J. Phys. Chem. 1981, 85,2531-2535
2531
Studies of the Reorientational Motion and Intermolecular Interaction of Dlmethyl Sulfoxide in Water by Depolarlzed Rayleigh Scattering Y. Higashigaki, D. H. Christensen,+ and C. H. Wang" Department of Chemistry. Unlversity of Utah, Sell Lake C@, Utah 84 112 (Received: March 3, 198 I; In Final Form: May 15, 198 1)
The reorientational process of dimethyl sulfoxide (MezSO)in pure liquid and in aqueous solutions is studied by depolarized Rayleigh scattering. The Rayleigh relaxation time is obtained as a function of temperature in solution of various MezSO concentrations. In pure MezSO, TWis found to follow the Stokes-Einstein-Debye (SED) equation, with the hydrodynamicvolume for reorientation being considerablysmaller than the molecular volume. In aqueous solutions TR~,,are found not to follow the SED equation and the data suggest the presence of a nonhydrodynamic effect. The nonhydrodynamic effect is most pronounced in solutions with small MezSO contents and vanishes gradually in high MezSO concentrations. From the vs. q/T plot we have obtained the hydrodynamic volume for MezSOreorientation as a function of concentration. The hydrodynamic volume of MezSOin aqueous solution is found to be quite similar to that found in the pure liquid and suggests that MezSOmolecules in aqueous solutions reorient without carrying H20 molecules with them. This result is in disagreement with the conclusion for reorientation of M e 8 0 molecules previously arrived at by using the NMR relaxation studies. The activation energy for reorientation is also obtained for the pure liquid and two aqueous solutions. The result is also found to disagree with the NMR results. The reason for the discrepancy between the NMR results and the depolarized Rayleigh scattering results is discussed.
Introduction The physical properties of the dimethyl sulfoxide (Me2SO)-water system have been investigated extensively in recent years by use of various experimental techniques. These studies are undertaken for the reason that the binary Me2So-HzO system is useful as a solvent and as a reaction medium, and, furthermore, the system exhibits interesting properties when it is brought to interact with proteins and other biological substances.' At ambient pressure, MezSO exists in the liquid state over a wide temperature range (fp 18 "C and bp 189 "C) and is completely miscible with water. The high boiling temperature has been attributed to the large dipole moment of the MezSO molecule (p = 4.3 D). This dipoledipole interaction is believed to be responsible for molecular association in the liquid state. Adding MezSO molecules to water causes a profound effect in the structure and dynamics of water. Not only the dipolar force due to MezSO will play a role, but also the hydrogen bonding of MezSO to a group of HzO molecules will be present. Raman2 and neutron3 scattering studies have shown that in solutions of high water content the interaction between Me2S0 and water causes a cooperative perturbation in the orientation of a group of hydrogen-bonded HzO molecules and results in rigidifying the water structure. On the other hand, when MezSO is present in large concentration it will tend to break up the water structure by formation of the hydrogen-bonded MezSO/HzO complexes, as evidenced by the results of IR, X-ray, and neutron-scattering s t ~ d i e s .From ~ ~ ~ thermodynamic considerations, Lindbergs has argued that the MezSO-HzO hydrogen bond is stronger than that between HzO and other oxygen-containing organic molecules; this conclusion is also supported by the IR stud^.^,^ The ability of MezSO to break down the water structure and to form hydrogen-bonded aggregates with water inhibits the formation of hexagonal ice from water at low temperature. This is believed to be responsible for the cryoprotective +Departmentof Chemical Physics, H. C. 0rsted Instituet, University of Copenhagen, 5, Universitetsparken, DK-2100 Copenhagen, Denmark.
