EPR Spectroscopy E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: EMR Books
• Sprache: Englisch

Seitenzahl: 2000

Veröffentlichungsjahr: 2018

Beschreibung

This unique, self-contained resource is the first volume on electron paramagnetic resonance (EPR) spectroscopy in the eMagRes Handbook series. The 27 chapters cover the theoretical principles, the common experimental techniques, and many important application areas of modern EPR spectroscopy. EPR Spectroscopy: Fundamentals and Methods is presented in four major parts: A: Fundamental Theory, B: Basic Techniques and Instrumentation, C: High-Resolution Pulse Techniques, and D: Special Techniques.

The first part of the book gives the reader an introduction to basic continuous-wave (CW) EPR and an overview of the different magnetic interactions that can be determined by EPR spectroscopy, their associated theoretical description, and their information content. The second provides the basics of the various EPR techniques, including pulse EPR, and EPR imaging, along with the associated instrumentation. Parts C and D builds on parts A and B and offer introductory accounts of a wide range of modern advanced EPR techniques, with examples of applications. The last two parts presents most of the new advances that do not appear in most of the classical EPR textbooks that focus on CW EPR.

EPR Spectroscopy: Fundamentals and Methods contains, in concise form, all the material needed to understand state-of-the-art EPR spectroscopy at the graduate school/research level, whilst the editors have ensured that it presents the topic at a level accessible to newcomers to the field and others who want to know its range of application and how to apply it.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

eMagRes Books

International Advisory Board

Contributors

Series Preface

Preface

Abbreviations and Acronyms

Part A: Fundamental Theory

Chapter 1: Continuous-Wave EPR

1.1 Introduction

1.2 Basic Design and Operation of a CW-EPR Spectrometer

1.3 Solution Spectra

1.4 The Effect of Motion on CW-EPR Spectra

1.5 Spectra of Solids

1.6 Concluding Remarks

Acknowledgment

Further Reading

References

Chapter 2: EPR Interactions –

g

-Anisotropy

2.1 Introduction

2.2 The Zeeman Interaction for Spin ½

2.3

g

-Anisotropy in the Solid State

2.4

g

-Anisotropy in the Liquid State

2.5 Fictitious Spin ½

2.6 The Origin of the

g

-Anisotropy

References

Chapter 3: EPR Interactions – Zero-field Splittings

3.1 Introduction

3.2 Effects of ZFS on the EPR Spectrum

3.3 Origins of ZFS

3.4 Examples of ZFS in Actual Systems

3.5 Conclusion

Acknowledgments

3.6 Recommended Reading

References

Chapter 4: EPR Interactions – Coupled Spins

4.1 Introduction

4.2 The Isotropic Exchange Interaction

4.3 Two Coupled Spins

s

i

: The Hamiltonian Matrix and EPR

4.4 Coupling Two General Spins

4.5 Coupling More than Two Spins

4.6 Relationship between Spin Hamiltonian Parameters of the Total Spin States and those of the Uncoupled Paramagnets

4.7 Breakdown of the Strong Exchange Limit

4.8 Other Terms in the Exchange Hamiltonian

4.9 Noncoincidence Effects between the Interaction Matrices:

g

,

A

,

J

,

D

,

J

dip

,

D

12

, etc

4.10 Exchange between Orbitally Degenerate Species

4.11 Concluding Comments

Acknowledgment

Further Reading

References

Chapter 5: EPR Interactions – Hyperfine Couplings

5.1 Introduction

5.2 Dipolar Interaction and Hyperfine Hamiltonian

5.3 EPR Transitions in the General High Magnetic Field Case

5.4 Examples of EPR Spectra in Liquid and Frozen Solution

5.5 Example of EPR Spectra at Multiple Frequencies

5.6 Mechanisms of Hyperfine Coupling

5.7 EPR Spectra in the Low-field Case

5.8 Transition Probabilities

Acknowledgment

Further Reading

References

Chapter 6: EPR Interactions – Nuclear Quadrupole Couplings

6.1 The Nuclear Quadrupole Interaction

6.2 Energy Levels and Spectra

6.3 Analysis of EFG Tensors

6.4 Experimental Examples – EPR Spectra

6.5 Experimental Examples – ENDOR/ESEEM Spectra

6.6 Conclusions

Acknowledgments

Further Reading

References

Chapter 7: Quantum Chemistry and EPR Parameters

7.1 Introduction

7.2 Theory

7.3 Electronic Structure Methods

7.4 Illustration: Hyperfine Couplings

7.5 Concluding Remarks

Acknowledgments

References

Chapter 8: Spin Dynamics

8.1 Introduction

8.2 The Single-Electron Spin System

8.3 Two Coupled Electrons

8.4 An Electron Coupled to a Nucleus with

I

= 1/2

Acknowledgments

Further Reading

References

Chapter 9: Relaxation Mechanisms

9.1 Introduction

9.2 Relaxation Processes

9.3 Relaxation of Triarylmethyl (Trityl) Radicals

9.4 Relaxation of Semiquinones

9.5 Relaxation of Nitroxides

9.6 Relaxation of Cu(II) Complexes

9.7 Relaxation of Iron–Sulfur Clusters

9.8 Relaxation in Irradiated Organic Solids

9.9 Metals with

S

> ½

9.10 Summary

Acknowledgments

References

Part B: Basic Techniques and Instrumentation

Chapter 10: Transient EPR

10.1 Introduction

10.2 Experimental Considerations

10.3 Applications of Transient EPR

10.4 Conclusions

Acknowledgments

References

Chapter 11: Pulse EPR

11.1 Introduction

11.2 Pulses and Spins

11.3 Basic Pulse Sequences

11.4 Bandwidths

11.5 Non-Coherent Effects

11.6 Spectrometer

11.7 Practical Aspects

11.8 Further Reading

Acknowledgments

References

Chapter 12: EPR Instrumentation

12.1 Introduction

12.2 Magnet Systems

12.3 Microwave Bridges

12.4 EPR Resonators

12.5 Sample Cryostats

12.6 Spectrometer Control and Signal Detection

12.7 Instrumentation for Special Applications

Further Reading

References

Chapter 13: EPR Imaging

13.1 Introduction

13.2 Brief History

13.3 Basics of Image Acquisition

13.4 Spin Probes

13.5 Imaging Instrumentation and Methodology

13.6 Applications

13.7 Conclusion

Acknowledgments

References

Chapter 14: EPR Spectroscopy of Nitroxide Spin Probes

14.1 Introduction

14.2 Types of Nitroxide Probes

14.3 The Hamiltonian Describing the Nitroxide Spectrum

14.4 Dynamics of the Nitroxide Probe Encoded in the Spectral Linewidth

14.5 Polarity and Proticity of the Nitroxide Microenvironment Extracted from

g

and

A

Tensor Parameters

14.6 Techniques to Monitor Water Accessibility Toward the Nitroxides

14.7 Concluding Remarks

Acknowledgments

References

Part C: High-Resolution Pulse Techniques

Chapter 15: FT-EPR

15.1 Introduction

15.2 Single Pulse,

S

= 1/2

15.3 Multipulse FT-EPR

15.4

S

> 1, Triplets and Interacting Radicals

15.5 Digital FT

15.6 FT-EPR Examples

Acknowledgments

Further Reading

References

Chapter 16: Hyperfine Spectroscopy – ENDOR

16.1 Introduction

16.2 Static Spin Hamiltonian and the Nuclear Frequencies

16.3 CW ENDOR

16.4 Pulse ENDOR Techniques

References

Chapter 17: Hyperfine Spectroscopy – ELDOR-detected NMR

17.1 Introduction

17.2 The EDNMR Experiment

17.3 EDNMR for

I

> 1/2

17.4 Experimental Considerations

17.5 EDNMR versus Pulse ENDOR

17.6 EDNMR versus ESEEM

17.7 Simulations

17.8 Two-dimensional Experiments

Acknowledgments

References

Chapter 18: Hyperfine Spectroscopy – ESEEM

18.1 Introduction

18.2 Basic 1D ESEEM Experiments

18.3 Four-pulse ESEEM

18.4 Improving the Performance of ESEEM

Acknowledgments

References

Chapter 19: Dipolar Spectroscopy – Double-resonance Methods

19.1 Introduction

19.2 Dilute Cluster Description of the Sample

19.3 Pulse Sequences

19.4 Expressions for Dipolar Evolution

19.5 Orientation Selection

19.6 Complications and Remedies

19.7 Spins

S

> ½

19.8 Data Analysis

19.9 Conclusions

Acknowledgments

Further Reading

References

Chapter 20: Dipolar Spectroscopy – Single-resonance Methods

20.1 Introduction

20.2 Basic Theoretical Aspects of PDS Methods

20.3 Double-quantum Coherence EPR, Six-pulse Sequence

20.4 Four- and Five-pulse ‘Single-quantum Coherence’ PDS Sequences

20.5 Other Single-frequency PDS Methods

20.6 2D-FT Orientation-correlation PDS

20.7 Relaxation and Instantaneous Diffusion

20.8 Conclusions

Acknowledgments

References

Chapter 21: Shaped Pulses in EPR

21.1 Introduction

21.2 Specific Types of Pulses

21.3 Instrumentation

21.4 Applications of Shaped Pulses in EPR

21.5 Future Outlook

Acknowledgments

References

Part D: Special Techniques

Chapter 22: Pulse Techniques for Quantum Information Processing

22.1 Spin Qubits

22.2 Decoherence

22.3 Measuring Gate Fidelities

22.4 High-fidelity Operations

22.5 Sensing

22.6 Multiple Qubits

22.7 Conclusions

Acknowledgments

References

Chapter 23: Rapid-scan EPR

23.1 Introduction

23.2 Advantages of Rapid Scan Relative to Conventional CW Spectroscopy

23.3 Why Does Rapid Scan Give Improved

S/N

?

