Relaxation phenomena in NMR
In general, relaxation processes in NMR are those responsible [1] for
a) Nonradiative energy exchanges between the spinsystem and any other degrees of freedom present in the sample (the latter constitute the socalled "lattice"), thus establishing thermodynamic equilibrium between the spin system and the rest. Most importantly, this establishes also the thermalequilibrium nuclear polarization without which there would be no signal.
b) Broadening of the spinsystem's energy levels related to a finite permanence time of the system in any particular quantum state. This is responsible for the homogeneous broadening of spectral peaks, as opposed to heterogeneous broadening due to a superposition of many sharp peaks.
There are many different relaxation mechanisms [113] which concur in an approximately additive way to the observed relaxation rate(s) of a spin system or, more correctly, of each quantum transition [9,15,16,18]. The additivity of relaxation rates with respect to the various contributing mechanisms is generally quite good, with typical deviations of the order of 1% (note that a relaxation rate is the inverse of a corresponding relaxation time which implies that relaxation times are not additive).
All such mechanisms (relaxation pathways) are characterized by i) an interaction involving a spin and ii) its random temporal variation due to a stochastic motion, generally activated by molecularlevel thermal effects. This picture can be used to classify the relaxation pathways by locating each one in a cell of a table whose columns are associated with various types of interactions and the rows with various types of motions. There are literally hundreds of relaxation pathways and each has its particular characteristics, above all the spin interaction which enables it, the random motion which drives it, a relaxation rate formula and its dependence on Larmor frequency, gamma ratio, motion parameters (correlation times) and their temperature dependence, etc.
On the theoretical side, the essential point is that each relaxation pathway is related to a stochastic component term in the Hamiltonian of the spin system, one which on the timescale of NMR relaxation rates averages to zero and therefore does not lead to distinct peak splittings in NMR spectra.
In practical situations, there is often one relaxation pathway which dominates over the others, though which one it is may depend upon many factors (including temperature). The best known case, at least in proton NMR in liquid state, is the dipolar relaxation which is enabled by direct dipoledipole interactions between nuclear spins and driven by the Brownian rotational tumbling of the whole molecule. It is to this case (and this case only) that one can apply the historic formulas of Bloembergen, Purcell and Pound (BPP) and those of Solomon [1417].
Scalar interaction as an example of a relaxationenabling pathway
By definition, scalar relaxation is enabled by the scalar interaction (also referred to as Jcoupling) j_{AB} between two nuclides A and B.
We will use it now to illustrate some of the principles mentioned above.
A scalar interaction is due to a simultaneous Fermi contact of both nuclei A and B with each bond electron and gives rise to a spinsystem Hamiltonian term of the form j_{AB}(I_{A}.I_{B}), where I_{X} is the spin vector of nuclide X and the dot denotes dot product between vectors. Actually, this is already a simplification since j_{AB} is really a tensor, being somewhat dependent upon the orientation of the local molecular fragment with respect to the field (this stems primarily from an extra contribution of the coupling between nuclear spins and the orbital momenta of bond electrons). However, since the orientation dependence is usually mild (from quantumtheoretical standpoint it is a second order effect) it can be neglected, leaving just the orientationindependent scalar part (hence the scalar interaction name). Just keep in mind that, should we keep it, the relaxation due to the anisotropic part of j_{AB} would be simply an additional relaxation pathway, akin to the well known and much more important relaxation pathway enabled by chemicalshift anisotropy (CSA).
However, even the isotropic (scalar) part of j_{AB} is not necessarily constant because its value depends to some extent on the stereoconformation of the whole molecule and is subject to random "modulations" due to internal molecular motions (vibrations, internal rotations, jumps between conformers, chemical exchange, ...). We must therefore consider j_{AB}(t) as a timedependent function and split it into two components:
1) The timeaveraged value J_{AB} = <j_{AB}(t)> which is the usual J listed in chemical reports and which manifests itself by splittings between spectral peaks, determining their multiplicity and positions.
2) The randomly varying remainder J_{AB}(t) = j_{AB}(t)  J_{AB} which averages to zero and, consequently, has no direct effect on the multiplicity and positions of spectral peaks. For those same reasons, however, this part contributes to (drives) the relaxation of nuclides A and B and, to some extent, also of any other nuclide coupled to them.
