Vibrational Relaxation of Normal and Deuterated Liquid Nitromethane

232

J. Phys. Chem. B 2008, 112, 232-241

Vibrational Relaxation of Normal and Deuterated Liquid Nitromethane† Shinsuke Shigeto, Yoonsoo Pang, Ying Fang, and Dana D. Dlott* School of Chemical Sciences, UniVersity of Illinois at Urbana-Champaign, Urbana, Illinois 61801 ReceiVed: May 26, 2007; In Final Form: June 22, 2007

Anti-Stokes Raman scattering is used to monitor vibrational energy redistribution in the ambient temperature liquids nitromethane (NM-h3) and perdeuterated nitromethane (NM-d3) after ultrafast IR excitation of either the symmetric or asymmetric CH- or CD-stretch transitions. The instantaneous populations of most of the fifteen NM vibrations are determined with good accuracy, and a global fitting procedure with a master equation is used to fit all the data. The pump pulses excite not only CH- or CD-stretches but also certain combinations of bending and nitro stretching fundamentals. The coupled vibrations that comprise the initial state are revealed via the instantaneous rise of the anti-Stokes transients associated with each vibrational fundamental. In contrast to many other polyatomic liquids studied previously, there is little energy exchange among the CH-stretch (or CD-stretch) excitations, which is attributed to the nearly free rotation of the methyl group in NM. The vibrational cooling process, which is the multistep return to a thermalized state, occurs in three stages in both NM-h3 and NM-d3. In the first stage, the parent CH- or CD-stretch decays in a few picoseconds, exciting all lower-energy vibrations. In the second stage, the midrange vibrations decay in 10-15 ps, exciting the lowerenergy vibrations. In the third stage, these lower-energy vibrations decay into the bath in tens of picoseconds. The initial excitations are thermalized in ∼150 ps in NM-h3 and there is little dependence on which CHstretch is excited. VC is somewhat faster in NM-d3 with more dependence on the initial CD-stretch, taking ∼100 ps with symmetric CD-stretch excitation and ∼120 ps with asymmetric CD-stretch excitation. Comparison is made with earlier nonequilibrium molecular dynamics simulations of VC [Kabadi, V. N.; Rice, B. M. Molecular dynamics simulations of normal mode vibrational energy transfer in liquid nitromethane. J. Phys. Chem. A 2004, 108, 532-540]. The simulations do a good job of reproducing the observed VC process and in addition they predicted the slow interconversion among CH-stretch excitations and the slower relaxation of the asymmetric CH-stretch now observed here.

1. Introduction In this paper we discuss measurements of vibrational relaxation (VR) and vibrational cooling (VC) of normal nitromethane (CH3NO2, NM-h3) and deuterated nitromethane (CD3NO2, NMd3) using an ultrafast IR-Raman technique.1-3 VR refers to energy relaxation out of an excited vibration, whereas VC refers to a process consisting of many VR steps, quite possibly including a vibrational cascade, where an initial high-energy excitation redistributes its excess energy within the molecule and into the bath until the system becomes thermalized.4,5 In the IR-Raman technique, an ultrafast IR pulse is tuned to excite a transition in the CH-stretch or CD-stretch region, depositing either ∼3000 or ∼2200 cm-1 of vibrational energy into the NM molecules. A probe pulse generates a time series of anti-Stokes Raman spectra that provide a quantitative measure of the instantaneous populations of all (or at least almost all) daughter vibrations in real time. The ability to probe the entire thermalization process, as opposed to the decay of the initial state alone, is an important feature of this technique. This paper extends previous IR-Raman work on NM1,6-9 through substantial advances in the experimental apparatus and through the comparison of NM-h3 to NM-d3. Vibrational energy dynamics of NM-d3 have not been studied previously. †

Part of the “James T. (Casey) Hynes Festschrift”. * Author to whom correspondence should be addressed. E-mail: [email protected].

NM is an interesting model system for understanding vibrational energy in molecular liquids. The other model systems that have been studied by this technique at a high level of detail so far are water,10-15 methanol,16,17 ethanol,18-20 chloroform,21 and acetonitrile.22,23 NM does not have the complications associated with the rapidly fluctuating network of hydrogen bonds in water, methanol, and ethanol. Chloroform turns out to be a relatively simple system that (along with bromoform24-26) appears to be almost completely understood,27 so NM represents the next level of complexity in polyatomic VR. Another important reason to study NM is that NM is a homogeneous high explosive, and one of the chemically most simple explosives that can detonate at a velocity of 6.3 km/s with an energy release of ∼5.7 kJ/cm3.28 Vibrational energy plays several roles in both shock initiation and detonation processes.29-37 When an initiating shock wave is present, before chemical reactions can begin, translational energy from the shock front must be driven into NM vibrations, a process termed “multiphonon up-pumping”.33,34 Once these reactions have been initiated and exothermic chemistry begins,38 energy from the excited vibrations of hot nascent molecular species such as39 HONO and HCN must be converted into translational energy to drive the detonation wave. Because of this practical application for NM vibrational energy transfer, theorists at the US Army Research Laboratory40-45 have devoted a great deal of effort to understanding molecular interactions in condensed phase NM.