23.4 Hardware and Software Used in Rapid Scan

23.5 Parameter Selection

23.6 Extending Rapid Scan to Wider Spectra

23.7 Rapid Frequency Scans

23.8 Future

Acknowledgments

References

Chapter 24: EPR Microscopy

24.1 Preface

24.2 Introduction

24.3 Pulse EPR Microscopy: Theory

24.4 Pulse EPR Microscopy: Practice

24.5 Experimental Examples for EPRM Applications

24.6 Conclusions and Future Prospects

References

Chapter 25: Optically Detected Magnetic Resonance (ODMR)

25.1 Introduction

25.2 Instrumentation

25.3 ODMR of Triplet States in Molecules and Crystals

25.4 ODMR of Half-integer Spin Systems

25.5 ODMR of Interacting Spin Pairs

Acknowledgments

References

Chapter 26: Electrically Detected Magnetic Resonance (EDMR) Spectroscopy

26.1 Introduction

26.2 A Brief History of EDMR Spectroscopy

26.3 Electronic Mechanisms that Cause EDMR Signals

26.4 Continuous-wave EDMR

26.5 Pulse EDMR

26.6 Radio Frequency EDMR

26.7 Conclusions and Outlook

Acknowledgments

References

Chapter 27: Very-high-frequency EPR

27.1 Introduction

27.2 Benefits of VHF-EPR

27.3 VHF-EPR Spectrometers

27.4 Further Reading and Outlook

Acknowledgments

References

Index

End User License Agreement

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Guide

Cover

Table of Contents

Preface

Begin Reading

List of Illustrations

Chapter 1: Continuous-Wave EPR

Figure 1.1 Schematic diagram of a simple CW-EPR spectrometer.

Figure 1.2 Tuning mode patterns for different coupling regimes: (a) undercoupled; (b) critically coupled; and (c) overcoupled

Figure 1.3 Magnetic and electric field intensity distribution in a TE

102

rectangular resonator

Figure 1.4 Field modulation of an EPR absorption line. (a) A Gaussian absorption line and the modulated signal at several positions in the line. (b) The resulting derivative-like line shape produced by the lock-in amplifier

Figure 1.5 The influence of the field-modulation amplitude on the intensity and shape of the EPR spectrum. The spectra are of a sample of DPPH (2,2-diphenyl-1-picrylhydrazyl) measured with the field-modulation amplitudes given above each trace

Figure 1.6 The EPR signal intensity and line shape as a function of the microwave power. (a) Saturation curve calculated using equation (1.6) with

P

1/2

= 24.8 mW and

b =

1.44. (b) Spectra calculated using the steady-state solution of the Bloch equations with

T

1

=

T

2

= 50 ns and

ω

res

/

ħg

e

μ

B

= 352 mT

Figure 1.7 Calculated spectra of a nitroxide radical showing the effect of molecular motion as a function of the rotational correlation time

τ

c

. The spectra have been calculated using EasySpin

13

and the principal

g

values (

g

xx

= 2.0089,

g

yy

= 2.0061,

g

zz

= 2.0027) and

14

N hyperfine couplings (

A

xx

= 14.6 MHz,

A

yy

= 14.6 MHz,

A

zz

= 86.9 MHz) are taken from the literature

18

Figure 1.8 Characteristic powder patterns for

S

= 1/2 and the different possible symmetries of the

g

-tensor

Figure 1.9 Calculated spectra of copper(II) porphyrin showing the contribution of hyperfine coupling. (a) The spectrum in the absence of hyperfine coupling. (b) The spectrum with hyperfine coupling to the

I

= 3/2 copper nucleus included. (c) The spectrum with hyperfine coupling to Cu and four equivalent coordinating N atoms from the porphyrin ring. Principal

g

values of Cu(II): 2.03, 2.03, 2.19; principal values of the Cu hyperfine coupling (MHz): 75, 75, 608; N hyperfine coupling (isotropic) 40 MHz

Figure 1.10 Rhombogram and example spectrum for

S

= 3/2. The spectrum is calculated for

E/D

= 0.32 and

D

= −1.1 × 10

5

MHz. Microwave frequency = 9.75 GHz. The features at

g

eff

= 5.4 arise from the

m

S

= ±1/2 transition with the field along the

x

-direction and the

m

S

= ±3/2 transition with the field along

z.

The peak at

g

eff

= 1.93 is from

m

S

= ±1/2,

y

and

m

S

= ±3/2,

y

and

x.

The feature at

g

eff

= 1.43 is from

m

S

= ±1/2,

z

Chapter 2: EPR Interactions –

g

-Anisotropy

Figure 2.1 (a) Experimental continuous-wave EPR spectrum of azurin recorded at 20 K and at 275 GHz; (b) to (d). Simulated fields of resonance at 275 GHz as a function of the orientation of the magnetic field in the principal planes of the

g

-tensor, the

yz

-, the

xy

-, and the

zx

-plane with parameters:

g

x

= 2.0393,

g

y

= 2.0568,

g

z

= 2.273, intrinsic linewidth = 5 mT

Figure 2.2 Continuous-wave EPR spectra of the nitroxide radical TEMPOL in a frozen water solution at different microwave frequency/magnetic field settings. Spectra are plotted relative to the fixed

g

z

value.

Figure 2.3 (a) Wulffnet projection of the orientation of the principal axes of the

g

-tensors of the azurin molecules in the P2

1

2

1

2

1

unit cell with respect to the crystallographic

a, b,

and

c

axes. The open symbols indicate the directions of the

x

axes, the shaded symbols those of the

z

axes. The different symbols refer to the four molecules in the asymmetric unit, and a total of 16 molecules in the unit cell. All points lie at the front side of the globe. The direction of the crystallographic axes has been derived from the direction of the

g

-tensor axes as obtained by EPR. (b) The direction of the

g

-tensor axes in the copper site. The copper is bound to two histidine nitrogens, a cysteine sulfur, and a methionine sulfur

Figure 2.4 Continuous-wave EPR spectra at 275 GHz of TEMPONE in H

2

O/glycerol (50/50% v/v) as a function of temperature.

Figure 2.5 Simulations of the EPR spectrum of an

S

= ½ radical in the slow-motion regime at 275 and 36 GHz using Easyspin.

7

Conditions

g

x

= 2.0083,

g

y

= 2.0058,

g

z

= 2.0030, intrinsic linewidth = 0.1 mT; values of the rotation–correlation time,

τ

c

, are given in the figure

Figure 2.6 Multifrequency EPR spectra of 131R2-labeled T4 Lysozyme recorded in 25 w/v% Ficoll solution at 2, 12, 22, and 32 °C; experimental data (solid lines), simulations (dashed lines).

