# HYSCORE on Photoexcited Triplet States

## Abstract

HYperfine Sublevel CORrElation spectroscopy (HYSCORE) is one of the most widely applied EPR techniques for the investigation of hyperfine couplings in a wide range of spin systems; here, it is applied for the first time to a photoexcited triplet state. The analytical expressions describing the electron spin echo envelope modulations in a HYSCORE experiment on a triplet state coupled to spin ½ nuclei and spin 1 nuclei with weak nuclear quadrupole interactions were derived employing the density matrix formalism and used to determine the characteristic features of this experiment when applied to a triplet state. Experimental HYSCORE spectra recorded on the photoexcited triplet state of a *meso*-substituted zinc porphyrin are used to investigate the ^{14}N hyperfine interactions in this system. The results are compared to one-dimensional three-pulse Electron Spin Echo Envelope Modulation (ESEEM) experiments and the comparison clearly shows the advantage of the HYSCORE experiment for investigation of the hyperfine couplings of a spin 1 nucleus for triplet states in disordered powder samples. The experimental HYSCORE data are discussed and interpreted in light of the properties of triplet state HYSCORE spectra identified using our theoretical approach.

## 1 Introduction

Photoexcited triplet states are of interest in a range of research fields, including natural and artificial photosynthesis [1, 2, 3, 4, 5, 6, 7, 8, 9], development of electrophosphorescent organic light-emitting devices [10], photodynamic therapy [11, 12] and, as shown recently, also for protein structure resolution using Double Electron–Electron Resonance (DEER) [13]. The characterisation of their electronic structure is of considerable importance for all this wide range of applications.

Information on the hyperfine couplings of nuclei to paramagnetic systems can be obtained using Electron Nuclear DOuble Resonance (ENDOR) and one- and two-dimensional Electron Spin Echo Envelope Modulation (ESEEM) experiments [14]. Photoexcited ENDOR and ESEEM have been extensively applied to the study of a series of triplet states [15, 16, 17, 18]; in particular of chlorophylls and carotenoid molecules in proteins involved in photosynthesis, where the results have shed light on energy transfer processes in natural photosynthesis [5, 19, 20, 21, 22, 23, 24, 25]. For organic radicals and transition metal centres, the two-dimensional ESEEM experiment HYSCORE has provided important advantages with respect to one-dimensional two- and three-pulse ESEEM [14, 26, 27, 28]. However, to the best of our knowledge, no published account of its application to photoexcited organic triplet state molecules exists.

A theoretical description of ESEEM experiments based on the derivation of analytical expressions describing the electron spin echo envelope modulation function has proven fundamental in the past for the interpretation of experimental data and has provided insight into the main features and characteristics of these experiments [29, 30, 31]. Several approaches to a numerical calculation of ESEEM and HYSCORE spectra have been developed and are usually employed for the simulation of experimental data and extraction of the hyperfine and nuclear quadrupole coupling parameters [32, 33, 34, 35]. However, analytical expressions, although often only applicable for special cases where simplifying assumptions are possible, can often provide further insight and understanding of the experimental results and help to optimise the experimental conditions for data acquisition. The explicit analytical expressions for the echo modulation in a number of systems and for different pulse sequences have been derived following Mims’ density matrix formalism [30]. Specifically, analytical expressions for two-pulse and three-pulse ESEEM and HYSCORE have been derived for *S* = ½, *I* = ½ and *I* = 1 systems [30, 36, 37]. As far as *S* > ½ spin systems are concerned, expressions for the echo modulation in the two-pulse ESEEM experiment have been derived for *S* = 1, *I* = ½ systems [15] and the *S* = Open image in new window case (Mn^{2+} complexes) has been analysed in some detail [38, 39, 40]. Sloop et al. [15, 16] have derived expressions for the echo modulation in two-pulse ESEEM experiments on *S* = 1, *I* = ½ systems and have found that the triplet state can be approximated to a fictitious *S* = ½ system assuming transition selectivity.

