Rotation of Nonspherical Molecules in Dense Fluids: A Simple Model

236

J. Phys. Chem. B 1997, 101, 236-243

Rotation of Nonspherical Molecules in Dense Fluids: A Simple Model Description A. P. Blokhin and M. F. Gelin* Institute of Molecular and Atomic Physics, Academy of Sciences of Belarus, F. Skarina AVenue 70, Minsk 220072, Belarus ReceiVed: June 6, 1996; In Final Form: October 4, 1996X

Generalized Langevin equations (GLEs) are used to calculate angular velocity correlation functions (AVCFs) and orientational correlation functions (OCFs) of asymmetric top molecules in the high friction limit. Two exponential memory functions are shown to fit simulated AVCFs satisfactorily. These memory functions consist of an intense but quickly relaxed contribution (in-cage rotation) and a small but slowly decaying contribution (due to collective modes). An exact analytical expression for an arbitrary rank asymmetric top OCF is obtained up to terms of order O(t6). This result allows establishment of the correspondence between the short-time value of the memory function and the averaged magnitude of the external torque. Explicit formulas are derived for symmetric top OCFs when arbitrary (from slip to stick) boundary conditions are presumed. These formulas are proved to reduce to the conventional first cumulant expressions provided that molecular rotation around the symmetry axis is driven by a Gaussian stochastic process. The experimentally observed non-Gaussian behavior is suggested to be attributed to the preferential angular velocity reorientation due to a collision, and a convenient analytical model is proposed to describe this effect.

Introduction Characteristic features of molecular rotation transparently manifest itself in the angular velocity correlation functions (AVCFs).1 In dense fluids, AVCFs have pronounced negative lobes and, not infrequently, decay in an oscillatory manner. This kind of behavior reflects the quasilibrational motion of molecules in the temporary cages formed by their neighbors. This situation differs markedly from the gaslike rotation that is primarily governed by rarely occurring binary collisions and results in the quasiexponential decay of AVCFs. There exists a number of theoretical approaches being of power to describe (at least, qualitatively) the aforementioned features.1-13 The molecule under consideration is assumed to be placed in the multilevel potential forming by its neighbors and to suffer some kind of external torques.5-10 Another way is to regard the neighbor cage as slowly rotating local structure(s) interacting with the molecule of interest2-4,11-13 or introduce external torques in the stochastic Liouville equation.11 However, an explicit evaluation of correlation functions (CFs) in these models require more or less extensive numerical calculations, even in the simplest case of linear and/or spherical molecules. So, the analysis is often restricted to the case of one-dimensional rotation, which allows us to study the problem by analytical tools.2,3-7,11 Fortunately, the high-density fluids are the strong torque systems, so that inertial rotation can be regarded as perturbation and one can successfully run cumulant expansions1,14 in order to arrive at the analytically manageable results. The typical representatives of the lowest cumulant approaches are the Gaussian cage model15,16 and the damped librator model.17,18 However, the models are formulated for a vector fixed to a molecule parallel to its symmetry axis, and generalization of the approaches to study orientational relaxation of an arbitrary vector fixed to an asymmetric top is not straightforward. On the other hand, rotational motion of nonspherical molecules can be conveniently treated in the framework of Langevin equations for angular velocities. Once again, one can invoke the cumulant expansion formalism or similar techniques to derive explicit X

Abstract published in AdVance ACS Abstracts, November 15, 1996.

S1089-5647(96)01673-2 CCC: $14.00

formulas for AVCFs and orientational CFs (OCFs) in the high friction limit. This approach had been suggested by Steele for spherical tops19 (the so-called friction model) and was also generalized to symmetric19-22 and asymmetric23,24 top molecules. The Langevin equations imply instantaneous δ-correlated torques. So, it is not entirely unexpected that the equations lead to strictly positive and monotonically decaying AVCFs and are inadequate in describing molecular rotation in dense fluids. To overcome this limitation, it is logical to employ generalized Langevin equations (GLEs) for angular velocities.2,20,25 The outlined (or very similar) scheme was applied to the simplest molecular tops, viz. planar,26,27 spherical, and linear.1,26,28-33 The present investigation deals with symmetric and asymmetric top molecules. The GLEs for angular velocities are introduced and solved in the high friction limit. In turn, this induces the lowest cumulant formulas for OCFs. It is demonstrated how the GLE-based description can be related to the microscopic picture of molecular rotation, and a simple relationship is established between the short-time value of the memory function and the averaged magnitude of the external torque initially acting on an asymmetric top. Symmetric top molecules are considered in more details. The explicit formulas are derived for OCFs when arbitrary (from slip to stick) boundary conditions are presumed. These formulas are proved to reduce to the conventional first cumulant expressions provided that molecular rotation around the symmetry axis is driven by a Gaussian stochastic process. Therefore, the experimentally observed nonGaussian behavior2,33 cannot be explained within a simple GLE scheme. The behavior is suggested to be attributed to the preferential angular velocity reorientation due to a collision, and a convenient analytical model is proposed to describe this effect. Note that dimensionless variables are used throughout this article: time is measured in units of τR ) (I/kBT)1/2 and angular velocities in τR-1, with T being the rotational temperature and I being a characteristic moment of inertia of the molecule, so that τR is the averaged period of free rotation. © 1997 American Chemical Society

