Skip to main content
SpringerLink
Log in

HYSCORE on Photoexcited Triplet States

  • Published:
Applied Magnetic Resonance Aims and scope Submit manuscript

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 14N 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. R. Uphaus, J. Norris, J. Katz, Biochem. Biophys. Res. Commun. 61, 1057–1063 (1974)

    Article  Google Scholar 

  2. H. Levanon, J. Norris, Chem. Rev. 78, 185–198 (1978)

    Article  Google Scholar 

  3. D. Budil, M. Thurnauer, Biochim. Biophys. Acta 1057, 1–41 (1991)

    Article  Google Scholar 

  4. H. Levanon, K. Möbius, Annu. Rev. Biophys. Biomol. Struct. 26, 495–540 (1997)

    Article  Google Scholar 

  5. W. Lubitz, F. Lendzian, R. Bittl, Acc. Chem. Res. 35, 313–320 (2002)

    Article  Google Scholar 

  6. D. Carbonera, M. Di Valentin, R. Spezia, A. Mezzetti, Curr. Protein Pept. Sci. 15, 332–350 (2014)

    Article  Google Scholar 

  7. D. Gust, T. Moore, A. Moore, Acc. Chem. Res. 34, 40–48 (2001)

    Article  Google Scholar 

  8. M. Wasielewski, J. Org. Chem. 71, 5051–5066 (2006)

    Article  Google Scholar 

  9. S. Fukuzumi, K. Ohkubo, T. Suenobu, Acc. Chem. Res. 47, 1455–1464 (2014)

    Article  Google Scholar 

  10. L. Xiao, Z. Chen, B. Qu, J. Luo, S. Kong, Q. Gong, J. Kido, Adv. Mater. 23, 926–952 (2011)

    Article  Google Scholar 

  11. K. Plaetzer, B. Krammer, J. Berlanda, F. Berr, T. Kiesslich, Lasers Med. Sci. 24, 259–268 (2009)

    Article  Google Scholar 

  12. J. Zhao, W. Wu, J. Sun, S. Guo, Chem. Soc. Rev. 42, 5323–5351 (2013)

    Article  Google Scholar 

  13. M. Di Valentin, M. Albertini, E. Zurlo, M. Gobbo, D. Carbonera, J. Am. Chem. Soc. 136, 6582–6585 (2014)

    Article  Google Scholar 

  14. A. Schweiger, G. Jeschke, Principles of Pulse Electron Paramagnetic Resonance (Oxford University Press, Oxford, 2001)

    Google Scholar 

  15. D.J. Sloop, H.-L. Yu, T.S. Lin, S.I. Weissman, J. Chem. Phys. 75, 3746–3757 (1981)

    Article  ADS  Google Scholar 

  16. H.-L. Yu, D.J. Sloop, S.I. Weissman, J. Lin, J. Norris, M.K. Bowman, J. Phys. Chem. 86, 4287–4290 (1982)

    Article  Google Scholar 

  17. T.S. Lin, Chem. Rev. 84, 1–15 (1984)

    Article  Google Scholar 

  18. W. Buma, E.J.J. Groenen, J. Schmidt, R. de Beer, J. Chem. Phys. 91, 6549–6566 (1989)

    Article  ADS  Google Scholar 

  19. D.J. Singel, W.A.J.A. van der Poel, J. Schmidt, J.H. van der Waals, R. De Beer, J. Chem. Phys. 81, 5453–5461 (1984)

    Article  ADS  Google Scholar 

  20. A. De Groot, R. Evelo, A.J. Hoff, R. De Beer, H. Scheer, Chem. Phys. Lett. 118, 48–54 (1985)

    Article  ADS  Google Scholar 

  21. F. Lendzian, R. Bittl, W. Lubitz, Photosynth. Res. 55, 189–197 (1998)

    Article  Google Scholar 

  22. J. Niklas, T. Schulte, S. Prakash, M. van Gastel, E. Hofmann, W. Lubitz, J. Am. Chem. Soc. 129, 15442–15443 (2007)

    Article  Google Scholar 

  23. E. Salvadori, M. Di Valentin, C.M.W. Kay, A. Pedone, V. Barone, D. Carbonera, Phys. Chem. Chem. Phys. 14, 12238–12251 (2011)

