Short Time Scale Dynamics and the Correlation between Liquid and

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J. Phys. Chem. A 2010, 114, 5231–5241

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Short Time Scale Dynamics and the Correlation between Liquid and Gas Phase Vibrational Energy Relaxation Rates Steven A. Adelman† Department of Chemistry Purdue UniVersity West Lafayette, Indiana 47907-2084 ReceiVed: July 17, 2009; ReVised Manuscript ReceiVed: March 4, 2010

Although gas (g) and liquid (l) phase densities F and Fl can differ by 3 orders of magnitude, from experiment the corresponding vibrational energy relaxation (VER) rate constants are at the same temperature T, strongly correlated. Namely, these rate constants obey the empirical correlation equation k(Τ,Fl) ) (Fl/Fg)k(Τ,Fg)G where typically G ≈ 0.5-2.0. The rate correlation equation is usually explained by the isolated binary collision (IBC) hypothesis, which yields a theoretical result for G. However, the physical assumptions underlying the IBC hypothesis have often been criticized. Thus we propose a new purely mathematical hypothesis, which yields the correlation equation including a molecular formula for G. This hypothesis is that the relaxing molecule’s normalized vibrational force autocorrelation function is F-independent to order t2. The hypothesis is checked numerically for three model dihalogen solute/rare gas solvent VER systems. It is found to be obeyed for 280 thermodynamic states, usually within 0.02% to 0.30%. An inertial dynamics mechanism is proposed as a tentative physical explanation for our mathematical hypothesis. I. Introduction While liquid (l) and gas (g) phase densities Fl and Fg can differ by factors of thousands, surprisingly l and g phase vibrational energy relaxation (VER) times T1,l and T1,g, or rate constants k(T,Fl) ≡ T1,l-1 and k(T,Fg) ≡ T1,g-1, when measured at the same temperature T are strongly correlated.1-3 For example, Table 1 illustrates that k(T,Fl) and k(T,Fg) are typically linked by the empirical equation

k(Τ,Fl) )

()

Fl k(Τ,Fg)G Fg

(1.1)

where G is an order unity factor (tabulated in the right most column in Table 1). Equation 1.1 shows, remarkably, that over the vast (∼13 order of magnitude) range of observed VER rates,4 k(T,Fl)’s may be meaningfully estimated from k(T,Fg)’s simply by multiplying the latter by (Fl/Fg). Equation 1.1 is nearly always rationalized using the isolated binary collision (IBC) model,1-3,5 which is based on the following hypothesis.

IBC Hypothesis: The molecular motions that induce VER are identical in the l and g phases (1.2) Equation 1.2 implies that the faster observed l phase VER rates arise solely because l phase collision frequencies Zl exceed g phase frequencies Zg, thus yielding the IBC model rate equation

k(Τ,Fl) ) (Zl /Zg)k(Τ,Fg)

eq 1.1 with G ) 1. More realistically, taking Zl/Zg = Floc,l/Floc,g, where Floc,l(g) are the solvent densities local to the solute, and further approximating Floc,l(g) = Fl(g)gl(g)(R*) where gl(g)(R*) is the contact solute-solvent pair correlation function, yields the familiar Davis-Oppenheim5 IBC model rate equation

k(Τ,Fl) )

() [ ] Fl gl(R*) k(Τ,Fg) Fg gg(R*)

Comparing eqs 1.1 and 1.4 gives a theoretical estimate for G as G = gl(R*)/gg(R*). The IBC model has been both highly influential1-3 and pointedly criticized.6 The criticisms reviewed here are, in essence, the following two claims. Criticism 1. Unlike a g phase collision an l phase collision is a blurry concept, rendering the physical solidity of the IBC model dubious. Criticism 2. Criticism 1 aside, eq 1.2’s supposition of identical l and g phase energy transfer dynamics is implausible. Chesnoy and Gale7 (CG) and Herzfeld8 have responded, respectively, to criticisms 1 and 2. We next summarize their comments. CG’s Rebuttal of Criticism 1. CG have addressed criticism 1 by giving a time correlation function reformulation of the IBC hypothesis eq 1.2, which may be paraphrased as follows. CG IBC Hypothesis: The Normalized Fluctuating force Autocorrelation Function (faf) of the Relaxing Mode

C(t,F) ≡

(1.3)

Equation 1.1 is easily developed from eq 1.3 by suitably approximating Ζl/Ζg. For example, setting Zl/Zg = Fl/Fg yields † Phone: (765) 494-5277. Fax: (765) 494-0239. E-mail: [email protected].

