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Electron transfer through time-dependent bridges: tunneling by virtual transitions that break the Born–Oppenheimer approximation
Chemical Physics Letters 372 (2003) 224–231 www.elsevier.com/locate/cplett
Electron transfer through time-dependent bridges: tunneling by virtual transitions that break the Born–Oppenheimer approximation Spiros S. Skourtis
*
Department of Physics, University of Cyprus, Nicosia 1678, Cyprus Received 17 December 2002; in final form 29 January 2003
Abstract For bridge-mediated electron transfer through time-dependent bridges it is shown that the superexchange mechanism may not be the only relevant tunneling mechanism. Electronic tunneling from donor to acceptor could also be mediated by a type of breakdown of the Born–Oppenheimer adiabatic approximation. A time-dependent effective Hamiltonian is derived for this process. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction Long distance, bridge-mediated electron transfer (ET) is an important research area of chemical physics, biophysics and nanotechnology [1–5]. In recent years much of the research in this field focused on the effects of time-dependent external fields and of nuclear motion on the distance and energy gap dependence of ET reactions (e.g. [6–12] and for reviews [13–15]). Different regimes were identified that range between the limits of coherent tunneling (exponential distance decay) and incoherent thermal hopping (weak distance decay), ([16] for a recent review). In the tunneling limit the
*
Fax: +357-22-339060. E-mail addresses: [email protected], net.com.cy
cabrskou@cyta-
relevant ET mechanism is superexchange. The subject of this Letter is the influence of a timedependent-bridge energy spectrum on superexchange ET. We consider effects related to the breakdown of the Born–Oppenheimer (BO) adiabatic approximation [17]. An effective time-dependent Hamiltonian (EH) which incorporates perturbatively the effects of this breakdown is derived. It is shown that the donor–acceptor matrix element of this Hamiltonian can create an additional coupling between the states that differs from the superexchange coupling. In the following we use a generalized time dependent definition of the superexchange mechanism that applies to nonadiabatic and adiabatic ET [18]. The type of BO breakdown discussed below can be interpreted as a correction to this superexchange mechanism which incorporates additional effects of the bridge time dependence on
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00370-1
S.S. Skourtis / Chemical Physics Letters 372 (2003) 224–231
the ET probability. The derived EH (and corresponding Schr€ odinger equation) is defined in the space of donor and acceptor states. It is chosen to produce an ET probability that approximates the exact donor to acceptor probability, computed from the N-state Hamiltonian and Schr€ odinger equation in the space of donor, acceptor and bridge states. In this sense the reduced Hamiltonian is an accurate description of the transfer dynamics. The final aim of this approach it to construct, with the aid of electronic structure and molecular dynamics (MD) simulations, ÔrealisticÕ reduced Hamiltonian models for ET in time-dependent molecular systems [19]. Such Hamiltonians could be compared to the different two-state models that address the influence of Condon, BO breakdown and inelastic processes on tunneling [6–15], thus leading to improved time-dependent two-state approximations. The proposed approach is useful for the analysis of tunneling influenced by classical driving fields (e.g. electric fields or classical nuclear motions). Although this is a limitation compared to vibronic Hamiltonian methods that treat quantum nuclear motions, it can be applied to large systems where vibronic methods would require prohibitively large Hamiltonian matrices.
225
The Hamiltonian in the basis of localized donor, bridge and acceptor states states is given by H^ ðtÞ ¼ H^ da ðtÞ þ H^ br ðtÞ þ V^ ðtÞ;
ð1Þ
where H^ da ðtÞ ¼ j/1 iH11 ðtÞh/1 j þ j/N iHNN ðtÞh/N j þ ½j/1 iH1N ðtÞh/N j þ h:c:
ð2Þ
is the donor–acceptor part (donor: j/1 i, acceptor: j/N i), H^ br ðtÞ ¼
N 1 X
j/i iHii ðtÞh/i j
i¼2
þ
N 2 X N 1 X i¼2
½j/i iHij ðtÞh/j j þ h:c:
ð3Þ
j>i
the bridge part, and V^ ðtÞ ¼
N 1 X
½j/1 iH1i ðtÞh/i j þ j/N iHNi ðtÞh/i j
i¼2
þ h:c:
ð4Þ
is the donor (acceptor)-bridge coupling (Fig. 1a). The instantaneous system eigenstates and eigenenergies are denoted as jWm ðtÞi and Em ðtÞ, i.e., H^ ðtÞjWm ðtÞi ¼ Em ðtÞ (Fig. 1b).
Fig. 1. Energetics of the time-dependent donor–bridge–acceptor system in the j/i i basis (a) and the jWm ðtÞi basis (b). There is a sufficiently large energy gap between j/1 i=j/N i and the bridge j/i i so that only two eigenstates jW ðtÞi have large donor and acceptor components (Eq. (6)) and are energetically separated from jWm6¼ ðtÞi. All jWm ðtÞi are coupled by the elements Kmn ðtÞ and each Km ðtÞ (m 6¼ ) is assumed to be small with respect to ðEm ðtÞ E ðtÞÞ=h (Eq. (8)). Direct processes (denoted A) and indirect virtual transition processes (denoted B) couple jWþ ðtÞi to jW ðtÞi and thus j/1 i to j/N i.
