Theoretical Investigation of the Temperature Dependence of the Fifth

J. Phys. Chem. B 2006, 110, 3773-3781

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Theoretical Investigation of the Temperature Dependence of the Fifth-Order Raman Response Function of Fluid and Liquid Xenon† Russell DeVane Center for Molecular Modeling, Department of Chemistry, UniVersity of PennsylVania, Philadelphia, PennsylVania 19104

Christina Kasprzyk and Brian Space* Department of Chemistry, UniVersity of South Florida, 4202 East Fowler AVenue, SCA400, Tampa, Florida 33620-5250

T. Keyes Department of Chemistry, Boston UniVersity, Boston, Massachusetts 02215 ReceiVed: September 16, 2005; In Final Form: December 15, 2005

The temperature dependence of the fifth-order Raman response function, R(5)(t1,t2), is calculated for fluid xenon by employing a recently developed time-correlation function (TCF) theory. The TCF theory expresses the two-dimensional (2D) Raman quantum response function in terms of a two-time, computationally tractable, classical TCF. The theory was shown to be in excellent agreement with existing exact classical MD calculations for liquid xenon as well as reproducing line shape characteristics predicted by earlier theoretical work. It is applied here to investigate the temperature dependence of the fifth-order Raman response function in fluid xenon. In general, the characteristic line shapes are preserved over the temperature range investigated (for the reduced temperature points T* ) 0.5, 1.0, and 2.0); differences in the signal decay times and a large decline in intensity with decreasing temperature (and associated anharmonicity) are observed. In addition, there are some signature features that were not observed in earlier results for T* ) 1. The most dramatic difference in line shape is observed for the polarization condition, xxzzxx, that shows a vibrational echo peak. In contrast, the fully polarized signal changes mainly in magnitude.

1. Introduction Recently, a computationally tractable classical time-correlation function (TCF) theory was proposed for the twodimensional (2D) fifth-order Raman quantum response function. The method was tested against extant exact theoretical results and was shown to quantitatively reproduce that result as well as reproduce characteristics predicted by earlier theoretical work.1-5 The driving force behind the development of theories for such complicated spectroscopic techniques is to gain insight into the complex spectra that result from such experiments. These techniques yield information that is exceedingly difficult to interpret and lend themselves to theoretical investigation. The inability of techniques based on the third-order response function, such as optical Kerr effect spectroscopy (OKE), to identify the origins of line shape broadening put an emphasis on the development of higher order techniques, for example, fifth-order Raman spectroscopy.6 The fifth-order Raman experiment has received a lot of attention over the last several years. However, progress has been difficult as a result of the weak signal strength compounded with the issue of contamination from lower order experiments (third-order cascading).7-13 Much theoretical attention has been given as well.1,3-5,14-23 The need for a sound computationally tractable theory is clear; however, †

Part of the special issue “Michael L. Klein Festschrift”. * To whom correspondence should be addressed. E-mail: space@ cas.usf.edu.

this presents a considerable theoretical challenge. The quantum mechanical nature of the fifth-order response function makes it impractical to calculate exactly. The classical limit is challenging as well; replacing the commutators with Poisson brackets is an approach that is only practical for relatively small simple systems.4,5,24,25 On the other hand, a classical TCF theory, similar to the approach successful in describing linear and related thirdorder nonlinear spectra, allows the description of the fifth-order response function in terms of fully anharmonic molecular dynamics (MD) calculations supplemented by a suitable spectroscopic (dipole and polarizability) model. Despite the fact that multidimensional nonlinear spectra cannot be expressed exactly in terms of classical TCFs,21 we proposed approximate TCF theories of nonlinear spectroscopy that are computationally tractable for complex molecular condensed phase systems.23,26 We have demonstrated the capability of these theories to accurately describe both intermolecular and intramolecular vibrational spectra for both the fifth-order Raman response1,23 and the third-order dipole response (responsible for 2D infrared (2DIR) spectra).26 Included in this demonstration of effectiveness were results in which R(5) xxxxxx(t1,t2) was calculated for liquid xenon using the TCF theory and compared to exact results previously reported by Ma and Stratt.1,4 Exact calculations are limited as a result of the computational burden.4,27,28 It was demonstrated that the TCF theory of R(5)(t1,t2) quantitatively reproduced the 2D Raman spectrum of liquid xenon, providing support for its further

