Calculation of orientational correlation functions for

THE JOURNAL OF CHEMICAL PHYSICS 130, 134105 共2009兲

Calculation of orientational correlation functions for free anisotropic rotational diffusion revisited Yuri P. Kalmykova兲 Laboratoire de Mathématiques, Physique et Systèmes, Université de Perpignan, 52, Avenue Paul Alduy, 66860 Perpignan Cedex, France

共Received 7 January 2009; accepted 18 February 2009; published online 3 April 2009兲 A simple matrix method for evaluating the orientational correlation functions of arbitrary rank j pertaining to free noninertial anisotropic rotational diffusion of rigid Brownian particles is presented. The first- and second-rank correlation functions are calculated analytically for a diagonal diffusion tensor. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3096981兴 I. INTRODUCTION

Free 共i.e., in the absence of external fields兲 noninertial anisotropic rotational diffusion of a rigid asymmetric Brownian particle has many applications in the analysis of orientation relaxation of molecules in liquids by various spectroscopic methods such as dielectric and Kerr effect relaxations, NMR relaxation, fluorescent depolarization, dynamic light scattering, etc. 共see, e.g., Refs. 1–15兲. This model yields an appropriate description of molecular reorientations in fluids when inertial effects are negligible and the rotation of the molecule can be described by a random walk over small angular orientations.6,9 If these assumptions are not fulfilled, alternative models for reorientational motion such as extended rotational diffusion models,16–19 jump models,20–22 etc. should be used. In the theoretical description of orientational relaxation of Brownian particles, the relevant quantities are the equilibrium orientational correlation functions involving Wigner’s D functions defined as16 D M,M ⬘共⍀兲 = e−iM ␣d MM ⬘共␤兲e−iM ⬘␥ , J

J

where dJMM ⬘共␤兲 is a real function with various explicit forms given, for example, in Ref. 16, and ⍀ = 兵␣ , ␤ , ␥其 are the Euler angles, which determine the orientation of the molecular 共body-fixed兲 coordinate system xyz with respect to the laboratory coordinate system XYZ. In practical applications, corjⴱ j relation functions of the type 具Dn,m 关⍀共t兲兴Dn,m 关⍀共0兲兴典0 are ⬘ of most interest 关9,10兴 共the asterisk denotes the complex conjugate and the brackets 具 典0 denote an equilibrium ensemble average兲. Furthermore, in some applications more complicated correlation functions of the type 具F关⍀共t兲兴Fⴱ关⍀共0兲兴典0 for a particular function F关⍀共t兲兴 are involved 共see, e.g., Ref. 15兲. However, the latter problem may be reduced to the former. Typical examples are the correlation functions 具m共t兲 · m共0兲典0 and 具Tr兵a共t兲 · a共0兲其典0 appearing in the theory of infrared and Raman spectra,19 respectively, where m is the transition dipole moment vector, a = ␣ − E Tr兵␣其 / 3 is the symmetric second-rank tensor with a zero trace 共anisotropic part of the polarizability tensor ␣兲 and E is the unit tensor. a兲

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Traditionally, the calculation of the orientational correlation functions for the free anisotropic rotational diffusion is usually affected2–9 via the formally exact expression for the conditional orientational probability density distribution function W共⍀ , t 兩 ⍀0兲 关⍀0 = ⍀共0兲兴 of the orientations of asymmetric particles. This distribution function is the fundamental solution of the rotational diffusion 共Smoluchowski兲 equation,2,6,10 ˆ Jˆ W, ⳵tW = − Jˆ D

