Pis'ma v ZhETF, vol. 98, iss. 4, pp. 247-250 © 2013 August 25
Vibrational energy transport in molecular wires
V. A. Benderskii+, A. S. Kotkin+, I. V. Rubtsov*, E. I. Katsx + Institute of Problems of Chemical Physics of the RAS, 142432 Chernogolovka, Russia
* Department of Chemistry, Tulane University, LA 70118 New Orleans, USA x Landau Institute for Theoretical Physics of the RAS, 142432 Chernogolovka, Russia
Submitted 12 July 2013
Motivated by recent experimental observation (see, e.g., I.V. Rubtsov, Acc. Chem. Res. 42, 1385 (2009)) of vibrational energy transport in (CH2O)N and (CF2)N molecular chains (N = 4—12), in this paper we present and solve analytically a simple one dimensional model to describe theoretically these data. To mimic multiple conformations of the molecular chains, our model includes random off-diagonal couplings between neighboring sites. For the sake of simplicity we assume Gaussian distribution with dispersion a for these coupling matrix elements. Within the model we find that initially locally excited vibrational state can propagate along the chain. However the propagation is neither ballistic nor diffusion like. The time Tm for the first passage of the excitation along the chain, scales linearly with N in the agreement with the experimental data. Distribution of the excitation energies over the chain fragments (sites in the model) remains random, and the vibrational energy, transported to the chain end at t = Tm is dramatically decreased when a is larger than characteristic interlevel spacing in the chain vibrational spectrum. We do believe that the problem we have solved is not only of intellectual interest (or to rationalize mentioned above experimental data) but also of relevance to design optimal molecular wires providing fast energy transport in various chemical and biological reactions.
DOI: 10.7868/S0370274X13160078
Introduction. Recently [1-5] one of the author of this work (I.R.) together with his collaborators has observed transport of vibrational energy along (CH2O)N and (CF2)n molecular chains (N = 4—12). Transfer of the energy from optically excited "donor" site is detected by the retardation time t = Tm, when the excitation for the first time arrives to the "acceptor" fragment attached to the end of the chain. Experimental data show that Tm scales linearly with N, and the maximum acceptor site n = N population decays with a characteristic time scale > 10 ps, comparable with Tm. Our main motivation in this work is to present and to solve a simple one dimensional model to describe these experimental data. Within the model we compute Tm(N), and excitation energy distribution (or site n population (|a„|2)).
From the first sight the very existence of such directed energy transport can be easily qualitatively understood. Indeed couplings between neighboring chain fragments lead to delocalized vibrational modes [6]. Donor site excitation energy transfers to these modes, and the latter ones transport the energy to the acceptor site. However this mechanism for quasi-ballistic energy transport contradicts to common wisdom based on the radiationless transition theory (see e.g., [7]), which predicts very fast equipartition of the excitation energy over all N sites of the chain. Moreover in multi-atomic
fragments (like CH2O and CF2 investigated in [1-5]) there are several local modes per each site with non-identical coupling matrix elements with the local modes at neighboring sites. This phenomenon yields to dense quasi-random spectrum of vibrations (similarly to energy levels in large complex nuclei [8]). In this case one can expect mode localization [9, 10], or diffusion like behavior [11, 12] rather than ballistic-like plane wave propagation.
Another way to treat this problem is to study directly quantum dynamics of one-particle vibrational excitations. For a uniform chain with one impurity site in its center, the problem was investigated recently [13-15]. The results of these papers suggest that in certain conditions (one delocalized mode weakly coupled to other chain degrees of freedom) one can expect realization of the regimes for vibrational excitation propagation observed in the experiments [1-4]. Motivated by this expectation in what follows we study quantum dynamics of a vibrational excitation created at the donor impurity site d which is connected by a uniform chain of N identical fragments to the acceptor site a (where the excitation is detected).
