B), and the If) state (with energy E,) corresponds to the vibronic manifold of

Setting I(j@f)l’=V%‘(j; f), where V is the electronic coupling and F(j; f) is the Franck-Condon vibrational overlap integral, the microscopic ET rates are k,=(2rP/fi)FF(j;

f)S(Ej-Ef)

(3)

Irreversibility of the intramolecular charge separation (A) and charge recombination (B) in the isolated supermolecule is implicitly ensured by sequential decay of the final D’BAmanifold via radiative decay and non-radiative charge recombination in case (A) and via IR radiative decay and possible isomerization processes within S,(DBA) in case (B). Equation (3) has the “golden rule” form for a quantum mechanical transition probability. This relation is applicable provided that interference effects between neighbouring lj) states are negligible [9, lo]. Invoking a coarse graining procedure, based on the assumption of the equal occupation probability for the quasi-degenerate initial state within a (small) energy interval 6E, the energy-dependent rate constant k(E) at the initial energy E (Fig. 1) is given by [13, 14, 301 k(E) = (27&=/~)AFD(E)

(4)

The averaged Franck-Condon density (AFD) was calculated for a simple harmonic model system with two displaced nuclear potential surfaces Uj(g) and Udg), which are characterized by the same frequencies {w,, %, . . . , q} and by the (dimensionless) displacements of the equilibrium positions of the minima of the potential surfaces {Al, A*,. . ., A,}. The intramolecular reorganization energy is

(5)

1. Jortner, M. Bixon 1 Electron transfer and intramolecular radiationless transitions

where S,= AI212 is the mode-specific electron-nuclear coupling strength and Al=)lw,S, is the mode-specific reorganization energy for a distinct vibrational mode q. AE is the energy gap between the minima of the potential surfaces (Fig. 1). The initial and final vibronic states will be specified in terms of the occupation numbers of the vibrational modes b> = (il, jZ,. . . , j,} and If) = {fl, f,,} respectively. The averaged Franck$&ion density (AFD) is [30]

(6) F@; fi) are the Franck-Condon factors between the initial j, and final fi states of mode I Hj,; fi) = exp( - A2/2)

(7) where on the right-hand side of eqn. (7) we abbreviate & = A,, j=j, and f =A. The Kroneker delta function in eqn. (6) restricts the sums over degenerate initial and final states. The nonnalization factors for AFD (eqn. (6)) involve the number N(E) of the initial vibronic states in the energy range 6E around E. SE is chosen as the common divider of the vibrational frequencies and of the energy gap.

3. Vibrational intramolecular

energy dependence ET rates

(ii) Charge separation S,((DBA)*) -+S,(D+BA-) (Fig. 1). The excess vibrational energy E in the vibronic manifold of S,((DBA)*) (with an energy gap AE relative to S,(D*BA-)) is acquired by electronic-vibrational excitation (hv = AEcr + AE -tE) to the S,((DBA)*) excited localized electronic state. Our model calculations of k(E) attempt to mimic the dependence of the ET rates on the energetic parameters, e.g. the energy gap AE and the intramolecular reorganization energy, which constituted superposition A = Z,& (eqn. (5)). In the choice of the input mode-specific Al parameters we were guided by the resonance Raman data of Markel ef al. [28] for the hexamethylbenzenetetracyano-ethylene charge transfer complex. We have subdivided the molecular modes into lowfrequency (ol = 200 cm-‘), medium-frequency (y = 500 cm-l) and two high-frequency (y = 1200 cm-l and We= 1500 cm-‘) vibrations, with the corresponding dimensionless shifts S1=6, S2= 3-1.2, S3 = l-O.4 and S, = l-0.8. The total intramolecular reorganization energies were taken in the range A=5400-3400 cm-l, in accord with recent spectroscopic data for barrelene-based DBA supermolecules in hexane, which yield an intramolecular reorganization energy A of 4600 cnP1 [31]. The energy gap for charge recombination, which is approximately corrected for solvent perturbations of the electronically excited charge transfer state, is AEm==25 000-28 000 cm-’ for typical barrelene-based DBA molecules [31]. For DA charge tranfer complexes [28], L?&= 12 000 cm-l. For charge separation, we use recent data for barrelene-based DBA supermolecules, which give AE =2SOo-6000 cm-’ [31]. In Figs. 2 and 3, we present the results of model calculations of the excess vibrational energy dependence of the quantum AFD(E)s for charge recombination and charge separation. For low and moderate energy gaps ( - AE = 1000-4000 cm-‘), which correspond to charge separation, the quantum mechanical AFDs at low values of E reveal an irregular spread (Fig. 3), which reflects mode selectivity. These mode selectivity effects are damped at higher values of E (Fig. 3) and are rather unpronounced for large values of IAEl over the entire E domain (Fig. 2) for charge recombination. For small and moderate values of /bEI, the energy dependence of AFD(E) qualitatively changes with increasing Iml (Fig. 3), reflecting the “transition” from intramolecular normal ET ( - AE
1