10.1021/jp055275l CCC: $33.50 © 2006 American Chemical Society Published on Web 01/11/2006

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applications to more complex systems. It was also shown to produce line shape characteristics predicted in an earlier theoretical analysis.1-3 In this paper, results are presented for the fifth-order response of fluid xenon at various temperatures. As in previous papers,2 several polarization conditions are also reported here to demonstrate the rich source of physical information provided by nonlinear experiments. In section 2, the development of the TCF theory of R(5)(t1,t2) will be summarized; details are provided in our earlier works.1,2,23 Section 3 will discuss technical aspects of the MD simulations and the polarizability model employed. Section 4 will briefly discuss the similarities and differences of the (exact) Poisson bracket approach vs the (approximate but simpler) TCF approach. Results from the application of the TCF theory to fluid xenon will be discussed in section 5. Finally, conclusions will be given in section 6.

with the classical two-time TCF given by

gR(t1,t2) ) 〈∆Π(0)∆Π(t1)∆Π(t1 + t2)〉

(7)

Here, ∆Π ) Π - 〈Π〉, and the TCF is written in terms of the correlated polarizability fluctuations.23 Equations 6 and 7 represent our TCF theory of R(5)(t1,t2). For the current investigations, primarily of intermolecular dynamics, only the classical limit will be considered here. Note, while our reference system gives the lowest order response for a harmonic system, anharmonic systems can give a response for a system with polarizability only expanded to the first order. These effects appear to be important for some molecular systems25 but not in simple atomic systems. Note, the present theory does include fully anharmonic dynamics in calculating the fifth-order signal, and using a harmonic reference system would include the anharmonic pathways albeit not with the proper weighting.23

2. TCF Theory The development of the TCF theory of the fifth-order Raman experiment begins with the quantum mechanical expression for the electronically nonresonant fifth-order polarization where the material system is described by the response function22,27 (5) RR,β,γ,δ,,φ (t1,t2) ) (i/p)2Tr{ΠRβ(t1 + t2)[Πγδ(t1),[Πφ(0),F]]} (1)

In eq 1, F ) e-βH/Q for a system with Hamiltonian H and partition function Q at reciprocal temperature β ) 1/kT and k is Boltzmann’s constant; Tr represents a trace, square brackets denote commutators, Π is the system polarizability tensor, and the Greek superscripts denote the elements and thus polarization condition being considered. By expanding the commutators in eq 1, and taking the classical limit, the response function can be rewritten as

R(5)(t1,t2) ) -β2 ∂2/∂t12 gR(t1,t2) - 2

iβ ∂/∂t1 gI(t1,t2) (2) p

The functions gR and gI represent the real and imaginary parts, respectively, of the TCF, g; g(t1,t2) is given by

g(t1,t2) ) 〈P(t2)Π Π*(t1)〉

(3)

where the angle brackets represent an equilibrium average. Although no exact analytic relationship is possible for the real and imaginary parts of a two-time TCF, an approximate relationship between the real and imaginary parts of the quantum TCF can be found for a harmonic system with the polarizability expanded to second order (which we refer to as a reference system because it is used to relate the real and imaginary parts of the TCF) in the harmonic coordinate, Q

Π ) Π0 + Π′Q + 1/2 Π′′Q2

(4)

This leads to a relationship between the real and imaginary parts which is given by

gI(ω1,ω2) ) tanh(-βp(ω1/4 + ω2/2))gR(ω1,ω2)

(5)

where the subscripts denote the Fourier transform of the real or imaginary parts of the TCFs both of which are real functions of frequency. Through the use of this approximate relationship, the fifth-order response function, in the classical limit (pω , kT), takes the form

R(5)(t1,t2) ) -β2/2[∂2gR(t1,t2)/∂t12 - 2 ∂2gR(t1,t2)/∂t1∂t2] (6)

