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

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.

1 Introduction

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 Δg−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.

2 Derivation of 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, ω₀=gβB),

$$\begin{array}{}\text{(1)}& \widehat{H}={\mathit{\omega }}_{\mathrm{0}}\left(S_{\mathrm{1}z}+S_{\mathrm{2}z}\right)+a{\mathit{S}}_{\mathrm{1}}\mathit{I},\end{array}$$

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:

$$\begin{array}{}\text{(2)}& {\mathit{\rho }}_{\mathrm{ss}}\left(t\right)=\sum _{m=-I}^I{\mathit{\rho }}_{\mathrm{ss}}(t;m).\end{array}$$

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

$$\begin{array}{}\text{(3)}& \begin{array}{rl}& E_{\mathrm{1}}\left(m\right)=-{\displaystyle \frac{a}{\mathrm{4}}}-{\displaystyle \frac{{\mathit{\omega }}_{\mathrm{0}}}{\mathrm{2}}}+R_m,\\ & E_{\mathrm{2}}\left(m\right)=-{\displaystyle \frac{a}{\mathrm{4}}}-{\displaystyle \frac{{\mathit{\omega }}_{\mathrm{0}}}{\mathrm{2}}}-R_m,\\ & E_{\mathrm{3}}\left(m\right)=-{\displaystyle \frac{a}{\mathrm{4}}}+{\displaystyle \frac{{\mathit{\omega }}_{\mathrm{0}}}{\mathrm{2}}}+R_{m-\mathrm{1}},\\ & E_{\mathrm{4}}\left(m\right)=-{\displaystyle \frac{a}{\mathrm{4}}}+{\displaystyle \frac{{\mathit{\omega }}_{\mathrm{0}}}{\mathrm{2}}}-R_{m-\mathrm{1}},\end{array}\end{array}$$

where

$$\begin{array}{}\text{(4)}& \mathrm{2}{R}_m=\sqrt{{\mathit{\omega }}_{\mathrm{0}}^{\mathrm{2}}+a{\mathit{\omega }}_{\mathrm{0}}\left(\mathrm{2}m+\mathrm{1}\right)+a^{\mathrm{2}}{\left(I+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\right)}^{\mathrm{2}}}.\end{array}$$

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

$$\begin{array}{}\text{(5)}& \begin{array}{rl}& E_{\mathrm{1}}(m=\pm I)={\displaystyle \frac{aI}{\mathrm{2}}},\\ & E_{\mathrm{2}}(m=\pm I)=\pm {\displaystyle \frac{{\mathit{\omega }}_{\mathrm{0}}\mp a}{\mathrm{2}}}+R,\\ & E_{\mathrm{3}}(m=\pm I)=\pm {\displaystyle \frac{{\mathit{\omega }}_{\mathrm{0}}\mp a}{\mathrm{2}}}-R,\end{array}\end{array}$$

where

$$\begin{array}{}\text{(6)}& \mathrm{2}R=\sqrt{{\left\{{\mathit{\omega }}_{\mathrm{0}}\pm a\left(I-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\right)\right\}}^{\mathrm{2}}+\mathrm{2}{a}^{\mathrm{2}}I}.\end{array}$$

For each value of m from the range $-I<m<I$ the levels are degenerate in pairs in the zero field (E₁=E₃, E₂=E₄), which gives rise to the ubiquitous zero field line. In addition, for the inner blocks $\left|m\right|<I$ the levels E₁ and E₄ 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

$$\begin{array}{}\text{(7)}& {\mathit{\omega }}_{\mathrm{0}}^{*}=-{\displaystyle \frac{aI\left(I+\mathrm{1}\right)}{\mathrm{2}m}}.\end{array}$$

