the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# The effect of the zero-field splitting in light-induced pulsed dipolar electron paramagnetic resonance (EPR) spectroscopy

### Andreas Scherer

### Berk Yildirim

### Malte Drescher

Laser-induced magnetic dipole (LaserIMD) spectroscopy and light-induced double electron–electron resonance (LiDEER) spectroscopy are important techniques in the emerging field of light-induced pulsed dipolar electron paramagnetic resonance (EPR) spectroscopy (light-induced PDS). These techniques use the photoexcitation of a chromophore to the triplet state and measure its dipolar coupling to a neighboring electron spin, which allows the determination of distance restraints. To date, LaserIMD and LiDEER have been analyzed with software tools that were developed for a pair of two $S=\mathrm{1}/\mathrm{2}$ spins and that neglected the zero-field splitting (ZFS) of the excited triplet. Here, we explore the limits of this assumption and show that the ZFS can have a significant effect on the shape of the dipolar trace. For a detailed understanding of the effect of the ZFS, a theoretical description for LaserIMD and LiDEER is derived, taking into account the non-secular terms of the ZFS. Simulations based on this model show that the effect of the ZFS is not that pronounced in LiDEER for experimentally relevant conditions. However, the ZFS leads to an additional decay in the dipolar trace in LaserIMD. This decay is not that pronounced in Q-band but can be quite noticeable for lower magnetic field strengths in X-band. Experimentally recorded LiDEER and LaserIMD data confirm these findings. It is shown that ignoring the ZFS in the data analysis of LaserIMD traces can lead to errors in the obtained modulation depths and background decays. In X-band, it is additionally possible that the obtained distance distribution is plagued by long distance artifacts.

- Article
(3906 KB) -
Supplement
(2771 KB) - BibTeX
- EndNote

Pulsed dipolar electron paramagnetic resonance (EPR) spectroscopy (PDS) has become an important tool for
nanoscale distance determination in soft matter. Its applications include
the structural determination of biomacromolecules, like proteins
(Yee
et al., 2015; Yang et al., 2020; Giannoulis et al., 2020; Weickert et al.,
2020; Robotta et al., 2014; Ritsch et al., 2022) DNA
(Wojciechowski et al.,
2015; Takeda et al., 2004; Marko et al., 2011) and RNA
(Collauto et al., 2020), as well as synthetic
polymers (Jeschke et al.,
2010) and nanoparticles
(Hintze et al., 2015;
Bücker et al., 2019). PDS measures the dipolar coupling between two spin
centers within the molecule under investigation. Oftentimes, the spin
centers need to be introduced as spin labels via site-directed labeling,
with nitroxide spin probes as the most common example
(Hubbell et al., 2013;
Roser et al., 2016; García-Rubio, 2020). The most common PDS technique
is double electron–electron resonance (DEER, also called PELDOR)
spectroscopy (Milov et
al., 1981, 1984; Jeschke, 2012). Here, one of the spin labels is excited by
microwave pulses at an observer frequency to generate a refocused echo. The
excitation of the other spin label by a pump pulse at a second frequency
leads to an oscillation of the refocused echo, when the pump pulse is
shifted in the time domain. The frequency of this oscillation depends on the
inverse cubic distance between the spin labels *r*^{−3} and, thus, provides
distance information for the molecule under investigation
(Jeschke, 2012).

The recent years have seen the advent of a new type of spin label that is
in an EPR-silent singlet ground state but can be converted transiently to a
triplet state by photoexcitation and subsequent intersystem crossing
(Di Valentin et al., 2014; Bertran et
al., 2022a). In contrast to spin labels with a spin of $S=\mathrm{1}/\mathrm{2}$, like
nitroxides, these transient triplet labels are subject to an additional
zero-field splitting (ZFS). It is described by the ZFS parameters *D* and
*E*. By now, several transient triplet labels with different ZFS strengths
have been used. Examples are triphenylporphyrin (TPP)
(*D*=1159, $E=-\mathrm{238}$ MHz) (Di Valentin
et al., 2014), fullerenes (*D*=342, $E=-\mathrm{2}$ MHz)
(Wasielewski
et al., 1991; Krumkacheva et al., 2019; Timofeev et al., 2022), rose bengal
(*D*=3671, $E=-\mathrm{319}$ MHz), eosin Y
(*D*=2054, $E=-\mathrm{585}$ MHz), Atto Thio12
(*D*=1638, $E=-\mathrm{375}$ MHz)
(Serrer et al., 2019; Williams et al.,
2020) and erythrosin B (*D*=3486, $E=-\mathrm{328}$ MHz)
(Bertran et al., 2022b). The most common PDS
techniques for transient triplet labels are light-induced DEER (LiDEER) and
laser-induced magnetic dipole (LaserIMD) spectroscopy
(Di Valentin et al., 2014; Hintze et al., 2016).
They both allow the determination of distances between one permanent spin
label and one transient triplet label. LiDEER is a modification of DEER with
an additional laser flash preceding the microwave pulses (see
Fig. 1a). The permanent spin is excited by the
pump pulse, as it typically has an EPR spectrum that is narrower than
the one of the transient triplet label, which gives higher modulation
depths. The transient triplet label is observed because, despite its broader
EPR spectrum, it is still possible to generate strong echoes, as the
photoexcitation of the transient triplet label typically leads to a high
spin polarization (Di Valentin et al., 2014). In LaserIMD, on
the other hand, the permanent spin label is observed. During the evolution
of the observer spin, the transient triplet label is excited by a laser
flash (see Fig. 1b). The induced transition from
the singlet to the triplet state has an equivalent effect to the microwave
pump pulse in DEER and results in an oscillation of the echo of the observer
spin. An advantage of LaserIMD is that, in contrast to DEER, the bandwidth
of the laser excitation is neither limited by the width of the EPR spectrum
of the pump spin nor the resonator bandwidth. This gives virtually infinite
excitation bandwidths and promises high modulation depths, even in cases
where the microwave excitation bandwidth is smaller than the EPR spectra of
the invoked spins (Scherer et al., 2022a).

In previous works, LaserIMD and LiDEER data were analyzed under the
assumption that the ZFS of the transient triplet label can be ignored
(Di
Valentin et al., 2014; Hintze et al., 2016; Bieber et al., 2018; Dal Farra
et al., 2019a; Krumkacheva et al., 2019). Under this assumption, the dipolar
traces of LaserIMD and LiDEER have the same shape as those of DEER on a
label pair with two $S=\mathrm{1}/\mathrm{2}$ spins. However, as is shown below, this
assumption is only correct if all spin–spin interactions are much smaller
than the Zeeman interaction with the external magnetic field. Then, all
non-secular terms in the Hamiltonian can be dropped
(Manukovsky et al., 2017). The excited triplet state of
transient triplet labels with a total spin of *S*=1, however, can be subject
to a strong ZFS, reaching values of over 1 GHz in many cases
(Di Valentin et al., 2014; Williams et al., 2020).
For other high-spin labels like Gd^{III} or high-spin Fe^{III},
it is already known that the ZFS can have an effect on the recorded dipolar
trace and that it has to be included in the data analysis routine if
artifacts in the distance are to be avoided
(Maryasov
et al., 2006; Dalaloyan et al., 2015; Abdullin et al., 2019).

Here, we set out to investigate the effect of the ZFS in light-induced PDS.
Therefore, we are going to derive a theoretical description for
light-induced PDS, taking the *S*=1 spin state and the ZFS of the triplet state
into account. Section 3 will report the materials and methods used. In
Sect. 4, the theoretical model will be used for numerical simulations of
LaserIMD, and time-domain simulations performed for LiDEER will be reported. It will be shown that the effect of the ZFS can result in significant differences in the dipolar traces in both methods compared with the $S=\mathrm{1}/\mathrm{2}$ case where the ZFS is ignored; however, this effect is particularly pronounced in LaserIMD. In Sect. 5, experimental
LaserIMD and LiDEER traces are shown, and the influence of the ZFS is
discussed by comparing the model with the experimental data.

## 2.1 DEER

For the analysis of DEER data, one typically uses the assumption that both spins are of $S=\mathrm{1}/\mathrm{2}$ nature and that the system is in high-field and weak-coupling limit so that all pseudo- and non-secular parts of the spin Hamiltonian can be dropped (Jeschke et al., 2006; Worswick et al., 2018; Fábregas Ibáñez et al., 2020). In this case, there are two coherence transfer pathways that contribute to the DEER signal: one where the pump spin is flipped from the state with ${m}_{\mathrm{S}}=+\mathrm{1}/\mathrm{2}$ to ${m}_{\mathrm{S}}=-\mathrm{1}/\mathrm{2}$ and one where it is flipped from ${m}_{\mathrm{S}}=-\mathrm{1}/\mathrm{2}$ to ${m}_{\mathrm{S}}=+\mathrm{1}/\mathrm{2}$. The frequency of the dipolar oscillation of the refocused echo for the two coherence transfer pathways is as follows:

Here, *β*_{dip} is the angle between the dipolar coupling
vector and the external magnetic field, and *ω*_{dip} is the
dipolar coupling in radial frequency units. *ω*_{dip} depends on the distance *r*
between the two labels:

with the Bohr magneton *μ*_{B}, the reduced Plank constant
and the *g* values (*g*_{1} and *g*_{2}) of the two spin labels.
In experiments, one typically measures powder samples; thus, molecules with
all orientations with respect to the external field contribute to the
signal, and the weighted integral over all angles *β*_{dip}
must be taken (Pake, 1948; Milov
et al., 1998). In the high-temperature limit, which is often fulfilled in
experiments, the population of the spin states with ${m}_{\mathrm{S}}=+\mathrm{1}/\mathrm{2}$ to
${m}_{\mathrm{S}}=-\mathrm{1}/\mathrm{2}$ is virtually identical; therefore, both coherence transfer
pathways contribute equally to the signal (Marko
et al., 2013). In this case, the integral over all orientations is as follows:

Here, *t* is the time at which the pump pulse flips the pump spins. Due to a
limited excitation bandwidth and pulse imperfections, not all spins can be
excited by the pump pulse; therefore, a part of the signal is not modulated:

where the modulation depth *λ* depends on the fraction of excited
pump spins. The experimental signal is the product of this intramolecular
contribution *F*_{DEER}(*t*,*r*) and a contribution from
the intermolecular dipolar interactions *B*(*t*), which is typically termed
background. Finally, the contributions from all distances need to be
included by integrating over the distance distribution *P*(*r*):

The kernel *K*_{DEER}(*t*,*r*) describes the relation between the
distance distribution and the measured dipolar trace in DEER. In a sample
with a homogenous distribution of spins, the background function can be
obtained by integrating over all dipolar interactions within the sample,
which results in the following (Hu and Hartmann, 1974):

The decay constant *k* is proportional to the spin concentration and
modulation depth (Hu and Hartmann, 1974). By inverting Eq. (6), it is possible to extract the distance distribution *P*(*r*) from the
experimentally recorded signal *V*_{DEER}(*t*). Because this is an
ill-posed problem, this is typically done by advanced techniques like
Tikhonov regularization
(Bowman et al., 2004; Jeschke
et al., 2004) or neural networks
(Worswick et al., 2018;
Keeley et al., 2022).