Nonspherical Molecules in Dense Fluids

J. Phys. Chem. B, Vol. 101, No. 2, 1997 237

Rotational GLEs The conventional mathematical description of the rotational Brownian motion is based on the stochastic generalization of the Langevin-Euler equations:23,24,34-39

∂tωR(t) ) RβγIR-1Iβωβ(t)ωγ(t) - ξRωR(t) + FR(t)

(1)

〈FR(t) Fβ(t′)〉 ) δRβξRIR-1δ (t - t′)

(2)

Here ωR(t) are the components of the angular velocity in the molecular reference frame, IR are the main moments of inertia, ξR are the friction coefficients, and random torques FR(t) are described by δ-correlated Gaussian Markovian processes. From now on, repeated silent Greek indexes are to be summed over x, y, and z. Due to the nonlinearity of the Langevin-Euler equations, the direct implementation of the Mori procedure for constructing asymmetric top GLEs runs into obstacles.40,41 However, one can improve eqs 1 and 2 by invoking phenomenological GLEs of the kind

∂tωR(t) ) RβγIR Iβωβ(t) ωγ(t) -

A formal solution of eq 9 can be written as the time-ordered exponential1

∫0tdt′ ωR(t′)JjR}〉

Gj(t) ) 〈ex bp{-i

(11)

One can calculate eq 11 by using the cumulant expansion method.1,14,33 In the high friction limit, the leading contribution to (11) reads

Gj(t) ) exp{-JjRJjβ

∫0tdt′ (t - t′)CωRβ(t′)} + O(ξγ-3)

(12)

ω (t) is assumed to be an exact AVCF. However, one Here CRβ can safely substitute this exact AVCF for that given by eq 5. This is guaranteed by the accuracy of expressions 5 and 12. According to eq 5 one can rewrite (12) in the form of the diffusion equation with time-dependent diffusion coefficients:1

Gj(t) ) exp{-[JjR]2DR(t)}

∫0tdt′ (t - t′)ΨR(t′)

DR(t) ) IR-1

(13)

-1

∫0 dt′ ξRhR(t - t′) ωR(t′) + FR(t) t

〈FR(t) Fβ(t′)〉 ) ξRhR(t - t′)IR-1δRβ

(3) (4)

For symmetric and asymmetric tops, eqs 3 are nonlinear stochastic equations driven by colored noise (eq 4). In the high friction friction limit ξR . 1, one can drop nonlinear (inertial) terms completely and derive the following expression for the AVCF:

For a large enough time interval (t . τωR ), one naturally arrives at the conventional diffusion equation19,42-44 with the diffusion coefficients DR ) τωR /IR. Here

τωR ) {CωRR(0)}-1

(14)

is the AVCF relaxation time. Starting from eq 13, one can deduce the following shorttime behavior for the OCF: 4

CωRβ(t) ) 〈ωR(0) ωβ(t)〉 ) δRβIR-1ΨR(t) + O(ξγ-3) (5) Ψ ˜ R(s) ) {s + ξRh˜ R(s)}-1

R*β*γ

(7)

(8)

is governed by the following general equation:15,34-37

∂tGj(ω b (τ)) ) -iωR(t) JjRGj(ω b (τ))

(9)

(0 e τ e t). Here Ω(t) is the set of three Euler angles specifying molecular orientation, Dj(Ω(t)) is the Wigner D function, and JjR are the matrix elements of the angular momentum operators over D functions:35-37

(Jjx)kl ( i(Jjy)kl ) δk,l-1 {(j ( 1)(j - 1 + 1)}1/2 (Jjz)kl ) lδkl, - j e k,l e j (10) [JjR, Jjβ] ) -iRβγJjγ

tn/n!Gjn + O(t5) ∑ n)0

(15)

Here

OCF

b (τ))〉 Gj(t) ) 〈Dj(Ω(t)) Dj(Ω(0))-1〉 ) 〈Gj(ω

Gj(t) )

(6)

(Adopted is the notation χ˜ (s) for the Laplace transformation of an arbitrary function χ(t).) Closely following the general procedure that has been developed in papers23,39 for standard Langevin equations (eq 1), one can calculate higher order corrections to (5). This procedure results in the solution of eq 3 in terms of perturbation series. These series are perturbation expansions simultaneously on the inverse friction values ξR-1 and asymmetry parameters:

χR ) (Iβ - Iγ)IR-1,

∫0∞dt CωRR(t) ) Ψ˜ R(0) ) {h˜ R(0)ξR}-1 , 1

(10′)

Gj0 ) 1, Gj1 ) 0, Gj2 ) [JjR]2CωRR(0), Gj3 ) 0

(16)

Gj4 ) 3{[JjR]2CωRR(0)}2 + [JjR]2∂t2CωRR(t)|t)0 According to eqs 5 and 6 2 ω -1 CωRR(0) ) I-1 R , ∂t CRR(t)|t)0 ) IR ξRhR(0)

(17)

On the other hand, as was pointed out by Gordon,45 the first few terms in the Taylor expansion for the OCF can be calculated exactly, by averaging the corresponding moments of the Liouville operator. This procedure is fulfilled in the Appendix up to terms of order O(t6) for an arbitrary rank asymmetric top OCF. By comparing eqs 16 and A14, one sees that the first four terms (Gj0 - Gj3) of the exact short-time expansion for the OCF coincide with those from (16), but the fifth one (Gj4) differs. In the high friction limit ξR . 1 (more specifically, when 〈NR2 . |(χR2 - 1)IR2(IβIγ)-1|) one can neglect the difference and establish the relationship

ξRhR(0) ) 〈NR2〉IR-1

(18)

The equation allows one to find the correspondence between the model description of molecular rotation in terms of GLEs and the microscopic description in terms of the external potential of nearest neighbors. Note also that the initial short time development for the OCF is given by eq 16 correctly. As distinct from the Markovian instantaneous collision approaches

238 J. Phys. Chem. B, Vol. 101, No. 2, 1997

Blokhin and Gelin

(as the extended diffusion models24,46-48 or the rotational Fokker-Planck equations19-23,34-38) Gj3 ≡ 0 in eq 16. It is in agreement with the general statement that all the odd-numbered terms in the Taylor expansion of any CF must be identically zero. This is a direct consequence of the reversibility of equations of motion in the classical mechanics.45 However, starting from the sixth term, eq 15 possesses nonzero oddnumbered coefficients. This is due to the fact that (3) and (12) are model irreversible stochastic equations. Memory Functions To get more insight into the formulas derived, it is necessary to specify memory functions hR(t) in eqs 3 and 4. By analogy with ref 49, the most natural choice is

hR(t) ) µR exp{-µR|t|}

(19)

When substituted into (6), eq 19 gives rise to simple analytical expressions for AVCFs and time-dependent diffusion coefficients: + + + ΨR(t) ) (zR exp{-zR t} - zR exp{-zR t})/(zR - zR ) (20)

Figure 1. Ψx(t) time development, τωx ) 0.085. The two exponential memory function parameter µx ) 1, qxξx ) 8, νx ) 15; pxξx ) 0 (curve 1), 2 (2), 4 (3), 7 (4), 10 (5).

+ -2 DR(t) ) IR-1{t/ξR + ([zR (zR ) ] exp{-zR t} - -2 + + [z+ R (zR ) ] exp{-zR t})/(zR - zR )} 2 1/2 zR ) {µR (µR - 4ξRµR) }/2

(21)

If µR . 1, the results of the friction theory19-24 are rederived. By decreasing µR, one enables a description of the increasing portion of the librational character of molecular rotation. When µR g 4ξR, AVCFs decay monotonically. When µR < 4ξR, AVCFs exhibit increasingly oscillatory behavior. So, simple analytical formulas 19-21 allow us to qualitatively describe the transformation of molecular rotation from gas-like to liquidlike. Note that case µR < 4ξR is equivalent to the damped librator model17,18 per rotational degree of freedom, with the libration frequencies ΩR ) (4ξRµR - µR2)1/2. In the GLE approach, the appearance of these frequencies is not a result of some ad hoc assumption: ΩR naturally arises in the theory when characteristic times of random torque correlations become substantial. However, simple exponential memory functions (19) are insufficient to fit quantitatively simulated AVCFs. So, one has to select more sophisticated sets of memory functions.25,50,51 We use two-exponential memory functions52-54 per rotational degree of freedom:

hR(t) ) pRµR exp{-µR|t|} + qRνR exp{-νR|t|}

(22)

This is a pictorial model for describing molecular rotation in liquids. One can expect that memory function 22 consists of an intense but quickly relaxed contribution (in-cage rotation) and a small but slowly decaying contribution (due to surmounting the potential barrier of the local cage or relaxation of the local structure forming by the neighbor molecules). According to eq 6, memory function 22 leads to the three-exponential decay of the AVCF with the integral relaxation time (eq 14):

τωR ) (ξR(pR + qR))-1

(23)

Figure 2. Same as in Figure 1 but for Dx(t).