(1.4)

〈F˜(t)F˜〉0 〈F˜2〉0

(1.5)

at fixed T is G-independent, and thus is identical in the l and g phases. CG show that the usual IBC model results emerge from their hypothesis, thus validating it as a basis for the model. However,

10.1021/jp906783k  2010 American Chemical Society Published on Web 04/07/2010

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TABLE 1: Empirical Liqui-Gas VER Rate Correlationa molecule H2 HC1 HBr O2 H2 HC1 CO N2 Br2 C12 CO2 CC14 CS2 C6H6

temp (K) 30 188 206 150 127 163 85 80 270 273 273 293 298 293

solvent

Tliq 1 (s) -5

2.23 × 10 0.75 × 10-9 1.23 × 10-9 0.77 × 10-4 Ar Xe O2/Ar CH4/Ar

1.14 × 10-9 1.03 × 10-9 1.56 × 10-8 (6.4-21) × 10-11 2.13 × 10-9 0.22 × 10-9

(1/FT1)liq -18

3.09 × 10 6.94 × 10-14 4.97 × 10-14 9.8 × 10-19 8.3 × 10-17 8.4 × 10-14 3.3 × 10-18 8.3 × 10-17 7.35 × 10-14 7.81 × 10-15 5.03 × 10-15 (2.5-0.77) × 10-12 4.7 × 10-14 6.7 × 10-13

(1/FT1)gas -18

3.0 × 10 4.2 × 10-14 2.33 × 10-14 7.0 × 10-19 7.15 × 10-17 6.0 × 10-14 3.9 × 10-18 1.0 × 10-16 3.6 × 10-14 6.1 × 10-15 4.72 × 10-15 1.93 × 10-12 5.7 × 10-14 9.54 × 10-13

(FT1)gas/(FT1)liq 1.03 1.65 2.10 1.40 1.15 1.40 0.85 0.83 2.04 1.28 1.07 0.4–1.3 0.82 0.70

a Entries are reproduced from Table 1 of Chesnoy and Gale.1 If the rightmost column entries were unity, all gas and liquid phase T1’s, when scaled for differences in density, would be identical. These entries in fact are close to unity for 14 diverse systems, suggesting universality of the liquid-gas rate correlation (see text).

since C(t,F) is defined solely in terms of true l phase motions, the criticized concept of an l phase collision never enters the CG formulation,7 thus negating criticism 1. However, CG’s assumption that C(t,Fl) ) C(T,Fg) is, of course, still subject to criticism 2. Next we turn to Herzfeld’s8 comments on criticism 2. Herzfeld’s Rebuttal of Criticism 2. Herzfeld has attempted to refute criticism 2. His argument8 may be paraphrased as the following claims about VER inducing solute-solvent collisions. a. Ordinary thermal collisions are inefficient for VER and contribute negligibly to k(T,P). b. Rather VER is induced by efficient but infrequent collisions with energies of several kBT. c. Collisions involving two or more energized solvent molecules are so rare that the VER inducing collisions are binary. d. Because of their very high energies, the VER inducing collisions are dominated by gas phase repulsive forces, and thus are isolated (Herzfeld and Litowitz8a actually make these comments. “The main contribution to the energy transfer comes from a very few collisions with exceptionally high energy, just as in (high temperature) gases. In these collisions the molecule in question comes particularly close to and interacts almost exclusively with a single (solvent) molecule of the surroundings, not with the cell wall as a whole”.) The available evidence does not validate Herzfeld’s claims. Rather classical and quantum dynamics studies show that at the pertinent temperatures T j 300 K (Table 1), relaxation is dominated by low energy near thermal collisions.9 This is because at low T the Maxwell speed distribution is sufficiently sharply peaked near the mean thermal speed that Herzfeld’s energized collisions, while efficient individually, have negligible occurrence probabilities. Thus we have reached an impasse. The correctness of the empirical rate correlation eq 1.1 is certain. Yet since criticism 2 has not been successfully refuted, the standard IBC explanation of the correlation is problematic. Here we give numerical and analytical results that bear on these questions. Specifically, we numerically show for many thermodynamic states and several systems that to order t2 the faf C(t,G) is isothermally density-independent. Then we prove that this short time F-independence is sufficient for the first principles emergence of eq 1.1, including a theoretical formula for G.