## 2.2 LaserIMD

In LaserIMD, the spin system consists of a permanent spin label, which
serves as an observer spin, and a transient triplet label, which is excited
by a laser flash. In many cases, the permanent spin label is or can be
assumed to be a doublet with ${S}_{\mathrm{D}}=\mathrm{1}/\mathrm{2}$. Before the
photoexcitation, the transient label is still in its singlet state; therefore it interacts with neither the external field *B* nor the doublet
*S*_{D}. Thus, the Hamiltonian only contains the Zeeman interaction
of *S*_{D}:

Here the Zeeman frequency ${\mathit{\nu}}_{\mathrm{D}}=\frac{{g}_{\mathrm{D}}{\mathit{\mu}}_{\mathrm{B}}}{\mathrm{2}\mathit{\pi}\textcolor{red}{}}B$, where *g*_{D} denotes the *g* values of
*S*_{D}, which is assumed to be isotropic. The Hamiltonian is
written in units of radial frequencies. This Hamiltonian has two
eigenvalues:

When the laser flash excites the transient triplet label to the triplet
state *S*_{T}=1, the Zeeman interaction of *S*_{T}, the
ZFS between the two unpaired electrons that form the triplet
*S*_{T}, and the dipolar coupling between *S*_{D} and
*S*_{T} has to be included in the Hamiltonian:

Here, ${\mathit{\nu}}_{\mathrm{T}}=\frac{{g}_{\mathrm{T}}{\mathit{\mu}}_{\mathrm{B}}B}{\mathrm{2}\mathit{\pi}\textcolor{red}{}}$ is the Zeeman frequency of the spin *S*_{T} with its
isotropic *g* value (*g*_{T}). *S*_{D} and
*S*_{T} represent the vectors of the Cartesian spin operators
${\mathit{S}}_{\mathrm{D}}={\left({\widehat{S}}_{\mathrm{D},x},\phantom{\rule{0.125em}{0ex}}{\widehat{S}}_{\mathrm{D},y},\phantom{\rule{0.125em}{0ex}}{\widehat{S}}_{\mathrm{D},z}\right)}^{\mathrm{T}}$ and
${\mathit{S}}_{\mathrm{T}}={\left({\widehat{S}}_{\mathrm{T},x},\phantom{\rule{0.125em}{0ex}}{\widehat{S}}_{\mathrm{T},y},\phantom{\rule{0.125em}{0ex}}{\widehat{S}}_{\mathrm{T},z}\right)}^{\mathrm{T}}$. The
ZFS tensor **D** is described by the ZFS values
$D=\frac{\mathrm{3}}{\mathrm{2}}{D}_{z}$ and $E=\frac{{D}_{x}-{D}_{y}}{\mathrm{2}}$, where *D*_{x}, *D*_{y}
and *D*_{z} are the eigenvalues of the ZFS tensor (Telser, 2017). Its
orientation is described by the three Euler angles, *α*_{T},
*β*_{T} and *γ*_{T}, that connect the laboratory frame
with the molecular frame of the transient triplet label. In the point-dipole
approximation, the dipolar coupling tensor **T** is axial
with the eigenvalues ${T}_{x}={T}_{y}=-{\mathit{\omega}}_{\mathrm{dip}}$ and
*T*_{z}=2*ω*_{dip} (Schweiger and Jeschke, 2001). Its
orientation towards the external magnetic field is described by the angle
*β*_{dip}.
In the high-field and weak-coupling limit all non- and pseudo-secular terms
can be dropped from the Hamiltonian. The remaining secular Hamiltonian (see
Eq. S2 in Supplement S1) is already diagonal in the high-field basis with the energy
levels ${E}_{{m}_{\mathrm{D}},\phantom{\rule{0.125em}{0ex}}{m}_{\mathrm{T}}}^{\mathrm{sec}}$, where
*m*_{D} and *m*_{T} are the magnetic quantum numbers of
the doublet *S*_{D} and the triplet *S*_{T}, respectively. The exact
expressions for the energies ${E}_{{m}_{\mathrm{D}},{m}_{\mathrm{T}}}^{\mathrm{sec}}$ can be found in Eqs. (S4)–(S9) in Supplement S1. In
LaserIMD, the initial $\frac{\mathit{\pi}}{\mathrm{2}}$-pulse generates a coherence of the
observer spin *S*_{D}. Before the laser excitation, the coherence
evolves with a frequency of
${E}_{+\frac{\mathrm{1}}{\mathrm{2}},\mathrm{dark}}-{E}_{-\frac{\mathrm{1}}{\mathrm{2}},\mathrm{dark}}=\mathrm{2}\mathit{\pi}{\mathit{\nu}}_{\mathrm{D}}$; it is not influenced by the dipolar coupling because the
transient triplet label is still in a singlet state with *S*_{T}=0
and *m*_{T}=0. The excitation of the transient triplet label leads
to three different coherence transfer pathways, depending on which manifold,
*m*_{T}=1, 0 or −1, of the triplet the transient label is
excited to. Depending on the triplet state *m*_{T}, the coherence
will then continue to evolve with
${E}_{+\frac{\mathrm{1}}{\mathrm{2}},{m}_{\mathrm{T}}}^{\mathrm{sec}}-{E}_{-\frac{\mathrm{1}}{\mathrm{2}},{m}_{\mathrm{T}}}^{\mathrm{sec}}$.
The refocusing *π*-pulse generates an echo at the time 2*τ*. Due to
the different frequencies before and after the excitation at a variable time
*t*, the coherences are not completely refocused; however, depending on the time
of the laser flash, they will have gained a phase $\mathit{\varphi}={\mathit{\omega}}_{{m}_{\mathrm{T}}}^{\mathrm{sec}}t$, which depends on the LaserIMD frequency
${\mathit{\omega}}_{{m}_{\mathrm{T}}}^{\mathrm{sec}}$ of the corresponding triplet
manifold *m*_{T}. When only the secular terms are
considered in the Hamiltonian, the LaserIMD frequencies ${\mathit{\omega}}_{{m}_{\mathrm{T}}}^{\mathrm{sec}}$ do not depend on the ZFS, as its
secular terms cancel each other out, and the same expressions as those of
Hintze et al. (2016) are obtained:

When the transient triplet label is excited to *m*_{T}=1 or
${m}_{\mathrm{T}}=-\mathrm{1}$, the LaserIMD frequencies in secular approximation from
Eqs. (12) and (14) are identical to the DEER frequencies in Eqs. (1) and (2), respectively.
Here, the laser flash leads to a change in the magnetic quantum number of
$\mathrm{\Delta}{m}_{\mathrm{T}}=\pm \mathrm{1}$, which is equivalent to the effect of
the microwave pump pulse in DEER. In the case when the transient triplet
label is excited to the state *m*_{T}=0, however, the secular
approximation predicts that the echo is not oscillating, as – loosely
speaking – there is no change in the magnetic spin quantum number of the
transient triplet label, which means that the dipolar coupling is not
changed. As is the case in DEER, the measured signal is the average
over all orientations of the spin system. Whereas it is only
necessary to consider the orientation of the dipolar vector in DEER, the
orientation of the transient triplet label must also be taken into account in LaserIMD;
therefore, it is necessary to also integrate over the three corresponding
Euler angles *α*_{T}, *β*_{T} and *γ*_{T} (Bak and Nielsen, 1997). In the absence of
orientation selection, the orientation of the dipolar vector and the
transient triplet label are not correlated, and the integration over the
corresponding Euler angles can be done independently. This is often realized
in practical applications where flexible linkers are used to attach labels
to the studied molecule. As the triplet state of the transient label is
reached by intersystem crossing, the population of the three high-field
triplet states, ${m}_{\mathrm{T}}=+\mathrm{1}$, 0, −1, depends on the orientation
of the transient label with respect to the external magnetic field and the
populations *P*_{x}, *P*_{y} and *P*_{z} of the zero-field eigenstates
(Rose, 1995). The contribution of the three coherence transfer
pathways must be weighted by population of these high-field states; this
gives (still in secular approximation) the following three expressions:

Performing the integration over the orientations of the transient label
*α*_{T}, *β*_{T} and *γ*_{T} and
taking the sum gives the following expression (Williams et al., 2020):

In secular approximation, the first term of the LaserIMD signal is
equivalent to the trace *S*_{DEER}(t)
(Edwards and Stoll, 2018). The second term is an
additional non-modulated contribution. For the final expression for the
kernel ${K}_{\mathrm{LaserIMD}}^{\mathrm{sec}}(t,r)$, the quantum yield of the
triplet state is considered by an additional factor *γ*, and the
intermolecular interaction to other spins in the sample has to be considered
as background *B*(*t*):

This can be rewritten as follows:

with the modulation depth $\mathit{\lambda}=\mathrm{2}/\mathrm{3}\mathit{\gamma}$. The only difference between
LaserIMD in the secular approximation and DEER is that in LaserIMD, even for
a triplet yield of *γ*=100 *%*, there is coherence transfer pathway
with Δ*m*_{S}=0 that does not result in a dipolar
oscillation, which limits the maximum achievable modulation depth to
$\mathrm{66}.\stackrel{\mathrm{\u203e}}{\mathrm{6}}\phantom{\rule{0.125em}{0ex}}\mathit{\%}$. The calculations so far show that, if the secular
approximation can be employed, the ZFS has no effect on the LaserIMD trace,
and it is possible to analyze experimentally recorded LaserIMD data with the
same kernel that can be used for DEER.

Even though the ZFS has no effect in the secular approximation in LaserIMD, it cannot be taken for granted that the non-secular terms can be ignored because the ZFS of some transient triplet labels can be quite large (Williams et al., 2020). Here, we additionally consider the terms ${\widehat{S}}_{\mathrm{T},z}{\widehat{S}}_{\mathrm{T},+}+{\widehat{S}}_{\mathrm{T},+}{\widehat{S}}_{\mathrm{T},z}$ and ${\widehat{S}}_{\mathrm{T},-}{\widehat{S}}_{\mathrm{T},z}+{\widehat{S}}_{\mathrm{T},-}{\widehat{S}}_{\mathrm{T},z}$ from the ZFS interaction and the terms ${\widehat{S}}_{\mathrm{D},z}{\widehat{S}}_{\mathrm{T},+}$ and ${\widehat{S}}_{\mathrm{D},z}{\widehat{S}}_{\mathrm{T},-}$ from the dipolar coupling. They connect the adjacent triplet states |+1〉 and |0〉 and |0〉 and |−1〉 of the triplet manifold and shift their energy in second order (Hagston and Holmes, 1980). This is illustrated in Fig. 2. The details of this calculation are described in Supplement S1. For this calculation, the remaining ZFS terms ${\widehat{S}}_{\mathrm{T},+}^{\mathrm{2}}$ and ${\widehat{S}}_{\mathrm{T},-}^{\mathrm{2}}$ were ignored. They connect the triplet states |+1〉 and |−1〉, which have a larger energy difference than adjacent states. Therefore, the second-order energy shift caused by ${\widehat{S}}_{\mathrm{T},+}^{\mathrm{2}}$ and ${\widehat{S}}_{\mathrm{T},-}^{\mathrm{2}}$ is weaker than those of the considered terms. The terms ${\widehat{S}}_{\mathrm{D},+}{\widehat{S}}_{\mathrm{T},+}$; ${\widehat{S}}_{\mathrm{D},-}{\widehat{S}}_{\mathrm{T},+}$; ${\widehat{S}}_{\mathrm{D},+}{\widehat{S}}_{\mathrm{T},-}$; ${\widehat{S}}_{\mathrm{D},-}{\widehat{S}}_{\mathrm{T},-}$; ${\widehat{S}}_{\mathrm{D},+}{\widehat{S}}_{\mathrm{T},z}$; and ${\widehat{S}}_{\mathrm{D},-}{\widehat{S}}_{\mathrm{T},z}$ of the dipolar coupling were also ignored. They connect the spin states of different manifolds of the doublet spin, and the corresponding energies cannot be significantly shifted by the comparably weak dipolar coupling. It is shown in Supplement S2 that the included non-secular terms from Eq. (S3) are sufficient at the magnetic field strengths that are relevant for experimental conditions, and no further distortions are to be expected due to the omitted terms.

