Electron Spin Resonance spectroscopy for transition metal complexes in solution and 2H Nuclear Magnetic Resonance spectroscopy - line of theoretical analogies

Jagiellonian University

Faculty of Physics, Astronomy and Applied Computer Science

Institute of Physics

Electron Spin Resonance spectroscopy for transition

metal complexes in solution and

H Nuclear Magnetic

Resonance spectroscopy  line of theoretical analogies

Aleksandra Kubica-Misztal

Supervisor: dr hab. Danuta Kruk, prof. UWM

Uniwersytet Jagiello«ski

Wydziaª Fizyki, Astronomii i Informatyki Stosowanej

Instytut Fizyki

Spektroskopia Elektronowego Rezonansu Spinowego

dla kompleksów metali przej±ciowych w roztworach

i deuteronowa spektroskopia Magnetycznego

Rezonansu J¡drowego  analogie opisu teoretycznego

Aleksandra Kubica-Misztal

Promotor: dr hab. Danuta Kruk, prof. UWM

Wydziaª Fizyki, Astronomii i Informatyki Stosowanej Kraków, 22 grudnia 2016 Uniwersytet Jagiello«ski

O±wiadczenie

Ja ni»ej podpisana Aleksandra Kubica-Misztal (nr indeksu: 1001448) doktorantka Wydziaªu Fizyki, Astronomii i Informatyki Stosowanej Uniwersytetu Jagiello«skiego o±wiadczam, »e przedªo»ona przeze mnie rozprawa doktorska pt. Electron Spin

Resonance spectroscopy for transition metal complexes in solution and 2_{H Nuclear}

Magnetic Resonance spectroscopy  line of theoretical analogies jest oryginalna i przedstawia wyniki bada« wykonanych przeze mnie osobi±cie, pod kierunkiem dr hab. Danuty Kruk, prof. UWM. Prac¦ napisaªam samodzielnie.

O±wiadczam, »e moja rozprawa doktorska zostaªa opracowana zgodnie z Ustaw¡ o prawie autorskim i prawach pokrewnych z dnia 4 lutego 1994 r. (Dziennik Ustaw 1994 nr 24 poz. 83 wraz z pó¹niejszymi zmianami).

Jestem ±wiadoma, »e niezgodno±¢ niniejszego o±wiadczenia z prawd¡ ujawniona w dowolnym czasie, niezale»nie od skutków prawnych wynikaj¡cych z ww. ustawy, mo»e spowodowa¢ uniewa»nienie stopnia nabytego na podstawie tej rozprawy.

Author's publications

PAPER I: Quadrupole relaxation enhancement-application to molecular crys-tals

D. Kruk, A. Kubica, W. Masierak, A.F. Privalov, M. Wojciechowski, W. Medycki, Solid State Nuclear Magnetic Resonance, 40, 114 (2011).

My contribution to this paper is simulations of 1_{H spin-lattice relaxation}

dispersion proles for systems containing nuclei possessing quadrupole mo-ments (quadrupolar nuclei). Using a software developed by D. Kruk I have

simulated eects of1_{H spin-lattice relaxation enhancement caused by}

dipole-dipole interactions with quadrupolar nuclei of dierent spin quantum num-bers.

PAPER II: Nuclear quadrupole resonance lineshape analysis for dierent mo-tional models: stochastic Liouville approach

D. Kruk, K.A. Earlie, A. Mielczarek, A. Kubica, A. Milewska, J. Mo±cicki, Journal of Chemical Physics, 135, 224511 (2011).

I have participated in extending the existing software for Electron Spin Res-onance (ESR) lineshape calculations in the presence of Zero Field Splitting (ZFS) to Nuclear Quadrupole Resonance (NQR) lineshape analysis. I have implemented dierent models of rotational dynamics and performed simu-lations to investigate the inuence of the mechanism of rotational motion on NQR lineshapes.

PAPER III: NMR Relaxation and ESR Lineshape of Anisotropically Rotating Paramagnetic Molecules

A. Kubica, A. Milewska, M. Noinska, K.A. Earle, D. Kruk, Acta Physica Polonica A, 121, 2, 527 (2012).

In this work I have performed the ESR lineshape analysis.

PAPER IV: Translational diusion in paramagnetic liquids by H-1 NMR re-laxometry: Nitroxide radicals in solution

D. Kruk, A. Korpaªa, A. Kubica, R. Maier, E.A. Rössler, J. Mo±cicki, i

ii Author's publications

Journal of Chemical Physics, 138, 024506 (2013).

In this work I have contributed to the analysis of 1_{H spin lattice relaxation}

dispersion data meant to determine relative translational diusion coe-cients of the solvent and solute molecules.

PAPER V: Zero-eld splitting in nickel(II) complexes: a comparison of DFT and multi-congurational wavefunction calculations

A. Kubica, J. Kowalewski, D. Kruk, M. Odelius, Journal of Chemical Physics,138, 064304 (2013).

This work has been done in collaboration with Stockholm University. For this paper I have performed extensive calculations of ZFS parameters by means of Density Functional Theory (DFT) and multi-congurational wave-function methods. I have used the ORCA software. The calculations have been carried out for several dierent functionals.

PAPER VI: 1_{H relaxation dispersion in solutions of nitroxide radicals: inuence of}

elec-tron spin relaxation

D. Kruk, A. Korpaªa, A. Kubica, J. Kowalewski, E.A. Rössler, J. Mo±cicki, Journal of Chemical Physics, 138, 124506 (2013).

I have contributed to calculations of electron spin relaxation applying the Redeld relaxation theory.

PAPER VII: ESR lineshape and 1_{H spin-lattice relaxation dispersion in propylene glycol}

solutions of nitroxide radicals  Joint analysis

D. Kruk, S. K. Homann, J. Goslar, S. Lijewski, A. Kubica-Misztal, A. Korpaªa, I. Ogªodek, J. Kowalewski, E. A. Rössler, J. Moscicki, Journal of Chemical Physics, 139, 244502 (2013).

My contribution to this work is twofold: I have derived analytical, per-turbation formulae for ESR lineshapes for nitroxide radicals and checked their validity regimes by comparing the perturbation predictions with re-sults obtained by means of a general theory based on the Stochastic Liou-ville Equation (SLE), available in the literature. Then I have performed the theoretical analysis of the ESR lineshapes using the software available on http://www.acert.cornell.edu/index les/acert resources.php, which is based on the SLE approach.

PAPER VIII: Systematic theoretical investigation of the zero-eld splitting in Gd(III) com-plexes: Wave function and density functional approaches

S. Khan, A. Kubica-Misztal, D. Kruk, J. Kowalewski, M. Odelius, Jour-nal of Chemical Physics, 142, 034304 (2015).

Proting from my experience with DFT calculations (paper V) I have con-tributed to DFT calculations of ZFS parameters for systems of the electron spin quantum number S=7/2.

Author's publications iii Besides the outlined research tasks, I have also performed further studies pre-sented in the thesis. They are:

• Extending the existing perturbation model of ESR lineshape for S = 7/2

(implemented by D. Kruk) by including 4th order ZFS interactions.

• Performing a comparison of perturbation and SLE approaches to ESR

Abstract

I the thesis theoretical aspects of calculating Electron Spin Resonance (ESR) and Nuclear Quadrupole Resonance (NQR) lineshapes and spin interaction parame-ters are presented. The term Nuclear Quadrupole Resonance refers to Nuclear

Magnetic Resonance for nuclei possessing quadrupole moments (like 2_{H). It}

be-gins with presenting the Hamiltonian formalism of spin interactions relevant for paramagnetic species of dierent electron spin quantum numbers, S: Zeeman in-teraction, Zero Field Splitting (ZFS) interactions (S ≥ 1) and electron-nuclear hyperne interaction (S = 1/2). The concept of transforming spin Hamiltonians between dierent reference frames is introduced. On this basis, the main prin-ciples of spin relaxation theory are presented. In the rst step the concept of the Hilbert space (associated with Hamiltonians) and the Liouville space (asso-ciated with Liouville operators) are presented. Then the well-known expressions for spin relaxation rates, resulted from the perturbation theory of spin relaxation (referred to as the Redeld relaxation theory) are provided. The relaxation rates are expressed in terms of spectral density functions describing stochastic uctu-ations of spin interactions causing the relaxation process. As this approach has a limited range of applicability, determined by the validity conditions of the per-turbation theory, a general recipe (also valid beyond the range of applicability of the perturbation approach) for calculating ESR lineshape is presented. This introduction provides the background for the results presented in next chapters. They can be outlined as follows: Closed form expressions for ESR lineshapes for

S = 1/2(nitroxide radicals containing15Nand14Nisotopes) have been derived in

terms of the perturbation approach. The predictions of these formulae have been compared with lineshape calculations based on the general approach (using the Stochastic Liouville Equation (SLE) ) in order to establish the validity conditions of the perturbation models. The considerations are meant to demonstrate that the limitations of perturbation approaches may not be neglected and should be treated with high caution. As far as the perturbation approach is considered, the existing perturbation description of ESR lineshape for S = 7/2 has been extended by including 4th order ZFS terms.

vi Abstract

To calculate ESR and NQR lineshapes one has to know the parameters of spin interactions. A theoretical way to obtain them is applying quantum chemistry calculations. Examples of applying multi-congurational wavefunction methods and Density Functional methods to obtain ZFS parameters for selected param-agnetic systems are presented. The calculations are preceded by an introduction to quantum chemistry methods. The purpose of these studies is to compare both approaches and demonstrate how the results depend on the chosen functional. With the quantum chemistry tools providing access to spin parameters one can fully exploit the potential of the general approach to ESR lineshape calculations. The exibility of this approach to motional models and the inuence of dier-ent motional mechanisms on ESR lineshapes has been demonstrated. Evdier-entually, proting from the mathematical analogies between the Hamiltonian formalisms of ZFS and quadrupolar interactions, the SLE approach has been extended to NQR lineshape analysis. In this way a general, exible (with respect to models of molecular dynamics leading to uctuations of spin interactions) tool has been presented.