3. Models and Methods For the results presented here, microcanonical MD simulations were performed for neat metastable and stable liquid and supercritical fluid xenon systems consisting of 108 atoms. MD was used to generate roughly 5 000 000 configurations with a 10 fs time step. These configurations were then used to calculate the TCFs. The atoms interacted via a Lennard-Jones pair potential with σ ) 4.099 Å and ) 222 K. Three different state points were compared here: these include kT/ ) 0.5 (111 K), kT/ ) 1.0 (222 K), and kT/ ) 2.0 (444 K) all with the same reduced density, Fσ3 ) 0.8; the lowest temperature considered corresponds to a two-phase solid/gas when equilibrated. However, analysis of the radial distribution function and instantaneous normal mode density of states indicates that it is a (long-lived, metastable) liquid. The medium and highest temperatures represent a liquid and a supercritical fluid.29 The 222 K system was prepared by starting with an face centered cubic (fcc) lattice and melting the system. The system was then equilibrated to producing a stable liquid at 222 K. The 111 K and the 444 K systems were prepared by starting with the equilibrated 222 K system and cooling and heating the systems, respectively. No polarization forces were explicitly included in the MD calculations. However, full many-body polarization effects were included in the polarizability calculations (used in the TCF) via a point atomic polarizability approximation (PAPA) as well as a first-order approximation (FODID).30 The expression for the effective polarizability, R˜ i, for site (atom) i is given by n

R˜ i ) Ri + Ri

T(rij)R˜ j ∑ j*i

(1)

where Ri is the isotropic point polarizability for site i and T(rij) is the dipole field tensor between sites i and j in a system with n sites. For the calculations performed here, the point polarizability of fluid xenon is taken to be R ) 4.11 Å3.4 The total system polarizability is given by summing the effective polarizabilities for all sites n

Π)

R˜ i ∑ i)1

(2)

The two forms of the many-body polarizability model used in our calculations include truncating eq 1 to terms first order in T(rij): the first-order dipole-induced dipole (FODID) and the exact infinite-order evaluation of the dipole-induced dipole

Fifth-Order Raman Response Function of Xenon

J. Phys. Chem. B, Vol. 110, No. 8, 2006 3775

equations (MBP); eq 1 gives the equations explicitly for the polarizability (needed to calculate the TCF), and multiplying through by the field yields the induced dipoles. From a computational standpoint, the MBP model requires iteratively solving eq 1 or a matrix inversion (matrix inversion was employed in the present calculations). The FODID only requires a single iteration of eq 1. Althought exact classical calculations of R(5)(t1,t2) within the MBP model have been performed for other systems,25 only calculations within the FODID approximation have been performed for xenon to date.4 Consequently, exact calculations of R(5)(t1,t2) for xenon, using finite field methods,17,28,31,32 within the MBP model, is an area of active pursuit for the authors. Time derivatives of the TCFs were evaluated using numerical approximation methods.33 For the evaluation of the derivatives at the boundaries (e.g., along t2 ) 0), it was necessary to use lower-order algorithms, making the results less precise. This is a result of calculating only the positive quadrant (+t1,+t2) of the TCF. It should be noted here that although the effects of including higher-order DID interactions via the MBP model are important, a more realistic model that accounts for short-range interactions could be of equal importance especially at higher temperatures.34,35 Finally, the instantaneous normal mode (INM) density of states (DOS) are used below to assess the vibrational spectra of fluid xenon for the temperatures investigated here. The DOS for each temperature was calculated using 1000 statistically independent configurations of 108 Lennard-Jones particles from microcanonical MD. By the use of the configurations, the system’s force constant matrix was calculated and diagonalized.36-38 The square root of the eigenvalues represent the characteristic frequencies that were sorted into bins to produce the DOS. The imaginary frequencies are shown on the negative abscissa in the figures.