property of MezSO,which has been used to prevent injury to the living cells subject to low t e m p e r a t ~ r e . ~ In spite of extensive investigations of the MezSO-HzO system being conducted so far, information regarding the motion of Me2S0 molecules in aqueous solution has been inconclusive. For example, measurements of the deuteron spin-lattice relaxation time in the Me2SO-D20 system have suggested that the reorientation of the MezS0/3H20 complex as a unit is the dominant motion causing relaxation? Another nuclear spin-lattice relaxation time study has also concluded that a MezSO molecule reorients together with two HzO molecule^.^ This is in disagreement with the deuteron relaxation time datas but in agreement with excess heat of mixing, density, and viscosity data where the results also indicate a MezSO-2Hz0 complex. We report in this paper recent resulta of the depolarized Rayleigh scattering study of the reorientational motion of pure MezSO and Me2SO-water solutions as a function of concentration and temperature. The depolarized Rayleigh scattering spectrum of aqueous solutions of M e a 0 reflects the orientational fluctuation of MezSO molecules, due to the fact that the depolarized light-scattering intensity from water is negligible. From the spectral Iine width measurement we have obtained the reorientation time of Me2S0 directly. Thus, this light-scattering technique provides a direct determination of reorientation times of MezSO molecules free from complicating approximation needed in the NMR technique. Our basic conclusion in this study is that the Me2SO/Hz0 complexes are short lived, and in the time scale of the present measurement (1)A. J. Parker, Chem. Reu., 69, l(1969). (2)J: J. Lindberg and C. Majani, Acta Chern. Scand., 17,1477(1963). (3)G.J. Safford, P. C. Schaffer, P. S. Leung, G. F. Doebbler, G. W. Brady, and E. F. X. Lyden, J. Chern. Phys., 50, 2140 (1969). (4)G.Brink and M. Falk, J. Mol. Struct., 5 , 27 (1970). (5) J. J. Lindberg, Fin. Kemistsamf. Medd., 70,33 (1961);J. J. Lindberg and J. Kenttamaa, Suom. Kemist. B, 33, 104 (1960). (6)0.D.Bonner and Y. S. Choi, J. Phys. Chem., 78, 1723 (1974). (7)J. E.Lovelock and M. W. H. Bishop, Nature (London), 183,1394 (1959). (8)J. A. Glasel, J. Am. Chem. SOC.,92,372 (1970). (9)K. J. Packer and D. J. Tomlinson, Trans. Faraday Soc., 67,1302 (1971).
0022-365418112085-2531$0 1.2510 0 1981 American Chemical Society
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The Journal of Physical Chemistty, Val. 85, No. 17, 1981
H
TABLE I: Rayleigh Relaxation Times ( 7 ~and ~ Shear ~ ) Viscosity (q ) at Various Concentrations in the Me,SO-H,O System
____3(
+FSR
2
0
Higashigaki et al.
0
Figure 1. The depolarized Rayleigh scattering spectra at 25 OC of Me2SO-H20 solution at x(Me,SO) = 0.49 and of pure water under identical experimental conditions. The free spectral range indicated is 232 GHz.
s) MezSO molecules do not carry H20 molecules with them as reorientation takes place, despite the fact that the Me2SO-H,0 interaction plays an important role in affecting the rotational dynamic. Our result, therefore, does not support the conclusion previously reported in the NMR study.
