Inelastic Electron Tunneling Erases Coupling-Pathway

Optimization of Exciton Trapping in Energy Transfer Processes. .... of electric charge transfer pathways in porphyrin: a light-controlled nanoscale cu...
56 downloads 0 Views 112KB Size
Download PDF
J. Phys. Chem. B 2004, 108, 15511-15518

15511

Inelastic Electron Tunneling Erases Coupling-Pathway Interferences Spiros S. Skourtis,*,† David H. Waldeck,*,‡ and David N. Beratan*,§ Department of Physics, UniVersity of Cyprus, 1678 Nicosia, Cyprus, Department of Chemistry, UniVersity of Pittsburgh, Pittsburgh, PennsylVania 15260, and Departments of Chemistry and Biochemistry, Duke UniVersity, Durham, North Carolina 27708 ReceiVed: April 3, 2004; In Final Form: July 8, 2004

Theoretical analysis of nonadiabatic electron-transfer reactions in molecules usually assumes that electron amplitude propagates coherently from the reductant to the oxidant via covalent and noncovalent coupling pathways. We show that when the tunneling electron excites local bridge vibrations (inelastic tunneling), the excitation “labels” the physical pathway traversed. As such, the coherence among the bridge-mediated tunneling pathways is destroyed. We illustrate this effect using a simple model Hamiltonian and show how the donoracceptor interaction, and thus the electron-transfer rate, is modified by inelastic effects. Pathway coherence loss provides a mechanism to relax orbital-symmetry constraints on electron-transfer reactions. This effect may be of particular significance in macromolecules with destructively interfering pathways or low tunneling barriers. Pathway decoherence that arises from inelastic effects in molecules is analogous to coherence loss in mesoscopic “which way” interferometers and might provide an approach to gate electron flow in molecularscale devices.

Introduction Some electronic devices exploit the intrinsic electron-wave properties of the electron in a dramatic way.1-3 Examples of current interest include macroscopic or mesoscopic “which way” interferometers and diffractometers, superconducting quantum interference devices, Aharanov-Bohm interferometers, and very fine gratings.2 Toward the goal of developing ever smaller electronic devices, one is motivated to explore molecular analogies for some of the quantum-mechanical control mechanisms familiar in mesoscale devices.1a,2 Here, we explore the use of supramolecules as interferometers for electron waves. In particular, we analyze electron donor-bridge-acceptor (DBA) molecules from a perspective in which the chemical bridging unit has multiple electron-tunneling pathways, analogous to the slits in the familiar multislit electron transmission experiment. More specifically, we explore the nature of the interferences among these multiple-coupling pathways. We consider DBA molecules in which an electron tunnels from the electron donor to the electron acceptor by way of intervening atomic and molecular orbitals of the molecular bridge unit. In this case, the bridge-mediated coupling is a sum over all pathway contributions and these coupling interactions may interfere with one another.4,5 In the commonly used Condon approximation, the electron motion is assumed rapid compared to nuclear motion so that the bridge nuclei are “frozen” and the tunneling proceeds coherently and elastically. Here we consider the implications of allowing the bridge’s nuclear degrees of freedom to be perturbed by the electron-tunneling event. In particular, we formulate the problem in the context of quantized bridge oscillators. In molecular systems, coherence among the coupling pathways6 can be lost through a number of already well-understood * Correspondence may be addressed to any author. † University of Cyprus. E-mail: [email protected]. ‡ University of Pittsburgh. E-mail: [email protected]. § Duke University. E-mail: [email protected].

mechanisms. Most commonly discussed is decoherence of the electron wave function that occurs by nuclear wave packet motion following the electronic transition from donor to acceptor.6 For nonadiabatic electron transfer, this decoherence mechanism corresponds to nuclear motion away from the crossing region of the reactant and product potential surfaces. In addition, an electronic-dephasing mechanism arises from energy-gap fluctuations of the donor-acceptor localized states relative to the bridging states. These fluctuations in energy gap change the electron-wave’s phase as it propagates through the bridge, but no energy is left on the bridge as a result (the electronic motion is elastic).7a,b Alternatively, the electron wave can dephase by depositing energy on the bridge: inelastic tunneling. We discuss circumstances where this inelastic tunneling will have a significant influence on the donor-acceptor interaction and the electron-transfer rate. In mesoscopic systems, it is well-known that quantum interference among electron-transmission pathways can be controlled by introducing probes of the path “actually chosen” by the quantum particle.1b,c,2,3 Indeed, manipulation of pathway interferences in solid-state devices is observed in “which way” interferometers.2,3 Our goal is to explore the manipulation of pathway interference in electron tunneling through molecules. We show that inelastic tunneling causes changes in pathway interactions and causes effects analogous to those found in “which-way” interferometers. We consider DBA systems where inelastic tunneling leaves a “marker” along the electrontunneling pathway that was traversed by the electron wave; i.e., the bridge unit is left in a different vibrational state as a consequence of the electron wave’s transit. Our analysis uses a very simple two-pathway Hamiltonian to describe inelastic tunneling coupled to local bridge vibrations. Bridge Motion, Inelastic Tunneling, and Electron Transfer. Chemical and biochemical electron-transfer bridges contain a large number of physical tunneling pathways linking donor to acceptor. The elastic tunneling matrix element, TDA, is

10.1021/jp0485340 CCC: $27.50 © 2004 American Chemical Society Published on Web 09/11/2004

