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Depolarization of fluorescence of polyatomic molecules in noble gas solvents
Chemical Physics 272 (2001) 69±76
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Depolarization of ¯uorescence of polyatomic molecules in noble gas solvents A.P. Blokhin, M.F. Gelin *, I.I. Kalosha, V.V. Matylitsky, N.P. Erohin, M.V. Barashkov, V.A. Tolkachev Institute of Molecular and Atomic Physics, Academy of Sciences of Belarus, F. Skaryna Avenue 70, Minsk 220072, Belarus Received 9 April 2001
Abstract The collisional depolarization of ¯uorescence is studied for p-quarterphenyl (PQP) in He, Ar, Xe solvents, under pressures ranging from zero to nearly atmospheric. The results are interpreted within the Keilson±Storer model of the orientational relaxation and smooth rigid body collision dynamics. This allows us to estimate the rate of the angular momentum scrambling due to encounters of PQP with its partners. The collisions are shown to be neither strong nor weak, so that the averaged number of encounters giving rise to the PQP angular momentum randomization equals to 33 (PQP±He), 4.5 (PQP±Ar), and 2.1 (PQP±Xe). Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 05.20.Dd; 33.50.)j; 34.50.Ez
1. Introduction It is well known that intermolecular encounters dier in their strength in transferring angular momentum [1]. In the standard way, this fact is embodied in the quantity Zeff zc sJ , which is the product of the collision frequency and the angular momentum relaxation time. Physically, Zeff is the number of collisions that is necessary for the randomization of the angular momentum. The common sense dictates that Zeff 1 for collision partners of comparable masses (strong collisions) and Zeff 1 for collisions of heavy molecules with light buer species.
*
Corresponding author. Fax: +375-17-284-0030. E-mail address: [email protected] (M.F. Gelin).
Orientational relaxation (OR) in the hindered rotation limit is universally governed by the diffusion equation [1]. Within this approach, the in¯uence of a solvent on the OR is taken into account by introducing the corresponding diusion coecients. According to the Green±Kubo formulas, these are uniquely determined by the angular momentum relaxation times and are independent of the collision strength. However, it is apparent enough that the collision eciency must manifest itself in the OR of molecules in dilute enough solutions and/or rari®ed gases. Recently, this was non-ambiguously demonstrated by Zewail and coworkers [2,3], who performed the real time measurements of the anisotropy decay of iodine molecules in noble gas solvents under a broad scale of densities, ranging from isolated molecule to liquid. It is therefore reasonable to expect
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A.P. Blokhin et al. / Chemical Physics 272 (2001) 69±76
that the collision eciency aects the ¯uorescence depolarization rate for large polyatomic species in various buer gases. The collisional depolarization of the total ¯uorescence was measured for several aromatic molecules in the bulk [4] and in dierent (polyatomic) solvents [5]. The masses of solute and solvent species were comparable, and it was tentatively concluded that the situation nearly corresponded to Zeff 1. More recently, the reliable indications have been found that OR of aromatic molecules in noble gases proceeds with a solvent dependent Zeff , ranging from several units to several tens [6]. The present contribution gives an account of the combined theoretical and experimental studies of the in¯uence of the collision ef®ciency on the total ¯uorescence depolarization for polyatomic molecules. Our aim is to develop a reliable approach for the description of the anisotropy decay and to analyze which kind of information can be extracted from such experiments. The paper is organized as follows. The theory of the depolarization of total ¯uorescence for linear molecules is outlined in Section 2, in which some general trends are established for the behavior of the anisotropy vs. the frequency and strength of encounters. The results of the measurements of the ¯uorescence depolarization for p-quarterphenyl (PQP) in He, Ar, Xe are presented in Section 3, along with their interpretation in terms of the Keilson±Storer (KS) model and rigid body collision dynamics. Our main ®ndings are summed up in Section 4. Note from the outset that the dimensionless variables are used throughout the article. Time is measured in units of the averaged free rotation p period I=kT , where I is the main moment of inertia of a linear molecule, k is the Boltzmann constant, and T is the temperature.
2. The Keilson±Storer theory The anisotropy of the total molecular emission, r, is proportional to the integral over the time domain of the second rank orientational correlation function (OCF) G2 t of the absorption ~ labs and emission ~ lem dipole moments [7,8]:
r 2=5sem1 sdep ; Z 1 sdep dt expf t=sem g G2 t; 0
G2 t hP2 ~ labs 0~ labs ti:
1
Here P2 x is the second order Legendre polynomial and sem is the lifetime in an electronically excited state. The simplest way for incorporating the notion of the collision strength into a theory is to invoke the KS model [1,9±11]. In this model, the rotational motion is governed by the two parameters: zc , which is the collision frequency, and 1 6 c 6 1, which is the collision eciency. The two parameters constitute the quantity mJ sJ 1 zc 1
c
2
which is the angular momentum relaxation frequency, yielding Zeff zc sJ 1
1
c :
3
We shall further restrict ourselves to the calculation of the anisotropy for linear species. If the dipole moments involved are parallel to the axis of a linear molecule, the KS model allows one to calculate the second rank OCF through the simple three term recurrence relations [11]. By slightly adapting these results for the case of interest here, one ®nds sdep 1 2b0 sem ; 16 m 2 8m 10 8m 6 bm1 r2m1 bm r2m2 r2m2 r2m 4m bm 1 3sem dm0 ; 4 r2m where rm zc 1
cm sem1 :
5
The most popular and widespread gas-like models of the OR correspond to the cases of strong (Jdiusion) and weak (rotational Fokker±Planck equation (FPE)) collisions. The KS model contains both of these approaches as a special case. Namely, the J-diusion is recovered by putting c 0 and the FPE is reproduced by letting zc ! 1, c ! 1, zc 1 c ! mJ const [1,9±11]. Positive values of c allow one to describe a crossover from
A.P. Blokhin et al. / Chemical Physics 272 (2001) 69±76
the FPE to J-diusion. Negative values of c correspond to the angular momentum reorientation due to a collision, that is characteristic for molecular liquids. The J-coherence model proposed in papers [2,3] was proved to be very similar in spirit to the KS model [11]. In order to get an idea of how the total ¯uorescence anisotropy changes with the buer gas density, one can inspect Fig. 1, in which the ratio r0 =r is presented vs. zc for various values of collision eciency c. r0 is the ¯uorescence anisotropy under collisionless conditions. For linear rotors, r0 2=5 P2 02 1=10. We plot the inverse of the anisotropy r0 =r instead of the direct proportionality r=r0 since it is somewhat traditional [4,5], and also because such graphs are more sensitive to testing against experimental results. The dependencies have a characteristic bell-like form. Due to the collisional depolarization, the curves initially increase in parallel with the growth of the collision frequency. They further reach their maximums and start to decrease to the common value of 1/4. This behavior is a direct mirroring of the transformation from the free molecular rotation to the hindered (diusion) one. In this latter case, the anisotropy is described by the Levshin±Perrin 1 formula r 0:4 1 6sJ sem [7], which yields r
Fig. 1. r0 =r vs. zc for sem 50: (1) c 0:99, (2) c 0:9, (3) c 0 (J-diusion), (4) c 0:9, (5) c 0:99.