The shift in the energy levels also leads to a shift in the LaserIMD frequencies (see Supplement S1):

where

As can be seen from Eqs. (21)–(23), the frequencies ${\mathit{\omega}}_{+\mathrm{1}}^{\text{non-sec}}$ and ${\mathit{\omega}}_{-\mathrm{1}}^{\text{non-sec}}$ are the sum
of the unperturbed frequencies ${\mathit{\omega}}_{+\mathrm{1}}^{\mathrm{sec}}$ and ${\mathit{\omega}}_{-\mathrm{1}}^{\mathrm{sec}}$ and a frequency shift *δ*_{ZFS}sin (2*β*_{dip})*ω*_{dip}, which contains
the effect of the ZFS. Most notably, the coherence transfer pathway with
Δ*m*_{T}=0 does not lead to a vanishing LaserIMD
frequency, as was the case in the secular approximation. Instead, we find
that ${\mathit{\omega}}_{\mathrm{0}}^{\text{non-sec}}$ equals twice the negative of the
frequency shift that is experienced by the other two coherence transfer
pathways. The frequency shift scales with *δ*_{ZFS}, which
depends on the ZFS values *D* and *E*, the Zeeman frequency of the transient
triplet label *ω*_{T}, and the orientation of the
transient triplet label, described by *α*_{T}, *β*_{T} and *γ*_{T}. At a higher ZFS and a smaller
magnetic field, the shift in the LaserIMD frequencies will be larger, so
that larger disturbances in the LaserIMD trace can be expected in these
cases.

The powder average is more complex when the non-secular terms are included, as the LaserIMD frequencies now also depend on the orientation of the transient triplet label. Still assuming no orientation selection, this gives the following integrals:

The sum over these terms gives the final intramolecular contribution in LaserIMD:

By including incomplete excitation and the intermolecular dipolar interactions, one arrives at the final model:

Unlike the case for the secular approximation, the integrals are
difficult to solve analytically, and further insight into this expression will
be gained by numerical integrations in the next sections. However, it can
already be seen without further calculations that, with the non-secular terms,
the ZFS has an influence in LaserIMD and that the resulting kernel no longer
corresponds to the kernel *K*_{DEER}(*t*,*r*) of the
$S=\mathrm{1}/\mathrm{2}$ case.

## 2.3 LiDEER

In LiDEER, the transient triplet label is observed, and the permanent spin
label is pumped. For simplicity, we will derive the expressions within the
secular approximation first and afterwards turn to the case that includes
the non-secular terms. Due to the limited excitation bandwidth of the
observer pulse, either the transition between the states with
*m*_{T}=1 and *m*_{T}=0 or the states with
*m*_{T}=0 and ${m}_{\mathrm{T}}=-\mathrm{1}$ of the transient triplet label is
excited. If the transition between the states *m*_{T}=1 and
*m*_{T}=0 is excited, the excited coherence of the triplet spin
will either evolve with the frequency ${\mathit{\omega}}_{+\frac{\mathrm{1}}{\mathrm{2}},\phantom{\rule{0.125em}{0ex}}\mathrm{1}\leftrightarrow \mathrm{0}}^{\mathrm{sec}}={E}_{+\frac{\mathrm{1}}{\mathrm{2}},+\mathrm{1}}^{\mathrm{sec}}-{E}_{+\frac{\mathrm{1}}{\mathrm{2}},\mathrm{0}}^{\mathrm{sec}}$
or ${\mathit{\omega}}_{-\frac{\mathrm{1}}{\mathrm{2}},\phantom{\rule{0.125em}{0ex}}\mathrm{1}\leftrightarrow \mathrm{0}}^{\mathrm{sec}}={E}_{-\frac{\mathrm{1}}{\mathrm{2}},+\mathrm{1}}^{\mathrm{sec}}-{E}_{-\frac{\mathrm{1}}{\mathrm{2}},\mathrm{0}}^{\mathrm{sec}}$,
depending on whether the permanent spin label is in the state with
${m}_{\mathrm{D}}=\mathrm{1}/\mathrm{2}$ or ${m}_{\mathrm{D}}=-\mathrm{1}/\mathrm{2}$. Pumping the permanent spin
label at the time *t* will result in a transition from
${m}_{\mathrm{D}}=+\frac{\mathrm{1}}{\mathrm{2}}$ to ${m}_{\mathrm{D}}=-\frac{\mathrm{1}}{\mathrm{2}}$ (or vice
versa), and the frequency ${\mathit{\omega}}_{+\frac{\mathrm{1}}{\mathrm{2}},\phantom{\rule{0.125em}{0ex}}\mathrm{1}\leftrightarrow \mathrm{0}}^{\mathrm{sec}}$ or ${\mathit{\omega}}_{-\frac{\mathrm{1}}{\mathrm{2}},\phantom{\rule{0.125em}{0ex}}\mathrm{1}\leftrightarrow \mathrm{0}}^{\mathrm{sec}}$ with which the coherence evolves will change accordingly.
At the time of the echo, the coherence will have gained a phase $\mathit{\varphi}={\mathit{\omega}}_{\pm \frac{\mathrm{1}}{\mathrm{2}}\to \mp \frac{\mathrm{1}}{\mathrm{2}},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}+\mathrm{1}\leftrightarrow \mathrm{0}}^{\mathrm{sec}}t$, where ${\mathit{\omega}}_{\pm \frac{\mathrm{1}}{\mathrm{2}}\to \mp \frac{\mathrm{1}}{\mathrm{2}},+\mathrm{1}\leftrightarrow \mathrm{0}}^{\mathrm{sec}}$ denotes the LiDEER
frequencies of the two coherence transfer pathways:

When the other transition of the triplet spin from *m*_{T}=0 and
${m}_{\mathrm{T}}=-\mathrm{1}$ is excited by the observer pulse, the frequencies are
the same:

As those are the same frequencies as the ones in DEER with two $S=\mathrm{1}/\mathrm{2}$
spins, one eventually arrives at the same kernel *K*_{DEER}(*t*,*r*).
This means that, as was the case in LaserIMD, the secular terms of the
ZFS cancel each other out, and there is no effect of the ZFS on the LiDEER
trace. In contrast to LaserIMD in secular approximation, there are also no
coherence transfer pathways with Δ*m*_{D}=0, so that
the maximum achievable modulation depth in LiDEER is 100 %.

It seems obvious that the same non-secular terms that led to change in the LaserIMD frequencies are also relevant in LiDEER. Therefore, the LiDEER frequencies were also determined from the energy levels ${E}_{{m}_{\mathrm{D}},{m}_{\mathrm{T}}}^{\text{non-sec}}$ that include the effects of the ZFS:

It can again be seen that the ZFS leads to a shift in the dipolar frequencies. This shift is, besides the factor of 3, identical to the one that was obtained for the LaserIMD frequencies ${\mathit{\omega}}_{+\mathrm{1}}^{\text{non-sec}}$ and ${\mathit{\omega}}_{-\mathrm{1}}^{\text{non-sec}}$. From here, the next step is again the averaging over the orientations of the transient triplet label and the dipolar coupling vector that contribute to the LiDEER signal. However, this is even more complicated than it was in LaserIMD, where all orientations are evenly excited by the laser flash. In LiDEER, the triplet spins are also excited by microwave pulses which typically have a bandwidth that is much narrower than the EPR spectrum of the transient triplet label. For example, the frequently used porphyrin labels have an EPR spectrum that is over 2 GHz broad (Di Valentin et al., 2014) of which a typical rectangular microwave pulse with a length of 10 ns can only excite roughly 120 MHz (Schweiger and Jeschke, 2001). Therefore, not all orientations of the transient triplet labels contribute to the LiDEER signal, and it is rather tedious to even derive an expression for the integrals that describe the orientation averaging. To circumvent this problem, the LiDEER traces will be calculated by time-domain simulations with weak microwave pulses in the next sections.

## 3.1 Simulations

The powder averages for LaserIMD were performed by a numerical integration
of Eqs. (25)–(27) with custom MATLAB (version 2020b) scripts. For the
angle *β*_{dip}, a linear, equidistant grid from 0 to
$\frac{\mathit{\pi}}{\mathrm{2}}$ was used. Each value was weighted proportional to
sin(*β*_{dip}). For the orientation of the transient
triplet label, a grid with all three Euler angles, *α*_{T},
*β*_{T} and *γ*_{T}, including the
corresponding weights, was calculated according to the REPULSION approach
(Bak and Nielsen, 1997; Hogben et
al., 2011) with the Spinach (version 2.6.5625) software package
(Hogben et al., 2011). To check for a
sufficient convergence, a test run with an increasing numbers of points for
the two grids was simulated. The test run was stopped when the relative
change Δ*ϵ* in the simulated signal, when the number
of grids points was increased, was below 1 %. For *β*_{dip},
a grid size of 200 points was sufficient, whereas for *α*_{T}, *β*_{T} and *γ*_{T} 12 800
points were necessary. For details on the convergence behavior, see Supplement S3.

The time-domain simulations for LiDEER were performed with Spinach version 2.6.5625 (Hogben et al., 2011). The powder averaging was done with the same grids that were used for LaserIMD. For details, see Supplement S8. The source code for the LiDEER simulations can be downloaded from https://github.com/andreas-scherer/LiDEER_simulations.git, last access: 8 January 2023.