Figure 2.7 Continuous-wave EPR spectrum of a powder sample of at X-band and 8 K

Figure 2.8 Schematic representation of the atomic orbitals involved in the singly occupied molecular orbitals for the NO bond in the ground state and the excited state: the 2p

z

orbitals on nitrogen and oxygen and the lone-pair orbitals

n

and

n

′ on oxygen

Figure 2.9 Schematic representation of the atomic orbitals involved in the singly occupied molecular orbital for the CuS bond in the ground state: the 3d

xy

orbital on copper and the 3p

y

orbital on sulfur. For azurin, the copper is less than 0.1 Å above the NNS (

xy

)-plane, and the drawing is a projection on this plane

Chapter 3: EPR Interactions – Zero-field Splittings

Figure 3.1 X-band EPR spectrum of a powder of [Ni(NH

3

)

6

]I

2

(black trace) recorded at 77 K, 9.235 GHz, with simulation (red trace) generated using

S

′ = 1/2,

g

= 2.126,

W

= 800 MHz (Lorentzian, hwhm)

Figure 3.2 Relative energy levels in units of

D

of spin triplets through sextets as a result of rhombic ZFS (|

E

/

D

| = 0.1). The ordering is for positive

D

; for negative

D

, the order would be reversed. The energy values for triplets through quintets are exact; those for the sextet use second-order perturbation theory but are accurate to within a few percent of an exact calculation

Figure 3.3 Diagrams of energy levels (eigenvalues) as a function of external magnetic field for a spin triplet with the following spin Hamiltonian parameters:

S

= 1,

g

iso

= 2.00,

D

/

hc

= +0.15 cm

−1

, upper plot:

E

= 0; lower plot:

E

/

hc

= +0.025 cm

−1

. Black lines indicate energies with the magnetic field along the ZFS

z

-axis (parallel) in both plots; in the upper plot, blue traces indicate energies with the magnetic field along the ZFS

x

-axis (=

y-

axis; perpendicular); in the lower plot, green traces are for the field along the ZFS

x

-axis and red traces for the field along the ZFS

y

-axis. Arrows indicate allowed EPR transitions with a microwave quantum of 9.3 GHz (0.31 cm

−1

), with their colors corresponding to the transition orientation

Figure 3.4 X-band (9.3 GHz) EPR transition energies calculated using EasySpin for a spin triplet with

g

= 2.00,

D

/

hc

= 0.15 cm

−1

as a function of the angle

θ

between the external magnetic field and the ZFS tensor

z-

direction. The correspondence between transitions labels

z

1

(red trace) and

z

2

(blue trace) and Δ

m

S

= ±1 transitions (where this is a good quantum number, i.e., at

θ

= 0°) is indicated

Figure 3.5 Simulated EPR spectra for a spin triplet with

g

iso

= 2.00, axial ZFS,

D

/

hc

= +0.15 cm

−1

; with rhombic splitting (

E

/

hc

= +0.015 cm

−1

; green traces) and without rhombic splitting (red traces). Single-crystal Gaussian linewidths (hwhm) of 100 MHz were used. A typical X-band microwave frequency of 9.3 GHz (≈0.31 cm

−1

) was used. Both an absorption presentation (lower traces) and a first derivative (upper traces) are shown. The turning points are labeled using standard nomenclature for triplets.

29

The relation between splittings of some of the turning points and ZFS parameters is also indicated. These splittings are shown in magnetic field quantities, not in energy, and thus need to be converted to energy quantities using

gμ

B

as described in the text

Figure 3.6 Simulated EPR spectra (absorption presentation) for spin triplets with axial ZFS. The

D

/

hc

values (in cm

−1

) are indicated in the figure; isotropic

g

= 2.00; single-crystal Gaussian linewidths (hwhm) of 100 MHz; microwave frequency 9.3 GHz (≈0.31 cm

−1

). The spectra are calculated using a powder pattern with the single-crystal resonant fields along the canonical ZFS tensor axes indicated by red bars. The turning points are labeled using a standard nomenclature for triplets

Figure 3.7 Diagrams of energy levels (eigenvalues) as a function of external magnetic field for a spin quartet with the following spin Hamiltonian parameters:

S

= 3/2,

g

iso

= 2.00,

D/hc

= +0.15 cm

−1

, upper plot:

E

= 0; lower plot:

E/hc

= +0.025 cm

−1

. Black lines indicate energies with the magnetic field along the ZFS

z

-axis (parallel) in both plots; in the upper plot, blue traces indicate energies with the magnetic field along the ZFS

x

-axis (perpendicular); in the lower plot, green traces are for the field along the ZFS

x

-axis and red traces for the field along the ZFS

y

-axis. Arrows indicate allowed EPR transitions with a microwave quantum of 9.3 GHz, with their colors corresponding to the transition orientation

Figure 3.8 Simulated EPR spectra for a spin quartet with

g

iso

= 2.00, axial ZFS,

D

/

hc

= +0.10 cm

−1

; with rhombic splitting (

E

/

hc

= +0.010 cm

−1

; green traces) and without rhombic splitting (red traces). Single-crystal Gaussian linewidths (hwhm) of 100 MHz were used. A typical X-band microwave frequency of 9.3 GHz (≈0.31 cm

−1

) was used. Both an absorption presentation (lower traces) and a first derivative (upper traces) are shown. The turning points are labeled using standard nomenclature for quartets; green labels are unique to the rhombic case; red labels are unique to the axial case; and black labels are common to both. The asterisk indicates an off-axis turning point (Figure 3.9). The approximate relation between splittings of the turning points and ZFS parameters is also indicated. These splittings are shown in magnetic field quantities, not in energy, and thus need to be converted to energy quantities using

gμ

B

as described in Section 3.2.1.3

Figure 3.9 X-band (9.3 GHz) EPR transition energies calculated using EasySpin for a spin quartet with

g

= 2.00,

D

/

hc

= 0.10 cm

−1

as a function of the angle

θ

between the external magnetic field and the ZFS tensor

z

-direction. The horizontal dashed line indicates the angle (

θ

≈ 39°) where there is significant spectral intensity that does not correspond to either a parallel (

θ

= 0°) or a perpendicular (

θ

= 90°) turning point, i.e., an off-axis resonance, occurring at d

B

/d

θ

= 0, as is also the case at the canonical values. The correspondence between transitions labels

z

1

(green trace),

z

2

(red trace), and

z

3

(blue trace) and Δ

m

S

= ±1 transitions (where this is a good quantum number, i.e., at

θ

= 0°) is indicated. The unlabeled traces (cyan, violet, and orange) are less easily classified and give rise to the lower intensity transitions as seen at low field in Figure 3.8

Figure 3.10 Simulated EPR spectra (absorption presentation) for spin quartets with

g

iso

= 2.00, axial ZFS. The

D

/

hc

values (in cm

−1

) are indicated; single-crystal Gaussian linewidths (hwhm) of 100 MHz; microwave frequency 9.3 GHz (≈0.31 cm

−1

). The spectra are calculated using a powder pattern with the single-crystal resonant fields along the canonical ZFS tensor axes indicated by red bars. The turning points are labeled using standard nomenclature; an off-axis turning point (see text) is indicated by an asterisk

Figure 3.11 Simulated EPR spectra for a spin quartet with axial ZFS,

D

/

hc

= +3.0 cm

−1

(

E

= 0;

g

= 2.00) using both 35 GHz (green traces; 2 T field sweep) and 330 GHz (red traces; 20 T field sweep). Single-crystal Gaussian linewidths (fwhm) of 300 MHz were used for the 35-GHz spectrum and 3000 MHz for the 330-GHz spectrum. Both an absorption presentation (lower traces) and a first derivative (upper traces) are shown. All intensities are arbitrarily scaled. The turning points are labeled using standard nomenclature for quartets. The 35- GHz spectrum is not labeled; its perpendicular (lower field) feature is

xy

3

and the parallel feature (higher field) is

z

1

Figure 3.12 Diagrams of energy levels (eigenvalues) as a function of external magnetic field for a spin quintet with the following spin Hamiltonian parameters:

S

= 2,

g

iso

= 2.00,

D

/

hc

= +0.15 cm

−1

, upper plot:

E

= 0; lower plot:

E

/

hc

= +0.025 cm

−1

. Black lines indicate energies with the magnetic field along the ZFS

z

-axis (parallel) in both plots; in the upper plot, blue traces indicate energies with the magnetic field along the ZFS

x

-axis (perpendicular); in the lower plot, green traces are for the field along the ZFS

x

-axis and red traces for the field along the ZFS

y

-axis. Arrows indicate allowed EPR transitions with a microwave quantum of 9.3 GHz (≈0.31 cm

−1

), with their colors corresponding to the transition orientation

Figure 3.13 Simulated EPR spectra for a spin quintet with

g

iso

= 2.00, axial ZFS,

D

/

hc

= +0.10 cm

−1

; with rhombic splitting (

E

/

hc

= +0.010 cm

−1

; green traces) and without rhombic splitting (red traces). Single-crystal Gaussian linewidths (hwhm) of 100 MHz were used. A typical X-band microwave frequency of 9.3 GHz (≈0.31 cm