Contents

Author's publications i

Abstract v

Contents vii

1 Spin interactions and Electron Spin Resonance 3

1.1 Spin Hamiltonian . . . 3

1.2 Zeeman interaction . . . 4

1.3 Hyperne interaction . . . 5

1.4 Zero Field Splitting (ZFS) . . . 5

2 Relaxation theory and ESR lineshape 11 2.1 Hilbert and Liouville space formalism. . . 11

2.2 Redeld relaxation theory . . . 13

2.3 Lattice Dynamics . . . 16

2.4 Electron Spin Resonance (ESR) lineshape . . . 17

3 Perturbation description of ESR lineshape 21 3.1 Perturbation approach . . . 21

3.2 Perturbation theory of ESR lineshape for the electron spin quan-tum number S = 7/2 . . . 30

3.3 4th order ZFS term. . . 32

4 Quantum Chemistry Methods of calculating Zero Field Splitting 37 4.1 Introduction. . . 37

4.2 Electron correlation methods . . . 39

4.3 Density Functional Theory (DFT) . . . 41

4.4 Studies of ZFS in Nickel (II) complexes. . . 42

4.4.1 Methodology of calculating ZFS . . . 42

4.4.2 Computational details . . . 44 vii

viii CONTENTS

4.4.3 ZFS in Ni(H2O)4F2 . . . 45

4.4.4 ZFS in Ni(H2O)2+₆ . . . 50

4.4.5 ZFS in pentacoordinate nickel (II) complexes . . . 53

5 General theory of calculating ESR lineshape in the presence of ZFS 57 5.1 Free and jump diusion models . . . 57

5.2 Simulations of ESR spectra for S = 7 2 in terms of the Stochastic Liouuville Equation (SLE) . . . 58

6 Extension of the SLE theory to Nuclear Quadrupole Resonance (NQR) 61 6.1 Principles of NQR . . . 61

6.2 Similarities between ZFS and quadrupolar interaction . . . 62

6.2.1 SLE theory for NQR . . . 63

6.3 Examples of NQR spectra . . . 64

6.4 NQR lineshape for the free and jump diusion models . . . 64

7 Summary 67 Bibliography 71 A Matrix elements of Hamiltonians 81 B Matrix elements of 4th order ZFS 83 C The Cartesian atomic coordinates for nickel (II) complexes 91 C.1 The Cartesian atomic coordinates for Ni(H2O)4F2 . . . 91

D ORCA input and output les 93 D.1 Ni(H2O)2+6 , NEVPT2 in QZVP basis . . . 93

D.2 Ni(H2O)4F2, NEVPT2 in QZVP basis . . . 96

D.3 SOC and SSC - QDPT calculations, Matlab code . . . 98

List of Symbols and Abbreviations 101

List of Figures 103

List of Tables 107

CONTENTS 1

Introduction

The thesis is focused on presenting the theory and analysis of ESR and NQR line-shapes. It begins with a simple example of ESR lineshape calculations performed for nitroxide radicals in solution in terms of the perturbation approach (referred to

as the Redeld relaxation theory) [13]. 14_{N-containing and}15_N-containing

radi-cals are considered as they dier in the spin quantum number (1 for14_{N and} 1

2 for

15_{N). The electron spin relaxation is caused by the anisotropic part of the}

nitro-gen spin - electron spin hyperne interaction modulated by rotational dynamics of the paramagnetic molecule. The validity range of the perturbation theory of spin relaxation is determined, in this case, by a product of the anisotropic part of the hyperne coupling (in angular frequency units) and the rotational correlation time. The product has to be smaller than one to apply the perturbation treat-ment, otherwise a full solution of the Stochastic Lioville Equation (SLE) has to be considered to describe the lineshape. The approach based on the SLE has been developed for nitroxide radicals [4]; the corresponding software is freely available on www.acert.cornell.edu [5].

In Chapter 3 ESR lineshapes simulated using this software and the perturba-tion formulae are compared for dierent rotaperturba-tional correlaperturba-tion times to reveal the validity range of the perturbation approach. The results have been used

in the analysis of the ESR data for propylene glycol solutions of 14_{N and} 15_N

radicals [6]. Furthermore, the comparison presented in the thesis goes beyond the specic case presented in [6]. It clearly shows progressive deviations of both approaches with increasing the rotational correlation time. Staying for a while with the perturbation approach, in the next section of Chapter 3 a perturbation description of ESR spectra for high electron spin quantum number (S = 7/2

like for Gd+3_{) is outlined. In this case it is assumed that the molecule}

carry-ing the electron spin is immobilized (like for paramagnetic contrast agents). In consequence, the energy level structure of the electron spin is determined by a su-perposition of the Zeeman and Zero Field Splitting (ZFS) interactions, while the electronic relaxation is caused by stochastic uctuations of the ZFS tensor. This already existing description has been extended by including 4th order terms of

the ZFS. These parts of the thesis are high relevance for the calculations of 2_H

NQR (alternatively one can use the term: 2_{H NMR) spectra presented in}

Chap-ter 6. Before coming to this point, in ChapChap-ter 4 quantum chemical methods of calculating the ZFS parameters are presented. The considerations are illustrated by advanced multi-congurational wavefunction methods and Density Functional Theory (DFT) calculations for Ni complexes dissolved in water. The calculations have been performed for six- and ve-coordinated nickel(II) complexes of the elec-tron spin quantum number S = 1 by means of the ORCA program package [79]. Multi-congurational wavefunction calculations performed in terms of the second-order perturbation theory and the quasi-degenerate perturbation approach have

2 CONTENTS

been compared with DFT methods using dierent functionals. The results have been published in [10]. It has been demonstrated that the results obtained by the multi-congurational wavefunction methods are quite consistent and in reason-able agreement with experimental data. At the same time it has turned out that the results of DFT methods are strongly functional-dependent and considerably dierent from the experimental data.

In Chapter 5 I again turn attention to the theory of ESR lineshape. Dierent models of motion (jump diusion and free diusion) causing uctuations of the ZFS interaction are considered. The inuence of dierent motional models on the ESR spectrum is discussed. Eventually, in the last part of the thesis, proting from the mathematical analogy between the ZFS interaction for high electrons spins and quadrupole interaction for high nuclear spins, the general theory is

ap-plied to calculate2_{H NQR spectra for arbitrary motional conditions also applying}

dierent motional models [11]. The results obtained in the thesis are summarized in Chapter 7.

Chapter 1

Spin interactions and Electron

Spin Resonance

Molecules with unpaired electrons are paramagnetic. Nitroxide radicals and com-plexes of transition metal ions are examples of paramagnetic systems. Nitroxide radicals contain one unpaired electron that implies the electron spin quantum

number S = 1

2. Transition metal ions are characterized by the electron spin

quan-tum number ranging between S = 1 (Ni) and 7

2 (Gd). The paramagnetic species

experience, when placed in an external magnetic eld, Zeeman coupling. More-over, they are subjected to another type of spin interactions that is not related to the external magnetic eld, namely hyperne and ZFS interactions. Nitroxide radicals show a strong hyperne coupling that results from dipole-dipole

interac-tions between the electron spin S = 1

2 and nitrogen spin P = 1 (for14N) or P =

1 2

(for 15_{N). ZFS interactions are present for S ≥ 1. For transition metal ions they}

stems from second order eects of the spin-orbit coupling. These interactions lead to a splitting of electron spin energy levels even in the absence of an external magnetic eld; this explains the terminology (Zero Field Splitting).

1.1 Spin Hamiltonian

In this section spin interactions relevant for for an electron spin placed in an external magnetic eld are presented in terms of their Hamiltonians. The total

Hamiltonian HS, consists of three main terms [1,12]:

H_S = HZ(S) + Hhf(P, S) + HZF S(S ≥ 1) (1.1)

where HZ(S)and HZF S(S ≥ 1)denote the Zeeman and ZFS interactions of the

electron spin respectively, while Hhf(P, S)denotes the electron spin-nuclear spin

4 CHAPTER 1. SPIN INTERACTIONS AND ELECTRON SPIN RESONANCE

hyperne coupling (the index "P" has been used for nitrogen spin- in the thesis it will always denote a nuclear spin, but not necessarily of nitrogen). To describe the ZFS interaction in terms of Hamiltonian an approximated solution of the Schödinger equation for a many-body system has to be considered. In the case when the solution includes only the spin operators [12] one says that the electronic ground state is well dened, i.e. it is possible to describe the energy levels of the electron spin by a spin Hamiltonian.

