18560
J. Phys. Chem. B 2006, 110, 18560-18565
Analysis of the Effect of Translation-Rotation Coupling on Diffusion along the Molecular Axes A. Idrissi* and P. Damay Laboratoire de Spectrochimie Infrarouge et Raman (UMR CNRS A8516), Centre d’Etudes et de Recherches Lasers et Applications, UniVersite´ des Sciences et Technologies de Lille, 59655 VilleneuVe d’Ascq Cedex, France
S. Krishtal and M. Kiselev Institute of Solution Chemistry of the RAS, Akademicheskaya st.1, 153045 IVanoVo, Russia ReceiVed: February 24, 2006; In Final Form: August 7, 2006
The analysis of the microscopic diffusion processes of CO2 and H2O in the coordinate system defined by the molecular orientation allows a new criterion to be introduced which provides information on the short and long time behavior of the rotation-translation coupling as well as to quantify the strength of this coupling. We discuss the general conditions under which this affects the translation diffusion “seen” by the molecule along its molecular axes. The results show that the translation-rotation coupling is correlated to the local environment in shaping the longitudinal and transversal translation dynamics of a molecule at a microscopic level.
I. Introduction It was pointed out in previous studies1-9 that the analysis of the dynamics of molecular liquids with respect to a molecular frame provides more insight than that with respect to a fixed laboratory frame. Indeed, it clearly shows the anisotropy of the dynamics. In fact, most of the important information on the effect of the local anisotropy on the dynamical properties is lost due to the averaging over all orientations. In most of these studies, which involve different molecular systems, homo- and hetero-nuclear diatomic molecules, as well as CO2, CS2, OCS, H2O, CHCl3, CO(NH2)2, and CO(CH3)2, it was generally found that the diffusion along a particular axis is preferred. The favored displacement of molecules along a given axis has considerable implications for interpreting the spectroscopic measurements which bring into play the local environment. Examples are infrared and Raman measurements,10 quasi-elastic neutron scattering experiments11 (particularly for large values of the transfer moment q), NMR,12 and Rayleigh-Brillouin experiments.13 A fundamental understanding of this preferential displacement is a key step to the interpretation of diffusion-controlled chemical reactions,14-16 molecular motion on a surface,17-21 transport phenomena near biological macromolecules,22-24 as well as to provide useful information on the anisotropic diffusion of molecules in brain white matter.25 The aim of this paper is to show that it is the translation-rotation coupling (TRC) driven by an effective intermolecular interaction potential which induces a preferential diffusion along a particular axis and to analyze the microscopic changes which accompany the observed effect of the TRC on diffusion along molecular axes. In addition, theoretical results showed that the self-diffusion process must be coupled to the rotational dynamics26,27 for nonspherical molecules, and that an analysis of this coupling in liquids provides useful insight on the structure and dynamics of these liquids. There is a variety of statistical properties that can be used as criteria for the analysis of the TRC. Examples are the distribution
of displacements as a function of time28 t and the time integral of the cross-correlation function 〈ω(0)V(t)〉1,29 (where ω and V are the angular and translational velocities relative to the molecular frame). The variables used in theses approaches are the Cartesian components of the translational and angular velocities expressed in a laboratory frame. Following these lines of approach, one might ask how the TRC affects the diffusion along the molecular axes and what microscopic changes accompany this effect. The best tool to settle this question is molecular dynamics simulation. In this paper, we report results of a computer simulation aiming to study the effect of the TRC on the diffusion along the molecular axes of carbon dioxide and water molecules. A criterion is introduced to quantify this effect, as well as its time evolution. The “observer” in the present picture “sits” on one molecule and moves with it while studying the diffusion process. In section II, the definitions needed in the evaluation of the TRC are introduced. In section III, details on the intermolecular potentials and the simulations are given. The results of the TRC effect on the diffusion along the molecular axes are presented and discussed in section IV. Finally, we conclude in section V with the summary of our main finding. II. Tools for the Analysis of the Translation Rotation Coupling The time dependent auto-correlation function (acf) of the translation velocity vector V, is calculated with respect to the intrinsic axes attached to the molecule. To relate the molecular coordinates (x, y, z) to the simulation box frame (X, Y, Z), three sets of direction cosines are used. Thus, the orientation of each unit vector of the molecular coordinates with respect to the three axes of the simulation-box frame is specified by a set of nine direction cosine elements. In the simulation, the transformation is a rotation matrix R, which gives at each step the orientation of the molecular coordinates with respect to the simulation-
10.1021/jp061194t CCC: $33.50 © 2006 American Chemical Society Published on Web 08/31/2006
Analysis of Translation-Rotation Coupling
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Figure 1. Simulation-box frame (X, Y, Z) and molecular coordinates (x, y, z). The translational velocity vector within the simulation-box frame, given by V b and V m, is the equivalent vector in molecular coordinates at time t.