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S.S. Skourtis / Chemical Physics Letters 372 (2003) 224–231
am ðtÞ ¼
2. The effective Hamiltonian description The ET probability from donor to acceptor is given by P1N ðtÞ ¼ jh/N jWðtÞij2 ;
ð5Þ
where jWðtÞi (the state of the transferring electron at time t) is the solution of the time-dependent Schr€ odinger equation i hdjWðtÞi=dt ¼ H^ ðtÞjWðtÞi with the initial condition jWð0Þi ¼ j/1 i. We consider P1N ðtÞ under two conditions. First, the donor (j/1 i) and acceptor (j/N i) states should be be offresonant and weakly coupled to the bridge states (j/i6¼1;N i) at all times (Fig. 1a). Weak coupling means that, upon diagonalization, two eigenstates jW ðtÞi are separated energetically from the remaining eigenstates and have the largest donor and acceptor components X j /1 jWm6¼ ðtÞ j; jh/1 jW ðtÞij > m6¼
jh/N jW ðtÞij >
X j /N jWm6¼ ðtÞ j:
ð6Þ
m6¼
The second condition to be imposed relates to the matrix elements that couple the jWm ðtÞi d Kmn ðtÞ ¼ Wm ðtÞj Wn ðtÞ dt D E Wm ðtÞ dtd H^ ðtÞWn ðtÞ ¼ for m 6¼ n; ð7Þ En ðtÞ Em ðtÞ (Kmm ðtÞ ¼ 0 for real jWm ðtÞi). In terms of the parameters RðtÞ ¼ fRa ðtÞg that may drive the system (e.g. external electric fields or classical nuclear motions), Kmn ðtÞ ¼ Kmn ðRðtÞÞ ¼
X a
dRa ðtÞ ; famn ðRðtÞÞ dt
where famn ðRðtÞÞ ¼ hWm ðRðtÞÞj
o jWn ðRðtÞÞi oRa
are analogous to the nonadiabatic derivative couplings used to describe BO breakdown in time independent systems [20]. The second condition requires that, for all times
hKm ðtÞ <1 Em ðtÞ E ðtÞ
for
m 6¼ :
ð8Þ
In [18,19] MD simulations and electronic structure calculations on protein bridges were used to show that the ET probability P1N ðtÞ in the regime described by Eqs. (6) and (8) can be approximated (for a wide range of parameters) by an effective two-state probability su P1N ðtÞ ¼ jh/N jWsu ðtÞij2 :
ð9Þ su
In the equation above jW ðtÞi is the solution of a two-state time-dependent Schr€ odinger equation ihdjWsu ðtÞi=dt ¼ H^ su ðtÞjWsu ðtÞi with the initial condition jWsu ð0Þi ¼ j/1 i. H^ su ðtÞ is a donor–acceptor EH given by the matrix H~ su ðtÞ
H11 ðtÞ þ T11 ðtÞ H1N ðtÞ þ T1N ðtÞ ¼ HN 1 ðtÞ þ TN 1 ðtÞ HNN ðtÞ þ TNN ðtÞ
h/1 jW ðtÞi h/1 jWþ ðtÞi ¼ h/N jW ðtÞi h/N jWþ ðtÞi
0 E ðtÞ hW ðtÞj/1 i hW ðtÞj/N i : 0 Eþ ðtÞ hWþ ðtÞj/1 i hWþ ðtÞj/N i ð10Þ H~ su ðtÞ generalizes the superexchange model for time independent bridges to the case of time dependent bridges. It is nonperturbative in the donor (acceptor)-bridge coupling V^ ðtÞ because it is computed from exact eigenenergies and eigenstate su amplitudes. The effective part of H1N ðtÞ, T1N ðtÞ, (the time-dependent superexchange donor–acceptor matrix element), reduces to the perturbative expression ^br ðE ðtÞ; tÞV^ ðtÞj/N i T1N ðtÞ ’ h/1 jV^ ðtÞG
ð11Þ
^br is the for weak V^ ðtÞ. In the equation above G bridge GreenÕs function at time t h i 1 ^br ðE; tÞ ¼ E H^ br ðtÞ : G ð12Þ In terms of the eigenstates, the ET process desu scribed by P1N ðtÞ is caused by direct transitions between jWþ ðtÞi and jW ðtÞi induced by Kþ ðtÞ (process A in Fig. 1b). In addition we may consider virtual transitions between jW ðtÞi, i.e., jW ðtÞi ! jWm6¼ ðtÞi ! jW ðtÞi (process B in Fig. 1b). The
S.S. Skourtis / Chemical Physics Letters 372 (2003) 224–231
latter constitute a breakdown of the adiabatic BO approximation between the manifolds {jW ðtÞi} and {jWm6¼ ðtÞi} (the term adiabatic is used to distinguish it from the crude BO approximation [17]). The question to be addressed is how these virtual transitions influence superexchange. The virtual transitions can be taken into account to lowest order in Km6¼ , to obtain a corcsu rected superexchange probability, P1N ðtÞ, given by csu P1N ðtÞ ¼ jh/N jWcsu ðtÞij
2
ð13Þ
csu
(see Appendix A). jW ðtÞi is the solution of a twostate time-dependent Schr€ odinger equation i hdjWcsu ðtÞi=dt ¼ H^ csu ðtÞjWcsu ðtÞi with jWcsu ð0Þi ¼ ^ ðtÞ where j/1 i. H^ csu ðtÞ ¼ H^ su ðtÞ þ M ~ ðtÞ M
M11 ðtÞ M1N ðtÞ ¼ MN 1 ðtÞ MNN ðtÞ
h/1 jW ðtÞi h/1 jWþ ðtÞi ¼ h/N jW ðtÞi h/N jWþ ðtÞi !
eig eig hW ðtÞj/1 i hW ðtÞj/N i M
ðtÞ M þ ðtÞ eig eig hWþ ðtÞj/1 i hWþ ðtÞj/N i Mþ
ðtÞ Mþþ ðtÞ ð14Þ with eig Mab ðtÞ ¼ h2
X Kam ðtÞKmb ðtÞ ; E ðtÞ Eb ðtÞ m6¼ n
a; b ¼ :
ð15Þ
~ ðtÞ, the EH matrix for the BO-breakdown corM rection in the basis j/1 i=j/N i is obtained from ~ eig ðtÞ, the matrix for the correction in the basis of M eig jW ðtÞi. The elements Mab ðtÞ are the lowest order terms of a perturbative expansion in the parameters am ¼ hKm =ðEm E Þ (valid for am < 1 and ~ ðtÞ can be consistent with Eq. (8)). Therefore, M ~ eig ðtÞ to improved systematically by computing M higher order in am . The donor–acceptor element M1N ðtÞ is the correction to the superexchange coupling T1N ðtÞ. M1N ðtÞ may also be written in terms of the bridge ^br by carrying out perturbation GreenÕs function G ^ theory in V ðtÞ in Eq. (14). Different expressions for M1N ðtÞ are obtained in this case, depending on which part of H^ ðtÞ changes more rapidly with time. If it is V^ ðtÞ, we get
h2 dV^ M1N ðtÞ ’ h/1 j dt 2
!"
227
^br d2 G dE2
#
! dV^ j/N i: dt ð16Þ
If it is H^ br ðtÞ ! br ^ h2 d H ^br M1N ðtÞ ’ h/1 jV^ G 2 dt " # ! ^br d2 G dH^ br ^br ^ G V j/N i: dE2 dt
ð17Þ
The difference between the BO-breakdown mechanism and the time-dependent superexchange mechanism is more transparent in this perturbative limit for V^ ðtÞ. Comparison of Eqs. (16) and (17) to Eq. (11) shows that M1N ðtÞ depends on the rate of change with respect to time of state energies and interstate couplings in contrast to T1N ðtÞ. M1N ðtÞ decays approximately exponentially with bridge length as it is a function of d2 Gbr =d2 E and Gbr but its decay generally differs from that of T1N ðtÞ.