10.1021/jp074082q CCC: $40.75 © 2008 American Chemical Society Published on Web 08/09/2007

Vibrational Relaxation of Liquid Nitromethane Theorists can simulate VR of molecular liquids using either equilibrium or nonequilibrium classical methods, and each has advantages and drawbacks. In equilibrium simulations,24,26,46-57 a tagged rigid molecule in a bath is studied. The bath exerts fluctuating forces on this molecule. The rate of transitions between two levels with an energy splitting ηΩ is proportional to the Fourier component of the fluctuating force correlation function at frequency Ω.50-52 Knowing the transition rates between all pairs of states, the VC process can be simulated using a master equation that can be solved for any initial condition. Two important issues arise in this type of theoretical analysis. First the classical force correlation function must be multiplied by a quantum correction,58 which is frequently a large factor that for anharmonic systems is not known with certainty.59 Second this method is based on linear response theory, so the bath correlation function must be unaffected by the relaxation process. But in a real system, as energy is dissipated in successive steps from a parent excitation, the bath grows hotter and the bath correlation function can evolve in time. In a nonequilibrium simulation,25,40,60-62 a tagged flexible molecule is initially configured in a vibrationally excited state, and the molecule and its bath allowed to evolve in time. Although this accounts for violations of linear response and does not require a quantum correction, such simulations are extremely sensitive to details of the potential energy surface and the nature of the initial state. Our 1999 IR-Raman measurements of NM-h3 with CHstretch excitation8 showed that VC occurred in three stages. In the first stage the parent excitation decayed within ∼3 ps, simultaneously populating all lower-frequency vibrations. These excited daughter vibrations could be divided into a midrange tier (∼1600-1000 cm-1) and a lower-frequency tier (∼1000480 cm-1). In the second stage, excitations in the midrange tier decayed into the lower-frequency tier in about 20 ps. The lowerfrequency tier thus became excited in two stages, the first lasting ∼3 ps and the second ∼15 ps. In the third stage, excitations of the lower-frequency tier decayed into the bath in 30-50 ps. The initial CH-stretch excitation became thermalized on the 100 ps time scale. Although this point was not investigated in detail at the time, there seemed to be little dependence on which of the two CH-stretch transitions, νa(CH3) or νs(CH3), were pumped. Kabadi and Rice40 have modeled our 1999 results using nonequilibrium molecular simulations of flexible NM. The initial excitation was equally partitioned among all three CH-stretch transitions. Interconversion between νa(CH3) and νs(CH3) was observed to be slow. Energy dissipation from νs(CH3) or a νa(CH3) and νs(CH3) mixture was similar to what was seen in experiment, including the three stages of VC. However, energy dissipation from νa(CH3) was about 3 times slower than from νs(CH3), and νa(CH3) relaxation populated only a few rather than all the lower-energy daughter vibrations. These differences between experiment and theory suggest the need for a closer examination of the nature of the initially excited state and how its nature affects the VC process. Although IRRaman measurements provide unique information about the redistribution of energy during VC, it is perhaps not as well recognized that they also provide unique information about the nature of the initial state.63 It is well-known that a pump pulse intended to excite a CH-stretch actually pumps a “bright” state, which is the state that carries oscillator strength. The bright state might be an admixture of CH-stretch (or CD-stretch) plus other stretches and bends whose overtones and combinations are nearly resonant with the CH- or CD-stretch. In other words,

J. Phys. Chem. B, Vol. 112, No. 2, 2008 233

Figure 1. (a) IR absorption and (b) Stokes Raman spectra of liquid nitromethane at ambient temperature. The Raman spectra were measured with either a narrow-band laser (dotted curve) or the ultrafast laser (solid curve). The arrows in (a) indicate the frequencies of the mid-IR pump pulses (2960 and 3030 cm-1).

