the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Simple rules for resolved level-crossing spectra in magnetic field effects on reaction yields

### Victor A. Bagryansky

### Yuri N. Molin

In this work we derive conditions under which a level-crossing line in a magnetic field effect curve for a recombining radical pair will be equivalent to the electron spin resonance (ESR) spectrum and discuss three simple rules for qualitative prediction of the level-crossing spectra.

The spin-correlated nature of radical (ion) pairs arising as intermediates in many natural or induced chemical transformations gives rise to a host of “magnetic and spin effects” in chemical reactions. It all started with observing (Bargon, 1967; Ward and Lawler, 1967) and understanding (Closs, 1969; Kaptein and Oosterhoff, 1969) strange-looking “polarized” nuclear magnetic resonance (NMR) spectra and has evolved into a mature field in itself with a wide range of powerful experimental and theoretical techniques relying on magnetically manipulating spins in chemical processes (Salikhov et al., 1984; Steiner and Ulrich, 1989; Hayashi, 2004), culminating in the modern high-tech finesse of advanced hyperpolarized NMR (Ivanov et al., 2014).

This paper deals with a curious bridge between the most humble magnetic field effect (MFE) curves, i.e., dependence of reaction yield on applied static magnetic field, and hyperpolarized NMR: additional sharp resonance-like lines that may occur against the smooth background of MFE due to genuine level crossings in the spin system of the radical pair. The lines were first discovered in a zero magnetic field (Anisimov et al., 1983; Fischer, 1983) and attributed to interference of pair states in the higher, spherical, symmetry conditions of a zero external field similar to the Hanle effect in atomic spectroscopy (Hanle, 1924). The zero field line, or low-field effect, was then put to the front as the possible physical mechanism of magnetoreception, and the research that followed was plenty. However, this completely overshadowed the other, spectroscopic, aspect of the level-crossing lines possible in fields other than zero.

Level crossing (Dupont-Roc et al., 1969; Silvers et al., 1970; Levy, 1972;
Astilean et al. 1994) and avoided crossing, or anticrossing (Eck et al., 1963; Wieder and Eck, 1967; Veeman and Van der Waals, 1970; Baranov and
Romanov, 2001; Yago et al., 2007; Kothe et al., 2010; Anishchik and Ivanov,
2017, 2019), spectroscopy has long been an established
tool in atomic and molecular spectroscopy as well as solid-state physics, providing structural information from specific (anti)crossing lines in
nonzero fields, whose positions are determined by interactions shaping the
energy levels of the system. For radical pairs purely spin level crossings
at nonzero fields in MFE first appeared in calculations in an already cited paper (Anisimov et al., 1983), although they were not discussed as they were
not observed in the accompanying experiments on radiolytically generated
radical ion pairs. However, a year later this group published a theoretical
work (Sukhenko et. al, 1985) that specifically explored level crossings in
nonzero fields for radical pairs with equivalent nuclei in only one pair
partner and gave explicit expression for their position determined by the hyperfine coupling (HFC) constant. Such lines were later indeed
experimentally observed in several systems by two teams (Stass et al., 1995b; Saik et al., 1995; Grigoryants et al., 1998; Kalneus et al., 2006a).
Furthermore, in a subsequent paper (Tadjikov et al., 1996) it was suggested
and demonstrated in numerical simulations for several systems of simple
structure, and confirmed in a proof-of-principle experiment, that hyperfine
structure of the second pair partner may be revealed at the level-crossing lines. The earliest mention of the very possibility of observing a resolved structure on a level-crossing line for a radical pair was probably the paper on MFE in a Ge-containing pair induced by a large difference in *g* values of the pair partners (Shokhirev et al., 1991), where a level-crossing line
appeared in modeling. Later the Δ*g*−induced level-crossing spectra were theoretically explored in detail in a paper (Brocklehurst, 1999).

In this work we develop the ideas of Sukhenko et al. (1985), Tadjikov et
al. (1996), and Brocklehurst (1999) to explore how a resolved structure may appear in MFE curves containing lines due to level crossing, referred to as
magnetically affected reaction yield (MARY) spectra. The discussion is based on the properties of radiation-induced
radical ion pairs, created by continuous wave (CW) X-irradiation of nonpolar solutions of suitable electron donor and acceptor molecules and detected by luminescence produced by pair recombination from an electron spin singlet state. To avoid a lengthy introduction to the properties of such pairs, the reader is referred
to a review book chapter (Stass et al., 2011) where a detailed discussion of
such pairs, as well as an introductory discussion of conventional MFE curves
in terms of level (anti)crossings, can be found. For the purposes of this
work it will suffice to assume that the pair starts from and recombines to a
spin-correlated singlet state, its spin evolution is governed by a Hamiltonian including only isotropic Zeeman and hyperfine interactions in independent
pair partners, the recombination itself is not spin-selective, the
relaxation can be neglected, and the theoretical counterpart to experimental
observables is the Laplace transform of a singlet state population *ρ*_{ss} taken in the time domain, as a function of applied static magnetic field. We
will first show analytically that for a pair containing a spin-*I* nucleus with a large HFC constant and spin *I*>1 at one partner and a compact arbitrary hyperfine structure at the other partner a resolved electron spin resonance (ESR) spectrum
of the “narrow” partner is expected at the level-crossing line due to the “driving” partner with large HFC, and then we will use this result to derive and discuss several simple rules for the possible resolved level-crossing spectra.