Experimental Section The depolarized Rayleigh scattering spectra of pure MezSO and of aqueous solutions have been obtained with the same technique and instrumentation as those used for the recent study of the pyridine-water system.1° Pure MezSO (spectraquality grade) was distilled in a dust-free vacuum system and transferrred to a rectangular cell. The aqueous solutions were prepared by addition of redistilled water, which has been filtered repeatedly through 0.22-mm millipore filters. The mole fration of Me2S0 of the aqueous solutions were determined from the weight of pure Me2S0 and HzO. The experimental spectra were recorded on an x-y recorder. Two interferometer-free spectral ranges corresponding to 415 and 232 GHz were used. The overall instrumental finesse was over 60 throughout the measurements. The spectra were fit to overlapping Lorentzian functions plus a constant background. The background is due mainly to the Raman-scattering intensity. Results and Discussion Shown in Figure 1 are the depolarized Rayleigh spectra of Me2SO-HzO solution at x = 0.49 and of pure water under identical experimental conditions. One notes that the intensity of the depolarized Rayleigh scattered light from pure water is negligibly small compared to the intensity from the Me2S0 solution. Thus, we assume that the spectra from solution of Me2SO-H20 represent only the reorientation fluctuation of the Me2S0 molecules. The Rayleigh relaxation times, are obtained from the half-width at half-height (I' in Hz) of the fitted spectra according to Thy = (2ar)-1 Since r is measured to about 10% of accuracy, T~~ is also accurate to within 10%. The experimentally determined ?Ray for pure (x = 1) Me2S0 at various temperatures and for aqueous solutions at various concentrations at 25 "C and at other temperatures are summarized in Table I along (10)J. M.G.Cowie and P. M. Toporowaki, Can. J . Chen., 39,2240 (1961).
a
25 8.5 2.00 36 6.8 1.65 45 5.4 1.41 60 4.2 1.12 70 3.4 1.00 0.88 25 8.6 2.16 0.74 25 10.8 2.47 0.66 25 10.8 2.72 45 8.3 1.76 0.49 25 12.8 3.42 25 9.3 2.07 65 8.2 1.37 0.35 25 14.4 3.77 0.32 25 12.8 3.74 0.25 25 13.5 3.43 0.23 25 14.2 3.29 0.15 25 11.4 2.37 0.12 25 12.8 2.08 Reference 18,and references cited therein.
15
5
c
r
I l l l l / l j l / l
0
M E p
0.2
0.G
0.4
X
0.8
1
H20
Figure 2. The rRaydata at 25 OC plotted as a function of Me,SO concentration (empty circles). Also included in the plot are the orie a 0 and H20 deduced entational correlation time data (at 26 "C)for M from the NMR relaxation time study (ref 9). The designation for the NMR reorientation time data (solid circles for Me,SO and empty hexagons for H,O) given in ref 9 is in error, but the reversal in the designation will not affect the discussion as presented in the text.
with the shear viscosity data. The data at 25 "C are also plotted as empty circles as a function of Me2S0 concentration (Figure 2). Also included in Figure 2 for comparison are the reorientation correlation time data obtained by using the NMR t e ~ h n i q u e .One ~ notes that both r h Y and the NMR reorientation times exhibit a maximum at about x = 0.65. However, over the entire Me2S0 concenis systematically shorter than the retration range rRaY ported NMR reorientation time. At x = 0.66 the NMR reorientation time appears to be about 50% longer than TRa
&nce the depolarized Rayleigh scattering probes both the single particle and the pair correlated reorientation process, whereas NMR is sensitive only to the average rte of single-particle reorientation, in principle, the NMR relaxation time is not expected to be equal to r h The two times become equal only at infinite dilution,ljwhere
.
(11)S.A. Schichman and R. L. Amey, J.Phys. Chem., 75,98(1971). (12)C. H.Wang, S. C. Whittenburg, P.-A. Lund, and D. H. Christensen, J. Chem. Phys., 72,4228 (1980). (13)S. L. Whittenburg and C. H. Wang, J . Chern. Phys., 70,3141 (1979).
