Per-Olof Westlund*a and Håkan Wennerströmb
aDepartment of Chemistry, Biological Chemistry, Umeå University, SE 901 87 UMEÅ, Sweden. E-mail: per-Olof.westlund@chem.umu.se
bDivision of Physical Chemistry, Chemical Center, Lund University, P.O. Box 124, SE 221 00 Lund, Sweden. E-mail: Hakan.Wennerstrom@fkem1.lu.se
First published on 6th November 2009
The low field ESR lineshape and the electron spin–lattice relaxation correlation function are calculated using the stochastic Liouville theory for an effective electron spin quantum number S = 1. When an axially symmetric permanent zero field splitting provides the dominant relaxation mechanism, and when it is much larger than the rotational diffusion constant, it is shown that both electron spin correlation functions 〈S1n(0)S1n(t)〉 (n = 0,1) are characterized by the same relaxation time τS = (4DR)−1. This confirms the conjectures made by Schaefle and Sharp, J. Chem. Phys., 2004, 121, 5287 and by Fries and Belorizky, J. Chem. Phys., 2005, 123, 124510, based on numerical results using a different formalism. The stochastic Liouville approach also gives the paramagnetically enhanced nuclear spin relaxation time constants, T1 and T2, and the ESR lineshape function I(ω). In particular, the L-band (B0 = 0.035 T) ESR spectrum of a low symmetry Ni(II)-complex with a cylindrical ZFS tensor is shown to be detectable at sufficiently slowly reorientation of the complex. The analysis shows that the L-band spectrum becomes similar to the zero-field spectrum with a electron spin relaxation time τS = (4DR)−1.
The dominant relaxation mechanism for electron spin systems with S≥ 1 is due to the zero field splitting (ZFS).1 For octahedral hexa aquo complexes, [M2+(H2O)6], the ZFS interaction is zero by symmetry. However, there are fluctuations in the ZFS, caused by solvent collisions. For flexible paramagnetic complexes of low symmetry, the ZFS interaction has a fast fluctuating transient ZFS with a non-zero average. This residual ZFS is called the permanent ZFS, with a magnitude that changes through rotational diffusion. The collision induced distortions of the coordination sphere thus modulate the transient ZFS and are typically in the sub ps to ps time regime. These timescales of the transient ZFS interaction were recently demonstrated in an MD simulation of the fluctuating electric field gradient of the Gd(H2O)3+8 complex.9 The reorientation of the complex, on the other hand, is expected to take place on much a longer time scale of typically 30–100 ps. This difference in time scale makes it possible to decompose the ZFS Hamiltonian into permanent and transient ZFS terms. In recent developments of the PRE theory, both transient and permanent ZFS are taken into account.10–13
We have previously presented a PRE formalism based on the stochastic Liouville equation (SLE).14 Within this formalism one can account for slow motion effects in the electron spin system. It is straightforward to generalize the formalism to explicitly describe the electron spin–lattice and spin–spin relaxation processes. In the present paper we focus on the electron spin relaxation properties of rigid paramagnetic complexes (S = 1), with only a permanent cylindrical ZFS interaction. We focus on the electron spin–lattice relaxation and the ESR line shape for slow tumbling complexes in the low field regime, where the Zeeman interaction is small compared with the permanent ZFS interaction. In this regime the traditional Bloembergen–Morgan theory5 is not valid.
The problem of interest in this work emerged from the work by Schaefle and Sharp15 and Fries and Belorizky.16 Their theoretical descriptions differ from the SLE approach, and based on their numerical results, the authors of these two papers make an interesting conjecture. In the low field limit, when the ZFS interaction (in rad s−1) also dominates the rotational diffusion constant DR, the spin relaxation time obeys the relation τS≈ (4DR)−1. The aim of this paper is to use the stochastic Liouville approach17 in the Fokker–Planck form19 to rationalize this result. We note that for the transverse relaxation, Lynden-Bell made a similar observation using the SLE approach already in 1971.18
The SLE formalism determines the orientation dependent spin density matrix operator, σ(Ω,t), from which several observables may be determined: (a) the ESR-lineshape function; (b) the electron spin–lattice correlation function, C0(t); and (c) the spectral density determining the PRE nuclear spin relaxation rates. All these measurable quantities are given by different matrix elements of a Liouville matrix. This illustrates an important strength of the SLE formalism. The calculations of the electron spin–lattice relaxation at low field are compared with the results of Schaefle and Sharp15 and Fries and Belorizky.16 In particular, the SLE calculations of C0(t) are obtained in the Fourier–Laplace space as 0(ω) = ∫∞0C0(t)eiωt dt, and compared with the Fourier–Laplace of the analytical approximation obtained in ref. 16.
