Chemically Induced Dynamic Electron Polarization

BY P. J. HORE, C. G. JOSLIN, AND K. A. McLAUCHLAN

1 Introduction

This review provides an account of the experimental and theoretical investigation of chemically induced electron spin polarization (CIDEP) in solution. The calculations which form the basis for its understanding are not repeated, but their assumptions and limitations are discussed, as is the extent to which they have been tested. Although new ways of interpreting the theories are sometimes presented, to expose their fuller implications, no basic changes are made. Emphasis is on the physical basis of CIDEP and on the analysis of results from diverse experiments; chemical results are reported but will be discussed more fully in a forthcoming review.

Spin polarization in solution, observed in transient radicals, arises from two basically different mechanisms. In the triplet mechanism (TM) polarization originates during inter-system crossing in the molecule, which eventually reacts to produce the radicals observed. In the radical-pair mechanism (RPM), polarization results from radical encounters, as does nuclear polarization in the chemically induced dynamic nuclear polarization (CIDNP) process. CIDEP involves an interplay of molecular motions and interactions and has yielded the spin-lattice relaxation times of both radicals and triplets, as well as radical and triplet reaction rate constants. It can be used to provide a label for following a radical reaction pathway and, as experiment improves, may allow the electron-exchange interaction to be investigated.

CIDEP has been reviewed previously, with particular reference to theory, and to chemical applications; two books contain valuable accounts of both CIDEP and CIDNP. In this review the two mechanisms are developed from simple physical models to their fullest extents. We proceed to the analysis of experiments through the Bloch equations and enquire of the extent to which the theories are proven; a brief account of experimental methods is included. Secondary polarization and relaxation receive due attention, and a concise record is provided of the results in chemistry and biology before some final speculation is made on possible further processes which affect observed polarizations.

The theory of CIDEP is intimately concerned with the populations of states, and as such is particularly amenable to the density-matrix formulation. For a time-dependent wavefunction as a complete set of basis functions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

an observable quantity is given by the ensemble average

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where Ô is the corresponding operator. The matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

can be thought of as a representation of an operator [??](t) and is known as the density matrix:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Diagonal elements of the matrix consequently represent occupancies of the various states, and any element is zero if either Cm or c*n is zero or their ensemble average is zero. From equation (2), the observable quantity is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

2 The Triplet Mechanism

The physical basis of the TM is summarized in Figure 1;it is observed only in irradiated systems. On irradiation of a precursor of less than cubic symmetry, polarization arises in the inter-system crossing (ISC) from the initially formed singlet into the triplet state; the low symmetry of the chromophore leads to unequal populations of the triplet sub-levels. This polarization appears in the radical products of the triplet reaction if it is quenched within its spin-lattice relaxation time, and may be large. The TM has received detailed theoretical analysis, which we recount before providing a physical interpretation.

Theory.- Initially, theories were of static arrays of triplets and yielded numerical calculations for molecules of any symmetry and a first-order perturbation expression for the magnetization of an ensemble of cylindrically symmetric triplets:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the magnetic field, D the zero-field splitting parameter, and P[perpendicular to] and P[??] are the normalized transition probabilities into the zero-field eigenstates [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [??]T[??]>(=Tz). This calculation demonstrated the two essentials for polari:zation, i.e. P[perpendicular to] ≠ P[?? and D≠0, but took no account of molecular rotation or triplet quenching.

Atkins and Evans 14 formulated their theory specifically to include these effects. They wrote the time-evolution of the triplet density matrix as the sum of terms representing its motion under the spin Hamiltonian

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

and the chemical reaction (assumed to occur at equal rates for the three sub-levels), and the differential ISC rates to the triplet sub-levels:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where i, j sum over x-z, kT is the pseudo-first-order quenching rate, k is the trace of the ISC rate matrix, and ks is the singlet quenching rate (negligible for aromatic carbonyls and quinones). The third term was expressed in this way for mathematical convenience. The two tensors Dij (the zero-field coupling) and Kij (the anisotropic ISC rate) can only be diagonalized simultaneously in cases of sufficiently high symmetry. Transformation to the rotating frame (the interaction representation), one stage of iteration, and ensemble averaging yields equation (8), where R is the Redfield tetradic.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The latter was incorrectly calculated originally and has been since corrected; the origin of the discrepancy lies in the reduction [??]of by virtue of Heisenberg level-broadening associated with the finite lifetime of the triplet state. The ensemble averaging assumed a random array of upper singlets, and would be inappropriate for irradiation with plane-polarized light (see below).

The radical polarization is deduced from equation (8) by Laplace transformation, possible because radical spin-lattice relaxation is much the slowest process involved. Defining the polarization as the difference in the populations of the spin states divided by their sum, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in Section 4, Bloch equations] , where is defined by equation (9) and S*Z, as defined by equation (10), is the initial triplet magnetization. Here D and E have their usual significance as zero-field terms, T1 is the triplet

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (l0)

spin-lattice relaxation time, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for axial symmetry), and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for axial symmetry. The two correlation times Tc and T for tumbling a bout the [??] and [perpendicular to] axes were taken to be Brownian, but the result is valid for any motion with an exponential correlation function. In the zero-motion limit, where E, kT, and τ'-1c, t'-1c [right arrow] 0, equations (9) and (10) yield equation (5).