box frame. Mathematically, this implies a relation of the form
V mi )
∑j RijV bj i ) x, y, z and j ) X, Y, Z
(1)
where V bj is the jth component of the translational velocity vector V b in the simulation box frame and V mi is the ith component of the equivalent vector V m in the molecular coordinates (See Figure 1). The Rij’s are the components of the rotation matrix R. The following correlation functions were then calculated:
Ωb(t) ) 〈V b(t).V b(0)〉,
(2)
Ωbj (t) ) 〈V bj (t).V bj (0)〉, where j ) X, Y, Z
(3)
Ωm(t) ) 〈Vm(t).Vm(0)〉
(4)
Ωmi (t) ) 〈V mi (t).V mi (0)〉 )
∑j 〈Rij(t)V bj (t).Rij(0)V bj (0)〉,
(5)
where i ) x, y, z and j ) X, Y, Z. It should be noted that first ΩbX(t), ΩbY(t) and ΩbZ(t) are statistically equal because the system is isotropic, second, that Ωm(0) and Ωb(0) are equal to (3kbT/m) and finally, that Ωbj (0) (j )X, Y, Z) and Ωmi (0) (i ) x, y, z) are equal to (kbT/m) (kb is the Boltzmann constant, T is the temperature, and m is the mass of the molecule) because the module of the velocity is conserved in both referential systems. If translation and rotation motions along a chosen axis were statistically independent, eq 5 writes the following:
∑j 〈Rij(t).Rij(0)〉〈V bj (t).V bj (0)〉
(6a)
It is then straightforward, to quantify the coupling between the translation and rotation along a chosen axis by calculating the difference:
∆RTC (t) ) i
∑j 〈Rij(t)V bj (t).Rij(0)V bj (0)〉 ∑j 〈Rij(t).Rij(0)〉〈V bj (t).V bj (0)〉,
Figure 2. (a) Carbon dioxide at 247 K: The quantities ∆RTC (t) (i ) x, i y, z) are defined by eq 6b. Ωmi (t) are the velocity correlation functions along the molecular axes (i ) x, y, z). (b) Carbon dioxide at 303 K.
motions, and therefore, they provide a tool to characterize the time evolution of the coupling. III. Simulation Details In these simulations, the carbon dioxide and water systems have been studied with the use of N ) 864 particles, which interact via a Lennard-Jones potential and Coulomb interactions between atomic partial charges. Newton’s equations are integrated by the leapfrog algorithm in time steps of 2.5 fs for carbon dioxide and 0.5 fs for water. During the equilibration process, the velocity rescaling is used to thermalize the system. The separation on a time scale of the rotational and translational motions depends on the density, temperature, and the molecular shape, as well as on the form of the intermolecular potential. It is then possible to analyze the TRC by varying the first two parameters. The equilibria were established at two temperatures for each system, T ) 247 K and 303 K for CO2; T ) 300 K and T ) 673 K for H2O. The interaction potential model of CO2 was that of Murthy et al.,30 while for H2O, the TIP4P model was used.31 Typical run lengths were (0.5-1) × 106 steps, depending on the density. These run times are usually considered to be sufficient for the determination of single-particle properties. IV. Results and Discussions
(6b)
where i ) x, y, z and j ) X, Y, Z. These functions are expected to vanish identically for uncorrelated translational and rotational
∆RTC (t) (i ) x, y, z) difference functions are given in Figure i 2a and b. A systematic difference between the motion along the longitudinal and transversal axes is observed. The function ∆RTC z (t) at 247 K (Figure 2a) shows a well-defined positive peak at short time followed by a broad negative peak above
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Idrissi et al.
Figure 4. Microscopic density Fm(r).