15512 J. Phys. Chem. B, Vol. 108, No. 40, 2004 computed routinely and it includes constructive and destructive contributions from the many coupling paths.4,5 The last fifteen years have witnessed a growth in our understanding of how orbital symmetry and energetics of molecular coupling pathways influence electron-transfer rates and currents.1a,4-10 Simple pathway models describe the relative role of through-bond vs through-space tunneling propagation,4 molecular orbital views include multiple tunneling pathways and their interferences,5 and dynamical studies incorporate the influence of coupling pathway fluctuations on the rate.7,8,9a,20 Importantly, it has been shown recently that fluctuations can remove destructive pathway interferences and, in some cases, may be essential for promoting biological electron transfer.8b The inelastic processes introduced in the present paper provide an additional mechanism to “erase” destructive interferences among coupling pathways. Although electron transport need not be coherent, electron tunneling via covalent bridges is often assumed to be coherent, and the electronic coupling between the donor and acceptor groups is usually described by a coherent sum of contributions from multiple coupling pathways.5-10 The influence of inelastic events on electron-transfer, including the impact on the FranckCondon factor and electronic coupling9,11 is the subject of great current interest. Our particular focus is on the dephasing of pathway contributions to the donor-acceptor interactions that arise from inelastic tunneling associated with localized bridge vibrational modes. Coherence among Parallel Tunneling Pathways. Two slit1b,c and modern mesoscale electron-transport experiments2,3 demonstrate how markers of the route traversed by an electron destroy pathway coherence. In the inelastic tunneling regime, an energy mismatch often exists between the donor and acceptor electronic states, and energy conservation is achieved by depositing (some of) the excess energy into one of these local vibrations. We will show that the inelastic event labels the pathway traversed by the electron, eliminating contributions from other coupling routes to the donor-acceptor interaction and the electron-transfer rate.

Skourtis et al.

Figure 1. Model donor (D)-bridge (U,L)-acceptor (A) system. The donor (acceptor)-bridge interaction elements are tij. The local oscillators are linearly coupled to the U and L orbitals.

respectively. γU (γL) describes the strength of the electronoscillator linear vibronic interaction on the upper (lower) pathway. All orbitals are orthonormal. Figure 1 illustrates the essence of the model. Each oval represents one of the four entities comprising the DBA molecule (donor, acceptor, and two bridge pathways) and their respective energies are described by the site Hamiltonian H ˆ esite. The springs represent the model’s inclusion of a vibrational mode for each bridging unit, described by the Hamiltonian H ˆ n. Such vibrations are often called inducing modes (e.g., see ref 11g) because they influence the donor-acceptor matrix element. The H ˆ e/n Hamiltonian describes the electron-nuclear coupling that gives rise to the excitation of the springs and the strength of the coupling is controlled by the parameter γ. The dashed arrows in Figure 1 represent the electronic coupling of the donor and acceptor units with the two bridge pathways, described by the hopping Hamiltonian H ˆ ehop. Bridge-Mediated Coupling Matrix Elements. To relate to the well-known expression for the tunneling matrix element TDA ) 〈φD|Vˆ G ˆ bridgeVˆ |φA〉,1a the Hamiltonian in eq 1 is written

H ˆ )H ˆ (0) + Vˆ

(2)

where Elastic and Inelastic Tunneling with Local Bridge Vibrations Model Hamiltonian. The simple model Hamiltonian for the system contains: site (H ˆ esite) and hopping (H ˆ ehop) electronic interactions among the donor, bridge, and acceptor, a bridge vibrational Hamiltonian (H ˆ n), and a vibronic interaction between the electron and the bridge oscillator (H ˆ e/n). For a linear vibronic interaction on the bridge sites, the equilibrium position of the local oscillator is offset and the energy is lowered when the electron resides on a bridge site.

H ˆ esite

) RD|φD〉〈φD| + RA|φA〉〈φA| + RL|φL〉〈φL| + RU|φU〉〈φU|

H ˆ ehop ) tDU|φD〉〈φU| + tDL|φD〉〈φL| + tUA|φU〉〈φA| + tLA|φL〉〈φA| + h.c. H ˆ n ) (bU+bU + 1/2)pωU + (bL+bL + 1/2)pωL H ˆ e/n ) γU(bU+ + bU)|φU〉〈φU| + γL(bL+ + bL)|φL〉〈φL| (1) D(A) indicates the donor (acceptor) orbital, U (L) is the upper (lower) bridge orbital coupled to a local oscillator, and the tij elements couple the donor (acceptor) to the upper or lower bridge orbital (h.c. denotes hermitian conjugate). b+ and b are the harmonic oscillator creation and destruction operators,

ˆ esite + H ˆn + H ˆ e/n H ˆ (0) ) H

Vˆ ) H ˆ ehop

The zero-order Hamiltonian is partitioned into bridge and “donor-acceptor” parts, H ˆ (0) ) H ˆ bridge + H ˆ da where

H ˆ bridge ) RU|φU〉〈φU| + RL|φL〉〈φL| + H ˆn + H ˆ e/n

(3)

H ˆ da ) RD|φD〉〈φD| + RA|φA〉〈φA|

(4)

and

H ˆ bridge does not contain off-diagonal coupling elements tij between the φU and φL bridge orbitals (U and L pathways). Further, H ˆ da does not contain a direct through-space interaction between the donor and acceptor orbitals. We do not anticipate qualitative changes in physical behavior as these assumptions are relaxed. Consider an initial (final) state of the system |φD;i〉 (|φA;f〉) where i (f) is the product of vibrational states of the U and L oscillators

|i〉 ) |n(yU) m(yL)〉

|f〉 ) |q(yU) r(yL)〉

(5)

(yU and yL are the coordinates of the oscillators). The tunneling matrix element between the initial and final state is T ifDA ) 〈φD;i|Tˆ DA|φA;f〉 where Tˆ DA ) H ˆ ehopG ˆ bridge(Etun)H ˆ ehop and

Pathway Coherence

J. Phys. Chem. B, Vol. 108, No. 40, 2004 15513

G ˆ bridge(Etun) ) [Etun - H ˆ bridge]-1 is the bridge Green’s function (this result is perturbative in the H ˆ ehop interaction). From eqs 1-5, if nm,qr T DA ) TDA ) 〈φD;n(yU) m(yL)|Tˆ DA|φA;q(yU) r(yL)〉 )

In eq 11, the classical accepting modes (their classical coordinates denoted R) modulate the donor and acceptor energies and the reorganization energy is λ.1a,4 The quantum bridge oscillator states U and L alter the donor-acceptor energy gap and energy degeneracy is achieved when

〈φD|H ˆ ehop|φU〉〈φU;n(yU) m(yL)|G ˆ bridge|φU;q(yU) r(yL)〉〈φU|H ˆ ehop|φA〉 + 〈φD|H ˆ ehop|φL〉〈φL;n(yU) m(yL)|G ˆ bridge|φL;q(yU) r(yL)〉〈φL|H ˆ ehop|φA〉 (6)