Figure 1. Chesnoy and Weis’10 results for the F-dependence of C(t) of eq 1.5. C(t), in their notation f(t)/f(0), is plotted in their Figure 2 for three reduced densities F* ) 0.3, 0.9, and 1.09 where F* ) Fσ3 with σ ) 340 pm. The plots show that C(t) is F-independent at the shortest times only, in accord with our order t2 F-independence hypothesis.

Our development is free of the flaws of earlier arguments for eq 1.1. For example, Chesnoy and Weis’10 molecular dynamics (MD) faf’s, plotted in Figure 1, invalidate CG’s hypothesis that for all times C(t,F) at fixed T is F-independent. In contrast, from Figure 1 these MD faf’s accord with our hypothesis of short time isothermal F-independence. Similarly in section III we show that short time isothermal F-independence of C(t,F) holds for T’s as low as 150 K, where Herzfeld’s picture8 is dubious. In summary, we derive eq 1.1 from the single easily testable hypothesis that at fixed T as t f 0 C(t,F) becomes independent of F. The plan of this paper is as follows. In section II we derive from our short time hypothesis a molecular expression for k(T,Fl), eqs 2.7 and 2.8, which duplicates the form of the empirical eq 1.1. Next in section III, we numerically verify our short time assumption for many thermodynamic states of three model dihalogen/rare gas systems. We summarize our work in section IV, and especially suggest a possible physical basis for our fundamental hypothesis. Finally, the analysis of high frequency friction given in the Appendix provides the theoretical basis for the work in sections II-IV.

Liquid and Gas Phase VER Rates

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II. Equation 1.1 From C(t,G) In this section, from an analysis of C(t,F) we will derive a molecular formula similar to the empirical eq 1.1. Before starting, we note that all quantities we deal with depend upon both T and F. Sometimes for brevity, we will, however, suppress their T-dependencies. For example, we abbreviate C(t,T,F) by C(t,F). We start with some preliminaries. a. Background Definitions and Results. In addition to C(t,F) of eq 1.5, we will require its frequency spectrum

F(ω,F) ≡

∫0∞ cos ωt C(t,F) dt

(2.1)

Notice that the theoretical eq 2.7 is identical in form to the empirical eq 1.1, except that G in eq 2.7 is given by the formula of eq 2.8 while G in eq 1.1 is found from experiment. So it might seem that we have successfully derived the molecular analogue of eq 1.1. However, this is not yet so since eq 2.6 is false. This follows since from eq 2.1 that F(ω,F) is never F-independent, because C(t,F) is F-dependent (Figure 1). We next give a valid derivation of eqs 2.7 and 2.8. To do this, in accord with Figure 1,we replace the CG hypothesis by the assumption that C(t,F) is F-dependent to order t2 only. c. Molecular Analogue of Eq 1.1 from the Hypothesis of Short Time G-Independence of C(t,G). To start, we consider the Maclaurin series for C(t,F). (i) Maclaurin Series for C(t,G). Since C(t,F) is even in t, its Maclaurin expansion is

A closely related quantity is the friction kernel β(ω,F) defined by

C(t,F) ) C(0)(F) + C(2)(F) 〈F˜2〉0 F(ω,F) kBT

β(ω,F) )

or

k(T,F) ) β(ωl,F)

(2.3)

where ωl is the liquid phase vibrational frequency of the relaxing normal mode. For the systems of section III, ωl differs only slightly from the gas phase frequency ωg,11 and so here we will assume