## 3.2 Experiments and data analysis

LaserIMD and LiDEER measurements were performed on the two peptides
TPP–pAA_{5}–NO• and TPP–pAA_{10}–NO• shown in
Fig. 3. They were purchased from Biosynthan
(Berlin) as powder samples and used without further purification. They were
dissolved in MeOD $/$ D_{2}O ($\mathrm{98}/\mathrm{2}$ vol %) and, prior to freezing in liquid
nitrogen, they were degassed with three freeze–pump–thaw cycles. Light
excitation was performed at a wavelength of 510 nm by an Nd : YAG laser system
from EKSPLA (Vilnius) that was coupled into the resonator via a laser fiber.
EPR measurements were performed on a commercial Bruker ELEXSYS-E580 spectrometer:
X-band measurements in an ER4118X-MS3 resonator and Q-band measurements in
an ER5106QT-2 resonator. In X-band, the resonator was critically coupled to a
*Q* value of ≈ 900–2000, whereas it was overcoupled to a *Q* value
of ≈ 200 in Q-band. LaserIMD was recorded with the pulse sequence $\mathit{\pi}/\mathrm{2}-\mathit{\tau}-\mathit{\pi}-t$ − laser pulse − (*τ*−*t*) − echo
(Hintze et al., 2016). A two-step phase cycle was implemented
for baseline correction. Signal averaging was done by recording 10 shots per
point. The zero-time correction was performed by recording a short
refocused LaserIMD (reLaserIMD) (Dal Farra et al., 2019a) trace, as
reported in Scherer et al. (2022a). LiDEER
measurements were performed with the following pulse sequence: laser pulse – DAF –
$\mathit{\pi}/\mathrm{2}-{\mathit{\tau}}_{\mathrm{1}}-\mathit{\pi}-t-{\mathit{\pi}}_{\mathrm{pump}}-({\mathit{\tau}}_{\mathrm{1}}+{\mathit{\tau}}_{\mathrm{2}}-t)-\mathit{\pi}-{\mathit{\tau}}_{\mathrm{2}}-$ echo
(Di Valentin et al., 2014). The delay-after-flash (DAF) was
set to 500 ns, and *τ*_{1} was set to 400 ns. Nuclear modulation averaging was
performed by varying the *τ*_{1} time in eight steps with Δ*τ*_{1}=16 ns. Phase cycling was performed with an eight-step scheme ((*x*) [*x*] *x**p* *x*), as proposed by Tait and Stoll (2016).
The LiDEER data were analyzed with the Python DeerLab (version
0.13.2) software package (Fábregas Ibáñez et al., 2020) and
Python 3.9 with the DEER kernel *K*_{DEER}(*t*,*r*) and Tikhonov
regularization. A 3D homogenous background function was used, and the
regularization parameter was chosen according to the Akaike information
criterion (Edwards and Stoll, 2018). The validation
was performed with bootstrapping by analyzing 1000 samples
generated with artificial noise. The error was then calculated as the 95 %
confidence interval. Further details can be found in Supplement S7 and S10.

## 4.1 LaserIMD simulations

An initial simulation to study the effect of the ZFS in LaserIMD was
performed for X-band (*ν*_{T}=9.3GHz) with a dipolar
coupling that corresponds to a distance of *r*=2.2 nm, a ZFS of
*D*=1159 and $E=-\mathrm{238}$ MHz, and zero-field populations of
*P*_{x}=0.33, *P*_{y}=0.41 and *P*_{z}=0.26. The
ZFS and zero-field populations correspond to TPP, which is often used to
perform LaserIMD and LiDEER measurements
(Di
Valentin et al., 2014; Hintze et al., 2016; Di Valentin et al., 2016; Bieber
et al., 2018; Bertran et al., 2020). For simplicity, a complete excitation
of the transient triplet label (*γ*=1) was assumed, and no background
was added (*B*(*t*)=1). For a more detailed analysis, the contributions from
the three coherence transfer pathways with $\mathrm{\Delta}{m}_{\mathrm{T}}=\mathrm{1},\phantom{\rule{0.125em}{0ex}}\mathrm{0},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}-\mathrm{1}$, termed ${V}_{+\mathrm{1}}^{\text{non-sec}}\left(t\right)$, ${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$ and ${V}_{-\mathrm{1}}^{\text{non-sec}}\left(t\right)$, respectively, are simulated separately and presented in
Fig. 4 with their resulting sum
${V}_{\mathrm{LaserIMD}}^{\text{non-sec}}\left(t\right)$. They are also compared with
the corresponding traces from the secular approximation,
${V}_{\mathrm{LaserIMD}}^{\mathrm{sec}}\left(t\right)$, ${V}_{+\mathrm{1}}^{\mathrm{sec}}\left(t\right)$,
${V}_{\mathrm{0}}^{\mathrm{sec}}\left(t\right)$ and ${V}_{-\mathrm{1}}^{\mathrm{sec}}\left(t\right)$, where the ZFS is
ignored. The comparison of the traces including and excluding the ZFS
(${V}_{+\mathrm{1}}^{\text{non-sec}}\left(t\right)$ and
${V}_{-\mathrm{1}}^{\text{non-sec}}\left(t\right)$ with
${V}_{+\mathrm{1}}^{\mathrm{sec}}\left(t\right)$ and ${V}_{-\mathrm{1}}^{\mathrm{sec}}\left(t\right))$ in Fig. 4a and c shows that there is no
visible effect of the ZFS in the traces ${V}_{+\mathrm{1}}^{\text{non-sec}}\left(t\right)$ and ${V}_{-\mathrm{1}}^{\text{non-sec}}\left(t\right)$, and they look
virtually identical to ${V}_{+\mathrm{1}}^{\mathrm{sec}}\left(t\right)$ and
${V}_{-\mathrm{1}}^{\mathrm{sec}}\left(t\right)$. The frequency shift *δ*_{ZFS}sin (2*β*_{dip})*ω*_{dip} seems to be averaged out after integration for these terms.
The situation is different in the case of ${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$ and ${V}_{\mathrm{0}}^{\mathrm{sec}}\left(t\right)$ in
Fig. 4b. Whereas ${V}_{\mathrm{0}}^{\mathrm{sec}}\left(t\right)$ is a constant function of time and does not contribute to the echo
modulation, ${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$ shows a continuous decay of the
echo intensity with increasing time. This decay does not contain any
additional dipolar oscillations, and its shape does not seem to follow any
obvious simple mathematical law. For the full LaserIMD traces in
Fig. 4d, this means that, whereas the trace ${V}_{\mathrm{LaserIMD}}^{\mathrm{sec}}\left(t\right)$ looks
like a $S=\mathrm{1}/\mathrm{2}$ DEER trace with a modulation depth of $\mathit{\lambda}=\mathrm{66}.\stackrel{\mathrm{\u203e}}{\mathrm{6}}\phantom{\rule{0.125em}{0ex}}\mathit{\%}$ when not considering the
ZFS, the trace ${V}_{\mathrm{LaserIMD}}^{\text{non-sec}}\left(t\right)$ with the ZFS shows the same dipolar oscillations but on top of a
decay. Moreover, this means that, due to the coherence transfer pathway with
Δ*m*_{T}=0 also resulting in a variation in the echo
intensity, the modulation depth of LaserIMD is increased by the ZFS, and
values higher than $\mathrm{66}.\stackrel{\mathrm{\u203e}}{\mathrm{6}}\phantom{\rule{0.125em}{0ex}}\mathit{\%}$ can be reached.

The frequency shift caused by the non-secular terms of the ZFS in LaserIMD
depends not only on *D* and *E* but also on the zero-field populations
(*P*_{x}, *P*_{y} and *P*_{z}), the Zeeman frequency *ν*_{T} and
the distance *r* (see Eqs. 21–24). The influence of these parameters was
studied by simulating additional LaserIMD traces with different magnetic
field strengths, ZFS values, zero-field populations and distance
distributions (see Figs. 5 and
6). In Fig. 5a,
two LaserIMD traces in X- and Q-band (*ν*_{T}=9.3
and *ν*_{T}=34.0 GHz) with TPP as a transient triplet
label and a distance of *r*=2.2 nm are compared.
Figure 5b shows the comparison between the ZFS of
TPP (*D*=1159 and $E=-\mathrm{238}$ MHz) and a stronger ZFS of
*D*=3500 and E$=-\mathrm{800}$ MHz, as such high values are
possible for some labels like rose bengal and erythrosin B
(Williams et al., 2020; Bertran et al.,
2022b). Both simulations were performed in Q-band with *r*=2.2 nm.
Figure 5c shows three simulations with the
population of the zero-field triplet states being completely assigned to *P*_{x}, *P*_{y} or *P*_{z}. In Fig. 5d,
the effect of different distances of *r*=2.2 and
*r*=5.0 nm on ${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$ is shown for
TPP in Q-band. The simulations in Fig. 5 were all
done with a single distance. To study the influence of the width of the
distance distribution on ${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$,
additional simulations were performed with a Gaussian distance distribution
with a mean of 3.0 nm and different standard deviations *σ*
ranging from 0.05 to 3.0 nm. The results of these
simulations are shown in Fig. 6a and b for X-band and
Q-band, respectively.

Figure 5a, b and
c show that there are no visible
differences in the dipolar oscillations in ${V}_{+\mathrm{1}}^{\text{non-sec}}\left(t\right)$ and ${V}_{-\mathrm{1}}^{\text{non-sec}}\left(t\right)$ when the Zeeman
frequency, ZFS or zero-field populations are changed. This can also be seen
in the Supplement S4, S5 and S6, where the traces for different Zeeman
frequencies, ZFSs and distances are compared in more detail. This agrees
with the former results in Fig. 4: the
frequency shift due to the ZFS is virtually averaged out in a powder sample
for ${V}_{+\mathrm{1}}^{\text{non-sec}}\left(t\right)$ and
${V}_{-\mathrm{1}}^{\text{non-sec}}\left(t\right)$, so changing the involved
parameters should also have little effect. The situation is different for
${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$, which, as is shown in
Fig. 4c, is more strongly affected by the ZFS.
The previously mentioned, decay is faster for lower Zeeman frequencies (see
Fig. 5a) and a stronger ZFS (see
Fig. 5b). Because *δ*_{ZFS}
ultimately depends on the ratio of the ZFS to the Zeeman frequency, a higher
ZFS and a lower Zeeman frequency both increase the magnitude of the
frequency shift in ${\mathit{\omega}}_{\mathrm{0}}^{\text{non-sec}}$ in the same way,
leading to the same effect on the LaserIMD trace. The parameters that have the
least influence on the LaserIMD trace are the zero-field populations (see
Fig. 5c). Changing the populations of the
zero-field states does not seem to affect the dipolar oscillations, as
was the case for different ZFSs and magnetic field strengths. This time, the decay of ${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$ is also barely
affected by different zero-field populations. Figure 5d shows that shorter distances lead to a faster decay of
${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$. As can be seen in Eqs. (21)–(23), changing the distance *r* from 2.2 to 5.0 nm leads to an
increase in the LaserIMD frequencies ${\mathit{\omega}}_{+\mathrm{1}}^{\text{non-sec}}$,
${\mathit{\omega}}_{\mathrm{0}}^{\text{non-sec}}$ and ${\mathit{\omega}}_{-\mathrm{1}}^{\text{non-sec}}$ that
scales with *r*^{−3}. This distance dependence of the dipolar oscillations
(not shown in Fig. 5c) is used in PDS for the
calculation of the distance distributions. In the case of LaserIMD, the
steepness of the decay of ${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$ is an
additional feature that depends on the distance between the spin labels. As
can be seen in Fig. 6, the width of the distance
distribution also has an influence on the decay of
${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$. In X-band (see
Fig. 6a) and for small standard deviations of
*σ*=0.05 nm, ${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$ has a
sigmoid-like shape. Increasing the width has a twofold effect on the decay
of ${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$. Whereas the initial decay is
steeper, on a long scale, the decay of ${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$ is decreased for broader distance distributions. This can clearly
be seen in the case of *σ*=3.0 nm: for *t*<1 µs, ${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$ decays faster for the
simulation with *σ*=3.0 nm than with *σ*=0.05 nm; for *t*>1 µs,
${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$ decays slower for *σ*=3.0 nm than for *σ*=0.05 nm. In Q-band, where the
decay of ${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$ is generally slower, the
simulations in Fig. 6b show that only the
first effect is of relevance here. It can be seen that the first part of the
decay of ${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$ is again steeper for
broader distance distributions, but the second part, where this behavior is
inverted, lies outside the time window. This means that, in Q-band, the
width of the distance distribution has a smaller influence on the decay of
${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$ than in X-band.