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

$$\begin{array}{}\text{(8)}& {\mathit{\rho }}_{\mathrm{ss}}\left(t\right)={\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}p\left(t\right)+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\mathit{Re}\left(e^{-i{\mathit{\omega }}_{\mathrm{0}}t}h\left(t\right)\right),\end{array}$$

where

$$\begin{array}{}\text{(9)}& {\displaystyle }\begin{array}{rl}p\left(t\right)=& \mathrm{1}-{\displaystyle \frac{a^{\mathrm{2}}}{\mathrm{2}I+\mathrm{1}}}\sum _{m=-I}^I\\ & {\displaystyle \frac{I(I+\mathrm{1})-m(m+\mathrm{1})}{{\left(\mathrm{2}{R}_m\right)}^{\mathrm{2}}}}\\ & \left[\mathrm{1}-\cos \left(\mathrm{2}{R}_{m}t\right)\right],\end{array}\text{(10)}& {\displaystyle }\begin{array}{rl}h\left(t\right)=& {\displaystyle \frac{\mathrm{1}}{\mathrm{4}\left(\mathrm{2}I+\mathrm{1}\right)}}\sum _{m=-I}^I\\ & \left[\left(\mathrm{1}+D_m\right)e^{i{R}_{m}t}+\left(\mathrm{1}-D_m\right)e^{-i{R}_{m}t}\right]\\ & \left[\left(\mathrm{1}+D_{m-\mathrm{1}}\right)e^{i{R}_{m-\mathrm{1}}t}+\left(\mathrm{1}-D_{m-\mathrm{1}}\right)e^{-i{R}_{m-\mathrm{1}}t}\right],\end{array}\text{(11)}& {\displaystyle }{D}_m={\displaystyle \frac{{\mathit{\omega }}_{\mathrm{0}}+a\left(m+\frac{\mathrm{1}}{\mathrm{2}}\right)}{\mathrm{2}{R}_m}}.\end{array}$$

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

$$\begin{array}{}\text{(12)}& M\left(s,{\mathit{\omega }}_{\mathrm{0}}\right)=\underset{\mathrm{0}}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}{\mathit{\rho }}_{\mathrm{ss}}\left(t\right)\mathrm{d}t,\end{array}$$

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

$$\begin{array}{}\text{(13)}& \begin{array}{rl}& sM\left({\mathit{\omega }}_{\mathrm{0}},s\right)={\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}\left\{\mathrm{1}-{\displaystyle \frac{a^{\mathrm{2}}}{\mathrm{2}I+\mathrm{1}}}\sum _{m=-I}^I\right.\\ & \left.{\displaystyle \frac{I\left(I+\mathrm{1}\right)-m\left(m+\mathrm{1}\right)}{{\mathit{\omega }}_{\mathrm{0}}^{\mathrm{2}}+a{\mathit{\omega }}_{\mathrm{0}}\left(\mathrm{2}m+\mathrm{1}\right)+a^{\mathrm{2}}{\left(I+\frac{\mathrm{1}}{\mathrm{2}}\right)}^{\mathrm{2}}}}{\displaystyle \frac{\mathrm{1}}{s^{\mathrm{2}}+{\left(\mathrm{2}{R}_m\right)}^{\mathrm{2}}}}\right\}\\ & +{\displaystyle \frac{\mathrm{1}}{\mathrm{8}\left(\mathrm{2}I+\mathrm{1}\right)}}\sum _{m=-I}^I[\left(\mathrm{1}+D_m\right)\left(\mathrm{1}+D_{m-\mathrm{1}}\right).\\ & \left.{\displaystyle \frac{s^{\mathrm{2}}}{s^{\mathrm{2}}+{\left(R_m+R_{m-\mathrm{1}}-{\mathit{\omega }}_{\mathrm{0}}\right)}^{\mathrm{2}}}}\right]\\ & +{\displaystyle \frac{\mathrm{1}}{\mathrm{8}\left(\mathrm{2}I+\mathrm{1}\right)}}\sum _{m=-I}^I[\left(\mathrm{1}+D_m\right)\left(\mathrm{1}-D_{m-\mathrm{1}}\right).\\ & \left.{\displaystyle \frac{s^{\mathrm{2}}}{s^{\mathrm{2}}+{\left(R_m-R_{m-\mathrm{1}}-{\mathit{\omega }}_{\mathrm{0}}\right)}^{\mathrm{2}}}}\right]\\ & +{\displaystyle \frac{\mathrm{1}}{\mathrm{8}\left(\mathrm{2}I+\mathrm{1}\right)}}\sum _{m=-I}^I[\left(\mathrm{1}-D_m\right)\left(\mathrm{1}+D_{m-\mathrm{1}}\right).\\ & \left.{\displaystyle \frac{s^{\mathrm{2}}}{s^{\mathrm{2}}+{\left(R_m-R_{m-\mathrm{1}}+{\mathit{\omega }}_{\mathrm{0}}\right)}^{\mathrm{2}}}}\right]\\ & +{\displaystyle \frac{\mathrm{1}}{\mathrm{8}\left(\mathrm{2}I+\mathrm{1}\right)}}\sum _{m=-I}^I[\left(\mathrm{1}-D_m\right)\left(\mathrm{1}-D_{m-\mathrm{1}}\right).\\ & \left.{\displaystyle \frac{s^{\mathrm{2}}}{s^{\mathrm{2}}+{\left(R_m+R_{m-\mathrm{1}}+{\mathit{\omega }}_{\mathrm{0}}\right)}^{\mathrm{2}}}}\right].\end{array}\end{array}$$