∂2 〈Π(-t1 - t2)Π(-t2)Π(0)〉 ∂t1∂t2

4. TCF vs Poisson Brackets As noted in earlier work, the current TCF theory of the fifthorder Raman response function represents an approximation of the Poisson bracket contribution to the classical response function.23 Starting with eq 6 written as

R(5) )

[

-β2 ∂2 〈Π(0)Π(t1)Π(t1 + t2)〉 2 ∂t 2 1 2

]

∂2 〈Π(0)Π(t1)Π(t1 + t2)〉 (1) ∂t1∂t2

2

(2)

where we have used the stationary property of the TCFs to rearrange the time arguments. Applying the derivatives of the TCF and then rewriting in terms of positive time arguments gives

∂2 〈Π(-t1)Π(0)Π(t2)〉 ) 〈Π ¨ (0)Π(t1)Π(t1 + t2)〉 ∂t12

Applying the derivatives gives

∂ [-〈Π˙ (-t1 - t2)Π(-t2)Π(0)〉 ∂t1 ¨ (-t1 - t2)Π(- t2)Π(0)〉 + 〈Π(-t1 - t2)Π˙ (-t2)Π(0)〉] ) 〈Π 〈Π˙ (-t1 - t2)Π˙ (-t2)Π(0)〉 (5) Rewriting the function in terms of positive time arguments, again using the stationary property of TCFs, gives

∂2 〈Π(0)Π(t1)Π(t1 + t2)〉 ) 〈Π ¨ (0)Π(t1)Π(t1 + t2)〉 + ∂t1∂t2 〈Π˙ (0)Π˙ (t1)Π(t1 + t2)〉 (6) We can now rewrite eq 6 as

R(5) )

β2 〈Π ¨ (0)Π(t1)Π(t1 + t2)〉 + β2〈Π˙ (0)Π˙ (t1)Π(t1 + t2)〉 2 (7)

where it is now in a form similar to that reported by Saito et al.18

R(5) ) β2〈Π˙ (0)Π˙ (t1)Π(t1 + t2)〉 β〈Π(t1 + t2){Π(t1), Π˙ (0)}〉 (8) The term β2〈Π˙ (0)Π˙ (t1)Π(t1 + t2)〉 is present in both approaches leaving us with an apparent approximation to the Poisson bracket term, -β〈Π(t1 + t2){Π(t1),Π˙ (0)}〉, given by ¨ (0)Π(t1)Π(t1 + t2)〉. β2/2〈Π With the methods employed in our calculations, it is straightforward to calculate the Poisson bracket approximation, 〈Π ¨ (0)Π(t1)Π(t1 + t2)〉, using ∂2/∂t12 〈Π(0)Π(t1)Π(t1 + t2)〉. The only contribution from this derivative combination is the approximation to the Poisson bracket as shown in eq 3. As seen in Figure 1, the Poisson bracket term for 222 K has a sizable contribution, and apparently (in liquid xenon within the FODID model where the TCF result showed quantitative agreement), this TCF is an excellent approximation to the Poisson bracket. Further, formal exploration is called for the approximation of a Poisson bracket expression via a TCF. 5. Results for Fluid and Liquid Xenon

the first term can be rewritten as

∂ 〈Π(-t1)Π(0)Π(t2)〉 ∂t12

(4)

(3)

where the dot indicates a time derivative, ∂/∂t Π ) Π˙ , and the double dots indicate second derivatives ∂2/∂t2 Π ) Π ¨ . In a similar fashion, the second term of eq 1 is rewritten as

Results for the squared fully polarized fifth-order response 2 function, |R(5) xxxxxx(t1,t2)| , within the FODID approximation, are shown in Figure 2 for the temperatures 111 K (top), 222 K (middle) (previously reported),1,2 and 444 K (lower). The signals are nonzero along t1 ) 0 for certain values of t2 and zero along t2 ) 0. For 111 K, the signal is characterized by a broadened peak (relative to 222 and 444 K). Along t2, the signal decays to zero by 0.9 ps and by 0.6 ps along t1. The signal for 222 K shows earlier rise times compared to 111 K and less broadening of the signal. However, the 222 K signal is characterized by a relatively slow decay along t2 that is nonzero beyond the 1.0 ps shown here; along t1, the signal is zero by approximately 0.6 ps; only the 222 K system (a fluid just over the critical temperature) shows the characteristic t2 plateau that was seen in other molecular systems. An interesting feature of the 444 K signal is the small peak near the origin. This feature is barely evident in the 222 K signal. The 444 K signal lacks the relatively