−1

) was used. Both an absorption presentation (lower traces) and a first derivative (upper traces) are shown. The turning points are labeled using standard nomenclature for quintets; green labels are unique to the rhombic case; red labels are unique to the axial case; and black labels are common to both. The asterisks indicate off-axis turning points (Figure 3.14). The approximate relation between splittings of the turning points and ZFS parameters is also indicated. These splittings are shown in magnetic field quantities, not in energy, and thus need to be converted to energy quantities using

g

μ

B

as described in the text

Figure 3.14 X-band (9.3 GHz) EPR transition energies calculated using EasySpin for a spin quintet with

g

= 2.00,

D

/

hc

= 0.10 cm

−1

as a function of the angle

θ

between the external magnetic field and the ZFS tensor

z

-direction. The horizontal dashed lines indicate the angles (

θ

≈ 18° (transition 3, cyan trace) and 59° (transition 2, red trace)) where there is significant spectral intensity that does not correspond to either a parallel (

θ

= 0°) or a perpendicular (

θ

= 90°) turning point, i.e., an off-axis resonance, occurring at d

B

/d

θ

= 0, as is also the case at the canonical values. The correspondence between transitions labels

z

1

(green trace),

z

2

(red trace),

z

3

(cyan trace), and

z

4

(blue trace) and Δ

m

S

= ±1 transitions (where this is a good quantum number, i.e., at

θ

= 0°) is indicated

Figure 3.15 Simulated EPR spectra (absorption presentation) for spin quintets with

g

iso

= 2.00, axial ZFS. The

D

/

hc

values (in cm

−1

) are indicated; single-crystal Gaussian linewidths (hwhm) of 100 MHz; microwave frequency 9.3 GHz (≈0.31 cm

−1

). The spectra are calculated using a powder pattern with the single-crystal resonant fields along the canonical ZFS tensor axes indicated by red bars. The turning points are labeled using standard nomenclature; an off-axis turning point (see text) is indicated by an asterisk

Figure 3.16 Diagrams of energy levels (eigenvalues) as a function of external magnetic field for a spin sextet with the following spin Hamiltonian parameters:

S

= 5/2,

g

iso

= 2.00,

D

/

hc

= +0.15 cm

−1

, upper plot:

E

= 0; lower plot:

E

/

hc

= +0.025 cm

−1

. Black lines indicate energies with the magnetic field along the ZFS

z

-axis (parallel) in both plots; in the upper plot, blue traces indicate energies with the magnetic field along the ZFS

x-

axis (perpendicular); in the lower plot, green traces are for the field along the ZFS

x

-axis and red traces for the field along the ZFS

y

-axis. Arrows indicate allowed EPR transitions with a microwave quantum of 9.3 GHz (≈0.31 cm

−1

), with their colors corresponding to the transition orientation

Figure 3.17 X-band (9.3 GHz) EPR transition energies calculated using EasySpin for a spin sextet with

g

= 2.00,

D

/

hc

= 0.10 cm

−1

as a function of the angle

θ

between the external magnetic field and the ZFS tensor

z

-direction. The horizontal dashed lines indicate the angles (

θ

≈ 69° (transition 2, cyan trace), 36° (transition 3, green trace), and 10° (transition 4, not shown)) where there is significant spectral intensity that does not correspond to either a parallel (

θ

= 0°) or a perpendicular (

θ

= 90°) turning point, i.e., an off-axis resonance, occurring at d

B

/d

θ

= 0, as is also the case at the canonical values. The correspondence between transitions labels

z

1

(orange trace),

z

2

(cyan trace),

z

3

(green trace),

z

4

(red trace), and

z

5

(blue trace) and Δ

m

S

= ±1 transitions (where this is a good quantum number, i.e., at

θ

= 0°) is indicated

Figure 3.18 Simulated EPR spectra for a spin sextet with

g

iso

= 2.00, axial ZFS,

D/hc

= +0.04 cm

−1

, and with rhombic splitting (

E/hc

= +0.004 cm

−1

; green trace) and without rhombic splitting (red trace). Single-crystal Gaussian linewidths (fwhm) of 100 MHz were used. A typical X-band microwave frequency of 9.3 GHz was used. Both an absorption presentation (lower traces) and a first derivative (upper traces) are shown. The turning points are labeled using standard nomenclature; green labels are unique to the rhombic case; red labels are unique to the axial case, and black labels are common to both. The asterisk indicates an off-axis feature. The approximate relation between splittings of the turning point and ZFS parameters is also indicated. These splittings are shown in magnetic field quantities, not in energy, and thus need to be converted to energy quantities using

g

μ

B

as described in the text

Figure 3.19 Simulated EPR spectra (absorption presentation) for spin sextets with

g

iso

= 2.00, axial ZFS. The

D

values (in cm

−1

) are indicated in the figure; single-crystal Gaussian linewidths (hwhm) of 100 MHz; microwave frequency 9.3 GHz (≈0.31 cm

−1

). The spectra are calculated using a powder pattern with the single-crystal resonant fields along the canonical ZFS tensor axes indicated by red bars. The turning points are labeled using standard nomenclature; the most prominent off-axis features (see text) are indicated by an asterisk

Figure 3.20 X-band (

ν

= 9.68 GHz) EPR spectrum (black trace) of the triplet state of a semisynthetic chlorophyll

a

derivative, zinc methyl 3-ethyl-pyrochlorophyllide, with simulation (red trace). The simulation uses

S

= 1,

g

= 2.00285,

D

/

h

= 916 MHz,

E

/

h

= 110 MHz (

D

/

hc

= 0.0306 cm

−1

E

/

hc

= 0.0037 cm

−1

), peak-to-peak linewidth = 1.494 mT, with the absorption/emission appearance, a consequence of the triplet's polarization, achieved using the experimental temperature function of EasySpin.

Figure 3.21 HFEPR spectra measured for a six-coordinate Ni(II) complex (see the text for description) as a powder at 10 K with the microwave frequencies 216.0 and 295.2 GHz. The simulations (colored traces) use

S

= 1,

g

x

= 2.138,

g

y

= 2.156,

g

z

= 2.188, |

D

/

hc

| = 3.262 cm

−1

, and |

E

/

hc

| = 0.1094 cm

−1

. Simulations with positive (blue traces) and negative (red traces)

D

and

E

values are shown, which indicate a better match of intensities using negative ZFS parameters. Resonances marked with O

2

are due to solid dioxygen adsorbed on the powder sample, which is typically observed in HFEPR below 30 K. DQ marks the ‘double-quantum transition’ (see Section 3.2.1.3), characteristic of Ni(II) systems.

117

Figure 3.22 HFEPR spectrum measured for a six-coordinate Mn(IV) complex (see the text for description) as a powder at 4.5 K with the microwave frequency 151.2 GHz. Experimental (black trace) and simulated (colored traces). The simulations (colored traces) use

S

= 3/2,

g

iso

= 1.98, |

D

/

hc

| = 0.98 cm

−1

, and |

E

/

hc

| = 0.056 cm

−1

. Simulations with positive (blue traces) and negative (red traces)

D

and

E

values are shown, which indicate a better match of intensities using negative ZFS parameters. The spectra are approximately normalized to the amplitude of the two central, strongest resonances.

Figure 3.23 HFEPR spectra of powder

[

Cr(N(TMS)

2

)

2

(L)

2

], L = py (left) and L = thf (right) (black traces) with their simulations (colored traces) at 4.2 K with the frequencies indicated. Red traces are spectra simulated using positive

D

values, while blue traces are spectra simulated using negative

D

values. Simulation parameters for L = py:

S

=2, |

D

/

hc

| = 1.80 cm

−1

, |

E

/

hc

| = 0.020 cm

−1

,

g

x

= 1.97,

g

y

= 1.98; for L = thf:

S

= 2, |

D

/

hc

| = 2.00 cm

−1

, |

E

/

hc

| = 0.025 cm

−1

,

g

x

= 1.98,

g

y

= 1.99. Precise values for

g

z

could not be determined but the range 1.0 <

g

z

< 2.0 gave acceptable fits.

Figure 3.24 Experimental (black traces) and simulated (red traces) multifrequency EPR spectra for frozen aqueous Na[Fe(EDTA)] at pH 5, 40 K (10 K for 275 GHz spectrum), and the microwave frequencies indicated. (a) Simulations with the standard

D

-strain approximation built into EasySpin. (b) Simulations with the distributions of the ZFS parameters optimized from the grid-of-errors method.

73

Spin Hamiltonian parameters for the simulations:

S

= 5/2,

g

x

= 2.005,

g

y

= 2.007,

g

z

= 2.004;

D

x

/

h

= 0.75 GHz,

D

y

/

h

= 14.45 GHz,

D

z

/

h

= −15.20 GHz (corresponding to

D

/

h

= −22.80 GHz (

D

/

hc

= 0.76 cm

−1

),

E

/

h

= −6.85 GHz (

E

/

hc

= 0.23 cm

−1

)). The strain in the ZFS was taken into account as Gaussian distributions in the

D

and

E

parameters with fwhm of 4.3 and 1.7 GHz, respectively.