ωl ) ωg

(2.4)

Next from eqs 2.2-2.4, k(T,Fl) ) (〈F˜2〉0,l/kBT) F(ωg,Fl) and k(T,Fg) ) (〈F˜2〉0,g/kBT) F(ωg,Fl), which immediately yield

k(T,Fl) )

〈F˜2〉0,l 〈F˜2〉0,g

[

k(T,Fg)

F(ωg,Fl) F(ωg,Fg)

]

(2.5)

CG Hypothesis

where C[2n](F) ≡ [d2nC(t,F)/dt2n]t)0 for n ) 0, 1, 2, .... Equation 2.9 may be easily recast into the following standard form

t2 t4 C(t,F) ) 1 - [〈F˙˜ 2〉0 /〈F˜2〉0]F + [〈F¨˜ 2〉0 /〈F˜2〉0]F - ... 2! 4! (2.10) The F-dependent parameters [〈F˜˙ 2〉0/〈F˜2〉0]F, [〈F˜¨ 2〉0/〈F˜2〉0]F, ... are all positive, rendering eq 2.10 an alternating series. Comparing eqs 2.9 and 2.10 then yields that C(0)(F) ) 1 is positive, C(2)(F) is negative, C(4)(F) is positive, and so on. Concisely stated C(2n) ) (-)n|C(2n)|. With these comments, eq 2.9 becomes

C(t,F) ) 1 - |C(2)(F)|

(2.6)

t2 t4 + |C(4)(F)| - ... 2! 4!

(2.11) Next we define the time parameters τ2n(F) by

τ2-2(F) ) [〈F˜˙ 2〉0 /〈F˜2〉0]F

b. Molecular Analogue of Eq 1.1 from the Hypothesis of Full G-Independence of C(t,G). Given the CG hypothesis eq 1.5 and eq 2.1, it follows that F(ω,F) is F-independent and thus

F(ω, Fl) ) F(ω, Fg)

(2.9)

(2.2)

All modern theories of VER rates11,12 are based on formulas that connect T1 or k(T,F) ) T1-1 to β(ω,F). In the simplest approximation11 the connecting formulas are

T1 ) β-1(ωl,F)

t2 t4 + C(4)(F) + ... 2! 4!

τ4-4(F) ) [〈F˜¨ 2〉0 /〈F˜2〉0]F,... (2.12)

From eqs 2.9-2.12, we may view C(t,F) as a function of t and the τ2n’s and C(2n)(F) as a function of τ2n. Denoting these functions by C[t;τ2(F),τ4(F),...] ≡ C(t,F) and C(2n)[τ2n(F)] ≡ C(2n)(F), eq 2.11 becomes

Given eq 2.6 the term [...] in eq 2.5 is unity yielding

k(Τ,Fl) )

()

Fl k(Τ,Fg)G Fg

C[t;τ2(F),τ4(F),...] ) 1 - |C(2)[τ2(F)]|

(2.7)

where

G)

Fl-1〈F˜2〉0,l Fg-1〈F˜2〉0,g

(2.8)

t2 + 2!

|C(4)[τ4(F)]|

t4 - ... (2.13) 4!

(ii) Introduction of Short Time G-Independence into the Maclaurin Series. The order t2 F-independence of C(t,F) is introduced by assuming that τ2(F) ≡ τ is density-independent. This assumption yields that |C(2)[τ2(F)]| is F-independent and equal to τ-2, which, in turn, reduces eq 2.13 to

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Adelman

1 t2 t4 (4) + |C [τ (F)]| - ... 4 2! τ2 4! (2.14)

C[t;τ,τ4(F),...] ) 1 -

which shows explicitly that, to order t2, C(t,F) is F-independent. (iii) Frequency Spectrum G[ω;τ,τ4(G),...]. Similarly, the spectrum F(ω,F) may be viewed as a function of ω, τ, τ4(F), ..., which we will denote by F[ω;τ,τ4(F),...] ≡ F(ω,F). Using this notation, eq 2.1 becomes

