Rotational motions of biradical metal chelates in polar solvents

Chemical Physics 45 (1980) 85-90

0 North-Holland Publishing Company

ROTATIONAL MOTIONS OF BIRADICAL METAL CHELATES IN POLAR SOLVENTS Giorgio MORO *, Pier Luigi NORDIO

and Ulderico SEGRE

Istituto di Chimica Ftiica. Universith df Padova, Padua. Italy

Received 22 June

1979

ModeIs of motion continuousIy varying from random jumps to rotationa diffusion have been tested to simulate the ESR Iineshapes of biradiul metal chelates spin probes in the slow-motion region. The evidence of a diffusional regime for the molecular reorientations contrasts with the anomalous temperature dependence of the process.

1. Introduction Very recently, the ESR spectral profiles of metal chelates in the triplet state have been investigated in the entire range of molecular mobility, by using solutions which give rigid glasses at low temperatures [I]. The spectra were simulated with reasonable accuracy by a method suggested by Norris and Weissman, which is essentially equivalent to a fmite difference solution of the stochastic equation, when a diffusion mode1 is adopted for the molecutar reorientation and the non-secular terms are neglected in the hamiltonian [2,3] _These assumptions appear to be quite reasonable, since the zero field splitting in all the chelates is relatively small, and the probe molecules are large compared with the bath molecules so that they may be expected to behave as brownian particles. It is therefore surprising that the temperature dependence of the rotational correlation time does not obey the Stokes-Einstein-Debye relation, which should hold in the hydrodynamic limit [4] : in fact the molecular reorientation is indeed found to be described by an activated process, but its activation energy is much greater than the activation energy for the viscosity of the pure solvent [l] . Before considering other possible explanations for ibis discrepancy, one should check carefully how much the secular approximation is reliable at the different tumbling * Present address: Department of Chemistry, Backer Laboratory, Ithaca, NY, USA.

frequencies, and if it may happen that the spectra be reproduced equally well by using molecuIar reorientation models different from the brownian diffusion. A possibility one might consider is that reorientations occur through jumps of relativery large size. A modei for this motion is one in which the molecules are unable to move unless their environment undergoes a gross rearrangement, so that the rotational correlation time is essentially given by the lifetime T of the microstructures in the liquid which hold the molecuie at a fmed orientation. For this type of motion, there is no reason why the activation energy should be the same as for the macroscopic viscosity_ We have therefore reinvestigated the motions of spin probes in triplet state by assuming jump models for the molecular rotations. hi these studies, one takes advantage of the fact that exact solutions have been obtained for the stochastic equation io the limit of jumps of arbitrary size (strong-coilision limit) [5,6]. Situations intermediate between the reorientation through small angular steps (diffusional limit) and the strong-collision limit, have also been analyzed according to a procedure proposed by Alexander et al. [7]. Our analysis leads to the conclusion that the average reorientation angle for the probe molecules cannot be larger than 15”, and so the assumption of brownian diffusion should be consistent with the experimental line&apes_ In the fmsl section we &tall therefore present some comments on other effects which may be invoked to explain the anomalous temperature dependence of the motional narrowing of the

G.

86

ESR spectra, such as microscopic fects and solvation equilibria.

_&for0 t?td/ROtutiOlm~

boundary

layer ef-

2. Theory The ESR spectral shape, under condition of low Gcrowave power, is given by the linear response theory as [2,3]: I(w) = n-l

w) ~ (I) ( I where S,(Q, w) is the Fourier-Laplace transform of the anguhn dependent spin operator S&2, t), which is assumed to obey the stochastic L.iouviIIe equation:

(d/dt)S,(Q,t)=

ReTr

S&QSX(Q,

[iZf(Qy

- R(sl)]S,(sZ,

motions of biradicak

state, the spherical tensor operators Stism) up to the ranki = 3s provide the complete set of spin operators. Another possible choice is given by the transition operators [M) vM’[, where M denotes the projection quantum number of the total spin angular momentum. It is worth pointing out that an exact solution can be obtained in the ‘strong collision” limit [5,6]. One obtains: I(w) = R-I ReTr{S,(l

- a/rr)-IaS,},

where it has been introduced Q(sZ,w) defmed as: a(CZ,w)=

the superoperator

-H(S2)93-1,

{l/~r+i[w

and the upper bar means an orientational

t)

(4)

(5) average:

C%J) =~dQI’GNX~.~).