A numerical experiment demonstrates that for positive ω₀ resonance-like peaks in M(ω₀) are produced only by the terms

$$\begin{array}{}\text{(14)}& \left(\mathrm{1}+D_m\right)\left(\mathrm{1}+D_{m-\mathrm{1}}\right){\displaystyle \frac{s^{\mathrm{2}}}{s^{\mathrm{2}}+{\left(R_m+R_{m-\mathrm{1}}-{\mathit{\omega }}_{\mathrm{0}}\right)}^{\mathrm{2}}}}\end{array}$$

at fields satisfying the condition

$$\begin{array}{}\text{(15)}& R_m+R_{m-\mathrm{1}}-{\mathit{\omega }}_{\mathrm{0}}=\mathrm{0},\end{array}$$

which is immediately seen to reproduce the level-crossing condition E₁=E₄ 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

$$\begin{array}{}\text{(16)}& A\left(I,m\right)=\mathrm{4}{\displaystyle \frac{{\left(I\left(I+\mathrm{1}\right)-m^{\mathrm{2}}\right)}^{\mathrm{2}}-m^{\mathrm{2}}}{I^{\mathrm{2}}{\left(I+\mathrm{1}\right)}^{\mathrm{2}}-m^{\mathrm{2}}}},\end{array}$$

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

$$\begin{array}{}\text{(17)}& W\left(I,m\right)={\displaystyle \frac{s}{\mathrm{2}}}\left({\displaystyle \frac{I\left(I+\mathrm{1}\right)}{m^{\mathrm{2}}}}-{\displaystyle \frac{\mathrm{1}}{I\left(I+\mathrm{1}\right)}}\right),\end{array}$$

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-I₂ nucleus at the other partner of the pair. The corresponding counterpart to Eq. (8) would read as

$$\begin{array}{}\text{(18)}& {\mathit{\rho }}_{\mathrm{ss}}\left(t\right)={\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}{p}_{\mathrm{1}}\left(t\right)p_{\mathrm{2}}\left(t\right)+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\mathit{Re}\left(h_{\mathrm{1}}\left(t\right)h_{\mathrm{2}}^{*}\left(t\right)\right),\end{array}$$

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

$$\begin{array}{}\text{(19)}& \begin{array}{rl}& \left(\mathrm{1}\pm D_{\mathrm{1},m}\right)\left(\mathrm{1}\pm D_{\mathrm{1},m-\mathrm{1}}\right)\left(\mathrm{1}\pm D_{\mathrm{2},n}\right)\left(\mathrm{1}\pm D_{\mathrm{2},n-\mathrm{1}}\right)\\ & \exp \left[i\left(\pm R_{\mathrm{1},m}\pm R_{\mathrm{1},m-\mathrm{1}}\mp R_{\mathrm{2},n}\mp R_{\mathrm{2},n-\mathrm{1}}\right)t\right],\end{array}\end{array}$$

and again numerical experiment demonstrates that for positive fields the only resonance-like contributions to the Laplace transform M(ω₀) come from the terms

$$\begin{array}{}\text{(20)}& \begin{array}{rl}& \left(\mathrm{1}+D_{\mathrm{1},m}\right)\left(\mathrm{1}+D_{\mathrm{1},m-\mathrm{1}}\right)\left(\mathrm{1}+D_{\mathrm{2},n}\right)\left(\mathrm{1}+D_{\mathrm{2},n-\mathrm{1}}\right)\\ & {\displaystyle \frac{s^{\mathrm{2}}}{s^{\mathrm{2}}+{\left(R_{\mathrm{1},m}+R_{\mathrm{1},m-\mathrm{1}}-R_{\mathrm{2},n}-R_{\mathrm{2},n-\mathrm{1}}\right)}^{\mathrm{2}}}},\end{array}\end{array}$$