The influence of electron spin relaxation on the paramagnetically enhanced water proton spin T1 NMR dispersion profile is often predominant. However, the electron spin relaxation time relation; τS≈ (4DR)−1, was not evident in the NMRD profiles first calculated.13,14 It is thus interesting to note that the relation τS≈ (4DR)−1 was revealed in numerical calculations of electron spin–lattice relaxation in the time domain but no attempt was made to rationalize the observation.15,16 Instead it turns out that the simple relation can be clearly explained within the SLE formalism. Different theoretical approaches may shed a different light on the same phenomenon.
A comparison between SLE14 and other simulation approaches15,16 has recently been presented in the context of paramagnetically enhanced nuclear spin relaxation.20 The low-field PRE data is complicated by the presence of a strong correlation between electron spin relaxation and molecular reorientation present in the electron spin dipole–nuclear spin dipole coupling. Thus using PRE data to study the electron spin relaxation may be complicated by this correlation effect. In fact, in some cases, it can contribute as much as 40% to the total relaxation enhancement.21 Consequently, the decomposition approximation12 is a prerequisite for making a direct relation between electron spin relaxation processes and PRE data.
The paper is organized as follows: Firstly we review the Liouville theory and show how spin dynamic information is related to the actual Liouville matrix. Then, the numerical results are presented and discussed in terms of electron spin correlations functions at zero field and the ESR lineshapes at low and intermediate fields. From the Liouville matrix we show that the relation between the electron spin relaxation time τS at low fields and the reorientation diffusion constant, τS = (4DR)−1, is obtained from a perturbation treatment of the Liouville matrix. Finally, we show that an L-band ESR lineshape of a rigid low-symmetric and slow tumbling Ni(II)-complex (where the rhombic term E = 0) may give rise to a detectable narrow ESR spectrum.
(1) |
σ(0) = i[(Ω) − 1ω](ω,Ω), | (2) |
(3) |
(Ω,ω) = [i0−iω1 + iZFZ(Ω) −D∇2Ω]−1σ(Ω,0) ≡ [−iω1]−1σ(Ω,0). | (4) |
(5) |
The molecular reorientational space is spanned by the symmetric top eigenfunctions.13,18
(6) |
The SLE equation of eqn (4) is expressed in this representation, where the expansion coefficients of the density matrix are {n(ω)}:
(7) |
(8) |
(9) |
The paramagnetically enhanced nuclear spin lattice relaxation rate is determined by a rather complex electron spin-nuclear spin dipole–dipole spectral density KDD1,−1(ωI).12–14,23 It is obtained from a 3 × 3 submatrix of the Liouville matrix defined by the basis operators {|1,p)|2,1 −p)} (p = 1,0,−1):
(10) |
Here ℏ is the Planck constant divided by 2π, ωI is the proton Larmor frequency, rIS is the I–S interspin distance and γI, γS and μ0 are the nuclear magnetogyric ratio, the electron magnetogyric ratio, and the permeability of space, respectively. The quantity, Tr{(S1qD20,1−q(Ω))*e−iτS1pD20,1−p(Ω)} is the composite lattice electron spin-reorientation correlation function.
When the spin dynamics and the reorientation dynamics are uncorrelated one can apply the decomposition approximation:
(11) |
However, when B0≈ 0, the electron spin is trapped in a molecule-fixed principal frame of the ZFS-tensor. In the absence of an additional collisionally modulated ZFS tensor, and f0τR≥ 1, the reorientational motion and the spin dynamics become strongly correlated. Consequently, the decomposition approximation, (cf.eqn (11)), is not applicable in this case. In PRE calculations of slowly reorienting low symmetric paramagnetic complexes it has been shown that the reduced spectral density, , was equal to unity in the whole low field region (f0τR > 1).13,14,23 = 1, represents the case when the effective dipole–dipole correlation time is equal to τR≡ (6DR)−1. This result is independent of the electron spin quantum number S≥ 1.13,14,23 We may estimate the correlation contribution by assuming for a moment that the spin relaxation and the molecular reorientation of the dipole–dipole correlation function are decomposed. Then the effective correlation time of eqn (11) may be defined as 1/τeff = 1/(τS(=τ(2)R) + 1/τ(2)R) which yields = ⅗. Consequently, the correlation effect in the dipole–dipole correlation function contributes with ⅖ or 40%.