71
0:4 for zc 1 (or, equivalently, sJ 1). In the opposite limit zc 1, Eq. (4) describes a crossover from the constant value r0 =r 1;
zc sem1 1
6
to the linear dependence r0 =r zc sem W; sem1 zc 1:
7
Here W
6b0 ;
32b1
10b0 3 r2 ;
rm 1
cm 8
and the coecients bm obey recurrence relationships (4) in which one should substitute rm ! rm and take m 1; 2; 3 . . . Expressions (6) and (7) can be combined into r0 =r 1 zc sem W:
9
This equation is precisely of the form proposed by Gordon for linear molecules in binary collision solvents [8], so that Eq. (8) provides the explicit expression for the ¯uorescence quenching cross-section in terms of the KS parameters. Note however that OCF G2 t does not decay exponentially in the rari®ed gas limit zc 1 [1,9±11]. This is a direct consequence of the correlation of molecular angular momenta before and after a collision. It is only in case of strong collisions (c 0, the J-diusion) that the pre- and post-collisional angular momenta are uncorrelated. Then G2 t 1=4 expf 3=4zc tg [1] and Eq. (9) with W 3=4 is exact for an arbitrary zc . As to the in¯uence of the collision eciency on the anisotropy decay, one noti®es the following (see Fig. 1). When zc is small enough, one arrives at the intuitively expected conclusion: the smaller is jcj (the stronger are collisions) the higher is depolarization. This is precisely the situation which corresponds to the experimental conditions described in Section 3. So, c 1 gives the lower boundary for the depolarization rate (in fact, no collisions occur) and c 0 gives the upper one (strong collisions). It is remarkable that the anisotropy is independent of the sign of parameter c for small zc . When the density (and therefore zc ) is further increased, the degeneracy on c disappears and the curves corresponding to various
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A.P. Blokhin et al. / Chemical Physics 272 (2001) 69±76
1 6 c 6 1 dier signi®cantly. It should also be pointed out that the ¯uorescence depolarization is quite sensitive to the value of the lifetime in the electronically excited state. Evidently, the smaller is sem , the smaller is the number of collisions which experiences a molecule at the electronically excited state and, as a result, the weaker is depolarization. The general trends in the ¯uorescence depolarization directly re¯ect the well known behavior for the integral OR times [1] Z 1 sX;j dt Gj t: 10 0
The point is that if sX;2 sem (this is so for all but very small (sJ 1) and very high (sJ 1) densities) one can approximately write sX;2 sdep and, therefore, r 0:4sX;2 =sem .
3. Experimental results and discussion We have performed measurements of the ¯uorescence depolarization for PQP in He, Ar, Xe solvents under pressures ranging from zero to nearly atmospheric. Our experimental setup was described in detail elsewhere [4±6]. In short, it consists of the cuvette compartment, the vacuum post, the excitation source, and the recording sys-
Fig. 2. Block diagram of setup for recording polarized ¯uorescence.
tem (see Fig. 2). The studied species are placed in the specially designed fused silica cuvette. It consists of the branch piece and the optical part which are connected to the vacuum post. The branch piece is placed into the oven, which governs the PQP concentration. To avoid the PQP absorption on the cuvette sides, the temperature of the optical part of the cuvette was kept 20±30 K higher than that of the branch piece. After the pre-evacuation up to 10 5 Torr, the PQP molecules in the branch piece are heated to the required temperature. The optical excitation is performed by the excimer laser (308 nm wavelength, 20 ns pulse duration). To keep of the saturation eects, its irradiation strength was controlled to be less than 1 MW/cm2 . At the maximum of the ¯uorescence spectrum, the two orthogonally polarized emission intensities are recorded by the photomultipliers. The signals are further processed with a computerized installation on the basis of CAMAC-PC. The experimental error in measuring polarization degrees was estimated to be 1%. Our experimental results are presented in Fig. 3 as the plots r0 =r vs. the noble gas concentration q,
Fig. 3. Experimental dependencies r0 =r vs. q and their linear ®ts: ( ) PQP±He, ( ) PQP±Ar, ( ) PQP±Xe.
A.P. Blokhin et al. / Chemical Physics 272 (2001) 69±76
along with the linear ®ts to these dependencies. Before embarking at the interpretation of these anisotropies in terms of the theory developed in Section 2, one must realize the following. First, PQP is a considerably prolate asymmetric top with the main moments of inertia Ix 7672, Iy 7632, 2 . In so far as the absorption and Iz 340 amu A and emission dipole moments for PQP are directed along the axis of its smallest moment of inertia [12], one can approximately treat it as a linear rotor with the moment of inertia I Ix Iy =2 2 . Second, it is necessary to recalculate 7654 amu A the buer species concentration q into the collision frequency zc . By invoking Eq. (11) (see below) one concludes that the maximal experimentally achieved values of concentration qmax correspond to the dimensionless collision frequency zc of the order 0.07 (PQP±He), 0.03 (PQP±Ar), and 0.024 (PQP±Xe). Eq. (9) is correct for such small values of zc , and this gives an additional support to the validity of the approximation of the experimental anisotropies by linear ®ts (see Fig. 3). Third, it is necessary to emphasize that the ¯uorescence depolarization of a linear molecule in the KS model is governed by the following four characteristic times: the free rotation period sr , the intercollision time sc zc 1 , the angular momentum relaxation time sJ mJ 1 , and the lifetime in an electronically excited state sem . As it has been mentioned in the Section 1, it is convenient to utilize the dimensionless variables, by using sr 4:16 ps as the unit of time (T 533 K). For PQP, sem 1:4 ns [13]. In principle, sem can depend on q due to the collisional ¯uorescence quenching [4±8]. The latter was not detected for PQP under our experimental conditions, so that the reduced lifetime sem 337 can indeed be considered constant. So, at q qmax , a PQP molecule experiences, averagely, 24 (PQP±He), 10 (PQP±Ar), and 8 (PQP± Xe) collisions during sem . The advantage of the KS model in interpreting experimental data will dramatically increase if one will be able to calculate the remaining two dynamic parameters, zc and c, from some ``®rst principles''. It is logical to surmise that the main source of the PQP angular momentum change in its collision with a noble gas atom is due to the short range repulsion. Therefore, it is reason-
73
Fig. 4. The PQP spherocylinder and the noble gas sphere. All the sizes are given in angstroms.