3. Numerical examples and discussion Consider a simple nearest neighbour model for H^ ðtÞ in Eq. (1), where the intra-bridge, donorbridge and acceptor-bridge couplings are given by Hij ðtÞ ¼ di;i 1 b and H1N ðtÞ ¼ HN 1;N ðtÞ ¼ V . The donor and acceptor energies are set equal and time independent, H11 ðtÞ ¼ HNN ðtÞ ¼ 0 , and the bridge site energies are set equal and time dependent, i.e., Hii ðtÞ ¼ br ðtÞ with br ðtÞ ¼ þ d sinðxtÞ. By making donor and acceptor time independent we isolate the effects of the bridge time dependence. For the above parameters we compute four difsu ferent ET probabilities: (i) P1N ðtÞ, (ii) P1N ðtÞ, (iii) csu sta P1N ðtÞ, and (iv) P1N ðtÞ, where this latter probability refers to the case of a time independent ÔstaticÕ sta bridge (br ðtÞ ¼ ). P1N ðtÞ is first compared to P1N ðtÞ to identify the effects of the bridge time dependence on the electron transfer probability. The su csu ability of P1N ðtÞ and P1N ðtÞ to capture these effects is probed by comparing the two approximate probabilities to P1N ðtÞ. The comparisons are made as a function of bridge driving frequency x and bridge length Nbr ¼ N 2. In Figs. 2–5
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Fig. 2. P1N ðtÞ for a 12-site time-dependent bridge as a function sta of bridge driving frequency versus P1N ðtÞ for a time independent bridge with the same energy parameters. The bridge frequencies are: (a) x ¼ 0:00025 fs 1 , (b) x ¼ 0:0025 fs 1 , (c) x ¼ 0:025 fs 1 , (d) x ¼ 0:25 fs 1 and (e) x ¼ 2:5 fs 1 .
0 ¼ 10 eV, b ¼ 4:5 eV and V ¼ 1:4 eV [21]. For the time dependence of the bridge site energies, d ¼ 1 eV and x ¼ 2:5 10 5 2:5 fs 1 . These bridge oscillation parameters are not supposed to approximate a particular physical situation but rather they are used to demonstrate general trends within the simplest possible model of a few bridge states. The only limitation imposed on the parameters is that there are no resonances between the donor and acceptor energies and the bridge eigenenergies. Also the upper limit to the frequency (x ¼ 2:5 fs 1 ) is chosen to approach the order of electronic excitations where it is known that the BO approximation breaks down. Fig. 2 is an example of a bridge with twelve states. Each plot shows P1N ðtÞ for a different bridge driving frequency: (2a) x ¼ 0:00025 fs 1 , (2b) x ¼ 0:0025 fs 1 , (2c) x ¼ 0:025 fs 1 , (2d) x ¼ 0:25 fs 1 and (2e) x ¼ 2:5 fs 1 . Superimposed on each P1N ðtÞ
su csu Fig. 3. P1N ðtÞ versus the approximate P1N ðtÞ and P1N ðtÞ as a function of bridge driving frequency (same systems as in Fig. 2).
sta ðtÞ whose period (denoted ssta is P1N Rabi ) gives the approximate Rabi time for the time independent system (ssta Rabi ¼ 4 ps). For a wide range of x, (Figs. 2a and c–e), the bridge oscillations do not affect sta the shape of the ET probability (P1N ðtÞ and P1N ðtÞ are approximately sinusoidal) but rather its period of oscillation 1. We denote the approximate period of P1N ðtÞ by sRabi and the period of bridge oscillations by sbr ¼ 2p=x. For Fig. 2a, sbr ¼ 25 ps > ssta Rabi and ET is slowed down in the sense that sta sRabi > ssta Rabi . For Figs. 2c–e, sbr < sRabi and ET is sta enhanced, i.e., sRabi < sRabi . The only exception to the sinusoidal behavior of P1N ðtÞ is the special case
1
The approximately sinusoidal behavior is not general. It is observed in these examples because donor and acceptor are time independent and resonant.
S.S. Skourtis / Chemical Physics Letters 372 (2003) 224–231
Fig. 4. The superexchange matrix element T1N ðtÞ and its BObreakdown correction M1N ðtÞ plotted for approximately one period of bridge oscillation sbr ¼ 2p=x. (a) x ¼ 0:25 fs 1 (same system as in Fig. 3d) and (b) x ¼ 2:5 fs 1 (same as in Fig. 3e).
Fig. 5. T1N ðtÞ versus M1N ðtÞ for an 18-site bridge with the same parameters as the 12-site bridge of Fig. 4a (x ¼ 0:25 fs 1 ).
of Fig. 2b for which sbr ’ ssta Rabi (a gating effect). These plots demonstrate that bridge oscillations can have significant effects on the ET probability for a wide range of bridge frequencies.
229
su csu ðtÞ and P1N ðtÞ as a Fig. 3 compares P1N ðtÞ to P1N function of the bridge driving frequency (same systems as in Fig. 2). For Figs. 3a–c (x < 0:25 fs 1 ) the two approximate probabilities recsu su produce P1N ðtÞ. Also P1N ðtÞ and P1N ðtÞ are equal because the magnitude of the correction matrix ~ ðtÞ is negligible compared to H~ su ðtÞ. Therefore, M in cases 3a–c, the proposed mechanism does not play a role in ET although the bridge time dependence has a significant effect on the probability (as shown in Figs. 2a–c). These effects are captured by the time-dependent superexchange approximation H~ su ðtÞ. For Fig. 3d (x ¼ 0:25 fs 1 ) the two approximate probabilities begin to deviate su csu from each other. Compared to P1N ðtÞ, P1N ðtÞ is a slightly better approximation to P1N ðtÞ (it reproduces better sRabi ). For Fig. 4e (x ¼ 2:5 fs 1 ), csu su csu P1N ðtÞ 6¼ P1N ðtÞ and only P1N ðtÞ approximates P1N ðtÞ. Therefore, in this case, the reduction of the effective Rabi time (observed in Fig. 2e) is caused by transitions that break the BO approximation. This is also seen by comparing the donor–acceptor superexchange matrix element T1N ðtÞ and its correction M1N ðtÞ. Fig. 4 shows plots of T1N ðtÞ and M1N ðtÞ for x ¼ 0:25 fs 1 (4a) and x ¼ 2:5 fs 1 (4b). M1N ðtÞ becomes much larger than T1N ðtÞ at the higher frequency. The breakdown of the BO approximation is expected for x ¼ 2:5 fs 1 because hx becomes of the order of electronic excitations. For the parameters used in the examples above, x ¼ 2:5 fs 1 csu csu is the limit of validity of P1N ðtÞ, i.e., P1N ðtÞ 6¼ P1N ðtÞ