The Journal of Physical Chemlstry, Vol. 85, No. 17, 198 1 2533
Study of Me2SO-H20 Interaction
the pair correlation factor vanishes.14 However, the observed difference in the present depolarized Rayleigh and NMR results is probably caused by the incorrect approximations which are introduced in deriving the NMR reorientation time. The difficulty in the NMR study of the reorientation process of M e a 0 in aqueous solution lies in the separation of several mechanisms which are known to make contribution to the relaxation of nuclear spin. Despite the elaborated proton dilution technique introduced in the MezSO + MezSO-d6+ DzO system to eliminate the intermolecular relaxation process, two intramolecular mechanisms corresponding to the methyl group reorientation and the end-over-end reorientation of the MezSO molecule cannot be easily separated. Both are expected to contribute to the proton relaxation in MezSO. In deducing the reorientation time from NMR the authors of ref 9 have neglected the contribution due to methyl group reorientation by assuming methyl group rotation to be infinitely fast and thus ineffective in causing relaxation. While the validity of this approximation is in question, it should be pointed out that in the other limit of rigid methyl groups, the derived NMR reorientation time for MezSOwould be shorter by a factor of 3.7 than that given in Figure 2. In this limit of rigid methyl groups the reorientation time would become less than the Rayleigh relaxation time. The realistic situation probably lies between these two limits, considering the fact that methyl groups in the liquid state undergo hindered rotation instead of free rotation, the activation energy for the methyl rotation in MezSO is the order of 1.4 kcal/mol.ls If the proper dynamics of methyl reorientation is taken into account better agreements between the NMR results and rRa would be obtained. $he rRayvs. MezSO concentration curve, as shown in Figure 2, exhibits a maximum at about x = 0.65. This maximum point coincides with the maximum in the solution viscosity vs. concentration plot. This maximum has been interpreted as due to formation of the MezS0/2Hz0 complex. Although much experimental evidence has supported the presence of such complex, the average lifetime of the complex is not clearly known. If indeed the MezSO/HzO complex is long lived and reorients as such over the time scale of the Rayleigh scattering experiment, it is expected that the effective hydrodynamic volume of the reorienting species in pure MezSO would be different from that in the aqueous solution. Thus comparison of the effective volume in pure liquid and aqueous solution should provide the evidence for the existence of the complex formation. Although in this work we have not studied the pair correlation factor in detail experimentally, rRayis found to follow the modified SED equation Thy
=
v7 + 70 kT
(2)
where V in the absence of orientational pair correlation corresponds to the effective hydrodynamic volume for reorientation. For small molecules V is considerably less than the molar volume (or, more precisely, the volume of revolution) of the reorienting molecule, due to the fact that the dynamics of reorientation of small molecules follows more closely slip rather than stick boundary conditions.16 r orepresents a nonvanishing intercept often found in the r h y vs. q/T plot. Both V and ro may be affected by the (14)T.Keyea and D. Kivelson, J. Chern. Phys., 56, 1057 (1972). (15)H.Versmold, private communication. (16) C.-M. Hu and R. Zwanzig, J. Chern. Phys., 60,4354 (1974).
15
n I
I /' 0
0
I
.'
I
5
10
$1
1051
15
P K-'
Flgure 3. The Raylelgh relaxation time Is plotted as a functlon of q / T . The empty circles are data taken for varlous concentrations at 25 O C ; the numeral lndlcates the mole percent of Me2S0in aqueous solutions. The solid circles are the pure Me2S0 data taken at varlous temperatures. The solid triangles are data taken at x = 0.49 and empty squares are data at x = 0.66.