+ d~‘{Ir(,rL’,R)S,(St’,t) s

(6)

In actual systems, an intrinsic linewidth (l/T.&,, which may be angular dependent too, must be included in the definition of Q, which thus becomes:

- It’@, XZ’)S,(Q, t)), (2)

with initial condition S&2,0) = S>(Q). In eq. (2), s2 denotes the Euler m-&es (crJ.7) for the transformation relating the moiecule axis system to the fxed laboratory reference frame [S], P(a) is the equilibrium orientation distribution which is a constant in isotropic fluids, R(Q) a stochastic operator defmed in terms of IV(Q, C), the transition rate for the -change of orientation from C? to G?‘. The detailed form of IV@, 4’) depends upon the model chosen to describe the rotational motion of the molecules [9] _ By specializing iV(!& SL’) for the ease in which the reorientation occurs through a sequence of infmitesimahy smaII angular steps, *he diffusion model is recovered. On the other hand, if jumps of any arbitrary size are allowed to occur, the following expression for the transition rate holds: I%@, s2’) = P(!z)frr,

(3)

where 1; is the residence time in a given orientation, and so the “strong collision” (or random-jump) model is obtained_ In the case of diffusional motion, the solution of the stochastic equation is usttahy performed either by the fmite difference technique or by expanding S,(S2,r) on a complete basis of spin operators and angular functions. A complete set of angular functions is provided by the Wigner functions [S] _For a spin S

Q(Q,w)=

{S + i[w - H(sL)x])-‘,

(7)

6 = l/rr f (l/T&_

(8.)

If we denote by IN(CZ)>the eigenvectors hamiltonian, then H(Q) IAV-0) = E,,(Q) IN(Q)) = g

of the spin

MQ)),

(9)

c~l,,I(fi)IM).

(IO)

In terms of the base operators spectrum is given by

Vi = IM) (M’I the

I(W) = n-l Re iZIk (S,. Vi) (Ui, (1 -iS /?,>-I q)

,*

x @. a &)($,

&)a

where (Vi, UJ s TrL$Uj 3are expressed as [lo]:

(11) and the matrix elements

(l~~oM’f,lil~)(t’l)=~~,~~~~)

of

G. More et al_fRotational 3.

Intermediate reorientation

schemes

The strong-co&ion and the brownian diffusion model are two lirnit.ing~cases for the molecular reorientation process. The average value of the cosine of the reorientation angle E = I Cl - Q’I is zero and one respectively for the two cases, and intermediate reorientation modeis will be characterized by vaIues of (cos E>lying in between these limits. In his random walk calculation, Ivanov [1 l] has shown that the correlation functions of spherical harmonics of different rank decay in time with similar rates, as the value of (cos 19 becomes vanisldngly small. Another generalized diffusion model has been proposed by Alexander et al. [7]. which suggest the folIowing expression for the transition rate of an axialiy symmetric molecule in an isotropic liquid: W(Q S2’)= (IV/r) exp(A cos E),

(13)

where P is the mean lifetime between jumps and N is a normalization factor. The average value of cos E computed from this expression is given by the Langevin function L(A). which varies between 0 and 1 as h varies from zero to hrfmity, and so this generalized model changes continuously from the strong collision to the diffusion limit. The transition rate of eq. (13) can be expressed in terms of spherical harmonics by using a Rayleightype expansion [7,12] : .

(14)

where i&A) are modified spherical Bessel functions. When this expansion is put into eq. (2), the resulting stochastic equation can be solved by the Neumann perturbation method, since the kernel (14) is completely continuous [13]. By using this procedure, the lineshape is obtained as a power series iu the expansion parameter r-l, I(a) = ~~&=J)(W,

(15)

with the first term &(w) giving the powder spectrum. The power series (15) converges quickly only when the motion’is very slow, but it can be analytically continued to large values of I--’ by forming its Padi approximants [13] _This method is found to be very

81

motions Of bircdicab

powerful, a& only the first few coefficients Jr(w). ’ are needed to ensure convergence even in the motion:: al narrowing (Redfield) limit. No matrix inversion or diagonalization must be performed in this way, and this is the defmite advantage of the Neumann-Pad6 method over the function expansion commonly employed.