the bright state is likely to be initially a coherently coupled admixture of several stretching and bending excitations. In an IR-Raman experiment, this admixture is directly seen as an instantaneous excitation of the vibrational fundamentals. For instance, in the case of a 2:1 Fermi resonance between CHstretch and CH-bend, an anti-Stokes probe would observe an instantaneous rise in the CH-stretch via the fundamental transition V ) 1 f 0 near 3000 cm-1 and in the bend via the V ) 2 f 1 transition near 1500 cm-1.17,21 Thus in an IR-Raman measurement the anti-Stokes transients of the daughter vibrations contain both an instantaneous component that provides information about the nature of the initial state and a delayed rise (and decay) that provides information about the daughter vibration’s VR pathways and about the VC process. In the rest of this paper we discuss the vibrational spectroscopy of NM-h3 and NM-d3, and we present VC data on both systems with νs(CH3) and νa(CH3), or νs(CD3) and νa(CD3) excitation. The IR-Raman method allows us to quantitatively determine the instantaneous populations of all NM vibrations. Because NM has 15 normal modes, a master equation involves a large number of kinetic parameters. To reduce the number of parameters to a manageable level, a model is developed of the VC process where only vibrational fundamentals (V ) 1) are considered in the kinetic scheme and the overtone or combination bands of these levels are present only in the initial state. There are three stages as discussed above with tiers of vibrations having equal VR lifetimes, and this model is used to generate a simplified master equation. A global fit of all NM-h3 and NMd3 data is then used to determine a matrix of state-to-state transition rates. The nature of the initial state and the dependence of the VC process on the initial state are discussed and a comparison is made to the Kabadi-Rice simulations. 2. Vibrational Spectroscopy of Normal and Deuterated Nitromethane Vibrational spectra of normal and deuterated nitromethane have been studied extensively,64-69 so here we focus on summarizing what is needed to understand the VR measurements. Figures 1 and 2 show FTIR and Raman spectra of NMh3 and NM-d3. The arrows in the IR spectra indicate the center frequencies of the pump pulses used in these experiments, intended to pump either the symmetric or asymmetric CH- or CD-stretching transitions. The Raman spectra were measured in two ways, using either a 532 nm narrow-band continuous wave (CW) laser (dotted line)

234 J. Phys. Chem. B, Vol. 112, No. 2, 2008

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Figure 2. (a) IR absorption and (b) Stokes Raman spectra of liquid deuterated nitromethane at ambient temperature. The Raman spectra were measured with either a narrow-band laser (dotted curve) or the ultrafast laser (solid curve). The arrows in (a) indicate the frequencies of the mid-IR pump pulses (2190 and 2280 cm-1).

TABLE 1: Assignments and Frequencies of Raman Transitions of Normal and Deuterated Nitromethane CH3NO2 (NM-h3) assignment νa(CH3) νs(CH3) νa(NO2) δa(CH3) νs(NO2) δs(CH3) F(CH3) ν (CN) δ(NO2) F(NO2)

frequency (cm-1)

CD3NO2 (NM-d3) assignment

frequency (cm-1)

νa(CD3) νs(CD3) νa(NO2)

2282 2170 1532

νs(NO2)

1380

δs(CD3) δa(CD3) ν (CN) δ(NO2) F (NO2)

1070 1040 890 620 540/428

3050 2970 1550 1430 1400 1377 1100 918 655 607/480

or the 0.8 ps duration 532 nm pulse (solid line). Figures 1b and 2b show that the resolution with the short pulses is quite close to what is obtained with the narrow-band laser. This is a substantial improvement over earlier work where excess laser bandwidth resulted in considerable broadening and loss of resolution. For instance in previous work we could not resolve the band structure in Figure 1b in the transitions located near 1500 cm-1. The assignments65,66,69 and frequencies of the Raman transitions of NM-h3 and NM-d3 are summarized in Table 1. Tuning the pump pulses into CH- or CD-stretch transitions might additionally excite overtones or combinations of the lower-energy vibrations. In NM-h3 there are several possibilities for 2:1 Fermi resonances, because both CH-bending and NO2stretching modes have energies in the 1400-1600 cm-1 range. Two new features are associated with NM-d3: the CD-stretch and CD-bend transitions both red shift and the NO2-stretching does not, and we do not observe F(CD3), which we presume is now buried under δ(CD3). Thus, although the possibility of a 2:1 Fermi resonance remains between CD-stretch and CD-bend excitations, NO2-stretching is too high in energy for a 2:1 resonance with CD-stretch. However, in NM-d3 there is also the possibility of a resonance between CD-stretching and the combination νs(NO2) + ν(CN). In Figure 3 we have listed the states in the second tier of excitations, consisting of binary combinations or first overtones, which have energies close to the CH-stretch or CD-stretch parent transitions.