1.2 Zeeman interaction

The rst term in equation 1.1, i.e. the electron spin Zeeman Hamiltonian, is

dened as follows:

HZ(S) = γSS · B0= −

µB

~ S · g · B0 (1.2)

where B0 denotes an external magnetic eld (vector), g is the electronic g-tensor,

µB denotes Bohr magneton, while ~ is the Planck constant divided by 2π. The

Zeeman interaction is a linear interaction of the electron spin with the external magnetic eld and it is parameterized by the g-tensor. For free electron, the Zee-man interaction is isotropic with a scalar g = 2.0023193. The ZeeZee-man interaction

for S = 1

2 leads to two energy levels corresponding to the parallel state (mS = −

1 2)

and the antiparallel state (mS = 1₂), where mS denotes the magnetic spin

quan-tum number. The energy dierence between the two states, ∆EZ, determines the

resonance frequency and it is given as [13,14]:

∆EZ = gµBB0∆mS = γSB0∆mS (1.3)

where γS denotes the electronic gyromagnetic factor. The g-tensor is strongly

dependent on the local structure around the electron spin. When the system is characterized by high local symmetry, g-tensor becomes a scalar (constant). It is

also very common for the g-tensor to have axial symmetry (gk= gzz, g⊥= gxx=

gyy) and then its value takes the form [15,16]:

g2 = g_⊥2 sin2θ + g_k2cos2θ (1.4)

where θ is the angle between the symmetry axis of the tensor and the direction of the magnetic eld. The Zeeman Hamiltonian for an axially symmetric g-tensor is given as [15,16]:

HZ(S) = µB[g⊥(BxSx+ BySy) + gkBzSz] (1.5)

For rhombic symmetry the g-tensor can be represented as an ellipsoid with three principal directions. The eective value of the g-tensor for an arbitrary orientation is then given by:

1.3. HYPERFINE INTERACTION 5

Here θx, θy, θz are the angles between the x, y, z axes and the magnetic eld

direction. The Zeeman Hamiltonian for this case takes the form [17]:

H_Z(S) = µB(gxBxSx+ gyBySy+ gzBzSz) (1.7)

1.3 Hyperne interaction

The second term in equation 1.1 represents the hyperne interaction that is

de-scribed by the Hamiltonian [12]:

Hhf(P, S) =P · A · S (1.8)

where A denotes the hyperne coupling tensor. Hyperne interactions are two-spin interactions - they include the electron two-spin (S) and a nuclear two-spin (P). It is important to stress that the hyperne coupling does not depend of the external magnetic eld. This interaction leads to an additional splitting of the electronic energy levels. In the simplest example when both the electron and the nuclear

spins are equal to 1/2 (P = S = 1

2) each electron Zeeman energy level splits

into two levels corresponding to dierent magnetic spin quantum numbers of the

nucleus: mP = ±1₂.

1.4 Zero Field Splitting (ZFS)

The last term in equation 1.1 represents the ZFS. This interaction characterizes

systems with electron spin quantum numbers S ≥ 1, like transition metal ions.

Examples of them are given in Table1.1. The electron spin quantum number S of

these ions varies from S = 1 to S = 7

2.

Table 1.1: Ground-state properties of free dn _{ions [17].}

Number n of Orbital Degeneracy Term Symbol Examples (3dn₎

d electrons 1 5 2_D 3/2 T i3+, Cr5+ 2 7 2_F 3 T i2+, Cr4+ 3 7 4_F 3/2 T i+, Mn4+ 4 5 5_D 0 Cr2+, Mn3+ 5 1 6_S 5/2 M n2+, Co4+ 6 5 5_D 4 M n+, F e2+ 7 7 4_F 9/2 Co2+, Ni3+ 8 7 3_F 4 Co+, Ni2+ 9 5 2_D 5/2 N i+, Cu2+

As already mentioned, ZFS for transition metal complexes is caused by the

6 CHAPTER 1. SPIN INTERACTIONS AND ELECTRON SPIN RESONANCE

can become very large when energy levels resulted from the full Hamiltonian, including spin-orbit interaction, are not far apart. Under this condition, the spin Hamiltonian approach can break down. Problems with the spin Hamiltonian arise also in the case of orbitally degenerated ground state, when the rst-order

spin-coupling eects do not vanish [1,12]. The ZFS spin Hamiltonian is dened

as [18,19]:

HZF S =S · D · S (1.9)

where D is the ZFS tensor. The tensor can be split into two components, axial and rhombic ones, dened as:

D = Dzz−

1

2(Dxx+ Dyy) (1.10)

E = 1

2(Dxx− Dyy) (1.11)

Using tensor notation the Hamiltonian can be written as [1,12,2023]:

H(P )_{ZF S}(S) = r 2 3D 2 X m=−2 (−1)mV−m2(P )Tm2(S) = 2 X m=−2 (−1)mV˜−m2(P )Tm2(S) (1.12)

In the principal (P) axes system of the D-tensor, the V2(P )

m components take the

form: V2(P ) 0 = 1, V 2(P ) ±1 = 0, V 2(P ) ±2 = q 3 2 E

D, where the index (P) explicitly refers

to the (P) frame (the (P) frame should not be confused with the nuclear spin quantum number, P), while:

T₀2(S) = √1 6[3S 2 Z− S(S + 1)] (1.13) T_±12 (S) = ∓1 2[SZS±+ S±SZ] (1.14) T_±22 (S) = 1 2S±S± (1.15)

To obtain the form of the ZFS Hamiltonian in the laboratory (L) frame

(deter-mined by the direction of the external magnetic eld vector B0), the Vm2(P ) tensor

components have to be transformed to the (L) frame using the relationship [1]:

V−m2(L) =

2

X

k=−2

V_k2(P )D_k,−m2 (ΩP L) (1.16)

where ΩP Ldescribes the orientation of the (P) frame with respect to the (L) frame,

while D2

k,−m(ΩP L) denote Wigner rotation matrices [1,12,24,25]. When the (P)

frame changes its orientation with respect to the (L) frame due to, for instance, tumbling (rotation) of the paramagnetic molecules in solution, the Hamiltonian

1.4. ZERO FIELD SPLITTING (ZFS) 7 H(L)_{ZF S}(S)(t) = r 2 3D 2 X m=−2 (−1)m( 2 X k=−2 V_k2(P )D_k,−m2 (ΩP L(t)))Tm2(S) (1.17)

Actually for paramagnetic complexes in solution an even more advanced descrip-tion of the ZFS interacdescrip-tion is needed. Every Hamiltonian can be expressed as a sum of two components, H(t) = hH(t)i+(H(t)−hH(t)i), where hi denotes time average of, thus hH(t)i is a results of a long time averaging of H(t). hH(t)i is static (time independent) in a molecule-xed frame, but in the laboratory frame it can stochastically uctuate in time due to, for instance, molecular tumbling that changes the orientation of the (P) frame with respect to the (L) frame. Momentar-ily deviations of the total Hamiltonian H(t) from its averaged part (H(t)−hH(t)i)

is referred to as a transient part of the interaction [1,12,26]. The origin of the

transient part can be internal dynamics of the molecule carrying the electron spin (vibrations of the complex or any other local motion) or its collisions with solvent molecules. Coming back to the ZFS interaction, its static (in a molecule-xed

frame) part can be described in the (L) frame by the Hamiltonian [1,12,26]:

HS(L)_{ZF S}(S)(t) = r 2 3DS 2 X m=−2 (−1)mV_−m2S(L)(t)T_m2(S) (1.18) where: V_−m2S(L)(t) = r 2 3DSD 2 0,−m(ΩPSL(t)) + ES[D 2 −2,−m(ΩPSL)(t) + D 2 2,−m(ΩPSL)(t)] (1.19)

The angle ΩPSL between the principal axis system of the static (permanent) part

of the ZFS Hamiltonian (PS) and the (L) frame is marked in Figure 1.1 (in the

case of slow rotation the angle becomes time independent), while DS and ES

denote the axial and rhombic components of the static part of the ZFS tensor. The transient part of the ZFS Hamiltonian needs two transformations to be expressed in the laboratory frame. The rst one is between the principal axis

system of the transient part (PT) and the (PS) frame. The form of the HT (P_{ZF S}T)(S)

Hamiltonian in the ((PT)frame) is [1,12,26]:

HT (PT) ZF S (S) = r 2 3DT 2 X m=−2 (−1)mV2T (PT) −m Tm2(S) (1.20) where: V2T (PT) 0 = 1 V 2T (PT) ±1 = 0 V 2T (PT) ±2 = 4 √ 6 ET DT

8 CHAPTER 1. SPIN INTERACTIONS AND ELECTRON SPIN RESONANCE

Laboratory frame (L) Principal axis system

of the static ZFS P_S

Principal axis system of the transient ZFS P_T

Ω_PSPT(t) Ω_PSL(t)

S

S T

Figure 1.1: Schematic view of the axes systems described in the text.

with DT and ET denote the axial and rhombic components of the transient ZFS

tensor. Applying the transformation of equation 1.16 the form of the transient

ZFS Hamiltonian in the (PS) frame yields [1,12,26]:

HT (PS) ZF S (S)(t) = (1.21) r 2 3DT 2 X m=−2 (−1)m[ 2 X k=−2 V2T (PS) k D 2 k,−m(ΩPTPS(t))]T 2 m(S) = 2 X m=−2 (−1)mV˜2T (PS) −m Tm2(S) where: ˜ V2T (PS) −m = r 2 3DTD 2 0,−m(ΩPSPT(t)) + ET[D 2 −2,−m(ΩPSPT)(t) + D 2 2,−m(ΩPSPT)(t)] (1.22)