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Figure 1. Upper figure shows the contribution to the fifth-order response function from the t1 derivatives, ∂2/∂t12 〈Π(0)Π(t1)Π(t1 + t2)〉, that represent an approximation to the Poisson bracket contribution. The lower figure shows ∂2/∂t1t2 〈Π(0)Π(t1)Π(t1 + t2)〉. The data were calculated for the 222 K system.

slow decay time seen in the 222 K signal but retains the dominant peak seen in both the 111 and 222 K signals. The decay time along t1 is roughly the same in the 444 K signal dying out by t1 ) 0.6 ps. Another striking feature is the temperature dependence of the overall magnitude; while the line shapes show relatively modest changes, the intensity is falling almost 2 orders of magnitude over the temperature range and is growing faster as the temperature increases. This is consistent with the fifth-order Raman signal being sensitive to dynamical anharmonicity.4,19,20,25 2 Figure 3 shows time slices of |R(5) xxxxxx(t1,t2)| along t2 with t1 ) 0 for the temperatures 111 K (dashed), 222 K (dotted), and 444 K (solid). The peaks are all (including all time slices presented in this paper) normalized to unity for the sake of comparison. The inset shows the details of the slices for 222 and 111 K before 0.175 ps (444 K was intentionally left out of the inset). The inset shows the existence of the small peak that occurs between 0 and 0.15 ps not only for the 222 K signal as suggested above but also for the 111 K. The dominant peak shifts out in time with decreasing temperature, while the smaller peak near the origin remains fixed in time regardless of temperature. It is not surprising the dominant peak shifts to shorter times with increasing temperature because the (normalized) vibrational DOS is moving to higher frequenciessthus, enhancing the short time signal. The origin of the smaller temperature-independent feature is less obvious. 2 Figure 4 shows diagonal slices (t1 ) t2) of |R(5) xxxxxx(t1,t2)| for the temperatures 111 K (dashed), 222 K (dotted), and 444 K (solid). All three signals decay to zero well within the time range shown (less than 0.5 ps). The inset shows the region between 0 and 0.09 ps to show the magnitude of the small peaks near the origin for 111 and 222 K. For the diagonal time slices, the peaks are not evenly spaced as in the slice along t1 ) 0 (Figure 3); the signals for 444 and 222 K are nearly superimposed, while the signal for 111 K is shifted significantly in time. Again, as

(5) Figure 2. Results for |Rxxxxxx (t1,t2)|2, using the FODID approximation for xenon: 111 K (upper), 222 K (middle), and 444 K (lower).

Figure 3. Time slices of the FODID results for the fifth-order response (5) function. Shown are slices along t2 with t1 ) 0, |Rxxxxxx (t1 ) 0,t2)|2, for the temperatures 111 K (dashed), 222 K (dotted), and 444 K (solid).

in the slice along t1 ) 0, the small peaks near the origin do not shift in time with increasing temperature. 2 Figure 5 compares results for |R(5) xxxxxx(t1,t2)| using the MBP models (they are predictions of fifth-order experiments). The spectra shown include 111 K (upper), 222 K (middle), and 444 K (lower). All three temperatures display very similar structures. The magnitude of the spectra increases with increasing temperature as is expected given that the anharmonicity of the


Figure 4. t1 ) t2 time slices of the FODID results for the fifth-order (5) response function, |Rxxxxxx (t1 ) t2)|2, for the temperatures 111 K (dashed), 222 K (dotted), and 444 K (solid).


Figure 6. Time slices of the MBP results for the fifth-order response (5) function. Shown are slices along t2 with t1 ) 0, |Rxxxxxx (t1 ) 0,t2)|2, for the temperatures 111 K (dashed), 222 K (dotted), and 444 K (solid).

Figure 7. t1 ) t2 time slices of the MBP results for the fifth-order (5) response function, |Rxxxxxx (t1 ) t2)|2, for the temperatures 111 K (dashed), 222 K (dotted), and 444 K (solid).

(5) Figure 5. Results for |Rxxxxxx (t1,t2)|2 in xenon, using the MBP model 111 K (upper), 222 K (middle), and 444 K (lower).