orientational pair correlation. Shown in Figure 3 is the plot of the Rayleigh relaxation time as a function of q / T for pure MezSO and for aqueous solutions at several Me2S0 concentrations and temperatures. It is evident that eq 2 is followed closely for pure MezSO for which we obtain a zero intercept ( T = ~ 0) and V = 10.4 cm3 mol-'. One notes that this hydrodynamic volume is considerably smaller than the specific molar volume of pure MezSO (V = 71 cm3mol-' at 20 OC'O). This result is consistent with the slip boundary conditions in spite of molecular associations arising from the dipoledipole interaction which is believed to be present in the pure M e a 0 liquid. In the presence of water the variations of rRaydo not follow the modified SED equation. Significant departure occurs at low MezSO concentrations, the 7RaY volumes for dilute solutions being somewhat greater than that calculated for the pure liquid at the same q/T. A greater rhy value corresponds to a slower rate of reorientation fluctuations, which may occur because of the specific interaction between MezSO and HzO molecules in the dilute solutions. It thus appears that reorientation dynamics of MezSO at low concentration is not governed by the hydrodynamic effect alone. The specific interaction is believed to be closely associated with the water structure. However, this nonhydrodynamic effect becomes less pronounced as the MezSO concentration is increased beyond x = 0.66. Apparently at this concentration the water structure starts to be broken down by the MezSO molecules and the reorientational process of such molecules is no longer affected by the water structure. As a result the usual hydrodynamic effect dominates the reorientational process. This behavior of MezSO in low and high concentrations is quite analogous to that found previously in the pyridine-water system.12 However, there exist important differences between the two systems. In the pyridine-water system the hydrodynamic volume changes with the water concentration, whereas in the present one, with exception of the solution with low MezSO concentration ( x < 0.3), the effective hydrodynamic volume for the reorienting species is found to be unchanged as the MezSO concentration is increased beyond x = 0.3, as shown in Figure 4. According to eq 2 the rhYq-' is proportional to the hydrodynamic volume of the reorientating species. A constancy in rhYq-' at a fixed temperature is an indication of a fmed hydrodynamic
2534
The Journal of Physical Chemistry, Vol. 85, No. 17, 198 1
I
2 t 0
Hlgashigakl et al.
I I I I I I / I I I I
0 H20
0.2
0.4 XaMe,SO)
0.6
0.0
1
M e p
Flgure 4. Plot of TRaYq-' vs. Me2S0 concentration at 25 OC.
volume. As mentioned previously if the Me2SO/H20 complex undergoes reorientation as a unit the hydrodynamic volume should be expected to increase when HzO is added to Me2S0, a situation which is found in the pyridine-water system.12 From Figure 4 it is evident that MezSO molecules in aqueous solutions reorient without carrying water molecules with them. This result is in disagreement with the assertion made by Glase18based on the deuteron relaxation time measurements. In ref 9 Packer and Tomlinson also arrived at the same conclusion that the MezSO/HzO complex reorients as a unit. Packer and Tomlinson arrived at their conclusion based on the observation that H20and M e a 0 molecules in the aqueous solutions have similar reorientational correlation time throughout the entire concentration range. However, as pointed out previously the derivation of the Me2S0 reorientation time in ref 9 is based on the assumption that methyl groups undergo free rotation in aqueous solutions, and thus the reorientation times for Me2S0 are probably too large. On the other hand, the reorientation time deduced for HzO are probably trustworthy as these values are obtained through a curve-fitting procedure to the observed T1 and T2data. Therefore, based on Figure 2, the reorientation time for Me2S0 is probably shorter than that for H20. Therefore, an alternate picture for the Me2S0 reorientational process in aqueous solution is called for. Although there is no doubt that the Me2SO/H20complexes exist in the aqueous solutions the lifetimes of the complexes are believed to be short compared with T& (=lo-" s at 25 "C). The short lifetimes of Me2SO/H20complexes are consistent with the existing evidence that associated complexes in both pure MezSO and in Me2SO-water solutions are thermolabile. In fact, Parker has provided evidence that the associated species in pure Me2S0 probably break down between 40 and 60 "C or when a proton donor is added.l' Cowie and Toporowskilo have also concluded that a Me2SO/H20 hydrate such as the Me2S0/2H20 complex is thermolabile and is stable only at low temperature. Additional support for such short-lived water/Me2S0 complexes comes from the kinetic study of alkaline hydrolysis reactions in various Me2SO-water mixtures.18 Therefore, we conclude that at 25 OC the Me2SO/H20complex probably exists only in a time longer than s, in order to be consistent with the observed additional structure in the IR absorption spectra in the ,~~~ 1000-1100-cm-' (Le., the S-0 stretching) r e g i ~ n but shorter than s, to be consistent with the present Rayleigh scattering results. At low Me2S0 concentrations, the Me2S0 molecule perturbs the motion of H20 molecules (17)A. J. Parker, Q.Reu. Chem. SOC.,16,163 (1962). (18)E.Tommila and M. Murto, Acta Chem. Scand., 17,1947(1963).