mincj.n (- l)+~r(Al~)‘+f-” 1 r!(j-r)!(f-r)! r=ll

of

We present model calculations of the energy dependence of the microscopic AFD(E), which determine k(E) (eqn. (4)) for charge recombination and charge separation in isolated, solvent-free, jetcooled supermolecules, where optical selection of vibronic states is not blurred by thermal sequence congestion effects [8] and no collisional damping prevails. recombination (i) Charge &(D’BA-) + S,(DBA) (Fig. 1). The excess vibrational energy E in the vibronic manifold of &(D+BA-) (with an energy gap AE, relative to !&(DBA)) is acquired by the electronic-vibrational optical excitation (hv,=AE,+E) to the SI(D+BA-) charge transfer band. This absorption band is broad and structureless for supermolecules and DA complexes in solution [28, 311 and in isolated jetcooled DA complexes [32], manifesting strong electron-phonon coupling.

-I

.I. Jortner, M. Bixon I Elechon

8

transfer and intramolecular radiationless transitions energy dependence of AFD(E) for charge recombination (Fig. 2) corresponds to intramolecular inverted ET which is characterized by: (i) increasing AFD(E) (for a fixed AIZ) with increasing E over the relevant domain, and (ii) decreasing AFD(E; AI!?) (for a fixed E) with increasing Iml. Feature (ii) reflects the energy gap law for intramolecular ET.

4. Energy

Fig, 2. Model calculations of the excess vibrational energy dependence of the average Franck-Condon density (AFD(E)) for intramolecular charge recombination. E is the excess energy above the zero-point energy in the S,(D+BA-) vibronic manifold; n =4; the characteristic frequencies are o, =200 cm-‘, ~=500 cm-‘, y= 1200 cm-’ and o,= 1500 cm-‘; the reduced displacements areS,=6,SZ=3,S~=1andS,=landh=5400cm-’.Theenergy gaps -AEcr in the range 12 000-27 000 cm-’ are marked on the curves and SE= 100 cm-‘.

gap law for intramolecular

ET

To assess the general characteristics of the energy gap law for ET in isolated supermolecules, we present in Fig. 4 the energy gap ( -A.!?) dependence of AFD(E) at the electronic origin (E = 0) and at a moderate excess vibrational energy (E =2000 cm-‘), which corresponds to a rough estimate for the onset of intramolecular vibrational redistribution in large molecules. At low \hEl values, which correspond to the intramolecular normal region, AFD(0) increases with increasing AI?, reaching a maximum value for intramolecular activationless ET. On the other hand, for excess vibrational excitation (E =2000 cm-‘), AFD is practically constant over the normal and activationless domain, the due to relation AFD(0) > AFD(E) in the vicinity of - AE = A. With a further increase in -aE, AFD decreases, reflecting the energy gap relation Charge recombination

0

0

,,:,,,,~,1,,,~~~~~~11”“““’ 4000 2000

6000

E cm-'

Fig. 3. Model calculations of the excess vibrational energy dependence of the average Franck-Condon density (AFD(E)) for intramolecular charge separation. E is the excess energy above the zero-point energy of S,((DBA)*). n =4, with the characteristic frequencies and displacements being the same as in Fig. 2. 6E=lOO cm-‘. *, -AE=lOOO cm-’ and E,=900 cm-‘; 0, -AE=4000 cm-’ and EA=91 cm-‘.