In this paper, we describe the application of HYSCORE to photoexcited triplet states from a theoretical and an experimental point of view, by extending the derivation of analytical expressions to *S* = 1, *I* = 1 systems with weak nuclear quadrupole interactions. The most important features of the HYSCORE experiment as applied to triplet states are discussed based both on the derived expressions and on experimental HYSCORE spectra recorded for the photoexcited triplet state of a *meso*-substituted porphyrin. We also discuss the advantages of using HYSCORE rather than two- or three-pulse ESEEM for the determination of hyperfine couplings of spin 1 nuclei, such as ^{14}N, in triplet state systems.

## 2 Experimental

[5,15-Bis(3,5(trihexylsilyl)phenyl)-10,20-bis ((tri-hexylsilyl)ethynyl)porphyrinato] zinc(II) (see Fig. 3a for the structure) was synthesised according to the published procedure [41]. Samples for EPR measurements were prepared in MeTHF:pyridine 10:1 and degassed using several freeze-pump thaw cycles.

X-band three-pulse ESEEM and HYSCORE experiments were performed on an X-band Bruker Elexsys 680 spectrometer equipped with a helium gas-flow cryostat from Oxford instruments. Q-band HYSCORE measurements were performed on a Bruker Elexsys 580 Q/S-band spectrometer with a 3 W TWT amplifier. Photoexcitation with a 10 Hz Nd:YAG laser at 532 nm with a power of 15 mJ was used to generate the photoexcited triplet state. The experiments were performed at 20 K.

*t*

_{π/2}= 16 ns,

*t*

_{π}= 16 ns and

*τ*= 120, 160 and 200 ns, and starting times

*t*

_{1}=

*t*

_{2}= 64 ns with increments

*dt*= 16 ns (data matrix 256 × 256).

^{14}N matched HYSCORE experiments were performed with matched pulses instead of the second and last pulse of the standard HYSCORE sequence. The matched pulses were applied at high microwave power and the pulse lengths were optimised for maximum modulation depth in three-pulse ESEEM experiments with varying pulse lengths (optimum length = 24 ns).

Q-band HYSCORE was performed with pulse lengths of *t*_{π/2} = 30 ns, *t*_{π} = 30 ns and *τ* = 100 and 212 ns, and starting times *t*_{1} = *t*_{2} = 64 ns with increments *dt* = 16 ns (data matrix 256 × 256). Q-band matched HYSCORE was performed at the *Z*^{−} field position with 74 ns matched pulses and 8 ns increments; *τ* values of 100 and 160 ns were used. A 4-step phase cycle was used to remove unwanted echoes. The experimental data were processed in Matlab; the time traces were baseline corrected by a third-order polynomial, apodised with a Hamming window and zero-filled to 1024 points in each dimension. A 2D Fourier transform was applied and the absolute-value spectrum was calculated.

X-band three-pulse ESEEM experiments were performed with the pulse sequence shown in Fig. 1a with mw pulses of lengths of *t*_{π/2} = 16 ns and an initial *T* value of 64 ns with increments *dt* = 8 ns (1024 data points were collected). A four-step phase cycle was used to remove unwanted echoes. The ESEEM traces were recorded for three different *τ* values (100, 160 and 200 ns) and the FT spectra were summed to avoid distortions due to blind spots. The experimental data were processed with a home-written Matlab program, the time traces were baseline corrected with a stretched exponential, apodised with a Hamming window and zero-filled to 2048 data points. The frequency spectrum was calculated using the cross-term averaging procedure implemented in Easyspin [42, 43].

The analytical expressions describing the time domain of the HYSCORE experiments were derived in Mathematica. The calculated HYSCORE time domain data were processed as described above to yield the HYSCORE spectra in the frequency domain.

## 3 Theory

The complete analytical expressions describing the four-pulse HYSCORE experiment for an *S* = 1 system coupled to *I* = ½ and *I* = 1 nuclei are derived here based on the density matrix formalism, first introduced for the analysis of ESEEM by Rowan, Hahn and Mims [29] and later generalised and described in detail by Mims [30].

