Calculation of ESR Spin-Spin Relaxation Times ( 1 / T _ 2 ) Transition Metal Ion Complexes : A DFT Application

Only a limited experimental ESR research had been carried out in this field because high values of spin orbit constants of transition metal ions which provide an important energy transfer mechanism would affect the values of ESR parameters (especially Aten) of their complexes. Therefore, theoretical predictions were useful. DFT implemented in ADF: 2012.01 was applied by giving a set of commands like Single Point, LDA, Default, Spin Orbit, ZORA, Unrestricted, None, Collinear, Nosym using TZP or TZ2P Basis sets in its ESR/EPR/EFG/ZFS Program after optimization of each one of 141 complexes to obtain their ESR parameters: g11, g22, g33, giso, a11, a22, a33, Aten. ESR SpinSpin Relaxation Times (1/T_2) whose values, to the best of our knowledge, were never reported before were, then, calculated from the giso values of the complexes.


INTRODUCTION
Magnetic resonance was associated with a typical a problem not encountered in higher energy forms of spectral techniques as the two spin levels were nearly equally populated [1][2][3] as at 298K with value: N upper /N lower =0.9986 at 3000G.But even this slight excess population in the lower level would lead to energy absorption.In order to maintain a population excess in the lower level, electrons from upper level gave up energy to return to lower level (Maxwell-Boltzmann law).This energy releasing process was called spin relaxation process.The relaxation processes were two types: spin-lattice relaxation and spin-spin relaxation.
Spin-lattice relaxation [1][2][3] Spin-lattice implied the interaction between the species with unpaired electrons, called "spin system" and the surrounding molecules, known as "lattice".The energy was dissipated within the lattice as vibrational, rotational or translational energy.The spin lattice relaxation characterized by a relaxation time T_ 1 , was the time for the spin system to lose 1/e th of its excess energy.Spin-lattice rate constant was equal to 1/T_ 1 .Rapid dissipation of energy (short T_ 1 ) was essential to maintain the population difference of the spin states.Slow spin-lattice relaxation, which frequently occurred in systems containing free radicals, especially at low temperatures, might cause saturation of the spin system which implied that the population difference of the upper and lower spin states approached zero and EPR signal would cease.Systems with a long T_ 1 were weakly coupled to the surroundings and thus would be easily saturated while those with shorter T_ 1 were more difficult to saturate.The effect of saturation was interpreted by a set of macroscopic equations proposed by Flex Bloch (1946) to calculate nuclear magnetization (M) as a function of relaxation times T_ 1 and T_ 2 .
As spin-orbit coupling provided an important energy transfer mechanism, it was found that odd-electron species with light atoms (organic radicals) possessed long T_ 1 s while those with heavier atoms(transition metal ions) had shorter T_ 1 s.

Spin-spin relaxation 1-3
In Spin-spin relaxation (Cross relaxation), energy exchange takes place between electrons in a higher energy spin state and nearby electrons or magnetic nuclei in a lower energy state, without transferring to the lattice.Analogously, the spin-spin relaxation was characterized by spin-spin relaxation time T_ 2 .
Ideally, both spin-spin and spin-lattice relaxations would contribute to the EPR signal.Resonance line width (∆H) or line width or half line width was the distance measured from the line's center to the point at which absorption value had half of maximal absorption value in the center of resonance line.It was represented as: ∆H 1/T_ 1 +1/T_ 2 ...(a) When T_ 1 >> T_ 2 , ∆H depends, primarily, on spin-spin interactions.
The following points were helpful to compare the two relaxation times: (a) Spin-Lattice (T_ 1 ) was known as longitudinal relaxation, or relaxation in the z-direction and Spin-Spin (T_ 2 ) was called transverse relaxation or relaxation in the x-y plane.Decreasing the spin-spin distance, which represented the spin concentration, T_ 1 would become very short i.e. less than 10 -7 second.Spin-lattice relaxation has a larger influence on the line width than spin-spin relaxation (b)In some cases, EPR lines were broadened beyond detection.When a spin system was weakly coupled to the lattice, i.e. the system possessed a long T_ 1 ; electrons had no time to return to the ground state.The population difference of two levels would tend to approach zero to decrease the intensity of EPR signal.This effect, called saturation, could be avoided by exposing the sample to low intensity microwave radiation.
Systems with shorter T_ 1 are more difficult to saturate.
(c)T_ 2 would represent the loss of phase coherence among nuclei.T_ 2 was less than or equal to T_ Short T_ 1 means NMR signal can be acquired faster.

Methodology
ESR technique was used to calculate ESR Spin-Spin Relaxation times [T_ 2 ] of the 1 st , 2 nd and 3 rd transition metal ion complexes with the help of ADF 2012.01 by installing it on Windows XP.

ESR Parameters 4-6
After optimization of complexes, the ADF software was run by using Single Point, LDA, Default, Spin Orbit, ZORA, Unrestricted, None, Collinear, Nosym using TZP or TZ2P Basis sets in ESR/EPR Program to obtain ESR parameters: g 11 , g 22 , g 33 , g iso .The g iso , values of metal ions (g M n+ ) were, then, used to calculate (1/T_2) values of complexes.Then from (5):

CONCLUSION
As expected, the spin-spin relaxation times of all the 141 complexes fall in picosecond range.