Vokune 35, number 3 CHEMICAL PHYSICS LETTERS 15 September 1975

RAPID COMPUTER SIMULATION OF ESR SPECTRB.

COWENTIONAL ESR OF AXIALLY SYMMETRIC “N-NITROXIDE SPIN LABELS

Patrick COFFEY Nieolrt Insimjment Corpomrion, Madison, WiscoMn 5371 I, USA

and

Bruce H. ROBINSON and Larry I?. DALTON * Dcpartmmh of Chemistry, Vanderbilt Univerxiry, Nashville, Tcnrressee 37235, USA

Received 6 Mzy 1975

A computer aorithn is presented for the simulation of the effect of mo!ecular tumblin,o on ESR spectra, and is applied LO simulation of the conventional ESR signal (the absorption sigzl detected at the first harmonic of the modulation frequency md in-phrse with the modulation frequency, h *he liiit or low microwa= and modulation power) Of axially symmetric 14N-nitroxide spin Ishe!s. The algorithm is extremely fast and is economical in terms of computer memory requirements.

1. Intraduction

The spin-labelling technique, introduced by McConnell [I ,2], has proved to be a valuable tool for the investiga- tion of bioiogical systems [3-S]. Wst biomolecules have no unpaired electrons azd are iherefore unobservable by ESR. If paramagnetic centers are introduced into the system, the ESR spectrum will reflect the effect of ihe biomolecular environment on those centers; very specific information may be obiained in this manner. In particular, changes in the molecular tumbling rate, due for example to formation of complexes, may cause marked changes in the spectrum.

2. Theory

Simulation of ESR lineshapes has been treated by a number of workers [B-l I]. Since the underlying theory has been presented elsewhere, only a brief review and the necessary equations for computer simulation will be dis- cussed here.

By expanding the elements of the spindensity matrix in a set of orthogonal functions, and by taking certain linea; combinations of the resultant equations, it is possible to reduce the problem tu the solution of a set of simul-

taneous real linear equations [6,8-l 11. Using the eigenfunctions of the isotropic and time-independent hamil- tonian as a basis set, we expand the elements of x, the deviation of the spin-density matrix from equilibrium, in a Fourier series over the tirnedependent microwave and modulation fields and in the elements of the Wigner rotation

matrices [12] over the orientational variables:

(1)

* Alfied ,D. Sio~n Fellow. Author to Morn correspondence should be addressed.

360.. -’

Volume 35, number 3 CHEMICAL PHYSICS LETTERS 15 September 1975

where Y and v’ refer to the nuclear eigenvalues, E and E’ to the electronic eigenvalues, and 52 to all orientational

variables. The microwave and modulation frequencies are denoted w0 and o, respectively; the ~(k, r, I, DZ, n) are complex expansion coeftkients. The conventional ESR signal is an absorption signal, detected at the first har- monic of the modulation frequency and in-phase with the modulation frequency. For an isotropic orientation& distribution, this signal is proportional to

I

Im C I>r”_‘,V,,,+,,2(1,1,0,0,O)+~V.IY/Z,tl,2(1,-1,0,0,0)~. v:-[

(3

In the limit of low (non-saturating) microwave power and low (non-distorting) modulation power, the problem is somewhat simplified, since all out-of-phase signals, all saturation effects, and all coupling to higher modulation

harmonics may be safely neglected. If we further neglect the nuclear Larmor frequency and assume that the tumbl- ing of the labelled molecules may be well represented by a brownian diffusion model, the following set of simul-

t’aneous linear equations results for each modulation harmonic r of an I = 1 system:

AX’ = F’ , (3)

where A is a supermatrix and X’ and F’ are supervectors cietined as

A=

40,o A0,2 0 0 a _ . .’

d2,0 A2,2 A2,4 ’ ’ - . -

0 AS,2 A,,, AS,6 0 - - -

0 0 A6,,! AG6 AG8 . . . 7

The indices in eqs. (4) and (5) are the I indices of the expansion of eq. (1). For an axially symmetric nitroxide with

the principal axes of the A and g tensors coincident, ITO coupling over nz indices occurs; coupling over [he IZ in- dices has been included in each e!ement of the supermatrix A. Furthermore, th” 0 Ith term couples only to I’ = 1 f 2.

Since we are ultimately interested only in the I = 0 term, only even 1 indices need be kept in the expansion, and

the superrnatrix A is tridiagonal. No coupling over microwave harmonics [the k index of eq. (l)] need be UXI- sidered, since the separation of harmonics is of the order of i0 11 Hz for X-band spectrometers. The individual

matrices A, , lti and the vectors X; and F; are.defined as

361

CHEM [CAL PHYFkS LETTERS 15 September 1975

0 C&i)

e4 tf,i) 0

0 0

0 0

F[ =

0 0

0 0

0 0

0 0

(77)

vihere the definitions of terms are as follows:

Volume 35. nanber 3 C’HEkfICAL PHYSICS LETTERS 1.5 September 1975

0

[(2)-“* + 11 Bx<l.j)

0

0 0

0 0

0 O \

0

0

%U,i)

0

0

0

@6V,i)

E1 (0 "j,o

e5U,i)

0

0

e3U,i)

0

0

0

e6U.i)

0

e,(U)

-Tz-~(O 6j,o

0

0

0

0

e,(l,i)

0

0

E’(r) 6j,o

eQ(l,i)

(6)

e6U,i)

0

08(l,i)

0

0

0

0

8 7U.i)

0

0

0

S9(l,il

@ (0 6 j,o ]

of(r,l) = im [z&,1) f Zi(+. r)] , di(r,I) = Re [Zi(F,l) + Zi(-r, I)] .