Rates and Bridge-Mediated Coupling Matrix Elements. For simplicity, the Hamiltonian in eq 1 does not include nuclear degrees of freedom of the system and the environment that couple only to the donor and acceptor electronic states (φD and φA)sthe so-called accepting modes. As discussed below, these modes influence the tunneling matrix element only through the tunneling energy Etun. To describe the effect of accepting modes, we follow the development of Troisi, Nitzan, and Ratner.11g The Golden Rule rate constant is

kET )

∑i ∑f Pi ki,fDA

(7)

2π if 2 |T | F p DA i,f

(8)

RD(R) + i ) RA(R) + f

(12)

The energy RD(R) + i is the tunneling energy Etun that appears in T ifDA through the bridge Green’s function (eq 6). Bridge Green’s Function. The donor-acceptor matrix element T ifDA that appears in the rate (eq 8) depends on both electronic and bridge nuclear degrees of freedom. Equation 6 shows that its dependence on the nuclear degrees of freedom is contained in the bridge Green’s function G ˆ bridge(Etun) ) [Etun H ˆ bridge]-1. If the bridge Hamiltonian (eq 3) is diagonalized, the bridge eigenstates are |φD;n(yU-y0U) m(yL)〉 and |φL;n(yU) m(yL-y0L)〉 with energy eigenvalues

RU + (n + 1/2)pωU - γU2/pωU + (m + 1/2)pωL RL + (n + 1/2)pωU + (m + 1/2)pωL γL2/pωL

n, m ) 0, 1, ... (13)

where

ki,f DA )

In the equations above, the index i (f) denotes the initial (final) bridge oscillator state as in eq 5 (i ) nm, f ) qr), and Pi is the population at thermal equilibrium of the initial oscillator state i. Fif is the Franck-Condon weighted density of states for the accepting modes (V and w),

Fi,f )

∑V PDV∑w (FC)Vwδ(RD + V + i - [RA + w + f])

(9)

G ˆ bridge(Etun) )

In eq 9 i and f are the energies of the |i〉 and |f〉 oscillator states in eq 5,

i ) (n + /2)pωU + (m + /2)pωL 1

1

f ) (q + /2)pωU + (r + /2)pωL 1

1

(10)

The sums of eq 9 are over all initial vibrational states V (energy V) of accepting modes that couple to the donor and over all final vibrational states w (energy w) of accepting modes that couple to the acceptor. (FC)Vw is the Franck-Condon factor between these states and PDV is the initial population at thermal equilibrium of state V when the electron is on the donor (the thermal probability for the total initial state of the system, including both bridge and accepting modes (i.e., |φD;i;V〉), is PDVi ) PDVPi). We assume that the accepting modes are of sufficiently low frequency so that they may be treated classically (KBT > V+1 - V, w+1 - w). In contrast, the inducing modes (U and L) are higher frequency (KBT < i+1 - i) and thus quantum mechanical. In the classical (high-temperature) regime with respect to the V and w states, we recover the well-known multichannel generalization of the Marcus formula for the FCweighted density of states:1a,9a

Fi,f )

1

x4πλKbT

where γL (γU) is the vibronic coupling parameter (see eq 1). The bridge eigenstates are products of the bridge electronic states and shifted or unshifted vibrational states. When the electron occupies one of the bridge sites, the equilibrium position of the oscillator coupled to that site is shifted (i.e., |n(yU)〉 f |n(yU-y0U)〉 for |φU〉 and |m(yL)〉 f |m(yL-y0L)〉 for |φL〉). The energy eigenvalue is lowered by -γ2/pω because of the linear vibronic interaction.11a In the eigenstate representation, the bridge Green’s function is a sum over states:

exp[-(RD + i - [RA + f] - λ)2/4λKBT] (11)

∑ n,m +

|φU;n(yU-y0U) m(yL)〉 〈φU;n(yU-y0U) m(yL)| Etun - (n + 1/2)pωU - (m + 1/2)pωL - (RU - γ2U/pωU)

∑ n,m

|φL;n(yU) m(yL-y0L)〉 〈φL;n(yU) m(yL-y0L)| Etun - (n + 1/2)pωU - (m + 1/2)pωL - (RL - γL2/pωL) (14)

where Etun is defined as the energy of the system at initial statefinal state degeneracy (eq 12). The Green’s function in eq 14 is nonperturbative in the bridge vibronic interaction. The corresponding tunneling matrix element, T ifDA, obtained from eqs 6 and 14 is also nonperturbative in the bridge vibronic interaction. The above analysis, and the tunneling limit more generally, is valid when the energy gap between the tunneling energy (eq 12) and the bridge eigenvalues (eq 13) is greater than tDU, tDL, and KBT. Below we compare the cases of elastic and inelastic tunneling. It is assumed that the bridge oscillators are high frequency so that they are in their ground vibrational states when the electron is on the donor. It is further assumed that the energy of the acceptor state is such that RD ) RA + ppωU (p g 0). Then, in the rate equations (eqs 7 and 8),

|i〉 ) |0(yU) 0(yL)〉 |f〉 ) |p(yU) 0(yL)〉

(15)

15514 J. Phys. Chem. B, Vol. 108, No. 40, 2004

Skourtis et al.

Pi ) P00 ) 1 and Etun ) RD + i ) RA + f, where i ) pωU/2 + pωL/2 and f ) (p + 1/2)pωU + pωL/2. Elastic Tunneling. For elastic tunneling with p ) 0 (RD ) RA), the coupling matrix element is T 00,00 DA ) 〈φD;0(yU) 0(yL)|Tˆ DA|φA;0(yU) 0(yL)〉 and is given by (eqs 6 and 14)

T 00,00 DA )

∑n

tDUtUA

+ tDLtLA

〈0(yU)|n(yU-y0U)〉〈n(yU-y0U)|0(yU)〉 Etun - (n + 1/2)pωU - (pωL/2) - (RU - γU2/pωU) 〈0(yL)|m(yL - y0L)〉〈m(yL - y0L)|0(yL)〉

∑ m

Etun - (m + 1/2)pωL - (pωU/2) - (RL - γL2/pωL)

sign), the inelastic coupling mechanism allows the electrontransfer process to proceed. Perturbative Bridge Green’s Function. We now repeat the above calculations using an approximate bridge Green’s function that is perturbative in the bridge vibronic interaction. This analysis highlights the origin of the (non-Condon) pathway dephasing from a perspective linked to the well-known McConnell expression for the tunneling matrix element.1a The zero-order bridge Green’s function is defined as

G ˆ bridge (Etun) ) 0

∑ n,m

(16) If, in eq 16, we set Etun ) RD(A) + pωU/2 + pωL/2 and use the well-known expression for the Franck-Condon factor between shifted oscillator states,11a then

T 00,00 DA ) tDUtUA

e-SUSnU/n!