Figure 3. Energy level diagram of parent CH-stretch or CD-stretch excitations plus nearby overtones and combination bands of midrange vibrations that might be directly excited by the IR pump pulses for (a) NM-h3 and (b) NM-d3.

3. Experimental Section The laser apparatus for the IR-Raman experiment has been described previously.11 A tunable mid-IR pulse (∼0.7 ps duration, 25 cm-1 bandwidth, 370 µm diameter) pumped a freeflowing jet of reagent-grade NM-h3 (Sigma-Aldrich) or NM-d3 (99 atom % D, Sigma-Aldrich). The excitation frequencies for NM-h3 were 2960 cm-1 (λ ) 3.38 µm) for νs(CH3) and 3030 cm-1 (λ ) 3.30 µm) for νa(CH3). The excitation frequencies for NM-d3 were 2190 cm-1 (λ ) 4.57 µm) for νs(CD3) and 2280 cm-1 (λ ) 4.38 µm) for νa(CD3). As described previously,2 the NM samples were optically thick at these pump wavelengths, so despite the fact that the different pump wavelengths have different absorption coefficients, the number of excitations initially produced in the sample is always about equal to the number of photons in the IR pump pulse. Pulse energies in the mid-IR were ∼35 µJ for NM-h3 pumping and ∼20 µJ for NMd3 pumping, which correspond to bulk temperature jumps after thermalization ∆T < 20 K. A time-delayed 532 nm probe pulse (∼0.7 ps duration, 25 cm-1 bandwidth, 400 µm diameter) generated an anti-Stokes Raman spectrum, which was detected with a multichannel spectrograph. This is a substantial advance over our 1999 work where single-channel detection was used.8 Anti-Stokes spectra were obtained in a time series of measurements, and the order of delay time was random so that any long-term drift in the laser system would not affect the data. In a typical run we used a 1 min integration time per delay and obtained 28 data points. The liquid jet system used in the present study is worth mentioning. In previous IR-Raman measurements, a flat jet circulated by a mechanical pump was used.2 Here we used a cylindrical jet from an 82 µm diameter capillary attached to a syringe pump (Harvard Apparatus) with a 50 mL syringe and a flow rate of 1 mL/min. Due to the smooth operation of the syringe pump, jet stabilities were much better than obtained previously. Furthermore, the low sample volume and reduced evaporation rate made it economically feasible to study NMd3. Although the system does not recirculate, the ∼50 min run time was long enough to obtain an entire time series of spectra without refilling the syringe.

Vibrational Relaxation of Liquid Nitromethane

Figure 4. Transient anti-Stokes spectra of NM-h3 with 2960 cm-1 νs(CH3) pumping. At longer delay times the anti-Stokes signals result from the equilibrium temperature jump ∆T.


Figure 7. Transient anti-Stokes spectra of NM-d3 with 2280 cm-1 νa(CD3) pumping.

the parent. Keep in mind that the Raman cross-section for CHbending is small, so it is difficult to see instantaneous excitation of the bending transitions in Figures 4-7. B. Relative Populations. The experiments measure the change in anti-Stokes intensity of the ith Raman transition at frequency νi induced by the pump pulses, ∆IAS i (t), which is directly proportional to the instantaneous change in state population ∆ni(t),3 4 ∆IAS i (t) ) const × ∆ni(t)giσi(νL + νi)

Figure 5. Transient anti-Stokes spectra of NM-h3 with 3030 cm-1 νa(CH3) pumping.

Figure 6. Transient anti-Stokes spectra of NM-d3 with 2190 cm-1 νs(CD3) pumping.

4. Results A. Transient Anti-Stokes Spectra. Representative transient anti-Stokes spectra of NM-h3 with νs(CH3) and νa(CH3) pumping and NM-d3 with νs(CD3) and νa(CD3) pumping are shown in Figures 4-7. The ambient-temperature anti-Stokes spectrum obtained with the probe pulses at negative delay time has been subtracted off as a background. Lower signal-to-noise ratios below 700 cm-1 compared to those above 1000 cm-1 are due to fluctuations in the background ambient-temperature antiStokes signals. The spectra observed at longer delay times represent the changes in anti-Stokes intensities due to the thermalized temperature jump ∆T. It is useful to examine the t ) -1 ps spectra. Transitions seen in the anti-Stokes spectrum at this delay time are indicative of states that were directly pumped by the laser,63 which include the parent CH-stretch transitions plus any other states such as CH-bending or NO2-stretching that are coherently coupled with