able to invoke the rigid body dynamics for mimicking these encounters. The use of the J-coherence model for the interpretation of the anisotropy decay of iodine molecules in dierent noble gas buers [2,3] also corroborates this idea (see also paper [11]). More speci®cally, we represent the noble gas atoms by smooth rigid spheres and PQP molecules by rigid smooth spherocylinders (Fig. 4). We have chosen spherocylinders by the following reasons. First, a highly prolate spherocylinder, as distinct from an ellipsoid of revolution, has nearly a constant mass density per length, that is roughly so for PQP. Second, the PQP concentration in our experiments (2:4 1018 m 3 ) is much less than that of the buer species (typically, 1024 m 3 ). Since the radii of the noble gas atoms (see below), their concentration is eviare 1±2 A dently much less than the corresponding close packing concentration. So, we have a dilute solution of PQP molecules in rari®ed noble gases. This allows us to treat the rigid body dynamics within the Enskog approximation, to put the contact pair correlation function to unity, and to analytically evaluate the collision frequency and the angular momentum relaxation frequency [14,15] 1 Rs R p zc p vq L 1 k h 2 2 2 Rs R 1 k 1/ ;
11
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A.P. Blokhin et al. / Chemical Physics 272 (2001) 69±76
mJ p vq
Rs R p L 1 k 2 2
Rs R 1 k
h 1/ :
12
p Here v 8kT = pl is the relative speed of the colliding species, l MMs = M Ms is their reduced mass, q is the buer gas concentration, Rs is the radius of the ``noble gas'' sphere, R and L are, correspondingly, the radius of the spherical cap and the length of the cylindrical part of the spherocylinder (see Fig. 4), q k arcsin k1 lL2 p k ; / ; 4I k 13 p p ln k k 1 p : h k Evidently, the rotational energy exchange proceeds entirely due to molecular non-sphericity, i.e. L 6 0. When L 0, one gets mJ 0 and the 2 standard result zc q vp Rs R for smooth rigid spheres is recovered. Both zc and mJ increase with the growth of R and/or L. The higher is the spherocylinder length L, the stronger are collisions. Generally, Eqs. (2) and (11)±(13) predict the collision strength parameter 1 6 c 6 1, as it must be from its de®nition. Both zc and mJ are proportional to the concentration of the noble gas atoms and also to the relative thermal speed of the colliding particles. In addition, zc and mJ depend quite complicatedly on the parameters specifying the spherocylinder (M,R,L,I) and the buer sphere (Ms ,Rs ). It is natural to assume that the mass and the main moment of inertia of the spherocylinder are those of PQP, 2 . In adviz. M 306:4 amu and I 7654 amu A dition, the masses Ms and radii Rs of the noble gases are known [16]. These are 4 amu and 1.1 A for Ar, 131 amu and 2.5 for He, 40 amu and 1.8 A for Xe. So, the two parameters governing the A rotational exchange, R and L, remain unspeci®ed yet. Of course, if the rigid body collision dynamics is an adequate description for actual PQP±noble gas collisions, one would expect that R and L would be close to their counterparts taken from the structural formula of PQP. One can however follow more pragmatic way. Indeed, having se-
lected a particular value of R, one can vary L to arrive at the agreement between the theory and experiment. Since the KS theory predicts linear dependencies r0 =r vs. q at the pertinent rank of buer gas concentrations (zc 1, Eq. (8)), one can achieve a perfect correspondence of the theoretical and experimental curves. By proceeding along these lines, one gets an array of data R±L (R) for PQP±He, PQP±Ar, and PQP±Xe system (Fig. 5). It is logical to surmise that the parameters R and L are inherent to the spherocylinder which models the PQP molecule. Therefore, to make a theory self-consistent, one should expect the parameters to be the same for PQP in He, Ar, and Xe. By inspecting Fig. 5 one noti®es that the three curves intersect, roughly, in the vicinity of the point R 4 L 15 A. By noting that these values are deA, termined with the accuracy not higher than 1 A, and also keeping in mind that the hydrogen atom one concludes that radius is of the order of 0.5 A, the size of spherocylinder resembles rather well the L 13 A) obtained PQP dimensions (R 2:4 A, from its structural formula (see Fig. 4). So, our modeling of PQP molecules by hard smooth spherocylinders looks quite reasonable. After L 15 A into the insertion of values R 4 A,
Fig. 5. Fittings of experimental anisotropies in the spherocyl ( ) PQP±He, ( ) PQP±Ar, ( ) inder model: R vs. L (in A). PQP±Xe.