1 csu for x > 2:5 fs . The failure of P1N ðtÞ to describe the dynamics for higher x is due the perturbative ~ ðtÞ which is second order in nature of the matrix M am . As x increases, higher order corrections in am are needed to reproduce P1N ðtÞ and finally perturbation theory in am breaks down (the condition in Eq. (8) is violated). This situation does not involve tunneling (the type B transitions in Fig. 1b are not virtual) and it is thus irrelevant to our discussion. For the 12-site bridge considered here, the frequency region for which the proposed BO-breakdown mechanism is mediated by virtual transitions and also modifies superexchange tunneling is given by: 0:25 fs 1 < x < 2:5 fs 1 . Such frequency regions are a general trend observed in the tunneling regime (defined by Eqs. (6)
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S.S. Skourtis / Chemical Physics Letters 372 (2003) 224–231
and (8)) for different sets of oscillation parameters. The size of a region depends on bridge length. In general, for any driving frequency in the tunneling regime the BO-breakdown mechanism may mediate tunneling if the bridge is large enough. We can estimate the parameters for which the proposed mechanism begins to compete with superexchange, i.e., jM1N =T1N j ’ 1, by use of Eqs. (11) and (17). In the McConnel limit, where V =ð 0 Þ < 1 and b=ð 0 Þ < 1, the bridge GreenÕs function is ji jj
ji jj 1 Gbr ð þ d sin xt 0 Þ . Substitutij ðtÞ ¼ðbÞ ing Gbr ðtÞ into Eqs. (11) and (17), and keeping the ij lowest order terms in hx=ð 0 Þ and d=ð 0 Þ, gives 2 M1N hx d ’ Nbr ; ð18Þ T 0 0 1N where Nbr is the number of bridge sites. Therefore, Mffiffiffiffiffiffi if ½ hx=ð 0 Þ ½d=ð 0 Þ ’ 1= jp 1N ffij ’ jT1N j Nbr . As the bridge length increases at a fixed energy gap and fixed x the effect of the bridge time dependence on the BO-breakdown mechanism is enhanced. This trend is shown in Fig. 5 where T1N ðtÞ and M1N ðtÞ are plotted for an 18-site bridge with the same parameters as the 12-site bridge of Fig. 4a (x ¼ 0:25 fs 1 ). Comparison of the two figures shows that the increase in the bridge length causes a 4-fold enhancement of jM1N =T1N j. Eq. (18) implies that the BO-breakdown mechanism may be relevant for long distance ET mediated by tunneling (originally proposed in [6] using a time independent vibronic model). Consider a system where the gap oscillations are sufficiently small so that there are no temporary resonances between the bridge and donor and acceptor, e.g. d=ð 0 Þ ’ 0:5. Also, the bridge oscillations which couple to ET are low frequency with respect to the energy gap, e.g. hx=ð 0 Þ ’ 0:1. Then, from Eq. (18), M1N ’ T1N for Nbr ’ 400. This number corresponds to a very long bridge that may be interpreted as a long time-dependent tunneling pathway or, in the case of a three-dimensional system, as many interconnected pathways 2. 2 This is a very rough estimate of the length because, in ~ ðtÞ that are lowest order only in contract to the elements of M am , Eq. (18) is also lowest order in V^ and in the intra-bridge couplings.
In general, to determine whether this type of BO breakdown is important for tunneling in a particular system it is necessary to compute a re~ ðtÞ matrix from MD simulations coupled alistic M to semi-empirical electronic structure calculations. Given this information the EH method is useful for a systematic analysis of BO breakdown if ~ ðtÞ is coupled to bath degrees to obtain H~ su ðtÞ þ M thermal equilibration. This can be done in the context of the driven spin-boson formalism ([15] for a review). The system part of the spin-boson ~ ðtÞ. The system-bath Hamiltonian is H~ su ðtÞ þ M spectral density can be obtained from MD simulations (e.g. as in [22]). In the simplest limit the inclusion of a harmonic bath will lead to a nonadiabatic ET rate k / h½T1N ðt Þ þ M1N ðt Þ 2 i (t denotes the crossing time between donor and acceptor energies and h i denotes an ensemble average). This is the limit where the EH derived from MD-electronic structure calculations can be compared to two-state models which treat Franck– Condon and BO-breakdown effects on tunneling (e.g. [6–8,11]). 4. Conclusions Using a time-dependent effective Hamiltonian approach we described a tunneling mechanism caused by the breakdown of the adiabatic BO approximation. This type of tunneling is induced by oscillations of the bridge energy spectrum and it may be relevant to long distance electron transfer with relatively small energy gaps. The donor–acceptor tunneling matrix element for this mechanism is different from the time-dependent superexchange matrix element. The effective Hamiltonian approach can be used to study BO breakdown in long distance electron transfer with the aid of molecular dynamics and electronic structure calculations. Appendix A The eigenstates jW ðtÞi with the largest donor and acceptor components (Eq. (6)) make the largest contribution to the electron transfer probability. The dynamics of jW ðtÞi is derived from ihdjWðtÞi=dt ¼ H^ ðtÞjWðtÞi. This equation is written
S.S. Skourtis / Chemical Physics Letters 372 (2003) 224–231
in the eigenstate (jWm ðtÞi) basis, transformed to the interaction picture and projected onto the subspace of jW ðtÞi. The projected equation contains terms with amplitudes Wm6¼ ðtÞjWðtÞ . These are neglected because the jWm6¼ ðtÞi have negligible donor and acceptor components (Eq. (6)). The final result is d in in C ðtÞ ’ Kþ
ðtÞC in ðtÞ dt þ X in in þ Kþm ðtÞSmþ ðtÞ þ Kþm ðtÞSm ðtÞ m6¼
d in in C ðtÞ ’ K þ ðtÞCþin ðtÞ dt
X in in þ K m ðtÞSmþ ðtÞ þ Kþm ðtÞSm ðtÞ; m6¼
ðA:1Þ Rt
where C in ðsÞ R t ¼ exp½i 0 dsE ðsÞ=h hW ðtÞjWðtÞi, in Kmn ðtÞ ¼ exp½i dsðEm ðsÞ En ðsÞÞ= h Kmn ðtÞ and 0 Rt in Sm ðtÞ ¼ 0 dsKm ðsÞC in ðsÞ. The second condition (Eq. (8)) allows a stationary phase expansion of Sm ðsÞ. To derive the two-state Schr€ odinger equation with H^ csu ðtÞ, we keep in Eq. (A.1) the ð1Þ leading terms of this expansion, Sm ðtÞ ¼ in in
ihKm ðtÞC ðtÞ=½Em ðtÞ E ðtÞ , transform to the Schr€ odinger picture and then to the basis of donor and acceptor states j/1 i, j/N i.
References [1] J. Jortner, M. Ratner (Eds.), Molecular Electronics, Blackwell Science, Oxford, 1997. [2] A.M. Kuznetzov, J. Ulstrup, Electron Transfer in Chemistry and Biology, Wiley, Chichetser, 1999.