*I*expressed in angular frequency units is:

*g*-value in the high-field approximation, the zero-field splitting (ZFS) interaction, the nuclear Zeeman interaction, the hyperfine interaction between the electron spin and a nuclear spin

*I*and the nuclear quadrupole interaction, present for a nucleus with

*I*> ½. Schematic energy level diagrams for a triplet state coupled to an

*I*= ½ or

*I*= 1 nucleus are shown in Fig. 1c.

A series of simplifying assumptions was necessary to derive the analytical expressions. First of all, the derivations were based on the high-field approximation for the electron spin and ideal pulses were considered throughout. Further, since the spacing of the triplet state sublevels due to zero-field splitting is generally such that the microwave frequency cannot simultaneously excite transitions that share a common sublevel, transition selective microwave pulses were considered. In this case, the triplet system can be well approximated to a fictitious spin ½ system under the conditions that *gβ*_{e}*B*_{1} ≪ *D*, where *B*_{1} is the amplitude of the microwave field and *D* is the zero-field splitting parameter [15, 38].

The modulation formula for the HYSCORE experiment, describing the modulation of the echo amplitude with time in two dimensions, was previously derived for the *S* = ½ *I* = ½ system by Gemperle et al. [36] and later revised by Tyryshkin et al. [44]. It is usually written as a sum of two contributions to emphasise the different evolution pathways of the nuclear spins: \(E_{\bmod }^{\alpha \beta }\) (\(E_{\bmod }^{\beta \alpha }\)) describes the nuclear spins precessing with the frequencies of the *α* (*β*) manifold during *t*_{1} and of the *β* (*α*) manifold during *t*_{2}. Here, we extend their results by calculating the equivalent expressions for the *S* = 1, *I* = ½ and *S* = 1, *I* = 1 systems.

### 3.1 Systems with *S* = 1 and *I* = ½

*m*

_{S}= 0 to

*m*

_{s}= +1 transition in an

*S*= 1,

*I*= ½ system is:

*t*

_{1}and

*t*

_{2}in the expression for \(E_{\bmod }^{\alpha \beta }\). The \(\chi\) coefficients are defined in the "Appendix". An analogous expression is obtained for the second triplet state transition between

*m*

_{S}= 0 and

*m*

_{S}= −1. The results differ from those obtained for the corresponding

*S*= ½ system by the definition of the nuclear frequencies and the modulation depth parameters. The nuclear frequencies

*ω*

_{12}(

*m*

_{S}= +1),

*ω*

_{34}(

*m*

_{S}= 0) and

*ω*

_{56}(

*m*

_{S}= −1) (in angular frequency units) are obtained by diagonalisation of the nuclear spin sub-Hamiltonian [14, 30]:

*A*and

*B*are related to the isotropic hyperfine constant

*a*

_{iso}and to the dipolar hyperfine coupling \(A_{\text{dip}} = [ - T,\; - T,\;2T]\) as

\(\chi_{0}\) is an unmodulated term;

\(\chi_{d}\) gives rise to peaks along the diagonal (

*ω*_{ij},*ω*_{ij}) for each nuclear frequency if the mixing pulse deviates from an ideal*π*pulse (see below);terms of the type \(\chi_{ij} \cos \left( {\omega_{ij} t_{1/2} + \frac{{\omega_{ij} \tau }}{2}} \right)\) give rise to axial peaks (

*ω*_{ij}, 0)/(0,*ω*_{ij}), which are, however, usually eliminated by the baseline correction [14];terms of the type \(\chi_{ij,lm}^{ + / - } \cos \left( {\omega_{ij} t_{1} \pm \omega_{lm} t_{2} + \frac{{(\omega_{ij} \pm \omega_{lm} )\tau }}{2}} \right)\) give rise to cross-peaks (±

*ω*_{ij},*ω*_{lm}) in the (+, +) or (+, −) quadrant of the HYSCORE spectrum.

The positions of the cross-peaks allow determination of the nuclear frequencies and, hence, of the hyperfine couplings between electron and nuclear spin. As can be seen from Eq. 2, for an *S* = 1, *I* = ½ system only two cross-peaks are expected in each quadrant.

