Studies of rotational diffusion through the electron-electron dipolar

3119

STUDIESOF ROTATIONAL DIFFUSION

Studies of Rotational Diffusion Through the Electron-Electron Dipolar Interactionld by James R. Norrislb and S. I. Weissman Chemistry DepuTtment, Washington University, S t . Louis, Missouri 63130 (Received March 6 , 1969)

Density matrix equations are used to calculate esr spectra for triplet molecules in rotational diffusion. Excellent argeement between calculated and experimental spectra is obtained.

Effects of rotational diffusion on magnetic resonance spectra were recognized soon after the discovery of the magnetic resonance phenomenon. The spectra are most sensitive to the rotational motion when the rotational frequencies are of the order of the “rigid” line breadth, ie., the line breadths of a rigid array of randomly oriented molecules. Such line breadths in nuclear magnetic resonance are normally less than 10 or 20 kHz, and the range of rotational rates accessible to convenient study are correspondingly limited. I n electron spin resonance spectroscopy of triplet molecules where the rigid line breadths may be as large as 1O’O Hz, the range of accessible rates is increased. I n this paper we report the viscosity-dependent electron spin resonance spectra of a series of ground-state triplet molecules. We use a simple but surprisingly satisfactory theory of the relations between rotational diffusion and spectra for analysis of the data. Several treatments3-’ have recently become available. Our treatment is a simple numerical evaluation of the frequency-dependent magnetic susceptibility under steady-state, slow passage, low power conditions. It is a direct extension of the methods developed by Alexanders and Kaplang and appears to us to be a numerical approximation to Fixman’s3 path integral method. To apply the Alexander-Kaplan method to the rotational diffusion problem we represent the molecular orientation by a point on a sphere, divide the sphere into an appropriate number of areas, and calculate the dependence of susceptibility on the motion of the point among the areas. I n a molecular principal axis system in the presence of an external field the spin Hamiltonian is

-BgS- + DSz2 + E(#,’

- 8,’)- ‘/3DS2

(1)

We restrict ourselves to cases in which g is isotropic and E = 0 (axial symmetry). Our experiments were carried out a t frequencies near 1O‘O Hz. The largest D among the substances included in this study is about lo9 Hz. Accordingly, we have used the high-field approximation in which the eigen-

functions in the laboratory frame are independent of orientation and the eigenvalues include only the diagonal contributions from the dipolar Hamiltonian. Further, since the rotation frequencies are considerably lower than the Larmor frequency the nonsecular contributions to the relaxations are not included. Also excluded is the constant term -1/3D52 in our calculation since its omission does not affect the line shapes. Since the Hamiltonian is invariant under 6 -t 6 (a/2), we consider only the angular range 6 = 0 to 6 = a/2. The range is divided into N segments, the j t h segment having 6 = 6, and B j = Bj+l - E for J = 1,2,. . . N - 1 where 61 = 4 2 and E = ( a / 2 ) / N . The number of molecules associated with each 6, is n sin 6, = nj, where sin 0, arises from the isotropic distribution of population. Those molecules associated with a particular value of 6, constitute “site”j. For the process of rotational diffusion we assume first-order kinetics, that is, the number of molecules leaving s i t e j per second is proportional to the number of molecules in site j . We further assume the equilibrium kinetic scheme

+

where ki, is a first-order rate constant. Setting klz = k and assuming detailed balance, one can establish values for all the k2, in terms of n j and k. We list a few of them on the following page. (1) (a) This work has been supported in part by the National Science Foundation and in part by the Petroleum Research Fund of the American Chemical Society to whom grateful acknowledgment is made. (b) National Institutes of Health Predoctoral Fellow. (2) N. Bloembergen, E. M. Purcell, and R. V. Found, Phys. Rev., 73, 679 (1948). (3) M. Fixman, J . Chem. Phys., 48, 223 (1968). (4) M. Saunders and C. S. Johnson, Jr., ibid., 48, 634 (1968). (5) M. S. Itzkowitz, ibid., 46, 3048 (1967). (6) R. Gordon, private communication. (7) H. Sillescu and D. Kivelson, J . Chem. Phys., 48, 3493 (1968). (8) S. Alexander, ibid., 37, 967, 974 (1962). (9) J. J. Kaplan, ibid., 28, 278 (1958); 29, 462 (1958).

