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Electron Paramagnetic Resonance Theory E.C. Duin Wilfred R. Hagen Electron Paper: EPR spectroscopy as a probe of metal centres in Paramagnetic biological systems (2006) Dalton Trans. 4415-4434 Resonance Book: Theory Biomolecular EPR Spectroscopy (2009) Fred Hagen completed his PhD on EPR of metalloproteins at the University of Amsterdam in 1982 with S.P.J. Albracht E.C. Duin and E.C. Slater. 1 3 EPR, the Technique…. Spectral Simulations • Molecular EPR spectroscopy is a method to look at the • The book ‘Biomolecular EPR spectroscopy’ comes with a structure and reactivity of molecules. suite of programs for basic manipulation and analysis of EPR data which will be used in this class. • EPR is limited to paramagnetic substances (unpaired electrons). When used in the study of metalloproteins not the • Software: www.bt.tudelft.nl/biomolecularEPRspectroscopy whole molecule is observed but only that small part where the paramagnetism is located. Isotropic Radicals • This is usually the central place of action – the active site of Simple Spectrum Single Integer Signal enzyme catalysis. • Sensitivity: 10 μM and up. Hyperfine Spectrum GeeStrain-5 • Naming: Electron paramagnetic resonance (EPR), electron Visual Rhombo spin resonance (ESR), electron magnetic resonance (EMR) EPR File Converter EPR Editor 2 4 1 BYOS (Bring Your Own Sample) EPR Theory A Free Electron in Vacuo Learn how to use the EPR spectrometer and apply your new knowledge to obtain valuable Free, unpaired electron in space: information on a real EPR sample. electron spin - magnetic moment 5 7 Discovery A Free Electron in a Magnetic Field In 1944, E.K. Zavoisky discovered magnetic resonance. Actually it was EPR on CuCl2. • An electron with spin S = ½ can have two orientations in a E.K. Zavoisky’s first EPR system magnetic field B0 labeled by mS = +½ or mS = −½. • The unpaired electron will have a state of lowest energy when First EMR on CuCl2.2H2O the moment of the electron is aligned with the magnetic field 4.76mT @ 133MHz and a stage of highest energy when aligned against the magnetic field. 6 8 2 A Free Electron in a Magnetic Field A Free Electron in a Magnetic Field ℎ휈 = 푔푒훽퐵0 Two fundamental constants: - Planck’s constant: h - Bohr magneton: β Two experimental parameters: - Microwave frequency: ν • The energy of each orientation is the product of µ and B0. For - Magnetic field: B0 an electron µ = msgeβ, where β is a conversion constant called the Bohr magneton and ge is the spectroscopic g-factor of the free electron and equals 2.0023192778 (≈ 2.00). A constant of proportionality: g-value Therefore, the energies for an electron with ms = +½ and ms = Property of matter, for the free electron: g = ge = 2.00232 -½ are, respectively: ½ geβB0 and -½ geβB0 9 11 A Free Electron in a Magnetic Field Spin-orbit Coupling The electronic-Zeeman energies are E = mSgβB Resonance condition: hν = geβB0 When the electron is bound to one, or more nuclei, then a virtual observer on the electron would experience the nucleus (nuclei) as an +½: E = +½geβB0 orbiting positive charge producing a second magnetic field, δB, at the electron. hν = g β(B + δB) -½: E = -½geβB0 e e Since only the spectrometer value of B is known: The electron can change orientation by absorbing electromagnetic hν = (ge + δg)βB = gβB radiation which energy should exactly The quantity g = ge + δg contains the chemical information on the equal the state energy difference ΔE, nature of the bond between the electron and the molecule, the and this defines the resonance electronic structure of the molecule. condition: The value of g can be taken as a fingerprint of the molecule. ∆퐸 = ℎ휈 = 푔푒훽퐵0 10 12 3 Anisotropy Angular Dependency of g-Value Defining the orientation of the magnetic field (a vector) with respect to the coordinates of the molecule (and vice versa). Two polar angles, θ and ϕ, where θ is the angle between the vector B and the molecular Anisotropy: the fact that molecular z-axis, and ϕ is the angle properties, such as δg are angular between the projection of B onto dependent and reflect the 3D the xy-plane and the x-axis. electronic structure of the paramagnet. In practice so-called direction cosines are used: Example: compound with axial paramagnetic anisotropy. This will lx = sin θ cos ϕ have a different dg-value for different orientations dependent on ly = sin θ sin ϕ the alignment of B0 along the z-axis or the y- or x-axes. lz = cos θ 13 15 Powder Spectrum Angular Dependency of g-Value ℎ휐 A sample of realistic size Resonance: 퐵푟푒푠 = consists of randomly oriented 푔(푙푥, 푙푦, 푙푧)훽 molecules, resulting in a so- 2 2 2 2 2 2 called powder spectrum. with 푔 푙푥, 푙푦, 푙푧 = 푙푥푔푥 + 푙푦푔푦 +푙푧 푔푧 In the example of the compound In terms of the polar angles: with axial paramagnetic 2 2 2 2 2 2 2 2 anisotropy, the spectrum has 