(1)

where the proportionality constant depends on the experimental setup, gi is the degeneracy, σi is the Raman cross-section, and νL is the laser frequency. This proportionality factor encompasses factors such as light-gathering efficiency, apparatus spectral response, and so on, and it is approximately equal for all transitions. To determine the Raman cross-sections in eq 1, we performed a least-squares fit of the Stokes Raman spectrum to a sum of Voigt line profiles and then found the area under each profile. We did not assign any physical significance to the Gaussian and Lorentzian contributions of the Voigt line shape, this function was used solely for better estimation of the integrated area. We used the same Voigt fitting procedure for the transient anti-Stokes spectra to determine ∆IAS i (t). Figures 8 and 9 are the time-dependent state populations computed using the data in Figures 4-7 and eq 1. The smooth curves in Figures 8 and 9 result from a global fit based on a master equation analysis4,5 of the VR model discussed in the next section. C. Vibrational Populations of NM-h3. With νs(CH3) pumping (Figure 8a), the parent builds up instantaneously and decays with a time constant of T1 ) 1.8 ( 0.1 ps, and νa(CH3) builds up with a slight delay and then decays with a similar time constant T1 ) 1.6 ( 0.3 ps. The maximum population of νa(CH3) is about 15% of the pumped νs(CH3). With νa(CH3) pumping, as shown in Figure 8d, some unusual effects are seen. Both νs(CH3) and νa(CH3) appear to build up instantaneously, but even though we are pumping the νa(CH3) transition, there is actually more νs(CH3) than νa(CH3). The lifetimes of νs(CH3) and νa(CH3), T1 ) 2.9 ( 0.2 and 2.3 ( 0.1 ps, respectively, are also quite a bit slower with νa(CH3) pumping than with νs(CH3) pumping. The transient populations of the midrange daughter vibrations in the 1400-1600 cm-1 range are shown in Figure 8b,e. The rising edges of these transients are a mixture of instantaneous processes (i.e., processes whose rise appears limited by the duration of the pump pulse) and delayed processes whose rise matches the decay of νs(CH3) and νa(CH3). It is difficult to see the instantaneous processes in Figure 8b,e, but referring back to the t ) -1 ps data in Figures 4 and 5, it can be seen that


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Figure 8. Time dependence of relative populations of NM-h3 vibrations (a)-(c) with νs(CH3) pumping and (d)-(f) with νa(CH3) pumping. The smooth curves are the fit based on a master equation analysis using the scheme depicted in Figure 11.

Figure 9. Time dependence of relative populations of NM-d3 vibrations (a)-(c) with νs(CD3) pumping and (d)-(f) with νa(CD3) pumping. The smooth curves are the fit based on a master equation analysis using the scheme depicted in Figure 12.

with νs(CH3) pumping there is an instantaneous component in the rise of both the 1400 and 1550 cm-1 transitions. The 1400 cm-1 transient is a mixture of νs(NO2) plus small contributions of δa(CH3) and δs(CH3), and the 1550 cm-1 transient originates from νa(NO2). With νa(CH3) pumping, the t ) -1 ps spectrum in Figure 5 indicates much less instantaneous population of the CH-bend and NO2-stretch excitation than with νs(CH3) pumping.

The decays of all the transients in Figure 8b,e appeared similar, and in the spirit of trying to minimize the number of kinetic parameters, we judged they all have the same lifetime, T1 ) 15 ( 1 ps. Parts c and f of Figure 8 show the time-dependent populations of the lower-energy vibrations F(CH3), ν(CN), δ(NO2), and F(NO2). As observed previously,8 the rising edges of all lower-


Figure 10. Early time dynamics of parent CD-stretch and midrange vibrations in NM-d3 with (a) νs(CD3) pumping and (b) νa(CD3) pumping. The transients are normalized to the same peak height to facilitate comparison. With νs(CD3) pumping, δ(CD3) is the only daughter vibration with a rising edge having an instantaneous component. With νa(CD3) pumping, the daughter vibrations δ(CD3), νs(NO2) and ν(CN) (not shown) all evidence an instantaneous component in the rising edge.

