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Slow motion lineshapesAsher Baram aa Department of Chemical Physics, Weizmann Institute of Science,76100, Rehovot, IsraelPublished online: 23 Aug 2006.

To cite this article: Asher Baram (1981) Slow motion lineshapes, Molecular Physics: An InternationalJournal at the Interface Between Chemistry and Physics, 44:4, 1009-1019

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MOLECULAR PHYSICS, 1981, VOL. 44, No. 4, 1009-1019

Slow motion lineshapes

by A S H E R BARAM

Depar tment of Chemical Physics, Weizmann Insti tute of Science, 76100 Rehovot, Israel

(Received 11 May 1981 ; accepted 2July 1981)

A slow motion expansion about the characteristic features of the powder spectrum is presented. Analytic expressions for the lineshape function, modulated by slow rotational diffusion, are derived. It is shown that the slow motion limit is characterized by harmonic oscillator equations of motion, and the resulting spectrum is determined by harmonic oscillator eigenvalues. The essential features of the lineshape show up naturally, and in particular the axial lineshape diverges like r 1/a while there is only a weak motional correction to the logarithmic divergence of the non-axial lineshape. The dynamic frequency shifts converge to their static limits like r -1/2 for all cases.

1. INTRODUCTION

The analysis of spectral lineshapes has established itself as an important tool for studying microscopic phenomena. It provides much information about potential interactions at the molecular level, and about the rate and mechanism of reorientation processes in condensed media. In the fast motion regime the orientational dependence of the molecular interaction is effectively averaged by the fast tumbling, and second order time dependent per turbat ion theory is sufficient to describe the relaxation process [1]. Th e resulting lineshape is a lorentzian completely determined by a single parameter, its second moment , which is not sensitive to the symmetry of the potential and to the details of the microscopic molecular motion. Many cases of interest such as powders., glasses, viscous solvents and biophysical systems are characterized by slow reorientation processes, where the relaxation limit no longer applies. A variety of statistical mechanical techniques have been developed in order to treat the spectrum in this regime. These methods are based either on Monte Carlo simulations [2-4], or various types of perturbat ion expansions about the relaxation limit [5-11]. Th e perturbat ion techniques are basically equivalent to moment expansions [12], in the sense that a finite order perturbat ion theory or equivalently a finite t runcation of a basis set of fast motion eigenfunctions [5, 7] results in a spectrum which has n exact moments, where n is the order of the per turbat ion (truncation). A large number of moments is required to reproduce the characteristic sharp features of the slow motion lineshape. Therefore these methods tend to become very cumbersome and involve extensive numerical computations. Moreover it is impossible to deduce from this approach information about the analytic properties of the lineshape function in the slow motion regime. Th e asymptotic convergence to constant terms of the anisotropic potential, in the fast motion eigenfunctions representation, was utilized to formulate the lineshape

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problem in terms of an infinite Toeplitz-like matrix [12, 13]. Using the analytic properties of Toeplitz matrices a non-perturbative solution for the spectrum was obtained. The resulting expression for the lineshape is compact and valid for all the tumbling regime whenever the modulation process is of the finite size jump type. However for brownian rotational diffusion the method is less effective, since the representation of the motion does not converge to constant terms asymptotically.

The static powder spectrum is completely determined by the density of states, and is known for all potentials [14, 15 ]. As long as the motion is sufficiently slow the powder spectrum is slightly modified but retains its main characteristics. In this paper we make use of this fact to derive a slow motion approach, based on the powder spectrum as a starting point, to the lineshape problem. The derivation leads to simple analytical expressions for the spectrum, which naturally exhibit the interesting aspects of the problem.

The method is valid for the general case of anisotropy modulated by slow rotational diffusion. It can be applied to magnetic resonance, fluorescence depolarization, dielectric relaxation and superparamagnetism [16].

2. GENERAL METHOD

We assume a hamiltonian of the form

H = H o + HI(Q ) + H r.f.(t), (2.1)

where H o is the interaction with the static external field, Hr.f.(t ) is the interaction with the oscillating radio frequency field and

Hx(~/) = [½(3 cos 2 0 - 1 ) + ½,/sin z 0 cos 2 $]ko (2.2)

is the anisotropic part of the hamiltonian representing the interaction of the system operators F with the time dependent lattice variables ~(t). In (2.2) 0 and 4~ are the polar and azimuthal angles specifying the direction of the external field in the body frame of reference, and ~ is the asymmetry parameter. We assume here a secular approximation, non-secular contributions are negligible in the slow motion limit and can be ignored. The evolution of the density matrix p is given by the stochastic Liouville equation [5, 7]

dp(,,,dt t ) = ( - i / 4 + 1~- V2) p(~' t), (2.3)

where r is the correlation time, in units of the interaction, of the rotational diffusion. For an isotropically distributed system the spectrum is given by the rotationally invariant part of the Fourier transform of the density matrix