Figure 3. (a) Water at 300 K: The quantities ∆RTC (t) (i ) x, y, z) are i defined by eq 6b. Ωmi (t) are the velocity correlation functions along the molecular axes (i ) x, y, z). (b) Water at 673 K.
t ) 0.5 ps. Along the transversal axes, the difference ∆RTC x,y (t) is negative at short time (t < 0.7 ps) and vanishes at higher time. All peaks broaden and their maximum shifts toward higher time with increasing temperature (Figure 2b). The amplitude of the positive part of ∆RTC z (t) decreases with temperature, while the amplitude and the width of the negative peak increase. To determine if these observations are also applicable for nonlinear molecules, the same functions were calculated for water at T ) 300 K and T ) 673 K (Figure 3a and b). The molecular frame is that of the principal molecular moment of inertia (the three atoms are in the x and y). The salient fact is that the time persistence of positive and negative regions is very short (∼0.3 ps) in comparison to that observed for CO2. At T ) 673 K, the intensity of the positive region of ∆RTC z (t) decreases drastically. Based on theses results, one defines two time domains for (t) (i ) x, y, z). The the analysis of the time behavior of ∆RTC i short domain lies below 0.7 and 0.2 ps for CO2 and H2O, respectively. IV.1. Short Time Domain. From the previous results, we assume that the short time behavior of ∆RTC (t) (i ) x, y, z) is a i general behavior, that is to say, the rotation and translation motions, driven by the intrermolecular potential, ends up with RTC (t) (i ) x, y). A physical opposite values of ∆RTC z (t) and ∆i picture which emerges from the analysis of these results is that the diffusion along a chosen axis is two-dimensional motiondependent. The positive values of ∆RTC z (t) indicates an occurrence of a concerted rotation and translation motions and results
in a preferential diffusion along this direction (It means that molecules slide almost freely along the longitudinal direction, at least in the short time). Conversely, the negative values of (t)(i ) x, y) suggest an occurrence of an antagonist rota∆RTC i tion translation motions and results in a hindrance of the transversal diffusion. These findings are in agreements with the results of Chrzanowska et al.32,33 obtained for needles. IV.1. long-Time Domain. Several results34,35 have shown that the long-time dynamics of the velocity correlation function is a sign of a structural relaxation in a system associated with collective interactions. A detailed analysis of the behavior of ∆RTC (t) (i ) x, y, z) is then difficult to carry out and one can i only bring out a few tendencies but not defined rules. Intuitively, the behavior of ∆RTC (t) (i ) x, y, z) could be associated with i the average microscopic density in the region of nearestneighboring molecules. We have then calculated the microscopic density Fm(r)36 as follows:
Fm(r) )
N(r) 4 3 π(r - RC3) 3
(7)
where N(r) is the coordination number at distance r between two molecules given by the following xpression:
N(r) ) FM
∫0r g(r′)4πr′2dr′
(8)
where, FM is the macroscopic density and g(r) is the center of mass-center of mass radial distribution function. In eq 7, the term (4/3)πr3 is a volume of a sphere of radius r, and (4/3)πRc3 is the volume occupied by a probe molecule. Rc is estimated to be equal to 2.7 Å for CO2. As shown in Figure 4, Fm(r) is higher than the macroscopic density in the range of the first shell of neighbor molecules; this indicates that there exists a rather strong attraction between nearest neighbors, and one may expect a strong orientational correlation between these molecules.7,37,38 As a consequence, at short range, the TRC along each molecular axis may not be equally sensitive to the intermolecular interaction. This leads us to postulate that longitudinal and transversal velocities depend on the mutual orientation of the interacting molecules and is proportional to the orientational distribution of molecules. The question is then how to get unambiguous information on this orientational distribution? The radial distribution function is a major means by which structure is usually described in fluids systems. However, this statistical property does not provide sufficient details on the nature of the local environment in fluids systems in terms of first-, second- or higher-order neighbors distributed around a central particle in fluids. Thus, it does not
Analysis of Translation-Rotation Coupling
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provide unambiguous information on the local orientational distribution. To go beyond the description provided by the radial distribution function,we use the distribution functions corresponding to each subset of “neighbor-ship” as described by Mazur.39 In this approach, the neighbors of a central atom are sorted by distance into the first neighbors, second neighbors, etc. Separate radial distribution functions, pRβ(n,r) may be defined for each set of nearest neighbors atoms β (indicated by n), at distance r from the central atom R. It is obvious that the corresponding radial distribution gRβ(r) is equal to ∑npRβ(n,r). These functions are less averaged and contain more information than the corresponding gRβ(r). We have then introduced the calculation of the orientational distributions, qRβ(n,r,cos(θ)), as follows: We calculate cos (θ) for each distance r between the central molecule R and a molecule β in each class of neighboring