We start by quoting the key result of the original paper (Sukhenko et al., 1985) and recasting it in the form that is convenient for further
generalization. Given a radical pair having a single spin-*I* nucleus with HFC *a* in only one of the partners described by a Hamiltonian (setting =1, *ω*_{0}=*g**β**B*),

its eigenstates are divided into non-overlapping sets indexed by the total angular momentum projection ${\mathrm{\Sigma}}_{z}={S}_{\mathrm{1}z}+{S}_{\mathrm{2}z}+{I}_{z}$, and spin
evolution proceeds independently in state subspaces with different values *m* of Σ_{z} with maximum dimension 4. For a pair with a singlet initial state and observable recombination into a singlet state, the needed time-dependent probability *ρ*_{ss}(*t*) is a sum of partial probabilities over subspaces:

For a sub-ensemble of pairs with Σ_{z}=*m*, $\left|m\right|<I$, the subspace includes four states with eigenvalues

where

The states with maximum possible ${\mathrm{\Sigma}}_{z}=\pm \left(I+\mathrm{1}\right)$, i.e., electron spin-triplet states with maximum nuclear spin projection, are isolated eigenstates and are completely excluded from pair spin evolution. For the outermost blocks involved in spin evolution with ${\mathrm{\Sigma}}_{z}=\pm I$ there are only three states with eigenvalues

where

For each value of *m* from the range $-I<m<I$ the levels are degenerate in pairs in the zero field (*E*_{1}=*E*_{3}, *E*_{2}=*E*_{4}), which gives rise to the ubiquitous zero field line. In addition, for the inner blocks $\left|m\right|<I$ the levels *E*_{1} and *E*_{4} may become degenerate in non-zero fields as well, crossing in the sub-ensemble *m*<0 for *a*>0 and vice versa, occurring in the fields

For the outermost blocks the levels become degenerate only at zero field.
Thus, for a pair with a single spin-*I* nucleus there should be a zero field
line and, provided *I*>1, additional level-crossing extrema of Eq. (7) in “multiple” fields may be expected.

Picking up at this point, we take a different view of this problem. Taking advantage of results from works (Brocklehurst, 1976; Salikhov et al., 1984;
Stass et al., 1995c), the sought singlet state population for an initially
singlet radical pair with single spin-*I* nucleus in an arbitrary magnetic field can be written as

where

Assuming the simplest possible exponential recombination kinetics, the theoretical counterpart of the MARY spectrum is given by the Laplace transform of Eq. (8):

where the Laplace variable *s* has the meaning of recombination rate, or, more generally, the inverse lifetime of the spin-correlated state of the radical pair (Stass et al., 1995a). Direct evaluation of Eq. (12) with the substituted Eqs. (8)–(11) produces

A numerical experiment demonstrates that for positive *ω*_{0}
resonance-like peaks in *M*(*ω*_{0}) are produced only by the terms

at fields satisfying the condition

which is immediately seen to reproduce the level-crossing condition *E*_{1}=*E*_{4} of Eq. (7). All the other terms in Eq. (13) produce the smoothly varying background of the conventional magnetic field effect curve, related to
gradual change in the eigenbasis with variation of the applied magnetic field.

However, having now an explicit expression for MARY spectrum Eq. (13), we can be more quantitative in characterizing the level-crossing lines at “multiple fields” of Eq. (7). Evaluation of the prefactor $\left(\mathrm{1}+{D}_{m}\right)\left(\mathrm{1}+{D}_{m-\mathrm{1}}\right)$ in Eq. (14) at the crossing point of Eq. (7) produces the amplitude of the corresponding peak as

while developing Eq. (15) into Taylor series for a small deviation from the crossing point of Eq. (7) produces its Lorentzian width as

where the Laplace variable $s={\mathit{\tau}}^{-\mathrm{1}}$ is the inverse of the genuine exponential lifetime of the pair.

The formalism of Eq. (8) makes it very convenient to introduce a spin-*I*_{2} nucleus at the other partner of the pair. The corresponding counterpart to Eq. (8) would read as

where subscripts 1 and 2 relate the corresponding functions in Eqs. (9) and (10) to the first and second pair partners, with their respective nuclear spins *I*_{1,2} and coupling constants *a*_{1,2} introduced as appropriate, and “^{*}” stands for the complex conjugation. The summations in functions of Eqs. (9) and (10) run over all $\mathrm{2}{I}_{\mathrm{1},\mathrm{2}}+\mathrm{1}$ values of the respective nuclear spin projections.

The last term in Eq. (18) containing $\mathit{Re}\left({h}_{\mathrm{1}}\left(t\right){h}_{\mathrm{2}}^{*}\left(t\right)\right)$ now produces for each pair (*m*,*n*) 16 terms in *ρ*_{ss} of the form

and again numerical experiment demonstrates that for positive fields the
only resonance-like contributions to the Laplace transform *M*(*ω*_{0}) come from the terms

with positions of the maxima determined by the equation

while all other terms only contribute to the smooth background.

Equation (21) is equivalent to an eighth-order algebraic equation and does not lend itself to an exact analytic solution. To advance further, we shall now impose the assumption *a*_{2}≪*a*_{1} and focus on the vicinity of one of the crossing points of Eq. (7) for the “dominant” partner with the larger HFC. Proceeding in two steps now, we first note that these assumptions automatically place the second partner in the high-field limit *a*_{2}≪*ω*_{0}, which lets us develop the square roots *R*_{2,x} in Eq. (21) into linear forms in the small parameter ${a}_{\mathrm{2}}/{\mathit{\omega}}_{\mathrm{0}}$, similarly to a high-field approximation in conventional magnetic resonance, and convert Eq. (21) to a much simpler expression

This is equivalent to a cubic equation, which is linearized further by introducing a second small parameter ${a}_{\mathrm{2}}/{a}_{\mathrm{1}}$ to obtain the sought solution:

This is valid for each pair of nuclear spin projections (*m*,*n*), but since we consider the crossings in positive fields, as in Eq. (7), we should formally restrict *m* to be in the range $-I<m<\mathrm{0},$, while *n* can assume any of its 2*I*_{2}+1 possible values.

Tracing the two-step linearizing high-field assumption for the second partner back to the starting expression of Eq. (18), it is readily seen that if the second partner contains an arbitrary set of magnetic nuclei with HFC so small that the high-field limit is valid at fields of Eq. (7) for its entire ESR spectrum in a conventional sense, we can set from the beginning

which in the same order produces

where *a*_{k} and *n*_{k} are the HFC constants and spin projections for the *k*th nucleus. By the same token, an inhomogeneous spectrum, like a
“semiclassical” Gaussian shape (Schulten and Wolynes, 1978), can be used
in place of the sum ${\sum}_{k,{n}_{k}}{a}_{k}{n}_{k}$. Substituting
Eq. (25) into Eq. (21) as a result of the first step of linearization, we
obtain Eq. (22) with the term *n**a*_{2} changed for the sum ${\sum}_{k,{n}_{k}}{a}_{k}{n}_{k}$. Solving it by the second step of
linearization, we arrive at the result similar to Eq. (23):