F[ω;τ,τ4(F),...] )

∫0∞ cos ωtC[t;τ,τ4(F),...] dt (2.15)

(iW) Rate Equation. Using the new notation, eq 2.5 transcribes to

〈F˜2〉0,l

F[ωg;τ,τ4(Fl),...] k(T,Fl) ) 2 k(T,Fg) F[ωg;τ,τ4(Fg),...] 〈F˜ 〉0,g

(2.16)

Our central eqs 2.7 and 2.8 follow at once from eq 2.16 if we assume (as done earlier) that the frequency spectrum F[ωg;τ,τ4(F),...] is F-independent. The assumption is, of course, false. Next we derive eqs 2.7 and 2.8 from eq 2.16 without assuming that the spectrum is F-independent. (W) Wing Function. As in the Appendix, the limiting high frequency functional form of F[ω;τ,τ4(F),...] will be written as limωf∞ F[ω;τ,τ4(F),...], and will be called the wing function of F[ω;...]. The wing function is pertinent here since typical ωg’s are so high that one may replace with negligible error13 the spectrum F[ωg;...] by its simpler wing function limωgf∞ F[ωg;...]. Making this replacement in eq 2.16 yields

k(T,Fl) )

〈F˜2〉0,l 〈F 〉0,g ˜2

lim F[ωg;τ,τ4(Fl),...]

k(T,Fg)

ωgf∞

lim F[ωg;τ,τ4(Fg),...]

ωgf∞

(2.17) (Wi) G-Independent faf C0(t;τ). In the Appendix, we develop a faf C0(t;τ) that is related to but simpler than the true faf C[t;τ,τ4(F),...]. C0(t;τ) coincides with C[t;τ,τ(F)] to order t2 but differs from it at longer times. Most importantly, while the true faf is a function of both the F-independent time parameter τ and the F-dependent parameters τ4(F), τ6(F), ... and is thus F-dependent, C0(t;τ) depends only on τ and hence is F-independent. We will also require the F-independent spectrum F0(ω;τ) of C0(t;τ) defined as

F0(ω;τ) )

∫0∞ cos ωtC0(t;τ) dt

(2.18)

(Wii) Wing Function Theorem. As just noted,

C0(t;τ) * C[t;τ,τ4(F),...]

except at the shortest times (2.19)

Equations 2.15, 2.18, and 2.19 then apparently give

F0(ω;τ) * F[ω;τ,τ4(F),...]

(2.20)

However, in the Appendix we show that eq 2.20 may be replaced by a statement that is symmetrical with eq 2.19; namely,

F0(ω;τ) * F[ω;τ,τ4(F),...] except at the highest frequencies (2.21) A more precise statement of eq 2.21 is that the wing functions of F[ω;τ,τ4,(F),...] and F0(ω;τ) are identical or

lim F[ω;τ,τ4(F),...] ) lim F0(ω;τ)

ωf∞

(2.22)

ωf∞

Equation 2.22, which we call the wing function theorem, is proven in the Appendix. (Wiii) Proof of Eqs 2.7 and 2.8. Given eq 2.22, eq 2.17 reduces to

k(T,Fl) )

〈F˜2〉0,l 〈F˜2〉0,g

[

k(T,Fg)

lim F0(ωg;τ)

ωgf∞

lim F0(ωg;τ)

ωgf∞

]

(2.23)

Since the factor [...] just above is unity, eq 2.23 is identical to eqs 2.7 and 2.8. In summary, we have derived a molecular analogue of eq 1.1 from the hypothesis that to order t2 C(t,F) is F-independent, or from eq 2.14 that τ is F-independent. III. Confirmation of C(t,G)’s Short Time G-Independence for Three Simple VER Systems So far, support for this hypothesis consist solely of the MD faf plots in Figure 1. Next we give it much stronger support. Namely, for the three VER systems of Table 2 we show in Tables 3-5 that τ is nearly F-independent over T ranges of greater than 1000 K despite the fact that over these ranges τ is strongly T-dependent. We plotted some of the numerical results of Tables 3 and 4 in Figures 2 and 3. Figure 2 shows for the systems of Table 4 the F-independence of τ over a T range of 300-1500 K. Figure 3 shows the strong T-dependencies of τ (and also its F-independence) for the systems of Tables 3 and 4 over the same T range. a. Significance of τ. To make the content of Tables 3-6 clearer, we first note the significance of τ. (i) Time τ and VER Rates. Consider the relation

k(T,F) )