molecules, where θ is the angle between the unit vectors along the z molecular axes. Theses functions allow us to address, in an unambiguous way, the relevant short-range orientational distribution and analyzing the local anisotropy and its consequence on the dynamical properties such as a preferential diffusion along a longitudinal axis of the molecule. The orientational distribution, qRβ(n, r, cos(θ)) (where n ) 1-4), are reported in Figure 5, at T ) 247 K and T ) 303 K. These figures show that the first nearest neighbor shell tends to be parallel to the reference molecule. Increasing the temperature slightly affects this distribution. For the second nearest neighbor molecules, the parallel configuration remains the most likely. However, with increasing temperature, the orientation distribution tends to randomness. In the case of the third and fourth nearest neighbors shell, the perpendicular configuration dominates and the temperature effect is minor. Orientations of higher order shells remain slightly in favor of a perpendicular orientation but rapidly become random. The negative values of ∆RTC z (t) (see Figure 2a and b) for time longer than 0.7 ps indicate that the collective intermolecular interactions results in a negative coupling between translation and rotational motions. This means that the rotation tends to stem the translation along the longitudinal direction. The signature of this negative coupling is associated with the negative region at intermediate time of Ωmz (t) (see the inset of Figure 2b). It is worth noting that this negative region is neither present in Ωbj (t) (j ) X, Y, Z) nor in the angular velocity autocorrelation function. The negative region of ∆RTC z (t) deepens with increasing the temperature, which reveals that a particular orientational distribution of the nearest-neighboring molecules is established along the longitudinal direction. The most important change in the orientational distribution concerns that of the second nearest neighbor shell which becomes random (see Figure 5). It is then suggested that the particular orientation distribution between nearest neighbors results in the antagonistic effects between translational rotation motions along the longitudinal direction. These results show that the effect of the TRC on the diffusion along the molecular axes is anisotropic. To quantify this anisotropy, we first define the pseudo-diffusion Dmi along the molecular axes (i ) x, y, z) and we calculate the diffusion coefficient Db. These parameters are as follows:
Dmi )
∫0+∞ Ωmi (t)dt,
(9)
as indicated by eq 1, Ωmi (t) is the correlation function of the product of two dynamical properties, the translation velocity and the components of the rotation.
Figure 5. The orientational distribution. θ is angle between the z axis of the reference molecule and that of the first, second, third and fourth nearest neighbors.
18564 J. Phys. Chem. B, Vol. 110, No. 37, 2006
Db )
∫0+∞ Ωb(t)dt,
1 3
Idrissi et al.
(10)
where, Ωb(t) is the correlation function of the translational velocity, and it is not normalized. It should be noticed that the diffusion coefficients, D bi (i ) X, Y, Z), along the box frame axes are equal to Db. The pseudo-diffusion coefficient, D mi , is calculated as the integral of Ωmi (t) (see eq 9), the value of the pseudo-diffusion with respect to the molecular axes is weighted by the time evolution of the TRC on each axis. At each temperature, the resultant contribution of the TRC to the value of the diffusion along a given axis is a balance between the (t) (i ) x, y, z): the positive and the negative regions of ∆RTC i higher the surface of the positive peak is, the faster will be the pseudo-diffusion; conversely, the larger the surface of the negative region is, the slower is the pseudo-diffusion. One (t) (i ) x, y, z) should notice that the long time part of ∆RTC i contributes more to the value of D mi than the short time region. The large value of the difference between Db and D mi is a clear indication of the occurrence of a cooperative translation-rotation motion along the chosen axis. We quantify then the TRC along a chosen molecular axis by comparing the values of Dmi and Db. The values of Db as well as that of D mi along the molecular axes of CO2 and H2O are given in Table 1. It is clear that at T ) 247 K (at liquidlike density), the longitudinal pseudo-diffusion is preferred; however, at T ) 303 K (low density) the transversal diffusion is favored. It seems that this change of behavior has never been observed before. Our results suggest that this can be correlated to a particular orientation distribution between nearest neighbors which results in an antagonistic effect between translational rotation motions along the longitudinal direction, and conversely, it results in a cooperative effect between the tow motions along the transversal direction. It is tempting to deduce that a fast decaying of the reorientation correlation of a unit vector along a chosen axis will imply a smaller value of Dmi with respect to that of Db. To analyze the validity of this conjecture, we have calculated the reorientation correlation function of the unit vectors along the x, y, and z molecular axes (which form the molecular frame), using the following:
Ci(t) ) 〈ui(t).ui(0)〉,
(11)
Figure 6. Reorientation correlation functions Ci(t) (i ) x, y, z) of unit vector along the molecular axes and the translation velocity correlation function Ωb(t).