A.P. Blokhin et al. / Chemical Physics 272 (2001) 69±76
Eqs. (11) and (12), one can invoke Eqs. (2) and (3) for estimating the collision strength. One gets c 0:97, Zeff 33 (PQP±He), c 0:78, Zeff 4:5 (PQP± Ar), and c 0:53, Zeff 2:1 (PQP±Xe). So, one arrives at the intuitively expected conclusion: PQP±He collisions are the weakest, PQP±Ar collisions are of an intermediate strength, and the strongest are PQP±Xe encounters. On closing this section, we would like to touch upon possible improvements of the present theory. The fact that the dimensions of the prototype spherocylinder exceed slightly those obtained from the structural formula of PQP indicates that the spherocylinder±sphere model, although captures essential features of the PQP±noble gas collisions, still somewhat underestimates the rate of the rotation±translation exchange. (The situation is not uncommon for the smooth rigid body models [17].) Since the attractive forces are hardly responsible for this discrepancy, we suggest the following explanation for the phenomenon. First, let us consider a binary collision of arbitrary shaped smooth rigid bodies. Evidently, the two bodies experience an impulsive impact which is perpendicular to their surfaces at the point of contact. As a consequence, the component of the angular momentum of a cylindrically symmetric body along its symmetry axis remains unaected, and it is a conventional practice to put it to zero [14,15,17]. This actually transforms a symmetric top into a linear rotor. In reality, all the components of linear velocities and angular momenta are involved in the rotation±translation exchange due to the coupling of electron clouds of the collision partners. Within the rigid body dynamics, this effect can be taken into account by introducing the so-called ``roughness'', and the rough spheres are the classical objects of this kind [18]. So, it makes sense to generalize the present description by invoking ``rough rigid non-spherical convex bodies'' (see, e.g. Ref. [19]). Second, and appears to be the most crucial. PQP is actually an asymmetric planar rotor. Let us split its angular momentum into the in-plane and perpendicular-to-the-plane components. Evidently, the rate of change of the in-plane projection is signi®cantly greater that of the perpendicular-to-the-plane one. So, the collision dynamics of cylindrically symmetric smooth bodies
75
can dier substantially from that of smooth planar bodies. Work is in progress on developing Enskog-like theory for colliding planar and/or rough non-spherical species. Both of the aforementioned improvements of the collision dynamics require also a generalization of the KS model to non-spherical molecules. This can bee done along the lines developed in Refs. [20,21]. 4. Conclusion The main result of our studies can be formulated as follows: monitoring of the collisional depolarization of ¯uorescence of polyatomic species in rari®ed gases allows one to know about the ef®ciency of the angular momentum change. To extract such an information, we implemented the KS model in conjunction with the Enskog treatment of the smooth rigid body collision dynamics [14,15]. The modeling of PQP molecules by spherocylinders and noble gases by spheres allowed us to estimate the collision frequency zc and the angular momentum relaxation time sJ , and also to self-coordinately describe the ¯uorescence depolarization for PQP in He, Ar, and Xe solvents. This made it possible to determine the quantity Zeff (see Eq. (3)) for all the collision partners under study. It was found to be 33 (PQP±He), 4.5 (PQP±Ar), 2.1 (PQP±Xe). This conforms with a simple expectation that the heavier is a collision partner, the stronger is the collision. From that point of view, the J-diusion (Zeff 1) and the FPE (Zeff 1) are seen to be totally inadequate in reproducing our experimental data and mimicking ``actual'' collisions of polyatomic molecules with noble gases. A theory is required, (e.g., the KS theory) which permits one to describe collisions of variable eciency. Note ®nally that the study of the ¯uorescence depolarization in rari®ed buers gives only a small fraction of the general dependencies r0 =r vs. q, which extend over many orders of magnitude of the foreign species density and are very collision speci®c (see Fig. 1). The analyses of such curves would allow one to follow the transformation of molecular rotation from collisionless conditions towards a dilute gas with binary collisions and further
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A.P. Blokhin et al. / Chemical Physics 272 (2001) 69±76
to the hindered rotation limit. The available data on the depolarization of the laser emission of the active medium 1,4-bis[2(5-phenyloxazolyl)]benzene vs. concentration of pentane buer gas con®rm qualitatively the bell-like behavior predicted by Fig. 1 [22,23] but, unfortunately, a quantitative interpretation of these dependencies within the framework of the present theory is impossible. These results however show rather transparently that the detection of the ¯uorescence depolarization in a broad range of densities is as informative as more traditional NMR experiments, from which one gets knowledge about transformation of OR times (10) vs. the angular momentum relaxation time [1]. That is why the measurements on the ¯uorescence depolarization can equip one with the ecient tool for studying rotational and OR of polyatomic molecules. In connection with this, we expect that our theory (in essence, Eqs. (4) and (5)) is valid not only for rari®ed gases, but for much higher buer species concentrations. Acknowledgement This work was supported by the International Science and Technology Center, project no. B-441. References [1] A.I. Burshtein, S.I. Temkin, Spectroscopy of Molecular Rotations in Gases and Liquids, Cambridge University Press, Cambridge, 1994.
[2] J.S. Baskin, M. Gupta, M. Chachisvilis, A.H. Zewail, Chem. Phys. Lett. 275 (1997) 437. [3] J.S. Baskin, M. Chachisvilis, M. Gupta, A.H. Zewail, J. Phys. Chem. A 102 (1998) 4158. [4] N.A. Borisevich, S.P. Pliska, V.A. Tolkachev, Dokl. Akad. Nauk SSSR 261 (1981) 1109. [5] S.P. Pliska, V.A. Tolkachev, J. Appl. Spectr. 50 (1989) 47. [6] S.P. Pliska, V.A. Povedailo, V.A. Tolkachev, J. Appl. Spectr. 56 (1992) 155. [7] P.P. Feo®lov, The Physical Basis of Polarized Emission, Consultants Bureau Enterprises, Inc., 1961. [8] R.G. Gordon, J. Chem. Phys. 45 (1966) 1643. [9] R.A. Sack, Proc. Phys. Soc. B 70 (1957) 402, 414. [10] A.P. Blokhin, M.F. Gelin, Mol. Phys. 87 (1996) 455. [11] M.F. Gelin, J. Phys. Chem. A 104 (2000) 6150. [12] A.P. Blokhin, V.N. Kniukshto, V.A. Tolkachev, Izv. AN SSSR, Ser. Fiz. 42 (1978) 360. [13] T.F. Raychenok, J. Appl. Spectr. 41 (1984) 730. [14] M.P. Allen, G.T. Evans, D. Frenkel, B.M. Mulder, Adv. Chem. Phys. 83 (1993) 89. [15] G.T. Evans, D.R. Evans, J. Chem. Phys. 81 (1984) 6039. [16] S. Dashman, Scienti®c Foundations of Vacuum Technique, Wiley, NY, 1962. [17] G.T. Evans, R.G. Cole, D.K. Homan, J. Chem. Phys. 77 (1982) 3209. [18] S. Chapman, T.G. Cowling, The Mathematical Theory of Nonuniform Gases, Cambridge University Press, Cambridge, 1960. [19] B.J. Berne, J. Chem. Phys. 66 (1977) 2821. [20] A.P. Blokhin, M.F. Gelin, Khim. Phys. 13 (1994) 14; A.P. Blokhin, M.F. Gelin, Khim. Phys. 16 (1997) 39, 50. [21] A.P. Blokhin, M.F. Gelin, J. Phys. Chem. B 101 (1997) 236. [22] A.P. Blokhin, N.A. Borisevich, I.I. Kalosha, V.A. Tolkachev, Dokl. Akad. Nauk SSSR 238 (1978) 123. [23] N.A. Borisevich, J. Lumin. 18/19 (1979) 127.