    Article  Google Scholar 

  24. M. Di Valentin, E. Salvadori, V. Barone, D. Carbonera, Mol. Phys. 111, 2914–2932 (2013)

    Article  ADS  Google Scholar 

  25. A. Marchanka, W. Lubitz, M. Plato, M. van Gastel, Photosynth. Res. 120, 99–111 (2014)

    Article  Google Scholar 

  26. S. van Doorslaer, EPR newsletter 17, 9–12 (2008)

    Google Scholar 

  27. C. Calle, A. Sreekanth, M.V. Fedin, J. Forrer, I. Garcia-Rubio, I.A. Gromov, D. Hinderberger, B. Kasumaj, P. Léger, B. Mancosu, G. Mitrikas, M.G. Santangelo, S. Stoll, A. Schweiger, R. Tschaggelar, J. Harmer, Helv. Chim. Acta 89, 2495–2521 (2006)

    Article  Google Scholar 

  28. J. Harmer, G. Mitrikas, A. Schweiger, in High Resolution EPR, ed. by G. Hanson, L. Berliner (Springer Science, New York, 2009), pp. 13–61

    Chapter  Google Scholar 

  29. L.G. Rowan, E.L. Hahn, W.B. Mims, Phys. Rev. 137, A61–A71 (1965)

    Article  ADS  Google Scholar 

  30. W.B. Mims, Phys. Rev. B 5, 2409–2419 (1972)

    Article  ADS  Google Scholar 

  31. W.B. Mims, Phys. Rev. B 6, 3543–3545 (1972)

    Article  ADS  Google Scholar 

  32. J.J. Shane, L.P. Liesum, A. Schweiger, J. Magn. Reson. 134, 72–75 (1998)

    Article  ADS  Google Scholar 

  33. R. Szosenfogel, D. Goldfarb, Mol. Phys. 95, 1295–1308 (1998)

    ADS  Google Scholar 

  34. Z. Madi, S. Van Doorslaer, A. Schweiger, J. Magn. Reson. 154, 181–191 (2002)

    Article  ADS  Google Scholar 

  35. S. Stoll, R.D. Britt, Phys. Chem. Chem. Phys. 11, 6614–6625 (2009)

    Article  Google Scholar 

  36. C. Gemperle, G. Aebli, A. Schweiger, R.R. Ernst, J. Magn. Reson. 88, 241–256 (1990)

    ADS  Google Scholar 

  37. A. Pöppl, R. Böttcher, Chem. Phys. 221, 53–66 (1997)

    Article  ADS  Google Scholar 

  38. A.R. Coffino, J. Peisach, J. Chem. Phys. 97, 3072–3091 (1992)

    Article  ADS  Google Scholar 

  39. R.G. Larsen, C.J. Halkides, D.J. Singel, J. Chem. Phys. 98, 6704–6721 (1993)

    Article  ADS  Google Scholar 

  40. N.P. Benetis, P.C. Dave, D. Goldfarb, J. Magn. Reson. 158, 126–142 (2002)

    Article  ADS  Google Scholar 

  41. F.C. Grozema, C. Houarner-Rassin, P. Prins, L.D.A. Siebbeles, H.L. Anderson, J. Am. Chem. Soc. 129, 13370–13371 (2007)