TABLE 1: Diffusion Dband Pseudo-Diffusion Dm i (i ) x, y, z) along the Molecular Axes CO2 diffusion coefficient (10-9 m2s-1) Db Dmx Dmy Dmz
H2O
247 K
303 K
300 K
676 K
8.60 6.64 6.52 8.25
31.10 16.58 16.72 15.11
3.90 3.53 4.09 4.10
52.70 14.2 15.8 19.2
of both reorientational and translational correlation functions, though in a complex manner. V. Concluding Remarks
with i ) x, y, z. Figure 6a and b report the values of Ci(t) and Ωb(t) for CO2 at two temperatures. At T ) 247 K, the time decay of Cz(t) is faster than that of Ci(t) (i ) x, y). However, as shown in Table 1, the values of Dmi (i ) x, y, z) are lower than that of Db and, most important, the value of D mz is close to that of Db. For water systems (the results are not shown), at T ) 300 K and T ) 673 K, the time decay of Cz(t) is faster than that of Ci(t) (i ) x, y). In this case, the values of Dmz are also close to Db. These results clearly rule out the idea that a fast decaying reorientation along an axis results in a small value of Dmi (i ) x, y, z). However, for CO2 system at T ) 303 K, one observes an inversion between the values of D mi (i ) x, y) and that of Dmz . The comparison of the time behavior of Cz(t) and Ωb(t) shows that, at time lower than 0.7 ps, the time decay of Ωb(t) is the fastest; however, the time decay of Cz(t) becomes the fastest at longer times. This result suggests that the values of the pseudodiffusion coefficient are correlated to the relative time decay
In this paper we attempted to explain the behavior of the pseudo-diffusion along the axes of a molecule and to bring to light the influence of the translation-rotation coupling (TRC) on the diffusion along the molecular axes. The time evolution of this effect is evaluated from the difference between the velocity autocorrelation function calculated with respect to the coordinate system defined by the orientation of the molecule and that of same function calculated under the hypothesis that there is no statistical correlation between the translation and rotation motions. The resultant contribution of the TRC to the value of the pseudo-diffusion along an axis is determined by the combination of two conflicting influences of the TRC, associated with the positive and the negative regions of this difference. The positive regions of these functions provide the necessary pattern for the increase of the pseudo-diffusion, while the negative regions lead to the hindrance of the pseudodiffusion along a given axis. The results also show that the effect of translation-rotation coupling is correlated to the anisotropy of the local environment in shaping the longitudinal and transversal translation dynamics of a molecule at a microscopic
Analysis of Translation-Rotation Coupling level. Indeed, it was suggested by calculating the angular distribution for each subset of “neighbor-ship” distribution that when increasing the temperature, the most important change in the orientational distribution concerns that of the second near neighbor shell which becomes random. It is then suggested that this particular orientation distribution between nearest neighbors results in the antagonistic effects between translational rotation motions along the longitudinal direction. Acknowledgment. We thank G. Turrell for reading the manuscript and for helpful suggestions. The Institut du De´veloppement et des Ressources en Informatique Scientifique is acknowledged for the CPU time allocation. The Centre d’Etudes et de Recherches Lasers et Applications is supported by the following organizations: Ministe`re Charge´ de La Recherche, Re´gion Nord/Pas de Calais, and les Fonds Europe´ens de De´veloppement Economique des Re´gions. References and Notes (1) Ryckaert, J.-P.; Bellemans, A.; Ciccotti, G. Mol. Phys. 1981, 44, 979. (2) Madden, P. A.; Impey, R. W. Chem. Phys. Lett. 1982, 46, 513. (3) Evans, M. W. Phys. ReV. Lett. 1983, 50, 371. (4) Frattini, R.; Ricci, M. A.; Ruocco, G.; Sampoli, M. J. Chem. Phys. 1990, 92, 2540. (5) Idrissi, A.; Sokolic, F.; Perera, A. J. Chem. Phys. 2000, 112, 9479. (6) Idrissi, A.; Longelin, S.; Sokolic, F. J. Phys. Chem. B. 2001, 105, 6004. (7) Idrissi, A. J. Mol. Liq. 2003, 107, 29. (8) Samios, J.; Dellis, D.; Stassen, H. Chem. Phys. 1993, 178, 83. (9) Tildesly, D. J.; Madden, P. A. Mol. phys. 1983, 48, 129. (10) Evans, M. W.; Evans, G. J.; Coffey, W. T.; Grigolini, P. Molecular Dynamics; Wiley/Interscience: New York, 1982.