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Depolarization of ¯uorescence of polyatomic molecules in noble gas solvents A.P. Blokhin, M.F. Gelin *, I.I. Kalosha, V.V. Matylitsky, N.P. Erohin, M.V. Barashkov, V.A. Tolkachev Institute of Molecular and Atomic Physics, Academy of Sciences of Belarus, F. Skaryna Avenue 70, Minsk 220072, Belarus Received 9 April 2001
Abstract The collisional depolarization of ¯uorescence is studied for p-quarterphenyl (PQP) in He, Ar, Xe solvents, under pressures ranging from zero to nearly atmospheric. The results are interpreted within the Keilson±Storer model of the orientational relaxation and smooth rigid body collision dynamics. This allows us to estimate the rate of the angular momentum scrambling due to encounters of PQP with its partners. The collisions are shown to be neither strong nor weak, so that the averaged number of encounters giving rise to the PQP angular momentum randomization equals to 33 (PQP±He), 4.5 (PQP±Ar), and 2.1 (PQP±Xe). Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 05.20.Dd; 33.50.)j; 34.50.Ez
1. Introduction It is well known that intermolecular encounters dier in their strength in transferring angular momentum [1]. In the standard way, this fact is embodied in the quantity Zeff zc sJ , which is the product of the collision frequency and the angular momentum relaxation time. Physically, Zeff is the number of collisions that is necessary for the randomization of the angular momentum. The common sense dictates that Zeff 1 for collision partners of comparable masses (strong collisions) and Zeff 1 for collisions of heavy molecules with light buer species.
*
Corresponding author. Fax: +375-17-284-0030. E-mail address: [email protected] (M.F. Gelin).
Orientational relaxation (OR) in the hindered rotation limit is universally governed by the diffusion equation [1]. Within this approach, the in¯uence of a solvent on the OR is taken into account by introducing the corresponding diusion coecients. According to the Green±Kubo formulas, these are uniquely determined by the angular momentum relaxation times and are independent of the collision strength. However, it is apparent enough that the collision eciency must manifest itself in the OR of molecules in dilute enough solutions and/or rari®ed gases. Recently, this was non-ambiguously demonstrated by Zewail and coworkers [2,3], who performed the real time measurements of the anisotropy decay of iodine molecules in noble gas solvents under a broad scale of densities, ranging from isolated molecule to liquid. It is therefore reasonable to expect
0301-0104/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 1 ) 0 0 4 6 0 - 8
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A.P. Blokhin et al. / Chemical Physics 272 (2001) 69±76
that the collision eciency aects the ¯uorescence depolarization rate for large polyatomic species in various buer gases. The collisional depolarization of the total ¯uorescence was measured for several aromatic molecules in the bulk [4] and in dierent (polyatomic) solvents [5]. The masses of solute and solvent species were comparable, and it was tentatively concluded that the situation nearly corresponded to Zeff 1. More recently, the reliable indications have been found that OR of aromatic molecules in noble gases proceeds with a solvent dependent Zeff , ranging from several units to several tens [6]. The present contribution gives an account of the combined theoretical and experimental studies of the in¯uence of the collision ef®ciency on the total ¯uorescence depolarization for polyatomic molecules. Our aim is to develop a reliable approach for the description of the anisotropy decay and to analyze which kind of information can be extracted from such experiments. The paper is organized as follows. The theory of the depolarization of total ¯uorescence for linear molecules is outlined in Section 2, in which some general trends are established for the behavior of the anisotropy vs. the frequency and strength of encounters. The results of the measurements of the ¯uorescence depolarization for p-quarterphenyl (PQP) in He, Ar, Xe are presented in Section 3, along with their interpretation in terms of the Keilson±Storer (KS) model and rigid body collision dynamics. Our main ®ndings are summed up in Section 4. Note from the outset that the dimensionless variables are used throughout the article. Time is measured in units of the averaged free rotation p period I=kT , where I is the main moment of inertia of a linear molecule, k is the Boltzmann constant, and T is the temperature.
2. The Keilson±Storer theory The anisotropy of the total molecular emission, r, is proportional to the integral over the time domain of the second rank orientational correlation function (OCF) G2 t of the absorption ~ labs and emission ~ lem dipole moments [7,8]:
r 2=5sem1 sdep ; Z 1 sdep dt expf t=sem g G2 t; 0
G2 t hP2 ~ labs 0~ labs ti:
1
Here P2 x is the second order Legendre polynomial and sem is the lifetime in an electronically excited state. The simplest way for incorporating the notion of the collision strength into a theory is to invoke the KS model [1,9±11]. In this model, the rotational motion is governed by the two parameters: zc , which is the collision frequency, and 1 6 c 6 1, which is the collision eciency. The two parameters constitute the quantity mJ sJ 1 zc 1
c
2
which is the angular momentum relaxation frequency, yielding Zeff zc sJ 1
1
c :
3
We shall further restrict ourselves to the calculation of the anisotropy for linear species. If the dipole moments involved are parallel to the axis of a linear molecule, the KS model allows one to calculate the second rank OCF through the simple three term recurrence relations [11]. By slightly adapting these results for the case of interest here, one ®nds sdep 1 2b0 sem ; 16 m 2 8m 10 8m 6 bm1 r2m1 bm r2m2 r2m2 r2m 4m bm 1 3sem dm0 ; 4 r2m where rm zc 1
cm sem1 :
5
The most popular and widespread gas-like models of the OR correspond to the cases of strong (Jdiusion) and weak (rotational Fokker±Planck equation (FPE)) collisions. The KS model contains both of these approaches as a special case. Namely, the J-diusion is recovered by putting c 0 and the FPE is reproduced by letting zc ! 1, c ! 1, zc 1 c ! mJ const [1,9±11]. Positive values of c allow one to describe a crossover from
A.P. Blokhin et al. / Chemical Physics 272 (2001) 69±76
the FPE to J-diusion. Negative values of c correspond to the angular momentum reorientation due to a collision, that is characteristic for molecular liquids. The J-coherence model proposed in papers [2,3] was proved to be very similar in spirit to the KS model [11]. In order to get an idea of how the total ¯uorescence anisotropy changes with the buer gas density, one can inspect Fig. 1, in which the ratio r0 =r is presented vs. zc for various values of collision eciency c. r0 is the ¯uorescence anisotropy under collisionless conditions. For linear rotors, r0 2=5 P2 02 1=10. We plot the inverse of the anisotropy r0 =r instead of the direct proportionality r=r0 since it is somewhat traditional [4,5], and also because such graphs are more sensitive to testing against experimental results. The dependencies have a characteristic bell-like form. Due to the collisional depolarization, the curves initially increase in parallel with the growth of the collision frequency. They further reach their maximums and start to decrease to the common value of 1/4. This behavior is a direct mirroring of the transformation from the free molecular rotation to the hindered (diusion) one. In this latter case, the anisotropy is described by the Levshin±Perrin 1 formula r 0:4 1 6sJ sem [7], which yields r
Fig. 1. r0 =r vs. zc for sem 50: (1) c 0:99, (2) c 0:9, (3) c 0 (J-diusion), (4) c 0:9, (5) c 0:99.