    Article  Google Scholar 

  42. S. van Doorslaer, G. Sierra, A. Schweiger, J. Magn. Reson. 136, 152–158 (1999)

    Article  ADS  Google Scholar 

  43. S. Stoll, A. Schweiger, J. Magn. Reson. 178, 42–55 (2006)

    Article  ADS  Google Scholar 

  44. A.M. Tyryshkin, S.A. Dikanov, D. Goldfarb, J. Magn. Reson. Ser. A 105, 271–283 (1993)

    Article  ADS  Google Scholar 

  45. A.A. Shubin, S.A. Dikanov, J. Magn. Reson. 52, 1–12 (1983)

    ADS  Google Scholar 

  46. C.P. Slichter, Principles of Magnetic Resonance (Springer, Berlin, 1980)

    Google Scholar 

  47. S.A. Dikanov, L. Xun, A.B. Karpiel, A.M. Tyryshkin, M.K. Bowman, J. Am. Chem. Soc. 118, 8408–8416 (1996)

    Article  Google Scholar 

  48. B. Corzilius, E. Ramić, K.-P. Dinse, Appl. Magn. Reson. 30, 499–512 (2006)

    Article  Google Scholar 

  49. A.G. Maryasov, M.K. Bowman, J. Phys. Chem. B 108, 9412–9420 (2004)

    Article  Google Scholar 

  50. A.G. Maryasov, M.K. Bowman, J. Magn. Reson. 179, 120–135 (2006)

    Article  ADS  Google Scholar 

  51. F. Lendzian, R. Bittl, A. Telfer, W. Lubitz, Biochim. Biophys. Acta 1605, 35–46 (2003)

    Article  Google Scholar 

  52. M. Thurnauer, Rev. Chem. Intermediates 3, 197–230 (1979)

    Article  Google Scholar 

  53. J. Harmer, S. van Doorslaer, I.A. Gromov, A. Schweiger, Chem. Phys. Lett. 358, 8–16 (2002)

    Article  ADS  Google Scholar 

  54. G. Jeschke, R. Rakhmatullin, A. Schweiger, J. Magn. Reson. 131, 261–271 (1998)

    Article  ADS  Google Scholar 

  55. S. Van Doorslaer, G. Jeschke, B. Epel, D. Goldfarb, R.-A. Eichel, B. Kräutler, A. Schweiger, J. Am. Chem. Soc. 125, 5915–5927 (2003)

    Article  Google Scholar 

  56. C.E. Tait, P. Neuhaus, H.L. Anderson, C.R. Timmel, manuscript in preparation

Download references

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.

Author information

Authors and Affiliations

  1. Dipartimento di Scienze Chimiche, Università degli Studi di Padova, Via Marzolo 1, 35131, Padua, Italy

    Claudia E. Tait, Donatella Carbonera & Marilena Di Valentin

  2. Department of Chemistry, Centre for Advanced Electron Spin Resonance, University of Oxford, South Parks Road, Oxford, OX1 3QR, UK

    Claudia E. Tait & Christiane R. Timmel

  3. Department of Chemistry, Chemistry Research Laboratory, University of Oxford, 12 Mansfield Road, Oxford, OX1 3TA, UK

    Patrik Neuhaus & Harry L. Anderson

Authors
  1. Claudia E. Tait
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Patrik Neuhaus
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Harry L. Anderson
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Christiane R. Timmel
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Donatella Carbonera
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Marilena Di Valentin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Claudia E. Tait or Marilena Di Valentin.

Appendix

Appendix

List of the coefficients used in Eq. 2 of the main text:

$$\begin{aligned} \chi_{0} & = 1 - \frac{k}{16}\left\{\vphantom{\left( {1 - \sqrt {1 - k} } \right)} {10 + 2\gamma_{ - } - 4\cos \omega_{12} \tau - 4\cos \omega_{34} \tau - \gamma_{ - } \left[ {\left( {1 - \sqrt {1 - k} } \right)\cos (\omega_{12} + \omega_{34} )\tau - \left( {1 + \sqrt {1 - k} } \right)\cos (\omega_{12} - \omega_{34} )\tau } \right]} \right\} \\ \chi_{d} & = \frac{k}{4}\gamma_{ + } \cos \left[ {\omega_{12} (t_{1} + t_{2} + \tau )} \right]\left( {1 - \cos \omega_{34} \tau } \right) \\ \chi_{12} & = \frac{k}{8}\left[ {2\cos \left( {\frac{{\omega_{12} \tau }}{2}} \right) - \left( {\sqrt {1 - k} - 1} \right)\cos \left( {\omega_{34} \tau + \frac{{\omega_{12} \tau }}{2}} \right) + \left( {\sqrt {1 - k} + 1} \right)\cos \left( {\omega_{34} \tau - \frac{{\omega_{12} \tau }}{2}} \right)} \right] \\ \chi_{34} & = \frac{k}{8}\left[ {2\cos \left( {\frac{{\omega_{34} \tau }}{2}} \right) - \left( {\sqrt {1 - k} - 1} \right)\cos \left( {\omega_{12} \tau + \frac{{\omega_{34} \tau }}{2}} \right) + \left( {\sqrt {1 - k} + 1} \right)\cos \left( {\omega_{12} \tau - \frac{{\omega_{34} \tau }}{2}} \right)} \right] \\ \chi_{12,34}^{ \pm }& = - \frac{k}{4}\left( {\sqrt {1 - k} \pm 1} \right)\sin \left( {\frac{{\omega_{12} \tau }}{2}} \right)\sin \left( {\frac{{\omega_{34} \tau }}{2}} \right) \\ \end{aligned}$$