Lennard-Jones systems increases with increasing temperature. The growth in intensity with temperature is almost linear in contrast with the FODID result. Considering the 111 K result, the feature appearing approximately at t1 ) 1.0 ps and t2 ) 0.5 ps appears to be real and not an artifact of insufficient averaging. A peak in the same approximate location is discernible when that region of the spectra is highlighted for the other temperatures. Not only do the MBP results display less temperature dependence for the fully polarized condition than the FODID results but they also exhibit similar decay times along both time

axes and the diagonal. In addition, the peaks display a smaller shift in time with decreasing temperature. This is seen in Figure 2 6 which shows time slices of |R(5) xxxxxx(t1,t2)| along t2 with t1 ) 0 for the temperatures 111 K (dashed), 222 K (dotted), and 444 K (solid) within the MBP model. Diagonal slices (t1 ) t2) of 2 |R(5) xxxxxx(t1,t2)| are shown in Figure 7 for the temperatures 111 K (dashed), 222 K (dotted), and 444 K (solid) using the MBP model. As in the FODID results, we see a smaller difference in the location of the 222 and 444 K signals compared to the 222 and 111 K. Figure 8 presents the semi-polarized polarization conditions using the MBP model.15 Included are the spectra for 2 |R(5) zzxxxx(t1,t2)| at 111 K (upper), 222 K (middle), and 444 K (lower). All three signals display a dominant ridge along t2 with t1 ) 0. The ridge falls off faster with increasing t1 for higher temperatures. Also, the existence of a small, flat, short-lived section along the top of the ridge in 111 K does not appear in the signals for 222 and 444 K. This polarization condition seems to enhance the pathway which contains a cosine term that is dependent only on t1 (cos(ωt1)) as suggested by Fourkas et al.3 This represents one of two pathways, both identified as R(π(1)π(1)π(2)), that are enhanced by this polarization condition (the pathway notation is given explicitly in the references).1-3 It is unclear at this point what role the other contribution with the term cos[ω(t1 + 2t2)] plays in this polarization result.3 However, it would appear the signal is dominated by the pathway containing the cos(ωt1) term.


(5) Figure 8. Results for |Rzzxxxx (t1,t2)|2 in xenon using the MBP model 111 K (upper), 222 K (middle), and 444 K (lower).

2 Figure 9 shows |R(5) xxzzxx(t1,t2)| using the MBP model at 111 K (upper), 222 K (middle), and 444 K (lower). This result was previously reported for 222 K and is shown here again for comparison with new results for 111 and 444 K.1,2 The most notable feature of all three figures is the echo peak that exists along the diagonal at all three temperatures. The echo peak implies that an intermolecular mode, excited at time zero, is still oscillating at the time of the measurement one period later. The other feature present at all three temperatures is the peak near the origin. The relative magnitude of the echo and origin peaks vary with temperature: the echo peak dominating the spectrum at 111 K and the origin peak dominating the spectrum 2 at 444 K. Figure 10 shows slices of |R(5) xxzzxx(t1,t2)| along the diagonal with the inset of each plot showing the INM DOS for xenon. The main echo peaks are shifted closer to the origin with increasing temperature. All three temperatures decay back to the baseline at approximately the same time. Comparing these observations with the INM DOS insets, we see the positive frequency peaks all have a maximum at the same frequency (roughly 10 cm-1). This frequency corresponds to a time of approximately 0.56 ps (within a harmonic model) that is roughly the time at which the signal has decayed to zero for all three temperatures (with the exception of 111 K in which the signal decays substantially to reach a baseline before decaying to zero). Finally, the shifting of the width of the INM DOS to higher frequencies with higher temperatures corresponds to earlier rise

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(5) Figure 9. Results for |Rxxzzxx (t1,t2)|2 in xenon using the MBP model 111 K (upper), 222 K (middle), and 444 K (lower).