Figure 3.25 Experimental K

a

-band (

ν

≈ 34 GHz) and W-band (

ν

≈ 95 GHz) electron-spin-echo-detected EPR spectra of a Gd(III) complex (black traces) in frozen aqueous solution and simulations (red traces) with

D/gμ

B

= 37 and 39 mT (respectively, K

a

-band and W-band; their average is

D

/

hc

≈ 0.035 cm

−1

). The sharp peak is from the |7/2, ± 1/2 ↔ |7/2, 1/2 transition; the other fine structure transitions are broadened out owing to a distribution of the ZFS parameters (

D

-strain). (Reprinted with permission from A. M. Raitsimring, C. Gunanathan, A. Potapov, I. Efremenko, J. M. L. Martin, D. Milstein, and D. Goldfarb, J. Am. Chem. Soc., 2007, 129, 14138. Copyright 2007 American Chemical Society)

Chapter 4: EPR Interactions – Coupled Spins

Figure 4.1 (a) Total spin states arising from isotropic exchange between two

s

= ½ paramagnets according to Hamiltonian (4.1), and effect of an applied magnetic field. (b) States arising from an isosceles triangle of three interacting

s

= ½ paramagnets as drawn in the scheme and in Hamiltonian (4.8). The states are labeled according to |

s

12

S

; if

J

1

=

J

2

then the two

s

= ½ states are degenerate

Figure 4.2 X-band EPR of a powder of [ClCNSSS](AsF

6

), which exists as a head-to-head dimer in the solid state, at 290 to 180 K in 10 K increments. The spin triplet spectrum arises from the excited multiplet, and the singlet–triplet separation can be determined by modeling its intensity (e.g., the

z

orientation transitions marked *) as a function of temperature (giving 2

J

= −1900 cm

−1

). The central feature is due to

s

= ½ lattice defects (where half a dimer is missing; about 0.1%)

Figure 4.3 322 GHz CW EPR spectra of [Cr

2

Zn

2

(NCS)

4

(dea)

2

(Hdea)

2

] (Hdea = diethanolamine) as a powder. EPR transitions are observed within the

S

= 1 (labeled T for triplet), 2 (Q for quintet), and 3 (S for septet) excited multiplets of the pair (2

J

= −14 cm

−1

; X, Y, Z denote the molecular orientations, with respect to the magnetic field, giving rise to the transitions). At 5.5 K, only the lowest excited multiplet (

S

= 1) is observed due to Boltzmann depopulation effects.

Figure 4.4 349 GHz CW. EPR spectrum of a powder of [Mn

12

O

12

(MeCO

2

)

16

(H

2

O)

4

]. The molecule has an

S

= 10 ground state with

D

≈ −0.5 cm

−1

and at 5 K only the lowest

M

states (= ±10) within this multiplet are populated significantly. At 5 K, the lowest field transition corresponds to

M

= −10 → −9 for the

z

orientation of the molecule (defined by the ZFS tensor). On warming, higher

M

states are populated and the

M

= −9 → −8, etc., transitions become observed. In the high-field limit (which is not quite reached here), the

z

resonance fields would be given by where

M

is the upper state in the transition.

Figure 4.5 (a) Energy level diagrams for two weakly coupled

s

= ½ radicals with different

g

values (

g

1

>

g

2

), for a fixed magnetic field

B

, with allowed EPR transitions and energies. (b) The resulting EPR (stick) spectra. The left picture is for |

J

| = 0; the right picture is for |

J

| ≠ 0 but (

g

1

−

g

2

)

μ

B

B

Figure 4.6 (a) X-band EPR spectrum of a weakly coupled Cu…nitroxide complex in frozen solution with stick spectrum highlighting exchange splitting of the nitroxyl

g

z

features (dotted line: simulation). (Reprinted from J. Magn. Reson., 52, S. Eaton, K. M. More, B. M. Sawat, M. Sawant, P. M. Boumel and G. R. Eaton, Metal-nitroxyl interactions. 29. EPR studies of spin-labeled copper complexes in frozen solution, 435. © 1983, with permission from Elsevier) (b) Q-band spectrum of a powder of a {Cr

7

Ni}…Cu species at 5 K: experimental (top), and calculated for the uncoupled limit (bottom). The exchange splitting of the

g

x

,

y

features of the Cu and Cr

7

Ni resonances into doublets is highlighted by sticks. The cyclic {Cr

7

Ni} cluster (

s

= ½ ground state) is represented as an octagon

Figure 4.7 (a) Calculated 9.5 GHz EPR spectra for two dissimilar

s

= ½ spins (isotropic

g

1

= 2.1,

g

2

= 2.0) as a function of |

J

|. From bottom to top: |

J

|/Δ

gμ

B

B

= 0, 0.1, 0.5, 1, and 3 (

B

= 0.32 T). (b) Energy level diagram (as a function of the magnetic field) for |

J

|/Δ

gμ

B

B

= 0.1 (bottom) and 3 (top). The former is an AX spin system with four allowed EPR transitions (arrows) associated with

g

1

and

g

2

. The latter behaves as a spin singlet and triplet, with mixing induced by the Zeeman terms leading to formally forbidden transitions (dashed arrows)

Figure 4.8 Calculated X-band solution EPR spectra (2 mT linewidth) for two

s

= ½ radicals with

g

1

=

g

2

, a single

14

N nuclear spin each with isotropic hyperfine coupling constant of |

a

iso

| = 44 MHz (typical of a nitroxyl), and |

J

| of (a) 1 MHz, (b) 100 MHz, and (c) 1 GHz

Figure 4.9 Spin states at zero field for a coupled

s

= 5/2, with axial and positive

D

, and

s

= ½ pair with

J

= 0 (uncoupled; center), 2

J

= −5

D

(approaching the strong exchange limit, such that states can be grouped as total spin multiplets

S

; left), 2

J

= −

D

/5 (weak exchange; right). The parentheses show the degeneracies of the states.

Figure 4.10 Zeeman diagram for an antiferromagnetically coupled isosceles triangle of

s

= ½, with the magnetic field in the

xy

-plane. Dashed lines show the effect of a small

d

z

( |

J

|) antisymmetric exchange leading to mixing between the bottom two doublets. The EPR transition in the lowest Kramers doublet (arrows) gives a small effective

g

Chapter 5: EPR Interactions – Hyperfine Couplings

Figure 5.1 Interaction of two classical magnetic dipoles

μ

1

and

μ

2

at a finite distance

r

in an external magnetic field

B

0

. The field produced by

μ

1

is shown with field lines marked by arrows.

θ

is the angle between the vector

r

interconnecting the dipoles and the direction of the external field

Figure 5.2 Energy levels and resonance position of allowed (solid lines) and forbidden (dashed lines) EPR transitions in the high-field limit according to equations (5.16) and (5.17) for an

S

= ½ electron spin coupled to a

I

= ½ nuclear spin with positive

g

n

Figure 5.3 (a): CW EPR spectrum of the PNT radical in solution and (b) CW ENDOR spectrum.

Figure 5.4 (a): Structure of an artificial, guanine-rich oligonucleotide stabilized by a terminal tetra-pyridine complexation with a Cu(II) ion. (b) The nature of the metal coordination was provided by the characteristic X-band CW EPR spectrum recorded at 70 K. Spectral regions with visible hyperfine splittings are marked.

Figure 5.5 Derivatives of echo-detected EPR spectra of a 3-aminotyrosyl radical trapped during enzymatic reaction of

E. coli

ribonucleotide reductase recorded at three different EPR frequencies. Dashed lines indicate simulations with the hyperfine parameters reported in Ref. 14. Spectra are aligned such that

g

≈ 2.002 overlay. Buffer was exchanged to D

2

O and amino protons do not contribute in the spectrum. Two detected radical conformations are shown in the top right inset in black and gray, respectively. Orientations of the C

β

-protons with respect to phenol ring are shown schematically (right, middle). Dependence of the isotropic C

β

-proton hyperfine couplings on the C

α

—C

β

—C

1

—C

6

dihedral angle (

θ

C

β

) was calculated from a density functional theory DFT model for a 3-aminotyrosyl radical (right, bottom). Top left: Chemical structure of a 3-aminotyrosyl radical

Figure 5.6 Structure of the semidione radical intermediate formed on the ms timescale during the reaction of a E441Q-

α

mutant of

E. coli

RNR with substrate CDP and allosteric effector TTP. The box around the radical represents the active site pocket with the essential groups involved in chemistry. Numbering of the

13

C atoms is shown.