List of the coefficients used in Eq. 9 of the main text:

$$\begin{aligned} \chi_{0} & = 1 - \gamma_{ + } C_{0} - \gamma_{ - } C_{1} \\ & \quad + (\gamma_{ + } C_{2} + \gamma_{ - } C_{3} )\left[ {\cos \omega_{12} \tau + \cos \omega_{23} \tau + \cos \omega_{45} \tau + \cos \omega_{56} \tau } \right] + (3\gamma_{ + } C_{4} + \gamma_{ - } C_{5} )\left[ {\cos \omega_{13} \tau + \cos \omega_{46} \tau } \right] \\ & \quad + \gamma_{ - } \left[ {C_{6}^{ + } \left[ {\cos (\omega_{12} + \omega_{45} )\tau + \cos (\omega_{23} + \omega_{56} )\tau } \right] + C_{6}^{ - } \left[ {\cos (\omega_{12} - \omega_{56} )\tau + \cos (\omega_{23} - \omega_{45} )\tau } \right]} \right. \\ & \quad + C_{7}^{ + } \left[ {\cos (\omega_{12} + \omega_{56} )\tau + \cos (\omega_{23} + \omega_{45} )\tau } \right] + C_{7}^{ - } \left[ {\cos (\omega_{12} - \omega_{45} )\tau + \cos (\omega_{23} - \omega_{56} )\tau } \right] \\ & \quad + C_{8}^{ + } \left[ {\cos (\omega_{13} + \omega_{45} )\tau + \cos (\omega_{13} + \omega_{56} )\tau + \cos (\omega_{12} + \omega_{46} )\tau + \cos (\omega_{23} + \omega_{46} )\tau } \right] \\ & \quad + C_{8}^{ - } \left[ {\cos (\omega_{13} - \omega_{45} )\tau + \cos (\omega_{13} - \omega_{56} )\tau + \cos (\omega_{12} - \omega_{46} )\tau + \cos (\omega_{23} - \omega_{46} )\tau } \right] \\ & \left. \quad + C_{9}^{ + } \cos (\omega_{13} + \omega_{46} )\tau + C_{9}^{ - } \cos (\omega_{13} - \omega_{46} )\tau \right] \\ \end{aligned}$$
$$\begin{aligned} \chi_{d} & = \gamma_{ + } \left[ {\cos \left[ {\omega_{12} \left( {t_{1} + t_{2} + \tau } \right)} \right]\left[ {C_{2} - C_{16}^{ + } \cos \omega_{45} \tau - C_{16}^{ - } \cos \omega_{56} \tau - 2C_{4} \cos \omega_{46} \tau } \right]} \right. \\ & \quad + \cos \left[ {\omega_{45} \left( {t_{1} + t_{2} + \tau } \right)} \right]\left[ {C_{2} - C_{16}^{ + } \cos \omega_{12} \tau - C_{16}^{ - } \cos \omega_{23} \tau - 2C_{4} \cos \omega_{13} \tau } \right] \\ & \quad + \cos \left[ {\omega_{23} \left( {t_{1} + t_{2} + \tau } \right)} \right]\left[ {C_{2} - C_{16}^{ - } \cos \omega_{45} \tau - C_{16}^{ + } \cos \omega_{56} \tau - 2C_{4} \cos \omega_{46} \tau } \right] \\ & \quad + \cos \left[ {\omega_{56} \left( {t_{1} + t_{2} + \tau } \right)} \right]\left[ {C_{2} - C_{16}^{ - } \cos \omega_{12} \tau - C_{16}^{ + } \cos \omega_{23} \tau - 2C_{4} \cos \omega_{13} \tau } \right] \\ & \quad + \cos \left[ {\omega_{13} \left( {t_{1} + t_{2} + \tau } \right)} \right]\left[ {3C_{4} - 2C_{4} \cos \omega_{45} \tau - 2C_{4} \cos \omega_{56} \tau + C_{4} \cos \omega_{46} \tau } \right] \\ & \quad + \left. {\cos \left[ {\omega_{46} \left( {t_{1} + t_{2} + \tau } \right)} \right]\left[ {3C_{4} - 2C_{4} \cos \omega_{12} \tau - 2C_{4} \cos \omega_{23} \tau + C_{4} \cos \omega_{13} \tau } \right]} \right] \\ \end{aligned}$$