*k*. The modulation depth parameters \(k_{ + 1/ - 1}\) for the two triplet state transitions are defined as:

The differences in the modulation depth parameters are due to the dependence on the nuclear frequencies of the manifolds connected by the transition. In analogy to *S* = ½ systems, modulations are only observed in the presence of an anisotropic hyperfine interaction.

In general, cross-peaks can appear in any of the quadrants of the HYSCORE spectrum and are usually observed in the (+, +) quadrant for weak hyperfine couplings and in the (−, +) quadrant for strong hyperfine couplings (see Fig. 1d). The terms in the modulation formula accounting for peaks in the two quadrants are weighted by terms of the form \(\sqrt {1 - k_{ + 1/ - 1} } \pm 1\) (see expressions for \(\chi_{ij,lm}^{ + / - }\) in the "Appendix"). Due to the different definitions of the modulation depth parameter \(k_{ + 1/ - 1}\) for the two triplet transitions, the relative intensities of peaks in the two quadrants will be different. The modulation depth parameter only depends on the hyperfine coupling and the nuclear frequency of the *m*_{S} = +1 (*ω*_{12}) or the *m*_{S} = −1 (*ω*_{56}) manifold (see Eq. 6). Since the modulation depth parameter will be larger for the transition involving the smaller of the two nuclear frequencies, peaks in the (−, +) quadrant will be stronger for the corresponding transition with respect to the transition involving the other two manifolds. For example, for a positive hyperfine coupling, the nuclear frequency of the *m*_{S} = +1 manifold (*ω*_{12}) is smaller than the nuclear frequency for the *m*_{S} = −1 manifold (*ω*_{56}), and hence \(k_{ + 1} > k_{ - 1}\). The HYSCORE spectrum for the *m*_{S} = 0 to *m*_{S} = +1 transition will thus have more intense peaks in the (−, +) quadrant with respect to the spectrum for the *m*_{S} = −1 to *m*_{S} = 0 transition. The shift of the signals from the (+, +) quadrant to the (−, +) quadrant with increasing hyperfine interaction will also occur at different points.

### 3.2 Systems with *S* = 1 and *I* = 1

*I*= 1, comprising nuclear Zeeman, hyperfine and nuclear quadrupole interaction, is of considerable difficulty. If the nuclear quadrupole interaction can be assumed small with respect to the nuclear Zeeman and hyperfine interaction, as is generally legitimate for

*I*= 1 nuclei except

^{14}N [45], a simplified treatment can be used, in which the effect of the nuclear quadrupole interaction is taken into account in the nuclear frequencies, but neglected in the derivation of the modulation expression. If such an approach is taken, then the nuclear frequencies may be determined by diagonalisation of the nuclear sub-Hamiltonians without the nuclear quadrupole interaction and its consequent incorporation through first-order perturbation theory, yielding the following results:

*ω*

_{+1},

*ω*

_{0}and

*ω*

_{−1}are defined in Eq. 4.

*m*

_{S}= 0 and

*m*

_{S}= +1 manifold in an

*S*= 1,

*I*= 1 system derived in this manner yields the following result:

*t*

_{1}and

*t*

_{2}in the expression for \(E_{\bmod }^{\alpha \beta }\). The coefficients \(\chi\) are defined in the "Appendix".

A comparison of Eq. 9 (*I* = 1) with Eq. 2 (*I* = ½) shows that they contain the same type of terms leading to the same type of HYSCORE peaks as discussed for the simpler *S* = 1, *I* = ½ case. However, since there are now two single-quantum and one double-quantum nuclear frequencies for each electron spin manifold, the number of cross-peaks increases from 2 to 18 in each quadrant.