I(w) = Im S P(~), w) d~. (2.4)

Fourier transforming (2.3), assuming the linear response approximation and high temperature limit we obtain

[ - ~ o + H ~ ( ~ ) + ! V 2] p(~, ~')=P0, (2.5)

where oJ and 1/r are in units of the interaction and Po is a constant proportional

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Slow motion lineshapes 1011

to the strength of the radio frequency field. Equation (2.5) describes the motion of a rotator in an anisotropic potential field, with the correlation time r playing the role of an imaginary moment of inertia. In the slow motion limit (or strong field limit) the motion of the rotator is determined mainly by the properties of the potential. The potential function is parabolic in the variables O, q~ in the neighbourhood of its extrema and saddle points, which are the extrema of the static density of states, or the characteristic frequencies of the powder spectrum. Thus the rotator tends to librate about these points, and since the motion is sufficiently slow the range of orientations satisfying the inequality Am < 1/7 is small, and the librators are as a result uncoupled. The motional process can be described as a combination of independent local librations about the characteristic frequencies of the powder.

Let y ( ~ , o~)= p(~), ~o)x x/(sin O) then (2.5) reduces to

I d 1 d 2 1 1 )--~-~ sin 2 0 dq~ ~ F 4 sin2~------0 ~¼-i ' c ( -w+Hl(~) ) Y= - i ~ , / ( s i n 0), (2.6)

where it is assumed that po =1. The next step is to expand (2.6) about the characteristic orientations. We start by treating the axially symmetric hamiltonian, V = 0.

2.1. Axially symmetric hamiltonian Defining a new angular variable

and expanding (2.6) about 0=7r/2 we obtain

x y = - i cos . (2.8)

Since the motion at 0~7r/2 is uncoupled to the motion at 0~0, (2.8) is a slow motion expansion in ~/(1/r). To lowest order in ~/(1/r ) it reduces to an equation for a single linear harmonic oscillator. Therefore it is convenient to expand y in a complete set of harmonic oscillator wavefunctions, and the contribution to the spectrum is determined by their set of eigenvalues. The eigenvalues are given by

the 1/r correction, which is pure imaginary, results from the constant ½ term in (2.8) and from the quartic correction to the harmonic oscillator. The contribution to the lineshape is

e x p [ i@/8)] 2 a"Z{1-[(2n+l)/4~/(3"r)]} (2.10) Ix(~o) = 1 ~ I m (2a ~.)1/4 ,=0 - oJ + ~ - - ~ '

2M2

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where

IF 1 a"=]i2"(n/2)!2-] nneVenodd

the derivation of a,~ is presented in the Appendix. The spectrum is a super- position of overlapping lines, each of them consists of a lorentzian, whose width is equal to the imaginary part of ~,~(x), and a dispersion-like line. Th e ratio of their intensities is equal to cot g(rr/8)=2"414. Th e properties of the characterstic axial peak at oJ ~ - ½ are determined mainly by the lowest eigenvalue /~0(x). The maximum in the intensity is shifted by +-~X'(3/r) f rom the static limit, o~ = - ~ . The intensity of the peak diverges like r ~'4, and this power law dependence can serve as a direct measurement for the correlation time. It is

1 / easy to see that for Aoj ~ ~V (3/r) the spectrum is proportional to (A~o) -12, i.e. the static square root dependence is retained. The derivative I'(~o) diverges like r 3!4, the dispersion-like contr ibution distorts its symmetry, and as a result its low f requency component is larger than the high f requency one. As a result of its relatively high sensitivity to the correlation t ime the derivative is the best probe for r.