with positions of the maxima determined by the equation

$$\begin{array}{}\text{(21)}& R_{\mathrm{1},m}+R_{\mathrm{1},m-\mathrm{1}}-R_{\mathrm{2},n}-R_{\mathrm{2},n-\mathrm{1}}=\mathrm{0},\end{array}$$

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 a₂≪a₁ 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 a₂≪ω₀, 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

$$\begin{array}{}\text{(22)}& R_{\mathrm{1},m}+R_{\mathrm{1},m-\mathrm{1}}=R_{\mathrm{2},n}+R_{\mathrm{2},n-\mathrm{1}}={\mathit{\omega }}_{\mathrm{0}}+n{a}_{\mathrm{2}}.\end{array}$$

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:

$$\begin{array}{}\text{(23)}& {\mathit{\omega }}_{\mathrm{0}}^{*}=-a_{\mathrm{1}}{\displaystyle \frac{I_{\mathrm{1}}\left(I_{\mathrm{1}}+\mathrm{1}\right)}{\mathrm{2}m}}-a_{\mathrm{2}}n\left({\displaystyle \frac{I_{\mathrm{1}}\left(I_{\mathrm{1}}+\mathrm{1}\right)}{\mathrm{2}{m}^{\mathrm{2}}}}-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}{I}_{\mathrm{1}}\left(I_{\mathrm{1}}+\mathrm{1}\right)}}\right).\end{array}$$

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 2I₂+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

$$\begin{array}{}\text{(24)}& p_{\mathrm{2}}\left(t\right)=\mathrm{1},h_{\mathrm{2}}\left(t\right)=\exp \left[i\left({\mathit{\omega }}_{\mathrm{0}}+{\sum }_{k,n_k}{a}_k{n}_k\right)t\right],\end{array}$$

which in the same order produces

$$\begin{array}{}\text{(25)}& R_{\mathrm{1},m}+R_{\mathrm{1},m-\mathrm{1}}=R_{\mathrm{2},n}+R_{\mathrm{2},n-\mathrm{1}}={\mathit{\omega }}_{\mathrm{0}}+{\sum }_{k,n_k}{a}_k{n}_k,\end{array}$$

where a_k and n_k are the HFC constants and spin projections for the kth 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 na₂ 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):

$$\begin{array}{}\text{(26)}& \begin{array}{rl}{\mathit{\omega }}_{\mathrm{0}}^{*}=& -a_{\mathrm{1}}{\displaystyle \frac{I_{\mathrm{1}}\left(I_{\mathrm{1}}+\mathrm{1}\right)}{\mathrm{2}m}}-\left({\displaystyle \frac{I_{\mathrm{1}}\left(I_{\mathrm{1}}+\mathrm{1}\right)}{\mathrm{2}{m}^{\mathrm{2}}}}-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}{I}_{\mathrm{1}}\left(I_{\mathrm{1}}+\mathrm{1}\right)}}\right)\\ & {\sum }_{k,n_k}{a}_k{n}_k.\end{array}\end{array}$$

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(ω₀), 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 a₁ and g value g₁, while the second partner has no magnetic nuclei, but a shifted g value g₂, 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

$$\begin{array}{}\text{(27)}& p_{\mathrm{2}}\left(t\right)=\mathrm{1},h_{\mathrm{2}}\left(t\right)=\exp \left[i{\mathit{\omega }}_{\mathrm{0}}\left(\mathrm{1}+\mathit{\delta }\right)t\right],\end{array}$$

yielding

$$\begin{array}{}\text{(28)}& R_{\mathrm{2},n}+R_{\mathrm{2},n-\mathrm{1}}={\mathit{\omega }}_{\mathrm{0}}\left(\mathrm{1}+\mathit{\delta }\right)\end{array}$$

and two sets of solutions:

$$\begin{array}{}\text{(29)}& {\mathit{\omega }}_{\mathrm{01}}^{*}=-a_{\mathrm{1}}{\displaystyle \frac{I_{\mathrm{1}}\left(I_{\mathrm{1}}+\mathrm{1}\right)}{\mathrm{2}m}},{\mathit{\omega }}_{\mathrm{02}}^{*}=a_{\mathrm{1}}{\displaystyle \frac{I_{\mathrm{1}}\left(I_{\mathrm{1}}+\mathrm{1}\right)}{\mathrm{2}m}}+a_{\mathrm{1}}{\displaystyle \frac{m}{\mathit{\delta }}}.\end{array}$$

