Lecture 6. Two molecular states separated with a square potential According to the Fermi-Golden rule, the transition rate from k’ and k states is given by )(||2)(2''2''' kkkkkkkEHdEwEwρπρh==∫ where the coupling strength, Hk’k’, between the final and initial states, plays an important role in the transition probability. Because it depends on the actual interaction between the states, which is often difficult to calculate, it has been widely treated as a fitting parameter to experimental data. However, the dependence of Hk’k’ on the separation between two molecules (acceptors and donors, say) is usually an exponential function. Here is a simple argument why it should be the case. Let us consider a donor molecule is separated from an acceptor molecule with a bridge, either free space or insulating materials (molecular group). For simplicity, we treat the electron in the donor or acceptor molecules as free particle and the bridge as a square barrier (figure). U Fig. 6.1 A donor and acceptor separated with a square barrier. H0 H’ H U 0 L L -U t<0 t>0 donor acceptorWe can decompose the potential profile as two step potentials as shown in the figure. The first potential is essentially a step function, except that it drops to zero at a very large x. It describes the electron in the donor date. The second potential is the acceptor state and we treat it as a perturbation, and H’ is 0 for x<L, and –U for x>L. Basically the picture is: Initially an electron is in the donor state. At t=0, the acceptor state suddenly appears, so the electron has a probability to transfer into the acceptor state. We first determine the wavefunctions of the system by dividing the space into two regions, x<0 and x>0, respectively. Applying the Schrodinger equation to the two regions, we obtained the wavefunctions x<0: ikxikxIBeex−+=)(ψ(1) x>0: xIICexαψ−=)( , (2) where hmEk2= and h)(2 EUm −=α. The wavefunction in region I contains a plane wave propagating along +x and a reflective wave with amplitude B along –x. Region II is classically forbidden region since E<U (corresponding to a negative kinetic energy), and wavefunction in decays exponentially, meaning a finite penetration of electrons into this classically forbidden region. The coefficients, B and C, are determined by the following boundary conditions at x=0, )0()0(IIIψψ= (3) dxddxdIII)0()0(ψψ= . (4) Substituting Eqs. 1 and 2 into Eqs. 3 and 4, we find kiC/12α+= and kiB/121α+−= . (5)The initial k’ state is described by wavefunctions 1 and 2 (the electron is in the donor), and the final k state is described by a plan wave ikxe−~ (the electron is in the acceptor). The coupling, Hk’k’ is given by LikLLikLxLikxkkeUekkieikeUikkdxCeUekHkHαααααα−−−−−∞−+−=++=−>==<∫22'22)(|'|'), from which LLkkeeUkkHβαα−−∝+=22222')2(|| . (6) So the coupling strength is an exponential function of the separation between the acceptor and donor. Many recent works have focused on determining the electron transfer rate as a function of the donor-acceptor separation that is systematically controlled experimentally. If U~5eV (vacuum), then β~2 Å-1. Another example is that the donor and acceptor are connected with a saturated molecular bridge, β is found to be ~1 Å-1. In contrast, for highly conjugated organic bridges, β is much smaller, lying in the range of 0.2 – 0.6 Å-1. Figure. 6.2 shows a few examples.Fig. 6.2. Length dependence of conductance for saturated chains and conjugated molecules. The conductance in each system decreases exponentially with the length, but with a different slope (decay constant), reflecting different Hk’k. Superexchange A more realistic situation is that the donor and acceptor are separated by a molecular bridge, such as a series of σ-bonds, rather than a vacuum. In this case, a superexchange model has been proposed. In the model, the molecular orbitals of donor and acceptor are represented by |D> and |A>, and the bridge by |1>, |2>, …, |N>.Fig. 6.3 Superexchange model. The energies of these bridge states, En (n=1, 2, … N) are much higher than the energy of the electron in the donor or acceptor states. Another assumption is that the bridge states are localized in space, so only the coupling between nearest neighbors exists, i.e., 1'1'±±=nnnnnnVVδ(called tight-binding model). In this case, the transition matrix element, H’k’k, is replaced by TDA, where the T operator is defined as VGVVT+=, (7) where V is an operator that represents the couplings between the donor, acceptor and the bridge states. G in the above expression is the Green function of the bridge and given by Γ+−=iHEEG)2/1(1)( (8) where H is the Hamiltonian of the bridge and Γ-1 is the lifetime matrix of the bridge states. ∑∑==>><><<+=><+>>=<=<NnNnDADAAVnnGnnVDVAVGVDAVDATDT11'||''||||||||||. (9) For a chain-like molecule, we can assume that the donor state |D> couples to |1> and the acceptor state |A> couples to |N> only and the above equation can be simplified as NANDDADADAVGVVAVNNGVDVT11||||11||+=>><><<+=. (10)The first term of Eq. is the direct transition of an electron from the donor to the acceptor, or the so called through space transfer. This term is usually ignored when the molecular chain or bridge is very long. The second term represents a bridge-mediated electron transfer process. Using the tight binding approximation and in the weak coupling limit, max|V| << min(En-E), where E is the energy of the transmitted electrons and should be equal to ED=EA when electron transfer occurs, the Green function element is given by GEEEVEENNnnnnN11111(),=−−+=−∏. (11) For a model of identical bridge elements, such as alkane-chain, En and Vn,n+1 are constants and denoted by EB and VB, respectively. The Green function element takes the simple form of NBBBNBBBNEEVVEEVEEEG⎟⎟⎠⎞⎜⎜⎝⎛−=⎟⎟⎠⎞⎜⎜⎝⎛−−=−11)(11. (12) The electron transfer probability is proportional to ]exp[)ln(2exp||||||222122''dNaVEEaVEEEEVVVVTHBBNBBNBBBNADDAkkβ−=⎥⎦⎤⎢⎣⎡−−∝⎟⎟⎠⎞⎜⎜⎝⎛−∝⎟⎟⎠⎞⎜⎜⎝⎛−=→−, (13) where a is the length of each bridge element so the d=Na is the length of the entire bridge. So the superexchange model also predicts an exponential decay the electron transfer rate over the distance, just like electron tunneling through a square barrier. The coefficient, )ln(2BBVEEa−=β, is an