Flgure 5. Arrhenius plots of aqueous solutlons.
T~~
for Me,SO In pure llquld and two
cooperatively and causes a rigidifying effect on the water structure, as clearly shown by the sharpening of the intermolecular vibration band in the neutron-scattering spectra of the aqueous solutions relative to that of pure water.3 This cooperative interaction which leads to the sharpening effect is believed to involve direct bonding of Me2S0to the water structure.4 As Me2S0 molecules reorient the rigidifying effect in the hydrogen-bonded H 2 0 molecules will be relaxed and this facilitates the reorientation of the H20 molecules. Thus we do not expect MezSOand H20molecules to reorient as a unit; more likely the reorientation time of Me2S0 is shorter than that of HzO in solutions with low Me2S0 concentration. This pattern appears to continue into solutions of high Me2S0 contents, and over the entire concentration range the Me2S0 molecule reorients faster than the H 2 0 molecule does. The activation energy for the reorientation of the M e a 0 molecules in pure liquid and in aqueous solution can be obtained by measuring T~ as a function of temperature keeping a fixed Me2S0 concentration. Shown in Figure 5 are the Arrhenius plots of T~~ given by the equation 7 h Y= 7OhYexP(E,/RT)
(3)
The activation energy (E,) for pure liquid is found to be equal to 18.4 kJ/mol. This is considerably greater than the activation energy for reorientation in other liquids consisting of molecules of similar size. At x = 0.66 the activation energy E, is equal to 10.6 kJ/mol and at x = 0.49 E, is equal to 9.3 kJ/mol. The results for the aqueous solutions can only be taken as tentative due to the insufficient number of data points at these concentrations, but the smaller activation energies obtained for these solutions as compared to that of pure liquid is a definite result. The activation energy for 7hy is also consistent with that obtained for viscosity." However, E, for x = 0.66 is about a factor of two smaller than that deduced from the temperature dependence of the NMR Me2S0 reorientation timeaQIt is quite probable that the NMR result is in error for the reason mentioned above. The larger activation energy found in pure liquid may lead one to conclude that the dipolar interaction imposes a greater barrier for reorientation than the DMSO-H20 hydrogen-bonding interaction. However, the activation energy for viscous flow found for x = 0.8 is lower than that for x = 0.6 by 15%.11 In view of this, it is not clear whether the dipolar interaction indeed imposes a greater barrier or not. Clearly,
J. Phys. Chern. 1081, 85,2535-2542
it is a combination of the dipolar and the hydrogenbonding interaction which jointly plays the complex role in determiing the energy barrier for the molecular reorientation and for the process of viscous flow.