$$\begin{aligned} \chi_{12} & = 2C_{3} \cos \frac{{\omega_{12} \tau }}{2} + C_{10} \cos \left( {\frac{{\omega_{12} \tau }}{2} + \omega_{23} \tau } \right) - 4C_{11}^{ + } \cos \left( {\frac{{\omega_{12} \tau }}{2} + \omega_{45} \tau } \right) - 4C_{11}^{ - } \cos \left( {\frac{{\omega_{12} \tau }}{2} - \omega_{56} \tau } \right) \\ & \quad - C_{12}^{ + } \cos \left( {\frac{{\omega_{12} \tau }}{2} - \omega_{45} \tau } \right) - C_{12}^{ - } \cos \left( {\frac{{\omega_{12} \tau }}{2} + \omega_{56} \tau } \right) - C_{13}^{ + } \cos \left( {\frac{{\omega_{12} \tau }}{2} + \omega_{46} \tau } \right) - C_{13}^{ - } \cos \left( {\frac{{\omega_{12} \tau }}{2} - \omega_{46} \tau } \right) \\ & \quad - C_{11}^{ + } \left[ {2\cos \left( {\frac{{(\omega_{13} + \omega_{23} )\tau }}{2} - \omega_{45} \tau } \right) - \cos \left( {\frac{{(\omega_{13} + \omega_{23} )\tau }}{2} - \omega_{56} \tau } \right)} \right] \\ & \quad - C_{11}^{ - } \left[ {2\cos \left( {\frac{{(\omega_{13} + \omega_{23} )\tau }}{2} + \omega_{56} \tau } \right) - \cos \left( {\frac{{(\omega_{13} + \omega_{23} )\tau }}{2} + \omega_{45} \tau } \right)} \right] \\ & \quad - 2C_{9}^{ + } \cos \left( {\frac{{(\omega_{13} + \omega_{23} )\tau }}{2} + \omega_{46} \tau } \right) - 2C_{9}^{ - } \cos \left( {\frac{{(\omega_{13} + \omega_{23} )\tau }}{2} - \omega_{46} \tau } \right) \\ \end{aligned}$$
$$\begin{aligned} \chi_{13} & = 2C_{5} \cos \frac{{\omega_{13} \tau }}{2} + C_{10} \cos \left( {\frac{{\omega_{13} \tau }}{2} - \omega_{23} \tau } \right) - 2C_{9}^{ - } \cos \left( {\frac{{\omega_{13} \tau }}{2} + \omega_{46} \tau } \right) + 2C_{9}^{ + } \cos \left( {\frac{{\omega_{13} \tau }}{2} - \omega_{46} \tau } \right) \\ & \quad - C_{13}^{ + } \left[ {\cos \left( {\frac{{\omega_{13} \tau }}{2} + \omega_{45} \tau } \right) + \cos \left( {\frac{{\omega_{13} \tau }}{2} + \omega_{56} \tau } \right)} \right] - C_{13}^{ - } \left[ {\cos \left( {\frac{{\omega_{13} \tau }}{2} - \omega_{45} \tau } \right) + \cos \left( {\frac{{\omega_{13} \tau }}{2} - \omega_{56} \tau } \right)} \right] \\ & \quad - C_{14} \left[ {2\cos \left( {\frac{{(\omega_{12} - \omega_{23} )\tau }}{2} + \omega_{45} \tau } \right) + 2\cos \left( {\frac{{(\omega_{12} - \omega_{23} )\tau }}{2} - \omega_{56} \tau } \right) - \cos \left( {\frac{{(\omega_{12} - \omega_{23} )\tau }}{2} + \omega_{46} \tau } \right) - \cos \left( {\frac{{(\omega_{12} - \omega_{23} )\tau }}{2} - \omega_{46} \tau } \right)} \right] \\ & \quad - C_{15} \left[ {\cos \left( {\frac{{(\omega_{12} - \omega_{23} )\tau }}{2} - \omega_{45} \tau } \right) + \cos \left( {\frac{{(\omega_{12} - \omega_{23} )\tau }}{2} + \omega_{56} \tau } \right)} \right] \\ \end{aligned}$$
$$\begin{aligned} \chi_{12,45}^{ + } & = 2C_{7}^{ - } \cos \left( {\frac{{(\omega_{12} - \omega_{45} )\tau }}{2}} \right) - C_{12}^{ + } \cos \left( {\frac{{(\omega_{12} + \omega_{45} )\tau }}{2}} \right) \\ & \quad - C_{11}^{ + } \left[ {\cos \left( {\frac{{(\omega_{13} + \omega_{23} - \omega_{45} )\tau }}{2}} \right) + \cos \left( {\frac{{(\omega_{12} - \omega_{46} - \omega_{56} )\tau }}{2}} \right)} \right] \\ & \quad - C_{15} \left[ {\cos \left( {\frac{{(\omega_{13} + \omega_{23} + \omega_{45} )\tau }}{2}} \right) + \cos \left( {\frac{{(\omega_{12} + \omega_{46} + \omega_{56} )\tau }}{2}} \right)} \right] \\ & \quad + C_{11}^{ - } \cos \left( {\frac{{(\omega_{13} + \omega_{23} + \omega_{46} + \omega_{56} )\tau }}{2}} \right) \\ \end{aligned}$$