molecule is a symmetric top (Ix ) Iy) and dimensionless time τR ) (Ix/kBT)1/2 ≈ 0.5 ps. In the agreement with the aforementioned qualitative expectations, curves 3 in Figures 1 and 3 well resemble simulated Ψx(t) and Ψz(t) correspondingly. So, the two-exponential memory functions, probably, permit a satisfactory fitting of simulated AVCFs. In this case, the accuracy of eqs 13 is determined primarily by the accuracy of the lowest cumulant approach itself. It is interesting that AVCFs with negative lobes (like Ψx(t)) can be modeled by means of a sum of two-exponential memory functions 22, whereas positive AVCFs with slowly decaying tails (like Ψz(t)) require a difference of such exponentials. This fairly unexpected result can be clarified by assuming that the AVCF itself can be represented by the two-exponential function

ΨR(t) ) aR exp{-xRt} + (1 - aR) exp{-yRt} The corresponding ΨR(t) and DR(t) are depicted in Figures 1-4. We use the memory function parameters so that to bring the calculated AVCFs to the correspondence with Ψx(t) (Figure 1) and Ψz(t) (Figure 3) simulated for CH3CN at 291 K.33,55 This

(24)

Except for the short-time behavior, this is a reasonable trial expression for AVCFs. According to eq 6, the memory function for AVCF 24 reads


J. Phys. Chem. B, Vol. 101, No. 2, 1997 239 AVCF remains positive. As to Ψx(t)-like AVCFs, random torques act (in the average) at the same direction during a characteristic time interval, thus ensuring a preferential angular velocity reorientation and a negative lobe of the AVCF. In concluding this section, we note that one can conveniently investigate orientational relaxation of nonspherical molecules by constructing memory functions directly for OCFs.24,56-60 Symmetric Tops

Figure 3. Ψz(t) time development, τωz ) 0.14. The two-exponential memory function parameter µz ) 5.5, pzξz ) -7, νz ) 30; qzξz ) 10 (curve 1), 12 (2), 14 (3), 16 (4), 30 (5). Curve 6 corresponds to pz ) 0, νz ) qz f ∞, ξz ) 7.2 (exponential approximation for AVCF).

In the previous sections of this article we presented simple analytical formulas that are of the power to describe simultaneously both AVCFs and OCFs for high-torque ensembles of asymmetric tops. However, symmetric top molecules can be studied in more detail. In addition, new qualitative features of molecular rotation arise for such molecules. It is well-known that molecular rotation around the symmetry axis (spinning) is considerably less hindered than that around the perpendicular axis (tumbling).19,33,61 For instance, collisions of hard axially symmetric convex bodies do not change projections of the angular velocities on their symmetry axes.34 More generally, this is also the case when intermolecular potential is unaltered when molecules rotate about their symmetry axes.19 From the hydrodynamic point of view, this corresponds to “slip” boundary conditions.62 Therefore, it is interesting to investigate how the lowest cumulant formulas transform when one assumes the strongly hindered tumbling (ξx . 1) but no approximations are made concerning spinning (ξz is arbitrary). To do this, it is convenient to invoke the cumulant approach in the interaction representation.14 By factoring out the spinning motion, it is natural to seek for the solution of eq 9 in the following form:

Gj(ω b (τ)) ) exp{-i

∫0tdt′ ωz(t′)Jjz}Rj(ωb(τ))

(26)

So, Rj(ω b (τ)) obeys the equation

∂tRj(ω b (τ)) ) Mj(ω b (τ))Rj(ω b (τ))

(27)

where

b (τ)) ) ) -i exp{i Mj(ω

Figure 4. Same as in Figure 3 but for Dz(t).

hR(t) ) {aRxR + (1 - aR)yR}δ(t) aR(1 - aR)(xR - yR)2 exp{-[aRyR + (1 - aR)xR]t} (25) The magnitude of the normalized AVCF must not exceed unity, i.e., ∂tΨR(t)|t)0 ) -{aRxR + (1 - aR)yR} < 0. Therefore, the coefficient near the δ function in eq 25 must be nonnegative. To describe a negative lobe of the AVCF Ψx(t), one must assume different signs of terms ax and 1 - ax in eq 24, i.e., the coefficient near exponential in eq 25 is positive. The positive AVCF Ψz(t) exhibits an initial quick decay followed by a slow long-time tail. So, both az and 1 - az must be positive that results in a negative coefficient near exponential in eq 25. Memory function 25 is nothing else than the random torque correlation function 4. In case of Ψz(t)-like AVCFs, the fluctuating part of the torque tends to reverse its sign after some characteristic time interval. In the average, a positive increment of the angular velocity is followed by a negative one, no net angular velocity reorientation takes place, and the corresponding