Volume 73, Number 9 September 1969

3120

JAMES R. NORRISAND S. I. WEISSMAN

k23

-

nz nl = k ___ n2

kaz = le

n2

- nl

.___

n3

IC34

= IC

n3

- n2 + nl n3

The choice of 6, separated by a constant E and the above assumptions about the kinetics of the rotational process are consistent with the model of Debye.lo The rotational diffusion rate constants, le,,, are substituted for the Alexander and Kaplan “exchange” rate constants. Use of the Hamiltonian of eq 1 above results in a set of simultaneous linear algebraic equations in the steady-state, slow passage case. A typical equation for the time derivative of the relevant elements of the density matrix is

F-

t 2

Figure 1. Example of a random walk.

v)

z 0

I 5 0

C F I

W

0.1

I

I

70

80

I

I

90

100

ar

0.0 I

60

110

FFigure 3. Graph of F’s us. k,. F1, Fz, Fa, Fa, ~ / T zand , are in the same units such that D is normalized to 100.

k6

Figure 2. Esr first derivative triplet spectrum with E = 0. The Journal of Physical Chemistry

(10) P. Debye, “Polar .Molecules,” Dover Publications, Ino., New York, N. Y . , 1939,pp 77-89.

3121

STUDIES OF ROTATIONAL DIFFUSION

-EX PER1MEN TAL CALCULATED

__--CAI CULATED

I !

NazFL2

T = -108°C k, = 3.84 x 107sec-'

I/T2 =2.16g

Figure 6. Comparison of calculated and experimental esr spectra.

100 g

T = -160°C

Na2FL2

l/T2=4.32 g

k r = 1.92x104Sed1 1

Figure 4. Comparison of calculated and experimental esr spectra.

-EX PER I MENTAL - - - CALCULATED

I I I

I I

I

I I I I

I I

I I

NazFL2

I

l/T2

-Y 100 g

NazFL2

L:

T = -80°C

kr = 1.1 x 10'Sec'

2.16 g

Figure 7. Comparison of calculated and experimental esr spect'ra.

line-width parameter, and C is a constant that is a function of temperature, microwave power, etc. The 21 indicates this magnetization is associated with transi~ ~ transitions between states 12) and 11). ~ ( 3 2 ) for tions between states 13) and 12>,is found by

T = -120°C

'

. I

I/T2=2.I6g

k r = 1 . 6 3 IO7%-' ~

Figure 5. Comparison of calculated and experimental esr spectra.

P(32)j(A) =

Thus, the desired solution to the line-shape problem is PT =

CP(Wj(A)- CjP*W)j(-A) 3

wo is the angular microwave frequency. I n the absence of motion ~ ( 2 1 is) ~proportional to the microwave-induced magnetization of the j t h site in the coordinate frame rotating at the angular microwave frequency, 1/Tz is a

- P*(Zl)f(-A)

j

(3)

The total magnetization given by PT includes both absorptive and dispersive components. The absorption is the imaginary part of PT. The solution to such linear equations requires inverVolume 78,Number 9 September 1969

3122

JAMES R. NORRISAND S. I. WEISSMAN

sion of a tridiagonal matrix, a problem easily accomplished using the IBM 360 program GELB, even for a 50 ( N = 50) site problem. The kinetic scheme of this density matrix model is consistent with a random-walk process where l / k is the average time for a single random-walk step in terms of the 0 coordinate axis

-

(€yZ

= (n/2)/N

(4)

As illustrated in Figure 1, a typical randon-walk step has components in both spherical angles e and 4. 6 is the total angular length of the random-walk step. For infinitesimally small E , E’,and 6 the spherical surface can be considered planar so that

+

€2

=

e’2

62

Because we assume isotropic rotation

e2

-

=

e‘2

then -

-

-

2e2 =

-

62

(a2)’/2 and not ( E - ‘ 1 2 should be used as the measure of the average random-walk step when interpreting the calculated spectra. The square of the total angular distance OT2 a molecule wanders from an arbitrary axis is given by the usual random-walk formula