This is the central result of this work, and its interpretation is as follows: provided the entire ESR spectrum of the second partner is compact
enough in comparison to the hyperfine coupling in the dominant first partner, each characteristic level-crossing “line at multiple field” of Eq. (7) spells out the ESR spectrum of the second partner, scaled in field by a constant factor, which depends on the specific crossing and is given in
parentheses in Eq. (26), with intensity of Eq. (16) borrowed from the original crossing and distributed over the spectrum as in the conventional
ESR. We also note that the field-scaling factor in Eq. (26) is identical to the scaling factor for the homogeneous width in Eq. (17), as both are
ultimately determined by the relative slopes of the linearized crossing
levels, so the scaling is uniform from both homogeneous and inhomogeneous
perspectives. The sum ${\sum}_{k,{n}_{k}}{a}_{k}{n}_{k}$ can be substituted for any spectral shape function *F*(*ω*_{0}), provided that it is restricted to a linear, first-order spectrum in
terms of conventional ESR. Second-order conventional ESR spectra (Fessenden,
1962) would not carry transparently through the double-step linearization procedure and would have required a more careful treatment to second order
at both steps.

We finally note that the same formalism can be used to analyze the level
crossings driven by substantial difference in *g* values of the pair partners
together with HFC, mentioned in the introduction and studied in detail in
Brocklehurst (1999). Assuming that the first partner has one spin-*I* nucleus with HFC *a*_{1} and *g* value *g*_{1}, while the second partner has no magnetic nuclei, but a shifted *g* value *g*_{2}, and introducing relative
shift of *g* values $\mathit{\delta}=\frac{{g}_{\mathrm{2}}-{g}_{\mathrm{1}}}{{g}_{\mathrm{1}}}\ll \mathrm{1}$, we should set
for the second partner

yielding

and two sets of solutions:

While the first set coincides with Eq. (7) or Eq. (26) with *a*_{2} set to zero and with the lines at ${\mathit{\omega}}_{\mathrm{01}}^{*}$ understood as lines in weak fields where the difference in *g* values is not yet consequential, the
second set has a small parameter *δ* in the denominator and gives the same lines translated to high fields. Expressions of Eq. (29) were first derived
in Brocklehurst (1999) and are re-derived here only to show the equivalence of the employed approach, and the reader is referred to Brocklehurst (1999)
for a more in-depth discussion of Δ*g*-induced level crossings.

Several comments regarding the results of the previous section are now in
order. First of all, the “driving” crossings of Eq. (7) require a nucleus
with spin *I*>1 and substantial HFC that would furthermore not compromise the relaxation properties of the recombining pair. Although nuclei with
spins $\frac{\mathrm{3}}{\mathrm{2}}$ and higher, like ^{35,37}Cl($\frac{\mathrm{3}}{\mathrm{2}}$) (Bagryansky et al., 1998), ^{27}Al($\frac{\mathrm{5}}{\mathrm{2}}$), ^{69,71}Ga($\frac{\mathrm{3}}{\mathrm{2}}$), ^{113,115}In($\frac{\mathrm{9}}{\mathrm{2}}$) (Sergey et al., 2012), and ^{73}Ge($\frac{\mathrm{9}}{\mathrm{2}}$) (Shokhirev et al., 1991; Borovkov et al., 2003), occasionally occur in magnetic field effect experiments, so far the only resolved lines in multiple fields of Eq. (7) have been reported for systems containing sets of equivalent spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei, either protons or fluorines (Stass et
al., 1995b; Saik et al., 1995; Grigoryants et al., 1998; Kalneus et al., 2006a). The best results making them promising for such applications were obtained for radical anions of either hexafluorobenzene (six fluorines with *a*=13.7 mT) or octafluorocyclobutane (eight fluorines with *a*=15.1 mT) paired with a narrow partner radical cation. This means that the single spin-*I* nucleus would in most cases be an effective spin equal to one of the possible values of the total spin for a set of equivalent spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei, with the corresponding statistical distribution, and thus there would be a corresponding composite level-crossing spectrum with contributions from all possible values of the total nuclear spin.

It is a fluke that in the most common case of an even number of
spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei the crossings of Eq. (7) occur at simple integer multiples of the HFC constant *a* and mostly overlap to reinforce each other, but the downside is that the overlapping spectra have different field-scaling factors. However, in reality the latter does not create that much of a problem. Let us take hexafluorobenzene with its six equivalent fluorines as a typical example. The possible values of pairs $\left(I,\left|m\right|\right)$ to produce crossings of Eq. (7) would be (3,2), (3,1), and (2,1), producing the lines of Eq. (7) in the fields 3*a*, 6*a*, and 3*a*, respectively. The corresponding field-scaling factors from Eq. (26) would be $\frac{\mathrm{12}}{\mathrm{8}}-\frac{\mathrm{1}}{\mathrm{24}}$, $\frac{\mathrm{12}}{\mathrm{2}}-\frac{\mathrm{1}}{\mathrm{24}}$, and $\frac{\mathrm{6}}{\mathrm{2}}-\frac{\mathrm{1}}{\mathrm{12}}$, respectively. It can be seen that the two overlapping lines at 3*a* have different scaling factors, reflecting the different slopes of the intersecting energy levels. Now let us estimate their relative contributions. The statistical weights of sub-ensembles with total spin *I* for a set of an even number *n* of spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei *W*(*I*;*n*) can be taken from Bagryansky et al. (2000),

and in our example evaluate to $W(\mathrm{3};\mathrm{6})=\frac{\mathrm{7}}{\mathrm{64}}$ and $W(\mathrm{2};\mathrm{6})=\frac{\mathrm{25}}{\mathrm{64}}$. It can be seen that the overlapping crossing at 3*a* is statistically dominated by the smaller total spin *I*=2, while the higher total spin *I*=3 is responsible for the crossing at 6*a*. Omitting the small corrections of $\frac{\mathrm{1}}{\mathrm{24}}$ and $\frac{\mathrm{1}}{\mathrm{12}}$, the field-scaling factors for the crossings at 3*a* and 6*a* are 3 and 6, respectively, with the crossing at 3*a* being nearly 4 times stronger and twice narrower, which is critical in field modulation experiments.