共1兲

ˆ is the where Jˆ is the angular momentum operator23 and D rotational diffusion tensor. For a diagonal diffusion tensor, noting that Eq. 共1兲 is analogous to the Schrödinger equation for a free rigid asymmetric top, the conditional probability density W共⍀ , t 兩 ⍀0兲 is expressed via an infinite series involvj ing Wigner’s D functions, the eigenmodes e−Ekt and expanj sion coefficients am,n of the quantum asymmetric rotor eigenfunctions in terms of the symmetric rotor eigenfunctions 共methods of calculation of the eigenenergies Ekj and expanj are discussed in textbooks on quantum sion coefficients am,n mechanics and molecular spectroscopy, see, e.g., Refs. 24 and 25兲. The equilibrium correlation function 具Fⴱ关⍀共0兲兴F关⍀共t兲兴典0 can then be evaluated as 具Fⴱ关⍀共0兲兴F关⍀共t兲兴典0 =

冕冕 ⍀

⍀0

Fⴱ共⍀0兲F共⍀兲W0共⍀0兲

⫻W共⍀,t兩⍀0兲d⍀0d⍀, where W0共⍀0兲 = 1 / 8␲2 is the equilibrium distribution density. Although the application of the above procedure does not create insuperable difficulties in the evaluation of correlation functions, nevertheless the practical calculations are rather involved and tedious especially for large j. However, the treatment of the problem can be considerably simplified if one reduces it to the solution of differential-recurrence equations for the j-rank orientational correlation functions j j 共t兲 = 具Fⴱ关⍀共0兲兴Dn,m 关⍀共t兲兴典0. These differential-recurRn,m rence equations can then be solved by the matrix methods of linear algebra. The objective of the present note is to give the details of these matrix calculations.

130, 134105-1

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J. Chem. Phys. 130, 134105 共2009兲

Yuri P. Kalmykov

II. DIFFERENTIAL-RECURRENCE EQUATIONS j FOR Rn,m „t…

The differential-recurrence equations for the orientaj j tional correlation functions Rn,m 共t兲 = 具Fⴱ关⍀共0兲兴Dn,m 关⍀共t兲兴典0 can be derived using the Langevin approach as described in Refs. 26 and 27. On averaging the governing Langevin equations for anisotropic rotational diffusion over their realizations, one obtains26,27 关Eq 共1兲兴 d ⴱ j 具F 关⍀共0兲兴Dn,m 关⍀共t兲兴典0 dt ˆ Jˆ D j 关⍀共t兲兴典 , = − 具Fⴱ关⍀共0兲兴Jˆ D 0 n,m

共2兲

ˆ ˆ ˆ ˆ ˆ ˆ Jˆ = 兺 where Jˆ D r,s=x,y,zDrsJrJs. The components Jx, J y , and Jz 23 ˆ of J in the molecular coordinate system are defined as Jˆx = 共Jˆ−1 − Jˆ+1兲/冑2, where

Jˆy = − i共Jˆ−1 + Jˆ+1兲/冑2,

冋

Jˆz = Jˆ0 ,

册

⳵ ⳵ 1 ⳵ i ⫿i␥ Jˆ⫾1 = 冑2 e ⫾cot ␤ ⳵ ␥ + i ⳵ ␤ ⫿ sin ␤ ⳵ ␣ , ⳵ Jˆ0 = − i . ⳵␥

izing the inertia tensor Iˆ.6 Thus if the orientation of the diffusion tensor principal axes is unknown, the nondiagonalˆ can ized diffusion tensor must be used. The components of D be estimated in principle for particle of an arbitrary shape using the so-called hydrodynamic approach.1,6,10,28 Furthermore, one can evaluate the components Drs from the experimentally measured nuclear magnetic relaxation times of appropriate nuclei in the molecule.6 Noting that ˆ Jˆ = 1 兵共D + D 兲Jˆ 2 + 关2D − 共D + D 兲兴共Jˆ0兲2 Jˆ D xx yy zz xx yy 2 + 共Dxx − Dyy兲关共Jˆ+1兲2 + 共Jˆ−1兲2兴其 + iDxy关共Jˆ+1兲2 Dxz ˆ 0 ˆ −1 ˆ +1 ˆ −1 ˆ +1 − 共Jˆ−1兲2兴 + 冑2 关2J 共J − J 兲 − J − J 兴 −

iDyz

ˆ 0 ˆ −1 + Jˆ+1兲 − Jˆ−1 + Jˆ+1兴,

冑2 关2J 共J

and using known properties of the angular momentum operators Jˆ and Jˆ␯, viz.,23 j j Jˆ 2Dn,m 共⍀兲 = j共j + 1兲Dn,m 共⍀兲,