Summary and Conclusion We have studied the reorientational process of Me2S0 in pure liquid and in aqueous solutions by depolarized Rayleigh scattering. The depolarized Rayleigh scattering spectra of aqueous solution reflect mainly the orientation fluctuation of Me2S0 molecules; the contribution of scattering from water molecules is found to be negligible. From the spectral line width measurements we have obtained T~~ as a function of temperature in solution of various Me2S0 concentrations. The activation energy for reorientation of MezSOin pure liquid is found to be greater than that in solution. In the pure Me2S0 liquid, ~b,, is found to follow the SED equation with the hydrodynamic volume for reorientation being considerably smaller than the molecular volume. This result suggests that despite the strong intermolecular interaction slip boundary conditions are appropriate for describing the reorientation process of the pure Me2S0 liquid. In aqueous solutions, T~ values are found not to follow the SED equation and the data suggest the presence of a nonhydrodynamic effect. The nonhydrodynamic effect is found to be significantly greater in solutions with low Me2S0 concentration. This result lends support to the conclusion derived from the neutron- and X-ray-scattering studies that in aqueous solutions of low M e a 0 content, the M e a 0 molecules tend
2535
to regidify the water structure. The water structure, however, is found to break down at high Me2S0 concentration. From the T~ vs. q/T plot we have obtained the hydrodynamic vofume for the Me2S0 reorientation as a function of Me2S0 concentration. The hydrodynamic volume of Me2S0 in aqueous solution is found to be quite similar to that found in the pure liquid. This result suggests that in spite of the Me2SO/H20complex formation in solutions, the lifetimes of Me2SO/H20complexes are s, which corresponds to the characteristic shorter than time in Rayleigh scattering. Thus, our result is in apparent disagreement with the conclusion previously derived from the NMR relaxation time study that the Me2S0 molecule and H20 molecules in aqueous solution reorient together. The discrepancy between the two results is believed to be due to the invalid approximation introduced in the analysis of NMR relaxation time data. The present work illustrates the usefulness of the depolarized scattering for the study of the molecular reorientation process and interaction in the liquid state. Rayleigh scattering is a direct technique and is free from complicated assumptions which are needed in the analysis of experimental data obtained in other techniques. Acknowledgment. Acknowledgment is made to the donors of Petroleum Research Fund, administered by the American Chemical Society, for support of this research. Travel support to D.H.C. by the Danish Natural Science Research Council to Salt Lake City is also acknowledged.
Laser Intensity Induced Nonradiatlve Processes in Molecules Claude Needham and William Rhodes” Institute of Molecular Biophysics and Department of Chemistry, FlorMa State University, Tallahassee, Florida 32306 (Received: February 18, 198 1; In Final Form: April 28, 198 1)
The effects of the coupling strength (Le., the Rabi frequency) of a coherent radiation mode interacting with a multilevel molecule are considered. A doorway-state basis is used in which the radiative doorway state, carrying all of the radiative interaction strength from the ground (initial)state, is coupled by intramolecular (nonradiative) interaction to other excited states. The resulting coupling scheme involves an effective Hamiltonian formulation in an extended rotating basis. Quantitative results from the theory are obtained by computer simulation. It is shown how variation of the laser coupling strength can modify the dynamics of nonradiative transitions, thereby producing (1) decoupling of radiationless decay, (2) enhancement of radiationless decay, (3) selectivity of photophysical and photochemical processes, or (4)laser-induced isolation of states, depending on the conditions of the system. The case of two states coupled through a common manifold of states and the conditions for biexponential decay are also considered.
Introduction The theory of nonlinear optical effectsin multilevel systems has been severely limited by the intractable mathematics involved. The two-level system, on the other hand, has been well studied and gelds for arbitrarily intense excitation within the dipole approximation and rotating wave appr~ximation.l-~
Although there has been some success with detailed solutions for three-level systems,6 success with truly systems has been sporadic.”6 In this research we study multilevel coupling models with the primary objective of discerning laser-molecule coupling dynamics qualitatively different from the oscillation-dissipation characteristics of two-level systems. We
(1) J. H. Eberly and P. Lambropoulos, “Multiphoton Processes”, Wiley, New York, 1978. (2) J. I. Steinfeld, “Molecules and Radiation”, Harper and Row, New York, 1974. (3) J. D. Macomber, “The Dynamics of Spectroscopic Transitions”, Wiley-Interscience, New York, 1975.
(4) L. Allen and J. H. Eberly, “Optical Resonance and Two-Level Atoms”, Wiley, New York, 1975. (5) J. V. Moloney and W. J. Meath, Mol. Phys., 30, 171 (1976). (6) J. R. Ackerhalt, P. L. Knight, and J. H. Eberly, Phys. Reu. Lett., 30, 456 (1973). (7) M. Quack, J. Chern. Phys., 69, 1282 (1978). (8) S. Mukamel, J. Chern. Phys., 71, 2012 (1979).
0022-3654/81/2085-2535$01.25/0
0 1981 American Chemical Society
