Hopping and tunnelling processes are different physically, however share a basic characteristic: in order for an electron to hop from an atomic orbital to an atomic orbital of the neighbouring atom, the two orbitals must have a finite overlap. And so is with tunnelling: for an electron in a state to tunnel to a different state in a region geometrically separated from the region in which the initial state is confined, the two states must have a finite overlap. The term 'hopping' is often employed in the context of dealing with lattice models, where the direct space consists of a set of discrete points (one can think of an assembly of point-like atoms over which the available electrons can be distributed, with the continuum space between the atoms excluded from the direct space). In this context, movement of an electron over the lattice points can be viewed as a series of 'hops' or 'jumps', hence the term 'hopping'.

As for superexchange, I cannot do justice to the notion of superexchange without describing in some detail the notions of direct exchange and kinetic exchange. For this reason, I suffice here by referring you to the very excellent book (one of the best books ever written to my judgement) 'Lecture Notes on Electron Correlation and Magnetism', by Patrik Fazekas (World Scientific, Singapore, 1999). See in particular Sections 2.5.1 (Direct Exchange), 2.5.2 (Kinetic Exchange), and Chapter 5 (Mott Insulators); Superexchange is discussed in Sec. 5.2. With regard to 'hopping', see Eq. (2.131) in Sec. 2.5.2 (Kinetic Exchange), where the hopping integral t is defined: t is minus one-half of the expectation value of the Hamiltonian with respect to two specific configurations of atomic orbitals pertaining to two neighbouring atoms; one of the configurations, |ψ_4>, is the singlet configuration of two electrons, each residing on the neighbouring atoms, and the other configuration, |ψ_5>, is an even mixture of two singlet states of two electrons each corresponding to two electrons residing on the same atom. Given the nature of the two states |ψ_4> and |ψ_5>, it should be evident why the expectation value at issue determines the 'hopping' integral (or 'hopping' amplitude); it describes a process whereby one electron must have hopped to a neighbouring atom. One can explicitly show that t = 0 in the cases where the atomic orbitals of the neighbouring atoms do not overlap.

Hopping and tunnelling processes are different physically, however share a basic characteristic: in order for an electron to hop from an atomic orbital to an atomic orbital of the neighbouring atom, the two orbitals must have a finite overlap. And so is with tunnelling: for an electron in a state to tunnel to a different state in a region geometrically separated from the region in which the initial state is confined, the two states must have a finite overlap. The term 'hopping' is often employed in the context of dealing with lattice models, where the direct space consists of a set of discrete points (one can think of an assembly of point-like atoms over which the available electrons can be distributed, with the continuum space between the atoms excluded from the direct space). In this context, movement of an electron over the lattice points can be viewed as a series of 'hops' or 'jumps', hence the term 'hopping'.

As for superexchange, I cannot do justice to the notion of superexchange without describing in some detail the notions of direct exchange and kinetic exchange. For this reason, I suffice here by referring you to the very excellent book (one of the best books ever written to my judgement) 'Lecture Notes on Electron Correlation and Magnetism', by Patrik Fazekas (World Scientific, Singapore, 1999). See in particular Sections 2.5.1 (Direct Exchange), 2.5.2 (Kinetic Exchange), and Chapter 5 (Mott Insulators); Superexchange is discussed in Sec. 5.2. With regard to 'hopping', see Eq. (2.131) in Sec. 2.5.2 (Kinetic Exchange), where the hopping integral t is defined: t is minus one-half of the expectation value of the Hamiltonian with respect to two specific configurations of atomic orbitals pertaining to two neighbouring atoms; one of the configurations, |ψ_4>, is the singlet configuration of two electrons, each residing on the neighbouring atoms, and the other configuration, |ψ_5>, is an even mixture of two singlet states of two electrons each corresponding to two electrons residing on the same atom. Given the nature of the two states |ψ_4> and |ψ_5>, it should be evident why the expectation value at issue determines the 'hopping' integral (or 'hopping' amplitude); it describes a process whereby one electron must have hopped to a neighbouring atom. One can explicitly show that t = 0 in the cases where the atomic orbitals of the neighbouring atoms do not overlap.

Basically, summarizing and missing a lot of physicochemical concepts, Hopping mechanisms is the transmission of the proton/hole/electron, etc, charge thereafter, over the orbitals of the different species involved into the process, i.e, all the molecules that are between the donor and the acceptor. In the Superexchange the orbitals of all these species are needed to configure the suitable environment, but the charge is only localized in the donor and in the acceptor, not in the intermediate species.

Take a look to this book,

http://eu.wiley.com/WileyCDA/WileyTitle/productCd-3527407324.html

and to this review

http://pubs.acs.org/doi/abs/10.1021/cr400400p

Thanks Farid , detailed interpretation, and Sanchez-Martinez, and all your valuable clues! The concepts are new to me. I'd look into the books and article before asking for furthur details.

You are welcome Xin-Yu!

