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Biological Energy 5: Electron transfer theory
Jan 1, 2013
5.1 Introduction to biological electron transfer
This is the third lecture on energy transfer in photobiological systems, and is on the topic of electron transfer (ET) (Slide 1).
Slide 1 Electron transfer theory describes the movement of an electron between two molecular species. Quantum mechanics is needed to describe the process correctly. The Hamiltonian, wavefunction and energy eigenvalues are all key elements of the theory
In a nutshell, electron transfer is a process whereby an electron moves from one molecular species (i.e. the electron donor chromophore) to another molecular species, (i.e. theelectron acceptor chromophore) .
Generally speaking, electron transfer can occur through space, through solvent or through a bond – the latter signifying a region of heightened electron density. Through-space electron transfer occurs only over very short distances (< 0.5 nm), whereas electron transfer that occurs through a medium (i.e. a solvent or covalently attached bridge) may occur over much longer distances.
Long range ET between a given donor and acceptor site (via a bridge) may involve a direct coherent process (i.e. superexchange) or an incoherent, stepwise process. The type of process will depend on magnitudes of the electronic couplings between the donor and the acceptor, the energy gaps within the donor-bridge-acceptor system, and electronic relaxation processes.
Electron transfer is fundamental to photosynthesis. The role of theelectron transport chain is to stabilize the oxidized and reduced species. This must occur before the downstream steps resulting in long-term energy storage can occur.
The focus of this lecture will be on the theoretical principles of electron transfer (Slide 2), starting with a very general overview of the photophysical processes that occur in photosynthesis. and the overall charge separation process in photosynthesis.
The rest of the lecture will focus on theoretical foundations of electron transfer theory, building on a basic background in quantum mechanics. Here, we shall focus specifically on the Hamiltonian, the wavefunction and energy eigenvalues.
Of central importance will be the diabatic vs adiabatic representations and electronic couplings. Once these have been explained, the important concept of the ‘superexchange’ mechanism will be discussed and this will lead into the more advanced topics. Returning to the Fermi Golden Rule, we shall also encounter Marcus-Hush theory and reorganisation theory.
5.1.2 Possible decay pathways
As was established in Lecture 3, when an antenna molecule enters the excited state, several different processes may occur: there may be direct relaxation, energy may transfer to another chromophore, or the energy associated with the excited state can start a charge transfer reaction. See Slide 3 (which is a repeat of Slide 10 from Lecture 3).
Slide 4 (which is a repeat of Lecture 3 Slide 19) gives a schematic representation of the charge separation process that occurs in the photosynthetic reaction centre. Typically, a chlorophyll molecule is excited away from the reaction centre (where the charge transfer process occurs). The chlorophyll molecule enters an excited state and this exciton is transferred, via the resonance energy transfer mechanism, towards the reaction centre. This is shown schematically as transfer occurring from Ch l1 to Ch l2. However, many of these steps generally occur.
5.2 Electron transfer theory
5.2.1 The ET Hamiltonian
Here (Slide 5) we are introducing the formal quantum mechanical description of electron transfer. In general, electron transfer reactions take place in complex molecular systems and we must introduce assumptions so that we can describe the system quantum mechanically.
We are considering donor-bridge-acceptor (DBA) systems. These are systems where the donor and acceptor chromophores are covalently bound to an intervening bridge (more on this later). We consider only the single electron that is being transferred from the donor chromophore to the acceptor chromophore. To do this, we introduce the Born-Oppenheimer (BO) approximation which allows separation of the electronic and nuclear degrees of freedom (justified by the fact that the proton is approximately 2000 times more massive than the electron).
H = Hn + Hel
In Equation 2, corresponds to the total wavefunction, with R and r corresponding to the nuclear and electronic degrees of freedom. Mathematically, the BO approximation allows one to break the wavefunction up into a nuclear component, and an electronic component, where the total wavefunction is simply the product of the two.
Another assumption is that the energy levels of the donor and acceptor orbitals are close in energy compared to those of the bridge. This gives rise to the two-state approximation, where we focus on the electron moving between two states of similar energy (one associated with the donor and the other with the acceptor).