$$\begin{aligned} \chi_{12,56}^{ + } & = 2C_{6}^{ - } \cos \left( {\frac{{(\omega_{12} - \omega_{56} )\tau }}{2}} \right) - 4C_{11}^{ - } \cos \left( {\frac{{(\omega_{12} + \omega_{56} )\tau }}{2}} \right) \\ & \quad - 2C_{14} \left[ {\cos \left( {\frac{{(\omega_{12} + \omega_{46} + \omega_{45} )\tau }}{2}} \right) + \cos \left( {\frac{{(\omega_{13} + \omega_{23} + \omega_{56} )\tau }}{2}} \right)} \right] \\ & \quad - 2C_{11}^{ + } \left[ {\cos \left( {\frac{{(\omega_{12} - \omega_{46} - \omega_{45} )\tau }}{2}} \right) + \cos \left( {\frac{{(\omega_{13} + \omega_{23} - \omega_{56} )\tau }}{2}} \right)} \right] \\ & \quad + 4C_{9}^{ - } \cos \left( {\frac{{(\omega_{13} + \omega_{23} - \omega_{45} - \omega_{46} )\tau }}{2}} \right) \\ \end{aligned}$$
$$\begin{aligned} \chi_{12,46}^{ + } & = 2C_{8}^{ - } \cos \left( {\frac{{(\omega_{12} - \omega_{46} )\tau }}{2}} \right) - C_{13}^{ - } \cos \left( {\frac{{(\omega_{12} + \omega_{46} )\tau }}{2}} \right) \\ & \quad + C_{14} \cos \left( {\frac{{(\omega_{13} + \omega_{23} + \omega_{46} )\tau }}{2}} \right) - 2C_{9}^{ - } \cos \left( {\frac{{(\omega_{13} + \omega_{23} - \omega_{46} )\tau }}{2}} \right) \\ & \quad + C_{11}^{ + } \left[ {\cos \left( {\frac{{(\omega_{12} - \omega_{45} + \omega_{56} )\tau }}{2}} \right) - 2\cos \left( {\frac{{(\omega_{12} + \omega_{45} - \omega_{56} )\tau }}{2}} \right)} \right. \\ & \left. \quad + \cos \left( {\frac{{(\omega_{13} + \omega_{23} + \omega_{45} - \omega_{56} )\tau }}{2}} \right) \right] \\ \end{aligned}$$
$$\begin{aligned} \chi_{13,46}^{ + } & = 2C_{9}^{ - } \left[ {\cos \left( {\frac{{(\omega_{13} + \omega_{46} )\tau }}{2}} \right) + \cos \left( {\frac{{(\omega_{13} - \omega_{46} )\tau }}{2}} \right)} \right. \\ & \quad - \cos \left( {\frac{{(\omega_{12} - \omega_{23} + \omega_{46} )\tau }}{2}} \right) - \cos \left( {\frac{{(\omega_{12} - \omega_{23} - \omega_{46} )\tau }}{2}} \right) \\ & \quad - \cos \left( {\frac{{(\omega_{13} + \omega_{45} - \omega_{56} )\tau }}{2}} \right) - \cos \left( {\frac{{(\omega_{13} - \omega_{45} + \omega_{56} )\tau }}{2}} \right) \\ & \left. \quad + 2\cos \left( {\frac{{(\omega_{12} - \omega_{23} + \omega_{45} - \omega_{56} )\tau }}{2}} \right) \right] \\ \end{aligned}$$
$$\begin{aligned} \chi_{12,45}^{ - } & = 2C_{6}^{ + } \cos \left( {\frac{{(\omega_{12} + \omega_{45} )\tau }}{2}} \right) - 4C_{11}^{ + } \cos \left( {\frac{{(\omega_{12} - \omega_{45} )\tau }}{2}} \right) \\ & \quad - 2C_{11}^{ - } \left[ {\cos \left( {\frac{{(\omega_{13} + \omega_{23} + \omega_{45} )\tau }}{2}} \right) + \cos \left( {\frac{{(\omega_{12} + \omega_{46} + \omega_{56} )\tau }}{2}} \right)} \right] \\ & \quad - 2C_{14} \left[ {\cos \left( {\frac{{(\omega_{13} + \omega_{23} - \omega_{45} )\tau }}{2}} \right) + \cos \left( {\frac{{(\omega_{12} - \omega_{46} - \omega_{56} )\tau }}{2}} \right)} \right] \\ & \quad + 4C_{9}^{ + } \cos \left( {\frac{{(\omega_{13} + \omega_{23} + \omega_{46} + \omega_{56} )\tau }}{2}} \right) \\ \end{aligned}$$
$$\begin{aligned} \chi_{12,56}^{ - } & = 2C_{7}^{ + } \cos \left( {\frac{{(\omega_{12} + \omega_{56} )\tau }}{2}} \right) - C_{12}^{ - } \cos \left( {\frac{{(\omega_{12} - \omega_{56} )\tau }}{2}} \right) \\ & \quad + C_{11}^{ - } \left[ {\cos \left( {\frac{{(\omega_{12} + \omega_{46} + \omega_{45} )\tau }}{2}} \right) + \cos \left( {\frac{{(\omega_{13} + \omega_{23} + \omega_{56} )\tau }}{2}} \right)} \right] \\ & \quad - C_{15} \left[ {\cos \left( {\frac{{(\omega_{12} - \omega_{46} - \omega_{45} )\tau }}{2}} \right) + \cos \left( {\frac{{(\omega_{13} + \omega_{23} - \omega_{56} )\tau }}{2}} \right)} \right] \\ & \quad + C_{11}^{ + } \cos \left( {\frac{{(\omega_{13} + \omega_{23} - \omega_{45} - \omega_{46} )\tau }}{2}} \right) \\ \end{aligned}$$