The Euler angle ΩPSPT(t) describes the relative orientation of the (PT) and (PS)

frames as shown in Figure 1.1. The transient ZFS Hamiltonian HT (PS)

ZF S depends

on time in the (PS) frame, (in contrary to the permanent part). In the simplest

model the transient ZFS has a constant magnitude and a principal direction, which is not xed in the molecule [1,27]. For paramagnetic complexes in solution

1.4. ZERO FIELD SPLITTING (ZFS) 9 distortions of the ligand framework cause uctuations of the relative orientation of

the (PS) and (PT). The second transformation is between the (PS) and laboratory

(L) frames via the Euler angle ΩPSL(t) [1]:

HT (L)_{ZF S} = r 2 3DT 2 X m=−2 (−1)m[ 2 X n=−2 ( 2 X k=−2 V2T (PT) k D 2 k,−n(ΩPTPS(t)))D 2 −n,−m(ΩPSL(t))]T 2 m(S) (1.23) = 2 X m=−2 (−1)mV˜_−m2T (L)T_m2(S) where ˜ V−m2T (L)= 2 X n=−2 r 2 3DTD 2 0,−n(ΩPSPT)(t) + ET[D 2 −2,−n(ΩPSPT)(t) + D2_2,−n(ΩPSPT)(t)]D 2 −n,−m(ΩPSL)(t) (1.24)

In summary, the ZFS Hamiltonian consists of two components: a permanent one

(static) and a transient one [1,12,26]:

HZF S(S)(t) = HSZF S(S)(t) + HTZF S(S)(t) (1.25)

The permanent component of ZFS depends on the local geometry. For high sym-metry complexes, the permanent component vanishes. The transient component is always present. This is the situation for nickel (II) ions in water solution

(dis-cussed in Chapter4) Ni(H2O)2+6 . Oscillations and collisions, as well as variations

in the orientations of water molecules in the rst hydration sphere of this complex are the origin of the ZFS uctuations [28].

Chapter 2

Relaxation theory and ESR

lineshape

2.1 Hilbert and Liouville space formalism

This chapter is devoted to the principles of the relaxation theory. Generally, the phenomenon of relaxation describes how the spin system is restored back to the equilibrium state, after being perturbed. In the case of spins the non-equilibrium state means that populations of the spin energy levels do not follow the Boltzmann distribution. Spin relaxation is characterized by two relaxation times. The longi-tudinal (parallel to the external magnetic eld) component of the magnetization

evolves toward the equilibrium with a relaxation time T1 called the spin-lattice

relaxation time, while the transverse magnetization decays with relaxation time

T2 referred to as the spin-spin relaxation time. From the viewpoint of the spin

the relaxation is caused by its interaction with the rest of the system. This is

reected by the Hamiltonian Htot ("tot" refers to total) being a sum of the three

Hamiltonians [1]:

Htot = HS+ HL+ HSL (2.1)

The rst term in equation2.1is the spin Hamiltonian dened in equation1.1. The

second term represents the rest of the molecular system and includes all relevant degrees of freedom (like molecular dynamics and other spins) that are referred to as lattice. The last term describes the coupling between the spin and the lattice. It is convenient to describe relaxation processes using a matrix formalism. For this purpose a basis has to be dened. The basis, referred to as a Hilbert space

[1,2,16,20,22,25,2931] is constructed from eigenvectors |µi of a sum of the

Hamiltonians characterizing the spin and the lattice: H0 = HS + HL [1]. The

complete and orthonormal set of functions |µi covers all degrees of freedom of the 11

12 CHAPTER 2. RELAXATION THEORY AND ESR LINESHAPE

system. Furthermore, a density operator, ρ(t) characterizing the spin system, is dened [1]. The representation of the density operator in the |µi basis is given by

the elements ρµν(t) = hµ|ρ(t)|νi. Time evolution of the expectation value of an

operator O(t) can be calculated from the matrix operation:

hO(t)i = T rρ(t)O) (2.2)

where T r denotes "trace". A representation alternative to the Hilbert space

is Liouville space representation [1,2,16,20,22,25,2931]. The Liouville space

is created from pairs of the Hilbert space functions: |µihν| and the notation:

|µihν| = |µ, ν)is often used. In consequence, the dimension of the Liouville space

is N2 _{when the corresponding Hilbert space dimension is N.}

Operators acting on the Liouville space elements are called superoperators and

denoted as ˆO. The Liouville superoperator ˆL is generated by the corresponding

Hamiltonian H : ˆL = [H, ...]. Time evolution of the density operator, ρ(t), under

a Hamiltonian H is given as [1,20,29,30]:

d

dtρ(t) = −i[H, ρ(t)] (2.3)

The equation can be rewritten in the operator formalism as [1,20,29,30]:

d

dtρ(t) = −i ˆˆ L|ˆρ(t)) (2.4)

Equations 2.4 is referred to as the Liouville von Neumann equation; equation

2.3 is its counterpart in the Hilbert space. The main Hamiltonian, H0 = HS+

HL, (represented in the Liouville space formalism as ˆL0 = ˆLS+ ˆLL) determines

the energy level structure of the system. Transitions between the energy levels

are caused by the small perturbation: HSL( ˆLSL). To solve the Liouville von

Neumann equation the density operator, ρ(t), and the perturbating Hamiltonian,

H_SL, are transformed into the interaction representation, generated by the main

Hamiltonian, H0 [1,16,20,29]:

ρ0(t) = exp(−iH0t)ρ(t) exp(iH0t) (2.5)

H0_SL(t) = exp(−iH0t)HSL(t) exp(iH0t) (2.6)

Analogous transformation can be performed in the Liouville space, with |ˆρ0_(t))and

ˆ

L0_SL(t)being the Liouville space counterparts of ρ0(t)and H0_SL(t). It is important

to underline that now the perturbing spin-lattice coupling H0

SL(t)has two sources

of time dependence. The rst one is connected with motional degrees of freedom of the lattice, while the second one is connected with the transformation to the interaction representation. The transformation gives the following form of the

Liouville von Neumann equation [1,16,20,29]:

d

dtρ

0_{(t) = −i[H}

2.2. REDFIELD RELAXATION THEORY 13 and analogously, in the Liouville space:

d dt|ˆρ

0_{(t)) = −i ˆL}0 SL(t)| ˆρ

0_{(t)) (2.8)}

2.2 Redeld relaxation theory

The Redeld theory [13] , also referred as Wangsness-Bloch-Redeld(WBR) the-ory, is based on time dependent perturbation approach applied to describe the time evolution of the density operator. This approach requires that the system is described by a Hamiltonian containing a dominating, time-independent part and

a small, perturbative time-dependent term. On the basis of equation 2.7the time

evolution of the spin density operator, ρ0

S(t)can be written in the form [13]:

dρ0_S(t)

dt = −

Z ∞

0

h[H0_SL(t), [H_SL0 (t − τ ), ρ0_S(t)]]idτ (2.9)

where hi denotes averaging over many spins (a spin assemble). The equation can be expressed in a basis constructed from the eigenstates |αi of the spin

Hamilto-nian HS and a set of coupled dierential equations for the individual matrix

ele-ments of the spin density operator (ρ0

S(t))αα0 = hα|ρ0 S(t)|α 0_{ican be created [13]:} d(ρ0_S(t))αα0 dt = X ββ0 Γαα0_ββ0exp[i(ω_αα0− ω_ββ0)t](ρ0_S(t))_ββ0 (2.10)

referred to as Redeld relaxation equation [1,2,31]. The frequencies ωαα0 denote

transition frequencies between the eigenstates |αi and |α0_{i of the spin:}

ωαα0 = ω_α− ω_α0 (2.11)

The time independent coecients, Γαα0_ββ0, determine the spin relaxation rates.

The terms which oscillate in time with the frequencies ωαα0 − ω_ββ0, should

ef-fectively be averaged out to zero on the time scale characterizing the changes of the density matrix, i.e. the relaxation process. The time scale of the relaxation

process is determined by the amplitude of the Hamiltonian HSL and a correlation

time τcdescribing the time uctuations of this Hamiltonian. Thus, the oscillating

terms are averaged out when ωS  ωSL2 τc, where ωS and ωSL denote the

am-plitudes of the Hamiltonians HS and HSL in angular frequency units [1]. This

condition implies that only the terms for which ωαα0 = ω_ββ0remain in the

summa-tion of equasumma-tion2.11. Then the Redeld relaxation equations can be transformed

to the laboratory frame in which they take the form [1,2,16,29,31]:

dραα0(t) dt = −iωαα0ραα0(t) + X ββ0_,ω αα0=ωββ0 Γαα0_ββ0ρ_ββ0(t) (2.12)

14 CHAPTER 2. RELAXATION THEORY AND ESR LINESHAPE

The matrix elements ραα0 are referred to as coherences. The number of coherences

is N2 _{(N is the dimension of the Hilbert space). The coecients Γ}

αα0_ββ0 are given as [1,12]: Γαα0_ββ0 = R_αα0_ββ0− iL_αα0_ββ0 (2.13) = =αβα0_β0(ω_αβ) + =_αβα0_β0(ω_β0_α0) − δ_α0_β0 X γ =αγβγ(ωγβ) − δαβ X γ =_β0_γα0_γ(ω_β0_γ)

Here we are interested only in their real parts: Rαα0_ββ0, called relaxation

coe-cients. The imaginary part Lαα0ββ0, describes so called dynamical frequency shift.