times in the time plots of the diagonal slices which is consistent with what is observed. A direct correspondence is found as well for the peak near the origin (that appears prior to the echo peak) in the time plots and the tail of the INM DOS. Using the frequencies 60, 80, and 100 cm-1 to represent the point at which the INM DOS has decayed to zero for 111, 222, and 444 K, respectively, corresponds to the times 0.088, 0.066, and 0.053 ps. These times match the location (along the diagonal) of the peaks that appear prior to the echo peaks. Although the origin of these peaks is unclear, since the magnitude of the peaks grows with increasing temperature (and increasing anharmonicity of the system), this would suggest the peaks arise from anharmonicity in the system. The magnitude of the echo peak does not increase in magnitude as much suggesting that it does not arise from anharmoncity or to a lesser extent. This demonstrates the power of multidimensional spectroscopy in distinguishing line shape contributions. Figure 11 presents the remaining semi-polarized result of the 2 form xxxxzz.15 |R(5) xxxxzz(t1,t2)| is shown for 111 K (upper), 222 K (middle), and 444 K (lower). Again, the feature present in the 111 K at approximately t1 ) 1 ps and t2 ) 0.5 ps appears to be real and not a result of poor averaging. This might suggest this signature is not anharmonic in nature while the other significant features in this result are anharmonic. At lower temperatures, as the anharmonicity of the system decreases, such features would become more apparent. Because it has been suggested the pathway that would be enhanced by this polariza-



(5) Figure 11. Results for |Rxxxxzz (t1,t2)|2 in xenon using the MBP model 111 K (upper), 222 K (middle), and 444 K (lower).

(5) Figure 10. Slices of |Rxxzzxx (t1,t2)|2 in xenon along the diagonal (t1 ) t2) with the inset representing the INM DOS: 111 K (upper), 222 K (middle), and 444 K (lower).

tion geometry is zero in some models (e.g., a harmonic system), it is unclear if this result represents a mixture of other pathways or if the signal here is a result of a single pathway that is enhanced.3 Figure 12 presents the depolarized polarization condition of 2 the form xxxzxz.15 |R(5) xxxzxz(t1,t2)| is shown for 222 K (upper) and 444 K (lower). The signal-to-noise ratio is too low for the 111 K spectrum making it extremely difficult to attain a wellaveraged signal that obeys the limiting behavior required by the TCF theory, that is, not zero at the origin or along t2 ) 0. This is surprising because the higher temperature line shapes have magnitudes similar to the other polarization conditions;

(5) Figure 12. Results for |Rxxxzxz (t1,t2)|2 in xenon using the MBP model 222 K (upper) and 444 K (lower).

this may indicate the depolarized condition is especially sensitive to anharmonic dynamics (as might be expected given that it is


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(5) Figure 13. Results for |Rxzxxxz (t1,t2)|2 in xenon using the MBP model 222 K (upper) and 444 K (lower).

dominated by off diagonal polarizability elements). The signals appear to arise from a combination of pathway contributions with features present in the xxxxxx, xxxxzz, and zzxxxx (ridge) spectra. Figure 13 presents the depolarized polarization condition of 2 the form xzxxxz.15 |R(5) xzxxxz(t1,t2)| is shown for 222 K (upper) and 444 K (lower). Again, the signal-to-noise ratio is too low to attain a sufficient spectrum for the 111 K. This condition would be expected to highlight the echo signal pathway as in the xxzzxx result. However, the echo signal is not obvious in these results. The plots do suggest a slight elongation along the diagonal. Note this depolarized polarization is not as effective at distinguishing the pathway contributions.2 Figure 14 presents the remaining depolarized polarization 2 condition of the form xzxzxx.15 |R(5) xzxzxx(t1,t2)| is shown for 111 K (upper), 222 K (middle), and 444 K (lower). The 111 K signal, although noisy, does obey the limiting vanishing behavior expected at the origin and along t2 ) 0. It is interesting to note the ridge that runs antidiagonal in the 111 K signal is also evident in the 222 K signal to a smaller extent. This feature is not evident in the 444 K signal again suggesting that its origin may be related to anharmonicity of the system. Other than this feature, all three temperatures are similar to the fully polarized results. 6. Conclusions It is interesting to note the wealth of information provided, via the fifth-order Raman response, in a system as simple as fluid/liquid xenon as shown here. The combination of temperature-dependent and -independent features in the spectra is reassuring. This suggests multidimensional spectroscopy indeed does have the ability to provide new insight into even more complex systems. However, it comes at the cost of interpreting such complex spectra. The next step in understanding the spectra is interpreting the pathways and understanding what information they provide about the system. It is clear that multidimensional spectroscopy allows the separation of this information with the