ˆ is symmetric with six distinct comThe diffusion tensor D ponents Dxx, Dyy, Dzz, Dxy, Dyz, and Dxz whose values depend on the shape of the particle. The principal axes of the diffuˆ for an asymmetric top molecule may not coinsion tensor D cide, in general, with the principal axes of inertia diagonal-

j,m+␯ j j 共⍀兲 = − 冑 j共j + 1兲C j,m,1, Jˆ␯Dn,m ␯Dn,m+␯共⍀兲,

we can represent Eq. 共2兲 as a five-term differentialrecurrence relation for the equilibrium correlation functions j j 共t兲 = 具Fⴱ关⍀共0兲兴Dn,m 关⍀共t兲兴典0, viz., Rn,m

1 1 d j j j 共t兲 − 兵⌶ⴱ冑关j2 − 共m − 1兲2兴关共j + 1兲2 − 共m − 1兲2兴Rn,m−2 共t兲 R 共t兲 = − 关⌬m2 + j共j + 1兲/2兴Rn,m dt n,m ␶D 4␶D j j 共t兲 + ⌰共2m + 1兲冑 j共j + 1兲 − m共m + 1兲Rn,m+1 共t兲 + ⌰ⴱ共2m − 1兲冑 j共j + 1兲 − m共m − 1兲Rn,m−1 j + ⌶冑关j2 − 共m + 1兲2兴关共j + 1兲2 − 共m + 1兲2兴Rn,m+2 共t兲其.

23 Here C j,l j1,l1,j2,l2 are the Clebsch–Gordan coefficients, ␶D = 共Dxx + Dyy兲−1 is the characteristic relaxation time, and

⌬=

1 Dzz − , Dxx + Dyy 2

⌶=

共3兲

results. However, the Langevin equation approach has, in our opinion, the advantage that it allows one to derive Eq. 共3兲 in a simpler manner.

Dxx − Dyy + i2Dxy , Dxx + Dyy III. MATRIX FORMULATION

⌰=2

Dxz + iDyz Dxx + Dyy

are dimensionless parameters characterizing the anisotropy ˆ . The recurrence Eq. 共3兲 can also be of the diffusion tensor D obtained from the diffusion Eq. 共1兲. The Langevin and diffusion equation treatments are equivalent and yield the same

The recurrence Eq. 共3兲 with different j are decoupled so that the analysis is simplified allowing us to write Eq. 共3兲 in compact matrix form as d C j共t兲 = A jC j共t兲, dt

j = 1,2, . . . ,

with the initial condition C j共t = 0兲 = C j共0兲. Here

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共4兲

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Anisotropic rotational diffusion

冢 冣

IV. EXAMPLE: FIRST-RANK CORRELATION FUNCTIONS

j Rn,−j 共t兲

j 共t兲 Rn,−j+1

]

C j共t兲 =

j Rn,0 共t兲

共5兲

]

j Rn,j−1 共t兲 j Rn,j共t兲

A1 = −

and A j is a Hermitian five-diagonal system matrix with matrix elements given by −1 ⴱ 共a j,r␦r,s−2 + bⴱj,r␦r,s−1 + c j,r␦r,s + b j,r−1␦r,s+1 共A j兲r,s = − ␶D

+ a j,r−2␦r,s+2兲,

共6兲

where ␦r,s is Kronecker’s symbol,

冕 冕 冕 2␲

0

2␲

0

␲

j Fⴱ共⍀0兲Dn,m 共⍀0兲d⍀0 ,

1

⌶/2

0

0 1+⌬

冣

共10兲

.