The quantities =νν0_µµ0(ω) are referred to as spectral density functions and they

depend on the Hamiltonian HSL taken at times t and t + τ [1,12]:

=_µµ0_νν0(ω) = <

Z ∞

0

hhµ|HSL(t)|µ0ihν|HSL(t − τ )|ν0ii exp(−iωτ )dτ (2.14)

where |µi, |µ0_{i, |νi, |ν}0_{i are eigenstates of the spin system.}

The Hamiltonian HSL describing the spin-lattice interactions can be written as

a symmetrized product of lattice functions Fl

−m(t)and spin operators Sml [1,12]:

H_SL= ξSL

l

X

m=−l

(−1)mS_ml F_−ml (t) (2.15)

where ξSL is a constant describing the interaction strength (amplitude). Then

the spectral densities =µµ0νν0(ω) can explicitly be expressed as [1,12]:

=_µµ0_νν0(ω) = ξ2_SL l X m,m0_=−l (−1)m+m0hµ|S_ml |µ0ihν|Sl_−m0|ν0i (2.16) Z ∞ 0 hF_−ml∗ (t)F−ml 0(t − τ )i exp(−iωτ )dτ

In equation 2.16 the spin and the lattice variables have been separated. This is

allowed as the relaxation process occurs on a much longer time scale than the time scale of the lattice dynamics. It implies that one can dene a lattice correlation function (including only the lattice degrees of freedom) [1]:

C_m,ml 0(τ ) = hF_−ml∗ (t)F_−ml (t − τ )i = hF_−ml∗ (τ )F_−ml 0(0)i (2.17)

The last equality shows that the correlation function depends only on the time interval τ (it does not function depend of the initial time-point, t). Fourier trans-form of the lattice correlation function gives a lattice density function:

J_m,ml 0(ω) =

Z ∞

0

hFl∗

2.2. REDFIELD RELAXATION THEORY 15 The presented formalism can be applied to systems for which the lattice is treated classically and quantum-mechanically. Nevertheless, when the lattice contains (besides classical) quantum mechanical degrees of freedom, the description be-comes more complicated. For such cases it is more convenient to use the Liouville

space representation. Equation 2.9 expressed in the Liouville space formalism

yields: d| ˆρ0(t)) dt = − Z ∞ 0 h ˆL0_SL(t) ˆL0_SL(t − τ )| ˆρ0(t))i(dτ ) (2.19)

In analogy to equation 2.10, the superoperator ˆΓ = ˆR − i ˆK (where ˆR and ˆK)

denote the relaxation and dynamic frequency shifts operators, respectively) links

the time derivative of the spin density operator d| ˆρ0_(t))

dt to its current value [1]:

d| ˆρ0(t))

dt = bΓ|ρ

0_{(t)) (2.20)}

The expression for the superoperator ˆΓ takes the form: b Γ = − Z ∞ 0 T rL  b L0_SL(t − τ )| ˆρeq_L)  dτ (2.21) where |ˆρeq

Li denotes the equilibrium lattice density operator:

|ˆρeq_L) = exp(−HL/kBT )

T rLexp(HL/kBT ) (2.22)

while T rLdenotes trace over the lattice variables. The transformation of equation

2.20 back to the laboratory frame leads to the Liouville space counterpart of the

Redeld relaxation equation [13]: d| ˆρ(t))

dt = [−i ˆLS+ ˆΓ]| ˆρ(t)) (2.23)

where the superoperator ˆΓ is given as: ˆ Γ = − Z ∞ 0 T rL  ˆ LSL[exp(−i( ˆLS+ ˆLL)τ ) ˆL]|ˆρeq_L)  dτ (2.24)

The matrix representation of the ˆΓ superoperator in the Liouville space contains

the Γαα0ββ0elements.

Finishing this section it is important to stress that equations 2.122.14 apply

only when the condition (referred to as the Redeld condition): ωSL · τc  1

applies. This condition implies that the relaxation is much slower than the motion

(characterized by the correlation time, τc) causing the relaxation. In consequence

the spin and lattice variables can be split (equation 2.16) and relaxation rates

can be dened as linear combinations of lattice density functions. Otherwise, one cannot explicitly dene the relaxation rates and Stochastic Liouville Equation (SLE) has to be applied.

16 CHAPTER 2. RELAXATION THEORY AND ESR LINESHAPE

2.3 Lattice Dynamics

Let a function φ(Ω, t) which varies randomly in time, describe the stochastic uctuations of the lattice. The stochastic process φ(Ω, t) can be characterized by

a probability density function P (Ω, t). Let the omega variable take the value Ω0

at time t0, then a conditional probability density function P (Ω, t|Ω0, t0) can be

dened. The conditional probability describes the probability that the molecule

takes the position ω at time t if its initial position (at time t0) was Ω0. For

stationary Markov processes [3234], the conditional probability depends only

on the time interval (τ = t − t0); it does not depend of the unitial t0 value.

For instance, rotational dynamics is described by the Markov operator ˆΓΩ given

as [32,34]:

ΓΩ= −DR∆2Ω (2.25)

where DR denotes rotational diusion tensor. In this case the symbol Ω denotes

a set of Euler angles that describe the molecular orientation, while ∆2

Ω denotes

Laplace operator.

The most important property of the operator of the rotational diusion is that its

eigenfunctions are spherical harmonics YL

M, which are related to Wigner

rotational-matrices DL

K,M [1,35]:

ΓΩYML = −L(L + 1)YML (2.26)

where −L(L + 1) are the corresponding eigenvalues.

Stationary Markov operators are used to describe the lattice dynamics. The lattice Liouville operator is dened by a sum of the terms [36]:

b

L_L= bL_R+ bL_D+ cLS

ZF S+ bLTZF S (2.27)

The termsLb_R and Lb_D are related to Markov operators describing the

reorienta-tion (R) and distorreorienta-tion (D) of the paramagnetic molecule. The last two terms in

equation 2.27correspond to the ZFS interaction (its static and transient

compo-nents, respectively) described in Chapter1. The superoperator Lb_Ris dened, for

an axially symmetric rotation motion as [1,3638]: b

LR= iΓR= −iDR,⊥Lb2− i(D_R,k− D_R,⊥) bL2_z (2.28)

where DR,⊥ and DR,k are coecients of rotational diusion perpendicular and

parallel to the symmetry axis;LbandLb_z refer to the total and the z component of

the angular-momentum operator. The coecients are related to the corresponding reorientational correlation times [1,39]:

DR,⊥= 1 6τR,⊥ (2.29) D_R,k= 1 6τR,k (2.30)

2.4. ELECTRON SPIN RESONANCE (ESR) LINESHAPE 17

For isotropic rotation the superoperator Lb_R can be expressed as:

b

LR= iΓR= −iDR∇2ΩP L (2.31)

where DRis the rotational diusion coecient related to the rank-two rotational

correlation time [1]:

DR=

1

6τR (2.32)

For transition metal complexes the superoperator Lb_D describes the distortional

motion of the ligand framework. The distortional dynamics is modeled as isotropic pseudorotational diusion (therefore this concept of describing the distortional

motion is referred to as pseudorotational model) [26,26,36,4044]:

b

LD = iΓD = −iDD∇2ΩPTL (2.33)

where DD denotes a constant related to the distortional correlation time:

DD =

1

6τD (2.34)

2.4 ESR lineshape

To describe ESR spectra for arbitrary motional conditions and interaction strengths (also beyond validity regimes of the perturbation approach) one can use the

the-ory based on the SLE referred to as the general thethe-ory [26,36,37,41,4547].

The theory can be applied to paramagnetic molecules of any symmetry, for any electron spin quantum number S, at various magnetic elds, and with arbitrary magnitudes of the dierent interactions present, like static and transient ZFS. The complete ESR lineshape function L(ω) (the symbol "L(ω)" should not be mistaken with lattice) is determined by single-quantum transition of the electron

spin and it is described by the expression [1,26,36,37,41,45,46]:

L(ωS− ω) = Z ∞ 0 T rS  S+[exp(−i bLLτ )S+]ρeq_S  exp(−iωτ )dτ (2.35) ∝ [S+]+[ cMESR]−1[S+]

where the absorption and dispersion spectra are represented by the real and

imag-inary parts, respectively; T rSdenotes trace operation over the spin variables. The

superoperatorMc_ESR is dened as [1,36,41,48]:

c

MESR= −i[ cLZ(S) + cLSZF S+ cLTZF S+ bLR+ bLD+ b1ω] (2.36)

while [Mc_ESR]describes its matrix representation in a basis |O_i). The basis vectors

18 CHAPTER 2. RELAXATION THEORY AND ESR LINESHAPE

|Oi) = |ABC ) ⊗ |LKM ) ⊗ |Σσ ) (2.37)

where |ABC ) represents the distortional states , |LKM ) describes the rotational states, while |Σσ ) represents the S spin variables. The ESR lineshape is

de-scribed by only one element of the matrix [Mc_ESR]−1 corresponding to the state

|ABC ) |LKM ) |Σσ ) = |000 ) |000 ) |1 − 1 )[1,26,36,3942].