Model and technical details of its solution. For the sake of simplicity we use the units, where the uniform chain site energy En = 0 and nearest neighboring
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y. A. Benderskii, A. S. Kotkin, I. V. Rubtsov, E. I. Kats
site coupling constant Cn = 1. In these units for the donor and acceptor sites we get (Ed , Cd) and (Ea , Ca) respectively. These quantities are defined as the Hamil-tonian H matrix elements calculated with wave functions in site representation. They read in self-evident notations as
n\H\fat) =0; {fa\H\fa) = Sn> ,n+ifn
(1)
for a regular chain 1 < n < N connecting the donor and acceptor sites, and for the latter sites
{fa\H\fa) = Ed, {fa\H\fa) = Ea {fa\H\4i) = Cd, {4>n\h\fa) = Ca
(2)
Random numbers fn with {fn) = 1 and {(fn)2) = a are our new ingredient in this work. There are at least three sources for these random off-diagonal matrix elements. First for the chains with multi-atomic fragments studied in [1-5] there are 5-7 local modes per each site. Intersite couplings split the corresponding energy levels into a band with dense spectrum. All band levels contribute non-identically into the excitation energy transfer from the donor to acceptor. Similar roles are played by the conformational degrees of freedom of the molecular chain and also by surrounding solvent molecules. Although conformational and solvent reorganization times are much larger than time scales of the order of 10 ps we are interested in, it makes non-equivalent conditions for the excitation energy propagation. Thus the matrix Hamiltonian H with off-diagonal randomness describes effectively an ensemble of molecular chains. The secular (N + 2) x (N + 2) determinant corresponding to the formulated model has the following form
(3)
F(e) 1
- Ed Cd 0 0 0 0 0 0
Cd e fi 0 0 0 0 0
0 0 e fn 0 0 0 0
0 0 fn e fn+i 0 0 0
0 0 0 fn+i e fn+2 0 0
0 0 0 0 fn+2 e 0 0
0 0 0 0 0 fN -i e Ca
0 0 0 0 0 0 Ca e-E
and the secular equation to find eigen values e is F(e) = = 0. The Jacobi form determinant (3) can be written in a more compact form as
F(e) = (e - Ed)(e - Ea)DN(e) -- (e - Ed)C2DN-i(e) + C^Dn-2(e),
where Dn(x) = £^(-l)^^(fk)xn-2k, [y] is an integer part of y, and we introduce the following notations An = 1, An-2 = £n-i ffc, An-4 =
= £ k+2<k' <n-1 f2fl, and so on.
Within the same approach and the Hamiltonian we can expand an arbitrary state time dependent wave function as a linear superposition local site wave functions with time dependent amplitudes
N
^(t) = ad(t)0d + a,n(t)fa + aa(t)fa,
(5)
n=1
and then the Schroedinger equation for ty(t) can be formulated as the set of the dynamic equations for the amplitudes
iad = Edad - Cdai, ia 1 = -Cdad - a,2,
ia n = -an-i + an+i, iaN = -aN-i - Ca aa, ia a = -Ca aN + Ea aa,
(6)
where h = 1. These equations supplemented by the initial condition ad(0) = 1, an(0) = 0, and aa(0) = 0 can be solved by the Laplace transformation, and then after some algebra we end up with the following formally exact solution for a given realization of random numbers fn
ak
(t) = Y,eM^kt)Fk(ek)(^pj . (7)
i, \ / e=eu
Illustrative results. Expression (7) is our main result and it is ready for further inspection. In the simplest ideal single chain case, when there is no any randomness, a = 0, the maximum of the amplitude at the site n = N determines the first passage time Tm. Using Exp.
(7) and found in [14] representation for the amplitude
K(t)\ = ^ [Jn-i(i) + Jn+i(t)] ;2<n<N, (8)
valid for the time t < Tm (Jn is the 1-st kind Bessel function), we find that the model quantum dynamics describes the wave packet motion with an approximately constant (independent of N) group velocity vg ~ 2 (in our units). The wave packet formed by the amplitudes
(8) has very sharp front and weak oscillating tail behind the front.
For a = 0 we have to deal with an ensemble of chain realizations. Assuming Gaussian random distribution of the matrix elements fn, we have to average the solution (7). The results, obtained numerically by using standard Matlab software are presented in the Figs. 1-3. By visual inspection and numeric fitting of the plots we conclude.
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Vibrational energy transport in molecular wires
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Similarly to the ideal non-random chain, the site n = N population achieves its maximum after the first passage time Tm. As a function of N we can fit Tm as a linear function (see Fig. 1)
Fig. 1. Time dependent populations (|aN|2}(t). (a) - N = = 10, and a values for the curves from 1 to 5 are: 0, 0.11, 0.29, 0.43, 0.57. (b) - N = 2, and a values for the curves from 1 to 5 are: 0, 0.06, 0.15, 0.22, 0.30. Tm are shown by arrows. The insertion shows the first passage time Tm(N). The ensemble mean values are computed by averaging over 100 realizations
Fig. 2. Energy transfer efficiency characterized by the ratio (|aN(t = Tm,a)\2)/{\aN(t = Tm,a = 0)|2). The curves from 1 to 4 in the upper plot correspond to N = 5, 10, 20, 30. The exponential exp[-a2/as(N)] fitting is shown in the lower plot
excitation energy over all sites (Fig. 3b). By the numeric fitting of the computed amplitudes presented in the Figs. 2 and 3, we find
(I aN((r,Tm)\2) \aN(0,TmW
= exp[-a2 (N +1)],
(10)
Tm(N ) = Tm +
N ~2 '
(9)
where Tm « Cd is determined by the rate of the excitation transfer from the donor site to the chain.
There are site population oscillations at t > Tm related to the waves reflected from the acceptor site. These oscillations are strongly suppressed when the randomness
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