energy transients have a two-part structure, with the faster part occurring on the 2-3 ps time scale of CH-stretch decay and the slower part on the 15 ps time scale of the midrange vibration decay. There is no evidence for an instantaneous rise in any of the lower-energy transients. The decay time constants are difficult to fix accurately due to signal-to-noise constraints and the significant amplitude of the plateau for the lower-frequency vibrations, but the decays are quite similar for all lower-energy transients, T1 ) 40 ( 20 ps. D. Vibrational Populations of NM-d3. The time dependences of the relative populations for NM-d3 with νs(CD3) or νa(CD3) pumping are displayed in Figure 9. With νs(CD3) pumping (Figure 9a), the parent rises instantaneously and decays with a lifetime of 1.3 ( 0.1 ps. There is in addition a small amount of excitation of νa(CD3) and 2δs(CD3). Because both rise instantaneously in a coordinated fashion, we attribute these signals to a coupled excitation consisting of νs(CD3) plus a contribution from νa(CD3) and the bending overtone 2δs(CD3). With νa(CD3) pumping (Figure 9d), the parent νa(CD3) excitation rises instantaneously and decays with a lifetime T1 ) 2.9 ( 0.1 ps, which is slower than with νs(CD3) pumping. Even though we are pumping the νa(CD3) transition, there is also a great deal of νs(CD3) excitation. The subsequent decay of νs(CD3) occurs with T1 ) 4.3 ( 0.2 ps. The δs(CD3) transition also evidences a delayed build up similar to what is seen in νs(CD3) and a longer-duration decay with T1 ) 9 ps. In the δs(CD3) transient we associate the shorter-time signal with 2δs(CD3) and the longer-time tail with δs(CD3). We do not see any appreciable signals from 2δa(CD3). VR of the midrange vibrations νa(NO2), νs(NO2), and δ(CD3) shown in Figure 9b,e is more complicated than in NM-h3. In Figure 9b,e the population denoted δ(CD3) represents the sum of the contributions from δs(CD3) and δa(CD3) and their first overtones, which are difficult to resolve individually. To facilitate comparisons, in Figure 10 we plot the parent and midrange daughter transients on an expanded scale where all the amplitudes are normalized to the same value. With νs(CD3) pumping, only δ(CD3) is generated instantaneously, whereas with νa(CD3) pumping, both δ(CD3) and νs(NO2) are generated instantaneously. The buildup of νa(NO2) is slower in both cases, indicating that it is generated by VR of ν(CD3). It is interesting that the instantaneous rise of νs(NO2) cannot be due to a 2:1 Fermi resonance, but instead, as suggested by Figure 3, is


Figure 11. Energy level diagram showing the kinetic model with three stages of vibrational relaxation used to fit the NM-h3 data.

Figure 12. Energy level diagram showing the kinetic model with three stages of vibrational relaxation used to fit the NM-d3 data.

apparently due to a resonance between νa(CD3) and the combination band νs(NO2) + ν(CN). This assignment is supported by the observation of anti-Stokes signals from both νs(NO2) and ν(CN) in the t ) -1 ps spectrum in Figure 7. The lifetimes of νa(NO2) and νs(NO2) are T1 ) 36 ( 5 ps and T1 ) 11 ( 1 ps, respectively. These values were obtained in a global fitting of the time dependence of νa(NO2) and νs(NO2) for both pumping conditions. The lower-energy transients shown in Figure 9c,f show a twopart rise as observed with NM-h3, with the faster part attributed to parent ν(CD3) decay and the slower part to midrange vibration decay. The lifetimes of the lower-energy vibrations are again difficult to determine with high accuracy due to the high level of the longer-time thermal plateau, and our best estimate is that they are all about the same and T1 ) 25-30 ps. 5. Master Equation Analysis of the Data The kinetics of systems with multiple states are generally described by a master equation:

d n(t) ) K‚n(t) dt

(2)

where n(t) is a column vector of time-dependent populations, and K is a matrix of transition rate constants. In NM, the initial excitation may consist of a CH-stretch or CD-stretch fundamental plus some bending overtones or the νs(NO2) + ν(CN) combination band, but after the initial state decays the daughter vibrations are mainly fundamental excitations. But even if we consider only vibrational fundamentals in the n-vector, there would be 152 rate constants in K. Even though these are not all independent, there are simply too many to determine individually with the data we have, so we have devised a procedure to fit the data using a minimal number of adjustable parameters. We


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did this by assuming kinetic models for NM-h3 and NM-d3 that reflect the three-stage nature of VR, and by lumping together some of the nearby vibrational transitions that appear to have quite similar time dependences. The resultant matrix K for NM-h3 takes the form

(

-(k1 + k2 + k3 + k4 + k5) 0 k1 K ) k2 k3 k4 k5 where the n(t) vector is

0 -(k02 + k03 + k04 + k05) 0 k02 k03 k04 k05

(

0 0 -(k12 + k13 + k14 + k15) k12 k13 k14 k15

nν(CH3)(t) nνa(NO2)(t) nνs(NO2)+δa(CH3)+δs(CH3)+2δa(CH3)+2δs(CH3)(t)

n(t) ) nF(CH3)(t) nν(CN)(t) nδ(NO2)(t) nF(NO2)(t)

)

