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1

identity matrix

γI

nuclear magnetogyric ratio

γS

electron magnetogyric ratio

δ

chemical shift

δcs

contact shift

δdia

diamagnetic chemical shift

δhs

hyperfine shift

δpcs

pseudocontact shift

Δt2

mean squared fluctuation of the ZFS

Δνdfs

dynamic frequency shift

Δνdia

diamagnetic residual dipolar coupling

Δνrdc

paramagnetic residual dipolar coupling

Δν0EPR

overall breadth of the EPR spectrum

Δνhom

homogeneous linewidth of an EPR spin packet

Δσdip

dipolar shielding anisotropy

Δχ

magnetic susceptibility anisotropy tensor

Δχax

axial magnetic susceptibility anisotropy

Δχrh

rhombic magnetic susceptibility anisotropy

Δωp

paramagnetic chemical shift of solvent ligand nuclei

ΔΕZFS

energy shift due to zero-field splitting

ε

DNP enhancement factor

ζ

electron spin tilting angle due to electron–electron coupling in the ZQ space

ηα/β

branching angle

ηI(J)

nuclear Overhauser enhancement

μ0

permeability of the vacuum

μB

electron Bohr magneton

μI

nuclear magnetic moment

μS

electron magnetic moment

μS

electron induced magnetic moment

ξ

Overhauser effect coupling factor

ρ

electron spin density at the nucleus

ρI/ρS

longitudinal relaxation rate of spin I/S

σDIP

Dipolar shielding tensor

σFC

Fermi contact shielding tensor

σhs

hyperfine shielding tensor

σI(J)

cross-relaxation rate

σIS

electron–nuclear cross-relaxation rate

τ1e

longitudinal electron relaxation time

τ2e

transverse electron relaxation time

τc

correlation time

τD

diffusional correlation time

τe

electron relaxation time

τl

correlation time for the local motions

τM

exchange time

τr

reorientation correlation time

τv

correlation time for electron relaxation

χ

magnetic susceptibility per molecule

χiso

isotropic part of the magnetic susceptibility

χq

quadrupolar coupling constant

ωI

nuclear Larmor frequency times 2π

ωS

electron Larmor frequency times 2π

ω1S

electron Rabi frequency times 2π

ωµw

microwave frequency times 2π

ωZFS

zero-field splitting shift in units of frequency times 2π

ΩS

EPR frequency offset in the µw rotating frame

A

hyperfine coupling operator

ADIP

dipole–dipole operator

AFC

Fermi-contact operator/coupling constant

A

secular hyperfine coupling constant

B0

external magnetic field in tesla (magnetic induction)

B1

oscillating μw field in tesla (magnetic induction)

B

pseudo-secular hyperfine coupling constant

d

distance of closest approach

D

axial zero-field splitting constant

D

ZFS tensor

Dee

electron–electron coupling tensor

dee

electron–electron dipolar coupling constant

Ddee

diagonal e–e coupling element

D0ee

off-diagonal e–e coupling element in the ZQ space

DM/DL

diffusion coefficients of the paramagnetic molecule/ligand molecule

E

rhombic zero-field splitting constant

f

Overhauser effect leakage factor

fM

mole fraction of ligand nuclei in bound positions

g

g tensor

g

g factor

ge

free-electron g factor

ħ

Planck's constant divided by 2π

Hhc

hyperfine coupling Hamiltonian

HSE

solid effect Hamiltonian

HCE

cross effect Hamiltonian

HΔ

electron–electron ZQ Hamiltonian

HΣ

electron–electron DQ Hamiltonian

HZFS

zero-field splitting Hamiltonian

I

nuclear spin

Iz

expectation value of Iz

Iz0

expectation value of Iz in thermal equilibrium

1J

J-coupling

Jee

electron–electron exchange interaction constant

J(ω,τ)

spectral density function

k

Boltzmann constant

k

rate matrix

K

cross effect transition moment

KD

dissociation constant

kex

exchange rate

KM

Michaelis constant

koff

dissociation rate constant

kon

association rate constant

M

magnetization

mI

nuclear magnetic spin quantum number

mS

electron magnetic spin quantum number

n

orientation vector in the unit sphere

NA

Avogadro's constant

Nα/β

population number of spin ground/excited state for I

P

probability tensor

pA

weighting factor for A

q

hydration number

r

vector connecting the positions of the nuclear and electron spins

r1

longitudinal relaxivity

R

relaxation matrix

R1/R2

longitudinal/transverse nuclear relaxation rate

R1ρ

nuclear longitudinal relaxation rate in the rotating frame

R1dia/R2dia

diamagnetic longitudinal/transverse nuclear relaxation rate

R1M/R2M

paramagnetic enhancement of the nuclear longitudinal/transverse relaxation rate

R1MCurie/R2MCurie

longitudinal/transverse Curie spin relaxation rate

R1Mdip/R2Mdip

longitudinal/transverse dipole–dipole relaxation rate

R1MFC/R2MFC

longitudinal/transverse Fermi contact relaxation rate

R1p/R2p

paramagnetic enhancement of the longitudinal/transverse relaxation rate of ligand nuclei

R1pOS/R2pOS

paramagnetic outer-sphere longitudinal/transverse relaxation rate of ligand nuclei

Rex

exchange contribution to relaxation

s

EPR saturation factor

s

null average part of the electron spin

S

electron spin

SAccc

solvent accessibility

SC

thermal average of the electron spin

SLS

Lipari–Szabo parameter

SS

effective electron spin dyadic

Sz

expectation value of Sz

S/N

signal-to-noise ratio

s.t.

subject to

t

time

T

temperature

T1M

longitudinal nuclear relaxation time due to the electron–nucleus interaction

T1q

quadrupolar longitudinal relaxation time

TB

time constant for exponential build-up of polarization

Tr

trace

v

reaction rate

Vmax

maximal rate achievable at saturating concentration of the substrate

w0

electron–nuclear ZQ transition probability induced by fluctuation of hyperfine coupling

w1

nuclear SQ transition probability induced by fluctuation of hyperfine coupling

w2

electron–nuclear DQ transition probability induced by fluctuation of hyperfine coupling

nuclear SQ transition probability not induced by hyperfine coupling

wij

transition probabilities

Wmax

largest weight

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