$$\begin{aligned} \chi_{12,46}^{ - } & = 2C_{8}^{ + } \cos \left( {\frac{{(\omega_{12} + \omega_{46} )\tau }}{2}} \right) - C_{13}^{ + } \cos \left( {\frac{{(\omega_{12} - \omega_{46} )\tau }}{2}} \right) \\ & \quad - 2C_{9}^{ + } \cos \left( {\frac{{(\omega_{13} + \omega_{23} + \omega_{46} )\tau }}{2}} \right) + C_{14} \cos \left( {\frac{{(\omega_{13} + \omega_{23} - \omega_{46} )\tau }}{2}} \right) \\ & \quad + C_{11}^{ - } \left[ {\cos \left( {\frac{{(\omega_{12} - \omega_{45} + \omega_{56} )\tau }}{2}} \right) - 2\cos \left( {\frac{{(\omega_{12} + \omega_{45} - \omega_{56} )\tau }}{2}} \right)} \right. \\ & \left. \quad + \cos \left( {\frac{{(\omega_{13} + \omega_{23} + \omega_{45} - \omega_{56} )\tau }}{2}} \right) \right] \\ \end{aligned}$$
$$\begin{aligned} \chi_{13,46}^{ - } & = 2C_{9}^{ + } \left[ {\cos \left( {\frac{{(\omega_{13} + \omega_{46} )\tau }}{2}} \right) + \cos \left( {\frac{{(\omega_{13} - \omega_{46} )\tau }}{2}} \right)} \right. \\ & \quad - \cos \left( {\frac{{(\omega_{12} - \omega_{23} + \omega_{46} )\tau }}{2}} \right) - \cos \left( {\frac{{(\omega_{12} - \omega_{23} - \omega_{46} )\tau }}{2}} \right) \\ & \quad - \cos \left( {\frac{{(\omega_{13} + \omega_{45} - \omega_{56} )\tau }}{2}} \right) - \cos \left( {\frac{{(\omega_{13} - \omega_{45} + \omega_{56} )\tau }}{2}} \right) \\ & \left. \quad + 2\cos \left( {\frac{{(\omega_{12} - \omega_{23} + \omega_{45} - \omega_{56} )\tau }}{2}} \right) \right] \\ \end{aligned}$$
$$\begin{array}{*{20}l} {C_{0} = k\left( {\frac{2}{3} - \frac{3}{8}k} \right)} \hfill & {C_{6}^{ \pm } = \frac{{k^{2} }}{16}\left( {1 \pm \frac{1}{3}\sqrt {1 - k} - \frac{5}{6}k} \right)} \hfill & {C_{12}^{ \pm } = \frac{k}{6}\left( {1 \pm \sqrt {1 - k} - \frac{13}{8}k \mp \frac{7}{8}k\sqrt {1 - k} + \frac{5}{8}k^{2} } \right)} \hfill \\ {C_{1} = k\left( {1 - \frac{11}{16}k + \frac{3}{32}k^{2} } \right)} \hfill & {C_{7}^{ \pm } = \frac{k}{24}\left( {1 \mp \sqrt {1 - k} - k} \right)^{2} } \hfill & {C_{13}^{ \pm } = \frac{{k^{2} }}{96}\left( {2 \pm 2\sqrt {1 - k} + k} \right)} \hfill \\ {C_{2} = \frac{k}{2}\left( {\frac{1}{3} - \frac{1}{4}k} \right)} \hfill & {C_{8}^{ \pm } = \frac{{k^{2} }}{48}\left( {1 \mp \sqrt {1 - k} - \frac{1}{4}k} \right)} \hfill & {C_{14} = \frac{1}{96}k^{3} } \hfill \\ {C_{3} = \frac{k}{2}\left( {\frac{1}{3} - \frac{1}{3}k + \frac{1}{16}k^{2} } \right)} \hfill & {C_{9}^{ \pm } = \frac{{k^{2} }}{192}\left( {1 \mp \sqrt {1 - k} } \right)^{2} } \hfill & {C_{15} = \frac{{k^{2} }}{24}(1 - k)} \hfill \\ {C_{4} = \frac{1}{48}k^{2} } \hfill & {C_{10} = \frac{{k^{2} }}{2}\left( {\frac{1}{3} - \frac{1}{4}k} \right)} \hfill & {C_{16}^{ \pm } = \frac{k}{12}\left( {1 \pm \sqrt {1 - k} - k} \right)} \hfill \\ {C_{5} = \frac{{k^{2} }}{16}\left( {\frac{1}{3} + \frac{1}{2}k} \right)} \hfill & {C_{11}^{ \pm } = \frac{{k^{2} }}{48}\left( {1 \pm \sqrt {1 - k} - k} \right)} \hfill & {} \hfill \\ \end{array}$$