j ⴱ

j

1 1

2 2

Dl 1,l⬘共⍀兲Dl 2,l⬘共⍀兲d⍀ = 8␲2共2j1 + 1兲−1␦ j1 j2␦l1l2␦l⬘l⬘ ,

冢 冣 1 共t兲 Rn,0 1 Rn,1共t兲

can

共7兲

0

12

冢 冣 ␦m⬘,−1

1 ␦m⬘,0 = T1⌳1共t兲T−1 1 3

共12兲

,

␦m⬘,1

where the 3 ⫻ 3 matrices ⌳1共t兲, T1, and T−1 1 are given in 1 共t兲 can be obAppendix A. The explicit solutions for Rn,m tained by matrix multiplication in Eq. 共12兲 yielding 1ⴱ

1 关⍀共t兲兴Dn,m⬘关⍀共0兲兴典0 = 具Dn,0

␦0,m⬘ 3

1

e−t/␶0 ,

共13兲

1ⴱ

1 关⍀共t兲兴Dn,m⬘关⍀共0兲兴典0 具Dn,⫾1

where d⍀0 = sin ␤0d␤0d␣0d␥0. The solution of Eq. 共4兲 is then given by C j共t兲 = eA jtC j共0兲.

冕

1 Rn,−1 共t兲

c j,r = ⌬共j − r + 1兲2 + j共j + 1兲/2.

1 8␲2

0

⌶/2

Eqs. 共7兲 and 共8兲 then yield

and

j 共0兲 = Rn,m

冢

1+⌬ 0

共11兲

b j,r = 共⌰/4兲共2j − 2r + 1兲冑r共2j + 1 − r兲,

j j Rn,m 共0兲 = 具Fⴱ关⍀共0兲兴Dn,m 关⍀共0兲兴典0

␶D−1

Here we noted that ⌶ = ⌶ⴱ = 共Dxx − Dyy兲 / 共Dxx + Dyy兲 and ⌰ = ⌰ⴱ = 0. Using the orthogonality properties of the Wigner D functions, viz.,23

⍀

a j,r = 共⌶/4兲冑关j2 − 共j − r兲2兴关共j + 1兲2 − 共j − r兲2兴,

The initial conditions be evaluated as

The analysis may be simplified by assuming that the off-diagonal elements Dij vanish, i.e., Dxy = Dyz = Dxz = 0 共this assumption suffices in most cases兲. For a diagonal diffusion tensor, the system matrix A1 defined by Eq. 共6兲 becomes

共8兲

=

␦⫾1,m⬘ 6

1

1

共e−t/␶1 + e−t/␶−1兲 +

␦⫿1,m⬘ 6

1

1

共e−t/␶1 − e−t/␶−1兲, 共14兲

29

Now from matrix analysis, the matrix exponentials eA jt can be expressed explicitly in terms of the eigenvalues ␭kj and corresponding eigenvectors gkj of the system matrix A j as follows: eA jt = T j⌳ j共t兲T−1 j .

共9兲

Here ⌳ j is a 共2j + 1兲 ⫻ 共2j + 1兲 diagonal matrix with elements j 关⌳ j共t兲兴mn = ␦m,ne␭mt and the 共2j + 1兲 ⫻ 共2j + 1兲 matrix T j is formed from the eigenvectors gkj.29 Here the eigenvalues ␭kj of the Hermitian matrix A j are all real and negative. We remark in passing that the eigenvalues ␭kj of A j are related to the rotational eigenenergies Ekj of a quantum asymmetric rotor as ␭kj = −Ekj. For isotropic rotational diffusion 共i.e., when ⌬ = ⌶ = ⌰ = 0兲, the system matrix A j becomes diagonal, viz., A j = −Ij共j + 1兲 / 共2␶D兲, where I is the 共2j + 1兲 ⫻ 共2j + 1兲 identity matrix. Thus Eq. 共8兲 yields the known result10 j j Rn,m 共t兲 = Rn,m 共0兲exp关− j共j + 1兲t/共2␶D兲兴.