Given the effective spin-spin relaxation time Tz, the rotational correlation time rc, the pticipal values of the A

and g tensors, and the mkrowave frequency oO, ztl other terms may be determined, and the signal may be simulated as the field is swept through different values Ho. The symbols (‘1 ml $ 2;) are Wigner 34 symbols [12], Amtlytic form&e may be found in Edmond’s book 1131. The terms d, and do ;?cz proporticnat to the modulation and microwave field intensities respectively; 4 is related to the Boltzrnann equilibrium distribution over spin levels. For the purpose of calculating lineshapes under the assumption of low microwave and modulation power, the mag- nitude of these terms is immaterial, since any changes wit! oniy uniforrnty sale the spectrum amplitude. We may thus take “Z&, = 1, d, = 1, simplifying the equations. The observed cunventionel ESR signal is simply proportional to

363

Tolume 35, number 3 CHEMICAL~PHYSICS LETTERS 15 September 1975

Q1(1,0) + (3-23’2)Q3(1,0). (8)

Similar!y, the in-phase dispersion signal at the first harmonic signal may be obtained from the same combination ofdl(1 ,O) and d, (1 ,O). Dispersion detection is now commercially available on the new Century series of Varian

-‘spectrometers. The above form for the equations was chosen for two reasons: first, a large number of terms in the matrices

4!.l+j are zeros, making considerable savings in computer execution time possible; second, introduction of satura- tion phenomena is particularly simple in this form, as will be shown in a later publication [14]. These equ:ltions are of COUIX invariant under any linear transformation, and are re!ated to earlier equations developed by other workers. .

3. Solution

Solution of the equations is at first glance formidable_ If the expansion is truncated ai I = 30, for exanp!e, six- teen even 1 components are needed. Since there are twelve equations for each I component, this corresponds to

192 simultaneous eq:lations to be solved at both the zeroth and the first modulation harmonic and at each field value. If 200 field values are required for a smooth lineshape simulation, this means that the equations must be solved 400 times. Fortunately methods other than a brute force approach are possible. Gordon and Messenger [S]

rewrite the equations as an elgenvalue problem, allowing the signal to be computed for any number of field values with only one diagonalizaticn of a matrix of the same dimension as the full supermatrix. Robinson et al. [9] have adapted the Gauss-.Jordan elimination scheme for matrix inversion to the supermatrix problem. Execution times for both of these schemes increase as approximately the square of the number of terms kept in the expansion over Wigner rotation matr’x elements. The inethod presented in the following is extremely rapid, simpIe to program, and increases only linearly with the number of terms retained.

If +the expansion is terminated at I = l,, , the supermatrix ea. (4) may be factored into upper and iower diagonal supermatrices, and eq. (3) may be rewritten as [IS]

cyo A02 0

0 a2 &A

0 0 a4

0

0

A46

. \ \

I

The problem may be solved in two steps

UV =.i=’ , LX’=Y’.

The following recursion formulae may be straightforwardly deve!oped:

364

0

1

04

0

0

0

1

&

Volume 35, number 3 CHEM!CAL PHYSICS LEI-TERS 15 September 1975

c

D

E -

Fg. 1. Experimental (left column) 2nd clcuhted (right column) spectra or^24ox:4propane in glycerol. A jump diffusion model witi a = 53’ was assumed for nU calcuhted specka, and the values A,, =A, = 5.4 G, AZz = 34.8 G,gxx =g~r_v = 2.0068,~~~ = 2.0032, wo = 9.15 GHz were used. The followi= temperatures and correldion ttmes rc, correspond to each pair: (A), T = 16.5’C, ~c = 7 x 1O-‘o s; (B), T = 9.O”C, ~c = 1.5 X lo4 s; (C), T = 1.0% :c = 3 % IO-9 s: @I. 7 = -8.O”C, rc = 4 x 10-3 s; (E), 7-p= -11 J”C, I= = 5 x IO+ s; (F), T = -22”C, TV = 7 X LO’+ s. For spectra (A) and (B), T-2 was taken as 6.8 X 10mg s, while for spectra (C) through (F) Tz was taken s 3.4 x 10~~ 5. The experimental spectra of 2doxylpropane in glycerol were kindly protidti by P.C. Jost z.nd O.H. Griffith.