Taken together, variations in the ZFS parameter, the population of the ZFS states and the employed magnetic field (X- or Q-band) do not affect the dipolar oscillations in ${V}_{+\mathrm{1}}^{\text{non-sec}}\left(t\right)$ and ${V}_{-\mathrm{1}}^{\text{non-sec}}\left(t\right)$. They mostly have an effect on the decay of ${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$, such that larger ZFS parameters and lower magnetic fields will lead to a stronger additional decay in the LaserIMD trace. The additional decay also depends on the distance distribution between the spin labels: it is faster for shorter distances, and the shape of the decay also depends on the width of the distance distribution (in X-band more than in Q-band). The decay of ${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$ can, therefore, be used as an additional source of information for the calculation of the distance distribution.

So far, all simulations only showed a visible effect of the ZFS on
${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$, but no significant influence on
${V}_{+\mathrm{1}}^{\text{non-sec}}\left(t\right)$ and
${V}_{-\mathrm{1}}^{\text{non-sec}}\left(t\right)$ was observed. To check if and
when the ZFS also has an influence on ${V}_{+\mathrm{1}}^{\text{non-sec}}\left(t\right)$ and ${V}_{-\mathrm{1}}^{\text{non-sec}}\left(t\right)$, we performed
additional simulations where the effect of the ZFS is expected to be
stronger. This can be obtained by either lower Zeeman frequencies or higher
ZFS values. As the effect on *δ*_{ZFS} is the same in both
cases, the ratio of *D* and the Zeeman frequency of the triplet *ν*_{T} can be defined as follows:

For simplification, the ZFS was assumed to be axial with *E*=0. This
simplifies the expression of *δ*_{ZFS} to

The simulation in X-band with TPP from Fig. 4
corresponds to a ratio where *q* is approximately 0.13. Here, we tried
values for *q* of up to 1. Figure 7 shows the sum
of ${V}_{+\mathrm{1}}^{\text{non-sec}}\left(t\right)$ and
${V}_{-\mathrm{1}}^{\text{non-sec}}\left(t\right)$ of these simulations and
compares it to a trace where the effect of the ZFS has been ignored. It can
be seen that the traces are negligibly affected by the ZFS up to *q*=0.5.
For higher values, the dipolar oscillations start to get shifted to slightly
higher frequencies and are also smoothed out more quickly. Analyzed with the
oversimplified kernel *K*_{DEER}(*t*,*r*) of the $S=\mathrm{1}/\mathrm{2}$ model, this
would result in a shift to smaller distances and an artificial broadening of
the distance distribution. However, for experimentally relevant distance
distributions with a finite width, the oscillations typically fade out much
quicker, and cases where four oscillations can be resolved are scarce. In
such a case, the observed influence of the ZFS for high values of *q* can be
expected to be almost negligible. Furthermore, as *q*=1 is equivalent to a ZFS
that is of the same order of magnitude as the Zeeman frequency, this is not
relevant for most practical applications, as LaserIMD is typically performed
in X- or Q-band (*ν*_{T}=9.3 or *ν*_{T}=34.0 GHz), and all transient triplet labels used so
far have a ZFS value *D* below 4 GHz
(Dal Farra et al.,
2019b; Williams et al., 2020). Even in the most extreme case, this would
result in *q* values smaller than 0.5. Consequently, the effect of the
ZFS on ${V}_{+\mathrm{1}}^{\text{non-sec}}\left(t\right)$ and
${V}_{-\mathrm{1}}^{\text{non-sec}}\left(t\right)$ is not relevant for most
experiments and, even though the ${V}_{+\mathrm{1}}^{\text{non-sec}}\left(t\right)$
and ${V}_{-\mathrm{1}}^{\text{non-sec}}\left(t\right)$ can, in principle, be
influenced by the ZFS, it seems to be a safe assumption that the ZFS in
LaserIMD affects only the decay in ${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$ and not the dipolar oscillations in
${V}_{+\mathrm{1}}^{\text{non-sec}}\left(t\right)$ and
${V}_{-\mathrm{1}}^{\text{non-sec}}\left(t\right)$.

As previously stated, in the secular approximation, LaserIMD traces can be
analyzed with the kernel *K*_{DEER}(*t*,*r*) of the $S=\mathrm{1}/\mathrm{2}$ model. To
examine the extent to which this is true when the ZFS is not negligible, we
simulated LaserIMD traces that were subsequently analyzed with
*K*_{DEER}(*t*,*r*). To mimic experimental conditions more closely, we
assumed an incomplete excitation of the transient triplet label, and the
intermolecular dipolar background was also considered. TPP was used as a
transient triplet label with a distance to the permanent spin label of
*r*=2.2 nm and a modulation depth of *λ*=50*%*, which
roughly correspond to the values that can be typically achieved in
experiments. Simulations were performed in X- and Q-band with different
background decay rates varying between *k*=0.0 µs^{−1} (no background) and *k*=0.4 µs^{−1}. The resulting traces were then analyzed with
*K*_{DEER}(*t*,*r*) and Tikhonov regularization (see Supplement S7 for details).

The simulations and fitted distance distributions can be seen in
Fig. 8, and the background decay rates and
modulations depths that were obtained by the fits are shown in
Table 1. Figure 8 shows
that the fits agree well with the simulated data, and the main peak of the
distance distribution at *r*=2.2 nm is fitted appropriately in X-
as well as in Q-band. However, there can be additional artifact peaks in the
distance distributions, and the fitted modulation depths and background
decay rates can be erroneous (see Table 1). This is
particularly pronounced in X-band, which shows artifacts in the distance
distribution between 3.9 and 5.0 nm and at the higher-distance end. Moreover, the background decay rates and modulation depths
deviate significantly from the values that were originally used for the
simulations. The simulations in X-band are always fitted with a background
decay rate close to zero (*k*_{fit}≈0.0 µms^{−1}), even in the cases where the strongest
background was included (*k*=0.4 µms^{−1}) in
the simulation. The modulation depth was fitted with values from 47 % to
70 % and varies significantly for different background decays. In Q-band,
the fitted parameters are closer to the input values of the simulations. The
distance artifacts that appeared in X-band between 3.9 and 5.0 nm have
disappeared, and only those at the long distance limit remain. In Q-band, the
fitted background decay is always a bit larger than the true value. Except
for the case were the true background decay is set to *k*=0 µs^{−1}, the deviation of the fitted and the true
background decay is smaller in Q-band than in X-band. Only the obtained
modulation depths are less accurate than in X-band and fitted to values
between 32 % and 36 %. Although these simulations are only
anecdotal evidence and generalizations from these data must be taken with
caution, they show that it is possible to extract the main distance peak
correctly when LaserIMD data are analyzed with
*K*_{DEER}(*t*,*r*). Thus, analyzing LaserIMD traces with *K*_{DEER}(*t*,*r*) can
be an option in situations where the ZFS values and zero-field populations
of the transient triplet label are unknown and their effect cannot be
included in the analysis. However, this way of analyzing LaserIMD data can
give artifacts at higher distances as well as errors in the obtained
modulation depth and background decay rate. This is particularly pronounced
for low magnetic fields (e.g., X-band), and similar results can be expected
for transient triplet labels with higher ZFS values.

## 4.2 LiDEER simulations

In LaserIMD, transient triplet labels of all orientations are excited by the
laser flash and contribute to the signal; thus, an integration over all
orientations was performed (Eqs. 25–27) to calculate the LaserIMD
signal. In contrast, the transient triplet labels are
additionally excited by microwave observer pulses in LiDEER. As the spectrum of many
transient triplet labels exceeds the excitation bandwidth of these pulses
(Di Valentin et
al., 2014; Williams et al., 2020; Krumkacheva et al., 2019), only a small
number of orientations within the excitation bandwidth contribute to the
signal. Because the frequency shift *δ*_{ZFS} of the LiDEER
frequencies (Eqs. 34–37) depends on the orientation of the transient
triplet labels, the choice of the observer frequency influences the shape of
the LiDEER trace.

In experiments in which the commonly used nitroxides or other spin labels with
*g*_{D}≈2 are used as pump spin, the resonator bandwidth
allows one to use only the *Y*^{±} peaks as the observer position, as the
other parts of the EPR spectrum of the transient triplet label lie outside
the resonator bandwidth
(Bieber et al., 2018;
Bowen et al., 2021). Figure 9 shows the
orientations of the triplet label TPP that, in this case, contribute to the
LiDEER signal. The contribution of the orientations where the *Y* axis of
eigenframe of the ZFS is parallel to the external magnetic field (${\mathit{\beta}}_{\mathrm{T}}=\mathit{\pi}/\mathrm{2}$ and ${\mathit{\gamma}}_{\mathrm{T}}=\mathit{\pi}/\mathrm{2})$ is eponymous for
the *Y*^{±} peaks. For this orientation, the frequency shift *δ*_{ZFS}=0, and the ZFS has no effect on the LiDEER trace. However,
it can be seen that other orientations are also excited if the observer
pulses are placed on either of the *Y*^{±} peaks. For these contributions,
it cannot guaranteed that *δ*_{ZFS} is always zero, so that
there might still be an effect of the ZFS.

To study the effect of the ZFS in LiDEER, numerical time-domain simulations
for different ZFS values in X- and Q-band were performed. The microwave
pulses were placed on the *Y*^{+} peak of the EPR spectrum and had a finite
length, power and bandwidth so that only the orientations that are shown in
Fig. 9 contribute to the LiDEER signal, as is
the case in the experimental setup. A simulation for TPP as a transient
triplet label was performed in X- and Q-band, and an additional simulation
with a larger ZFS of *D*=3500 and $E=-\mathrm{800}\phantom{\rule{0.125em}{0ex}}$ MHz was
performed in X-band. The permanent spin label was included as a doublet spin
with an isotropic *g* value (*g*_{D}=2) and without any additional
hyperfine interactions. The distance was set to *r*=2.2 nm, and no
background from intermolecular spins was included. To check for artifacts
that occur in distance distributions if the ZFS is ignored in data analysis,
the simulated LiDEER traces were analyzed with *K*_{DEER}(*t*,*r*) and
Tikhonov regularization. The details of the calculation of the distance
distribution are given in Supplement S7, and the details of the simulations can be found in Supplement S8.