4. Non-secular corrections In all the biradical metal chelates studied in ref. [ 11, the g-factor and hyperfme coupling anisotropies are negligibly small compared to the dipole interaction between the unpaired spins, and this in turn is always less than one-tenth of the electronic Zeeman term. Under this circumstances it appears very reasonable the use of a secular approximation. Nevertheless, with a minor computational effort one can include second-order corrections, which allow to test the validity of the secular approximation at any value of the absorption frequency. In spherical tensor notation, the spin hamiltonian for a triplet system is

where the T&mJ are the spherical components of the electron dipole interaction in the laboratory reference frame. Due to the Dzd symmetry of the metal chelates, the interaction tensor is axially symmetric in the molecular principal axis system. Within a van Vlecktype perturbation scheme [14], one obtains the following expression giving, for the time evolution operator of eq. (2), results corrected up to second order: i [w + H$ + H~“)(i2)x] - R(i2) - mxo H~m$2)x + X [inzoo - R(Cl)]-1H$-“‘(C2)“.

(17)

The c‘non-secular” terms of this expression,correspending to m = il,i2, give rise to diagonal as well as off-diagonal elements on the basis of thetransition operators ]M)(M’I. Detailed calculations show that the off-diagonal corrections are vanishingly small except that in the motional narrowing region, where the line&ape &dyL sis is more conveniently carried out with the usual

88

Ci. Moro et al./Rorarional

morions of biradicals

Redfield theory [ 15,16]. _&pin, very simple expressions are obtained in the strong-collision limit. Onefmds: I(o) = P-I Re 5

[S(S + 1) - M(M - l)]

XP$f(i - pn$J1.

(la

The index

M denotes the alIowed ESR transitions, running from --St 1 to S, and the function PM is defmed as:

119)

I

T=173K

Fig. I. Experimental and c&x&ted

spectm for &@W’-P

t&d 1,4diaz&utadiene)2: o diffusion model, D/OR= 55;

X random jump model, Drr = 5.

(2-O)

5. Results and discussion Owing to the axial symmetry of the zero-field splitting in the tripIet systems which are considered, the computed lineshapes depend upon two fitting parameters only: the model parameter and the mean lifetime 5-which appear in the expression for the reorientational transition rate eq. (13). When X = 0, that is in the strong collision limit, r is the same as the residence time 7,. whereas for X > 0 it is convenient to defiie a generalized rotational correlation time [73:

7~ = r&J@)@&) - &@)l-l,

(21)

which yields an identical lineshape for aU jump models in the fast relaxation limit. As it was said above, the rotational diffusion model is recovered as h goes to intinity, and the rotational diffusion constant in this case is defined as OR = (67R)-I _In the hydrodynamic limit D, islinked to the solvent viscosity by the Stokes-Einstein-Debye relation: OR = IV_&= kT/81i2q2. The computed spectra with h = 0 and X = 200 (a

vaIue which ensures the convergence to the diffusional limit) were compared with the experimental

ones, by optimizing the correlation time rr_ At any degree of motion, the diffusion model (X = 200) gives the better reproduction of the lineshapes, and a typical result is shown in fig. 1 for the biradical chelate Zn(N,N’-p-tolyl l,4-diazabutadiene)2 in 2-methyltetrahydrofurane [I]_ A Ieast-squares fit over a temperature range of 60°C gives for log@&) versus T-l a straight line witt a correlation coefficient of 0.9994. The calculated activation energy is 1.9 kcal!mole, whereas that of the pure solvent is reported to be 0.9 kcal/mole 11’71.

021

I

a5

1.0 Cd./0

F&c 2. Effect of the reorientational model pammeter A on the computed ESR spectra for DT, = 3.3: dotted line, A = 1; dashed line, A = 10; solid line, h_= 100. All the spectra have been normalized to the same maxhnum height, and the secular approximation has been used.