Slid 6 is a schematic helping to describe Slide 5 visually. The orbitals of the donor and acceptor should be close in energy, while the bridge orbitals should have energies that are well separated in magnitude.
In the equation ΔE corresponds to the energy between the donor state (ED) and the acceptor state (EA). The energies of the bridge states are defined collectively as EB(i)
5.2.2 Diabatic and adiabatic states
A very important concept in electron transfer theory is the representation of states. In electron transfer theory, diabats or diabaticsurfaces are often used to describe the reactant and product states – the reactant state being the non charge-transfer state, and the product being the charge-transfer state. The term ‘diabatic’ relates to the distinction from an adiabatic surface, within which a state evolves with a constant energy. Adiabatic states are those which diagonalise the Hamiltonian.
It is important to realize that while these states (often depicted by parabolas as in Slide 7) are useful from a theoretical point-of-view, they are in general not real. The diabatic surfaces couple together to give rise to anavoided crossingwhere a ground and excited state surface is formed. It is these surfaces that correspond to the real states of the molecule; i.e. S0, S1, S2, .....
It is also important to realize that these surfaces in fact span the whole 3N-6 internal degrees of freedom of the molecule. However, they are often simplified by devising a generalised ET coordinate (Slide 7).
From Slide 7, we know that there are two types of representations. The diabatic representation, where the charge delocalized and charge transfer states are represented by intersecting parabola, and the adiabatic representation where there is a ground and excited state surface (Slide 8).
The principal assumption behind electron transfer theory is that of a two-state model within a molecular system. That is, the two electronic states involved in the process are very well separated from any other electronic states (which can be neglected). It is this that gives rise to the avoided crossing and hence two adiabatic states. This theory should not be confused with band gap theory, which generally deals with extended solid-state metals and semiconductors (rather than molecules). In band-gap theory delocalised electronic states form, giving rise to effectively two continua of densely packed states (the conduction and valence bands). This situation is quite different from the electron transfer processes occurring in biological systems.
Diabatic and adiabatic wavefunctions can be transformed between one another using a simple rotation (a unitary transformation) in the two-dimensional space represented by the basis states:
Here the rotation angle (sometimes called the mixing angle) is dependent on the nuclear geometry. For symmetric systems the angle is π/4.
5.2.3 The diabatic representation and electronic coupling
Working in the diabatic representation, coupling between the two states is governed by the electronic Hamiltonian (see Slide 9):
The adiabatic representation, however, requires calculation of the non-adiabatic coupling vector (which is outside the scope of this lecture). This makes working in the diabatic representation much more appealing. The major advantage of working within the diabatic representation is that the coupling is dominated by the electronic Hamiltonian (i.e. not by non-Born-Oppenheimer effects)
We can construct the electronic Hamiltonian by setting the diagonal elements to the site energies of the chromophores and setting the off-diagonal elements to the electronic couplings. The electronic couplings determine how strongly the diabatic states are coupled to one another.
If the coupling is very strong, then the electron transfer process is highly favourable; if it is weak, then electron transfer is significantly compromised. The rate of electron transfer is a very important parameter, and the rata parameter, kET, is proportional to the square of the electronic coupling:
By expanding out the rotation matrix (Slide 10), we can more easily see that the adiabatic states are linear combinations (mixtures) of the diabatic states, or vice versa:
This also means that there is a direct relationship between the electronic coupling and the energy gap between the ground and excited states at the avoided crossing. At the avoided crossing, the electronic coupling is exactly half the energy gap:
ΔE = 2Vif
This makes calculation of the electronic coupling at the avoid crossing quite straight-forward using ab initio wavefunction methods. By calculating the energy gap at the avoided crossing for a variety of systems, one can gain insight into which molecular systems are more likely to undergo electron transfer, also inferring the relative rates of electron transfer. Note however, that this only gives indications of relative values; it does not allow us to calculate absolute rates without calculating other parameters.
We can use a Molecular Orbital (MO) picture (Slide 11) to further understand coupling between chromophores. For simplicity, consider two p-orbitals separated in space. At very large distances they do not ‘feel’ each others’ presence and they are effectively degenerate. This means that, because there is no energy gap between the states, the electronic coupling must be zero.