While the first set coincides with Eq. (7) or Eq. (26) with a₂ 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 Δg-induced level crossings.

3 Even number of equivalent spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei to drive spin evolution in the pair

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 ^35,37Cl($\frac{\mathrm{3}}{\mathrm{2}}$) (Bagryansky et al., 1998), ²⁷Al($\frac{\mathrm{5}}{\mathrm{2}}$), ^69,71Ga($\frac{\mathrm{3}}{\mathrm{2}}$), ^113,115In($\frac{\mathrm{9}}{\mathrm{2}}$) (Sergey et al., 2012), and ⁷³Ge($\frac{\mathrm{9}}{\mathrm{2}}$) (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 3a, 6a, and 3a, 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 3a 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),

$$\begin{array}{}\text{(30)}& W\left(I;n\right)={\displaystyle \frac{{\left(\mathrm{2}I+\mathrm{1}\right)}^{\mathrm{2}}n\mathrm{!}}{{\mathrm{2}}^n\left(\frac{n}{\mathrm{2}}-I\right)\mathrm{!}\left(\frac{n}{\mathrm{2}}+I+\mathrm{1}\right)\mathrm{!}}},\end{array}$$

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 3a is statistically dominated by the smaller total spin I=2, while the higher total spin I=3 is responsible for the crossing at 6a. Omitting the small corrections of $\frac{\mathrm{1}}{\mathrm{24}}$ and $\frac{\mathrm{1}}{\mathrm{12}}$, the field-scaling factors for the crossings at 3a and 6a are 3 and 6, respectively, with the crossing at 3a 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 10a, 5a, and 10a/3 from I=4 in addition to already described lines at 3a and 6a. Again the strongest line at 3a is dominated by the contribution from the I=2 sub-ensemble with field-scaling factor 3 and swamps the much weaker nearby line at 10a/3 coming from I=4. The line at 6a is dominated by I=3 with the scaling factor of 6 and swamps the nearby line at 5a from I=4, and the only genuinely new line from I=4 is the line at 10a 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

$$\begin{array}{}\text{(31)}& \begin{array}{rl}\text{for}\left|m\right|& =\mathrm{1}\left\{{\displaystyle \frac{{\mathit{\omega }}_{\mathrm{0}}^{*}}{a_{\mathrm{1}}}}={\displaystyle \frac{I_{\mathrm{1}}\left(I_{\mathrm{1}}+\mathrm{1}\right)}{\mathrm{2}\left|m\right|}}\right\}\\ & =\left\{f={\displaystyle \frac{I_{\mathrm{1}}\left(I_{\mathrm{1}}+\mathrm{1}\right)}{\mathrm{2}{m}^{\mathrm{2}}}}\right\}={\displaystyle \frac{I_{\mathrm{1}}\left(I_{\mathrm{1}}+\mathrm{1}\right)}{\mathrm{2}}}.\end{array}\end{array}$$

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

Given a radical pair with n=2k 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 3a 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 3a 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.

4 Other possible configurations of the driving spins

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).