∑n

∑ m

∑ n,m

Etun - [RL + (n + 1/2)pωU + (m + 1/2)pωL]

RD(A) - [RL - γL /pωL + mpωL]

(17)

where S ) γ2/(pω)2. Inelastic Tunneling. Consider a channel that deposits p > 0 quanta into the upper oscillator. The tunneling matrix element, from eqs 6 and 14, is

and the vibronic interaction on the bridge (the perturbation) is

Applying perturbation theory in the vibronic interaction gives, to first order:

G ˆ bridge ) G ˆ bridge +G ˆ bridge Vˆ vibr G ˆ bridge + ... 0 0 0

tDUtUA

∑n

RA + ppωU - [RU - γU2/pωU + npωU]

T 00,00 DA ≈ + 0 (18)

In eq 18, RA + ppωU could also be replaced by RD, as required by energy conservation. The appearance of the zero on the right-hand side of eq 18 arises from the orthogonality relation 〈0(yU)|n(yU)〉〈n(yU)|p(yU)〉 ) 0, for p * 0. This result means that the L pathway does not contribute (as in eq 17), and depositing the Vibration in the upper pathway erases the quantum interference with the lower path. This elimination of interference is analogous to the double-slit transmission experiment when the electron transmission through one of the slits is monitored,1b,c thus destroying the interference pattern. Transitions that leave vibrational energy on the lower pathway (and none on the upper path) similarly show effective donor-acceptor interactions derived from a single pathway. For this model Hamiltonian, which assumes local vibronic-coupling interactions, the coupling elements associated with leaving vibrational energy on both pathways are identically zero (see eq 14). In the discussion above, we showed that inelastic tunneling can eliminate specific pathways from contributing to the donoracceptor interaction. Therefore inelastic tunneling can (1) “remove” destructively interfering pathways arising from orbitalsymmetry constraints (e.g., see ref 12) and (2) eliminate “accidental” destructive interference among pathways in lower symmetry species.8 We show below that when the elastic donor-acceptor interaction is symmetry-forbidden (e.g., nearestneighbor coupling interactions of either the donor or the acceptor, but not both, with U and L orbitals are of opposite

(21)

The donor-acceptor matrix element for elastic tunneling arises from the zeroth-order term in eq 21:

T 00,p0 ˆ DA|φA;p(yU) 0(yL)〉 ) DA ) 〈φD;0(yU) 0(yL)|T 〈0(yU)|n(yU-y0U)〉〈n(yU-y0U)|p(yU)〉

(19)

γLyLx2mωL/p|φL〉〈φL| (20)

e-SLSmL /m! 2

+

Etun - [RU + (n + 1/2)pωU + (m + 1/2)pωL] |φL;n(yU) m(yL)〉〈φLn(yU) m(yL)|

ˆ e/n ) γUyUx2mωU/p|φU〉〈φU| + Vˆ vibr ) H

+

RD(A) - [RU - γU2/pωU + npωU]

tDLtLA

|φU;n(yU) m(yL)〉〈φUn(yU) m(yL)|

[

tDLtLA tDUtUA + RD - RU RA - RL

]

(22)

This is the familiar McConnell-like coupling expression (RD ) RA as a result of energy conservation) for two parallel electronic pathways. Here, the RD and RA terms define the “tunneling energy” for the ET process.4 The inelastic transition element that deposits one vibration on the upper pathway is given by the first-order term in eq 21:

T 00,10 DA ≈

[

]

tDUγUtUA

) (RD - RU)(RD - [RU + pωU]) tDUγUtUA

[

(RD - RU)(RA - RU)

]

(23)

where we have used the fact that RD ) RA + pωU to derive the final expression. Similarly, for the lower pathway, we have

T 00,01 DA ≈

[

tDLγLtLA

]

(RD - RL)(RD - [RL + pωL])

[

)

tDLγLtLA

(RD - RL)(RA - RL)

]

(24)

Equations 23 and 24 are the McConnell-like couplings generalized for the inelastic pathways. As before, we find that the inelastic interaction erases the “elastic” pathway’s contribution to the coupling. The model used above is similar to that of Troisi, Ratner, and Nitzan,9e but differs in two ways: (1) the vibronic interaction is coupled to a localized bridge site, rather than to a hopping matrix element, and (2) the vibronic interaction acts locally in two separate regions of the bridge.

Pathway Coherence

J. Phys. Chem. B, Vol. 108, No. 40, 2004 15515

Magnitude of the Effect. This analysis indicates that the relative strength of the inelastic channels compared to the elastic channels is γU(L)/(RD(A) - RU(L)). Tunneling barriers in proteins and saturated organic-bridged molecules are of the order of electronvolts. Vibronically coupled modes contribute to an inner sphere reorganization energy λin, and this connection can be used to estimate the size of a vibronic parameter γ. γ is related to the inner sphere reorganization energy by λin ) γ2/pω.11a For high-frequency modes, pω is of the order 0.1 eV (RD RL). Because inner sphere reorganization energies are typically of the order 0.1 eV, γ is also of the order 0.1 eV. As such, γ/(RD - RL) is of the order 0.1; we expect inelastic rates to be about 10-100 times weaker than elastic rates except when interferences disfavor the elastic process. Elastic and Inelastic Contributions to the ElectronTransfer Rate. Both elastic and inelastic tunneling channels contribute to the overall electron-transfer rate (eqs 7 and 8). Let us describe the rate in terms of elastic and inelastic rate contributions assuming that the bridge oscillators are initially in their ground states. Further, we assume that all other (nonbridge) vibrational degrees of freedom are classical. We can treat the three electron-transfer reactive outcomes as distinct “product states” (D+A- with no excited bridge vibrations, D+Awith vibrational excitation on the upper arm, D+A- with vibrational excitation on the lower arm). Using eqs 7 and 8 with Pi ) P00 ) 1 we have

kET ) kDfVirt(U+L)fA +

p p′ + ∑kDfVirt(L)fA ∑p kDfVirt(U)fA p′

(25)