Figure 5.7 Spin energy levels for the ground-state electron in the hydrogen atom at low magnetic fields. The isotropic hyperfine constant is

a

iso

/

h

≈ 1.4 GHz

22

. In the high-field limit, level 1 will cross level 2 at a field around 16 T

Figure 5.8 CW EPR spectrum of the vanadyl acetylacetonate complex in toluene solution at room temperature at X-band frequencies. Note also the line width and intensity variation due to incomplete motional averaging.

24

Figure 5.9 Vector representations of the nuclear effective fields

ω

12

(in the electron spin manifold

α

e

) and

ω

34

(in the electron spin manifold

β

e

) and their direction with respect to the nuclear Zeeman quantization axis

z

from equation (5.29). The coordinate system refers to directions defined by the spin vector components and for the

α

e

manifold (a) and and for manifold

β

e

(center and right). For

β

e

,

ω

I

and

A

have different signs in equation (5.31), and two cases are distinguished, i.e.,

ω

I

>

A

/2 or

ω

I

<

A

/2 (b and c), respectively. In all cases, the effective fields are simply the vectorial sum of the field components along the

z

and

x

axes, respectively. The relevant angles, either

η

α

or

η

β

,

are those between the effective field and the

z-

axis

Chapter 6: EPR Interactions – Nuclear Quadrupole Couplings

Figure 6.1 Illustration of the relation between the nuclear quadrupole moment

Q

, the effective shape of the nuclear charge distribution, and the nuclear spin quantum number

I

, with examples of some

I

> 1/2 isotopes often investigated in EPR. The vertical dashed line indicates the axis of cylindrical symmetry

Figure 6.2 Illustration of the orientation dependence of the energy of a quadrupolar nucleus (here,

Q

> 0) in an external inhomogeneous electric field produced by a set of negatively charged particles (electrons). The arrows indicate the torque

Figure 6.3 Relative orientation of the nuclear quadrupole tensor and the effective magnetic field experienced by a nucleus. The ellipsoid illustrates the quadrupole tensor for

Q

> 0, and

x

,

y

, and

z

indicate its principal axes.

θ

is the angle between the

z

axis and the effective magnetic field direction

n

,

ϕ

is the angle between the

x

axis and the projection of the effective magnetic field onto the

xy

plane

Figure 6.4 Schematic illustration of first-order energy level shifts due to the quadrupole interaction in the high-field regime. All diagrams assume

P

> 0,

η

= 0, and a nucleus with

g

n

> 0. The magnetic field is assumed parallel to the

z

axis of the NQI tensor

Figure 6.5 Zero-field energy levels and transitions of a spin-1 nucleus (a), a spin-3/2 nucleus (b), and a spin-5/2 nucleus (c). The plot assumes

K

> 0 and 0 ≤

η

≤ 1. For

K

< 0, the order of the levels is inverted

Figure 6.6 Graphical method to determine the three nuclear transition frequencies for an

I

= 1 nucleus for a given orientation (

θ

,

ϕ

) under the assumption of an isotropic hyperfine coupling. Left: function

g

(

θ

,

ϕ

) = (3 −

η

cos 2

ϕ

)cos

2

θ

+

η

cos 2

ϕ

− 1, right: function with

x

=

ν

/

K

,

w

=

ν

eff

/

K

, , and

ν

eff

= |

ν

I

+

m

S

a

iso

|. All

g

(

θ

,

ϕ

) values lie between (−1 −

η

) and 2.

F

(

ν

/

K

) is defined for . Specific values used for the plot:

ν

I

= 1.08 MHz (

14

N at 350 mT),

a

iso

= 4 MHz,

m

S

= + 1/2,

K

= 0.7 MHz,

η

= 0.3. Top right: powder nuclear spectrum showing the two single-quantum and the double-quantum transitions. To determine the nuclear frequencies for a given (

θ

,

ϕ

), draw a horizontal line from the corresponding grid point on the left across

F

(

ν

/

K

) on the right. The nuclear frequencies are at the

ν

/

K

positions where this line intersects the

F

(

ν

/

K

) curve. This is shown for the three principal directions

x

,

y,

and

z

, as well as for (

θ

,

ϕ

) = (35

°

, 50

°

)

Figure 6.7 Valence orbitals used in the Townes–Dailey analysis for an uncoordinated imino nitrogen such as in imidazole or pyridine. (a) In-plane lone-pair orbital, (b) two in-plane σ-bonding orbitals, and (c) π orbital perpendicular to the paper plane. (d) Definition of local coordinate system

Figure 6.8 (a) The structure of Cu(dtc)

2

, (b) experimental and simulated X-band spectrum of Cu(dtc)

2

in pyridine.

20

(Reprinted with permission from Liczwek, D. L.; Belford, R. L.; Pilbrow, J. R.; Hyde, J. S., Evaluation of Copper Nuclear-Quadrupole Coupling in Thio Complexes by Completion of the Coordination Sphere. J. Phys. Chem. 1983, 87, 2509–2512. Copyright 1983 American Chemical Society) (c) Simulated 9.1077 GHz spectra using EasySpin

21

with (

g

x

,

g

y

,

g

z

) = (2.030 2.033 2.120), (

A

x

,

A

y

,

A

z

) = (−43, −76, 423) MHz, (

P

x

,

P

y

,

P

z

) = (−7.17, −7.17, +14.34) MHz, and all tensors were taken as collinear. The blue spectrum was calculated with (

P

x

,

P

y

,

P

z

) = (0,0,0)

Figure 6.9 (a) Schematic structure of Cu

2

RGT. The exchangeable protons are indicated in red. (b) The W-band echo-detected EPR spectrum of Cu

2

RGT measured at 6.5 K, (c)

2

H-Mims W-band ENDOR spectrum of Cu

2

RGT/D

2

O recorded at

g

||

(

B

0

= 3.386 T), labeled in (b), and the corresponding simulations (red lines) obtained with the parameters (

A

x

,

A

y

,

A

z

) = (−0.8, −0.66, −0.2) MHz,

e

2

qQ

/

h

= 220 kHz,

η

= 0.14 and the appropriate orientations of the hyperfine and quadrupole tensors. For convenience, the intensity was multiplied by −1. * marks an impurity signal.

Figure 6.10 (a) The Gd-DOTA complex. (b) W-band echo-detected EPR spectrum of Gd-DOTA in the region of the central transition recorded at 10 K. (c) W-band

14

N Mims ENDOR spectrum of Gd-DOTA, recorded at 10 K and a magnetic field of 3407 mT, with assignments of the hyperfine,

A

, and the quadrupolar, 3

P

, splittings. The spectral features marked with * correspond to contributions attributed to the

m

S

= ±3/2 electron spin manifolds, while all other features are due to the

m

S

= ±1/2 electron spin manifolds (central transition). The blue curve represents the simulation of the ENDOR frequencies due to the central transition with the parameters presented in the text, the green curve represents the simulation of the contribution from the

m

S

= ±3/2 electron spin manifolds (see text), and the red trace is their sum. For the simulations, the anisotropic hyperfine coupling,

T

⊥

= 0.30 MHz, was derived from an average Gd-N distance of 2.66 Å obtained from the crystal structure of Gd-DOTA,

27

the isotropic part was

a

iso

= −0.37 MHz,

e

2

Qq

/

h

= 4.31 MHz, and

η

= 0. At this field

ν

I

= 10.6 MHz.

Figure 6.11 Three-pulse ESEEM spectra from

Dv

[NiFe]-hydrogenase (top) and

Re

MBH (bottom) in the Ni

r

–B state, recorded at

g

= 2.16. Experimental conditions:

T

= 8 K, microwave frequency 9.7 GHz. From the frequencies of the peaks, the values

e

2

Qq

/

h

= 1.94 MHz,

η

= 0.38, and

A

= 1.4–1.6 MHz were derived for

Re

MBH and

e

2

Qq

/

h

= 1.90 MHz,

η

= 0.37, and

A

= 1.4–1.6 MHz for

Dv

H

2

ase. The

14

N Larmor frequency at this field is 0.99 MHz. Inset: Structure of histidine.