5 Introducing nuclei into the driving partner: crossings vs. anticrossings

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 a₂≪a₁. 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:

$$\begin{array}{}\text{(32)}& \begin{array}{rl}{E}_{\mathrm{1}}\left(m\right)& =\left(-{\displaystyle \frac{a}{\mathrm{4}}}+R_m\right)-{\displaystyle \frac{{\mathit{\omega }}_{\mathrm{0}}}{\mathrm{2}}},\left|{\mathit{\psi }}_{\mathrm{1}}\right(m)\rangle \\ & =\left({\cos }_m|\mathit{\alpha },m{\rangle }_{\mathrm{1}}+{\sin }_m|\mathit{\beta },m+\mathrm{1}{\rangle }_{\mathrm{1}}\right)|\mathit{\beta }{\rangle }_{\mathrm{2}},\\ E_{\mathrm{2}}\left(m\right)& =\left(-{\displaystyle \frac{a}{\mathrm{4}}}-R_m\right)-{\displaystyle \frac{{\mathit{\omega }}_{\mathrm{0}}}{\mathrm{2}}},\left|{\mathit{\psi }}_{\mathrm{2}}\right(m)\rangle \\ & =\left(-{\sin }_m|\mathit{\alpha },m{\rangle }_{\mathrm{1}}+{\cos }_m|\mathit{\beta },m+\mathrm{1}{\rangle }_{\mathrm{1}}\right)|\mathit{\beta }{\rangle }_{\mathrm{2}},\\ E_{\mathrm{3}}\left(m\right)& =\left(-{\displaystyle \frac{a}{\mathrm{4}}}+R_{m-\mathrm{1}}\right)+{\displaystyle \frac{{\mathit{\omega }}_{\mathrm{0}}}{\mathrm{2}}},\left|{\mathit{\psi }}_{\mathrm{3}}\right(m)\rangle \\ & =\left({\cos }_{m-\mathrm{1}}|\mathit{\alpha },m-\mathrm{1}{\rangle }_{\mathrm{1}}+{\sin }_{m-\mathrm{1}}|\mathit{\beta },m{\rangle }_{\mathrm{1}}\right)|\mathit{\alpha }{\rangle }_{\mathrm{2}},\\ E_{\mathrm{4}}\left(m\right)& =\left(-{\displaystyle \frac{a}{\mathrm{4}}}-R_{m-\mathrm{1}}\right)+{\displaystyle \frac{{\mathit{\omega }}_{\mathrm{0}}}{\mathrm{2}}},\left|{\mathit{\psi }}_{\mathrm{4}}\right(m)\rangle \\ & =\left(-{\sin }_{m-\mathrm{1}}|\mathit{\alpha },m-\mathrm{1}{\rangle }_{\mathrm{1}}+{\cos }_{m-\mathrm{1}}|\mathit{\beta },m{\rangle }_{\mathrm{1}}\right)|\mathit{\alpha }{\rangle }_{\mathrm{2}},\end{array}\end{array}$$

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 E₁=E₃, E₂=E₄ in the zero field and E₁=E₄ 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 V₁₃, V₂₄, and V₁₄ 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 $\left|{\mathit{\psi }}_{\mathrm{1},\mathrm{2}}\right(m-\mathrm{1})\rangle$:

$$\begin{array}{}\text{(33)}& \begin{array}{rl}& \langle {\mathit{\psi }}_{\mathrm{1}}\left(m\right);n\left|\widehat{V}\right|{\mathit{\psi }}_{\mathrm{1}}(m-\mathrm{1});n+\mathrm{1}\rangle \\ & ={\cos }_m{\sin }_{m-\mathrm{1}}\langle \mathit{\alpha },m;n\left|{\displaystyle \frac{b}{\mathrm{2}}}{S}_{\mathrm{1}+}{K}_{-}\right|\mathit{\beta },m;n+\mathrm{1}{\rangle }_{\mathrm{1}}\\ & ={\cos }_m{\sin }_{m-\mathrm{1}}{\displaystyle \frac{b}{\mathrm{2}}}\sqrt{(K+n+\mathrm{1})(K-n)},\end{array}\end{array}$$

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 E₁=E₄, we obtain

$$\begin{array}{}\text{(34)}& V_{\mathrm{11}}=b{\displaystyle \frac{n}{\mathrm{2}}}\left(\mathrm{2}{\cos }_m^{\mathrm{2}}-\mathrm{1}\right),V_{\mathrm{44}}=-b{\displaystyle \frac{n}{\mathrm{2}}}\left(\mathrm{2}{\cos }_{m-\mathrm{1}}^{\mathrm{2}}-\mathrm{1}\right).\end{array}$$

This moves the surviving active crossing in energy by

$$\begin{array}{}\text{(35)}& \mathrm{\Delta }{E}_{\mathrm{1},\mathrm{4}}=V_{\mathrm{11}}-V_{\mathrm{44}}=bn\left({\cos }_m^{\mathrm{2}}+{\cos }_{m-\mathrm{1}}^{\mathrm{2}}-\mathrm{1}\right)\end{array}$$

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 bn, 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 Δg-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 $\mathrm{\Gamma }={\mathrm{\Gamma }}_{\mathrm{i}}\cdot {\mathrm{\Gamma }}_{\mathrm{o}}\cdot {\mathrm{\Gamma }}_{\mathrm{c}}$.