While considering the behaviour at the second characteristic point it is convenient to define the angular variable

uz=(-air)l! 40=otzO. (2.11)

Expanding (2.6) about 0 = 0 we obtain

[~(_ai_t(d~rj\du2+~u2_l u2) +~rl (l+u")+i(c°-l)+O(r-a"'] y "

=-i /(sin~ I. (2.12) X / \ =z/

To lowest order in \/1/r (2.12) reduces to the equation of motion of a three- dimensional oscillator, with the generalized Laguerre polynomials as eigen- functions. The eigenvalues are

, ~ , , ( z ) = l - r ( 2 n + l ) ( l + i ) r(2n~+2n+l); (2.13)

the basic f requency is twice as large as that of k,(x), and as a result the broadening is more effective at ~o ~ 1. The reason for this is that the spectrum at co ~ - ½ is invariant to librations in the xy plane and therefore the relaxation effect is described by a linear oscillator, while at 0~ 0 the spectrum is sensitive to all types of librations, and the motion is described in terms of a three-dimensional oscillator. The pure imaginary 0 ( I / r ) contribution arises f rom the constant term and the quartic correction in (2.12). Th e cofltribufi0n to the lineshape is

L(w) 1 2 l+i{l+[(2n+l)/x~(3r)]} ~r ~/ (3r ) Im 2 (2.14) ,=0 - c o + ,~,,(z)

Here the intensities of the lorentzian and dispersion line are approximately

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equal. The dependence of I~(o~) on r is quite low. while the derivative diverges like r l!z. The total lineshape is the sum of the two local spectra. It should be noted that (2.6) for ~/=0 is an inhomogeneous spheroidal wavefunction [17]. whose asymptotic solutions are Legendre polynomials (fast motion) and generalized Hermite (oblate type) and generalized Laguerre polynomials (prolate type) for the strong field (slow motion) limit.

2.2. Non-axial hamiltonian The non-axial density of states has three characteristic features, at the saddle

point (7r/2, 0) and the extrema (~r/2, rr/2) (0, 0). We start by expanding about the saddle point. The slow motion equation of motion about the saddle point is equivalent to the equation for two non-interacting linear harmonic oscillators

[ /( i(3-~)~( d' - u ' ) + _ i '~ ( d' _v')+i(oJ+½_ ./( In) ~] \ 27 , \ du.' , , r , \ d% '

] t( +-r ½+"~-+'-~-) +0(7-'/z) s = - i c°su--x]'%/ (2.15)

where %=(- i r~7) x/4 sin 04 and % is slightly modified : % = [i/2(3--'~)T] 1/4. It is convenient here to expand y in a complete set of products of harmonic oscillator wavefunctions, with the eigenvalues

+j(i) A", , , (x )=-1(1 -~) ~ ( I X ' ( 3 - ~ ) ( 2 n + l ) ( 1 - i )

k/("q) (2m+ 1)(1 + i ) ) + / ( l + I n ( n + 1 ) + l m ( m + 1)). (2.16) - 2 r

The two basic frequencies correspond to two independent modulation processes. The first one is the axial-like out of plane modulation, while the second results from the motion in the xy plane. The two dynamic frequency shifts have opposite signs, since IV~(3-)1)> ~'(-q/2) the line is shifted towards the centre. For the special case -q = 1 the two frequencies are equal, and there is no dynamic frequency shift. The spectrum is given by the summation

2 a, 2 am2(1 +ib(x)) /x(co) rr211(3_~)r/r2]l,4 am y' , (2.17) . . . . 0 --oJqt- l~nm(X)

where b(x)=~ /(1"~{ 2 n + l 2 m + l )

"

To lowest order in X/(1/r) the weights of the eigenvalues are real, and therefore the line is a superposition of lorentzians. To the same order the line is r independent, motional effects arise from the 0(1/7) corrections. Thus the sensitivity of the intensity of the peak to changes in the rate of the rotational diffusion is quite low. As ~ decreases from one to zero the sensitivity of the peak to 7 increases. For ~ ~ 1/r a crossover to the axial form occurs, and the r 1/4 dependence is recovered. At the limit 1/7=0 the double summation in (2.17) reduces to the series representation of K(y), the complete elliptic integral of the first kind [18]. At the vicinity of the peak, ~o~ - 1 ( l - n ) , y converges

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to unity, and K(1) diverges logarithmically resulting in the typical two-dimensional characteristic of the non-axial powder spectrum [14],

K(1) I0(a, = - ½ + ½~) = - - (2.18)

As ~ decreases from one to zero the peak shifts from oJ = 0 to oJ = - 1 , and the singularity at the peak grows progressively till it changes its character to one over square root divergence for 7/= 0. The derivative I'x(eo ) is proportional to ~/r , and is the most convenient probe to simulate r.