71
0:4 for zc 1 (or, equivalently, sJ 1). In the opposite limit zc 1, Eq. (4) describes a crossover from the constant value r0 =r 1;
zc sem1 1
6
to the linear dependence r0 =r zc sem W; sem1 zc 1:
7
Here W
6b0 ;
32b1
10b0 3 r2 ;
rm 1
cm 8
and the coecients bm obey recurrence relationships (4) in which one should substitute rm ! rm and take m 1; 2; 3 . . . Expressions (6) and (7) can be combined into r0 =r 1 zc sem W:
9
This equation is precisely of the form proposed by Gordon for linear molecules in binary collision solvents [8], so that Eq. (8) provides the explicit expression for the ¯uorescence quenching cross-section in terms of the KS parameters. Note however that OCF G2 t does not decay exponentially in the rari®ed gas limit zc 1 [1,9±11]. This is a direct consequence of the correlation of molecular angular momenta before and after a collision. It is only in case of strong collisions (c 0, the J-diusion) that the pre- and post-collisional angular momenta are uncorrelated. Then G2 t 1=4 expf 3=4zc tg [1] and Eq. (9) with W 3=4 is exact for an arbitrary zc . As to the in¯uence of the collision eciency on the anisotropy decay, one noti®es the following (see Fig. 1). When zc is small enough, one arrives at the intuitively expected conclusion: the smaller is jcj (the stronger are collisions) the higher is depolarization. This is precisely the situation which corresponds to the experimental conditions described in Section 3. So, c 1 gives the lower boundary for the depolarization rate (in fact, no collisions occur) and c 0 gives the upper one (strong collisions). It is remarkable that the anisotropy is independent of the sign of parameter c for small zc . When the density (and therefore zc ) is further increased, the degeneracy on c disappears and the curves corresponding to various
72
A.P. Blokhin et al. / Chemical Physics 272 (2001) 69±76
1 6 c 6 1 dier signi®cantly. It should also be pointed out that the ¯uorescence depolarization is quite sensitive to the value of the lifetime in the electronically excited state. Evidently, the smaller is sem , the smaller is the number of collisions which experiences a molecule at the electronically excited state and, as a result, the weaker is depolarization. The general trends in the ¯uorescence depolarization directly re¯ect the well known behavior for the integral OR times [1] Z 1 sX;j dt Gj t: 10 0
The point is that if sX;2 sem (this is so for all but very small (sJ 1) and very high (sJ 1) densities) one can approximately write sX;2 sdep and, therefore, r 0:4sX;2 =sem .
3. Experimental results and discussion We have performed measurements of the ¯uorescence depolarization for PQP in He, Ar, Xe solvents under pressures ranging from zero to nearly atmospheric. Our experimental setup was described in detail elsewhere [4±6]. In short, it consists of the cuvette compartment, the vacuum post, the excitation source, and the recording sys-
Fig. 2. Block diagram of setup for recording polarized ¯uorescence.
tem (see Fig. 2). The studied species are placed in the specially designed fused silica cuvette. It consists of the branch piece and the optical part which are connected to the vacuum post. The branch piece is placed into the oven, which governs the PQP concentration. To avoid the PQP absorption on the cuvette sides, the temperature of the optical part of the cuvette was kept 20±30 K higher than that of the branch piece. After the pre-evacuation up to 10 5 Torr, the PQP molecules in the branch piece are heated to the required temperature. The optical excitation is performed by the excimer laser (308 nm wavelength, 20 ns pulse duration). To keep of the saturation eects, its irradiation strength was controlled to be less than 1 MW/cm2 . At the maximum of the ¯uorescence spectrum, the two orthogonally polarized emission intensities are recorded by the photomultipliers. The signals are further processed with a computerized installation on the basis of CAMAC-PC. The experimental error in measuring polarization degrees was estimated to be 1%. Our experimental results are presented in Fig. 3 as the plots r0 =r vs. the noble gas concentration q,
Fig. 3. Experimental dependencies r0 =r vs. q and their linear ®ts: ( ) PQP±He, ( ) PQP±Ar, ( ) PQP±Xe.
A.P. Blokhin et al. / Chemical Physics 272 (2001) 69±76
along with the linear ®ts to these dependencies. Before embarking at the interpretation of these anisotropies in terms of the theory developed in Section 2, one must realize the following. First, PQP is a considerably prolate asymmetric top with the main moments of inertia Ix 7672, Iy 7632, 2 . In so far as the absorption and Iz 340 amu A and emission dipole moments for PQP are directed along the axis of its smallest moment of inertia [12], one can approximately treat it as a linear rotor with the moment of inertia I Ix Iy =2 2 . Second, it is necessary to recalculate 7654 amu A the buer species concentration q into the collision frequency zc . By invoking Eq. (11) (see below) one concludes that the maximal experimentally achieved values of concentration qmax correspond to the dimensionless collision frequency zc of the order 0.07 (PQP±He), 0.03 (PQP±Ar), and 0.024 (PQP±Xe). Eq. (9) is correct for such small values of zc , and this gives an additional support to the validity of the approximation of the experimental anisotropies by linear ®ts (see Fig. 3). Third, it is necessary to emphasize that the ¯uorescence depolarization of a linear molecule in the KS model is governed by the following four characteristic times: the free rotation period sr , the intercollision time sc zc 1 , the angular momentum relaxation time sJ mJ 1 , and the lifetime in an electronically excited state sem . As it has been mentioned in the Section 1, it is convenient to utilize the dimensionless variables, by using sr 4:16 ps as the unit of time (T 533 K). For PQP, sem 1:4 ns [13]. In principle, sem can depend on q due to the collisional ¯uorescence quenching [4±8]. The latter was not detected for PQP under our experimental conditions, so that the reduced lifetime sem 337 can indeed be considered constant. So, at q qmax , a PQP molecule experiences, averagely, 24 (PQP±He), 10 (PQP±Ar), and 8 (PQP± Xe) collisions during sem . The advantage of the KS model in interpreting experimental data will dramatically increase if one will be able to calculate the remaining two dynamic parameters, zc and c, from some ``®rst principles''. It is logical to surmise that the main source of the PQP angular momentum change in its collision with a noble gas atom is due to the short range repulsion. Therefore, it is reason-
73
Fig. 4. The PQP spherocylinder and the noble gas sphere. All the sizes are given in angstroms.