The explicit denitions of the vectors are [1,26,36,41]:

|ABC) = |ABCihABC| = |ψA_BC) = r 2A + 1 8π2 D A BC(ΩPTPS) (2.38) |LKM ) = |LKM ihLKM | = |ψL_KM) = r 2L + 1 8π2 D L KM(ΩPSL) (2.39) |Σσ) =X m (−1)S−m−σ√2Σ + 1 S S Σ m + σ −m −σ ! |S, m + σihS, m| (2.40)

The elements of the matrix [Mc_ESR] are obtained using Wigher-Eckart theorem

[1,26,36,3942]. For the operators representing the discussed degrees of freedom,

they are given as [1,26,36,3942]:

(Oi| bLZ(S)|Oj) = (A0B0C0|(L0K0M0|(Σ0σ0| cLZ(S)|ABC)|LKM )|Σσ) (2.41) = δAA0δ_BB0δ_CC0δ_LL0δ_KK0δ_{M M}0δ_ΣΣ0δ_σσ0ω_Sσ (Oi| bLSZF S|Oj) = (A0B0C0|(L0K0M0|(Σ0σ0| bLSZF S|ABC)|LKM )|Σσ) (2.42) = δAA0δ_BB0δ_CC0√1 6(−1) σ+B0−C0 ˜ V2S(PS) |K−K0_|[(−1)Σ 0_+Σ − 1] p (2S + 3)(2S + 1)(S + 1)S(2S − 1)(2L0_{+ 1)(2L + 1)(2Σ}0_{+ 1)(2Σ + 1)} L0 2 L −K0 K0− K K ! L0 2 L −M0 M0− M M ! Σ0 2 Σ −σ0 B0− B σ ! ( Σ0 2 Σ S0 S S ) (2.43)

2.4. ELECTRON SPIN RESONANCE (ESR) LINESHAPE 19 (Oi| bLTZF S|Oj) = (A0B0C0|(L0K0M0|(Σ0σ0| bLTZF S|ABC)|LKM )|Σσ) (2.44) = (−1)σ+B0+K0−C0−M0V˜2S(PT) |B−B0_| 1 √ 6[(−1) Σ0+2+Σ_] p (2S + 3)(2s + 1)(S + 1)S(2S − 1)(2A0_{+ 1)(2A + 1)(2L}0_{+ 1)(2L + 1)(2Σ}0+1_{)(2Σ + 1)} A0 2 A −B0 B0− B B ! A0 2 A −C0 C0− C C ! L0 2 L −K0 C0− C K ! A0 2 A −M0 M0− M M ! Σ0 2 Σ −σ0 M − M0 σ ! Σ0 2 Σ S0 S S ! (Oi| bLR|Oj) = (A0B0C0|(L0K0M0|(Σ0σ0| bLR|ABC)|LKM )|Σσ) (2.45) = δAA0δ_BB0δ_CC0δ_LL0δ_KK0δ_{M M}0δ_ΣΣ0δ_σσ0iD_RL(L + 1) (Oi| bLD|Oj) = (A0B0C0|(L0K0M0|(Σ0σ0| bLR|ABC)|LKM )|Σσ) (2.46) = δAA0δ_BB0δ_CC0δ_LL0δ_KK0δ_{M M}0δ_ΣΣ0δ_σσ0iD_DA(A + 1)

Although the outlined method of modeling ESR lineshapes is specically dedi-cated to systems possessing ZFS interactions and undergoing rotational and dis-tortional dynamics, it captures the main concept of the SLE approach.

When the system can be described in terms of the Redeld relaxation theory, the calculations of ESR lineshape become simpler. The basis can be limited to spin variables. By diagonalizing the main (time-independent) Hamiltonian one

obtains the set of eigenfunctions, |ψαi, and eigenvalues, Eα, (energy levels) of the

electron spin. Pairs of the eigenfunctions form the Liouville space {|ψαihψβ|} of

the dimension (2S + 1)2_{. The matrix [S}

+]contains the representation coecients

of the operator S+ in the basis {|ψαihψβ|}, while the [Mc_ESR]matrix contains a

block of a dimension of (2S + 1) (so called population block) associated with spin-lattice electron spin relaxation and a diagonal part, associated with the spin-spin

relaxation [24,29]. The diagonal part (mainly determining the ESR lineshape)

consists of the elements:

[ cMESR]αβαβ = i(ω − ωαβ) + Rαβαβ α 6= β (2.47)

where Rαβαβ denotes the electron spin relaxation terms calculated in terms of

equation 2.13, while ωαβ = Eα− Eβ. The population block contains the Rαααα

Chapter 3

Perturbation description of

ESR lineshape

3.1 Perturbation approach

Analytical description of the ESR lineshape is possible only when the Redeld

limit is fullled [1,16,23,24,2931]. For nitroxide radicals this means that the

product of the amplitude of the anisotropic part of the hyperne scalar coupling

and the rotational correlation time is much smaller than 1: ωA,anisoτR  1. As

already explained, in the motional range in which the perturbation theory brakes

down, an approach based on the SLE can be used [22,26,36,37,41,4549]. In

this section the perturbation approach is applied to calculate ESR lineshape for nitroxide radicals in solution. Two types of nitroxide radicals are considered,

with isotope 15_{N and} 14_{N. For} 15_{N (P =} 1

2) ESR spectra consist of two lines,

while for 14_{N (P = 1) three lines are present. For nitroxide radicals in solution}

several electron spin relaxation mechanisms have been reported [5053], but the ESR lineshape function is mainly determined by the relaxation rates associated with the anisotropic part of the hyperne scalar coupling, while the isotropic part contributes to the energy level structure. The hyperne coupling is an eect of

the dipole-dipole interaction between the electron and nuclear spin of15_{N or}14_N.

For this system the Hamiltonian of the scalar isotropic hyperne interaction takes the form [5457]:

H_hf,iso(P, S) = HSC(P, S) = AP · S = A[PZSZ+

1

2(P+S−+ P−S+)] (3.1)

22 CHAPTER 3. PERTURBATION DESCRIPTION OF ESR LINESHAPE

A sum of the electron and nitrogen Zeeman interactions and the scalar coupling determines the energy levels of the P -S spin systems:

H0(P, S) = HZ(S) + HZ(P ) + HSC(P, S)

= ωSSZ+ ωPPZ+ A[PZSZ+

1

2(P+S−+ P−S+)] (3.2)

For P = 1/2 the Zeeman states |mS, mP >, where mS and mP denote the

mag-netic spin quantum numbers of the electron and the nucleus, respectively, can be

labeled as follows [5456,58,59]: |1i = |1 2, 1 2i |2i = | 1 2, − 1 2i |3i = | −1 2, 1 2i |4i = | − 1 2, − 1 2i (3.3)

Thus, the matrix representation of the static Hamiltonian in the Zeeman basis is given as (the nitrogen Zeeman interaction has been neglected because it is much

weaker than the electron spin Zeeman coupling) [5456,58,59]:

[H0(P, S)] =      ωS 2 + A 4 0 0 0 0 ωS 2 − A 4 A 2 0 0 A₂ −ωS 2 − A 4 0 0 0 0 −ωS 2 + A 4      (3.4)

The analytical expressions for the resulted eigenvalues Eα and eigenstates |ψαi

are presented in [54,58,60] and are given as follows:

|ψ1i = |1i = | 1 2i E1= ωS 2 + A 4 (3.5) |ψ₂i = a|2i + b|3i = a|1 2, − 1 2i + b| − 1 2, 1 2i E2 = q ω_S2 + A2 2 − A 4 (3.6) |ψ3i = −b|2i + a|3i = −b| 1 2, − 1 2i + a| − 1 2, 1 2i E3 = − q ω2 S+ A2 2 − A 4 (3.7) |ψ4i = |4i = | − 1 2, − 1 2i E4 = − ωS 2 + A 4 (3.8)

3.1. PERTURBATION APPROACH 23 where: a = 1 2[1 + ωS q ω2 S+ A ] 1/2 (3.9) b = 1 2[1 − ωS q ω_S2 + A ] 1/2

The operator S+ is represented in the Liouville space |ψαihψβ|, constructed from

pairs of these eigenfunctions, as [59]:

S+ ∝ a[|ψ1ihψ3| + |ψ2ihψ4|] − b[|ψ1ihψ2| − |ψ3ihψ4|] (3.10)

The ESR lineshape function L(ω) can be calculated from the expression:

L(ω) ∝ Re[S+]+[i∆ωαβ + Rαβαβ]−1[S+] (3.11)

where ∆ωαβ = ω+ωαβ while Rαβαβis the electron spin relaxation rates associated

with the coherence ραβ = |ψαihψβ|. In the high eld limit, A  ωSthe coecients

converge to a → 1, b → 0. The spectrum consists of two lines given as a sum of

two Lorentzian functions corresponding to the frequencies ω13 and ω24 which are

separated by A: LS(ω) ∝ T13 1 + (ω − ω13)2T132 + T24 1 + (ω − ω24)2T242 (3.12)

where T13 = R−11313, T24 = R−12424. The anisotropic part of the hyperne coupling

is represented in its principal axes system (P) by the Hamiltonian [55,56,59]:

H(P )_aniso= 2 X m=−2 (−1)mF_−m2 T_m2(P, S) (3.13) where F2 0 = 23[Azz − 1 2(Axx+ Ayy)], F±2 = 0, F±22 = 12(Axx− Ayy); Axx, Ayy, Azz

are Cartesian components of the hyperne tensor. The two-spin tensor operators,

T_m2 have the form [1,12,20,24,25]:

T₀2(P, S) = √1 6[2PZSZ− 1 2(P+S−+ P−S+)] T_±12 (P, S) = ∓1 2[PZS±+ P±SZ] (3.14) T±22 (P, S) = 1 2P±S±

To express the anisotropic term of the hyperne coupling in the laboratory frame: H(L)_aniso(P, S) =