As the two orbitals are brought closer together, coupling occurs giving rise to a p + p and p – p state. There is now an energy gap, ΔE between these states and consequently a coupling that is equal to half the energy gap.
(from Equation 9)
As the p orbitals are brought even closer together, the energy gap and coupling increases, with the rate of electron transfer increasing quadratically with increasing coupling.
(from Equation 7)
Note that in this hypothetical case, where the p orbitals are separated in empty space, through-space electron transfer between the orbitals could only occur where there is significant wave function overlap (distances less than about 0.6 nm).
5.2.4 Through-bond ET – superexchange mechanism
Bridge-mediated electron transfer (Slide 12) requires an intermediate bridge to help facilitate electron transfer. Broadly speaking there are two kinds of processes. The first is where the electron is actually localized within the bridge orbitals during its relocation from the donor to the acceptor. For this to happen, the bridge orbitals must be of similar energy. A schematic of this is given in the top picture. An example where this might occur would be in highly conjugated systems – like the chains on a carotene molecule. The process whereby the electron is localized within the bridge is sometimes referred to aselectron transport.
In the second case the orbital energies of the bridge units are well separated from the orbital energies of the donor and acceptor chromophores. This means that although the bridge orbitals facilitate electron transfer through the superexchangemechanism, the electron will never be seen in the bridge. The electron will either be localized on the donor, or else localized on the acceptor. This is commonly referred to as electrontransfer.
In superexchange (Slide 13), the orbitals of the bridge are not directly involved in the electron transfer process (i.e. the electron does not become localized in the bridge orbitals). However, the bridge orbitals are involved in coupling to the orbitals of the donor and acceptor chromophore, giving rise to a coupling over longer distances that are achievable via through-space coupling (i.e. coupling just between the donor and acceptor orbitals). Through-bond coupling and through bond-mediated ET are long range, > 1 nm.
Slide 14 summarises the main points so far. The rest of the lecture focuses on specific topics of electron transfer.
5.3 The Fermi Golden Rule
Non-adiabatic processes are those that involve more than one electronically excited state. The Fermi Golden Rule (Slide 15) can be derived from time-dependent perturbation theory – although the derivation is outside the scope of this lecture (see relevant text book, e.g. May & Kühn in the references listed above).
Fermi’s Golden Rule is at the heart of the vast majority of common processes associated with time-independent rates, and underpins almost the whole of spectroscopy from atomic to molecular spectroscopy. It very much is standard quantum mechanics and appears in many standard texts on quantum mechanics (see e.g. Davydov, p 400). The sense in which we use it in these lectures is exactly the same as these contexts.
The first term in Equation 10 corresponds to the electronic coupling, the second to the overlap of vibrational wavefunctions, and the final one is the density of states.
Equation 11 states that the rate of electron transfer is proportional to the electronic coupling squared, multiplied by the Franck Condon-weighted density of states, which implicitly accounts for molecular vibrations. (We state Equation 11 here without derivation. It is actually a complicated expression, but our aim here is to present a qualitative discussion.)
5.4 Marcus-Hush theory
Slide 16 shows the central features of Marcus-Hush Theory– the most famous semi-classical theory of electron transfer. The red parabola represents the charge delocalised species, and the blue parabola represents the charge localised, or charge transfer species. di and df represent the equilibrium geometries along the generalised charge transfer coordinate, ΔE† represents the activation energy for the reaction, ΔE0 represents the energy change of the reaction (note that for traditional reasons, this is known as the driving force of the reaction), and λ represents the reorganizationenergyof the charge transfer reaction.
The reorganization energy, λ, is the energy change associated with relaxation of the molecular structure after electron transfer occurs. This is calledinner sphere relaxation. The reorganization energy also includes relaxation of any surrounding solvent molecules after the electron transfer process; this is known as outer sphere reorganization. reduces the complicated behaviour of the many intramolecular and intermolecular vibrational degrees of freedom to just one. Q represents the generalized reaction coordinate for the electron transfer reaction (a projection down from the 3N-6 molecular degrees of freedom, where N is the number of atoms in the system).