Defining uu=[i/2(3+~)r]l/40 and vu=(ir~)l/4 sin 0q~ we obtain an equation similar to (2.15) describing the motion about oo = -½(1 + 7 )

+ - ½+u'~4 0(~ -a/2) y cos z - - f + + = - i . (2.19)

Again the motion is described in terms of two uncoupled linear harmonic oscillators. The set of eigenvalues is

N / ( 1 ) ~ / C 7) (2m + 1)](1 - i) A"'"~(Y) = - ½(1 + n) + r [½~/(3 + 7/)(2n + 1) +

i +- [l+½n(n+l)+½m(m+l)]+O(r-a/2), (2.20) T

the two real contributions to )'n,,(Y) are correlated resulting in a relatively large dynamic frequency shift. The lineshape is given by the double summation

2 Im • a"2amZ[1-i(l+b(Y))], (2.21) I"(~°) - 7r212~(3 + ~)r2] 1/4 . . . . 0 - w + A,~.,,,(y)

where

b(Y)=x/ \27)\ ~/~ ~--3-+-~)"

The weights of the lorentzian and dispersion contributions are approximately the same. The derivative I'u(~o ) at oJ,-~ -½(1 + ~7) is determined mainly by the lowest eigenvalue, and it behaves like ~/r . At the neighbourhood of ~o-- 1 it is convenient to define the two angular variables uz=[-(3-rl /2) ir]ll4 0cos ~, % = [ - ( 3 +~l/2)ir)] 1/4 0 sin $. The three-dimensional harmonic oscillator, (2.12) separates into two linear harmonic oscillators with eigenvalues

k / ( 1 ) A,,,,(z) = 1 - ~ r [~/(3-~7)(2n+l)+v~(3+~)(2m+l)](l+i)

i +- [l+½n(n+l)+½m(m+l)]+O(y-a/2). (2.22) T

The sensitivity to 7/is quite low, and it is easy to see that for ~ = 0 the expression in (2.13) is recovered. Note that n in (2.13) may assume any integer value, while here n and m are restricted to even integers. For 7/= 1 An,,,(z ) = - A~,,,*(y),

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I--4

2.0

10

/ I

- - .5

Slow motion lineshapes

I l

1 - - = .01 "Z"

0 ,5 1.0

co

I I I

1015

1__ = . 0 1 T

3 v

-10.

F igure 1.

[ 1 I I - .5 0 .5 1.0

cO

Calculated absorp t ion spec t rum and its der ivat ive ob ta ined f rom (2.10), (2.14) for r = 100. All rates and f requencies are in uni ts of the interact ion.

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1016 A. Baram

I.---I

1.0

.5

I I I I l

_ +

I I I I I -1.0 -.5 0 .5 1.0

Oa

5 .

.3 v

,-5.

I I I 1 I

1 _ = .01 T

" r / = . 8

I I I I I -I .0 -.5 0 .5 1.0

t.D

Figure 2. Calculated absorption spectrum and its derivative obtained from (2.17), (2.21), (2.23) for T= 100, 7=0-8.

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Slow motion lineshapes 1017

the dynamic frequency shifts are in opposite directions, while the linewidths are identical :

2 /.(o~)-7r2[(3_~)(3 +r/)r21114 Im ..... Y" a'*2a"2[l+i(l+b(z))]- co + A.,,,(z) , (2.23)

where / (1"~( 2 m + l 2 n + l b(z)=~/\71\~-~(3---~?) V/(3 + T/),/"

The total lineshape is the combination of the three independent components.

3. DISCUSSION

The derivation of the lineshape function, as represented in § 2, is based on the solution of a generalized anharmonic oscillator model. The perturbation series for the energy levels is an asymptotic expansion, because of the existence of an infinite number of branch points, which have a limit point at the origin, in the analytic continuation into the 1/r complex plane [19, 20]. Nevertheless as long as the perturbation is small the low lying energy levels are adequately described by first order perturbation theory, regardless of the divergent nature of the series [19, 21]. Therefore the expressions derived for the eigenvalues and their weights are valid for all the slow and intermediate motion regime. The eigenvalues of higher energy levels can be calculated either by WKB analysis [19] or by applying acceleration methods [21, 22] to the perturbation series. These eigenvalues exhibit a crossover towards rotator like eigenvalues, while their conjugate weights decay exponentially. It is sufficient therefore to sum over levels obeying the criterion of validity of first order perturbation theory, which here assumes the form n~/(1/r)< 1. The faster the motion the less the number of states required to determine the lineshape.

In figure 1 the lineshape due to an axially symmetric hamiltonian and its derivative are displayed. The spectra were calculated using (2.10) and (2.14) with the correlation time r=100. Similarly non-axial spectrum and its derivative, with ~=0.8 and r=100 , are shown in figure 2. The lines are broadened by the motion but still maintain the characteristic features of the powder spectrum. It is possible to recover the fast motion limit by applying an acceleration method [21] or Padg approximant [23] to the perturbation series of the ground state of the harmonic oscillator. Only a few terms in the series are required to obtain the rotator like eigenvalue. The excited states (n ¢ 0) do not contribute in this limit. Obviously it is easier to obtain the single lorentzian by using the conventional fast motion eigenfunctions representation.