2

X

m=−2

24 CHAPTER 3. PERTURBATION DESCRIPTION OF ESR LINESHAPE

the transformation given by equation 1.16 has to be applied. As a result one

obtains: F−m2(L)(t) = D20,−m(ΩP L)(t) r 2 3[Azz− 1 2(Axx+ Ayy)] (3.16) + (D_−2,−m2 (ΩP L) + D2,−m2 (ΩP L)) 1 2(Axx− Ayy) (3.17)

The angle ΩP L(t) describes the orientation of the principal axis system of the

anisotropic part of the hyperne coupling with respect to the laboratory frame,

which changes in time due to molecular rotation. The relaxation rates Rαβαβ

[1,24,29] can be calculated by means of the Redeld relaxation theory according

to equation 2.12[1,24,29,61]: R1313= R2424 = 1 6J (0) + 1 16J (ω12) + 1 8J (ω13) + 1 4J (ω14) + 1 24J (ω23) + 1 16J (ω34) (3.18) Assuming isotropic molecular tumbling the spectral density for the anisotropic hyperne coupling are given as:

J (ω) = 2 3[Azz− 1 2(Axx+ Ayy)] 2₊1 2(Axx− Ayy) 2 1 5 τR 1 + ω2_τ2 R (3.19) Due to the higher spin quantum number (P=1), the case of nitroxide radicals

with 14_{N isotope is dierent. The system is represented by six Zeeman states}

|mS, mPi: |1i = |1 2, 1i |2i = | 1 2, 0i |3i = | −1 2, −1i |4i = | − 1 2, 1i (3.20) |5i = | −1 2, 0i |6i = | − 1 2, −1i

In this case the matrix representation of the main Hamiltonian in the Zeeman

basis set is given as [55,56,59]:

[H0(P, S)] =           ωs 2 + A 2 0 0 0 0 0 0 ωs 2 A √ 2 0 0 0 0 √A 2 ωs 2 − A 2 0 0 0 0 0 0 −ωs 2 − A 2 A √ 2 0 0 0 0 +√A 2 − ωs 2 0 0 0 0 0 0 −ωs 2 + A 2           (3.21)

The energy levels Eα, and eigenvectors |ψαi are presented in [5860] and given

3.1. PERTURBATION APPROACH 25 |ψ1i = | 1 2, 1i E1= ωS 2 + A 2 (3.22) |ψ2i = a| 1 2, 0i + b| − 1 2, 1i E2= 1 2 r 9A2 4 + ω 2 S+ ωSA − A 4 (3.23) |ψ3i = c| 1 2, 0i + d| − 1 2, 1i E3= − 1 2 r 9A2 4 + ω 2 S+ ωSA − A 4 (3.24) |ψ4i = e| 1 2, −1i + f | − 1 2, 0i E4= 1 2 r 9A2 4 + ω 2 S− ωSA − A 4 (3.25) |ψ5i = g| 1 2, −1i + h| − 1 2, 0i E5= − 1 2 r 9A2 4 + ω 2 S− ωSA − A 4 (3.26) |ψ6i = | − 1 2, −1i E6 = − ωS 2 + A 2 (3.27)

with: a = (1 + α2₎1/2_{, b = αa, c = (1 + β}2₎−1/2_{, d = βc, e = (1 + γ}2₎−1/2_{, f = γe,}

g = (1 + δ2)−1/2, h = δg where: α = A + 2ωS− q 9A + 4AωS+ 4ω2_S 2√2A β = A + 2ωS+ q 9A + 4AωS+ 4ω_S2 2√2A γ = A − 2ωS+ q 9A − 4AωS+ 4ω2S 2√2A δ = A − 2ωS− q 9A − 4AωS+ 4ωS2 2√2A

The operator S+ represented in the Liouville space |ψαihψβ| takes the form [58

60]:

S+∝ b|ψ1ihψ2| + d|ψ1ihψ3| + e|ψ4ihψ6| + g|ψ5ihψ6|

+af |ψ2ihψ4| + ah|ψ2ihψ5| + cf |ψ3ihψ4| + ch|ψ3ihψ5| (3.28)

In the high eld limit, A  ωS the coecients a → d → e → h → 1, while

b → c → f → g → 0. In consequence the spectrum consists of three lines separated by A: L(ω) ∝ T14 1 + (ω − ωS− A)2T142 (3.29) + T25 1 + (ω − ωS)2T₂₅2 + T36 1 + (ω − ωS+ A)2T₃₆2

where the relaxation rates are given as the following combinations of spectral densities: R1414 = 2 3J A_{(0) +}1 4J A₍A 2) + 7 12J A_(ω S+ A 2) + 1 2J A_(ω S+ A) (3.30) R2525= 7 12J A_(ω S+ A 2) + 7 12J A_(ω S− A 2) + 3 8J A₍A 2) (3.31) R3636 = 2 3J A_{(0) +}1 4J A₍A 2) + 7 12J A_(ω S− A 2) + 1 2J A_(ω S+ A) (3.32)

26 CHAPTER 3. PERTURBATION DESCRIPTION OF ESR LINESHAPE

Below a comparison between the presented above perturbation formulae and the

SLE approach [4,5,6264] is presented. The rotational correlation time has been

progressively increased to reveal the validity range of the perturbation approach. As already explained the SLE approach is valid for arbitrary motional conditions, while for the perturbation approach it is required that the Redeld condition

is fullled, that implies the rotational correlation time τR < 2 · 10−9s (for the

case of nitroxide radicals). In Figures 3.2 and 3.1 ESR spectra for solutions of

14_Nand15_{Ncontaining nitroxide radicals obtained by means of the perturbation}

approach are compared with the predictions of the general theory of ESR line-shape for nitroxide radicals (S=1/2) [4](the software is available on the webpage: www.acert.cornell.edu). When the Redeld condition is not fullled, the ESR spectra can no longer be described as a superposition of a Lorentzian lines. The relationship between the lineshape and the spin and dynamical parameters are no longer straightforward. Details of the SLE approach for nitroxide radicals are

given in [4,5].

The simulations of X-band and Q-band spectra shown in Figure3.1correspond to

the range of correlation times: 2 · 10−9_{− 5 · 10}−9_{s. The spectra are in good}

agree-ment till the Redeld limit: τR = 2 · 10−9s. For τR ≈ 2.8 · 10−9s the lineshapes

are no longer in agreement as the perturbation approach breaks down.

0 , 3 3 8 0 , 3 3 9 0 , 3 4 0 0 , 3 4 1 0 , 3 4 2 0 , 3 4 3 τ_R= 2 * 1 0- 9 s τ_R= 5 * 1 0- 9 s τ_R= 3 . 5 * 1 0- 9 s τ_R= 2 . 8 * 1 0- 9 s τ_R= 2 . 3 * 1 0- 9 s M a g n e t i c F i e l d [ T ] 1 , 1 9 8 1 , 1 9 9 1 , 2 0 0 1 , 2 0 1 1 , 2 0 2 1 , 2 0 3 τR= 2 * 1 0 - 9 s τR= 5 * 1 0 - 9 s τR= 3 . 5 * 1 0 - 9 s τR= 2 . 8 * 1 0 - 9 s τR= 2 . 3 * 1 0 - 9 s M a g n e t i c F i e l d [ T ]

Figure 3.1: Comparison of ESR lineshapes for15_{N containing radicals in solution}

calculated by means of the perturbation and SLE approaches for dierent

rota-tional correlation times τR. Axx = 0.574mT, Ayy = 0.854mT, Azz = 4.676mT,

3.1. PERTURBATION APPROACH 27 0 , 3 3 7 0 , 3 3 8 0 , 3 3 9 0 , 3 4 0 0 , 3 4 1 0 , 3 4 2 0 , 3 4 3 τR= 2 * 1 0 - 9 s τR= 5 * 1 0 - 9 s τR= 3 . 5 * 1 0 - 9 s τR= 2 . 8 * 1 0 - 9 s τR= 2 . 3 * 1 0 - 9 s M a g n e t i c f i e l d [ T ] 1 , 1 9 6 1 , 1 9 7 1 , 1 9 8 1 , 1 9 9 1 , 2 0 0 1 , 2 0 1 1 , 2 0 2 1 , 2 0 3 τR= 5 * 1 0 - 9 s τR= 3 . 5 * 1 0 - 9 s τR= 2 . 8 * 1 0 - 9 s τR= 2 . 3 * 1 0 - 9 s τR= 2 * 1 0 - 9 s M a g n e t i c F i e l d [ T ]

Figure 3.2: Comparison of ESR lineshapes for14_{N containing radicals in solution}

calculated by means of the perturbation and SLE approaches for dierent

rota-tional correlation. Axx = 0.574mT, Ayy = 0.854mT,Azz = 4.676mT, g = 2.00,

B0= 0.34T(left), B0= 1.2T(right).

In Figure3.3experimental ESR spectra for propylene glycol solutions of 4 − oxo−

TEMPO − d16(black lines) are presented; the data are taken from [6]. Although,

the dominant mechanism of the electron spin relaxation in nitroxide radicals is the anisotropic hyperne coupling, the lineshape is also aected by g-tensor anisotropy, what could be seen from the dierent amplitudes of the lines. Figure

3.3 presents a comparison of lineshapes calculated using the perturbation

ap-proach and the general apap-proach. The perturbation treatment does not include the eect of g-tensor anisotropy. The perturbation theory does not work in this case anyway, due to the large amplitude of the hyperne coupling (40-50MHz

for 14_{N) combined with the rotational correlation time that is in the range of ns.}

Taking into account those facts the experimental results have been analyzed using the SLE approach.