0 0 0 -k6 0 0 0

0 0 0 0 -k7 0 0

0 0 0 0 0 -k8 0

0 0 0 0 0 0 -k9

)

(3)

(4)

In eq 3, the reciprocal of a diagonal element determines the lifetime of an individual state, whereas off-diagonal rate constants reflect the branching ratio for the parent into its daughters. In eq 4, we have grouped the two types of CH-stretch transitions into a single term. The form of eq 3 shows that these parent CH-stretch excitations can decay to two midrange states, νa(NO2) or the difficult-to-resolve triad νs(NO2), δa(CH3) and δs(CH3) which are grouped into a single term, plus the four lower-energy states. The midrange states can decay to the lower-energy states, and the lower-energy states disappear into the bath. This three-stage relaxation scheme, which has been described previously, is described by the schematic diagram Figure 11. For NM-d3, the K-matrix takes the form

(

-(k1 + k2 +k3 + k4 + k5 + k6) k1 k2 K ) k3 k4 k5 k6 with n(t) given by

0 -(k14 + k15 + k16) 0 0 k14 k15 k16

0 0 -(k24 + k25 + k26) 0 k24 k25 k26

( )

0 0 0 -k7 0 0 0

0 0 0 0 -k8 0 0

0 0 0 0 0 k9 0

0 0 0 0 0 0 -k10

)

(5)

nν(CD3)(t)

nνa(NO2)(t) nνs(NO2)

n(t) ) nδ(CD3) + 2δ(CD3)(t) nν(CN)(t) nδ(NO2)(t)

(6)

nF(NO2)(t)

In eq 6 we have grouped the two types of CD-stretching excitations into a single term and the two types of CD-bending excitations into a single term. The parent CD-stretch excitations can decay into three midrange vibrations, νa(NO2), νs(NO2) and δ(CD3) which combines both types of CD-bending, or into three lower-energy vibrations [recall we do not see F(CD3) in NM-d3]. As in the NM-h3 scheme above, the midrange states can decay to the lower-energy states, and the lower-energy states disappear into the bath. This three-stage relaxation scheme is depicted in Figure 12. We dealt with detailed balance in a simplified way. Technically the elements of n(t) should have the form, ni(t) - neq, but the neq term is significant only for the lower-energy vibrations which evidence a significant longer-time plateau, so we used the ni(t) - neq form for these states only. These values for neq for all vibrations are simultaneously determined by a single parameter, the temperature jump ∆T.7,70 In eqs 3 and 5, all the transitions are described as down-conversion processes71 only because up-conversion processes are negligible. In the VC process, we start with a higher-energy ∼3000 cm-1 excitation, which creates a vibrational cascade through a manifold of states that are not very dense, so that the energy spacings ∆E > kT. Thus in every VR step of the vibrational cascade,


Figure 13. Fitting the population transients of δ(CD3) midrange vibrations that are coherently coupled to the parent CD-stretch vibration of NM-d3. At shorter times the transient is due mainly to 2δ(CD3) and at longer times to δ(CD3), and these transitions are not resolved. The 2δ(CD3) component is modeled with an instantaneous rise and a fall that mirrors the parent CD-stretch decay (dotted curves), and the δ(CD3) component (dashed curves) is modeled with a rise that tracks the parent decay plus an exponential decay with the δ(CD3) lifetime T1.


Figure 14. Fitting the population transient of νs(NO2) with νa(CD3) pumping. The pump pulses excite a coherent superposition of νa(CD3) and νs(NO2) + ν(CN). The shorter-time component (dotted curve) is due to νs(NO2) + ν(CN) and the longer-time component (dashed curve) to νs(NO2) alone.

TABLE 2: Rate Constants Derived from Master Equation Analysis of VC in NM-h3 rate constanta (ps-1)