The cross-peak intensities for single-crystal ^{2}H HYSCORE spectra for an *S* = ½ system were discussed from a theoretical and experimental point of view in Ref. [37]. The authors showed that 16 of the cross-peak terms arise from Δ*m*_{I} = ±1 nuclear transitions and 20 cross-peaks from combinations of nuclear frequencies involving the Δ*m*_{I} = ±2 nuclear transitions, but that the intensity of most of them is too low to be detected experimentally. A comparison of the modulation factors \(\chi_{ij,lm}\) of different cross-peaks (given in the "Appendix") allowed the identification of only two (\(\chi_{12,45}^{ + }\) and \(\chi_{23,56}^{ + }\)), which depend linearly on the modulation depth parameter *k*, as observed also in the *S* = ½ case [37]. Hence, in analogy to the *S* = ½ case, only four of the eight theoretically possible cross-peaks arising in the first quadrant from Δ*m*_{I} = ±1 nuclear transitions relative to a single EPR transition are experimentally observable, namely the cross-peaks at (*ω*_{12}, *ω*_{45}), (*ω*_{45}, *ω*_{12}), (*ω*_{23}, *ω*_{56}) and (*ω*_{56}, *ω*_{23}). The intensities of the double-quantum cross-peaks (*ω*_{13}, *ω*_{46}) and (*ω*_{46}, *ω*_{13}) depend on the square of the modulation depth parameter and are thus predicted to be weaker.

The terms defining the intensities for signals in the (+, +) and (−, +) quadrants are more complicated in this case; however, they follow the same trends explained for the simpler *I* = ½ systems, i.e., the peaks in the (−, +) quadrant are stronger for the triplet transition with the smaller nuclear frequency and the shift of the signals from the (+, +) to the (−, +) quadrant will occur at different hyperfine couplings, depending on the relative magnitude of the nuclear frequencies of the *m*_{S} = +1 or *m*_{S} = −1 manifold compared to the nuclear Larmor frequency.

*S*> ½ centre, complications can arise due to the different nominal pulse angle with respect to the simpler

*S*= ½ case, since the nominal pulse angle depends on the

*m*

_{S}quantum numbers of the manifolds of the selected EPR transition [40]. In the case of a triplet state system both allowed EPR transitions require the same nominal pulse angle; the effective rotation for an

*S*= 1 system corresponds to \(\sqrt 2\) times the rotation for an

*S*= ½ system [15]. In case of incomplete inversion by the mixing pulse, diagonal peaks appear and the intensity of the cross-peaks is reduced, as can be seen from the multiplicative factors

*γ*

_{+}and

*γ*

_{−}in Eqs. 2 and 9.

*ω*

_{N}stands for the strength of the B

_{1}field and

*t*

_{pIII}is the length of the third (mixing) pulse. If complete inversion is assumed, the terms in the echo modulation expression giving rise to diagonal peaks are absent. Experimentally, the length of this pulse can be optimised for complete inversion in order to reduce diagonal peaks.

The derivation of an analytical expression for HYSCORE of a nuclear spin *I* = 1 with strong nuclear quadrupole interaction using this method was not possible. Analytical treatments of nuclear spin *I* = 1 systems are usually only possible under severely simplifying assumptions or for special cases [40, 47, 48]. Different alternatives used to describe the powder patterns generated in HYSCORE spectra based on the Cayley–Hamilton theorem or on catastrophe theory have also been proposed [49, 50]. Nevertheless, it is often necessary to resort to either numerical simulations, such as implemented in GAMMA [32, 34] or Matlab [35, 43], or consideration of just the frequency positions of the cross-peaks, as the relative intensities are often difficult to reproduce [26].