Figure 6.12 Correlation plot of

14

N quadrupolar parameters of

14

N interacting with protein-bound semiquinones as measured by ESEEM (black dots) and

14

N nuclear quadrupole resonance spectroscopy of amino acids and peptides (gray dots). (H/W) Indole, tryptophan, and histidine N

ε

nitrogen; (Q/N) glutamine and asparagine NH

2

nitrogen; (B) backbone nitrogen, peptide, di- and tripeptide, triglycine; (Am) and (Am′) NH

3

+

amino group nitrogen; (K) NH

3

+

lysine nitrogen; (P) proline nitrogen; (H) histidine N

δ

nitrogen; and (R) arginine N

ε

.

Figure 6.13 Inset, X-band (9.2325 GHz) EPR spectrum of [4Fe–4S]

+

aconitase in the presence of citrate. The field position and

g

value at which the ENDOR spectrum was recorded is indicated with an arrow. Main figure,

17

O ENDOR spectrum at

g

= 1.88 for this sample in H

2

17

O solution, recorded at 365 mT and 2 K.

Figure 6.14 (a) Structure of [Mo

17

O(SPh)

4

]

1−

. (b) Echo-detected EPR spectrum (MW frequency 29.372 GHz). (c) Cosine Fourier transforms (low-frequency part) of integrated four-pulse ESEEM obtained at magnetic fields A, B, and C, respectively (see (b)). The sum combination line is a quintet and indicated by . Dashed traces are the corresponding simulated spectra of the integrated four-pulse ESEEM. and correspond to the nuclear frequencies in the α and β electron spin manifolds for which Δ

m

I

= 1, and is their sum (see equation (6.24)).

33

Figure 6.15 W-band Davies

27

Al ENDOR spectrum of dehydrated Cu

2+

exchanged faujasite with Si/Al = 1 measured at the

g

||

field position (

g

= 2.351, 2.8799 T) and 5 K. (a) The blue and red sticks represent the two quintets of the two electron spin manifolds. (b) A schematic of the local environment of the Cu

2+

ion.

Figure 6.16 (a) Chemical structure of VO(acac)

2

with position X for an additional axial ligand and an X-band (10.297 GHz) echo-detected EPR spectrum (15 K) of VO(acac)

2

without X; (b) Davies ENDOR of VO(acac)

2

taken at the

m

I

= −7/2 hyperfine component (left) and the

m

I

= +7/2 hyperfine component (right), as indicated in the EPR spectrum in (a).

Figure 6.17 Energy level diagram for vanadyl (

S

= 1/2,

I

= 7/2), where

β

represents the EPR transition for

m

I

= −7/2 and a and d represent the associated ENDOR transitions. The EPR transition for

m

I

= +7/2 is represented by

α

, where b and c represent the associated ENDOR transitions.

Chapter 7: Quantum Chemistry and EPR Parameters

Figure 7.1 (a) Numbering scheme for a derivatized phenoxyl radical. (b) Spin-density distribution contoured at ±0.003 electrons/

a

0

(red = positive spin-density, yellow = negative spin-density)

Figure 7.2 (a) Lack of correlation between total s-orbital spin population and hyperfine coupling for nitrogen in a series of square-planar Cu(II)N

4

complexes. (b) Good correlation between calculated and experimental HFCs for the same series of complexes, thus demonstrating that the HFC is not determined by the spin population (iz = imidazole, py = pyridine, en = ethylene-diamine, gly = glycine)

Figure 7.3 Singly occupied molecular orbital of the CH

3

radical contoured at ±0.05 (electrons/

a

0

3

)

1/2

. (b) Spin density of the CH

3

radical contoured at ±0.003 electrons/

a

0

3

. Red = positive, yellow = negative. It is evident that there is negative spin density at the hydrogen positions although the SOMO has no amplitude there. (B3LYP/cc-pCVTZ)

Figure 7.4 Electronic structure of the H

2

O

+

cation radical in its

2

B

1

ground state. (a) Molecular orbital diagram of valence orbitals. (b) Optimized structure, (c) spin-density plot (contour levels for orbitals ±0.05 (electrons/

a

0

3

)

1/2

and for spin densities ±0.003 electrons/

a

0

3

)

Figure 7.5 Validity of the point-dipole approximation for a model system consisting of a metal d-orbital and a hydrogen 1s orbital. The black line is the correct quantum mechanical calculation and the broken red line represents the point-dipole prediction

Figure 7.6 Energy and property surface for the methyl radical along the out-of-plane ‘umbrella’ mode. The calculations were done with the B3LYP functional and the cc-pCVTZ basis set (black, full dots = relative total energy, red, open dots = hyperfine coupling)

Figure 7.7 Crystal structure of the Y

730

NH

2

Y-α2 mutant of

Escherichia coli

RNR (molecule A) showing the residues Y

731

, NH

2

Y

730

, and C

439

in the PCET pathway. The O—O distance and the O—S distance between the essential amino acid residues are marked. The inset shows the current model for the long-range (≈35 Å), reversible PCET starting at Y

122

in β2 and reaching C

439

in α2. Note that the position of Y

356

is unknown as the C-terminus is disordered in all X-ray structures of β2.

Figure 7.8

2

H-Mims ENDOR spectrum of NH

2

Y

730

•

in D

2

O assay buffer of the weakly coupled regions (black, from Figure 7.2) and simulation (red).

Figure 7.9 Combined EPR and DFT model for the hydrogen bond network around NH

2

Y

730

•

in α2. (a) (Side view): distances and out-of-plane angles of the hydrogen bonds and to a water molecule within the optimized DFT model 4. (b) Top view. Inset: location of the

g

tensor within the NH

2

Y

•

structure. The direction of the hyperfine tensor component

A

z

of the three hydrogen/deuteron couplings is also schematically drawn

Figure 7.10 Energy level diagram including the transition states for the PCET pathway residues Y

731

–Y

730

–C

439

. The reaction coordinate was investigated using large models with and without the water molecule

Figure 7.11 Structure of the Cu

1.5

…Cu

1.5

mixed valence model complex analyzed in Ref. 80.

Figure 7.12 Comparison of experimental and simulated

1

H-Davies W-band ENDOR spectra of [Cu

2

(RGT)]

3+

from Ref. 80.

Chapter 8: Spin Dynamics

Figure 8.1 The Cartesian representation of the magnetic moment vector of an electron described by a spin state |ψ(

t

) expressed in terms of the eigenstates |α and |β of its Hamiltonian of the form

Ĥ

= ω

0

Ŝ

z

(a) together with its energy level diagram. (b) The evolution of this magnetization vector in 3D space is described by equation (8.14)

Figure 8.2 The two basic pulse-EPR sequences that are applied on a single electron spin as discussed in Section 8.2: (a) the Hahn echo sequence and (b) the stimulated echo sequence

Figure 8.3 The energy level diagram of a two-electron spin system under ‘weak coupling’ conditions in (a) the laboratory frame and (b) the MW rotating frame in the case of a semiselective irradiation at the frequency of spin

e

B

. (c) A stick diagram of the spectrum obtained after a semiselective π/2 pulse at

ω

MW

=

ω

B

Figure 8.4 The matrix representations of the 16 product operators presented in equations (8.97)–(8.102) in the basis set of states defined by equation (8.96) and shown in Figure 8.3. These operators are used to express the Hamiltonian and the spin density operator of two-electron spin systems

Figure 8.5 (a) The energy level diagram of a two-electron spin system in the MW rotating frame defined by the irradiation frequency . The rotating frame Hamiltonian of this system is shown in equation (8.150) with Δ

ω

A

= −Δ

ω

B

. The eigenstates |1 and |4 are degenerate such that the transition frequencies form a symmetric spectrum with

ω

1−3

= −

ω

3−4

and

ω

1−2

= −

ω

2−4

. (b) The stick diagram of the spectrum obtained after a nonselective π/2 pulse composed of four lines with frequencies and amplitudes given in the Figure and defined in equations (8.152), (8.153), and (8.154)

Figure 8.6 The expression for the time evolution of the spin density operator of a two-electron spin system after a single nonselective π/2 pulse in terms of its fictitious spin-1/2 operators is given in equation (8.156). Using the expansions in equations (8.115), (8.116), these operators can be expressed in terms of the product operators . The resulting expression for is shown here schematically, where the time evolutions of the product operator terms are represented by arrows indicating oscillations at frequencies

ω

i

−

j

between pairs of terms the form

Figure 8.7 The two basic pulse EPR sequences that are applied on a two-electron spin system as discussed in Section 8.3: (a) the solid-echo pulse sequence and (b) the DEER/PELDOR pulse sequence

Figure 8.8 The matrix representations of the 16 product operators presented in equations (8.180)–(8.182) in the basis set of states defined by equation (8.183). These operators are used to express the Hamiltonian and the spin density operator of electron–nucleus spin systems