In order to illustrate the matrix method, the first-rank 1ⴱ 1 1 correlation functions Rn,m 共t兲 = 具Dn,m 关⍀共t兲兴Dn,m 关⍀共0兲兴典0 共ap⬘ pearing, e.g., in the theory of dielectric relaxation of fluids兲10,19 are calculated in the next section.

where the characteristic relaxation times

␶10 = 共Dyy + Dxx兲−1,

␶m1

are given by

1 ␶−1 = 共Dzz + Dyy兲−1,

␶11 = 共Dzz + Dxx兲−1 .

共15兲

Equations 共13兲–共15兲 coincide with the known results for the firs-rank correlation functions.10 The second-rank correlation 2ⴱ 2 关⍀共t兲兴Dn,m 关⍀共0兲兴典0 appearing, e.g., in the functions 具Dn,m ⬘ theory of NMR relaxation and Raman spectroscopy of molecular liquids,6,19 are calculated in Appendix B. V. CONCLUDING COMMENTS

Equation 共8兲 is a general result pertaining to the free anisotropic noninertial rotational diffusion. Now modern computer software, e.g., MATHEMATICA®, can evaluate matrix exponentials, hence Eq. 共8兲, in principle, allows one to calculate directly the equilibrium orientational correlation j j 共t兲 = 具Fⴱ关⍀共0兲兴Dn,m 关⍀共t兲兴典0 of any rank j. For functions Rn,m j 共t兲 example, using MATHEMATICA®, the calculation of Rn,m from Eq. 共8兲 requires a three line program 共the program is available on request兲. Two examples considered in the paper

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134105-4

J. Chem. Phys. 130, 134105 共2009兲

Yuri P. Kalmykov

共namely, the calculation of the first- and second-rank correlation functions兲 clearly demonstrate that the matrix method we developed is equivalent to the traditional Favro approach.2 The advantage of the matrix treatment is that it is not based on the quantum theory of a free asymmetric rotor 共which is rather a complicated problem兲24,25 and the calculations can be readily accomplished for any rank j. Moreover, previous results for the equilibrium orientation correlation functions may be regained from Eq. 共8兲 in a much simpler way than before. Furthermore, the new matrix method can be readily generalized to nonlinear dielectric and Kerr effect responses using perturbation theory.30

冢 冣 冢冣 ␦m⬘,−2

2 Rn,−2 共t兲

2 共t兲 Rn,−1

␦m⬘,−1

1 = e A2t 5

2 共t兲 Rn,0

2 共t兲 Rn,1 2 Rn,2共t兲

␦m⬘,0

␦m⬘,1

␦m⬘,2

2 共t兲 can be obtained from Eq. The explicit solutions for Rn,m 共B1兲 by calculating eA2t = T2⌳2共t兲T−1 2 just as for j = 1 in Appendix A. Thus one obtains the known results for second2ⴱ 2 关⍀共t兲兴Dn,m 关⍀共0兲兴典0, viz.,2,6 rank correlation functions 具Dn,m

⬘

2ⴱ 2 具Dn,0 关⍀共t兲兴Dn,m⬘关⍀共0兲兴典0

ACKNOWLEDGMENTS

=

I thank Professor W. T. Coffey for stimulating discussions and useful comments.

=

=

0

,

1

A1t

⌳1共t兲 =

T1 =

冢

冢

1

0

0

e

= 0 . 1

0

0

−1 0 1 0 1

冣

1 0 , 0 1

0 e

−t/␶11

T−1 1

10

1 = 2

冢

,

0 1

2

2

共e−t/␶1 + e−t/␶−1兲 +

␦⫿1,m⬘ 10

2

2

共e−t/␶1 − e−t/␶−1兲,

20

2

2

2

共2e−t/␶0 + e−t/␶2 + e−t/␶−2兲

␦0,m⬘⌶冑6 + ␦⫾2,m⬘2⌬ 20冑4⌬2 + 3⌶2

2

2

共e−t/␶2 − e−t/␶−2兲,

2 where the characteristic relaxation times ␶m are given by

␶20 =

−1 0 1

␦⫾2,m⬘

+

= T1⌳1共t兲T−1 1

冣

0

−t/␶10

2

2ⴱ

=

can be calcuThus the matrix exponential e lated by matrix multiplication of the matrices ⌳1共t兲, T1, and T−1 1 given by e−t/␶−1