For our second example of eight equivalent fluorines we would get the weights of $W(\mathrm{4};\mathrm{8})=\frac{\mathrm{9}}{\mathrm{256}}$, $W(\mathrm{3};\mathrm{8})=\frac{\mathrm{49}}{\mathrm{256}}$, and $W(\mathrm{2};\mathrm{8})=\frac{\mathrm{100}}{\mathrm{256}}$ and lines at 10*a*, 5*a*, and 10*a*/3 from
*I*=4 in addition to already described lines at 3*a* and 6*a*. Again the strongest line at 3*a* is dominated by the contribution from the *I*=2 sub-ensemble with field-scaling factor 3 and swamps the much weaker nearby line at 10*a*/3 coming from *I*=4. The line at 6*a* is dominated by *I*=3 with the scaling factor of 6 and swamps the nearby line at 5*a* from *I*=4, and the only genuinely new line from *I*=4 is the line at 10*a* with the scaling factor of 10. To generalize this, we note that the “dominant” lines come from pairs $\left(I,\left|m\right|\right)$ with all possible values of *I* in the range $\mathrm{1}<I\le n/\mathrm{2}$ and $\left|m\right|=\mathrm{1}$. Comparing the expressions for the positions ${\mathit{\omega}}_{\mathrm{0}}^{*}$ of the crossing peaks and the field-scaling factors *f* in Eqs. (26) and (17) and omitting the small correction ${\left(\mathrm{2}{I}_{\mathrm{1}}\left({I}_{\mathrm{1}}+\mathrm{1}\right)\right)}^{-\mathrm{1}}$ in the scaling factors, we see that

From this we derive our “Simple” rule of structure:

*Given a radical pair with* *n*=2*k* *equivalent spin-*$\frac{\mathrm{1}}{\mathrm{2}}$ *nuclei with a large HFC constant a in one partner to drive spin evolution and a compact relative to the HFC constant ESR spectrum in the other partner, expect in the magnetic field effect curve* *k*−1 *progressively weaker copies of the ESR spectrum of the narrow partner at fields* ${\mathit{\omega}}_{\mathrm{0}}^{*}={f}_{q}a$ *scaled in the field by* *f*_{q}, ${f}_{q}=q(q+\mathrm{1})/\mathrm{2}$, $q=\mathrm{2},\mathrm{\dots}k$. *The strongest copy is the lowest of them, for* *f*=3; *i.e., it is at “triple field” and is “triply scaled”.*

Although so far the only experimental observations of the ESR structure
using this approach have been the “spectrum” at 3*a* for an unresolved
inhomogeneous spectrum as a proof of principle in Tadjikov et al. (1996) and arguments based on the lack of the inhomogeneous spectrum at 3*a* in several
works on radiation chemistry (Tadjikov et al., 1997; Usov et al., 1997;
Sviridenko et al., 1998) from just one group, we hope that the current surge
of interest to the level (anti)crossing interpretation of magnetic resonance
will draw attention to this aspect of the humble magnetic field effect
experiments.

Although the case of an even number of driving spins-$\frac{\mathrm{1}}{\mathrm{2}}$
is the most convenient, it is not the only possible one. Still staying with
equivalent nuclei, one experimental case of three spins-$\frac{\mathrm{1}}{\mathrm{2}}$ has been reported, for the radical anion of 1,3,5-trifluorobenzene
complemented with a partner with a narrow ESR spectrum (Kalneus et al., 2006a). The only level-crossing line here comes from the effective total
spin $I=\frac{\mathrm{3}}{\mathrm{2}}$ for three fluorines, as expected, and could in principle be used as a vehicle to obtain the ESR spectrum of the partner. Furthermore, tracing back how the structure-bearing partner was introduced after linearization in Eqs. (24) and (25), we see that there is nothing special in
the single spin-*I* or equivalent spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei other than the possibility of treating them analytically and obtaining a well-defined level-crossing line if the HFC coupling is sufficiently strong. Of course,
this is a rather substantial “other than”, but it does not exclude other
possible spin systems as the driving partner if they appear.

Such systems do indeed exist. Several experimental reports of the resolved MARY spectra for systems with non-equivalent nuclei with large HFC constants, in all cases fluorines, have been published. These include radical anions of 1,2,3-trifluorobenzene (Kalneus et al., 2007), pentafluorobenzene (Kalneus et al., 2006b), and recently several fluorosubstituted diphenylacetylenes (Sannikova et al., 2019), again complemented with a radical cation with a narrow ESR spectrum. The spectra featured well-defined lines that were reproduced in simulations and were traced to clusters of level crossings in the spin system of the pair. Although the “multiplication of the ESR spectrum” of the possible pair partner would in this case not be very informative due to high concentration of close and overlapping level crossings, it would be rather important to at least keep in mind the inhomogeneous broadening of these lines due to hyperfine couplings in the second partner.

Analysis of the level-crossing spectra for systems with non-equivalent
nuclei also helped develop the concept of “active crossings” (Pichugina
and Stass, 2010) as a substitute for traditional selection rules for
transitions in conventional magnetic resonance. To put it simply, of all the
energy-level crossings present in the spin system of the pair, only those between levels reachable from the same initial (singlet) state of the pair may produce observable lines due to interference of coherently populated
eigenstates. In terms of the discussion of this work, the active crossings would be the crossings of levels from the same four-dimensional blocks with energies of Eq. (3), to which correspond the terms with fixed *m* in the sums
of Eq. (13).

The transparency of translating the ESR spectrum of the narrow partner to
the level-crossing line due to the partner with strong HFC is rather
amazing and is a consequence of separating these two roles and adding the new nuclei to the partner that originally just complements the pair. This
can be more easily understood using the language of wave functions rather
than the density matrix as follows. Suppose we have an active level crossing of Eq. (7) from a subspace of pair eigenstates of Eq. (3) spanning four
functions of the product basis |*S*_{1z},*I*_{z}〉_{1}|*S*_{2z}〉_{2} with projections
|*α*,*m*〉_{1}|*β*〉_{2},
$\left|\mathit{\alpha},m-\mathrm{1}{\rangle}_{\mathrm{1}}\right|\mathit{\alpha}{\rangle}_{\mathrm{2}}$,
|*β*,*m*〉_{1}|*α*〉_{2}, and $\left|\mathit{\beta},m+\mathrm{1}{\rangle}_{\mathrm{1}}\right|\mathit{\beta}{\rangle}_{\mathrm{2}}$. The only non-secular interaction in the pair is hyperfine coupling
in the first partner, which means that the eigenstates of the pair will be
of the form |*ξ*_{i}〉_{1}|*S*_{2z}〉_{2}, still remaining the products of functions for the two partners. The energies of the eigenstates will be the sums of energies for the two partners, and the nontrivial spin evolution leading to level-crossing lines is due to simultaneously projecting the starting singlet state onto several eigenstates at the moment of pair creation and back at the moment of recombination and beating due to different energies of the populated eigenstates that partially stops when some energies become equal; i.e., some levels cross.