∫0tdt′ ωz(t′)Jjz}{(ωx(t)Jjx + t ωy(t)Jjy} exp{-i∫0 dt′ ωz(t′)Jjz}

(28)

By further assuming that the relaxation time for the tumbling AVCF τωx ∝ ξx-1 , 1, one can state that ∫0tdt′ (t - t′)nΨx(t′) ∝ {τωx }n at t . τωx . So, the first cumulant solution of eq 27

b (τ))〉 ) exp{ 〈Rj(ω

∫0tdt′ ∫0t′dt′′ 〈Mj(ωb(t′)) Mj(ωb(t′′))〉} + O(ξx-3) (29)

can be simplified to

b (τ))〉 ) exp{〈Rj(ω

∫0tdt′ (t - t′)Ψx(t′)([JBj]2 - [Jjz]2}

(30)

Finally, by employing the explicit form 10 of the matrix elements of the angular momentum operators over D functions, one gets

Gjkm(t) ) Gjk(t)δkm

(31)

∫0tdt′ ωz(t′)}〉z

Gjk(t) ) exp{-[j(j + 1) - k2]Dx(t)}〈exp{-ik

(32)

240 J. Phys. Chem. B, Vol. 101, No. 2, 1997

Blokhin and Gelin

So, the original three-dimensional problem of the evaluation of the ordered exponential 11 reduces to the finding of the onedimensional averaging of the conventional exponential in eq 32. Provided that ωz(t) is governed by GLEs 3, it is a simple Gaussian process: for symmetric tops (Ix ) Iy, ξx ) ξy), the equation for ωz(t) decouples from those for ωx(t) and ωy(t). In this case, the averaging in (32) can be performed elementary with the result

Gjk(t) ) exp{-[j(j + 1) - k2]Dx(t) + k2Dz(t)}

(33)

Equation 33 is nothing else than the first cumulant formula 13. So the analysis given above proves the statement of the authors of ref 33 that the first cumulant result is correct for an arbitrary friction intensity ξz, provided that ωz(t) can be considered as a Gaussian variable. Within the approach embodied in eq 32, complete information about the spinning motion is contained in the function

∫0tdt′ ωz(t′)}〉

Φjk(t) ) 〈exp{-ik

(34)

As was suggested in ref 33, one can extract this information from eq 32 by means of the identity

ln Φjk(t) ) {ln Gjk(t) - [1 - k2/{j(j + 1)}]ln Gj0(t)} (35) Equation 33 predicts that the function

Sjk(t) ) ln{Φjk(t)}/k2 ) -Dz(t)

(36)

is j and k independent. On the contrary, molecular dynamics simulations33 reveal a significant dependence of Sjk(t) upon index k. In addition, it is clearly established that AVCFs themselves can exhibit considerably non-Gaussian behavior in liquids.2,25,63 The approach based on GLEs 3 and 4 failed to explain these features. To do so, molecular dynamics simulations in two different models (a planar Brownian rotor in a static cosine potential and non-Gaussian Kubo oscillator) were performed.33 Here a simple physical model is suggested, which ascribes the aforementioned behavior for AVCFs and OCFs to the preferential angular momentum reorientation due to collisions. Let us assume that the spinning motion is governed by the standard kinetic equation:

kind δ(ω + ω′). So, the magnitude of the angular velocity is preserved but its direction reversed due to a collision. Equation 37 with kernel 38 was suggested in refs 65 and 66 to investigate one-dimensional rotation. It contains, as a special case, the J diffusion model (γ ) 0) and the Fokker-Planck equation (zc f ∞, γ f 1, zc(1 - γ) f ξ). If 0 < γ < 1, the KS model interpolates smoothly between the two aforementioned extremes and generalizes these to investigate a physical situation when -1 e γ < 0. The latter situation corresponds to the preferential reorientation of the angular velocity in the course of collisions. Starting from eq 37, one can analytically calculate both the spinning AVCF and OCF.65,66 The former is a simple exponential function:

Cωz (t) ) exp{-t/τω}, τω ) {zc(1 - γ)}-1

∫-∞∞dω′ T(ω|ω′) F(φ,ω′,t)

and the Laplace image of the latter is expressed through the continued fraction:

2k2/(1 - χ) k2/(1 - χ) nk2/(1 - χ) 1 + + + . . . + s s + zc(1 - γ) s + z (1 - γ2) s + z (1 - γn) c

F(φ,ω,0) ) δ(φ - φ0) δ(ω - ω0)

(39)

Φ ˜ jk(s) )

∂tF(φ,ω,t) ) -ω∂φF(φ,ω,t) - zcF(φ,ω,t) + zc

Figure 5. Sjk(t) vs t dependencies. 1, γ ) -0.95; k ) 2; 2, γ ) -0.9, k ) 2; 3, γ ) -0.8, k ) 2; 4, γ ) -0.95, k ) 1; 5, γ ) -0.9, k ) 1; 6, γ ) -0.8, k ) 1; 7, γ ) 0, k ) 1; 8, γ ) 1, k ) 1.