## 4 Results and Discussion

Pulse EPR of triplet state systems is usually strongly orientation-selective, as the microwave pulses only excite a small fraction of the EPR spectrum typically broadened significantly by the zero-field splitting interaction. This leads to the contribution of only a limited number of orientations to the consequently often single-crystal-like ENDOR, ESEEM and HYSCORE spectra. In the most general case of \(0 < E < D/3\), this is strictly true for the outermost (*Z*) transitions, while more orientations contribute to the *X* and *Y* transitions. In the special case of \(E = D/3\), the *Z* and *Y* transitions coincide and orientations of the molecule with either of these axes aligned with the magnetic field are excited, while the orientation selection for *X* is similar to the general case. In the case of an axial ZFS tensor (\(E = 0\)), pure orientation selection is again obtained for the *Z* canonical transition, while the orientations on the whole plane perpendicular to *Z* are excited for the other canonical transition. Hence in this case, single-crystal-like spectra are only obtained for the *Z* orientation, unless the hyperfine tensor is also axial and collinear with the ZFS tensor. However, in all other cases even if multiple orientations are excited for the *X* and *Y* canonical transitions, typically the orientations lying along the ZFS tensor axes contribute preferentially to the hyperfine spectrum due to increased relaxation for non-canonical orientations [51], leading to a larger distribution with respect to the *Z* transition, but smaller still compared to the absence of orientation selection. In the case of the hyperfine techniques, this means that measurements close to the canonical positions, corresponding to the three axes of the ZFS tensor, can be considered to reflect the hyperfine couplings along those directions in the molecular frame to a good approximation. HYSCORE spectra of triplet state systems are thus expected to exhibit single peaks instead of the broad ridges typical of many *S* = ½ systems in most cases.

In addition to orientation selection, triplet state pulse EPR is also characterised by transition selection, i.e., only the transition between two of the triplet state sublevels is selected depending on the magnetic field position. HYSCORE spectra, hence, only contain nuclear frequencies of two of the three manifolds (*m*_{S} = +1 and *m*_{S} = 0, or *m*_{S} = −1 and *m*_{S} = 0). Both contain the nuclear frequencies of the *m*_{S} = 0 manifold, determined exclusively by the nuclear Zeeman and nuclear quadrupole interaction (for *I* = 1) (see Fig. 1c). In triplet state ENDOR, this transition selectivity allows the determination of the sign of the hyperfine coupling, if the sign of the ZFS parameter *D* is known [52]. It will be shown that the sign of the hyperfine couplings can also be determined using triplet state HYSCORE.

### 4.1 Theoretical Results

In the case of spin ½ nuclei, such as for example ^{1}H, only two cross-peaks are observed for each of the two triplet state transitions corresponding to a single orientation of the molecule with respect to the external magnetic field. The cross-peaks are given by (*ν*_{12}, *ν*_{34}) and (*ν*_{34}, *ν*_{56}) (frequencies in MHz), where for a positive hyperfine coupling *ν*_{12} < *ν*_{34} < *ν*_{56}, while for a negative hyperfine coupling the order is reversed. The position of the cross-peaks with respect to the nuclear Larmor frequency can thus be used to determine the sign of the hyperfine coupling. If more than a single orientation were selected, there would be a distribution of nuclear frequencies in the *m*_{S} = −1 and *m*_{S} = +1 manifolds, but not in the *m*_{S} = 0 manifold, leading to ridges parallel to the frequency axes. This is a specific feature of triplet state HYSCORE.

For spin 1 nuclei with weak nuclear quadrupole couplings spectra were calculated for a hyperfine coupling comparable to (cancellation regime, Fig. 2 centre) or larger than the nuclear Zeeman frequency ν_{I} (Fig. 2 right). In the cancellation regime (*A* ≈ *ν*_{I} for a triplet state) in addition to the nuclear frequencies of the *m*_{S} = 0 manifold, the nuclear frequencies of one of the other manifolds also do not depend on the hyperfine coupling and reduce to the pure nuclear quadrupole frequencies (*m*_{S} = −1 for *A* < 0 and *m*_{S} = +1 for *A* > 0). The nuclear frequencies of the two manifolds will be very similar in many cases, one depending only on the nuclear quadrupole interaction and the other on the nuclear quadrupole and nuclear Zeeman interaction, and hence a single peak on or close to the diagonal will be observed in the HYSCORE spectrum for that transition (see Fig. 2, bottom centre). All the information on the hyperfine coupling is in this case contained in the HYSCORE spectrum of the triplet state transition involving the other *m*_{S} manifold, for which intense cross-peaks are observed (see Fig. 2, top centre).