00,p0 where kDfVirt(U+L)fA ) k00,00 DA , kDfVirt(U)fA ) kDA , and kDfVirt(L)fA ) k00,0p′ DA . The first term in eq 25 arises from elastic tunneling and the last two terms from inelastic tunneling, where p and p′ are the number of vibrational quanta deposited in the bridge. The elastic term in eq 25 is

kDfVirt(U+L)fA ) 2π |〈φD;0(yU) 0(yL)|Tˆ DA|φA;0(yU) 0(yL)〉|2F00,00 (26) p and the inelastic terms are p ) kDfVirt(U)fA 2π |〈φD;0(yU) 0(yL)|Tˆ DA|φA;p(yU) 0(yL)〉|2F00,p0 (27) p

and p′ ) kDfVirt(L)fA 2π |〈φD;0(yU) 0(yL)|Tˆ DA|φA;0(yU) p′(yL)〉|2F00,0p' (28) p

In the equations above, the Franck-Condon weighted densities of states are

F00,00 ) F00,p0 )

and

1

x4πλkbT

1

x4πλkbT

exp[-(RD - RA - λ)2/4λKbT] (29)

exp[-(RD - [RA + ppωU] - λ)2/4λKbT] (30)

F00,0p′ )

1

x4πλKbT

exp[-(RD - [RA + p'pωL] - λ)2/4λKbT] (31)

When the vibronic interaction is retained to lowest order, the only allowed values for p and p′ are (1. Erasing Destructive Pathway Interference with Inelastic Coupling to Local Bridge Modes. The above relations indicate that rate studies as a function of reaction free energy could assist in unmasking inelastic electron-transfer pathway effects.9,11 A more profound experimental signature is evident, however, because of the qualitatively different pathway amplitude contributions to the effective donor-acceptor coupling in the elastic and inelastic regimes.9e Electron transfer in an orbitally forbidden donor-bridge-acceptor system with a small tunnelingenergy gap should be promoted by inelastic tunneling. This approach requires that the reaction free energy exceed the vibrational energy spacing of the bridge oscillators so that the inelastic channels are accessible. Reaction free energies significantly larger than high-frequency modes in organic molecules, typically ∼0.1 eV, are readily accessible. Figure 2 illustrates this concept. Using symmetry-forbidden electron transfer, with and without vibronically coupled highfrequency bridge modes, may allow probing of the inelastic channel. The DBA molecule shown on the left side of the figure was studied by Zeng and Zimmt12 and was shown to have a weak electronic coupling because of the relative electronic symmetries of the ET active electron donor and electron acceptor groups. (The donor’s electronic orbital symmetry is a′′ and the acceptor’s orbital symmetry is a′ for this molecule with Cs pointgroup symmetry). The analogous molecule on the right possesses mirror plane symmetry but also has CdN groups that may be vibronically coupled to the tunneling electron. If the electronnuclear coupling is sufficiently large, the symmetry constraint should be relaxed and the electron-transfer rate will be enhanced. Tight-binding Hamiltonian parameters take on the sign of the overlap between neighboring orbitals. As such, introduction of a nodal plane in the donor-localized orbital (e.g., a simple p-orbital) introduces a different sign for the t elements associated with L and U pathways. In the framework of our simple Hamiltonian (eq 1), orbitally forbidden donor-acceptor interactions are modeled by setting tDU ) -tDL and tUA ) tLA, which gives zero elastic coupling (see eqs 17 and 22). To estimate the size of the elastic and inelastic tunneling matrix elements in molecules such as those in Figure 2, we parametrize our four-orbital tight-binding model accordingly. For tunneling across about seven bonds from donor to acceptor, simple pathway analysis4a suggests a wave function decay of about (0.6)7 ) 0.028. Because our tight binding model has only two superexchange “steps,” we use a decay constant per step of 0.16 to reproduce this decay. Assuming an energy difference of 1 eV between donor (acceptor) and bridge orbitals, the McConnell approximation indicates that hopping elements in the Hamiltonian of about 0.16 eV can generate the appropriate decay. The inner sphere reorganization energy (associated with reducing the bridge local mode), the vibronic parameter, and the vibrational energy quantum energy are related by λ ) γ2/ pω. We choose tDU ) tUA ) tDL ) 0.2 eV, tLA ) (0.2 eV, pωL ) pωU ) 0.1 eV, and RU(L) - RD ) 1 eV. Figure 3 shows the elastic and inelastic tunneling matrix elements as a function of γ for the two sign choices of tLA. The negative sign for tLA (symmetry-forbidden electron transfer) causes the elastic tunneling matrix element to be identically zero (destructively

15516 J. Phys. Chem. B, Vol. 108, No. 40, 2004

Skourtis et al.

Figure 2. (Left) DBA structure that is anticipated to have a weak electronic coupling from symmetry constraints on the donor and acceptor electronic states. (Right) analogous system that has CdN groups that may couple to the electron tunneling. The orbitally forbidden interaction can be modeled in the Hamiltonian of eq 1 by setting tDU ) -tDL.