(37)

Here φ is the angle of rotation about the symmetry axis, zc is the collision frequency, and the subscripts z in ωz, φz, Iz, etc., are omitted for clarity. To specify the collision process, we invoke the Keilson-Storer (KS) kernel:64

TKS(ω|ω′) ) TKS(ω - γω′) ) [2πI(1 - γ2)]-1/2 × exp{-(ω - γω′)2/[2I(1 - γ2)]} (38) According to this kernel, the averaged angular velocity after a collision 〈ω〉 ) γω′, so that the parameter -1 e γ e 1 determines the collision efficiency. When γ ) 1, the angular velocity is conserved, T(ω|ω′) reduces to δ(ω - ω′) and (38) describes a free rotation in the plane. Provided that γ ) 0, intermolecular interactions are so strong that a single collision is enough to establish a Boltzmann equilibrium distribution. By letting γ ) -1, one also arrives at the δ function, but of the

c

-1 e χ ) 1 - Iz/Ix e 1 (40) The corresponding Sjk(t) functions (eq 36) are depicted in Figure 5 for k ) 1, 2 and different γ. We choose χ ) 0.94 (highly prolate top) and τωz ) 0.14, as those in the computer simulation of liquid CH3CN at 291 K.33,55 It is nonambiguously seen that the curves with different k distinguish considerably. To put it differently, when γ is close to -1 the curves fan out more markedly. The curves with γ = -0.9 are in a sufficiently good agreement with their counterparts calculated by molecular dynamics.33,55 So that the KS model presents a reasonable explanation for the observed non-Gaussian effects. By accepting this explanation, one has to conclude that the angular velocity reorientation due to a collision is quite high in the liquid CH3CN. On the other hand, rotational motion of the NO3- ions from the molecular dynamics simulation of LiNO3 corresponds to Sjk(t) functions that lie below the first cumulant approxima-



tion Dz(t), i.e., Sjk(t) e Dz(t) for some values j and k.67 This result is not consistent with the KS model introduced here. By keeping parameters τω, j, and k fixed, one sees from Figure 5 that when γ increases, the value of Sjk(t) decreases, so that Sjk(t) g Dz(t). A likely explanation for this discrepancy is that spinning function 34 is not rigorously extracted from OCFs Gjk(t) by means of formula 35. This latter formula is derived under assumption that the first cumulant result for the tumbling motion is accurate enough. This is evidently not the case for NO3-.67 In the present context it is pertinent to make two remarks. First, the points of inflexion in the Sjk(t) vs t curves appear for negative γ for an arbitrary value of the symmetry parameter χ. Therefore, the angular velocity reorientation due to a collision manifests itself as points of inflexion in logarithmic OCFs. On the other hand, when γ is near 1 (more precisely, in the limit zc f ∞, γ f 1, zc(1 - γ) f {τω}-1 ) constant), eq 37 corresponds to the FPE and describes a Gaussian process. So, eq 40 gives rise to formula 36 for δ-correlated torques. In this case, Sjk(t) is independent of index k, and no points of inflexion are observed. Second, the KS model predicts the purely exponential AVCF Ψz(t) (eq 39), even for negative γ. In case of CH3CN, this AVCF is positive and more or less exponential (Figure 3), although a long-time tail still exists. Generally, the spinning AVCFs in liquids can exhibit oscillatory behavior with some negative portions.2,67,68 OCFs in high-torque systems appear to have a relatively minor dependence on a particular form of the corresponding AVCFs. Indeed, OCFs in the diffusion limit are uniquely determined by the diffusion coefficients (eq 13). These coefficients are completely specified by angular velocity integral relaxation times (eq 14), irrespectively of specific forms of AVCFs. Nonetheless, to generalize eq 37 to properly describe AVCFs, it is logical to resort to the non-Markovian master equation with memory.69,70 For our problem, one can suggest the following equation:

Figure 6. Sjk(t) vs t dependencies for γ ) -0.95. k ) 1, unprimed letters; k ) 2, primed letters. Curves a, b, and c correspond to AVCFs 5, 3, and 6 from Figure 3.