Now let us introduce nuclei to the second partner, i.e., augment its eigenstate |*S*_{2z}〉_{2} to include the indices of the newly
introduced nuclear spin projections to $|{S}_{\mathrm{2}z},{n}_{\mathrm{1}z},\mathrm{\dots},{n}_{kz}{\rangle}_{\mathrm{2}}$. Since we are in the conditions of the high-field limit for the second partner, as in conventional ESR, the augmented eigenfunctions will in fact be products of electron and nuclear functions $\left|{S}_{kz}{\rangle}_{\mathrm{2}}\right|{n}_{\mathrm{1}z},\mathrm{\dots},{n}_{kz}\rangle $ splitting in energy by the corresponding secular contribution $\pm \frac{\mathrm{1}}{\mathrm{2}}{\sum}_{k,{n}_{k}}{a}_{k}{n}_{k}$. The states of the newly introduced nuclei in this approximation are not affected by spin evolution in the pair, and thus the nuclear function $|{n}_{\mathrm{1}z},\mathrm{\dots},{n}_{kz}\rangle $ effectively becomes a new conserved multi-index, by which the state space for the pair augmented with new nuclei is partitioned. The original four-dimensional subspaces housing the active crossings are multiplied
into copies differing only by the new multi-index, each giving the same
active crossing, but at a correspondingly shifted energy and with a
proportionally reduced intensity borrowed from the original crossing. The
varying scaling with the field comes from the different relative slopes of
the linearized crossing levels, which now differ from the $\pm {\mathit{\omega}}_{\mathrm{0}}/\mathrm{2}$ of conventional ESR and become progressively more shallow with an increasing external field, spreading the same vertical shift in energy to a progressively wider horizontal scaling with the field.
The same can be said about the scaling of the homogeneous contribution to the linewidth of Eq. (17), which converts the same width of the energy levels due to the finite lifetime into the width along the field axis. Since
this multiplication of state subspaces is entirely due to the second
partner, this discussion applies to any hyperfine structure of the driving
partner, provided its HFC constants are sufficiently high.

The situation with adding the new nuclei to the first, driving, partner is
quite different. Now the function augmented with additional nuclear spins is
not of the high-field limit case, and effectively a new interaction is added into a coupled spin system. Let us again turn to the wavefunction illustration, first for single nuclear spin-$\frac{\mathrm{1}}{\mathrm{2}}$ and just one added spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nucleus with a small HFC constant *a*_{2}≪*a*_{1}. The original functions $|{S}_{\mathrm{1}z},{I}_{z}{\rangle}_{\mathrm{1}}$ for the first partner are now augmented to functions $|{S}_{\mathrm{1}z},{I}_{z},n{\rangle}_{\mathrm{1}}$, which do not factor into a simple product, and the newly introduced index *n* is not just an external conserved quantity. Instead we have the introduction of (weak)
additional interactions into a system of crossing levels, which leads to
anticrossings. Since the total spin projection is conserved for each
radical, e.g., functions $|\mathit{\alpha},\mathit{\alpha},\mathit{\beta}{\rangle}_{\mathrm{1}}$, $|\mathit{\alpha},\mathit{\beta},\mathit{\alpha}{\rangle}_{\mathrm{1}}$, $|\mathit{\beta},\mathit{\alpha},\mathit{\alpha}{\rangle}_{\mathrm{1}}$ now fall into one sub-block of the
Hamiltonian and are mixed together, and we note that without the added
nucleus the first of them and the two other were in different blocks and would have contributed to different active crossings. Now addition of a weak
new coupling introduces an anticrossing, possibly between different blocks,
instead. The key questions are now what anticrossings are being introduced and whether the original crossings turn into anticrossings upon addition of
the new interaction. This situation must be familiar to experts in
hyperpolarized NMR in the form of level anticrossings in three-spin systems,
where one nucleus is *J* coupled to two other nuclei (Miesel et al., 2006;
Pravdivtsev et al., 2013). Another close example is a three-spin system
biradical ion/radical ion with an exchange interaction within the biradical and hyperfine interaction with a nucleus in either partner (Lukzen et al., 2002;
Verkhovlyuk et al., 2007), where a nucleus in the biradical ion produces an
anticrossing near the main line of *J* resonance in the biradical, while a
nucleus in the radical partner produces a crossing. A similar dichotomy is also observed in magnetic effects in a biradical/stable radical complex with
different distributions of inter- and intra-partner exchange interactions
(Magin et al., 2004, 2005, 2009).

To analyze the resulting changes in eigenstructure, let us review Eq. (3). The expressions for energies are clearly of the form $\left(-\frac{a}{\mathrm{4}}\pm {R}_{m}\right)\pm \frac{{\mathit{\omega}}_{\mathrm{0}}}{\mathrm{2}}$ and are the sums of the energies of two independent partners. One of them has coupled electron and nuclear spins and corresponds to the first term, which is the
familiar Breit–Rabi expression (Breit and Rabi, 1931) for arbitrary nuclear
spin *I*. The other partner has just electron spin. We also require that the
states of Eq. (3) be reachable from the same electron spin singlet state. To
obtain the pair state subspace with total spin projection Σ_{z}=*m*, we thus need to combine two states of the first partner with total
projection ${M}_{z}=m+\frac{\mathrm{1}}{\mathrm{2}}$ spanning the product basis
states $|\mathit{\alpha},m{\rangle}_{\mathrm{1}}$, $|\mathit{\beta},m+\mathrm{1}{\rangle}_{\mathrm{1}}$ with the
|*β*〉_{2} state of the second partner and two states of the first partner with total projection ${M}_{z}=m-\frac{\mathrm{1}}{\mathrm{2}}$
spanning the product basis states $|\mathit{\alpha},m-\mathrm{1}{\rangle}_{\mathrm{1}}$, $|\mathit{\beta},m{\rangle}_{\mathrm{1}}$ with the |*α*〉_{2} state of the
second partner. So, the energies of Eq. (3) correspond to the following functions:

where trig notation was adopted for the mixing coefficients in the Breit–Rabi functions.