For S = 1 and axial symmetry, Fries and Belorizky16 investigated the low field regime, B0 = 0, using their direct simulation approach. The electron spin correlation function was calculated with ZFS interaction ranging from D (≡√()f0) = 0.01, 0.1, 1.0 cm−1 and a reorientational correlation time τR(≡6DR−1) = 100 ps. These illustrative cases correspond to the Kubo parameter f0τR = 0.15, 1.54 and 15.4 which means that two cases correspond with slow-motion regime. The electron spin correlation function decays exponentially for D = 0.01 cm−1, whereas for D = 0.1 and D = 1 cm−1, it displays a damped oscillation. According to the authors of ref. 16, the electron spin correlation function can be fitted to the static limit expression with an exponential relaxation term reading:16
(12) |
(13) |
When f0τR≫1, (ωD is the frequency of the ZFS system with splitting D) and setting the electron spin–lattice relaxation time to τS = τR, eqn (13) displays a dispersion at ωτR≥ 1, and a Lorenzian peak centered at ωD, which has a full width at half height equal to 2/τR. Fig. (1a) displays the spin–lattice spectral density 0(ω), which is obtained from the SLE theory (cf.eqn (9)) at low field, B0 = 0.0035 T and τR = 100 ps. It may be compared with b), which displays the approximate spectral density eqn (13). In (c) the dispersion is shown, thus confirming that in the rigid limit, eqn (13) becomes identical with SLE. When the static field is increased to 0.35 T (X-band), the marked peak at ωD almost disappears in the SLE calculation (not shown). In the limit of large ZFS interaction we could also confirm that for electron spin quantum number S = , S = and S = 7/2, the electron spin relaxation time approaches τR. Consequently, the analytical form of eqn (13) for the electron spin–lattice spectral density only conforms to the SLE calculations in the zero field limit when the condition, ωDτR≫ 1, also applies.
Fig. 1 The spectral density 0(ω) is displayed for f0 = 0.1 cm−1 and τR = 100 ps. In (a) the SLE result (cf.eqn (9)) and in (b) eqn (13) and in (c) the function . |
Fig. 2 displays the calculated ESR-lineshape function I(ω) for S = 1 corresponding to X-band, L-band and B≈ 0 together with the absorption lineshape for f0 = 0.1 cm−1 and the reorientational correlation time τR = 100 ps. Interestingly, the ESR- and the absorption lineshapes become very complex as we pass from X-band (0.35 T) in (2a), L-band (0.035 T) in (2b) to the zero field cases shown in (2c) and (2d). In Fig. 2d the reorientational correlation time is 500 ps and the central ESR signal now displays a nice Lorentzian. These findings may open up the possibility of recording the ESR signal of rigid low symmetric Ni(II) complexes. Generally Ni(II) complexes are not detectable at high fields because of their large ZFS interaction (i.e. >1 cm−1). Consequently, for low field ESR spectra, such as L-band, we have HZFS≫HZeem and an ESR spectrum should be similar to a zero field spectrum. Fig. 3 displays a L-band ESR spectrum, calculated for f0 = 1.0 cm−1 and τR = 10 ns. It is centered at the ZFS frequency ωD. In principle, when the ZFS is expected to be much larger than the Zeeman interaction, an otherwise unobservable ESR spectra may be detected if it is possible to slow down the reorientational dynamics of the reorientation modulated ZFS sufficiently to yield a narrow Lorenzian line shape. When f0τR increases, the L-band ESR signal of Fig. 3 becomes more and more Lorenzian with a full at half height approaching .
Fig. 2 (a–d) The ESR spectra and the corresponding absorption lineshapes are displayed for S = 1. In (a) the X-band (B0 = 0.33 T) and in (b) the L-band (B0 = 0.035 T) spectra. In (c) and (d) the zero field spectra all calculated for the reorientation correlation time τR = 100 ps and the ZFS interaction f0 = 0.1 cm−1 except for (d) where τR = 500 ps. |
Fig. 3 The L-band ESR and absorption spectra (S = 1) are displayed centered around the ZFS frequency ωQ for the interaction f0 = 1.0 cm−1 and τR = 1.0 ns. |
In the zero field limit the electronic states remain unaffected in the molecular frame. This remains true as long as one does not consider additional relaxation mechanisms for the ZFS degrees of freedom. Such generalizations have been made listing a variety of methods.24–27 In the model used in the present paper, as well as in ref. 15 and 16 it is only when the spin system is seen from the laboratory fixed frame that there are spin relaxation dynamics. This is due to the rotational diffusional motion. Consequently, the electron spin relaxation is directly determined by the rotational diffusion constant.18
It is more involved to see the origin of the proportionality factor ¼. The underlying mechanism can be traced back to the fact that the ZFS Hamiltonian leads to a coupling between second and first rank irreducible tensor operators. In her paper from 1971, Lynden-Bell gives the relevant matrix elements using ‘low field’ basis operators. Owing to the overall rotational symmetry of the combined ZFS and diffusion operators, the corresponding matrix is block-diagonal in the limit of zero magnetic field. For a cylindrical symmetric ZFS-tensor one obtains a 3 × 3 matrix.18 This matrix is decomposed into a ZFS(F) and a diffusion D component: M = F + D. Now F is diagonalized by a transformation T. In the slow reorientation regime, when the ZFS is much larger than DR, the effect of the diffusion can be treated as a first order perturbation,
M−1ii = (T−1MT)ii≈ (T−1FT)ii + (T−1DT)ii, | (14) |
This journal is © the Owner Societies 2010 |