∂tF(φ,ω,t) ) -ω∂φF(φ,ω,t) -

∫0tdt, ∫-∞∞dω′ T(ω′,ω,t-t′) F(φ,ω,t) + ∫0tdt′, ∫-∞∞dω′ T(ω,ω′,t-t′) F(φ,ω′,t) (41)

In the absence of a better choice, we assume the factorization:

T(ω,ω′,t) ) hz(t) zcTKS(ω - γω′)

(42)

By taking the Laplace transform of eq 41 with kernel 42, it is a simple matter to demonstrate that the AVCF is given now by “correct” formula

Ψ ˜ z(s) ) {s + h˜ z(s)/τωz }-1

(6′)

and expression 40 for OCF holds true provided that zc f zch˜ z(s). The Markovian kinetic equation 37 is recovered in the limit hz(s) f 1, i.e., µz,νz f ∞ for memory function 22. Representative calculations of OCFs that correspond to AVCFs depicted in Figure 3 are presented in Figures 6 and 7. Despite the fact that exponential “Markovian” AVCF 6 and the “correct” one 3 in Figure 3 distinguish considerably, the difference between the corresponding OCFs b, b′, and c, c′ in Figures 6 and 7 is less pronounced. This gives additional support to the validity of Markovian master equation 37 in describing OCFs, provided that corresponding AVCFs are more-or-less gas-like, i.e., positive and monotonically decaying. On the other hand, curves a, a′ and c, c′ exhibit a pronounced difference (the former curves correspond to sign-alternating AVCF 5). Qualitatively, eq 41

Figure 7. Same as in Figure 6 but for γ ) -0.9.

with kernal 46 appears to self-consistently describe both AVCFs and OCFs. It can be useful in studying OCFs for spinning motion, when AVCFs exhibit oscillatory behavior with negative excursions. However, a quantitative agreement between “improved” curves b, b′ and those calculated by molecular dynamics33 is even a bit worse than for “Markovian” curves c, c′ (points of inflection are less pronounced). A likely explanation for this discrepancy is that factorization assumption 42 is crude enough so that more comprehensive analyses of the problem is required (see, e.g., refs 71 and 72). Another possible source for this quite unexpected behavior, as mentioned above, is an approximate nature of eq 35. In any case, factorization assumption 42 deserves further testing. Conclusion We attempt to construct simple (ideally, analytical) models for describing rotational and reorientational motion of nonspherical molecules in dense fluids. For this purpose, we invoke GLEs for angular velocities and solve these equations in the

242 J. Phys. Chem. B, Vol. 101, No. 2, 1997 high friction limit. The corresponding AVCFs are used to derive the lowest cumulant approximation for OCFs. So, GLEs present an opportunity to investigate both the angular velocity and orientational relaxation processes simultaneously, because the cumulant expansion for an arbitrary OCF is also a perturbative one, with a small parameter proportional to the inverse friction value. An exact expression for the asymmetric top OCF is obtained up to terms of order O(t6). This result allows us to establish the correspondence between the short-time values of the GLE memory functions and the averaged magnitude of the initial torque components. The two-exponential memory functions per rotational degree of freedom are shown to fit simulated AVCFs satisfactorily. In line with physical expectations, these memory functions consist of an intense but quickly relaxed contribution (in-cage rotation) and a small but slowly decaying contribution (due to surmounting the potential barrier of the local cage or relaxation of the local structure forming by the neighbor molecules). Symmetric top molecules are considered in more details. The explicit formulas are derived for OCFs in the case when one assumes the strongly hindered tumbling motion, but no approximations are made concerning spinning (arbitrary boundary conditions, from slip to stick). These formulas are demonstrated to reduce to the conventional lowest cumulant expressions provided that molecular rotation around the symmetry axis is driven by a Gaussian stochastic process. We suggest that the experimentally observed non-Gaussian behavior in the spinning motion is the manifestation of the preferential angular velocity reorientation due to a collision. To describe this effect, we invoke a rotational kinetic equation with the collision kernel by Keilson and Storer. A non-Markovian generalization of this equation is also introduced. This allows one to self-consistently incorporate both oscillatory and/or sign alternating behavior for AVCFs and also collisional angular velocity reorientation. In this paper, we do not intend to go beyond the lowest cumulant approach, although pertinent contributions could be calculated, in principle, by the method outlined in refs 1, 14, 23, and 39. For spherical tops, this procedure was followed in refs 28-33 and 35 but for less symmetric molecules explicit calculations are becoming very cumbersome. However, this is not a unique argument against the usefulness of the computation of higher order corrections. Starting from GLEs 3 for spherical tops (Ix ) Iy ) Iz), Pomeau and Weber28 have proved that the quantity ln{Gj(t)}/[j(j + 1)] is j independent up to the sixthorder cumulant contribution; only the eighth-order cumulant introduces a j-dependent term to this expansion. That is why the summation of higher order cumulants will hardly improve the situation for nonspherical molecules. In addition, the spinning GLE for a symmetric top (as also spherical and linear top GLEs) is a Gaussian process that is in disagreement with experimental observations. To overcome these drawbacks, one has to resort to more sophisticated approaches, e.g., nonlinear and non-Markovian stochastic equations,2 kinetic equations with memory (like those used in the present article to study the spinning motion), or Fokker-Planck equations with memory.71-75 Appendix The short-time behavior for OCFs was studied in a number of papers. The first few coefficients in the Taylor expansion were derived for linear,45,76,77 spherical,77,78 symmetric77,79 top OCFs, and also for asymmetric top OCFs of the first80,81 and second82 rank. Here we present a convenient operator method for calculating the Taylor expansion of the asymmetric top OCF of an arbitrary rank. The rotational Liouville operator for the asymmetric top molecule subjected to the external torque N B (Ω) reads34,77