J. Phys. Chem. B, Vol. 110, No. 37, 2006 18565 (11) Farone, A.; Liu, L.; Chen, S. H. J. Chem. Phys. 2003, 119, 6302. (12) Holtz, M.; Mao, X.; Seiferling, D.; Sacco, A. J. Chem. Phys. 1996, 104, 6669. (13) Wang, C. H. J. Chem. Phys. 1979, 70, 3796. (14) Gupta, S. J.; Khakhar, D. V. J. Chem. Phys. 1998, 108, 5626. (15) Svishchev, I. M.; Kusalik, P. G. Phys. ReV. Lett. 1995, 75, 3289. (16) Dichtl, M. A.; Sackmann, E. New. J. Phys. 1999, 18, 1. (17) Ueba. H.; Wolf, M. Science 2005, 310, 1774. (18) Backus, E. H. G.; Eichler, A.; Kleyn, A. W.; Bonn, M. Science 2005, 310, 1790. (19) Yang, X.; Xu, Z.; Zhang, C. J. Colloid Interface Sci. (doi:101016/ j.jcis.2005.10.002) 2006. (20) Yang, X.; Zhang, C. Chem. Phys. Lett. 2005, 407, 427. (21) Morison, J. I. L.; E. Galloue¨t; T. Lawson; G. Cornic; Herbin, R.; Baker, N. R. Plant. Physiol. 2005, 139, 254. (22) Imparato, A.; Shillcock, J. C.; Lipowsky, R. Eur. Phys. J. E. 2003, 21. (23) Makarov, A. M.; Feig, M.; Andrews, B. K.; Pettitt, B. M. Biophys. J. 1998, 75, 150. (24) Liger-Belair, G.; Topgaard, D.; Voisin, C.; Jeandet, P. Langmuir 2004, 20, 4132. (25) Le Bihan, D. Nature 2003, 4, 469. (26) Chandler, D. J. Chem. Phys. 1975, 62, 1358. (27) Ryckaert, J.-P.; Bellemans, A. Chem. Phys. Lett. 1975, 30, 123. (28) Steele, W. A. Mol. Phys 1980, 40, 723. (29) Evans, G. T. Mol. Liq. 1978, 36, 65. (30) Murthy, C. S.; Singer, K.; McDonald, I. R. Mol. Phys. 1981, 44, 135. (31) Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D. J. Chem. Phys. 1983, 79, 926. (32) Chrzanowska, A.; Ehrentraut, H. Phys. ReV. E 2002, 66, 012201. (33) Chrzanowska, A. J. Chem. Phys. 2004, 120, 2857. (34) Alder, B. J.; Wainwright, E. Phys. ReV. Lett. 1970, 1, 1. (35) Allen, M. P. Phys. ReV. Lett. 1990, 65, 2881. (36) Tucker, S. C. Chem. ReV. 1999, 99, 391. (37) Fedechenia, I. I.; Schro¨der, J. J. Chem. Phys. 1997, 106, 7749. (38) Terzis, A. F.; Smulski, E. T. Chem. Phys. Lett. 1996, 251, 157. (39) Mazur, S. J. Chem. Phys. 1992, 97, 9276.