Figure 3. Inelastic 00 f 10 (solid lines) and elastic 00 f 00 (dotted lines) tunneling matrix elements vs the vibronic coupling parameter (γU,L) for symmetry-forbidden (top) and for symmetry-allowed (bottom) tunneling. The parameters in (top) are tDU ) tUA ) tDL ) 0.2 eV, tLA ) -0.2 eV, pωU ) pωL ) 0.1 eV, and γL ) γR. (Bottom) same as in (top), except tLA ) 0.2 eV. The weak increase in the magnitude of the elastic coupling (bottom) as a function of γU,L arises from the lowering of the tunneling barrier associated with the bridge orbital energy term -γU,L2/pω.

interfering paths, Figure 3 (top)), whereas the positive sign (symmetry-allowed electron transfer) causes the two coupling pathway contributions to add (constructive interference, Figure 3 (bottom)). Discussion We have shown that pathway-mediated electronic coupling in molecules can be dephased by inelastic events, just as electronic propagation through the arms of an Aharanov-Bohm

interferometer can be dephased by interactions localized on one arm. In solid-state devices, dephasing of electron transmission may be controlled with inelastic scattering, applied magnetic fields, electron-electron scattering, and coupling to external structures (e.g., wires and dots).1-3 Understanding how to control and switch the coherence among parallel electronic coupling pathways in molecules may assist the design and assembly of future nanoscale electronic devices. When is pathway dephasing likely to be of consequence in molecular electron-transfer reactions? The simplest case seems to be that of orbital-symmetry-forbidden electron transfer. The inelastic mechanism provides a channel for “vibronically allowed” processes that would be forbidden on the basis of pure electronic symmetry arguments.16 A number of cases have been described where donoracceptor couplings around equilibrium are weak, and geometric fluctuations are thought to be required to access geometries with coupling elements of a sufficient magnitude to support electron transfer. Systems of this kind pose another opportunity to examine inelastic coupling, because removal of an interfering pathway contribution may arise from inelastic tunneling. Examples include DBA model systems,8e,17-20 DNA electrontransfer systems in some geometries and energy regimes,8c,21 synthetically modified proteins in some geometries,8d and the bacterial photosynthetic reaction center primary to secondary quinone electron-transfer reaction.8b More generally, the sensitivity of donor-acceptor couplings and electron-transfer reorganization energies to molecular conformations and geometries is becoming apparent from current studies.17-22 In most prior studies of destructively interfering pathways, the theoretical analysis used classical molecular-dynamics simulations coupled to quantum chemical calculations of bridgemediated donor-acceptor interactions (the donor-acceptor interactions fluctuate along the molecular dynamics trajectory). This hybrid approach assumes classical nuclear motion and it is valid for a wide range of tunneling matrix element magnitudes and fluctuation time scales if the matrix element is computed nonperturbatively7a,b (so that non-Condon effects are well described). Further, the approach can be improved by adding effects of Born-Oppenheimer breakdown7c induced by bridge motion. Noncondon effects, such as inelastic tunneling, are expected to be of increasing significance as the tunneling traversal time approaches the vibrational time scale of the inelastic bridge modes.10 Indeed, as donor-acceptor energies are brought more nearly into energetic resonance with bridge states, the tunneling traversal time will increase and the probablility of inelastic tunneling will rise as well. The assumption of fully localized modes to describe the vibronic interaction is a simplification. Analysis of inelastic tunneling problem in a normal mode basis, going beyond the assumption of vibronically isolated modes, is needed to quantify further

Pathway Coherence the structure dependence of the effects illustrated above. In this regard, the normal-mode analysis of inelastic tunneling performed by Troisi, Ratner, and Nitzan may prove helpful.9e It will also be useful to explore inelastic tunneling effects beyond the perturbative regime used in our study. Using classical dynamics to describe the nuclear motion, it is conceptually straightforward (although computationally demanding) to examine the influence of (classical) bridge conformational fluctuations on pathway interferences, and the effects of pathway dephasing on the electron-transfer rate. Dephasing effects are reflected in the correlation function 〈TDA(t) TDA(0)〉. A recent study examined fluctuations of the tunneling matrix element for solvent-mediated electron transfer, and showed that the time decay of 〈TDA(t) TDA(0)〉 can be as fast as 0.1 ps.20 In general, the decay of 〈TDA(t) TDA(0)〉 must be compared with the decay of the ensemble-averaged timedependent overlap between the nuclear wave packets of the promoter modes propagated on the different (donor/acceptor) electronic surfaces. Prezhdo and Rossky6e have shown that the electron-transfer rate can be approximated by an integral over time of the product of 〈TDA(t) TDA(0)〉 with this averaged overlap (semiclassical Franck-Condon factor), whose decay with time arises from the time-dependent solvation dynamics at the crossing point between donor and acceptor electronic surfaces. If the nuclear factor decays more rapidly than does 〈TDA(t) TDA(0)〉, the nuclear factor6f determines the rate (and dephasing from Franck-Condon transitions is more important than dephasing from tunneling matrix element fluctuations). That is, the rate in this regime is proportional to 〈|TDA(0)|2〉 and TDA time correlations do not enter the electron-transfer rate. This may be the case for azurin ET, where semiclassical calculations show that dephasing from solvent/protein matrix relaxation could be as fast as 2.4 fs.6f One limitation of all calculations that compute 〈TDA(t) TDA(0)〉 based on bridge conformational sampling from classical molecular dynamics is that the results will not include a mechanism for energy exchange between the tunneling electron and the bridge vibrations. Methods of this kind cannot describe inelastic tunneling; the computed decay of 〈TDA(t) TDA(0)〉 based on classical molecular dynamics simulations is that associated with ensemble averaging over dynamically fluctuating pathway phases. Troisi, Nitzan, and Ratner described a systematic approach to obtain the electrontransfer rate when the tunneling electron couples to (quantum) bridge vibrational modes11g (generalizing an earlier study of Stuchebrukhov9a). The formal difference between classical and quantum views of nuclear dynamics is that the donor-acceptor tunneling matrix element becomes an operator in the space of the vibrational states of the bridge oscillators, giving rise to inelastic rate channels that label the pathway traversed by the tunneling electron. They also computed inelastic contributions to electronic currents for relatively small molecules between electrodes where the vibrational modes are largely delocalized on the molecule.9e Their inelastic tunneling analysis showed that the vibrational modes can enhance orbitally forbidden electron transfer. Our model focuses on dephasing between interfering pathways when different localized quantum vibrations are coupled to each pathway. This inelastically induced pathway dephasing mechanism may, similarly, enhance electron transfer that is orbitally forbidden in the elastic regime. Further, the present model illustrates how, within the general Golden Rule framework, loss of interference between competing pathways can influence the electron-transfer rate. Recent scanning-tunneling microscope experiments illustrate the richness of mechanism that can be accessed by introducing