20冑4⌬2 + 3⌶2

2 关⍀共t兲兴Dn,m⬘关⍀共0兲兴典0 具Dn,⫾2

1

g13

= 1 , 0

␦⫾1,m⬘

␭13 = − 1/␶11 ,

0

g12

␦⫾2,m⬘⌶冑6 − ␦0,m⬘4⌬

2ⴱ

冢 冣 冢冣 冢冣 −1

2

2 关⍀共t兲兴Dn,m⬘关⍀共0兲兴典0 具Dn,⫾1

1 have been given by Eq. 共15兲兴 and 关the relaxation times ␶m

g11

10

2

共e−t/␶2 + e−t/␶−2兲 + 2

The eigenvalues ␭1k and corresponding eigenvectors g1k of the matrix A1 defined by Eq. 共6兲 are given, respectively, by ␭12 = − 1/␶10,

␦0,m⬘

⫻共e−t/␶2 − e−t/␶−2兲,

APPENDIX A: CALCULATION OF eA1t

1 ␭11 = − 1/␶−1 ,

共B1兲

.

␶D ˆ 兲−1 , = 共3Dzz + TrD 3 + 4⌬

2 ␶⫾1 =

冣

2 0 . 0 1

2 ␶⫾2 =

␶D ˆ 兲−1 , = 共3Dxx ⫾ TrD 3 + ⌬ ⫾ 3⌶/2 ␶D

1

ˆ

= 共TrD ⫾ ␬兲 3 + 2⌬ ⫾ 冑4⌬2 + 3⌶2 2 ˆ =D +D +D TrD yy xx zz

Here

= 冑共Dxx − Dyy兲 + 共Dzz − Dyy兲共Dzz − Dxx兲.

−1

.

and

APPENDIX B: SECOND-RANK CORRELATION FUNCTIONS

F. Perrin, J. Phys. Radium 5, 497 共1934兲; 7, 1 共1936兲. D. L. Favro, Phys. Rev. 119, 53 共1960兲; in Fluctuations Phenomena in Solids, edited by R. E. Burgess 共Academic, New York, 1965兲, p. 79. 3 J. H. Freed, J. Chem. Phys. 41, 2077 共1964兲. 4 D. Ridgeway, J. Am. Chem. Soc. 88, 1104 共1966兲. 5 R. Pecora, J. Chem. Phys. 50, 2650 共1969兲. 6 W. T. Huntress, Jr., in Advances in Magnetic Resonance, edited by J. Waugh 共Academic, New York, 1970兲, Vol. 4, p. 1; J. Chem. Phys. 48, 3524 共1968兲. 7 H. Brenner and D. W. Condiff, J. Colloid Interface Sci. 41, 228 共1972兲. 8 T. J. Chuang and K. B. Eisenthal, J. Chem. Phys. 57, 5094 共1972兲. 9 B. J. Berne and R. Pecora, Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics 共Dover, New York, 2000兲. 10 J. R. McConnell, Rotational Brownian Motion and Dielectric Theory 共Academic, New York, 1980兲. 1

For j = 2 and Dxy = Dyz = Dxz = 0, the matrix A2 from Eq. 共6兲 is given by

A2 = −

␶D−1

冢

3 + 4⌬

0

冑3/2⌶

0

0

0

3+⌬

0

3⌶/2

0

0

3

0

冑3/2⌶

0

3+⌬

0

0

3 + 4⌬

冑3/2⌶ 0

3⌶/2

0

0

冑3/2⌶

␬

2

冣

.

Using the orthogonality properties of the Wigner D functions Eq. 共11兲, Eq. 共8兲 then becomes

2

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134105-5 11

J. Chem. Phys. 130, 134105 共2009兲

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