able to invoke the rigid body dynamics for mimicking these encounters. The use of the J-coherence model for the interpretation of the anisotropy decay of iodine molecules in dierent noble gas buers [2,3] also corroborates this idea (see also paper [11]). More speci®cally, we represent the noble gas atoms by smooth rigid spheres and PQP molecules by rigid smooth spherocylinders (Fig. 4). We have chosen spherocylinders by the following reasons. First, a highly prolate spherocylinder, as distinct from an ellipsoid of revolution, has nearly a constant mass density per length, that is roughly so for PQP. Second, the PQP concentration in our experiments (2:4 1018 m 3 ) is much less than that of the buer species (typically, 1024 m 3 ). Since the radii of the noble gas atoms (see below), their concentration is eviare 1±2 A dently much less than the corresponding close packing concentration. So, we have a dilute solution of PQP molecules in rari®ed noble gases. This allows us to treat the rigid body dynamics within the Enskog approximation, to put the contact pair correlation function to unity, and to analytically evaluate the collision frequency and the angular momentum relaxation frequency [14,15] 1 Rs R p zc p vq L 1 k h 2 2 2 Rs R 1 k 1/ ;
11
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A.P. Blokhin et al. / Chemical Physics 272 (2001) 69±76
mJ p vq
Rs R p L 1 k 2 2
Rs R 1 k
h 1/ :
12
p Here v 8kT = pl is the relative speed of the colliding species, l MMs = M Ms is their reduced mass, q is the buer gas concentration, Rs is the radius of the ``noble gas'' sphere, R and L are, correspondingly, the radius of the spherical cap and the length of the cylindrical part of the spherocylinder (see Fig. 4), q k arcsin k1 lL2 p k ; / ; 4I k 13 p p ln k k 1 p : h k Evidently, the rotational energy exchange proceeds entirely due to molecular non-sphericity, i.e. L 6 0. When L 0, one gets mJ 0 and the 2 standard result zc q vp Rs R for smooth rigid spheres is recovered. Both zc and mJ increase with the growth of R and/or L. The higher is the spherocylinder length L, the stronger are collisions. Generally, Eqs. (2) and (11)±(13) predict the collision strength parameter 1 6 c 6 1, as it must be from its de®nition. Both zc and mJ are proportional to the concentration of the noble gas atoms and also to the relative thermal speed of the colliding particles. In addition, zc and mJ depend quite complicatedly on the parameters specifying the spherocylinder (M,R,L,I) and the buer sphere (Ms ,Rs ). It is natural to assume that the mass and the main moment of inertia of the spherocylinder are those of PQP, 2 . In adviz. M 306:4 amu and I 7654 amu A dition, the masses Ms and radii Rs of the noble gases are known [16]. These are 4 amu and 1.1 A for Ar, 131 amu and 2.5 for He, 40 amu and 1.8 A for Xe. So, the two parameters governing the A rotational exchange, R and L, remain unspeci®ed yet. Of course, if the rigid body collision dynamics is an adequate description for actual PQP±noble gas collisions, one would expect that R and L would be close to their counterparts taken from the structural formula of PQP. One can however follow more pragmatic way. Indeed, having se-
lected a particular value of R, one can vary L to arrive at the agreement between the theory and experiment. Since the KS theory predicts linear dependencies r0 =r vs. q at the pertinent rank of buer gas concentrations (zc 1, Eq. (8)), one can achieve a perfect correspondence of the theoretical and experimental curves. By proceeding along these lines, one gets an array of data R±L (R) for PQP±He, PQP±Ar, and PQP±Xe system (Fig. 5). It is logical to surmise that the parameters R and L are inherent to the spherocylinder which models the PQP molecule. Therefore, to make a theory self-consistent, one should expect the parameters to be the same for PQP in He, Ar, and Xe. By inspecting Fig. 5 one noti®es that the three curves intersect, roughly, in the vicinity of the point R 4 L 15 A. By noting that these values are deA, termined with the accuracy not higher than 1 A, and also keeping in mind that the hydrogen atom one concludes that radius is of the order of 0.5 A, the size of spherocylinder resembles rather well the L 13 A) obtained PQP dimensions (R 2:4 A, from its structural formula (see Fig. 4). So, our modeling of PQP molecules by hard smooth spherocylinders looks quite reasonable. After L 15 A into the insertion of values R 4 A,
Fig. 5. Fittings of experimental anisotropies in the spherocyl ( ) PQP±He, ( ) PQP±Ar, ( ) inder model: R vs. L (in A). PQP±Xe.