J. Phys. Chem. B, Vol. 108, No. 40, 2004 15517 inelastic bridge channels. These STM studies involve small molecules, such as carbon monoxide and acetylene, on Cu, Ni, and Pd surfaces. Experiments identified several intriguing characteristics of inelastic tunneling in the STM apparatus: (1) inelastic tunneling via specific high-frequency molecular normal modes (and not others) can be observed and understood using symmetry-based selection rules,13a,b (2) molecular vibration mediated negative-differential resistance has been observed to arise from inelastic tunneling,13d and (3) energy deposited on molecular adsorbates during inelastic tunneling has been observed to drive lateral molecular hopping on surfaces.13c The corresponding challenge now exists for the design and synthesis of molecular electron-transfer structures that exploit the control that may be enabled by inelastic tunneling. That is, we are challenged to construct molecular “which way” dephasing structures that dephase tunneling pathways by leaving inelastic markers in the molecular electron-tunneling bridge. Inelastic tunneling effects may be particularly important in larger chemical bridged donor-acceptor structures. Inelastic coupling pathways effectively lower the bridge symmetry to make symmetry-forbidden transitions allowed (as discussed in section 3). A second important case will arise when “accidental” destructive interference (small pathway coherence parameter8b) slows the rate. This may be the case that arises in DNA and some protein electron-transfer reactions. Inelastic coupling may be particularly important in DNA electron transfer (see refs 14 and 15, and references therein) because the smaller tunneling energy gap is expected to increase the inelastic tunneling cross section. As such, inelastic tunneling may contribute to some of the richness of mechanism yet to be fully understood in DNA electron-transfer reaction.14,15 Acknowledgment. This project is supported by the Keck Center for Nanoscale Electronics and a grant from the NEDO Foundation (Japan). D.N.B. acknowledges a J. S. Guggenheim Foundation Fellowship, a Hirschmann Fellowship (University of Pennsylvania), and a Ronneberg Fellowship (University of Chicago) during which this research was initiated. D.N.B. and D.H.W. thank the NSF for support through the NER program and the US-Israel Binational Science Foundation. S.S.S. thanks the University of Cyprus Grant GPHDNA. We also thank M. Galperin, R. Naaman, M. A. Ratner, and M. B. Zimmt for helpful discussions. References and Notes (1) (a) Jortner, J.; Ratner, M. A., Eds. Molecular Electronics; Blackwell Science: Oxford, U.K., 1997. (b) Greenstein, G.; Zajonc, A. G. The Quantum Challenge; Jones and Bartlett Publishers: Boston, 1997. (c) Feynman, R. P.; Leighton, R. B.; Sands, M. The Feynman Lectures on Physics; Addison-Wesley: New York, 1964; Vol. 3, Chapter 1. (d) Tegmark, M.; Wheeler, J. A. Sci. Am. February, 2001. (2) (a) Datta, S. Electronic Transport in Mesoscopic Systems; Cambridge Press: New York, 1997. (b) Imry, Y. Mesoscopic Physics; Oxford Press: New York, 1997. (c) Takagi, S. Macroscopic Quantum Tunneling; Cambridge Press: New York, 2002. (d) Schuster, R.; Buks, E.; Heiblum, M.; Mahalu, D.; Umansky, V.; Shtrikman, H. Nature 1997, 385, 417-420. (3) (a) Buks, E.; Schuster, R.; Heiblum, M.; Mahalu, D.; Umansky, V. Nature 1998, 391, 871-874. (b) Haule, K.; Bonca, J. Phys. ReV. B 1999, 59, 13087-13093. (c) Aleiner, I. L.; Wingreen, N. S.; Meir, Y. Phys. ReV. Lett. 1997, 79, 3740-3743. (4) For reviews: (a) Skourtis, S. S.; Beratan, D. N. AdV. Chem. Phys. 1999, 106, 377-452. (b) M. D. Newton AdV. Chem. Phys. 1999, 106, 303375. (c) Regan, J. J.; Onuchic, J. N. AdV. Chem. Phys. 1999, 107, 497553. (d) V. Balzani, Ed. Electron Transfer in Chemistry; Wiley: Weinheim, Germany, 2001; Vols. 1-5. (5) (a) Larsson, S. J. Am. Chem. Soc. 1981, 103, 4034-4040. (b) Beratan, D. N.; Hopfield, J. J. J. Am. Chem. Soc. 1984, 106, 1584-1894. (c) Onuchic, J. N.; Beratan, D. N. J. Am. Chem. Soc. 1987, 109, 67716778. (d) Liang, C. X.; Newton, M. D. J. Phys. Chem. 1992, 96, 28552866. (e) Shephard, M. J.; Paddon-Row, M. N.; Jordan, K. D. J. Am. Chem.