are of interest for the elucidation of the experimental excess energy dependence of intramolecular ET, explored by Verhoeven and coworkers [15-191 in jet-cooled supermolecules. The situation for the

IO -” IO -I2

IO-‘3 ‘o-‘O

’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ 10000 15000 20000 25000 5000 -AE

cm-’

Fig. 4. The energy gap (-AE) dependence of AFD(E) for the electronic origin (full curve) and for E = 200 cm-’ (broken curve). n =4, with the frequencies and reduced displacements as in Fig. 2 and SE=100 cm-l. Note the exponential decrease in AFD with increasing -A.!? at large energy gaps, which corresponds to the energy gap law (eqn. (8)).

J. Jottnet, M. B&on / Electron tmnsferand intramolecular radiationless transitioon~

AFD(E

-A

exp( - @ZSl)

(8)

for both E = 0 and E= 2000 cn-‘, with the constants A and y exhibiting E dependence. The overall E dependence of AFD(E) at E =. 2000 cm- ’ roughly corresponds to a constant value at low - AE followed by eqn. (8) at high - AJZ. For the intramolecular ET from the electronic origin (E = 0) for both charge separation and recombination (Fig. 4), the AE dependence of AFD(0) corresponds to a poissonian distribution, which in the limit of large IA.51 reduces to relation (8). The poissonian IA/Z1dependence of AFD(0) and of k(0) constitutes a quantum mechanical relation for radiationless ET in an isolated supermolecule. There is a basic physical difference between the ET in a solvent-free supermolecule and conventional ET in solution. The isolated molecule ET does not take place at the crossing point (region) of the two nuclear potential surfaces, in contrast with conventional ET in solution, for which the crossing region (where ET occurs) becomes accessible by solvent-induced vibrational excitation. Rather, both for the normal intramolecular region (IhE h) ET occurs via nuclear tunnelling. For a large energy gap /aEl, where the exponential energy gap law (eqn. (8)) applies, the lowest crossing point of the nuclear potential surfaces is high in energy, whereupon the intramolecular ET in the solvent-free supermolecule occurs via nuclear tunnelling from the low-lying, optically excited vibronic states of D+BAto the final DBA manifold. Adopting the terminology of ET theory [l-5] and of intramolecular dynamics [g-11], we consider nuclear tunnelling in the inverted region for ET, which is isomorphous with an intramolecular radiationless transition in the weak coupling limit

WI. The (nearly) exponential energy gap law [32] (eqn. (8)) for ET in the isolated supermolecule distinct from the (at large IAEI) IS ’ q uantitatively thermal Marcuss gaussian dependence. The gaussian (high IAEI) tail of the energy gap dependence in a dense medium at finite temperatures can be traced to the implication of thermal activation to the (lowest) crossing point of the potential surfaces, which is characterized by the energy E, = (AI? + h)‘/ 4Ak,T above the origin of the initial potential surface, and the thermally averaged ET rate is k(7’) a exp( -E,&~. In contrast, ET in the isolated large molecule occurs via nuclear tunnelling in a system characterized by fmite (or high) frequency vibrational modes, with fiwj > k, T. A similar situation pertains to the applicability of the ex-

9

ponential energy gap law for other non-radiative electronic-vibrational relaxation processes, e.g. internal conversion and intersystem crossing in isolated and solvated large molecules [lo, 111, f-f relaxation in lanthanide complexes [lo, 331, lowspin-high-spin relaxation of transition metal complexes [34] and ET in metal-polypyridine complexes [35-371. All these non-radiative processes are characterized by a significant contribution from a finite number of high-frequency “intramolecular” or “intracomplex” vibrational modes, which are manifested in the exponential energy Igap law. This study builds a bridge between ET and radiationless transitions in isolated molecules and reflects on the universal unifying features of intramolecular and medium-induced non-radiative relaxation.

Acknowledgments We are grateful to Professor J.W. Verhoeven for illuminating discussions, correspondence and for prepublication information. This research was supported by a grant from the Bundesministerium tir Forschung und Technologie (BMFT) (Contract No. 317-4003-0328955A) under the auspices of the Technical University of Munich.

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