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[3] J. Jortner, M. Bixon (Eds.), Advances in Chemical Physics, vols. 106–107, 1999. [4] V. May, O. Kuhn, Charge and Energy Transfer Dynamics in Molecular Systems, Wiley-VCH, Berlin, 2000. [5] V. Balzani, P. Piotrowiak, M.A.J. Rodgers, J. Mattay, D. Astruc, H.B. Gray, S. Fukuzumi, T.E. Mallouk, Y. Haas, A.P. de Silva, I.R. Gould (Eds.), Electron Transfer in Chemistry, vols. I–V, Wiley-VCH, Weinheim, 2001. [6] D.N. Beratan, J.J. Hopfield, J. Chem. Phys. 81 (1984) 5753. [7] A.M. Kuznetsov, M.D. Vigdorovich, J. Ulstrup, Chem. Phys. 176 (1993) 539. [8] J. Tang, J. Chem. Phys. 98 (1993) 6263. [9] Y. Dakhnovskii, R.D. Coalson, J. Chem. Phys. 103 (1995) 2908. [10] I.A. Goychuk, E.G. Petrov, V. May, J. Chem. Phys. 103 (1995) 4937. [11] E.S. Medvedev, A.A. Stuchebrukhov, J. Chem. Phys. 107 (1997) 3821. [12] I.A. Balabin, J.N. Onuchic, Science 290 (2000) 114. [13] M.D. Newton, in: V. Balzani, Piotrowiak, M.A.J. Rodgers (Eds.), Electron Transfer in Chemistry, vol. I, Wiley-VCH, Weinheim, 2001, p. 3. [14] S.S. Skourtis, D.N. Beratan, in: V. Balzani, P. Piotrowiak, M.A.J. Rodgers (Eds.), Electron Transfer in Chemistry, vol. I, Wiley-VCH, Weinheim, 2001, p. 109. [15] M. Grifoni, P. H€anggi, Phys. Rep. 304 (1998) 229. [16] A. Nitzan, Ann. Rev. Phys. Chem. 52 (2001) 681. [17] C.J. Ballhausen, A.E. Hansen, Ann. Rev. Phys. Chem. 23 (1972) 15. [18] Q. Xie, G. Archontis, S.S. Skourtis, Chem. Phys. Lett. 312 (1999) 237. [19] S.S. Skourtis, Q. Xie, G. Archontis, J. Chem. Phys. 115 (2001) 9444. [20] G. Fischer, Vibronic Coupling. The Interaction between Electronic and Nuclear Motions, Academic Press, New York, 1984. [21] S.S. Skourtis, S. Mukamel, Chem. Phys. 197 (1995) 367. [22] D. Xu, K. Schulten, Chem. Phys. 182 (1994) 91.
Electron transfer through time-dependent bridges: tunneling by virtual transitions that break the Born–Oppenheimer approximation Spiros S. Skourtis
*
Department of Physics, University of Cyprus, Nicosia 1678, Cyprus Received 17 December 2002; in final form 29 January 2003
Abstract For bridge-mediated electron transfer through time-dependent bridges it is shown that the superexchange mechanism may not be the only relevant tunneling mechanism. Electronic tunneling from donor to acceptor could also be mediated by a type of breakdown of the Born–Oppenheimer adiabatic approximation. A time-dependent effective Hamiltonian is derived for this process. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction Long distance, bridge-mediated electron transfer (ET) is an important research area of chemical physics, biophysics and nanotechnology [1–5]. In recent years much of the research in this field focused on the effects of time-dependent external fields and of nuclear motion on the distance and energy gap dependence of ET reactions (e.g. [6–12] and for reviews [13–15]). Different regimes were identified that range between the limits of coherent tunneling (exponential distance decay) and incoherent thermal hopping (weak distance decay), ([16] for a recent review). In the tunneling limit the
*
Fax: +357-22-339060. E-mail addresses: [email protected], net.com.cy
cabrskou@cyta-
relevant ET mechanism is superexchange. The subject of this Letter is the influence of a timedependent-bridge energy spectrum on superexchange ET. We consider effects related to the breakdown of the Born–Oppenheimer (BO) adiabatic approximation [17]. An effective time-dependent Hamiltonian (EH) which incorporates perturbatively the effects of this breakdown is derived. It is shown that the donor–acceptor matrix element of this Hamiltonian can create an additional coupling between the states that differs from the superexchange coupling. In the following we use a generalized time dependent definition of the superexchange mechanism that applies to nonadiabatic and adiabatic ET [18]. The type of BO breakdown discussed below can be interpreted as a correction to this superexchange mechanism which incorporates additional effects of the bridge time dependence on
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00370-1
S.S. Skourtis / Chemical Physics Letters 372 (2003) 224–231
the ET probability. The derived EH (and corresponding Schr€ odinger equation) is defined in the space of donor and acceptor states. It is chosen to produce an ET probability that approximates the exact donor to acceptor probability, computed from the N-state Hamiltonian and Schr€ odinger equation in the space of donor, acceptor and bridge states. In this sense the reduced Hamiltonian is an accurate description of the transfer dynamics. The final aim of this approach it to construct, with the aid of electronic structure and molecular dynamics (MD) simulations, ÔrealisticÕ reduced Hamiltonian models for ET in time-dependent molecular systems [19]. Such Hamiltonians could be compared to the different two-state models that address the influence of Condon, BO breakdown and inelastic processes on tunneling [6–15], thus leading to improved time-dependent two-state approximations. The proposed approach is useful for the analysis of tunneling influenced by classical driving fields (e.g. electric fields or classical nuclear motions). Although this is a limitation compared to vibronic Hamiltonian methods that treat quantum nuclear motions, it can be applied to large systems where vibronic methods would require prohibitively large Hamiltonian matrices.
225
The Hamiltonian in the basis of localized donor, bridge and acceptor states states is given by H^ ðtÞ ¼ H^ da ðtÞ þ H^ br ðtÞ þ V^ ðtÞ;
ð1Þ
where H^ da ðtÞ ¼ j/1 iH11 ðtÞh/1 j þ j/N iHNN ðtÞh/N j þ ½j/1 iH1N ðtÞh/N j þ h:c:
ð2Þ
is the donor–acceptor part (donor: j/1 i, acceptor: j/N i), H^ br ðtÞ ¼
N 1 X
j/i iHii ðtÞh/i j
i¼2
þ
N 2 X N 1 X i¼2
½j/i iHij ðtÞh/j j þ h:c:
ð3Þ
j>i
the bridge part, and V^ ðtÞ ¼
N 1 X
½j/1 iH1i ðtÞh/i j þ j/N iHNi ðtÞh/i j
i¼2
þ h:c:
ð4Þ
is the donor (acceptor)-bridge coupling (Fig. 1a). The instantaneous system eigenstates and eigenenergies are denoted as jWm ðtÞi and Em ðtÞ, i.e., H^ ðtÞjWm ðtÞi ¼ Em ðtÞ (Fig. 1b).
Fig. 1. Energetics of the time-dependent donor–bridge–acceptor system in the j/i i basis (a) and the jWm ðtÞi basis (b). There is a sufficiently large energy gap between j/1 i=j/N i and the bridge j/i i so that only two eigenstates jW ðtÞi have large donor and acceptor components (Eq. (6)) and are energetically separated from jWm6¼ ðtÞi. All jWm ðtÞi are coupled by the elements Kmn ðtÞ and each Km ðtÞ (m 6¼ ) is assumed to be small with respect to ðEm ðtÞ E ðtÞÞ=h (Eq. (8)). Direct processes (denoted A) and indirect virtual transition processes (denoted B) couple jWþ ðtÞi to jW ðtÞi and thus j/1 i to j/N i.