The question now is how large can be the two components of scalar interactions.
First of all, we know  both theoretically and empirically  that j_{AB} is only appreciable when the nuclides A and B are structurally close, i.e., separated by a limited number of covalent bonds (simultaneous Fermi contact at two different nuclides requires the bonding electron wavefunction to extend all the way from one to the other). Stereo vicinity, as opposed to structural vicinity, is in this case irrelevant.
We therefore distinguish nbond J's (n = 1, 2, ..., 9) and often write the n as a leading superscript (examples: ^{1}J_{AB}, ^{2}J_{AB}, ^{3}J_{AB}, ...)
As far as numeric values are concerned, a general discussion is nearly impossible since the J values range from barely measurable fractions of a Hz up to hundreds of kHz! The largest values are all related to one and two bond scalar couplings between heavy nuclides (statistically, the Fermi contact term grows sharply with the atomic number of the nuclide) and thus pertain to inorganic chemistry. Such cases also exhibit high quantumbased variability and must be studied casebycase.
The situation is simpler to handle in diamagnetic organic molecules where relaxations are dominated by the protonproton interactions, the anisotropy of the J's is quite small, and the scalar couplings are relatively easy to classify:
Onebond protonproton J exists only in the hydrogen molecule and is therefore of little interest in this context.
Twobond protonproton scalar couplings (^{ 2}J_{HH}) are ubiquitous, primarily between the two geminal protons of methylene groups (CH_{2}). In those cases where the two protons are chemically and magnetically equivalent, the geminal coupling may be unobservable in the spectrum, but it is still there and may contribute to relaxation. The ^{2}j couplings are almost always negative and most fall in the interval between 20 Hz to 15 Hz (though values up to +40 Hz crop up occasionally [19]). What is more important in the relaxation context is that their values are little sensitive to internal conformations of the molecule which hardly ever induce variations greater than 2 Hz. This makes the ^{2}j's rather ineffective as a scalar relaxation pathway  a feature which applies to most ^{1}j and ^{2}j scalar couplings (even those between heteronuclei).
Threebond scalar couplings (^{ 3}J_{HH}) between vicinal protons (>CHCH<) are also very common in organic molecules and they are those exhibiting the largest variations upon changes in molecular geometry; in particular, they depend strongly upon the dihedral angle φ defined by the three bonds. Semiempirically, this dependence is well known (see the Mestrelab utility [20] based on the Karplus formula [21,22] and its more recent analogs) and covers the approximate range of 0 to 16 Hz. It is frequently used to study molecular stereochemistry in those cases where the dihedral bond rotamers are sufficiently persistent on NMR timescale (not very common) or where the dihedral angle is constrained to a fixed value by furtheraway bonds (cyclic compounds with bridges and the like). When the dihedral rotamers are interconverting too fast to give separate spectra, one observes only an average, temperature dependent ^{2}J coupling, while the random component ^{2}J(t) affects just relaxation, with a correlation time given approximately by the lifetime of the individual rotamers. Given the Karplus relations, the mean amplitude of ^{2}J(t) never exceeds 8 Hz (5 Hz is probably the most likely value).
Scalar couplings of appreciable amplitude over four or more bonds are more rare. They fall mostly in the interval of 4 to +4 Hz, but there do exist surprising exceptions [19], particularly in rigid, extensively conjugated fragments and in sterically severely strained molecules (for example, there exists a ^{4}J_{HH} of 18 Hz). Because of the rigidity of such molecules, and because of the small longrange coupling values in flexible molecules, the amplitude of the variable parts ^{n}J(t) for n ≥ 3 probably hardly ever exceeds 1 Hz (though, admittedly, this is just the Author's conjecture and there might exist a few exceptions).
Orderofmagnitude estimates of scalar relaxation rates
The above estimates of the amplitudes of the variable parts of proton scalar couplings in organic molecules enable us to compare them to the direct dipoledipole interactions D_{AB}. Under rapid isotropic tumbling these average to zero, but their timevariable amplitude is very large.
For geminal protons, in particular, the mean amplitude of D_{AB} is about 30 kHz. Due to inverse thirdpower dependence of D_{AB} on internuclear distance, the D_{AB} values for vicinal proton pairs are smaller (of the order of 10 kHz) and they drop still more for protons pairs separated by three and more bonds.