Now let us introduce an additional nucleus with spin *K* with a weak hyperfine coupling into the first partner by building product functions of the form $\left|{\mathit{\psi}}_{i}\right(m)\rangle |n\rangle $ and treating the new hyperfine interaction as perturbation $\widehat{V}=b{\mathit{S}}_{\mathrm{1}}\mathit{K}$. We recall that the original active crossings were the ones within the blocks of Eq. (32) for
*E*_{1}=*E*_{3}, *E*_{2}=*E*_{4} in the zero field and *E*_{1}=*E*_{4} in the fields of Eq. (7). Now we note that the perturbation is diagonal with
respect to the second electron spin and thus has zero matrix elements *V*_{13}, *V*_{24}, and *V*_{14} between the required functions and conclude that the original active crossings all survive and do not turn into anticrossings.

Non-vanishing matrix elements can be obtained between functions of adjacent blocks of Eq. (32), e.g., $\left|{\mathit{\psi}}_{\mathrm{1},\mathrm{2}}\right(m)\rangle $ and $\left|{\mathit{\psi}}_{\mathrm{1},\mathrm{2}}\right(m-\mathrm{1})\rangle $:

with similar results for all four combinations of indices 1 and 2 for the two functions and all four combinations for indices 3 and 4 of the other two functions. This implies anticrossings if the original functions corresponded
to crossing energy levels. Looking at expressions for energies of Eq. (32),
this in turn implies ${R}_{m}=\pm {R}_{m-\mathrm{1}}$, which is indeed possible in the zero field in the variant ${R}_{m}={R}_{m-\mathrm{1}}$. Thus, for each pair of adjacent
four-dimensional blocks of Eq. (32) with Σ_{z}=*m* and ${\mathrm{\Sigma}}_{z}=m-\mathrm{1}$, we had four crossings in the zero field for ${E}_{i}\left(m\right)={E}_{i}(m-\mathrm{1})$, $i=\mathrm{1},\mathrm{\dots},\mathrm{4}$ that turn into the respective anticrossings. Furthermore, these anticrossings stitch together all the subspaces of the
initially partitioned state space. Note that the original crossings were not
active: they corresponded to different subspaces, and thus their states would not be populated simultaneously and interfere. However, as opposed to the
interference effects of genuine level crossings, the anticrossings just
reshape the energy-level layout and do not require the simultaneous population of the contributing states, and thus the added weak hyperfine
interaction in the first partner turns dormant crossings into acting
anticrossings in the zero field. We further note that the specific hyperfine structure for the added nuclei is not important: it just suffices that they
provide the nonzero couplings of Eq. (33) between the zero-field states of
the adjacent blocks.

Since the newly introduced weak interaction does not affect the original
active crossings, we may evaluate its effect on energy levels to first order
by evaluating the average values of the perturbing interaction for product
functions $\left|{\mathit{\psi}}_{\mathrm{1}}\right(m)\rangle |n\rangle $, and for the interesting case of the crossing *E*_{1}=*E*_{4}, we obtain

This moves the surviving active crossing in energy by

evaluated at the crossing field of Eq. (7), which is then converted to shift
in the field by the field-scaling factor due to differential slopes taken from Eq. (26). Such evaluation yields some unwieldy expression, hardly qualifying for a “simple rule” and of little practical utility, but note that the
shift in the field is simply proportional to *b**n*, with the scaling factor depending only on the properties of the “driving” spins and particular
crossing, and again this linearity means that the spectrum corresponding to
the added weaker hyperfine structure will be spelled out at the original
crossing point of Eq. (7), as was the case for Eq. (26). In practice this
means that additional nuclei with smaller HFC constants in the driving
partner at least contribute an inhomogeneous broadening to the level-crossing line, complicating its experimental observation, as is the case for the radical anion of 1,2,4,5-tetrafluorobenzene containing two protons with
smaller couplings in addition to four equivalent fluorines (Kalneus et al., 2006a). The reason for the noted linearity is of course the applicability of first-order perturbation theory due to survival of the active crossings. Such simple considerations can sometime help advance in a problem that seems otherwise overwhelming (Stass, 2019).

Similar issues of “localization of interaction” in pair partners also
arise in the discussion of Δ*g*-induced resonances (Brocklehurst,
1999) and in the discussion of the zero field line in the magnetic field effect curve, where it was mentioned several times that distribution of HFC over
both partners as opposed to their concentration in one partner decreases the
magnitude of the effect (Timmel et al., 1998; Kalneus et al., 2005; Woodward
et al., 2008). In our picture it appears as arising of anticrossings at the zero field that spreads and counteracts the active crossings originally present there, additionally washing away the well-defined partitioning into state
subspaces with pronounced state interference. Another place where the
crossing vs. anticrossing discussion is very relevant is the so-called
*J* resonance in radical pairs (Hamilton et al., 1989; Shkrob et al., 1991) or linked donor–acceptor dyads (Weller et al., 1984; Ito et al., 2003; Wakasa et al., 2015; Steiner et al., 2018), where exchange coupling between the two
partners shifts the triplet electron spin manifold relative to the singlet, and at a certain magnetic field the singlet term crosses with one of the triplet
sublevels. In many cases these crossings turn into an anticrossing due to additional weaker interactions, such as HFC with magnetic nuclei, but
traditionally the situation is often still referred to as “ST_{−} crossing”, even though the technical discussion clearly identifies it as anticrossing.