The Marcus-Hush theory of electron transfer can be represented mathematically as follows:
which leads to
and hence (see Slide 17)
5.5 Special topics
5.5.1 Reorganization energy
One can see from Equations 13 and 14 that the rate of electron transfer depends on several parameters. The rate is proportional to the square of the electronic coupling. There is an additional pre-exponential and exponential factor that also affects the rate of electron transfer. The important parameters are temperature, the reorganization energy λ and the driving force for the reaction (Slide 18).
Reorganisation energy and the other terms, such as the driving force and electronic coupling can in principle be obtained both experimentally and theoretically (through quantum chemical calculations) . The reorganization energy can be analysed by doing experiments in different polar solvents. A controllable change of ΔE0 and Vifcan be obtained by altering details of the chemical structure of the complex (e.g. chromophore orientation or bridge geometry).
5.5.2 Marcus-Hush inverted region
There is a very important relationship between the driving force and the reorganization energy. According to the Marcus-Hush equation (Equation 13), the rate of electron transfer is exponentially dependent on the square of the driving force minus the reorganization energy squared in the exponent. This gives rise to three regimes of electron transfer as shown in Slide 19:
· the normal region, where the driving force ΔE0 is less than the reorganization energy λ:
λ > ΔE0
· the activationless region, where the driving force is equal to the reorganization energy. This is situation that gives rise to the most rapid electron transfer.
λ = ΔE0
· theMarcus inverted region, where the reorganization energy is less than the driving force ΔE0.
λ < ΔE0
Slide 20 shows how the rate of electron transfer depends on the driving force (here in terms of free energy ΔG). One can clearly see that the maximum rate of electron transfer occurs when the driving force is equal to the reorganization energy. When the driving force becomes greater than the reorganization energy, we enter the Marcus inverted region.
Note that we have changed from ΔE to ΔG in this slide because the driving force also includes entropic changes:
ΔG = ΔH - TΔS
For the sake of simplicity, entropy has been ignored in the previous slides that discussed the theory.
It is interesting to note that the inverted region was proposed by Marcus in the 1950’s, but was not experimentally verified until the 1980s. Marcus won the Nobel Prize in Chemistry in 1992.
5.5.3 ET in proteins
Through-bond (superexchange) electron transfer can occur over distances greatly exceeding 120 nm. However the rate of electron transfer falls off exponentially with distance:
KET = k0 exp(-βR)
The parameter β is called the decay or attenuation coefficient. For most molecules, this falls in the range of 8-12 nm-1 and for proteins it is about 10 nm-1 (Slide 21).
In conclusion, Slide 22 summarises the discussion of energy transfer in photobiological systems from this and the previous two lectures.
About the author
David L. Andrews is Professor at the University of East Anglia, where he leads research on fundamental photonics and molecular energy transport, optomechanical forces, quantum and nonlinear optics. He has approximately 300 research papers and twelve books to his name. His current research focuses on light-harvesting, optical angular momentum, and mechanisms for optical nanomanipulation and switching. Andrews is a Fellow of the Institute of Physics, the SPIE, and the Royal Society of Chemistry, and he is 2013 Chair of the OPTO Symposium at Photonics West. Garth Jones a physical chemist who uses computational and theoretical techniques, and is principally interested in excited state dynamics of molecules and light-molecule interactions. Recently, he has developed a coupled electron-nuclear dynamics approach that has been used to model the photodissociation of water, and is currently being used to model important atmospheric and interstellar reactions. The major focus of his work now is the development of density matrix approaches for modelling electronic energy transfer in biological and artificial systems.
P.W. Atkins and R. Friedman, Molecular Quantum Mechanics, 5thedn. University Press, Oxford, 2011
V. Balzani (ed.) Electron Transfer in Chemistry Wiley-VCH, Weinheim, 2001
R.E. Blankenship, Molecular Mechanisms of Photosynthesis Blackwell, Oxford, 2002
A.S. Davydov, Quantum mechanics (2nd ed.) Pergamon Press, 1976
V. May and O. Kühn, Charge and Energy Transfer Dynamics in Molecular Systems, Wiley-VCH, Weinheim, 2004