I found a paragraph in "Peptide-Mediated Intramolecular Electron Transfer: Long-Range Distance Dependence" (Chem. Rev. 1992, 92, 381-394 , Page 382), might be related to superexchange in "biological " context:

"

Intramolecular electron transfer may or may not use the intervening bridge molecular orbitals to accomplish effective electronic coupling between the donor and acceptor. When the electron transfer proceeds directly from the edge of the donor to the edge of the acceptor by mechanisms not involving the orbitals of the bridging peptide it is referred to as a through-space (or noncovalently linked) pathway. Since all our studies are in aqueous solution, the phrase "through space" does not imply 'through vacuum" since the pathway may involve solvent, ion pairing, and/or other noncovalent interactions. If on the other hand the orbitals of the bridging group assist in carrying the charge transfer between the donor and acceptor, the pathway is referred to as a through-bond pathway and the coupling between the donor and acceptor is therefore achieved through the orbitals of the bridging groups, in addition to other medium effects. In these cases the mechanisms of electron transfer have been discussed by a number of authors and is referred to as a superexchange mechanism. 27-30

"

There must be a correspondence between the descriptions here and that in quantum mechanics.

Any suggestions?

Dear Xin-Yu, if you consider Fig. 5.1 of the book by Fazekas (to which I have referred earlier on this page), displaying the superexchange between two cations via an intermediate anion, you will immediately see the analogy.

Dear Farid, I noticed the figure. What I can see is that bridging groups is analogous to an anion. Then I cannot go further.

In fact, the problem is I have difficulty in understanding a sentence:

In the article mentioned in the main question, it says "Thus, we can rule out the possibility of direct two-step electron hopping.......Thus, we conclude that the electron tunneling from the isoalloxazine ring to substrates must be mediated by adenine through a super-exchange mechanism."(Page 8108)

Dear Xin-Yu, as I pointed out in my very first response on this page, hopping, similar to tunnelling, is not possible when the relevant orbitals/wavefunctions do not overlap. In Fig. 5.1 of the book by Fazekas, one sees that the d-orbitals on the cations A and B have zero overlap, so that direct hopping of the spin-up electron on cation B to cation A is impossible (by the same reasoning, direct exchange is also ruled out). Because the p-orbital on the bridging anion in the middle is doubly occupied, it would seem that by the Pauli principle the above-mentioned spin-up electron cannot hop onto the anion (it costs a lage on-site energy to occupy the higher-lying state on this anion) and then onto the cation A. What happens instead, is what is explained in the text around Fig. 5.1. The key word is 'covalent mixing' of the p- and d-orbitals. The amount of mixing, denoted by b, is expressed in (5.55), which is a first-order perturbative expression. One clearly sees that if the p- and d-orbitals also do not overlap, the covalent mixing is vanishing (note the numerator of the expression for b -- clearly, the H, of which the pd-matrix element is calculated, cannot affect the ranges of the relevant p- and d-orbitals). A non-vanishing covalent mixing of the p- and d-orbitals, brought about by the non-vanishing operlap of the p- and d-orbitals, gives rise to an effective coupling between the spins on the cations, known as superexchange.

You may wish to consult the following original publication by PW Anderson where the ideas of covanelnt mixing and superexchange originate from:

http://journals.aps.org/archive/abstract/10.1103/PhysRev.79.350

What the authors of the paper you are referring to seem to suggest is along the lines described above; the "two-step electron hopping" they refer to, can be viewed as a process of covalent mixing as explained above. Note that the first step of the "two-step" hopping process in Fig. 5.1 of the book by Fazekas is just the hopping of  electrons between the cations and the anion inbetween; this hopping is possible because of a non-vanishing opevrlap between the neighbouring p- and d-orbitals, which in turn translates into a non-vanishing value of b, the amount of covalent mixing. Note further that for b=0, both energies in Eqs (5.56) and (5,58) become equal (both equal to 2εp), whereby the effective exchange coupling in Eq. (5.59) becomes vanishing.

Incidentally, would you kindly provide me with a copy of the paper you are referring to?

Thanks Farid , very detailed interpretation and clues! New things for me to study.

The paper, sure, I have sent it to you via message.

You are welcome Xin-Yu. And thank you for the paper which I have just received. Incidentally, as I wrote in my first message on this page, to appreciate superexchange one cannot bypass appreciating direct exchange and kinetic exchange; the effective exchange Jeff that one obtains in dealing with superexchange, is to be understood in the light of direct exchange.

Oh. Finally I found the exact answer in this great TUTORIAL REVIEW

Electron transfer in peptides and proteins

by

Meike Cordes and Bernd Giese 

@

http://pubs.rsc.org/en/content/articlehtml/2009/cs/b805743p

The keypoints are

"that an overlap between donor and acceptor orbitals allows electron tunneling or super-exchange between the redox partners"

"the separating matrix is regarded as a bridge between the two redox partners, with bridge orbitals involved in electronic communication between donor and acceptor,

thereby lowering the potential barrier and facilitating electron tunneling"

"Since ET rates drop exponentially with increasing distance, a practical upper limit for reasonably fast ET through proteins exists"

"Distances that exceed the upper limit for fast electron tunneling reactions can be overcome in a multistep hopping process"

"the main characteristics of sequential ET (hopping) are the occurrence of chemical intermediates"

"superexchange ET shows a strong exponential distance dependence, whereas electron hopping exhibits only weak distance dependence"