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S.S. Skourtis / Chemical Physics Letters 372 (2003) 224–231
am ðtÞ ¼
2. The effective Hamiltonian description The ET probability from donor to acceptor is given by P1N ðtÞ ¼ jh/N jWðtÞij2 ;
ð5Þ
where jWðtÞi (the state of the transferring electron at time t) is the solution of the time-dependent Schr€ odinger equation i hdjWðtÞi=dt ¼ H^ ðtÞjWðtÞi with the initial condition jWð0Þi ¼ j/1 i. We consider P1N ðtÞ under two conditions. First, the donor (j/1 i) and acceptor (j/N i) states should be be offresonant and weakly coupled to the bridge states (j/i6¼1;N i) at all times (Fig. 1a). Weak coupling means that, upon diagonalization, two eigenstates jW ðtÞi are separated energetically from the remaining eigenstates and have the largest donor and acceptor components X j /1 jWm6¼ ðtÞ j; jh/1 jW ðtÞij > m6¼
jh/N jW ðtÞij >
X j /N jWm6¼ ðtÞ j:
ð6Þ
m6¼
The second condition to be imposed relates to the matrix elements that couple the jWm ðtÞi d Kmn ðtÞ ¼ Wm ðtÞj Wn ðtÞ dt D E Wm ðtÞ dtd H^ ðtÞWn ðtÞ ¼ for m 6¼ n; ð7Þ En ðtÞ Em ðtÞ (Kmm ðtÞ ¼ 0 for real jWm ðtÞi). In terms of the parameters RðtÞ ¼ fRa ðtÞg that may drive the system (e.g. external electric fields or classical nuclear motions), Kmn ðtÞ ¼ Kmn ðRðtÞÞ ¼
X a
dRa ðtÞ ; famn ðRðtÞÞ dt
where famn ðRðtÞÞ ¼ hWm ðRðtÞÞj
o jWn ðRðtÞÞi oRa
are analogous to the nonadiabatic derivative couplings used to describe BO breakdown in time independent systems [20]. The second condition requires that, for all times
hKm ðtÞ <1 Em ðtÞ E ðtÞ
for
m 6¼ :
ð8Þ
In [18,19] MD simulations and electronic structure calculations on protein bridges were used to show that the ET probability P1N ðtÞ in the regime described by Eqs. (6) and (8) can be approximated (for a wide range of parameters) by an effective two-state probability su P1N ðtÞ ¼ jh/N jWsu ðtÞij2 :
ð9Þ su
In the equation above jW ðtÞi is the solution of a two-state time-dependent Schr€ odinger equation ihdjWsu ðtÞi=dt ¼ H^ su ðtÞjWsu ðtÞi with the initial condition jWsu ð0Þi ¼ j/1 i. H^ su ðtÞ is a donor–acceptor EH given by the matrix H~ su ðtÞ
H11 ðtÞ þ T11 ðtÞ H1N ðtÞ þ T1N ðtÞ ¼ HN 1 ðtÞ þ TN 1 ðtÞ HNN ðtÞ þ TNN ðtÞ
h/1 jW ðtÞi h/1 jWþ ðtÞi ¼ h/N jW ðtÞi h/N jWþ ðtÞi
0 E ðtÞ hW ðtÞj/1 i hW ðtÞj/N i : 0 Eþ ðtÞ hWþ ðtÞj/1 i hWþ ðtÞj/N i ð10Þ H~ su ðtÞ generalizes the superexchange model for time independent bridges to the case of time dependent bridges. It is nonperturbative in the donor (acceptor)-bridge coupling V^ ðtÞ because it is computed from exact eigenenergies and eigenstate su amplitudes. The effective part of H1N ðtÞ, T1N ðtÞ, (the time-dependent superexchange donor–acceptor matrix element), reduces to the perturbative expression ^br ðE ðtÞ; tÞV^ ðtÞj/N i T1N ðtÞ ’ h/1 jV^ ðtÞG
ð11Þ
^br is the for weak V^ ðtÞ. In the equation above G bridge GreenÕs function at time t h i 1 ^br ðE; tÞ ¼ E H^ br ðtÞ : G ð12Þ In terms of the eigenstates, the ET process desu scribed by P1N ðtÞ is caused by direct transitions between jWþ ðtÞi and jW ðtÞi induced by Kþ ðtÞ (process A in Fig. 1b). In addition we may consider virtual transitions between jW ðtÞi, i.e., jW ðtÞi ! jWm6¼ ðtÞi ! jW ðtÞi (process B in Fig. 1b). The
S.S. Skourtis / Chemical Physics Letters 372 (2003) 224–231
latter constitute a breakdown of the adiabatic BO approximation between the manifolds {jW ðtÞi} and {jWm6¼ ðtÞi} (the term adiabatic is used to distinguish it from the crude BO approximation [17]). The question to be addressed is how these virtual transitions influence superexchange. The virtual transitions can be taken into account to lowest order in Km6¼ , to obtain a corcsu rected superexchange probability, P1N ðtÞ, given by csu P1N ðtÞ ¼ jh/N jWcsu ðtÞij
2
ð13Þ
csu
(see Appendix A). jW ðtÞi is the solution of a twostate time-dependent Schr€ odinger equation i hdjWcsu ðtÞi=dt ¼ H^ csu ðtÞjWcsu ðtÞi with jWcsu ð0Þi ¼ ^ ðtÞ where j/1 i. H^ csu ðtÞ ¼ H^ su ðtÞ þ M ~ ðtÞ M
M11 ðtÞ M1N ðtÞ ¼ MN 1 ðtÞ MNN ðtÞ
h/1 jW ðtÞi h/1 jWþ ðtÞi ¼ h/N jW ðtÞi h/N jWþ ðtÞi !
eig eig hW ðtÞj/1 i hW ðtÞj/N i M
ðtÞ M þ ðtÞ eig eig hWþ ðtÞj/1 i hWþ ðtÞj/N i Mþ
ðtÞ Mþþ ðtÞ ð14Þ with eig Mab ðtÞ ¼ h2
X Kam ðtÞKmb ðtÞ ; E ðtÞ Eb ðtÞ m6¼ n
a; b ¼ :
ð15Þ
~ ðtÞ, the EH matrix for the BO-breakdown corM rection in the basis j/1 i=j/N i is obtained from ~ eig ðtÞ, the matrix for the correction in the basis of M eig jW ðtÞi. The elements Mab ðtÞ are the lowest order terms of a perturbative expansion in the parameters am ¼ hKm =ðEm E Þ (valid for am < 1 and ~ ðtÞ can be consistent with Eq. (8)). Therefore, M ~ eig ðtÞ to improved systematically by computing M higher order in am . The donor–acceptor element M1N ðtÞ is the correction to the superexchange coupling T1N ðtÞ. M1N ðtÞ may also be written in terms of the bridge ^br by carrying out perturbation GreenÕs function G ^ theory in V ðtÞ in Eq. (14). Different expressions for M1N ðtÞ are obtained in this case, depending on which part of H^ ðtÞ changes more rapidly with time. If it is V^ ðtÞ, we get
h2 dV^ M1N ðtÞ ’ h/1 j dt 2
!"