From the practical viewpoint the important difference between crossings and
anticrossings is that the former partially block spin evolution due to state interference and thus lock the pair in its initial state, while the latter accelerate spin evolution and assist in leaving the initial state.
Furthermore, using the settings of this work as an example, while the
crossings produce sharp lines with widths of the order of inverse lifetime
*τ*^{−1} separated by intervals of the order of introduced interaction
*V*, the anticrossings produce much broader lines of the opposite phase with
widths on the order of $\sqrt{{\mathit{\tau}}^{-\mathrm{2}}+{V}^{\mathrm{2}}}$. If we introduce three parities, to indicate the initial state ${\mathrm{\Gamma}}_{\mathrm{i}}=+\mathrm{1}$ for singlet and −1 for triplet, the observation state ${\mathrm{\Gamma}}_{\mathrm{o}}=+\mathrm{1}$ for singlet and −1 for triplet, and the type of crossing ${\mathrm{\Gamma}}_{\mathrm{c}}=+\mathrm{1}$ for crossing and −1 for anticrossing, then we can derive our even simpler Rule of signs:

*The sign of a feature in a level-crossing spectrum is given by* $\mathrm{\Gamma}={\mathrm{\Gamma}}_{\mathrm{i}}\cdot {\mathrm{\Gamma}}_{\mathrm{o}}\cdot {\mathrm{\Gamma}}_{\mathrm{c}}$.

Creation of a spin-correlated radical (ion) pair is a shock excitation for a radical pair Hamiltonian, and, as any shock-excited quantum system, the pair “rings” at its eigenfrequencies (Salikhov, 1993). Since for the pairs of this work the Hamiltonian of the pair is a sum of independent Hamiltonians for the two partners, the ring frequencies must be some linear combinations of the eigenfrequencies for the pair partners. It is clear that creating the pair in a singlet state with a given nuclear configuration must select some subset of the possible ring frequencies, and the examples discussed in this work provide some very useful insight regarding this selection.

Let us again review the expressions of Eq. (32) for energies/functions of
the typical subspace of a radical pair spin system. The condition *E*_{1}=*E*_{4} for the level-crossing line can be trivially rearranged as

where on the left-hand side we have the difference of energies of two eigenstates of the first pair partner and on the right-hand side an equivalent difference for the second partner. Now we look at the corresponding functions *ψ*_{1,4}
and recall that our pair starts from and recombines to a singlet state with
nuclear spin projection *m*, which is the function $\left(|\mathit{\alpha},m{\rangle}_{\mathrm{1}}|\mathit{\beta}{\rangle}_{\mathrm{2}}-|\mathit{\beta},m{\rangle}_{\mathrm{1}}|\mathit{\alpha}{\rangle}_{\mathrm{2}}\right)/\sqrt{\mathrm{2}}$. We note the following correlation
between “transitions” between functions *ψ*_{1,4} and changes in the
spin states of the individual radicals:

The two latter relations mean an allowed ESR transition in the first radical
and simultaneously an opposing allowed ESR transition in the second radical,
at the same frequency given by the differences in the energies of the
corresponding true eigenstates of each of the pair partners. The statement
about “allowed ESR transition” should be understood as transition induced
by nonzero matrix element of electron spin operator, e.g., *S*_{1x} (for the
first partner), between the factors of the functions pertaining to this
partner, and for functions *ψ*_{1,4} this is reduced to nonzero matrix
element between the functions of Eq. (37). Therefore in this case the level-crossing line appears in the field where such a flip-flop energy conserving
transition in the pair partners can occur. Inspection of functions in Eq. (32) demonstrates that the same also turns out to be true for the two level crossings in the zero field that correspond to conditions *E*_{1}=*E*_{3} and *E*_{2}=*E*_{4}.

Now let us consider the case of compact ESR structure at the second partner, for which the level-crossing condition of Eq. (25) can again be slightly rearranged to give

The functions for the subspace with the conserving nuclear configuration of the narrow partner are now given by expressions of Eq. (32) with all functions multiplied by the conserved multi-index $|{n}_{\mathrm{1}z},\mathrm{\dots},{n}_{kz}\rangle $. The previous paragraph can be repeated nearly word for word with the conclusion that the level-crossing lines appear in the field where simultaneous flip-flop energy conserving allowed ESR transitions in the pair partners can occur, between the entangled electron-nuclear energy levels of the first partner and between the conventional high-field limit decoupled energy levels in the second radical.

Finally, let us consider the case of both partners containing a single
nucleus with arbitrary spin without any assumptions on the relative sizes of
their HFC constants *a*_{1,2}. The level-crossing condition for this case is given by Eq. (21), which can again be rearranged into

expressing the equality of transition frequencies for the two partners. To build the functions for a sub-ensemble with ${\mathrm{\Sigma}}_{z}=m+n$ reachable from the same singlet state $\left(|\mathit{\alpha},m{\rangle}_{\mathrm{1}}|\mathit{\beta},n{\rangle}_{\mathrm{2}}-|\mathit{\beta},m{\rangle}_{\mathrm{1}}|\mathit{\alpha},n{\rangle}_{\mathrm{2}}\right)/\sqrt{\mathrm{2}}$ similar to Eq. (32), we need to combine the Breit–Rabi functions for the first partner with total projection ${M}_{z}=m+\frac{\mathrm{1}}{\mathrm{2}}$ with the Breit–Rabi functions for the second partner with total projection ${N}_{z}=n-\frac{\mathrm{1}}{\mathrm{2}}$ and, vice versa, functions with ${M}_{z}=m-\frac{\mathrm{1}}{\mathrm{2}}$ with functions with ${N}_{z}=n+\frac{\mathrm{1}}{\mathrm{2}}$, to obtain function sets

The energy matching condition of Eq. (39) corresponds to the following functions:

We see that again the level-crossing line corresponds to a simultaneous energy-conserving flip-flop transition

in the two pair partners that correspond to allowed ESR transitions in the opposite directions.

We should not try to generalize these observations beyond what can be established from results derived in this work, but the pattern is quite obvious, and therefore we suggest for further consideration and discussion a provisional Rule of resonances:

*The level-crossing lines appear in the fields where simultaneous energy-conserving ESR allowed flip-flop transitions can proceed in the pair partners.*

The idea that simultaneous transitions in spin systems of pair partners can lead to level-crossing lines probably goes back to work (Brocklehurst, 1999), and a similar result was also obtained for interference of ESR transitions in the ESR (RYDMR) spectra of radical pairs in Salikhov et al. (1997), Tadjikov et al. (1998).

In this section we present several figures illustrating typical resolved
level-crossing spectra that could be reasonably expected in experiment. For
all figures the driving partner with large HFC constants mimics
hexafluorobenzene radical anion and has six equivalent spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei with HFC constant *A*, while the second partner contains two
equivalent or nonequivalent spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei, as indicated. All parameters, i.e., the external magnetic field, the smaller
couplings in the second partner, and the recombination parameter *s*, are
measured in the units of *A*.