To solve the ESR problem at a particular field value, the following procedure may be employed: (i) Construct the matrices “1, through co, PI I through Pz [eps. (I2)] -

$i) CalculaZ! the Y” supervecior [eqs. (13)J. since Fi through FFm are all 0 [eq. (7)], the only non-zero v&E is Y, = *l_F$ =X0

(iii) Calcu!ate x8 . 2 through X& [eq. (14)]. (iv) Construct the supervector F1 from X0 [eq. (7)]. (v) Calculate V,t through Yd [eqs. (1361. Since Y(j = XJ ,

no further elements%f X1 need be co,nputed. and since ail observed sign& are in Xd [eq. (8)],

Execution times may be reduced 5y taking advantage of the symmetry of the problem. From the symmetry properties of the 3J’symbols 112,131 it may be shown that

e&i) = e&+i, -i) , d,(r,i) = e,(l+i. 4 , $(Li) = e,(W, -13 - (15) It follows that the matrices A& are syrnmcFc,+and thar A!,[+ = A~*j,,l. Since OLD,, = Aim, and sinde “1-2 =

Al-2 1-2 --Al-, rPr =AI-2 1-2 -A;-i,rqr AI-2,19 all matrices Q! must also be symmetric. These matrices may

the&me be stor’ed and maiipulated in upper triangular form. Run-times may also be greatly reduced ‘0~ taking

advantage of the many zero elements of the matrices in matrix muItipiication routines- Approximately 650(:, f i) computer muItipkations are required far each fie!d value using this algorithm.

365

Yoluma 35, number 3 CHEMICAL PHYSICS LETTERS 15 September 1975

4. &rnulations and disc&on

Fig. 1 displays ex,p&nentai [16] and calculated spectra for 2doxylpropane in glycerol. Rather than assume a brown&l diffusional. model, wherein the solute molecule undergoes an infinitesimal reorientation with each co!- iision, we have used ,the jump diffusional model proposed by Egelstaff [ 171. This latter model is more suitable when solvent and solute molecules are of comparable size. T&e solute molecule is assumed to remain fixed for a rnezn time T and then to instantly reorient through some mean angle g. The time T and the mean jump angle care related thrcueh e2 = T/T~. This model has been treated in solme detail by Goldman et ai. [18]. The only alteration in the equati&s described in this work is the redefinition of TF1(/) be!ow eq. (7) as TF’(o = _Tzl f I(I+I){6r,[l + (7/6~J1(1+1)] l-l,

‘IThe agreement between the experimentd spectra and the simulations is quite good, supporting the approxirna- tions mnde in deriving the equations_

We wish to thank PC. Iost and OH Griffith for their assistance and comments. Acknowledgement is made to

?he Donors of the Pekoleur;l Research Fund, administered by the American Chemical Society, to the Researckl Corporation, and to the National Science Foundation (GP42998X) for partial support of this research.

References

[l] S. Ohtihi and H.M. McConneU, J. Am. Chem. Sot. 87 (196.5) 2293. -[2] T. Stone, T. Buckman, P. Nor&o and H.M. hlcConneU, Proc. Nat]. Acad. Sci. US 54 (196.5) 1010. [3] H.bf. McConnell and B.C. hicFa&nd, Quart. Rev. Biophys. 3 (1970) 91. [41 X.P. Smith, in: Biological applications of electron spin resonance spectroscopy, cds. H.M. Swartz, J.R. Bolton and DC. Borg

Wiky-LnteKcienc:, New York, 1972) p. 433. 151 P. lost and O.H. Griffith, in: Methods in pharmacology, Vol. 2, ed. CF. ChigneU (Appleton-Century-Crofts, New York,

1972)p. 223. [6] J.H. Freed, G.V. Bruno and CF. Polnaszek, I. Phys. Chem. 75 (1971) 3385. [ 71 R.C. M&alley, E.J. Shhnshick anA Hhi. McCormell. C&m. Phys. Letters 13 (i972) 11.5.

IS] R.G. Gordon and T. hiessenger, in: Electron spin relaxation in liquids, eds. LT. hfuus and P.W. Atkins (Plenum Press, New York, 1972) ch. 13,~. 341.

[g] B.H. Robinson, L.R. Dalton, LA. Dalton and A.L. Kwiram, Chem. Phys. Letters 29 (1974) 56. [lOI L.R. Dalton, _p. Coffey, LA. Dalton, B.H. Robbson and A.D. Keith, Phys. Rev. All (1975) 468. [I 1] L.R. Dalton, B.H. Robinson, L.A. D&on and P. Coffey, in! Advances in magnetic resonance, Vol. 8, ed. J.S. Waugh (Acdemic

Press, New York, 2975), to ‘oe pubIished. [ 121 E.P. W&e;, Group theory and its application to the quantum mechanics of atomic spectra (Aademic Press, New York,

1959).

[13! A.R. Edmonds, Arpdar momentum in qu~~~rium mechanics (Princeton Univ. Press, Princeton, 1957). [la] P. Coffey, B.H. Robinson and L.R. Dalton, to be published. 1151 E. ISXCSOII and HB. Keller, Arslysb of numerical methods (Wi!ey, New York, 1966) p. 58. [ 161 P.C. lost znd O.H. Griffith, private communication. 1171 Pd. Egehtaff, J. Chem. Phys. 53 (1971) 2590. 1181 S.A. Goldmul, G.V. Bruno, CP. Potnxzek and J.H. Freed, J. Chem. Phys. 56 (1572) 716.

366