If the hyperfine coupling is larger than *ν*_{I}, at least four cross-peaks are observed for each of the transitions, at positions determined by the single-quantum frequencies. In this case of weak nuclear quadrupole coupling, the double-quantum cross-peaks are predicted to have negligible intensity. In the presence of a distribution of hyperfine and nuclear quadrupole splittings due to selection of more than one orientation again ridges approximately parallel to the axes are expected, since the changes in hyperfine coupling are usually more significant with respect to those of the nuclear quadrupole interaction.

The simulations shown in Fig. 2 were performed assuming ideal pulses, leading to the absence of artefact peaks along the diagonal, as discussed in Sect. 3. Experimentally, this is often difficult to achieve and peaks along the diagonal with varying intensity compared to the cross-peaks will be observed.

### 4.2 Experimental Results

^{14}N HYSCORE spectra of the zinc porphyrin depicted in Fig. 3a were recorded at field positions corresponding to orientations along the in-plane

*X*and out-of-plane

*Z*ZFS axes after laser excitation at 532 nm and are shown in Fig. 3c. Measurements along the second in-plane axis

*Y*gave similar spectra as along

*X*, indicating an approximately axial hyperfine tensor.

^{14}N nuclei in porphyrin-like systems are typically characterised by a nuclear quadrupole coupling of about 2–3 MHz [19, 53] and the assumption of weak nuclear quadrupole coupling is no longer valid. For

*S*= ½ systems,

*I*= 1 nuclei with weak nuclear quadrupole coupling (

^{2}H) were shown to give strong single-quantum peaks in the HYSCORE spectrum and weak double-quantum peaks, as also predicted here for the triplet state. On the other hand,

*I*= 1 nuclei with strong nuclear quadrupole coupling interactions such as

^{14}N give rise to HYSCORE spectra with strong double-quantum features and weaker single-quantum peaks [28]. This implies that the intensities of the HYSCORE features due to the

^{14}N nuclei will not be accurately predicted by Eq. 9; however, a discussion of the theoretical and experimental data side by side is very instructive as it shall be shown that despite its shortcomings and approximations, the theoretical considerations above predict many features observed experimentally in this porphyrin system.

The strongest cross-peaks observed in the ^{14}N HYSCORE spectra have been attributed to double-quantum cross-peaks. Double-quantum frequencies do not depend on the nuclear quadrupole coupling to first order and can be used to obtain an estimate of the hyperfine couplings [28]. In almost all of the spectra strong peaks on the diagonal are observed, which are mostly due to non-ideal mixing pulses.

The HYSCORE spectra recorded at field positions corresponding to the two triplet state transitions for the *X* direction of the ZFS tensor (*X*^{+} and *X*^{−}) exhibit strong cross-peaks at about (6.1, 3.2) MHz and (3.2, 6.1) MHz for *X*^{−} [cross-peaks marked as (*dq*_{−1}, *dq*_{0})] and cross-peaks very close to the diagonal for *X*^{+} [cross-peaks marked as (*dq*_{0}, *dq*_{+1})]. This resembles what was observed in the simulations for weak nuclear quadrupole coupling in the exact cancellation regime (Fig. 2, centre). In this regime, the difference between the two double-quantum frequencies approximately corresponds to two times the hyperfine coupling for that orientation; based on calculations, the error of neglecting the contribution of the nuclear quadrupole coupling amounts to an underestimate of about 20 %. The in-plane hyperfine coupling for the pyrrolic nitrogen nuclei can thus be estimated to be about 1.7 MHz. The sign of the hyperfine coupling is positive, since the larger nuclear frequencies are observed for the *X*^{−} transition and the *D* value of the porphyrin molecule is positive (see Fig. 1).

The HYSCORE spectra recorded for the out-of-plane orientation are more difficult to interpret. First of all, the double-quantum cross-peaks for the *Z*^{−} transition could only be clearly resolved using matched HYSCORE, which leads to an increase in the intensity of multiple quantum features [54]. Secondly, instead of single-crystal-like spectra, broad ridges parallel to the frequency axes were observed. The hyperfine coupling estimated from the centre of these ridges is about 6.5 MHz, which is far from the cancellation regime where HYSCORE spectra are easier to interpret.