Even so, if we compare these values with mean amplitudes of the scalar coupling variations, we find that the latter are typically about 3 orders of magnitude smaller. Only in extremely exceptional cases of exceptionally large longrange scalar couplings could the magnitudes of the two entities become remotely comparable. Considering that relaxation is a secondorder effect and the relaxation rates correlate with the square of the variable component of the enabling interaction, it would appear that scalar relaxation should be virtually always inferior to dipolar relaxation by a factor of about 10^{6} (give or take an order of magnitude).
This, however, is not exactly true because we must take into account also the correlation times τ of the random motions which drive the respective relaxation. For dipolar relaxations it is the characteristic time τ_{r} of the molecular rotational tumbling, while for a scalar interaction it is the mean lifetime τ_{c} of the involved conformers. The correlation time appears in relaxation rate formulas inside factors known as spectral density functions which, at least for low values of ωτ (ω is the Larmor frequency) are proportional to τ. Rather than discussing the details of the specific formulae, it is sufficient for the present purposes to understand that when τ_{c} is substantially larger than τ_{r}, it can compensate for the difference in the mean amplitudes of the interactions.
The τ_{r} values vary from tens of picoseconds (~10^{11}) for small molecules to tens of microseconds (~10^{5}) for large proteins. On the other hand, τ_{c} values can vary from values just as low as τ_{r} (in which case the scalar relaxation pathway is certainly negligible) up to values comparable to the relaxation times (seconds), depending upon the internal rotational barriers and upon temperature.
This means that there indeed exist cases where the scalar relaxation handicap due to interaction amplitudes is all but offset by a very large value of the ratio τ_{c}/τ_{r}. In particular, this can happen in small molecules with one or more internalrotation degrees of freedom characterized by high energy barriers. In such cases, should τ_{c} become even longer than relaxation times, the rotamers would start appearing in the NMR spectrum as independent entities. Before reaching that point, however, we are in a situation in which the rotamer spectra are still coalesced into a single, averageHamiltonian spectrum and only the linewidths (T_{2}) and the T_{1}'s indicate that we are not very far from the coalescence point. When this happens, scalar relaxation may dominate over dipolar relaxation.
For two different reasons, a situation of this kind is much less likely to occur in large molecules. First, the reorientation times τ_{r} are much longer so that it becomes difficult to reach a large value of τ_{c}/τ_{r} while still having coalesced rotamer spectra. Second, in molecules characterized by extensive steric constraints (such as nondenaturated proteins), local rotamers are all but ruled out (at most, there can be librations and vibrations about some average positions).
Conclusions
Analyzing the case of protonproton scalar and dipolar relaxation pathways, we have seen how their relative importance depends upon a delicate balance of two factors:
(1) the ratio of the amplitudes of the timevariable components of the interactions which enable the relaxations
(2) the ratio of the correlation times of the respective random motions which drive the relaxations.
The feasible ranges for both of these factors being very large (several orders of magnitude), all kinds of situations can occur. For protons in organic molecules, factor (1) is very much biased in favor of dipolar relaxation but in small molecules with strongly expressed rotamers, and at temperatures close to the coalescence of the rotamer spectra, factor (2) can offset the bias and overturn it in favor of scalar relaxation. This is unlikely to happen in large molecules where dipolar relaxation reigns supreme.
In systems with heteronuclei, particularly inorganic ones, the situation can be very different. Given the smaller γ values and larger distances, direct dipoledipole interactions are generally much weaker, while scalar couplings are often much larger. However, remember that it is not the value of the average J that matters, rather than the amplitude of its variable part J(t)  and that is a very unpredictable entity. The final outcome is therefore hard to predict and a casebycase analysis is called for.
Final note: Though there was a mention of chemical exchange as a source of the random modulations of the scalar couplings, I have later always mentioned only the modulation by means of internal rotations. This was intentional, since I was trying to keep the argumentation as simple and fluid as possible. But chemical exchange is certainly a very frequent reason behind J modulation and should be kept in mind. Fortunately, as long as J is indeed subject to random variations, their exact cause does not really matter and referring to one or to the other does not alter the above discussion.