15518 J. Phys. Chem. B, Vol. 108, No. 40, 2004 Soc. 1994, 116, 5328-5333. (f) Paulson, B. P.; Curtiss, L. A.; Bal, B.; Closs, G. L.; Miller, J. R. J. Am. Chem. Soc. 1996, 118, 378-387. (g) Napper, A. M.; Read, I.; Waldeck, D. H.; Head, N. J.; Oliver, A. M.; Paddon-Row, M. N. J. Am. Chem. Soc. 2000, 122, 5220-5221. (h) May, V.; Ku¨hn, O. Charge and Energy Transfer Dynamics in Molecular Systems, A Theoretical Introduction; Wiley-VCH: Weinheim, 1999. (6) (a) Wynne, K.; Hochstrasser, R. M. AdV. Chem. Phys. 1999, 107, 263-309. (b) Ratner, M. Nature 1999, 397, 480-481. (c) Davis, W. B.; Wasielewski, M. R.; Ratner, M. A. J. Phys. Chem. B 2000, 104, 38173829. (d) Nitzan, A. Annu. ReV. Phys. Chem. 2001, 52, 681-750. (e) Prezhdo O. V.; Rossky, P. J. J. Chem. Phys. 1997, 107 5863-5878. (f) Lockwood, D. M.; Cheng, Y.-K.; Rossky, P. J. Chem. Phys. Lett. 2001, 345, 159-165. (g) Bixon, M.; Jortner, J. Chem. Phys. 2002, 281, 393408. (h) Marquardt, F.; Bruder, C., Phys. ReV. B 2003, 68, 195305(1)195305(17). (7) (a) Xie, Q.; Archontis, G.; Skourtis, S. S. Chem. Phys. Lett. 1999, 312, 237-246. (b) Skourtis, S. S.; Archontis, G.; Xie, Q, J. Chem. Phys. 2001, 115, 9444-9462. (c) Skourtis, S. S. Chem. Phys. Lett. 2003, 372, 224-231. (8) (a) Wolfgang, J.; Risser, S. M.; Priyadarshy, S.; Beratan, D. N. J. Phys. Chem. B 1997, 101, 2986-2991. (b) Balabin, I. A.; Onuchic, J. N. Science 2000, 290, 114-117. (c) Tong, G. S. M.; Kurnikov, I. V.; Beratan, D. N. J. Phys. Chem. B 2002, 106, 2381-2392. (d) Kawatsu, T.; Kakitani, T.; Yamato, T. J. Phys. Chem. B 2002, 106, 11356-11366. (e) Zimmt, M. B.; Waldeck, D. H. J. Phys. Chem. A 2003, 107, 3580-3597. (9) (a) Medvedev, E. S.; Stuchebrukhov, A. A. J. Chem. Phys. 1997, 107, 3821-3831. (b) Wingreen, N. S.; Jacobsen, K. W.; Wilkins, J. W. Phys. ReV. B 1989, 40, 11834-11850. (c) Haule, K.; Moncˇa, J., Phys. ReV. B 1999, 59, 13087-13093. (d) Galperin, M.; Nitzan, A. J. Chem. Phys. 2001, 115, 2681-2694. (e) Troisi, A.; Ratner, M. A.; Nitzan, A. J. Chem. Phys. 2003, 118, 6072-6082. (f) Wolfram, T., Ed. Inelastic Electron Tunneling Spectroscopy; Springer-Verlag: New York, 1978. (10) (a) Nitzan, A.; Jortner, J.; Wilkie, J.; Burin, A. L.; Ratner, M. A.; Jortner, J. J. Phys. Chem. B 2000, 104, 5661-5665. (b) Peskin, U.; Galperin, M.; Nitzan, A. J. Phys. Chem. B 2002, 106, 8306-8312. (11) (a) Onuchic, J. N.; Beratan, D. N.; Hopfield, J. J. J. Phys. Chem. 1986, 90, 3707-3721. (b) Onuchic, J. N.; da Gama, A. A. S. Theor. Chim.

Skourtis et al. Acta 1986, 69, 89-100. (c) Beratan, D. N.; Onuchic, J. N.; Hopfield, J. J. J. Chem. Phys. 1987, 86, 4488-4498. (d) Stuchebrukhov, A. A. Proc. Natl. Acad. Sci. U.S.A. 1997, 94, 3703-3708. (e) Olson, M.; Mao, Y.; Windus, T.; Kemp, M.; Ratner, M.; Leon, N.; Mujica, V. J. Phys. Chem. B 1998, 102, 941-947. (f) Tang, J. J. Chem. Phys. 1993, 98, 6263-6266. (g) Troisi, A.; Nitzan, A.; Ratner, M. A. J. Chem. Phys. 2003, 119, 5782-5788. (12) Zeng, Y.; Zimmt, M. B. J. Am. Chem. Soc. 1991, 113, 51075109. (13) (a) Gaudioso, J.; Lauhon, L. J.; Ho, W. Phys. ReV. Lett. 2000, 85, 1918-1921. (b) Lorente, N.; Persson, M. Phys. ReV. Lett. 2000, 85, 29973000. (c) Komeda, T.; Kim, Y.; Kawai, M.; Persson, B. N. J.; Ueba, H. Science 2002, 295, 2055-2058. (d) Lorente, N.; Persson, M.; Lauhon, L. J.; Ho, W. Phys. ReV. Lett. 2001, 86, 2593-2596. (e) Wang, W. Y.; Lee, T.; Reed, M. A. Physica E 2003, 19, 117-125. (14) Lewis, F. D.; Liu, J. Q.; Weigel, W.; Rettig, W.; Kurnikov, I. V.; Beratan, D. N. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 12536-12541. (15) For recent reviews of this active field see: (a) Schuster, G. B., Ed. Topics in Current Chemistry: Long-Range Charge Transfer in DNA; Springer: Berlin, 2004. (b) Enders, R. G.; Cox, D. L.; Singh, R. P. ReV. Mod. Phys. 2004, 76, 195-214. (16) (a) Reimers J. R.; Hush, N. S.; Sammeth, D. M.; Callis P. R. Chem. Phys. Lett. 1990, 169, 622-626. (b) Reimers J. R.; Hush, N. S. Chem. Phys. 1990, 146, 105-114. (17) Ungar, L. W.; Newton, M. D.; Voth, G. A. J. Phys. Chem. B 1999, 103, 7367-7382. (18) Yip, W. T.; Levy, D. H.; Kobetic, R.; Piotrowiak, P. J. Phys. Chem. A 1999, 103, 10-20. (19) Paddon-Row, M. N.; Jones G. A.; Paddon-Row, M. N.; Carpenter, B. K.; Piotrowiak, P. J. Phys. Chem. A 2002, 106, 5011-5021. (20) Troisi, A.; Ratner, M. A.; Zimmt, M. B. J. Am. Chem. Soc. 2004, 126, 2215-2224. (21) Troisi, A.; Orlandi, G. J. Phys. Chem. B 2002, 106, 2093-2101. (22) (a) Shin, Y.-G. K.; Newton, M. D.; Isied, S. S., J. Am. Chem. Soc. 2003, 125, 3722-3732. (b) Siriwong, K.; Voityuk, A. A.; Newton, M. D.; Rosch, N. J. Phys. Chem. B 2003, 107, 2595-2601.