-

8T2

=

-

NTP = ~ N T E ~

Table I Organic compound

Decacyclene 1,3,5-Triphenylbenzene 9-Fluorenone 2,9-Dimethyl-4,7diphenyl- 1,lOphenanthroline 2,2‘-Biquinoline

Free radical

D’,G

DECA TPB

Na2+DECA2Kz+TPB’-

238 483.4

FL DMDPP

N a z + (FL-)2 Zn2+(DMDPP-)2

107.5 97

BQ

Zn2+(BQ-)2

104.7

Abbrev

The spectra were measured over a range of temperatures. Survival of the compounds was checked by repetition of the observations a t low temperature subsequent to heating a sample to the highest temperature at which it was studied. Concentrations were varied between and 10-3 M . No dependence of spectra on concentration in this range was observed. For the dianion triplets the initial reduction formed only monoradical, and the rigid medium esr spectrum consisted of only a single resonance centered near g = 2. Further reduction produced a triplet spectrum superimposed on the monoradical peak. I n the chelates the triplet signal was always present at the first reduction stage. For the paramagnetic chelates the concentration of monoradical was much smaller than the monoradical of the triplets. The “monoradical” resonance

where NT is the total number of steps taken by the random walker. Using eq 4 from above

eT2 =

I I

(~ZT/~)~NT/N~

Therefore, for a typical molecule to diffuse *n/2 radians away from an axis of arbitrary direction requires on the average N 2 random-walk steps. Each step takes the average time l / k ; therefore, the time in which a typical molecule diffuses 4 a / 2 radians is

I I I I

-

I I I

rr = N 2 / k (or IC, = IC/N2)

I

and defines the rotational correlation time of this paper. This interpretation is consistent with Debye’s model.

Experimental Section The triplet molecules studied included dianions of symmetrical aromatic hydrocarbons and pairs of monoanions tightly bound to each other by metal ions. The parent organic compounds and the paramagnetic species formed from them and the abbreviations used for their designation are listed in Table I. All the paramagnetic species were produced in 2methyltetrahydrofuran (MTHF), a solvent which forms a rigid glass a t low temperatures. Standard high-vacuum methods for preparations and manipulation of the substances were used. The Journal of Physical Chemistry

Na2FL2

I/Tp

EXPER IM ENT A L

--- CALCULATED

T = -66OC

2.1 6 9

kr = 3.19 x IO8 Set"

Figure 8. Comparison of calculated and experimental esr spectra.

STUDIES OF ROTATIONAL DIFFUSION

-EXPERIMENTAL

--- CALCULATED

c ZnBQ2

4

100 g

T=

- 127.5

I I T 2 = 4. I 9 g

OC

k r = 7.45 x IO6 s e d '

Figure 9. Comparison of calculated and experimental esr spectra.

is from either monoradicals or exchange narrowed paramagnetic microcrystals. In no instance was a pure triplet spectrum uncontaminated by a resonance centered near g = 2 obtained. The temperature dependence of ZnBQz was also checked using Apiezon-N stopcock grease as a solvent. Apiezon-N (previously degassed by heating and stored behind a break-seal) was added t o an MTHF solution of the radical ZnBQz. After removing the MTHF by evaporation to a trap a t liquid nitrogen temperature the spectrum obtained at 0" in the Apiezon-N was essentially the same as obtained in pure MTHF a t -lGOo, indicating the zero-field parameters (D and E ) , for this compound a t least, are not a function of temperature of solvent. The matching between observed and computed spectra was accomplished by the following procedure. The observed spectra are characterized, when possible, by a set of parameters F1, Fz, FB, and F4 (Figure 2 ) , I n the fast motion limit where only a single line is observed only one parameter, Fz, is assigned. A rotational rate constant, IC,, is assigned by matching the observed F's with computed ones. I n addition to the k, and D' parameters (D' is measured directly from the rigid medium spectrum) a line breadth parameter l/Tz associated with the breadth for each individual orientation of molecular axis is required. As the curves in Figures 3a and b indicate, the rates extracted from a given spectrum are not very sensitive t o choice of l/Tz. At the various rotational rates a match can be made which is almost independent of l/Tz. Finally, when k, and 1/Tz have been assigned, the entire computed spectrum may be compared with the observed one. Figures 4-10 illustrate typical computed and experi-

