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

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

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-

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-

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

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

The formalism of Eq. (8) makes it very convenient to introduce a spin-

The last term in Eq. (18) containing

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

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

We finally note that the same formalism can be used to analyze the level
crossings driven by substantial difference in

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

It is a fluke that in the most common case of an even number of
spin-

For our second example of eight equivalent fluorines we would get the weights of

Although so far the only experimental observations of the ESR structure
using this approach have been the “spectrum” at 3

Although the case of an even number of driving spins-

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

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

Now let us introduce nuclei to the second partner, i.e., augment its eigenstate

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-

To analyze the resulting changes in eigenstructure, let us review Eq. (3). The expressions for energies are clearly of the form

Now let us introduce an additional nucleus with spin

Non-vanishing matrix elements can be obtained between functions of adjacent blocks of Eq. (32), e.g.,

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

Similar issues of “localization of interaction” in pair partners also
arise in the discussion of

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

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

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

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

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 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-

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.

The review MFE curve for a pair with six equivalent spin-

Figures 2 and 3 show in more details the regions of the level-crossing lines at 3

Closeup of the spectrum from Fig. 1 in the vicinity of

Closeup of the spectrum from Fig. 1 in the vicinity of

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-

Region in the vicinity of

Region in the vicinity of

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-

Region in the vicinity of

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.