Figure 10a shows the simulated LiDEER traces, and
Fig. 10b presents the obtained distance distributions. The
differences in the LiDEER traces for different ZFS and Zeeman frequencies
are smaller than they are in LaserIMD (see Fig. 4). This is because, in LiDEER, there is no equivalence for the coherence
transfer pathway with Δ*m*_{T}=0 that showed the
strongest dependency on the ZFS and magnetic fields in LaserIMD (see
Fig. 5). The distance distribution for TPP in Q-band shows a narrow peak at 2.20 nm with a full width at half
maximum (FWHM) of 0.004 nm. This fits to the
2.20 nm (FWHM = 0 nm) that was used for
the simulation. In X-band, the distance distribution with TPP is also
centered at 2.20 nm but gets broadened to a FWHM of
0.014 nm. This trend increases for the large ZFS with
*D*=3500 and $E=-\mathrm{800}\phantom{\rule{0.125em}{0ex}}$ MHz in X-band. Here, the
distance distribution gets even broader with an FWHM of 0.028 nm
and is now also shifted to a center of ≈2.22 nm. This
behavior fits with the results of LaserIMD in Fig. 7, where the shifts in the dipolar oscillation also get larger when the ZFS
is large compared with the Zeeman frequency. However, it must also be stated
that the observed shifts in the distance distribution are still rather small
here and should be below the resolution limit that is relevant in most
experiments. Additional traces in which the observer pulse was set
off-resonance to the canonical peaks were also performed and are presented
in Supplement S9. Here, the effect of the ZFS can clearly be seen, and the LiDEER trace
of the simulation with *D*=3500 and $E=-\mathrm{800}\phantom{\rule{0.125em}{0ex}}$ MHz in X-band shows strong deviations from the other traces that were simulated
with a smaller ZFS. The dipolar oscillations fade out much faster, which
also leads to a stronger broadening of the distance distributions. However,
for experimentally relevant cases with distance distributions of a finite
width, the oscillations in the dipolar trace fade out much faster anyway. It
is to be expected that, in these cases, the effect of the ZFS on the LiDEER
trace are rather small and that artifacts in the distance
distribution are, therefore, not so pronounced, even in the case when the observer
pulses are set to a non-canonical orientation.

This means that, in general, the ZFS has an effect on LiDEER and the LiDEER
trace changes when different parts of the EPR spectrum of the transient
triplet label are used for excitation by the observer pulses. However, in
the special case when either of the *Y*^{±} peaks is used as the position for
the observer pulse, the effect of the ZFS can be suppressed and LiDEER
traces can be analyzed with the *K*_{DEER}(*t*,*r*) kernel without
introducing significant artifacts in the distance distribution. This is
particularly valid for TPP – and other transient triplet labels with a
similar ZFS – in Q-band.

## 4.3 Experiments

To experimentally confirm the theoretical finding that the ZFS has an
influence on the shape of the LaserIMD trace, LaserIMD measurements were
performed at different magnetic field strengths in X- and Q-band and with
two model systems with shorter and longer distances between the labels.
This should result in scenarios were the ZFS has either a weak effect on the
trace (high magnetic field strength and long distance) or a strong effect on the
trace (low
magnetic field strength and short distance). The LaserIMD experiments were
simulated with the newly derived model that includes the ZFS. The distance
distributions and background decay rates that were used for these
simulations of the LaserIMD traces were determined with LiDEER. The
measurements were performed with the peptides TPP–pAA_{5}–NO•
and TPP–pAA_{10}–NO•. They contain TPP as a transient triplet
label and the nitroxide 2,2,6,6-tetramethylpiperidine-1-oxyl-4-amino-4-carboxylic acid (TOAC) as permanent spin label. Both labels are
separated by a rather rigid helix consisting of l-alanine and
*α*-isobutyric acid (Di Valentin et al., 2016).

So far, the LaserIMD simulations that were described above mostly only
invoked a single delta-like distance. To simulate LaserIMD for an entire
distance distribution in a fast way, the dipolar kernel
${K}_{\mathrm{LaserIMD}}^{\text{non-sec}}\left(t,r\right)$ needs to be
calculated. Therefore, we implemented a C$++$ software tool that can
perform the numerical integration of Eqs. (25)–(27) to calculate
${S}_{\mathrm{LaserIMD}}^{\text{non-sec}}\left(t,r\right)$. It allows the
user to specify different ZFS values, zero-field populations and Zeeman
frequencies. The background decay and modulation depth can then be included
afterwards to obtain the full kernel
${K}_{\mathrm{LaserIMD}}^{\text{non-sec}}\left(t,r\right)$ (see Eq. 29).
The obtained kernel can, for example, be used in combination with the DeerLab software
(Fábregas Ibáñez et al., 2020) to
analyze experimental LaserIMD traces. The program, including its source code,
is available at GitHub (https://github.com/andreas-scherer/LaserIMD_kernel, last access: 21 December 2022). Here,
it was used to calculate the kernel that corresponds to the experimentally
determined parameters for TPP of the peptides TPP–pAA_{5}–NO•
and TPP–pAA_{10}–NO• (ZFS values of *D*=1159 and
$E=-\mathrm{238}$ MHz and zero-field populations of *P*_{x}=0.33,
*P*_{y}=0.41 and *P*_{z}=0.26; Di Valentin
et al., 2014) at the Zeeman frequencies that correspond to the used
magnetic field strengths (*ν*_{T}=9.28 and *ν*_{T}=9.31 GHz in X-band and *ν*_{T}=34.00 GHz in Q-band; see also Supplement S10). The distance
distributions of TPP–pAA_{5}–NO• and
TPP–pAA_{10}–NO• that were used for the LaserIMD simulations
were obtained by LiDEER measurements. LiDEER traces were recorded in Q-band
with the observer pulse placed on the *Y*^{−} peak and analyzed with
*K*_{DEER}(*t*,*r*) and Tikhonov regularization, as the simulations in
Sect. 4.2 showed that no artifacts are to be expected in this case. More
details on the experiments and distance calculations can be found in Supplement S7 and
S10. The results of the LiDEER measurements are shown in
Fig. 11, and the extracted distance distributions
exhibit a narrow peak at 2.2 nm for TPP–pAA_{5}–NO•
and at 3.5 nm for TPP–pAA_{10}–NO•, as expected
(Bieber et al., 2018; Di Valentin et
al., 2016). As the LaserIMD and LiDEER measurements have different
modulation depths, the modulation depth of LiDEER (*λ*_{LiDEER}) cannot be used for the simulation of the LaserIMD. This
makes the modulation depth of the LaserIMD traces (*λ*_{LaserIMD}) the only parameter that is missing for the
simulations. Therefore, the simulated LaserIMD traces were fitted to the
measured ones by rescaling the modulation depth. As the background decay
rate depends linearly on the modulation depth (Hu
and Hartmann, 1974; Pannier et al., 2000), it must be rescaled together with
the modulation depth. For LaserIMD, we assume that coherence transfer
pathways with Δ*m*_{T}=0 do not contribute to the
background, as the decay of the echo intensity is on a much longer timescale
than the dipolar oscillations that constitute the main contribution of the
intermolecular background. Therefore, we additionally reduce the rescaled
background decay rate by a factor of $\mathrm{2}/\mathrm{3}$:

The simulated LaserIMD trace ${V}_{\mathrm{LaserIMD}}{\left(t\right)}_{{\mathit{\lambda}}_{\mathrm{LaserIMD}}}$ was fitted to the experimental LaserIMD data by
varying the modulation depth *λ*_{LaserIMD} so that the
root-mean-square displacement of the simulated and experimental traces was
minimized. Simulations without the effect of the ZFS were also performed in
order to clearly see the difference between them and the simulations with the ZFS. For the
simulations without the ZFS, the modulation depth of the LaserIMD
simulations with the ZFS was taken because it was determined by the fit and
reduced by a factor of $\mathrm{2}/\mathrm{3}$, as the coherence transfer pathway with
Δ*m*_{T}=0 no longer contributes to the echo
modulation.

The results of the LaserIMD measurements and the corresponding simulations
are shown in Fig. 12. It can be clearly seen that the shape of the experimental traces changes depends on whether they were recorded in X- or Q-band: in X-band, the traces have a stronger decay than in Q-band. This is a first strong indication of the effect of the ZFS, as predicted by
the simulations (see Fig. 5). The influence of
the ZFS shows itself clearly in the differences between the experimental
data and the simulations where the effect of the ZFS was ignored. In
particular, the experimental LaserIMD traces show a stronger decay than the
background decay of simulations without the ZFS. This difference is more
pronounced in TPP–AA_{5}–NO• than in TPP–AA_{10}–NO•
and also stronger in X-band than in Q-band. Thus, for
TPP–AA_{5}–NO• in X-band, the deviation between the simulations
without the ZFS and the experiments is the largest, whereas it is nearly absent in the case of
TPP–AA_{10}–NO• in Q-band. This additional
decay of the experimental traces cannot be explained without considering the
effect of the ZFS, but it is understandable with a model that includes
the ZFS. The stronger decay of the experimental traces can be assigned to
the coherence transfer pathway with Δ*m*_{T}=0,
which leads to an additional contribution to the LaserIMD trace
${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$ with a continuously decaying
signal (see Fig. 4). As shorter distances and
lower magnetic fields lead to a stronger decay of
${V}_{\mathrm{0}}^{\text{non-sec}}\left(t\right)$, this also explains why the
additional decay in the experimental data is stronger for
TPP–AA_{5}–NO• than for TPP–AA_{10}–NO• and stronger
in X-band than in Q-band. It is noteworthy that the model with the ZFS provides
not only a qualitative but also a quantitative agreement between the
experimentally recorded LaserIMD traces and the corresponding simulations.

To see how the additional decay of the ZFS affects the analysis of
experimental LaserIMD traces, the recorded data were analyzed with Tikhonov
regularization; the results that are obtained with a LaserIMD kernel
that includes the ZFS are compared to those obtained by a DEER kernel that
ignores the ZFS (see Supplement S11 for a detailed overview of the results). The
comparison of the obtained distance distributions shows that, even when the
ZFS is ignored, the main distance peak is obtained correctly in all cases.
For the measurements in Q-band, the entire distance distributions turn out
to be virtually identical, regardless of whether the ZFS is included in the
analysis routine or not (see Fig. S13c–d). The situation is different in
X-band. For TPP–AA_{5}–NO• in X-band, the strong additional
decay is interpreted as an additional artifact peak at around
5.0 nm if the ZFS is ignored (see Fig. S13a). This peak
disappears when the ZFS is considered. For TPP–AA_{10}–NO• in X-band, the analysis that ignores the ZFS also shows an additional peak
around 7.0 nm. However, this artifact is not as pronounced as the
one of TPP–AA_{5}–NO• and disappears in the validation.
For the modulation depths and the background decay rates, there are notable
differences when the ZFS is considered or omitted (see Tables S5 and S6 in Supplement S11).
In all cases, ignoring the ZFS leads to a reduced modulation depth. In Q-band, the modulation depth is reduced by a factor of $\approx \mathrm{2}/\mathrm{3}$,
meaning that the additional decay is completely assigned to the intermolecular
background. In accordance with that, the background decay rates are larger
when the ZFS is ignored. In X-band, these effects are not as pronounced. As
the additional decay is partially fitted by introducing distance artifacts
when ignoring the ZFS, the modulation depth is only reduced by a factor of
0.72 for TPP–AA_{10}–NO• and by a factor of 0.84 for
TPP–AA_{5}–NO•.

These results show that ignoring the ZFS for the analysis of LaserIMD leads to artifacts in the obtained results. For TPP as transient spin label, the artifacts are not as prominent in Q-band. There, the additional decay mostly leads to a stronger background decay and reduced modulation depth, and the distance distribution remains virtually unchanged. In X-band, however, artifact peaks in the distance distribution can occur if the ZFS is ignored.