3123 mental spectra. The NazFlzseries represents the best, and the ZnBQz represents the worst. The discrepancy over the complete spectrum between the two is in some cases smaller than the noise in the experimental spectra (Figure 4) and even in the worst case (Figure 9) it is not very large. Because of the labor involved, full lineshape comparisons were not made in all cases. Because of the chemical properties of the substances the expected dependence of k, on viscosity could not be easily checked through use of a variety of solvents. Instead temperature dependence in MTHF was studied for all substances. Only the ZnBQzhas been studied in more than one solvent. Its spectrum in Apiezon-N at 0" is identical with the one in MTHF at - 160". The temperature dependence of Zn(DMDPP)z, a typical example, in MTHF is given in Figure 11 as a plot of log k,/T against 1/T. The plot is linear within the errors of assignment of k,. The linearity is consistent with a simple temperature dependence of viscosity 7 = Ae-B'T and the Debye relation between rotational correlation time, viscosity, temperature, and molecular size. The four other compounds exhibit similar linear plots. The parameters which characterize them are given in Table 11. The independence of the tumTable I1 Log ( k , / T ) =, B / T

Compound

-C

B X deg-1

-600 -626

NazDECA NaAFLh

KzTPB Zn(DMDPP)2 zn(BQ)z

C

8 .8 9.1 9.1 8.6 9.9

-642 -598 -756

-

EX PER IM E N T A L

--- CALCULATED

ZnBQ2

T = -116.5OC

I / T 2 = 3. I 4 g

k,. = 2.05 x IO7 sec-l

Figure 10. Comparison of calculated and experimental esr spectra. Volume 73, Number 9

September 1069

3124

JAMES R. NORRISAND S. I. WEISSMAN

Zn DMDPP TUMBLING RATE AS A FUNCTION

OF TkMPERATURE METHOD OF LEAST SQUARES EMPLOYED

7.20

3 20

024

I

0.32

I

I

0.40 0.48

I

I

0.56

0.64

I

0.72

I

0.80

088

I / T x IOp2

Figure 12. Zn(BQ)2and Zn(DMDPP)2 esr spectrum at the same temperature.

Figure 11. Temperature dependence of k, for Zn(DMDPP)2.

bling rate and spin-coupling parameters will be noted. At - 104” the averaging for ZnBQ is more effective than for the large ZnDMDPP despite the smaller D value for the latter (Figure 12).

Discussion Our model, despite its naivete, reproduces the complete spectrum through major adjustment of only one parameter, the rotational frequency, and minor adjustment of the single orientation line breadth (1/Tz). The values for these parameters, although not yet checked by independent observation, are reasonable. We have tried many other mechanisms of rotation besides the one described in this paper. We mention as one extreme an “all site” model which permits jumps with equal probability between any two equal areas on the sphere, independent of their distance from each other.” This model fails completely; no choice of rate constant produces a fit with the spectrum. Several other mechanisms have been tried; none was successful. The sensitivity of the results to the specific choice of

Ths Journal of Physical Chemistry

mechanisms leads us to believe that we have accomplished something more than a convenient one-parameter fitting of a complicated spectral function. We note also that the successful mechanism is completely consistent with Debye’s treatment of rotational diffusion. We have, however, not included consideration of departures from isotropic rotation. Our treatment has neglected the nonsecular contributions to the line breadths. Their magnitude should be determined by direct TI measurements and by observations at lower fields. Other effects which should become important a t low fields are associated with the nonadiabaticity of the behavior of the spins under rotation. I n the limit of zero external field the spin eigenstates for stationary molecules are quantized along molecular axes, yet the gyroscopic properties of the spin angular momenta tend to maintain their alignment along laboratory axes. A familiar case of this effect is seen in the numerous observations of the vanishing of dipolar interactions in liquids in the absence of external fields. (11) J. Norris, Chem. Phys. Lett., 1, 333 (1967).