227
^br d2 G dE2
#
! dV^ j/N i: dt ð16Þ
If it is H^ br ðtÞ ! br ^ h2 d H ^br M1N ðtÞ ’ h/1 jV^ G 2 dt " # ! ^br d2 G dH^ br ^br ^ G V j/N i: dE2 dt
ð17Þ
The difference between the BO-breakdown mechanism and the time-dependent superexchange mechanism is more transparent in this perturbative limit for V^ ðtÞ. Comparison of Eqs. (16) and (17) to Eq. (11) shows that M1N ðtÞ depends on the rate of change with respect to time of state energies and interstate couplings in contrast to T1N ðtÞ. M1N ðtÞ decays approximately exponentially with bridge length as it is a function of d2 Gbr =d2 E and Gbr but its decay generally differs from that of T1N ðtÞ.
3. Numerical examples and discussion Consider a simple nearest neighbour model for H^ ðtÞ in Eq. (1), where the intra-bridge, donorbridge and acceptor-bridge couplings are given by Hij ðtÞ ¼ di;i 1 b and H1N ðtÞ ¼ HN 1;N ðtÞ ¼ V . The donor and acceptor energies are set equal and time independent, H11 ðtÞ ¼ HNN ðtÞ ¼ 0 , and the bridge site energies are set equal and time dependent, i.e., Hii ðtÞ ¼ br ðtÞ with br ðtÞ ¼ þ d sinðxtÞ. By making donor and acceptor time independent we isolate the effects of the bridge time dependence. For the above parameters we compute four difsu ferent ET probabilities: (i) P1N ðtÞ, (ii) P1N ðtÞ, (iii) csu sta P1N ðtÞ, and (iv) P1N ðtÞ, where this latter probability refers to the case of a time independent ÔstaticÕ sta bridge (br ðtÞ ¼ ). P1N ðtÞ is first compared to P1N ðtÞ to identify the effects of the bridge time dependence on the electron transfer probability. The su csu ability of P1N ðtÞ and P1N ðtÞ to capture these effects is probed by comparing the two approximate probabilities to P1N ðtÞ. The comparisons are made as a function of bridge driving frequency x and bridge length Nbr ¼ N 2. In Figs. 2–5
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S.S. Skourtis / Chemical Physics Letters 372 (2003) 224–231
Fig. 2. P1N ðtÞ for a 12-site time-dependent bridge as a function sta of bridge driving frequency versus P1N ðtÞ for a time independent bridge with the same energy parameters. The bridge frequencies are: (a) x ¼ 0:00025 fs 1 , (b) x ¼ 0:0025 fs 1 , (c) x ¼ 0:025 fs 1 , (d) x ¼ 0:25 fs 1 and (e) x ¼ 2:5 fs 1 .
0 ¼ 10 eV, b ¼ 4:5 eV and V ¼ 1:4 eV [21]. For the time dependence of the bridge site energies, d ¼ 1 eV and x ¼ 2:5 10 5 2:5 fs 1 . These bridge oscillation parameters are not supposed to approximate a particular physical situation but rather they are used to demonstrate general trends within the simplest possible model of a few bridge states. The only limitation imposed on the parameters is that there are no resonances between the donor and acceptor energies and the bridge eigenenergies. Also the upper limit to the frequency (x ¼ 2:5 fs 1 ) is chosen to approach the order of electronic excitations where it is known that the BO approximation breaks down. Fig. 2 is an example of a bridge with twelve states. Each plot shows P1N ðtÞ for a different bridge driving frequency: (2a) x ¼ 0:00025 fs 1 , (2b) x ¼ 0:0025 fs 1 , (2c) x ¼ 0:025 fs 1 , (2d) x ¼ 0:25 fs 1 and (2e) x ¼ 2:5 fs 1 . Superimposed on each P1N ðtÞ
su csu Fig. 3. P1N ðtÞ versus the approximate P1N ðtÞ and P1N ðtÞ as a function of bridge driving frequency (same systems as in Fig. 2).
sta ðtÞ whose period (denoted ssta is P1N Rabi ) gives the approximate Rabi time for the time independent system (ssta Rabi ¼ 4 ps). For a wide range of x, (Figs. 2a and c–e), the bridge oscillations do not affect sta the shape of the ET probability (P1N ðtÞ and P1N ðtÞ are approximately sinusoidal) but rather its period of oscillation 1. We denote the approximate period of P1N ðtÞ by sRabi and the period of bridge oscillations by sbr ¼ 2p=x. For Fig. 2a, sbr ¼ 25 ps > ssta Rabi and ET is slowed down in the sense that sta sRabi > ssta Rabi . For Figs. 2c–e, sbr < sRabi and ET is sta enhanced, i.e., sRabi < sRabi . The only exception to the sinusoidal behavior of P1N ðtÞ is the special case
1
The approximately sinusoidal behavior is not general. It is observed in these examples because donor and acceptor are time independent and resonant.
S.S. Skourtis / Chemical Physics Letters 372 (2003) 224–231
Fig. 4. The superexchange matrix element T1N ðtÞ and its BObreakdown correction M1N ðtÞ plotted for approximately one period of bridge oscillation sbr ¼ 2p=x. (a) x ¼ 0:25 fs 1 (same system as in Fig. 3d) and (b) x ¼ 2:5 fs 1 (same as in Fig. 3e).
Fig. 5. T1N ðtÞ versus M1N ðtÞ for an 18-site bridge with the same parameters as the 12-site bridge of Fig. 4a (x ¼ 0:25 fs 1 ).
of Fig. 2b for which sbr ’ ssta Rabi (a gating effect). These plots demonstrate that bridge oscillations can have significant effects on the ET probability for a wide range of bridge frequencies.