the likelihood of an up-conversion is small. The functions used to fit the data in Figures 8 and 9 were determined by solving eq 2 for a delta-function initial condition of ν(CH3) or ν(CD3) excitation followed by convolution of n(t) with a Gaussian function to represent the instrument response. To limit the number of adjustable parameters, we fit the data in three steps: (1) The time dependences of the parent vibrations ν(CH3) and ν(CD3) were fitted (see Figures 8a,d and 9a,d), which gives k1 + k2 + k3 + k4 + k5 ) (1.8 ps)-1 for 2960 cm-1 νs(CH3) pumping and (2.9 ps)-1 for 3030 cm-1 νa(CH3) pumping in NMh3, and k1 + k2 + k3 + k4 + k5 + k6 ) (1.3 ps)-1 for 2190 cm-1 νs(CD3) pumping and (4.3 ps)-1 for 2280 cm-1 νa(CD3) pumping in NM-d3. (2) The time dependences of the midrange transitions shown in Figures 8b,e and 9b,e were fitted. In NM-h3, we assumed a single lifetime for all the midrange vibrations so that our fitting procedure obtained k02 + k03 + k04 +k05 ) k12 + k13 + k14 + k15 ) (15 ps)-1. For NM-d3, as can be seen by comparing Figure 9b,e, all the midrange vibration lifetimes are similar for a given parent, but the lifetimes are quite a bit shorter when νs(CD3) is pumped rather than νa(CD3), and our fitting procedure gave k14 + k15 + k16 ) (11 ps)-1 and k24 + k25 + k26 ) (36 ps)-1. We also added an ad hoc correction to describe the initial state as a coherent superposition of CH-stretch or CD-stretch and midrange overtones or combinations, leading to a part of some midrange transients that rises instantaneously. The couplings of this type that need to be considered are indicated in the schemes in Figures 11 and 12. This correction was small in the case of NM-h3 but more significant in NM-d3. To illustrate how this was done, we will describe the procedure in detail for NM-d3. With both νs(CD3) and νa(CD3) pumping there is instantaneous generation of 2δ(CD3) plus delayed generation of δ(CD3), and the fundamental and overtone transitions are not resolved. The 2δ(CD3) component rises instantaneously and decays with the CD-stretch lifetime whereas the δ(CD3) component rises with the CD-stretch lifetime and decays with the δ(CD3) lifetime. Thus the only adjustable parameter in this correction procedure is the ratio of fundamental to overtone amplitude at t ) 0, which was varied to achieve the fits in Figure 13. The same procedure was used to fit the shorter-time part of the νs(NO2) transient due to coherent coupling between νa(CD3) and the combination νs(NO2) + ν(CN), as shown in Figure 14. (3) The lower-energy daughters (Figures 8c,f and 9c,f) were fitted with a two-part rise using the rate constants for parent and midrange vibration decay given above, by varying the off-

notation

νs(CH3) pumping

νa(CH3) pumping

k1 k2 k3 k4 k5 k02 ()k12) k03 ()k13) k04 ()k14) k05 ()k15) k6 k7 k8 k9

0.31 (56) 0.02 (4) 0.04 (8) 0.08 (14) 0.10 (18)

0.14 (40) 0.02 (6) 0.04 (13) 0.06 (16) 0.09 (25)

VC stage (I)

(II)

(III)

0.020 (30) 0.016 (25) 0.007 (11) 0.023 (34) 0.053 0.024 0.02 0.018

a Values in parentheses indicate the percentage of each rate constant to the sum of rate constants that gives the time constant of the relevant VC stage.

diagonal rate constants to fit the relative amplitudes of the two parts of the rise. The decay lifetimes of all the lower-energy NM-h3 vibrations were taken as T1 ) 40 ps and the decay lifetimes of the lower-energy NM-d3 vibrations were taken to be T1 ) 25 ps. The values of all the rate constants used for global fits of the NM-h3 data are given in Table 2, and the values used for global fits of NM-d3, in Table 3. Even a brief inspection of Figures 8 and 9 shows that this master equation analysis does an excellent job in generating a global fit of these data sets. 6. Discussion We have presented VR and VC data on NM and its perdeuterated isotopomer, and we believe this is the most detailed study so far of vibrational energy of a polyatomic liquid. In this section we discuss three issues, the nature and decay of the parent state, the nature of the subsequent VC process and its dependence on the initial state, and the comparison to simulation. A. CH-Stretch and CD-Stretch Excitation and Relaxation. IR-Raman experiments with CH-stretch pumping of methyl -CH3 or methylene -CH2- groups are usually interpreted as indicating a faster redistribution among the CH-stretch excitations and a slower VR process. This model was originally proposed by Laubereau et al.,3,19,72 to explain why Raman-active CH-stretch vibrations in molecules such as CH3I and ethanol19 showed a prompt rise after IR pumping. Later the model was extended to propose that fast redistribution should lead to a quasiequilibrium within the CH-stretch manifold.18 In a 2000


Shigeto et al.

TABLE 3: Rate Constants Derived from Master Equation Analysis of VC in NM-d3 rate constanta (ps-1) VC stage (I)

(II)

(III)

notation

νs(CD3) pumping

νa(CD3) pumping

k0 k1 k2 k3 k4 k5 k14 k15 k16 k24 k25 k26 k7 k8 k9 k10

0.11 (15) 0.48 (64) 0.09 (12) 0.05 (6)

Vibrational Relaxation of Normal and Deuterated Liquid Nitromethane

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