6 Level-crossing lines as flip-flop resonance in pair partners

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 E₁=E₄ for the level-crossing line can be trivially rearranged as

$$\begin{array}{}\text{(36)}& \left(-{\displaystyle \frac{a}{\mathrm{4}}}+R_m\right)-\left(-{\displaystyle \frac{a}{\mathrm{4}}}-R_{m-\mathrm{1}}\right)={\mathit{\omega }}_{\mathrm{0}},\end{array}$$

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:

$$\begin{array}{}\text{(37)}& \begin{array}{rl}& {\mathit{\psi }}_{\mathrm{1}}\leftrightarrow {\mathit{\psi }}_{\mathrm{4}}\text{means}|\mathit{\alpha },m{\rangle }_{\mathrm{1}}\leftrightarrow |\mathit{\beta },m{\rangle }_{\mathrm{1}}\\ & \text{and}|\mathit{\beta }{\rangle }_{\mathrm{2}}\leftrightarrow |\mathit{\alpha }{\rangle }_{\mathrm{2}}.\end{array}\end{array}$$

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 E₁=E₃ and E₂=E₄.

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

$$\begin{array}{}\text{(38)}& \left(-{\displaystyle \frac{a}{\mathrm{4}}}+R_m\right)-\left(-{\displaystyle \frac{a}{\mathrm{4}}}-R_{m-\mathrm{1}}\right)={\mathit{\omega }}_{\mathrm{0}}+{\sum }_{k,n_k}{a}_k{n}_k.\end{array}$$

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

$$\begin{array}{}\text{(39)}& \begin{array}{rl}& \left(-{\displaystyle \frac{a_{\mathrm{1}}}{\mathrm{4}}}+R_{\mathrm{1},m}\right)-\left(-{\displaystyle \frac{a_{\mathrm{1}}}{\mathrm{4}}}-R_{\mathrm{1},m-\mathrm{1}}\right)\\ & =\left(-{\displaystyle \frac{a_{\mathrm{2}}}{\mathrm{4}}}+R_{\mathrm{2},n}\right)-\left(-{\displaystyle \frac{a_{\mathrm{2}}}{\mathrm{4}}}-R_{\mathrm{2},n-\mathrm{1}}\right),\end{array}\end{array}$$

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

$$\begin{array}{}\text{(40)}& \begin{array}{rl}& \left\{\begin{array}{c}{\cos }_m|\mathit{\alpha },m{\rangle }_{\mathrm{1}}+{\sin }_m|\mathit{\beta },m+\mathrm{1}{\rangle }_{\mathrm{1}}\\ -{\sin }_m|\mathit{\alpha },m{\rangle }_{\mathrm{1}}+{\cos }_m|\mathit{\beta },m+\mathrm{1}{\rangle }_{\mathrm{1}}\end{array}\right\}\\ & \times \left\{\begin{array}{c}{\cos }_{n-\mathrm{1}}|\mathit{\alpha },n-\mathrm{1}{\rangle }_{\mathrm{2}}+{\sin }_{n-\mathrm{1}}|\mathit{\beta },n{\rangle }_{\mathrm{2}}\\ -{\sin }_{n-\mathrm{1}}|\mathit{\alpha },n-\mathrm{1}{\rangle }_{\mathrm{2}}+{\cos }_{n-\mathrm{1}}|\mathit{\beta },n{\rangle }_{\mathrm{2}}\end{array}\right\},\\ & \left\{\begin{array}{c}{\cos }_{m-\mathrm{1}}|\mathit{\alpha },m-\mathrm{1}{\rangle }_{\mathrm{1}}+{\sin }_{m-\mathrm{1}}|\mathit{\beta },m{\rangle }_{\mathrm{1}}\\ -{\sin }_{m-\mathrm{1}}|\mathit{\alpha },m-\mathrm{1}{\rangle }_{\mathrm{1}}+{\cos }_{m-\mathrm{1}}|\mathit{\beta },m{\rangle }_{\mathrm{1}}\end{array}\right\}\\ & \times \left\{\begin{array}{c}{\cos }_n|\mathit{\alpha },n{\rangle }_{\mathrm{2}}+{\sin }_n|\mathit{\beta },n+\mathrm{1}{\rangle }_{\mathrm{2}}\\ -{\sin }_n|\mathit{\alpha },n{\rangle }_{\mathrm{2}}+{\cos }_n|\mathit{\beta },n+\mathrm{1}{\rangle }_{\mathrm{2}}\end{array}\right\}.\end{array}\end{array}$$