In light-induced PDS, the ZFS interaction of the transient triplet label is
a crucial parameter that can alter the shape of the dipolar traces. This
implies that, in contrast to the former assumption, the spin system in
LaserIMD and LiDEER cannot be treated in the secular approximation where the
spin system behaves as if it would consist of two $S=\mathrm{1}/\mathrm{2}$ spins. A
theoretical description of LaserIMD and LiDEER that also includes
non-secular terms was developed, and it was shown that the dipolar
frequencies depend on the magnitude of the ZFS and the Zeeman frequency
(i.e., the external magnetic field). Time-domain simulations showed that, in
LiDEER, this effect of the ZFS can be suppressed by exciting either of the
*Y*^{±} peaks with the observer pulses and by using transient triplet
labels whose ZFS is small compared with the Zeeman frequency, such as TPP in Q-band. For experimental LiDEER data that are recorded under such
conditions, the effect of the ZFS is negligible and a standard DEER kernel
that does not consider the ZFS can be employed for data analysis.

In LaserIMD, simulations and experiments confirmed that there is an
influence of the ZFS on the dipolar trace. It virtually does not affect the
dipolar oscillation of the coherence transfer pathways with $\mathrm{\Delta}{m}_{\mathrm{T}}=\pm \mathrm{1}$, but it is manifested in an additional decay of
the LaserIMD trace. This decay is caused by the third coherence transfer
pathway with Δ*m*_{T}=0, which was formerly believed
not to contribute to the signal. The strength of this additional decay
primarily depends on the ratio of the ZFS to the Zeeman frequency as well as on
the distance between the transient and permanent spin label: it is stronger
for a larger ZFS, lower magnetic fields and shorter distances. A software tool
for the calculation of LaserIMD kernels that considers the influence of the ZFS
was developed. It is available at GitHub (https://github.com/andreas-scherer/LaserIMD_kernel) and
allows one to specify different ZFS values, zero-field populations and Zeeman
frequencies. The feasibility of the new kernel was proven by experimentally
recorded LaserIMD traces. A DEER kernel that ignores the ZFS cannot fit
these traces correctly, and strong derivations between the experimental data
and simulations can be observed. However, with the newly developed model
that considers the ZFS, excellent fits of the experimental data were
produced. The analysis of the experimental and simulated LaserIMD data with
Tikhonov regularization showed that ignoring the ZFS compromises the
obtained results. For transient triplet labels with a ZFS of ≈1 GHz, like TPP, this is no that problematic in Q-band. There, only
the obtained modulation depths and background decay rates are affected if
the ZFS is ignored; the distance distribution remains unchanged. In X-band,
however, ignoring the ZFS is more severe and can additionally lead to
artifact peaks in the distance distributions. This shows that the ZFS can
have a significant impact in LaserIMD and should be considered when
experimental data are analyzed.

The source code for the LaserIMD kernel can be downloaded from https://doi.org/10.5281/zenodo.7576913 (Scherer et al., 2023a; https://github.com/andreas-scherer/LaserIMD_kernel). The source code for the time-domain LiDEER simulations can be downloaded from https://doi.org/10.5281/zenodo.7580933 (Scherer et al., 2023b; https://github.com/andreas-scherer/LiDEER_simulations.git).

The raw data can be downloaded from https://doi.org/10.5281/zenodo.7283499 (Scherer et al., 2022b).

The supplement related to this article is available online at: https://doi.org/10.5194/mr-4-27-2023-supplement.

AS and MD conceived the research idea and designed the simulations and experiments. AS performed the analytical calculations, and AS and BY conducted the simulations and experiments and analyzed the results. AS prepared all the figures and wrote the draft manuscript. All authors discussed the results and revised the manuscript.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank Joschua Braun and Stefan Volkwein for helpful discussions concerning the numerical integration for the LaserIMD kernel. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation program (grant no.772027; SPICE, ERC-2017-COG). Andreas Scherer gratefully acknowledges financial support from the Konstanz Research School Chemical Biology (KoRS-CB).

This research has been supported by the Horizon 2020 (SPICE – Spectroscopy in cells with tailored in vivo labeling strategies and multiply addressable nano-structural probes; grant no. 772027).

This paper was edited by Stefan Stoll and reviewed by Gunnar Jeschke and one anonymous referee.

Abdullin, D., Matsuoka, H., Yulikov, M., Fleck, N., Klein, C., Spicher, S., Hagelueken, G., Grimme, S., Luetzen, A., and Schiemann, O.: Pulsed EPR Dipolar Spectroscopy under the Breakdown of the High-Field Approximation: The High-Spin Iron(III) Case, Chem. Eur. J., 2, 8820–8828, https://doi.org/10.1002/chem.201900977, 2019.

Bak, M. and Nielsen, N. C.: REPULSION, A Novel Approach to Efficient Powder Averaging in Solid-State NMR, J. Magn. Reson., 125, 132–139, https://doi.org/10.1006/jmre.1996.1087, 1997.

Bertran, A., Henbest, K. B., De Zotti, M., Gobbo, M., Timmel, C. R., Di Valentin, M., and Bowen, A. M.: Light-Induced Triplet–Triplet Electron Resonance Spectroscopy, J. Phys. Chem. Lett., 12, 80–85, https://doi.org/10.1021/acs.jpclett.0c02884, 2020.

Bertran, A., Barbon, A., Bowen, A. M., and Di Valentin, M.: Chapter Seven – Light-induced pulsed dipolar EPR spectroscopy for distance and orientation analysis, in: Methods in Enzymology, Vol. 666, edited by: Britt, R. D., Academic Press, 171–231, https://doi.org/10.1016/bs.mie.2022.02.012, 2022a.

Bertran, A., Morbiato, L., Aquilia, S., Gabbatore, L., De Zotti, M., Timmel, C. R., Di Valentin, M., and Bowen, A. M.: Erythrosin B as a New Photoswitchable Spin Label for Light-Induced Pulsed EPR Dipolar Spectroscopy, Molecules, 27, 7526, https://doi.org/10.3390/molecules27217526, 2022b.

Bieber, A., Bücker, D., and Drescher, M.: Light-induced dipolar spectroscopy – A quantitative comparison between LiDEER and LaserIMD, J. Magn. Reson., 296, 29–35, https://doi.org/10.1016/j.jmr.2018.08.006, 2018.

Bowen, A. M., Bertran, A., Henbest, K. B., Gobbo, M., Timmel, C. R., and Di Valentin, M.: Orientation-Selective and Frequency-Correlated Light-Induced Pulsed Dipolar Spectroscopy, J. Phys. Chem. Lett., 12, 3819–3826, https://doi.org/10.1021/acs.jpclett.1c00595, 2021.

Bowman, M. K., Maryasov, A. G., Kim, N., and DeRose, V. J.: Visualization of distance distribution from pulsed double electron-electron resonance data, Appl. Magn. Reson., 26, 22, https://doi.org/10.1007/BF03166560, 2004.

Bücker, D., Sickinger, A., Ruiz Perez, J. D., Oestringer, M., Mecking, S., and Drescher, M.: Direct Observation of Chain Lengths and Conformations in Oligofluorene Distributions from Controlled Polymerization by Double Electron–Electron Resonance, J. Am. Chem. Soc., 124, 1952–1956, https://doi.org/10.1021/jacs.9b11404, 2019.

Collauto, A., von Bülow, S., Gophane, D. B., Saha, S., Stelzl, L. S., Hummer, G., Sigurdsson, S. T., and Prisner, T. F.: Compaction of RNA Duplexes in the Cell, Angew. Chem. Int. Ed., 59, 23025–23029, https://doi.org/10.1002/anie.202009800, 2020.

Dal Farra, M. G., Richert, S., Martin, C., Larminie, C., Gobbo, M., Bergantino, E., Timmel, C. R., Bowen, A. M., and Di Valentin, M.: Light-induced pulsed EPR dipolar spectroscopy on a paradigmatic Hemeprotein, Chem. Phys. Chem., 20, 931–935, https://doi.org/10.1002/cphc.201900139, 2019a.

Dal Farra, M. G., Ciuti, S., Gobbo, M., Carbonera, D., and Di Valentin, M.: Triplet-state spin labels for highly sensitive pulsed dipolar spectroscopy, Mol. Phys., 117, 2673–2687, https://doi.org/10.1080/00268976.2018.1503749, 2019b.

Dalaloyan, A., Qi, M., Ruthstein, S., Vega, S., Godt, A., Feintuch, A., and Goldfarb, D.: Gd(iii)-Gd(iii) EPR distance measurements – the range of accessible distances and the impact of zero field splitting, Phys. Chem. Chem. Phys., 17, 18464–18476, https://doi.org/10.1039/C5CP02602D, 2015.

Di Valentin, M., Albertini, M., Zurlo, E., Gobbo, M., and Carbonera, D.: Porphyrin Triplet State as a Potential Spin Label for Nanometer Distance Measurements by PELDOR Spectroscopy, J. Am. Chem. Soc., 136, 6582–6585, https://doi.org/10.1021/ja502615n, 2014.

Di Valentin, M., Albertini, M., Dal Farra, M. G., Zurlo, E., Orian, L., Polimeno, A., Gobbo, M., and Carbonera, D.: Light-Induced Porphyrin-Based Spectroscopic Ruler for Nanometer Distance Measurements, Chem. Eur. J., 22, 17204–17214, https://doi.org/10.1002/chem.201603666, 2016.

Edwards, T. H. and Stoll, S.: Optimal Tikhonov regularization for DEER spectroscopy, J. Magn. Reson., 288, 58–68, https://doi.org/10.1016/j.jmr.2018.01.021, 2018.

Fábregas Ibáñez, L., Jeschke, G., and Stoll, S.: DeerLab: a comprehensive software package for analyzing dipolar electron paramagnetic resonance spectroscopy data, Magn. Reson., 1, 209–224, https://doi.org/10.5194/mr-1-209-2020, 2020.

García-Rubio, I.: EPR of site-directed spin-labeled proteins: A powerful tool to study structural flexibility, Arch. Biochem. Biophys., 684, 108323, https://doi.org/10.1016/j.abb.2020.108323, 2020.

Giannoulis, A., Feintuch, A., Barak, Y., Mazal, H., Albeck, S., Unger, T., Yang, F., Su, X.-C., and Goldfarb, D.: Two closed ATP- and ADP-dependent conformations in yeast Hsp90 chaperone detected by Mn(II) EPR spectroscopic techniques, P. Natl. Acad. Sci. USA, 117, 395–404, https://doi.org/10.1073/pnas.1916030116, 2020.

Hagston, W. E. and Holmes, B. J.: Matrix methods for spin Hamiltonians of low symmetry, J. Phys. B, 13, 3505–3519, https://doi.org/10.1088/0022-3700/13/18/005, 1980.

Hintze, C., Schutze, F., Drescher, M., and Mecking, S.: Probing of chain conformations in conjugated polymer nanoparticles by electron spin resonance spectroscopy, Phys. Chem. Chem. Phys., 17, 32289–32296, https://doi.org/10.1039/C5CP05749C, 2015.

Hintze, C., Bücker, D., Domingo Köhler, S., Jeschke, G., and Drescher, M.: Laser-Induced Magnetic Dipole Spectroscopy, J. Phys. Chem. Lett., 7, 2204–2209, https://doi.org/10.1021/acs.jpclett.6b00765, 2016.

Hogben, H. J., Krzystyniak, M., Charnock, G. T. P., Hore, P. J., and Kuprov, I.: Spinach – A software library for simulation of spin dynamics in large spin systems, J. Magn. Reson., 208, 179–194, https://doi.org/10.1016/j.jmr.2010.11.008, 2011.