A.P. Blokhin et al. / Chemical Physics 272 (2001) 69±76
Eqs. (11) and (12), one can invoke Eqs. (2) and (3) for estimating the collision strength. One gets c 0:97, Zeff 33 (PQP±He), c 0:78, Zeff 4:5 (PQP± Ar), and c 0:53, Zeff 2:1 (PQP±Xe). So, one arrives at the intuitively expected conclusion: PQP±He collisions are the weakest, PQP±Ar collisions are of an intermediate strength, and the strongest are PQP±Xe encounters. On closing this section, we would like to touch upon possible improvements of the present theory. The fact that the dimensions of the prototype spherocylinder exceed slightly those obtained from the structural formula of PQP indicates that the spherocylinder±sphere model, although captures essential features of the PQP±noble gas collisions, still somewhat underestimates the rate of the rotation±translation exchange. (The situation is not uncommon for the smooth rigid body models [17].) Since the attractive forces are hardly responsible for this discrepancy, we suggest the following explanation for the phenomenon. First, let us consider a binary collision of arbitrary shaped smooth rigid bodies. Evidently, the two bodies experience an impulsive impact which is perpendicular to their surfaces at the point of contact. As a consequence, the component of the angular momentum of a cylindrically symmetric body along its symmetry axis remains unaected, and it is a conventional practice to put it to zero [14,15,17]. This actually transforms a symmetric top into a linear rotor. In reality, all the components of linear velocities and angular momenta are involved in the rotation±translation exchange due to the coupling of electron clouds of the collision partners. Within the rigid body dynamics, this effect can be taken into account by introducing the so-called ``roughness'', and the rough spheres are the classical objects of this kind [18]. So, it makes sense to generalize the present description by invoking ``rough rigid non-spherical convex bodies'' (see, e.g. Ref. [19]). Second, and appears to be the most crucial. PQP is actually an asymmetric planar rotor. Let us split its angular momentum into the in-plane and perpendicular-to-the-plane components. Evidently, the rate of change of the in-plane projection is signi®cantly greater that of the perpendicular-to-the-plane one. So, the collision dynamics of cylindrically symmetric smooth bodies
75
can dier substantially from that of smooth planar bodies. Work is in progress on developing Enskog-like theory for colliding planar and/or rough non-spherical species. Both of the aforementioned improvements of the collision dynamics require also a generalization of the KS model to non-spherical molecules. This can bee done along the lines developed in Refs. [20,21]. 4. Conclusion The main result of our studies can be formulated as follows: monitoring of the collisional depolarization of ¯uorescence of polyatomic species in rari®ed gases allows one to know about the ef®ciency of the angular momentum change. To extract such an information, we implemented the KS model in conjunction with the Enskog treatment of the smooth rigid body collision dynamics [14,15]. The modeling of PQP molecules by spherocylinders and noble gases by spheres allowed us to estimate the collision frequency zc and the angular momentum relaxation time sJ , and also to self-coordinately describe the ¯uorescence depolarization for PQP in He, Ar, and Xe solvents. This made it possible to determine the quantity Zeff (see Eq. (3)) for all the collision partners under study. It was found to be 33 (PQP±He), 4.5 (PQP±Ar), 2.1 (PQP±Xe). This conforms with a simple expectation that the heavier is a collision partner, the stronger is the collision. From that point of view, the J-diusion (Zeff 1) and the FPE (Zeff 1) are seen to be totally inadequate in reproducing our experimental data and mimicking ``actual'' collisions of polyatomic molecules with noble gases. A theory is required, (e.g., the KS theory) which permits one to describe collisions of variable eciency. Note ®nally that the study of the ¯uorescence depolarization in rari®ed buers gives only a small fraction of the general dependencies r0 =r vs. q, which extend over many orders of magnitude of the foreign species density and are very collision speci®c (see Fig. 1). The analyses of such curves would allow one to follow the transformation of molecular rotation from collisionless conditions towards a dilute gas with binary collisions and further
76
A.P. Blokhin et al. / Chemical Physics 272 (2001) 69±76
to the hindered rotation limit. The available data on the depolarization of the laser emission of the active medium 1,4-bis[2(5-phenyloxazolyl)]benzene vs. concentration of pentane buer gas con®rm qualitatively the bell-like behavior predicted by Fig. 1 [22,23] but, unfortunately, a quantitative interpretation of these dependencies within the framework of the present theory is impossible. These results however show rather transparently that the detection of the ¯uorescence depolarization in a broad range of densities is as informative as more traditional NMR experiments, from which one gets knowledge about transformation of OR times (10) vs. the angular momentum relaxation time [1]. That is why the measurements on the ¯uorescence depolarization can equip one with the ecient tool for studying rotational and OR of polyatomic molecules. In connection with this, we expect that our theory (in essence, Eqs. (4) and (5)) is valid not only for rari®ed gases, but for much higher buer species concentrations. Acknowledgement This work was supported by the International Science and Technology Center, project no. B-441. References [1] A.I. Burshtein, S.I. Temkin, Spectroscopy of Molecular Rotations in Gases and Liquids, Cambridge University Press, Cambridge, 1994.
[2] J.S. Baskin, M. Gupta, M. Chachisvilis, A.H. Zewail, Chem. Phys. Lett. 275 (1997) 437. [3] J.S. Baskin, M. Chachisvilis, M. Gupta, A.H. Zewail, J. Phys. Chem. A 102 (1998) 4158. [4] N.A. Borisevich, S.P. Pliska, V.A. Tolkachev, Dokl. Akad. Nauk SSSR 261 (1981) 1109. [5] S.P. Pliska, V.A. Tolkachev, J. Appl. Spectr. 50 (1989) 47. [6] S.P. Pliska, V.A. Povedailo, V.A. Tolkachev, J. Appl. Spectr. 56 (1992) 155. [7] P.P. Feo®lov, The Physical Basis of Polarized Emission, Consultants Bureau Enterprises, Inc., 1961. [8] R.G. Gordon, J. Chem. Phys. 45 (1966) 1643. [9] R.A. Sack, Proc. Phys. Soc. B 70 (1957) 402, 414. [10] A.P. Blokhin, M.F. Gelin, Mol. Phys. 87 (1996) 455. [11] M.F. Gelin, J. Phys. Chem. A 104 (2000) 6150. [12] A.P. Blokhin, V.N. Kniukshto, V.A. Tolkachev, Izv. AN SSSR, Ser. Fiz. 42 (1978) 360. [13] T.F. Raychenok, J. Appl. Spectr. 41 (1984) 730. [14] M.P. Allen, G.T. Evans, D. Frenkel, B.M. Mulder, Adv. Chem. Phys. 83 (1993) 89. [15] G.T. Evans, D.R. Evans, J. Chem. Phys. 81 (1984) 6039. [16] S. Dashman, Scienti®c Foundations of Vacuum Technique, Wiley, NY, 1962. [17] G.T. Evans, R.G. Cole, D.K. Homan, J. Chem. Phys. 77 (1982) 3209. [18] S. Chapman, T.G. Cowling, The Mathematical Theory of Nonuniform Gases, Cambridge University Press, Cambridge, 1960. [19] B.J. Berne, J. Chem. Phys. 66 (1977) 2821. [20] A.P. Blokhin, M.F. Gelin, Khim. Phys. 13 (1994) 14; A.P. Blokhin, M.F. Gelin, Khim. Phys. 16 (1997) 39, 50. [21] A.P. Blokhin, M.F. Gelin, J. Phys. Chem. B 101 (1997) 236. [22] A.P. Blokhin, N.A. Borisevich, I.I. Kalosha, V.A. Tolkachev, Dokl. Akad. Nauk SSSR 238 (1978) 123. [23] N.A. Borisevich, J. Lumin. 18/19 (1979) 127.