*Z*

^{−}are observed more clearly; while for

*Z*

^{+}, they are closer to the diagonal, confirming that the average hyperfine coupling is about 6.5 MHz and has a positive sign. The observation of ridges instead of single-crystal-like peaks expected on account of orientation selection, which should be strongest for the

*Z*orientation, was attributed to a distribution in hyperfine couplings, in analogy to the results in previous papers [53, 55]. The origin for the distribution might be the conformational flexibility of the porphyrin plane leading to changes in hyperfine couplings, particularly in the out-of-plane orientation [56]. The coordinated pyridine molecule was identified as a possible cause for distortions of the porphyrin plane and a closer analysis of the solvent interaction and the influence of pyridine on the ESEEM and HYSCORE data will be the subject of further investigation.

*m*

_{S}= 0 manifold; some peaks in the low frequency region are also present, but their exact assignment was difficult. The signals corresponding to the double-quantum nuclear frequencies of the other two manifolds are too broad to be clearly identified due to the distribution of hyperfine couplings. The inhomogeneous hyperfine broadening that affects the line shapes in three-pulse ESEEM is refocused by the mixing pulse in HYSCORE, achieving better resolution and allowing the detection of broad peaks [14]. The nitrogen hyperfine couplings could not have been unambiguously determined from the ESEEM data alone; the HYSCORE data with the clearly resolved double-quantum features were essential for the characterisation of the investigated porphyrin system in terms of the

^{14}N hyperfine interaction.

^{14}N nuclei, numerical simulations implemented in Easyspin [35, 43] were employed. The HYSCORE spectra were simulated for a set of hyperfine and nuclear quadrupole parameters simultaneously optimised for X- and Q-band HYSCORE and three-pulse ESEEM data. The simulation parameters were

*A*

_{x}= 1.55 ± 0.15 MHz,

*A*

_{y}= 1.65 ± 0.15 MHz,

*A*

_{z}= 6.95 ± 1.35 MHz,

*Q*= 2.35 ± 0.05 MHz and

*η*= 0.75 ± 0.10, where the standard deviations indicate the width of the distribution around the central values considered to simulate the observed ridges. The tensor orientations were defined based on DFT calculations [56]. The resulting simulated HYSCORE spectra are shown in Fig. 6. The agreement between simulation and experiment is satisfyingly close; in particular, the peak positions are well reproduced. The strong peaks on the diagonal in the experimental data are absent in the simulations, confirming that they are due to incomplete inversion by the mixing

*π*pulse. The simulated relative intensities of the different types of cross-peaks also agree quite well with experiment, considering that an accurate simulation of HYSCORE peak intensities can often be challenging [26, 34].

## 5 Conclusions

The features of HYSCORE spectra of photoexcited triplet states coupled to spin ½ and spin 1 nuclei were discussed based on the analytical expressions derived in the framework of the density matrix formalism. It was shown that the orientation and transition selection in these systems allows the determination of the hyperfine coupling in different orientations of the molecular frame. In addition to this, the sign of the hyperfine coupling can also be determined, assuming the sign of the zero-field splitting parameter *D* is known. The conclusions drawn based on the example calculations were used to interpret experimental HYSCORE spectra of the triplet state of a porphyrin molecule.

## Notes

### Acknowledgments

The Q-band EPR measurements were performed at the National EPR Research Facility at the University of Manchester with the help of Dr. Alistair Fielding. The authors are grateful to Dr. Jeffrey Harmer for helpful discussions and useful comments on the manuscript. The authors gratefully acknowledge the financial support of this work by EPSRC. Marilena Di Valentin and Donatella Carbonera express their gratitude to Prof. Giovanni Giacometti for laying the foundations of research on photoexcited states, and in particular on triplet states, at the University of Padova, for introducing them to this research field and his longstanding collaboration and support in their research. This paper is dedicated to him on the occasion of his 85th birthday.

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