Blokhin and Gelin

L(ω,Ω) ) ωRJR(Ω) - iRβγIRIγ-1ωRωβ∂ωγ - iNR(Ω)/IR∂ωR (A1) where Jˆ R(Ω) are the angular momentum operators. The asymmetric top OCF

Gj(t) ) 〈Dj(Ω(t)) Dj(Ω(0))-1〉 )

∫dωb ∫dΩ Gj(ωb,Ω,t) Dj(Ω)-1

(A2)

obeys the equation

b ,Ω,t) ) -iL(ω,Ω) Gj(ω b ,Ω,t) ∂tGj(ω

(A3)

Equilibrium initial conditions are assumed:

b ,Ω,0) ) FB(ω b ) FB(U(Ω)) Dj(Ω) Gj(ω FB(ω b ) ) Zω-1 exp{-IRωR2/2}

(A4)

Zω ) (2πkBT)-3/2(IxIyIz)1/2

FB(U(Ω)) ) ZU-1 exp{-U(Ω)}, ZU )

∫dΩ exp{-U(Ω)}

(A5)

Notice

NR(Ω) ≡ -iJR(Ω) U(Ω)

(A6)

It is further convenient to introduce the Fourier transformed function

b,Ω,t) ) Hj(u

∫-∞∞dωb e-iubωbGj(ωb,Ω,t)

exp{uR2/(2IR)}[FB(U(Ω))]-1

(A7)

(compare with ref 38). Evidently

Gj(t) )

∫dΩ FB(U(Ω)) Hj(0,Ω,t) Dj(Ω)-1

(A8)

According to eq A3, function A7 is governed by the equation

b,Ω,t) ) {A + C}Hj(u b,Ω,t), ∂tHj(u

Hj(0,Ω,0) ) Dj(Ω) (A9)

where

A ) A 1 + A2 + A3 A1 ) Jˆ R(Ω)IR-1∂uR,

A2 ) iRβγIγ-1uR∂uβ∂uγ, A3 ) -iNR(Ω)/IR∂uR (A10)

C ) C1 + C2,

C1 ) -JjRuR, C2 ) iRβγIRIγ-1uRuβ∂uγ

The formal solution of (A9) reads ∞

Hj(u b,Ω,t) )

(tn/n!)Hjn(u b,Ω), ∑ n)0 b,Ω) ≡ {A + C}nDj(Ω) (A11) Hjn(u

By invoking an analogy with quantum mechanics in the second quantization, operators Ai in (A10) can be regarded as “annihilation operators”, Ci as “creation operators”, and 1 as a “vacuum state”. It is meant that by acting on a function of the kind

uRx uβy uγz

with R + β + γ ) N

(A12)



Ai lowers N by 1 and Bi raises N by 1. According to eq 10′, operators A10 obey rather complicated commutation rules, but the aforementioned analogy is useful. Indeed, the initial value Hj(0,Ω,0) ) D(Ω) is of the “vacuum state” type. In addition, according to (A8), calculation of the OCF Gj(t) is also equivalent to finding a “vacuum average” (u b ) 0). So, one can immediately write nonzero contributions to the Taylor components of OCFs, e.g., H j1(0,Ω) ) 0, H j2(0,Ω) ) (A1 + A3)C1Dj(Ω), etc. Keeping these arguments in mind, using permutation rules (eq 10′), and employing the integration by parts:

∫dΩ FB(U(Ω)){JˆR(Ω) Φ(Ω)} Ξ(Ω) ) ∫dΩ FB(U(Ω)) Φ(Ω){JˆR(Ω) Ξ(Ω)}

(A13)

(Φ(Ω) and Ξ(Ω) are arbitrary functions), it is possible to calculate the first few terms in the Taylor expansion of the OCF: ∞

Gj(t) )

Gj2mt2m/2m! ∑ m)0

(A14)

The first three terms read

Gj0 ) 1,

Gj2 ) -[JjR]2IR-1

(A15)

Gj4 ) 3{[JjR]2IR-1}2 + [JjR]2{(IxIyIz)-1IR(χR2 - 1) + 〈NR2〉/IR2} Here χ are given by eq 7:

〈NR2〉 )

∫dΩ FB(U(Ω))NR2(Ω)

(A16)

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JP9616730

Rotation of Nonspherical Molecules in Dense Fluids: A Simple Model

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