G. More et aL/Rotational

In fig.2 is shown the effect of the intermediate models for rhe motion: when the model parameter h is greater than 30, and the amplitude of fhe meau rotation jump is less than lS~,~the results are practically undistinguishable from those obtained under diffusional regime. The secular approximation has been tested for various values of the model parameter h, in the slow motion region_ No marked difference between approximate and exactly computed spectra has been found as long as the ratio D/u0 is less than 0.2, and therefore the secular model can be safely adopted in our cases, where D/an is about 0.1. It is also interesting to note that the strong collision spectra are found to he more sensitive to the nonsecular corrections than the diifusional ones. This result is easily understood, since the average value of the reorientational angle increases as the X parameter goes to zero, and therefore the change in the direction of the quantization axis for each reorientational jump is more pronounced in the strong collision model. In conclusion, the experimental ESR line&rapes of these biradicals in the SIOWmotion regime are well reproduced when a reorientation model with angular jumps lesser than 15” is adopted, and this practically confirms the original hypothesis of dif fusional motion through small angular steps As a consequence, the temperature dependence of the correlation time should not differ from the Stokes law appreciably. Some comments on other possible reasons of the anomalous temperature dependence of the diffusion coefficients are therefore in order. The effects on the observed value of the diffusion coefficients DB due to the rotor-solvent interactions have been analyzed by several authors [4,18]. Partial Yip” boundary conditions lead to an observed value of DB which is larger than that obtained from the molecular dimensions through the Stokes-Einstein-Debye relation. However, for the metal chelates under investigation, the actual conditions (probe size iarge compared to the solvent molecules, irregular shape, surface roughness due to solvent coordination) all favour “stick” boundary conditions. Besides, the corrections of eq. (22) introduced by %ip’* effects are expected to be slowly varying with the temperature. One might question if

motions of birodicak

89.

the Einstein relation DB = kT/,$ is true for irregularly shaped. molecules, where coupling between the translational and rotational motions are certainl,: present. It has been ‘shown, however, that correlation’ functions calculated from a coupled translationalrotational diffusion equation reduce to the simple Debye form under conditions which are relevant to magnetic resonance ‘expermrents [19]. The last possibility to be examined is the effect of salvation. Ey using the temperature dependence of the solvent viscosity reported in the literature [17] we fmd that the temperature dependence of (D,/T) can be fitted if a 30% reduction of the apparent molecular radius is assumed to occur in a 60” temperature range. This might indicate a solvent coordination at the lowest temperatures. Further experimental work on less polar solvent is required to test this hypothesis_

Acknowledgement We thankfully acknowledge Professors Corvaja’and Pasimeni for supplying their experimental results. This work has been supported by the Italian National Research Council through its Centro Studi sugli Stati Molecolari Radicalici ed Eccitati.

References [I] L. Pasirueni, F. Bortolotto and C. Conraja, Chem. Phys. 38 (1979) 199. [2] R.G. Gordon and T. Messenger, in: Elecnon spin retaxation in liquids, eds L.T. Muus and P-W_ Atkins (Pienum Press, New York, 1972) ch. 13. [33 J.H. Freed, in: Electron spin relaxation in liquids, op. cit., ch. 14: Spin labeling, theory and applicatious. ed. L.J. Berliner (Academic Press, New York, 1976) ch. 3. [4] D. Hoe1 and D. Kivelson, I. Chem. Phys. 62 (1975) 1323. [5] I. Roberts and R.M. Lynden-Bell, Mol. Phys 21 (197 1) 689. [6] S. Dattagupta and M. Blume, Phys. Rev. BiO (1974) 4540; Phys. Rev. Al4 (1976) 480. [7] S. Alexander. 2. Lux. Y. Faor and R. Poupko, MoL Phys. 33 (1977) 1119. 181 MB. Rose, Elementary theory of anguku momentum (Wiley, New York, 1957). [V] R.I. Cukier, J. Chem. Phys. 60 (1974) 734_

90

G. Man, et aI./Rotational motions of biradicals

[IO] L-T. Muus, in: Elecuon spin relaxation in liquids, op. cit. ch. 1_ [ll] EN. Ivanov, Sov. Phys. JETP I8 (1964) 1041. [12] hi. Abramowitz and A. Stegun edr, Handbook of mathematical funstiovs (Dover, New York, 1972)_ 1131 J.S.R. Chisholm, in: The Padk approximant in theoretical physics, edr, G_ Baker and G. Gammell (Academic Press. New York. 1970). 1141 A. Baram, Z. Luz and S. Alexander, J. Chem. Phys. 58 (1973) 4558.

[15] G-R. Luckhmst, R.Poupko and C. Zannoni, M?l_ Phys. 30 (1976) 499. [I61 P-L. Nordio and U. Segre, in: The molecular physics of liquid crystals, eds. G.R. Luckhurst and G.W. Gray (Academic Press, London). to be pubIished. 1171 D. Nicholls, C. Sutphen and M. Szwarc, J. Phys. Chem. 72 (1968) 1021. [18] J-T. Hynes, R. Kapral and hf. Weinberg, J. Chem. Phys 69 (1978) 2725. [I91 J-A_ Mongomery Jr_ and BJ. Berne, 3. Chem_ Phys 67 (1977) 4589.