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

$$\begin{array}{}\text{(41)}& \begin{array}{rl}& \left({\cos }_m\right|\mathit{\alpha },m{\rangle }_{\mathrm{1}}+{\sin }_m|\mathit{\beta },m+\mathrm{1}{\rangle }_{\mathrm{1}})\\ & \times (-{\sin }_{n-\mathrm{1}}|\mathit{\alpha },n-\mathrm{1}{\rangle }_{\mathrm{2}}+{\cos }_{n-\mathrm{1}}|\mathit{\beta },n{\rangle }_{\mathrm{2}}),\\ & (-{\sin }_{m-\mathrm{1}}|\mathit{\alpha },m-\mathrm{1}{\rangle }_{\mathrm{1}}+{\cos }_{m-\mathrm{1}}|\mathit{\beta },m{\rangle }_{\mathrm{1}})\\ & \times \left({\cos }_n\right|\mathit{\alpha },n{\rangle }_{\mathrm{2}}+{\sin }_n|\mathit{\beta },n+\mathrm{1}{\rangle }_{\mathrm{2}}).\end{array}\end{array}$$

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

$$\begin{array}{}\text{(42)}& |\mathit{\alpha },m{\rangle }_{\mathrm{1}}\leftrightarrow |\mathit{\beta },m{\rangle }_{\mathrm{1}}\text{and}|\mathit{\beta },n{\rangle }_{\mathrm{2}}\leftrightarrow |\mathit{\alpha },n{\rangle }_{\mathrm{2}}\end{array}$$

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).

7 Compendium of typical resolved spectra

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.

Figure 1 The review MFE curve for a pair with six equivalent spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei with HFC constant A in one partner and two equivalent spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei with HFC constant $A/\mathrm{10}$ in the second partner, recombination parameter $s=A/\mathrm{100}$.

Figures 2 and 3 show in more details the regions of the level-crossing lines at 3A and 6A 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.

Figure 2 Closeup of the spectrum from Fig. 1 in the vicinity of B₀=3A for two values of the recombination parameter $s=A/\mathrm{100}$, $s=A/\mathrm{400}$.

Figure 3 Closeup of the spectrum from Fig. 1 in the vicinity of B₀=6A for two values of the recombination parameter $s=A/\mathrm{100}$, $s=A/\mathrm{400}$.

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.

Figure 4 Region in the vicinity of B₀=3A for a pair with six equivalent spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei with HFC constant A in one partner and two non-equivalent spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei with HFC constant $A/\mathrm{10}$ and $A/\mathrm{40}$ in the second partner, recombination parameter $s=A/\mathrm{100}$, $s=A/\mathrm{400}$.

Figure 5 Region in the vicinity of B₀=6A for a pair with six equivalent spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei with HFC constant A in one partner and two non-equivalent spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei with HFC constant $A/\mathrm{10}$ and $A/\mathrm{40}$ in the second partner, recombination parameter $s=A/\mathrm{100}$, $s=A/\mathrm{400}$.

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.

Figure 6 Region in the vicinity of B₀=3A for a pair with six equivalent spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei with HFC constant A in one partner and two equivalent spin-$\frac{\mathrm{1}}{\mathrm{2}}$ nuclei with a relatively large HFC constant $A/\mathrm{5}$ in the second partner, recombination parameter $s=A/\mathrm{100}$, $s=A/\mathrm{400}$.

8 Conclusions

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.