Hu, P. and Hartmann, S. R.: Theory of spectral diffusion decay using an uncorrelated-sudden-jump model, Phys. Rev. B, 9, 1–13, https://doi.org/10.1103/PhysRevB.9.1, 1974.

Hubbell, W. L., López, C. J., Altenbach, C., and Yang, Z.: Technological advances in site-directed spin labeling of proteins, Curr. Opin. Struct. Biol., 23, 725–733, https://doi.org/10.1016/j.sbi.2013.06.008, 2013.

Jeschke, G.: DEER Distance Measurements on Proteins, Annu. Rev. Phys. Chem., 63, 419–446, https://doi.org/10.1146/annurev-physchem-032511-143716, 2012.

Jeschke, G., Panek, G., Godt, A., Bender, A., and Paulsen, H.: Data analysis procedures for pulse ELDOR measurements of broad distance distributions, Appl. Magn. Reson., 26, 223, https://doi.org/10.1007/BF03166574, 2004.

Jeschke, G., Chechik, V., Ionita, P., Godt, A., Zimmermann, H., Banham, J., Timmel, C. R., Hilger, D., and Jung, H.: DeerAnalysis2006 – a comprehensive software package for analyzing pulsed ELDOR data, Appl. Magn. Reson., 30, 473–498, https://doi.org/10.1007/BF03166213, 2006.

Jeschke, G., Sajid, M., Schulte, M., Ramezanian, N., Volkov, A., Zimmermann, H., and Godt, A.: Flexibility of Shape-Persistent Molecular Building Blocks Composed of p-Phenylene and Ethynylene Units, J. Am. Chem. Soc., 132, 10107–10117, https://doi.org/10.1021/ja102983b, 2010.

Keeley, J., Choudhury, T., Galazzo, L., Bordignon, E., Feintuch, A., Goldfarb, D., Russell, H., Taylor, M. J., Lovett, J. E., Eggeling, A., Fabregas Ibanez, L., Keller, K., Yulikov, M., Jeschke, G., and Kuprov, I.: Neural networks in pulsed dipolar spectroscopy: a practical guide, J. Magn. Reson., 338, 107186, https://doi.org/10.1016/j.jmr.2022.107186, 2022.

Krumkacheva, O. A., Timofeev, I. O., Politanskaya, L. V., Polienko, Y. F., Tretyakov, E. V., Rogozhnikova, O. Yu., Trukhin, D. V., Tormyshev, V. M., Chubarov, A. S., Bagryanskaya, E. G., and Fedin, M. V.: Triplet Fullerenes as Prospective Spin Labels for Nanoscale Distance Measurements by Pulsed Dipolar EPR, Angew. Chem. Int. Ed., 585, 13271–13275, https://doi.org/10.1002/ange.201904152, 2019.

Manukovsky, N., Feintuch, A., Kuprov, I., and Goldfarb, D.: Time domain simulation of Gd3+–Gd3+ distance measurements by EPR, J. Chem. Phys., 147, 044201, https://doi.org/10.1063/1.4994084, 2017.

Marko, A., Denysenkov, V., Margraf, D., Cekan, P., Schiemann, O., Sigurdsson, S. T., and Prisner, T. F.: Conformational Flexibility of DNA, J. Am. Chem. Soc., 133, 13375–13379, https://doi.org/10.1021/ja201244u, 2011.

Marko, A., Denysenkov, V., and Prisner, T. F.: Out-of-phase PELDOR, Mol. Phys., 111, 2834–2844, https://doi.org/10.1080/00268976.2013.807369, 2013.

Maryasov, A. G., Bowman, M. K., and Tsvetkov, Yu, D.: Dipole-dipole interactions of high-spin paramagnetic centers in disordered systems, Appl. Magn. Reson., 30, 683, https://doi.org/10.1007/BF03166227, 2006.

Milov, A. D., Salikhov, K. M., and Shirov, M. D.: Use of the double resonance in electron spin echo method for the study of paramagnetic center spatial distribution in solids, Fiz. Tverd. Tela, 23, 975–982, 1981.

Milov, A. D., Ponomarev, A. B., and Tsvetkov, Yu, D.: Electron-electron double resonance in electron spin echo: Model biradical systems and the sensitized photolysis of decalin, Chem. Phys. Lett., 110, 67–72, https://doi.org/10.1016/0009-2614(84)80148-7, 1984.

Milov, A. D., Maryasov, A. G., and Tsvetkov, Y. D.: Pulsed electron double resonance (PELDOR) and its applications in free-radicals research, Appl. Magn. Reson., 15, 107–143, https://doi.org/10.1007/BF03161886, 1998.

Pake, G. E.: Nuclear Resonance Absorption in Hydrated Crystals: Fine Structure of the Proton Line, J. Chem. Phys., 16, 327–336, https://doi.org/10.1063/1.1746878, 1948.

Pannier, M., Veit, S., Godt, A., Jeschke, G., and Spiess, H. W.: Dead-Time Free Measurement of Dipole–Dipole Interactions between Electron Spins, J. Magn. Reson., 142, 331–340, https://doi.org/10.1006/jmre.1999.1944, 2000.

Ritsch, I., Lehmann, E., Emmanouilidis, L., Yulikov, M., Allain, F., and Jeschke, G.: Phase Separation of Heterogeneous Nuclear Ribonucleoprotein A1 upon Specific RNA-Binding Observed by Magnetic Resonance, Angew. Chem. Int. Ed., 61, e202204311, https://doi.org/10.1002/anie.202204311, 2022.

Robotta, M., Gerding, H. R., Vogel, A., Hauser, K., Schildknecht, S., Karreman, C., Leist, M., Subramaniam, V., and Drescher, M.: Alpha-Synuclein Binds to the Inner Membrane of Mitochondria in an á-Helical Conformation, Chem. Bio. Chem., 15, 2499–2502, https://doi.org/10.1002/cbic.201402281, 2014.

Rose, M. E.: Elementary theory of angular momentum, 3rd Edn., John Wiley & Sons, Courier Corporation, ISBN: 978-0-471-73524-3, 1995.

Roser, P., Schmidt, M. J., Drescher, M., and Summerer, D.: Site-directed spin labeling of proteins for distance measurements in vitro and in cells, Org. Biomol. Chem., 14, 5468–5476, https://doi.org/10.1039/C6OB00473C, 2016.

Scherer, A., Yao, X., Qi, M., Wiedmaier, M., Godt, A., and Drescher, M.: Increasing the Modulation Depth of Gd(III)-Based Pulsed Dipolar EPR Spectroscopy (PDS) with Porphyrin-Gd(III) Laser-Induced Magnetic Dipole Spectroscopy, J. Phys. Chem. Lett., 13, 10958–10964, https://doi.org/10.1021/acs.jpclett.2c02138, 2022a.

Scherer, A., Yildirim, B., and Drescher, M.: Raw data for “The effect of the zero-field splitting in light-induced pulsed dipolar electron paramagnetic resonance (EPR) spectroscopy”, Zenodo [data set], https://doi.org/10.5281/zenodo.7283499, 2022b.

Scherer, A., Yildirim, B., and Drescher, M.: LaserIMD kernel for “The effect of the zero-field splitting in light-induced pulsed dipolar electron paramagnetic resonance (EPR) spectroscopy”, Zenodo [code], https://doi.org/10.5281/zenodo.7576913, 2023a.

Scherer, A., Yildirim, B., and Drescher, M.: LiDEER simulations for “The effect of the zero-field splitting in light-induced pulsed dipolar electron paramagnetic resonance (EPR) spectroscopy”, Zenodo [code], https://doi.org/10.5281/zenodo.7580933, 2023b.

Schweiger, A. and Jeschke, G.: Principles of Pulse Electron Paramagnetic Resonance, Oxford University Press on Demand, ISBN: 0-19-850634-1, 2001.

Serrer, K., Matt, C., Sokolov, M., Kacprzak, S., Schleicher, E., and Weber, S.: Application of commercially available fluorophores as triplet spin probes in EPR spectroscopy, Mol. Phys., 117, 2688–2699, https://doi.org/10.1080/00268976.2019.1608379, 2019.

Tait, C. E. and Stoll, S.: Coherent pump pulses in Double Electron Electron Resonance spectroscopy, Phys. Chem. Chem. Phys., 18, 18470–18485, https://doi.org/10.1039/C6CP03555H, 2016.

Takeda, K., Takegoshi, K., and Terao, T.: Dynamic nuclear polarization by electron spins in the photoexcited triplet state: I. Attainment of proton polarization of 0.7 at 105 K in naphthalene, J. Phys. Soc. Jpn., 73, 2313–2318, 2004.

Telser, J.: EPR Interactions – Zero-Field Splittings, eMagRes, 6, 207–234, https://doi.org/10.1002/9780470034590.emrstm1501, 2017.

Timofeev, I. O., Politanskaya, L. V., Tretyakov, E. V., Polienko, Y. F., Tormyshev, V. M., Bagryanskaya, E., Krumkacheva, O. A., and Fedin, M. V.: Fullerene-based triplet spin labels: methodology aspects for pulsed dipolar EPR spectroscopy, Phys. Chem. Chem. Phys., 24, 4475–4484, https://doi.org/10.1039/D1CP05545C, 2022.

Wasielewski, M. R., O'Neil, M. P., Lykke, K. R., Pellin, M. J., and Gruen,
D. M.: Triplet states of fullerenes *C*_{60} and *C*_{70}, Electron paramagnetic
resonance spectra, photophysics, and electronic structures, J.
Am. Chem. Soc., 113, 2774–2776, 1991.

Weickert, S., Wawrzyniuk, M., John, L. H., Rüdiger, S. G. D., and Drescher, M.: The mechanism of Hsp90-induced oligomerizaton of Tau, Sci. Adv., 6, eaax6999, https://doi.org/10.1126/sciadv.aax6999, 2020.

Williams, L., Tischlik, S., Scherer, A., Fischer, J. W. A., and Drescher, M.: Site-directed attachment of photoexcitable spin labels for light-induced pulsed dipolar spectroscopy, Chem. Commun., 56, 14669–14672, https://doi.org/10.1039/D0CC03101A, 2020.

Wojciechowski, F., Groß, A., Holder, I. T., Knörr, L., Drescher, M., and Hartig, J. S.: Pulsed EPR spectroscopy distance measurements of DNA internally labelled with Gd3+-DOTA, Chem. Commun., 51, 13850–13853, https://doi.org/10.1039/C5CC04234H, 2015.

Worswick, S. G., Spencer, J. A., Jeschke, G., and Kuprov, I.: Deep neural network processing of DEER data, Sci. Adv., 4, eaat5218, https://doi.org/10.1126/sciadv.aat5218, 2018.

Yang, Y., Chen, S.-N., Yang, F., Li, X.-Y., Feintuch, A., Su, X.-C., and Goldfarb, D.: In-cell destabilization of a homodimeric protein complex detected by DEER spectroscopy, P. Natl. Acad. Sci. USA, 117, 20566–20575, https://doi.org/10.1073/pnas.2005779117, 2020.

Yee, E. F., Diensthuber, R. P., Vaidya, A. T., Borbat, P. P., Engelhard, C., Freed, J. H., Bittl, R., Möglich, A., and Crane, B. R.: Signal transduction in light–oxygen–voltage receptors lacking the adduct-forming cysteine residue, Nat. Commun., 6, 10079, https://doi.org/10.1038/ncomms10079, 2015.