The remaining coefficients \(\chi\) can be derived from the ones given above by substitution of the frequency indices according to the following table:

$$\begin{array}{*{20}c} {\text{Coefficient}} & {\text{Initial coefficient}} & {\text{Substitution of frequency indices}} \\ {\chi_{23} } & {\chi_{12} } & {12 \leftrightarrow 23,\;45 \leftrightarrow 56} \\ {\chi_{45} } & {\chi_{12} } & {12 \leftrightarrow 45,\;23 \leftrightarrow 56,\;13 \leftrightarrow 46} \\ {\chi_{56} } & {\chi_{12} } & {12 \leftrightarrow 56,\;23 \leftrightarrow 45,\;13 \leftrightarrow 46} \\ {\chi_{46} } & {\chi_{13} } & {12 \leftrightarrow 45,\;23 \leftrightarrow 56,\;13 \leftrightarrow 46} \\ {} & {} & {} \\ {\chi_{23,45}^{ + } } & {\chi_{12,56}^{ + } } & {12 \leftrightarrow 23,\;45 \leftrightarrow 56} \\ {\chi_{23,45}^{ - } } & {\chi_{12,56}^{ - } } & {12 \leftrightarrow 23,\;45 \leftrightarrow 56} \\ {\chi_{23,56}^{ + } } & {\chi_{12,45}^{ + } } & {12 \leftrightarrow 23,\;45 \leftrightarrow 56} \\ {\chi_{23,56}^{ - } } & {\chi_{12,45}^{ - } } & {12 \leftrightarrow 23,\;45 \leftrightarrow 56} \\ {\chi_{23,46}^{ + } } & {\chi_{12,46}^{ + } } & {12 \leftrightarrow 23,\;45 \leftrightarrow 56} \\ {\chi_{23,46}^{ - } } & {\chi_{12,46}^{ - } } & {12 \leftrightarrow 23,\;45 \leftrightarrow 56} \\ {\chi_{13,45}^{ + } } & {\chi_{12,46}^{ + } } & {12 \leftrightarrow 45,\;23 \leftrightarrow 56,\;13 \leftrightarrow 46} \\ {\chi_{13,45}^{ - } } & {\chi_{12,46}^{ - } } & {12 \leftrightarrow 45,\;23 \leftrightarrow 56,\;13 \leftrightarrow 46} \\ {\chi_{13,56}^{ + } } & {\chi_{12,46}^{ + } } & {12 \leftrightarrow 56,\;23 \leftrightarrow 45,\;13 \leftrightarrow 46} \\ {\chi_{13,56}^{ - } } & {\chi_{12,46}^{ - } } & {12 \leftrightarrow 56,\;23 \leftrightarrow 45,\;13 \leftrightarrow 46} \\ \end{array}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tait, C.E., Neuhaus, P., Anderson, H.L. et al. HYSCORE on Photoexcited Triplet States. Appl Magn Reson 46, 389–409 (2015). https://doi.org/10.1007/s00723-014-0624-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00723-014-0624-5

Keywords

Search

Navigation