푔 휃, 휙 = 푔푥푠푛 휃푐표푠 휙 + 푔푦푠푛 휃푠푛 휙 + 푔푧 푐표푠 휃 axial EPR absorption. (Higher chance of having the B- vector anywhere in the xy-plane than parallel to the z-axis.) 14 16 4 Angular Dependency of g-Value g-Value 2 2 2 2 For axial spectra: 푔푎푥 휃 = 푔푥푦푠푛 휃 + 푔푧 푐표푠 휃 ℎ휈 = 푔훽퐵0 Two fundamental constants: - Planck’s constant: h - Bohr magneton: β For X-band: 휈 (푀퐻푧) 푔 = 0.7145 퐵 (퐺푎푢푠푠) Plot of the angle θ and the axial g-value versus the resonance field (g = 2.40 and g = 2.05, ν = 9500 MHz) z xy 17 19 Line Shape of EPR Spectra Correlation Between Line Shape and Structure? • Does the shape of the EPR signal, ‘isotropic’, ‘axial’, or ‘rhombic’ reflect the symmetry of a coordination site in a metalloprotein? • Most of the time the answer is: No! • Example: the Cu(II) spectrum of plastocyanin is virtually axial (gx ≈ gy) even when recorded at higher frequency for increased g-value resolution. • Crystallographic analysis, however, reveals a highly distorted tetrahedral site essentially with no symmetry at all! Note: g// = gz and g = gxy 18 20 5 S=1/2 Systems Hyperfine Interactions +½: E = +½geβB0 Additional splitting can be observed in EPR signals: Hyperfine interaction -½: E = -½geβB0 Interaction of the electron spin with the nuclear spin of the metal ion nucleus Super hyperfine interaction Interaction of the electron spin with first coordinate sphere ligands nuclei For X-band: Spin-spin interaction Resonance condition: Interaction of the electron spin with other 흂 (푴푯풛) electron spins within 10 Å distance. ∆퐸 = ℎ휈 = 푔푒훽퐵0 품 = ퟎ. ퟕퟏퟒퟓ 푩 (푮풂풖풔풔) 21 23 What is this !?! Hyperfine Interactions • Interaction of the electron spin (S = ½) with the nuclear spin of the metal ion nucleus (I = ½) • Four different situations: Four new energy levels 22 24 6 Hyperfine Interactions Quantum Mechanical Description • With the four new energy • For an isolated system with a single unpaired electron levels there are two field and no hyperfine interaction the only relevant interaction positions where the is the electronic Zeeman term, so the spin Hamiltonian is resonance conditions are met. 퐻푠 = 훽퐵(푔푥푙푥푆푥 + 푔푦푙푦푆푦 + 푔푧푙푧푆푧) A shorter way of writing this is • The signal is in principle split in two. 퐻푠 = 훽퐵 • 푔 • 푆 Solving this we get the equation we saw earlier for the • The original, unsplit, peak angular dependency of the g-value would have been exactly in the middle of the two ℎ휐 = 푔2푙2 + 푔2푙2 +푔2푙2 훽퐵 hyperfine lines. 푥 푥 푦 푦 푧 푧 25 27 Quantum Mechanical Description Quantum Mechanical Description • A full quantum mechanical description of the • More terms can be added to the Hamiltonian when spectroscopic EPR event is not possible due to the needed. complexity of the systems under study. • For hyperfine interactions Hs becomes • In EPR we use the concept of the spin Hamiltonian. 퐻푠 = 훽퐵 • 푔 • 푆 + 푆 • 퐴 • 퐼 This describes a system with an extremely simplified from of the Schrödinger wave equation that is a valid where A is the anisotropic hyperfine tensor. description only of the lowest electronic state of the • For multi-electron (high-spin) systems Hs becomes molecule plus magnetic interactions. 퐻푠 = 훽퐵 • 푔 • 푆 + 푆 • 퐷 • 푆 where D is the zero-field interaction. 퐻푠휓푠 = 퐸휓푠 • When both are present, both terms will have to be With: Hs, spin Hamiltonian; ys, the spin functions; E, added! energy values of the ground state spin manifold. 26 28 7 Quantum Mechanical Description Quantum Mechanical Description • These simplified wave equations will sometimes, under • Using these assumptions the resonance condition strict conditions, give analytical solutions. becomes ℎ휐 = 푔훽퐵0 + ℎ퐴푚퐼 • It is important to realize that a lot of the tools and simulations software used in biomolecular EPR where A is called the Hyperfine Coupling Constant spectroscopy can only be used when certain conditions and mI is the magnetic quantum number for the nucleus. are met. • This describes most hyperfine patterns we will • In most systems we will encounter, we can use these encounter. tools without any problem. There are specific cases, • Exceptions can be found for example for Cu-ion spectra however, where we cannot use these tools. (A-values of 30-200 Gauss) measured at lower frequencies (L-band). In some Cu spectra the g and A tensors are not linear. • Other examples are Mn2+ spectra where D is small. 29 31 Quantum Mechanical Description Hyperfine Interactions • Hyperfine interaction: 퐻푠 = 훽퐵 • 푔 • 푆 + 푆 • 퐴 • 퐼 • Hyperfine coupling constant A Interaction with I = 1/2 • Assumption 1: the Zeeman interaction is much larger A (two orders of magnitude or more) than the hyperfine interaction which can therefore be treated as a perturbation of the larger Zeeman interaction. Interaction • Assumption 2: Just like the Zeeman interaction, the with I = 1 hyperfine interaction will be anisotropic.
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