Figure3.4shows the results of applying the SLE theory to a large set of ESR

spectra for 4 − oxo − TEMPO − d16−15N and 4 − oxo − TEMPO − d16−14N

in propylene glycol [6]. The obtained components of the hyperne tensor and

the g-tensor for 4 − oxo − TEMPO − d16−15Nare: Axx = Ayy = 14.85MHzand

Azz= 150.2MHzand for 4 − oxo − TEMPO − d16−14N: Axx= Ayy = 10.6MHz,

Azz= 107.3MHz, while the components of the g-tensor are: gxx = 2.009, gyy =

28 CHAPTER 3. PERTURBATION DESCRIPTION OF ESR LINESHAPE

Figure 3.3: ESR spectra for

4 − oxo − TEMPO − d16−15N

and 4 − oxo − TEMPO − d16−

14_{N dissolved in propylene glycol.}

Black lines-experimental data for 278 K, red lines  corresponding spectra obtained using the SLE approach, blue lines  spectra calculated using the perturbation approach (not including g-tensor ansisotropy). 0 . 3 4 1 0 . 3 4 2 0 . 3 4 3 0 . 3 4 4 0 . 3 4 5 0 . 3 4 6 1 4 N M a g n e t i c F i l d [ T ] 1 5 N 0 , 3 4 1 0 , 3 4 2 0 , 3 4 3 0 , 3 4 4 0 , 3 4 5 M a g n e t i c F i e l d [ T ] 2 9 4 K 2 8 5 K 2 8 3 K 2 7 8 K 2 7 3 K 2 6 8 K 2 6 3 K 2 5 8 K 2 5 3 K 2 4 8 K 0 , 3 4 1 0 , 3 4 2 0 , 3 4 3 0 , 3 4 4 0 , 3 4 5 2 9 2 . 3 K 2 8 4 . 5 K 2 7 8 K 2 7 1 . 5 K 2 6 4 K 2 6 0 K 2 5 4 K 2 5 0 K M a g n e t i c F i e l d [ T ]

Figure 3.4: Left: ESR spectra for 4 − oxo − TEMPO − d16−15N in propylene

glycol. Black lines-experimental data, green lines-ts using the SLE approach.

Right: ESR spectra for 4 − oxo − TEMPO − d16−14N in propylene glycol. Black

lines-experimental data, blue lines-ts using the SLE approach. Reprinted with permission from Kruk et al., J. Chem. Phys., 139, 244502 (2013), license number 4006190874784, AIP Publishing LLC.

3.1. PERTURBATION APPROACH 29

The rotational correlation times for 4-oxo-TEMPO-d16with isotopes14N and

15_{N in propylene glycol solutions obtained from the ESR lineshape analysis are}

presented in the Table 3.1[6]:

Table 3.1: The rotational correlation times for 4-oxo-TEMPO-d16 with isotopes

14_{N and} 15_{N in propylene glycol solutions [6]. Reprinted with permission from}

Kruk et al., J. Chem. Phys., 139, 244502 (2013), license number 4006190874784, AIP Publishing LLC. 15_N 14_N Temp (K) τR(s) Temp (K) τR(s) 294 1.39 · 10−10 _{292 1.41 · 10}−10 285 2 · 10−10 284.5 2.10 · 10−10 283 2.70 · 10−10 _{278 3.50 · 10}−10 278 4.09 · 10−10 _{271.5 5.53 · 10}−10 273 6.05 · 10−10 _{264 9.37 · 10}−10 268 7.98 · 10−10 _{260 1.21 · 10}−9 263 1.21 · 10−9 254 1.97 · 10−9 258 1.79 · 10−9 250 2.77 · 10−9 253 2.64 · 10−9 248 3.56 · 10−9

As expected, the correlation times are very similar  this shows the consistency of the analysis.

30 CHAPTER 3. PERTURBATION DESCRIPTION OF ESR LINESHAPE

3.2 Perturbation theory of ESR lineshape for the

elec-tron spin quantum number S = 7/2

In this section results of applying the perturbation theory to electron spin S = 7/2 in the presence of ZFS are presented. The basis appropriate for the spin quantum

number S = 7

2 consists of (2S+1)=8 functions formed by the Zeeman states:

|1i = | − 7₂i, |2i = | − 5₂i, |3i = | − 3₂i, |4i = | − 1₂i, |5i = |1₂i, |6i = |3₂i,

|7i = |5

2i, |8i = |

7

2i. The perturbation approach is valid in two situations. The

rst one is when the rotational correlation time is very long, which implies that the permanent part of the ZFS interaction contributes to the energy level structure of the spin, while the transient part acts as a relaxation mechanism:

H = H0+ H0(t) = HZ+ HSZF S+ HZF ST (t) (3.33)

As said in the previous section an analytical description of the ESR lineshape is possible only when the spin system is within the Redeld limit. This implies that for this case the product of the amplitude of the transient ZFS and the distortional

correlation time has to be smaller than one, ωT

ZF SτD  1. The matrix elements

of the static Hamiltonian H0(S) including the Zeeman coupling as well as the

static ZFS in the Zeeman basis set are explicitly given in Appendix A. The static

ZFS included into the main Hamiltonian changes the energy level structure of the spin (eigenvalues) and the corresponding eigenfunctions. Now they are given as

linear combinations of the Zeeman functions |mSi, with coecients depending on

the molecular orientation. In consequence the spin-spin and spin-lattice electron spin relaxation cannot be treated as independent processes. The second case describes the situation when the molecule rotates fast. Then the energy levels are determined only by the Zeeman interaction and the relaxation processes are caused by the transient as well as static ZFS:

H0(t) = HS_{ZF S}(t) + H_{ZF S}T (t) (3.34)

The Redeld condition is fullled for both, the static and transient ZFS

com-ponents (ωS

ZF SτR  1, ωZF ST τD  1). In Figure 3.5 a comparison between the

perturbation treatment and the SLE approach is presented. The agreement is very good as long as the Redeld condition holds.

3.2. PERTURBATION THEORY OF ESR LINESHAPE FOR THE ELECTRON

SPIN QUANTUM NUMBER S = 7/2 31

3 . 3 9 3 3 . 3 9 6 3 . 3 9 9 3 . 4 0 2 3 . 4 0 5 τR= 2 2 0 p s τR= 1 7 0 p s τR= 1 3 0 p s τR= 9 0 p s M a g n e t i c F i e l d [ T ] τR= 4 0 p s 3 . 4 0 4 3 . 4 0 5 3 . 4 0 6 τD= 1 9 0 p s τD= 1 5 0 p s τD= 1 1 0 p s M a g n e t i c F i e l d [ T ] τD= 7 0 p s

Figure 3.5: Comparison of ESR spectra for S = 7/2 calculated by using the SLE theory (black lines) and the perturbation approach (red lines); for the case

of fast rotation. Left: DS= 0.03cm−1, DT= 0.035cm−1, τD = 20ps. Right:

32 CHAPTER 3. PERTURBATION DESCRIPTION OF ESR LINESHAPE

3.3 4th order ZFS term

In this section the inuence of higher order terms on ESR lineshape is considered within the perturbation approach. For high spin systems, S ≥ 2, the fourth order ZFS term appears. The 4th order contribution to the ZFS Hamiltonian can be expressed in the molecular frame as follow [65]:

H4th_{ZF S} = a(4)₀ T₀(4)+ a(4)₂ (T₊₂(4)+ T₋₂(4)) + a(4)₄ (T₊₄(4)+ T₋₄(4)) = 4 X m=−4 (−1)ma(4)_−mT_m(4) (3.35)

The rank-four spin tensor components are dened as [66]:

T₀(4) = 1 2√70{35S 4 Z+ (5[5 − 6S(S + 1)])SZ2+ 3[−2S(S + 1) + S2(S + 1)2]} (3.36) T_±2(4)= 1 2√7S 2 ±(7SZ2 ± 14SZ+ [9 − S(S + 1)]) (3.37) T_±4(4)= 1 4S 4 ± (3.38) T_±3(4) = ∓ 1 2√2S 3 ±{2Sz± 3} (3.39) T_±1(4) = ∓ 1 2√14S±{14S 3 Z± 21SZ2 + [19 − 6S(S + 1)]SZ± 3[2 − S(S + 1)]} (3.40)

Again to calculate the Hamiltonian elements in the Zeeman basis, one has to apply the transformation from the molecular frame to the laboratory frame:

H4th(L)_{ZF S} = 4 X m=−4 (−1)mb(4)_−mT_m(4) (3.41) b(4)−m= 4 X k=−4 D(4)_k−m(ΩP L) (3.42)

where ΩP L denotes the orientation of the principal axis system of the ZFS tensor

with respect to the laboratory frame. The explicit forms of the Wigner rotation

matrices and the rank-four Hamiltonian elements are given in Appendix B. In

Figure 3.6 and Figure 3.7 examples of ESR spectra obtained by means of the

perturbation approach, including the rank-four contribution to the static ZFS Hamiltonian for the case of very slow rotational dynamics are presented. One can clearly see that the lineshapes are sensitive to the 4th order parameters of the ZFS tensor.