229
su csu ðtÞ and P1N ðtÞ as a Fig. 3 compares P1N ðtÞ to P1N function of the bridge driving frequency (same systems as in Fig. 2). For Figs. 3a–c (x < 0:25 fs 1 ) the two approximate probabilities recsu su produce P1N ðtÞ. Also P1N ðtÞ and P1N ðtÞ are equal because the magnitude of the correction matrix ~ ðtÞ is negligible compared to H~ su ðtÞ. Therefore, M in cases 3a–c, the proposed mechanism does not play a role in ET although the bridge time dependence has a significant effect on the probability (as shown in Figs. 2a–c). These effects are captured by the time-dependent superexchange approximation H~ su ðtÞ. For Fig. 3d (x ¼ 0:25 fs 1 ) the two approximate probabilities begin to deviate su csu from each other. Compared to P1N ðtÞ, P1N ðtÞ is a slightly better approximation to P1N ðtÞ (it reproduces better sRabi ). For Fig. 4e (x ¼ 2:5 fs 1 ), csu su csu P1N ðtÞ 6¼ P1N ðtÞ and only P1N ðtÞ approximates P1N ðtÞ. Therefore, in this case, the reduction of the effective Rabi time (observed in Fig. 2e) is caused by transitions that break the BO approximation. This is also seen by comparing the donor–acceptor superexchange matrix element T1N ðtÞ and its correction M1N ðtÞ. Fig. 4 shows plots of T1N ðtÞ and M1N ðtÞ for x ¼ 0:25 fs 1 (4a) and x ¼ 2:5 fs 1 (4b). M1N ðtÞ becomes much larger than T1N ðtÞ at the higher frequency. The breakdown of the BO approximation is expected for x ¼ 2:5 fs 1 because hx becomes of the order of electronic excitations. For the parameters used in the examples above, x ¼ 2:5 fs 1 csu csu is the limit of validity of P1N ðtÞ, i.e., P1N ðtÞ 6¼ P1N ðtÞ
1 csu for x > 2:5 fs . The failure of P1N ðtÞ to describe the dynamics for higher x is due the perturbative ~ ðtÞ which is second order in nature of the matrix M am . As x increases, higher order corrections in am are needed to reproduce P1N ðtÞ and finally perturbation theory in am breaks down (the condition in Eq. (8) is violated). This situation does not involve tunneling (the type B transitions in Fig. 1b are not virtual) and it is thus irrelevant to our discussion. For the 12-site bridge considered here, the frequency region for which the proposed BO-breakdown mechanism is mediated by virtual transitions and also modifies superexchange tunneling is given by: 0:25 fs 1 < x < 2:5 fs 1 . Such frequency regions are a general trend observed in the tunneling regime (defined by Eqs. (6)
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S.S. Skourtis / Chemical Physics Letters 372 (2003) 224–231
and (8)) for different sets of oscillation parameters. The size of a region depends on bridge length. In general, for any driving frequency in the tunneling regime the BO-breakdown mechanism may mediate tunneling if the bridge is large enough. We can estimate the parameters for which the proposed mechanism begins to compete with superexchange, i.e., jM1N =T1N j ’ 1, by use of Eqs. (11) and (17). In the McConnel limit, where V =ð 0 Þ < 1 and b=ð 0 Þ < 1, the bridge GreenÕs function is ji jj
ji jj 1 Gbr ð þ d sin xt 0 Þ . Substitutij ðtÞ ¼ðbÞ ing Gbr ðtÞ into Eqs. (11) and (17), and keeping the ij lowest order terms in hx=ð 0 Þ and d=ð 0 Þ, gives 2 M1N hx d ’ Nbr ; ð18Þ T 0 0 1N where Nbr is the number of bridge sites. Therefore, Mffiffiffiffiffiffi if ½ hx=ð 0 Þ ½d=ð 0 Þ ’ 1= jp 1N ffij ’ jT1N j Nbr . As the bridge length increases at a fixed energy gap and fixed x the effect of the bridge time dependence on the BO-breakdown mechanism is enhanced. This trend is shown in Fig. 5 where T1N ðtÞ and M1N ðtÞ are plotted for an 18-site bridge with the same parameters as the 12-site bridge of Fig. 4a (x ¼ 0:25 fs 1 ). Comparison of the two figures shows that the increase in the bridge length causes a 4-fold enhancement of jM1N =T1N j. Eq. (18) implies that the BO-breakdown mechanism may be relevant for long distance ET mediated by tunneling (originally proposed in [6] using a time independent vibronic model). Consider a system where the gap oscillations are sufficiently small so that there are no temporary resonances between the bridge and donor and acceptor, e.g. d=ð 0 Þ ’ 0:5. Also, the bridge oscillations which couple to ET are low frequency with respect to the energy gap, e.g. hx=ð 0 Þ ’ 0:1. Then, from Eq. (18), M1N ’ T1N for Nbr ’ 400. This number corresponds to a very long bridge that may be interpreted as a long time-dependent tunneling pathway or, in the case of a three-dimensional system, as many interconnected pathways 2. 2 This is a very rough estimate of the length because, in ~ ðtÞ that are lowest order only in contract to the elements of M am , Eq. (18) is also lowest order in V^ and in the intra-bridge couplings.
In general, to determine whether this type of BO breakdown is important for tunneling in a particular system it is necessary to compute a re~ ðtÞ matrix from MD simulations coupled alistic M to semi-empirical electronic structure calculations. Given this information the EH method is useful for a systematic analysis of BO breakdown if ~ ðtÞ is coupled to bath degrees to obtain H~ su ðtÞ þ M thermal equilibration. This can be done in the context of the driven spin-boson formalism ([15] for a review). The system part of the spin-boson ~ ðtÞ. The system-bath Hamiltonian is H~ su ðtÞ þ M spectral density can be obtained from MD simulations (e.g. as in [22]). In the simplest limit the inclusion of a harmonic bath will lead to a nonadiabatic ET rate k / h½T1N ðt Þ þ M1N ðt Þ 2 i (t denotes the crossing time between donor and acceptor energies and h i denotes an ensemble average). This is the limit where the EH derived from MD-electronic structure calculations can be compared to two-state models which treat Franck– Condon and BO-breakdown effects on tunneling (e.g. [6–8,11]). 4. Conclusions Using a time-dependent effective Hamiltonian approach we described a tunneling mechanism caused by the breakdown of the adiabatic BO approximation. This type of tunneling is induced by oscillations of the bridge energy spectrum and it may be relevant to long distance electron transfer with relatively small energy gaps. The donor–acceptor tunneling matrix element for this mechanism is different from the time-dependent superexchange matrix element. The effective Hamiltonian approach can be used to study BO breakdown in long distance electron transfer with the aid of molecular dynamics and electronic structure calculations. Appendix A The eigenstates jW ðtÞi with the largest donor and acceptor components (Eq. (6)) make the largest contribution to the electron transfer probability. The dynamics of jW ðtÞi is derived from ihdjWðtÞi=dt ¼ H^ ðtÞjWðtÞi. This equation is written
S.S. Skourtis / Chemical Physics Letters 372 (2003) 224–231
in the eigenstate (jWm ðtÞi) basis, transformed to the interaction picture and projected onto the subspace of jW ðtÞi. The projected equation contains terms with amplitudes Wm6¼ ðtÞjWðtÞ . These are neglected because the jWm6¼ ðtÞi have negligible donor and acceptor components (Eq. (6)). The final result is d in in C ðtÞ ’ Kþ
ðtÞC in ðtÞ dt þ X in in þ Kþm ðtÞSmþ ðtÞ þ Kþm ðtÞSm ðtÞ m6¼
d in in C ðtÞ ’ K þ ðtÞCþin ðtÞ dt
X in in þ K m ðtÞSmþ ðtÞ þ Kþm ðtÞSm ðtÞ; m6¼
ðA:1Þ Rt
where C in ðsÞ R t ¼ exp½i 0 dsE ðsÞ=h hW ðtÞjWðtÞi, in Kmn ðtÞ ¼ exp½i dsðEm ðsÞ En ðsÞÞ= h Kmn ðtÞ and 0 Rt in Sm ðtÞ ¼ 0 dsKm ðsÞC in ðsÞ. The second condition (Eq. (8)) allows a stationary phase expansion of Sm ðsÞ. To derive the two-state Schr€ odinger equation with H^ csu ðtÞ, we keep in Eq. (A.1) the ð1Þ leading terms of this expansion, Sm ðtÞ ¼ in in
ihKm ðtÞC ðtÞ=½Em ðtÞ E ðtÞ , transform to the Schr€ odinger picture and then to the basis of donor and acceptor states j/1 i, j/N i.
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