Figure 8.9 The energy level diagram of an electron–nucleus spin system with a hyperfine coupling frequency smaller than the nuclear Larmor frequency in (a) the laboratory frame and (b) in the MW rotating frame with

ω

MW

=

ω

0

− Δ

ω

0

. The definitions of Ω

e

and Ω

en

are given in equations (8.194), (8.195). (c) A stick diagram of the electron spectrum obtained after a single non-selective π/2 pulse showing the line splittings of 2Ω

e

and 2Ω

en

around Δ

ω

e

. The amplitudes of these lines can be found in equations (8.203), (8.204)

Figure 8.10 The three basic pulse EPR sequences that are applied on an electron–nucleus spin system as discussed in Section 8.4: (a) the Hahn echo sequence resulting in the ESEEM modulations, (b) the Davies ENDOR pulse sequence, and (c) the Mims ENDOR pulse sequence

Chapter 9: Relaxation Mechanisms

Figure 9.1 Methods to measure

T

1

. (a)

Saturation recovery

. The length of the saturating pulse is increased until there is no further increase in the recovery time constant. A picket fence of pulses can be used in place of the single saturating pulse. (b)

Inversion recovery

. A π pulse is used to invert the spins. A π/2–

τ

–π–

τ

–echo spin-echo sequence is used to monitor the magnetization along the

z

-axis after delay time

T

. (c)

Echo-detected saturation recovery

. A long low-power pulse is used to saturate the spin system. A π/2–

τ

–π–

τ

–echo spin-echo sequence is used to monitor the magnetization along the magnetic

z

-axis after delay time

T

Figure 9.2 Temperature dependence of

T

1

at X-band for 0.2 mM solutions in 1 : 1 water : glycerol for trityl-CD

3

(

I

) (•), trityl-CH

3

(

II

) (), OX31 (

III

) (), and OX63 (

IV

) (). The dashed line is a fit to the experimental data including contributions from the Raman process and local-mode process.

T

m

is shown for trityl-CD

3

(O), trityl-CH

3

(), OX31 (◊), and OX63 (Δ). The solid line connects the points for

T

m

.

41

Figure 9.3 Frequency dependence of 1/

T

1

for trityl-CD

3

(

I

) (•), trityl-CH

3

(

II

) (), OX63 (

III

) (), and OX31 (

IV

) () in water at 293 K. The fit lines (____) include contributions from the local mode plus modulation of intermolecular and intramolecular electron–proton interactions by molecular tumbling and solvent motion.

25

Figure 9.4 Temperature dependence of 1/

T

1

at X-band for 2,5tBSQ (

VII

) in 1 : 1 ethanol : glycerol (O), DDBSQ (

V

) in 1 : 1 water : glycerol (Δ), and TCSQ (

VI

) in 1 : 1 triacetin : hexamethyl-phosphoramide (). The solid lines are the fits, which are the sum of contributions from the direct, Raman, and local-mode processes. The contributions to the fit line from the Raman process (----) and a local-mode process (-·-·-) for TCSQ and DDBSQ and direct process (…) for TCSQ are also given.

42

Figure 9.5 Temperature dependence of spin-echo dephasing rates in triethanolamine at X-band for 2,5-PSQ (

X

) (Δ), THSQ (

IX

) (×), 2,5-tASQ (

XII

) (O), 2,6-tBSQ (

X

I) (), and 2,5-tBSQ (

VII

) (∇).

42

Figure 9.6 Temperature dependence of X-band spin–lattice relaxation rates for 0.2 mM trityl-CD

3

(

I

) in 1 : 1 water : glycerol (+), 1 mM tempone (

XIII

) in 1 : 1 water : glycerol (◊), and 1 mM tempone (

XIII

) in glassy sucrose octaacetate (Δ). The data for tempone were obtained in the center of the spectrum. The solid lines are the fits to the data including contributions from the direct, Raman, and local-mode processes. The dashed lines are the values calculated based only on the direct and Raman processes.

51

Figure 9.7 Temperature dependence of

T

m

in 1 : 1 water : glycerol at X-band for the spin label MTSSL (

XIV

) (•) and for a nitroxide in which the methyl groups were replaced by spirocyclohexyl groups (

XV

) ().

Figure 9.8 Temperature dependence of spin–lattice relaxation rates at X-band for dithionite reduced [4Fe-4S]

+

in

Rhodobacter sphaeroides

electron-transfer flavoprotein ubiquinone oxidoreductase. The fit line is the sum of contributions from the Raman and Orbach processes. The energy of the low-lying excited state was Δ

E

= 210 K (146 cm

−1

).

72

Figure 9.9 Temperature dependence of the recovery rate constant obtained by (×) X-band inversion recovery, (+) X-band echo-detected saturation recovery, (Δ) X-band saturation recovery or ELDOR, (O) S-band saturation recovery, and () X-band saturation recovery values for 4-Me-2,6-t-Bu-phenoxy radical (

XVI

). The significance of contributions from spectral diffusion varies with temperature and detection method. Fits to the X-band (__) or S-band (_ _ _) data are based on a thermally activated process. Rates by various methods that were shorter than that from ELDOR were not included in the fitting.

4

Chapter 10: Transient EPR

Figure 10.1 Schematic drawing on how trEPR data sets are collected. EPR time profiles are detected as a function of time

t

after pulsed photoexcitation of a sample. The instant of the laser pulse defines

t

= 0. Time-dependent EPR signals are recorded in the presence of a weak microwave magnetic field

B

1

. The typical time resolution of a spectrometer operating at X-band microwave frequencies is in the range 1–10 ns and does not suffer from any dead-time problems

Figure 10.2 Schematic diagram of a conventional CW-EPR spectrometer with the additional components required to examine short-lived photoexcited paramagnetic species with high time resolution. A laser and a fast transient recorder or a digital oscilloscope are required to record the trEPR response as a function of time

t

following a laser pulse. The digital oscilloscope is triggered either by a TTL pulse provided by the laser electronics or by a photodiode that detects reflections of the laser beam and converts them into a fast trigger signal. To record trEPR in a phase-sensitive mode, a detection system with suitably fast response time is required (dashed rectangle). Its components are described in more detail in Figure 10.3

Figure 10.3 Typical phase-sensitive detection scheme used for trEPR at X-band microwave frequencies. A commercially available microwave bridge can be modified such that recording of trEPR data at high time resolution (all three electromechanical switches in position ) is possible, or, alternatively, the accumulation of standard CW-EPR using a diode (all three electromechanical switches in position ). For a description of the components, see text

Figure 10.4 trEPR transients of the triplet state (

S

= 1) of a single crystal of 0.1% pentacene in

para

-terphenyl. A: enhanced absorption and E: emission. The oscillations in the time domain are the so-called Torrey or

B

1

oscillations of the magnetization vector

M

about

B

1

, whose frequency

ω

1

is proportional to the square root of the microwave power

Figure 10.5 trEPR spectra from the secondary radical pair of photosynthetic charge separation in plant photosystem I recorded at different spectrometer frequencies: X-band (9.706 GHz, c), K-band (24.269 GHz, b), and W-band (95.462 GHz, a). The dashed lines show spectral simulations. Higher frequency trEPR provides better spectral resolution for systems with high

g

-anisotropy. Hence, the resonances of the two radicals, P

700

⋅+

and A

1

⋅−

, of the coupled radical pair are best separated at W-band frequency.

Figure 10.6 Energy level diagram of a triplet state with dipolar coupling between the individual electron spins

S

1

and

S

2

. The zero-field eigenstates are denoted T

X

, T

Y

, and T

Z

, where

X

,

Y

, and

Z

are the principal axes of the dipolar coupling tensor

D

. T

+

, T

−

and T

0

are the triplet eigenstates in the presence of an external magnetic field

B

. The population of the zero-field eigenstates is typically sublevel specific and depends on the symmetries of the excited singlet and the triplet wavefunctions. In the present example, the zero-field populations, indicated by the filled black circles, are

p

Z

= 0 and

p

X

=

p

Y

= 0.5. The nonthermal population of T

X

, T

Y

, and T

Z

is converted into electron-spin polarization at high magnetic field, i.e., the high-field eigenstates T

+

, T

−

, and T

0

are also populated in a nonthermal manner (see filled black circles; their diameters indicate the extend of population of a certain state), depending on the orientation of the principal axes of

D

with respect to

B

. This results in (enhanced) absorptive (A) EPR transitions (selection rule Δ

M

S

= ±1), see, e.g., the high-field transition T

−

↔ T

0

in the

B

||

Z

case (

p

−

>

p

0

), or emissive (E) EPR transitions, see, e.g., the low-field transition T

+

↔ T

0

in the

B

||

Z

case (

p

+

>

p

0