Figure 1 shows a review spectrum for a pair with equivalent nuclei in both partners that can be calculated analytically in the full field range from zero to well past the level-crossing lines. In this case the smaller couplings are taken as one-tenth of the large ones, and the spectrum fully conforms to expectations as discussed in this work.

Figures 2 and 3 show in more details the regions of the level-crossing lines at 3*A* and 6*A* for the parameters used in Fig. 1, as well as for a 4-fold reduced recombination parameter, i.e., for a longer lived pair, to increase
resolution. Note the familiar 1-2-1 pattern for two equivalent
spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei with splittings equal to 3$A/\mathrm{10}$ and 6$A/\mathrm{10}$, as
expected. Also note a pair of lower-intensity lines with half the splitting
in Fig. 2, corresponding to the minor contribution of the sub-ensemble with total nuclei spin of the “driving” partner *I*=3 with the scaling factor
$\mathrm{12}/\mathrm{8}$ instead of 3 to the level-crossing line at triple HFC constant.

When non-equivalent nuclei need to be introduced into the second partner the full MFE curve can no longer be calculated analytically, and only the regions of the level-crossing lines can be described assuming compactness of the ESR structure of the second partner. Figures 4 and 5 show these regions for a pair that has two spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei with different HFC constants, equal to $A/\mathrm{10}$ and $A/\mathrm{40}$, in the second partner. Again the familiar “doublet of doublets” pattern with the expected splittings is clearly seen in both figures. More busy spectra for systems with a more complicated hyperfine structure could have been readily generated, but they bring no new insight and would hardly ever be obtained in experiment, and thus are not included here.

Finally, Fig. 6 shows what happens if the smaller HFC constant becomes not that small and the linearizing assumptions of this work are pushed too far. The figure, which was obtained by analytic calculation of the full MFE curve, illustrates the region in the vicinity of the level-crossing line at triple HFC constant for a pair that has two spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei with HFC constant $A/\mathrm{5}$ in the second partner. The 1-2-1 pattern becomes distorted, the lines are no longer equidistant, and the spectrum for a longer-lived pair demonstrates that the central line of the triplet is split. All these features are of course familiar from conventional second-order ESR spectra and are due to violation of the high-field approximation. It can be reasonably claimed that to stay within the linearized paradigm of this work the upper limit for the HFC constants in the second partner is about one-tenth of the HFC constant of the driving partner. Given the couplings in the actual available experimental systems of 13.7 mT (hexafluorobenzene radical anion) and 15.1 mT (octafluorocyclobutane radical anion), there is hope in resolving couplings of the order of milliTesla, which is quite typical for organic radical ions.

In this work we have provided a full justification for the term “MARY ESR” introduced in Tadjikov et al. (1996) by showing that under the claimed conditions the level-crossing lines will indeed recover an arbitrary ESR spectrum without limitation to the simple cases discussed originally in Tadjikov et al. (1996). We also hope that the discussed parallels between level-crossing spectroscopy and conventional magnetic resonance spectroscopy can help bridge the existing conceptual and perceptional gap between the two fields. Although the discussion relied on the properties of a specific class of systems, radiation-induced radical ion pairs in nonpolar solutions, it may well be that similar approaches could be more easily realized on other correlated spin systems. Given that the language of level (anti)crossings also becomes a unifying language in hyperpolarized magnetic resonance (Sosnovsky et al., 2016), the suggested approaches may come more naturally to experts in spin chemistry and magnetic resonance today than they were 20 years ago and thus may be more useful now rather than alien as they looked originally.

On the more sober side, though, it is clear that many real experimental systems will be more complicated than discussed here. In particular this will be true for photoinduced radical pairs, for which pair partners often cannot be treated as independent electron spins, and additional electron spin-spin interactions like dipolar and exchange must be accounted for. Furthermore, the longer lifetimes of the pairs one is often interested in bring such factors as relaxation and chemical reactivity of the radicals into picture, which also complicates the matters considerably. These factors have received significant attention in the context of the level-crossing line in the zero field, related to tentative magnetoreception (see, e.g., Efimova and Hore, 2008, 2009; Lau et al., 2010; Kattnig et al., 2016a, b; Worster et al., 2016; Kattnig and Hore, 2017; Keens et al., 2018; Babcock and Kattnig, 2020), and so far the feeling is that their due account is anything but “simple”. Additional interactions destroy the neat partitioning of state space into manageable subspaces similar to introduction of additional nuclei in the “crossing vs. anticrossing” section above, and relaxation further adds to this complexity. There is no reason to expect that things will become much easier when moving from zero field crossings to level-crossing lines in non-zero fields, and probably comparable effort would be needed to analyze the consequences and implications of such additional complications. The more valuable then seem the simple and comprehensible insights elaborated in this work for a more sterile but still realistic model of a radiation-induced radical ion pair.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

DVS and YNM conceived the study. VAB derived the key result. DVS and VAB prepared the manuscript with contributions from all the authors.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Robert Kaptein Festschrift”. It is not associated with a conference.

The authors are grateful to the Russian Science Foundation (RSF) and to the paper by Kaptein (1971) for the inspiration for this work.

This research has been supported by the RSF (project no. 20-63-46034).

This paper was edited by Jörg Matysik and reviewed by two anonymous referees.

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- Abstract
- Introduction
- Derivation of resolved level-crossing spectra
- Even number of equivalent spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei to drive spin evolution in the pair
- Other possible configurations of the driving spins
- Introducing nuclei into the driving partner: crossings vs. anticrossings
- Level-crossing lines as flip-flop resonance in pair partners
- Compendium of typical resolved spectra
- Conclusions
- Data availability
- Author contributions
- Competing interests
- Special issue statement
- Acknowledgements
- Financial support
- Review statement
- References

- Abstract
- Introduction
- Derivation of resolved level-crossing spectra
- Even number of equivalent spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei to drive spin evolution in the pair
- Other possible configurations of the driving spins
- Introducing nuclei into the driving partner: crossings vs. anticrossings
- Level-crossing lines as flip-flop resonance in pair partners
- Compendium of typical resolved spectra
- Conclusions
- Data availability
- Author contributions
- Competing interests
- Special issue statement
- Acknowledgements
- Financial support
- Review statement
- References