(5) Figure 14. Results for |Rxzxzxx (t1,t2)|2 in xenon using the MBP model 111 K (upper), 222 K (middle), and 444 K (lower).

use of polarization conditions. The results demonstrate there are features that are both highly temperature dependent and others that are temperature independent. Further, different parts of the line shape are more or less sensitive to anharmonicity vs the presence of (sufficiently long-lived) nearly harmonic modes. TCF methods appear to be effective and have also been applied to R(3)(t1,t2) yielding a practical TCF theory of the 2DIR experiment.26 Acknowledgment. The research at USF was supported by an NSF grant (No. CHE-0312834) and a grant from the Petroleum Research Fund to Brian Space. The authors acknowledge the use of the services provided by the Research Oriented Computing Center at USF. The authors also thank the Space (Basic and Applied Research) Foundation for partial support. The research at BU was supported by an NSF grant (No. CHE0090975) to T. Keyes. References and Notes (1) DeVane, R.; Ridley, C.; Keyes, T.; Space, B. Phys. ReV. E 2004, 70, 50101. (2) DeVane, R.; Ridley, C.; Keyes, T.; Space, B. Application of a Time Correlation Function Theory for the Fifth Order Raman Response Function I: Atomic Liquids. J. Chem. Phys. 2005, in press. (3) Murry, R. L.; Fourkas, J. T. J. Chem. Phys. 1997, 107, 9726. (4) Ma, A.; Stratt, R. M. J. Chem. Phys. 2002, 116, 4962-4971. (5) Ma, A.; Stratt, R. M. Phys. ReV. Lett. 2000, 85, 1004-1007. (6) Tanimura, Y.; Mukamel, S. J. Chem. Phys. 1993, 99, 9496-9511.

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J. Phys. Chem. B, Vol. 110, No. 8, 2006 3781 (23) DeVane, R.; Ridley, C.; Keyes, T.; Space, B. J. Chem. Phys. 2003, 119, 6073-6082. (24) Saito, S.; Ohmine, I. Phys. ReV. Lett. 2002, 88, 207401-4. (25) Saito, S.; Ohmine, I. J. Chem. Phys. 2003, 119, 9073-9087. (26) DeVane, R.; Ridley, C.; Perry, A.; Neipert, C.; Keyes, T.; Space, B. J. Chem. Phys. 2004, 121, 3688-3701. (27) Saito, S.; Ohmine, I. Phys. ReV. Lett. 2002, 88, 207401-4. (28) la Cour Jansen, T.; Duppen, K.; Snijders, J. Phys. ReV. B 2003, 67, 134206. (29) Nicolas, J.; Gubbins, K.; Street, W.; Tildesley, D. J. Mol. Phys. 1979, 37, 1429. (30) Applequist, J.; Carl, J. R.; Fung, K.-K. J. Am. Chem. Soc. 1972, 94, 2952. (31) la Cour Jansen, T.; Snijders, J. G.; Duppen, K. J. Chem. Phys. 2001, 114, 10910. (32) la Cour Jansen, T. Theoretical Simulation Nonlinear Spectroscopy in the Liquid Phase. Thesis, The University Groningen, Groningen, The Netherlands, 2002. (33) Abramowitz, M.; Stegun, I. Handbook of Mathematical Functions; Dover Publications: New York, 1970. (34) Menander, N.; Tabisz, G. C.; Zoppi, M. J. Chem. Phys. 1986, 84, 3005-3013. (35) Ceccherini, S.; Moraldi, M.; Frommhold, L. J. Chem. Phys. 1999, 111, 6316-6321. (36) Keyes, T. J. Phys. Chem. 1997, 101, 2921. (37) Stratt, R. M. Acc. Chem. Res. 1995, 28, 201. (38) Ahlborn, H.; Ji, X.; Space, B.; Moore, P. B. J. Chem. Phys. 1999, 111, 10622-10632.

Theoretical Investigation of the Temperature Dependence of the Fifth

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