A later theoretical analysis due to Pedersen and Freed purported to provide a theory of wider validity, pertinent to any tumbling rate provided D was small. Their analytic treatment closely parallels that given, but the Laplace transform was taken before the Redfield approximation was made. Expansion of the density operator transform in terms of spherical harmonics reduced the problem of finding the triplet magnetization to a matrix equation whose solution was known. The analytic solution is, however, complex, and the authors only indicated their approach. After re-arrangement, their expression for the triplet magnetization is (11), where T1 is the Redfield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

spin-lattice relaxation time, which is inadequate, as before; the true relaxation rate is obtained with the substitution [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] This equation differs from equation (10), but the two become equivalent in the limit MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]and E = 0.

This is the same limit in which Pedersen and Freed acknowledge their approach to be valid, but it is in fact more restrictive than that for the modified Atkins and Evans equation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The numerical calculations of Pedersen and Freed which give the dependence of S*z on kT and τ-1c for some values of D and ω0 are exact and free of the errors generated in the analytical treatment by premature truncation of an expansion.

Both density-matrix treatments predict the same numerical values unless kT exceeds the greater of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and in all experiments to date[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], whilst D is often <ω0, and the condition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for general validity is satisfied. If quenching occurs by a faster process than molecular diffusion, the modified Atkins and Evans expression should be used.

Adrian recognized a further contribution to polarization which can arise on irradiation with plane-polarized light: because of the anisotropy of the transition moments of a radical precursor, only those molecules in the correct orientation with respect to the light beam are excited to upper singlets. If intersystem corssing into the triplets is fast compared with molecular rotation, ISC occurs preferentially to a specific triplet level; this condition usually pertains (k ≈[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Lifetime broadening effects were specifically recognized, and for an axially symmetric system the magnetization in the triplet is given by equation (12), where x is the angle between

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

the electric vector of the light and the applied field. α= 1 for a parallel transition and -0. 5 for a perpendicular one. For isotropic irradiation, cos2 x over a sphere is 1/3, and the orientational terms vanish, but for excitation with an unpolarized directional source cos2 x = 1/2, and S*z changes by less than 10%. From a polarized source in the fast motion limit, S*z could double, but in the more usual slow motion limit an increase of 25% is expected. This result was later generalized to the case [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](µ is the transition dipole moment), although only two diffusion constants were used. The effect suggested a useful test of the operation of the TM (see below).

Interpretation.- We now return to the basic features of these calculations, and attempt to provide a physical rationale of them. First we demonstrate why the ISC step is more favourable into some triplet sub-levels than others. ISC occurs under the action of the spin-orbit coupling Hamiltonian, which can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13b)

where ζ is a mean coupling constant for the two unpaired electrons produced in the process. The symmetric first term is the usual approximation for the non-degenerate case and has the singlet (S) and triplet (T) functions as its spin eigenstates,, but the second, antisymmetric term, which is important when level crossing occurs, mixes spin and orbital functions of opposite parity into the wavefunction. It allows, for example, change from a singlet to a triplet level:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

However, as a result of the product in equation (13), concurrently with the change in the spin angular momentum, a change in the orbital angular momentum occurs. In fact, since all components of the total angular momentum, J, commute with H so, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

no component of J changes during the transition, and changing the spin momentum about an axis by [??]T) [left arrow] [??]S) must lead to a change in orbital momentum about that axis. Thus the molecular frame state [??]Tx), [??]Ty>, or [??]Tz> that is populated preferentially in the ISC is that where incipient spin momentum is most readily balanced by the generation of orbital momentum. For example, in the halogen molecules the important transition is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for which ΔΛ = ± 1: to ensure ΔΩ = 0, Δ[SUMMATION]= [- or +] 1, which implies that the perpendicular spin levels [??]T±> = 2-1/2([??]Tx)> [- or +] i[??]Ty>) are populated.

For chromophores with C2v symmetry, such as the carbonyl group in ketones and quinones, Atkins, using Group Theory, showed that for the ISC matrix element from S to the state Tα(α = x , y , z) not to vanish

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

This implies, as did the previous discussion, that the final state has a component of the same total angular momentum transformation properties as the initial one. For example, for a 1ππ* [right arrow] 3nπ* transition, using spin-orbit representations which are the direct products of the orbital and spin representations, equations (17) and (18) are applicable, since the orbital triplet

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

states all transform as A2. Inserting the representations of the triplet spin states shows that ISC is allowed only to Tz; this is also obtained for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] transitions. Consequently, P[??] = 1 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] although vibronic coupling modifies this, the result P[??]>P[perpendicular to] remains.

The zero-field parameter D is generally negative for carbonyl compounds, and with P[??] > P[perpendicular to] the sign of D(P[perpendicular to] - P[??] is positive, suggesting emissive signals, as are usually observed.

We now turn to the problem of how a polarization that is established in the molecular frame can be manifest in the laboratory frame. Quite irrespective of the randomness or otherwise of the molecular array, no magnetization can arise in zero field, but in any e.s.r. experiment an applied field is present, and here the situation changes. We consider an axially symmetric system with D > 0 and refer to the states defined in Figure 2, with the zero-field situation shown on the left. We imagine a small magnetic field applied as a perturbation (gµBB [??] D); this lifts the degeneracy of the T[perpendicular to] levels but leaves T[??] unaltered in energy. If the molecular axes make angles θ and [empty set] with the field direction, the zeroth-order eigenstates are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19c)

where [??]1>, [??]0>), and [??]-1> are the Zeeman (high-field) states. The first-order energies are given in Figure 2.