The method can easily be extended to include a natural linewidth term, whose effect is to add a constant imaginary term tO the eigenvalues. Anisotropic motion can be included in the equation of motion, and it modifies slightly the line of the solution.

APPENDIX

The weight coefficient an is given by ~D

a , ,=(Vrr2nn! ) -1/2 j" exp (-½u2)H,,(u) du, --OD

(A 1)

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1018 A. Baram

where Hn(u ) is the hermite polynomial and the integrand is the normalized harmonic oscillator eigenfunction. Substi tuting the generating function for the hermite polynomials

$9"~ exp ( - s ~ + 2 s u ) = 2~ ~ H , ( u )

n=0

(~/~r2,~n I)-1/2 exp (s 2)

(A 2)

results in

and equating powers of s we obtain

o o

I exp ( - ~u 2) du = a, (A 3) --o0 n=0 ~

a-= U for even n and a m = 0 otherwise. polynomials

2uH,(u) = H,+l(u ) + 2nHn_l(U )

it is easy to obtain expressions for expectation values of powers of u.

<u~), , = (2n + 1),

( u ' ) n n = ~n(n + 1 ) + ~.

The generalized Laguerre polynomial is the eigenfunction of the three- dimensional harmonic oscillator. The weight coefficient here is given by

x/z exp ( - ½ u ' ) L , ( u 2) du. ( a 5) b- = - ~ ( °

Substi tut ing the generating function

u2t) exp -T -L- t j _

1 - t we find

Using the recursion relation of the hermite

(A 4)

In particular

tn n=0 ~ Ln(ug)

u e x p ~ + ~ u S du = ~, ~ b~ n ! ( 1 - t) 0 ,=0

(A 6)

(A 7)

implying b~ = ( - ) n ~ / 2 . The relevant expectation values are

(U~)nn = 2n + 1, <U4>nn = 6n z + 6n + 2.

REFERENCES [1] REDFIELD, A. G., 1957, I B M J I Res. Dev., 1, 19. [2] SAUNDERS, M., and JOHNSON, C. S., 1968, J. chem. Phys., 48, 534. [3] SILLESCU, H., 1971, J. chem. Phys., 54, 2110. [4] MCCALLEY, R. C., SHIMSHICK, E. J., and McCONNELL, H. M., 1972, Chem. Phys.

Lett., 13, 115. [5] FIXMAN, M., 1968, ft. chem. Phys., 48, 223. [6] Fm~ED, J. H., 1968, J. chem. Phys., 49, 376. [7] FREED, J. H., BRUNO, G. V., and POLNASZEK, C. F., 1971, ft. phys. Chem., 75, 3386. [8] ALBERS, J., and DEUTCH, J. M., 1971, J. chem. Phys., 55, 2613. [9] BARAM, A., Luz, Z., and ALEXANDER, S., 1973, J. chem. Phys., 58, 4558.

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[10] ATKIN$, e. W., and HtLLS, B. P., 1975, Molec. Phys., 29, 761. [11] FRIEDMAN, H. L., BLUM, L., and YuE, G., 1976, ft. chem. Phys., 65, 4396. [12] ALEXANDER, S., BARAM, A., and Luz, Z., 1974, J. chem. Phys., 61, 992. [13] BARAM, A., 1980, Molec. Phys., 41, 823. [14] BLOEMBERGEN, N., and ROWLAND, T. J., 1953, Acta metall., 1, 731. [15] COHEN, M. H., and REIF, F., 1957, Solid St. Phys., 5, 321, edited by F. Seitz and

D. Turnbull (Academic Press). [16] EISENSTEIN, I., and AHARONI, A., 1977, Phys. Rev. B, 16, 1278. [17] FLAMMER, C., 1957, Spheroidal Wave Functions (Stanford University Press). [18] ABRAMOWITZ, M., and STEGUN, I. A., 1970, Handbook of Mathematical Functions

(Dover Publications, Inc.), p. 590. [19] BENDER, C. M., and Wu, T. T., 1969, Phys. Rev., 184, 1231. [20] BENDER, C. M., 1970, J. math. Phys., 11, 796. [21] LUBAN, M. (to be published). [22] BARAM, A., and LUBAN, M., 1979, J. Phys. C, 12, L659. [23] BAKER, G. A., 1975, Essentials o[ Pad~ Approximant (Academic Press).

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