〈F˜2〉0 lim F (ω ;τ) kT ωgf∞ 0 gi

(3.1)

which follows from eqs 2.2-2.4 and eq 2.23. From eq 3.1 the significance of τ is evident. Namely, the variations of a system’s rate constant over its phase diagram strongly mirror the corresponding τ variations, like those illustrated in Tables 3-6. (ii) Gaussian Choice for C0(t;τ). The key eqs 2.7, 2.8, and 3.1 hold for all acceptable faf’s C0(t;τ). However, the content

Liquid and Gas Phase VER Rates

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TABLE 2: Lennard-Jones Potential Parameters from the Solute-Solvent (UV) and Solvent-Solvent (VV) Atomic Pairs Occurring in the Br2/Ar, I2/Xc, and I2/Ar Fluid Solutions Studied in This Paper solute-solvent atomic pair

σUV (Å)

εUV (K)

Br-Ar I-Xc I-Ar

3.51 3.94 3.63

143 324 201

solvent-solvent atomic pair

σVV(Å)

εVV(K)

Ar-Ar Xc-Xc

3.42 4.10

120 222

of Tables 3-6 is made optimally clear if one chooses C0(t;τ) to be the Gaussian faf so that

( )

C0(t;τ) ) CG(t) ≡ exp -

1 t2 2 τ2

(3.2)

From eq 2.18 the corresponding spectrum is

F0(ω;τ) ) FG(ω) )

( ) ( πτ2 2

1/2

1 exp - ω2τ2 2

)

(3.3)

Comparing eq 3.1 and 3.3 yields the Gaussian rate constant

( )

πτ2 kG(T,F) ) 2

1/2 〈F ˜ 2〉

1 exp - ωg2τ2 2

(

0

kBT

)

(3.4)

Equation 3.4 makes explicit the link between k(T,F) and τ just noted and thus renders transparent the centrality of τ in VER theory. We next turn to our method for evaluating τ. b. Integral Equation Method. Our evaluation of τ is made from the relation

τ)

[ ] 〈F˜2〉0

〈F˙2〉0

1/2

(3.5)

which, since τ ≡ τ2(F), follows from eq 2.12. We compute τ from eq 3.5 using a method developed elsewhere11c and summarized in Ssection 2B of ref 11b. In brief, since 〈F˜2〉0 is an equilibrium property and since 〈F˙2〉0 may be reduced to an equilibrium property, both quantities may be expressed as potential energy integrals over solute-solvent pair correlation functions. (As an example, for 〈F˜2〉0, such an expression is given in eqs 2.11-2.14 of ref 11b.) The pair correlation functions are computed as solutions to the PercusYevick integral equation. Then the potential energy integrals ˜˙2〉0, and hence τ. are found numerically to obtain 〈F˜2〉0, 〈F The method is viable whenever the Percus-Yevick (or another) integral equation is valid. Consequently, the method can yield τ over broad portions of the solvent’s phase diagram, as is illustrated in Tables 3-6. Next we turn to Tables 3-5. c. Tables 3-5. Tables 3-5 list τ values over broad ranges of thermodynamic states (280 states total) for three model dihalogen solute/rare gas solvent VER systems. These are meant to roughly simulate the Br2/Ar, I2/Xe, and I2/Ar fluid solutions.

All potentials are taken as additive atom-atom Lennard-Jones interactions, with the parameters given in Table 2. The Br2 and I2 bond lengths are taken as bBr2 ) 228.11 pm and bI2 ) 266.63 pm. (i) The Temperature Ranges. In Tables 3-5, T ) 1500 K is the highest temperature studied. This choice of maximum T is arbitrary. The lowest temperatures, however, are not arbitrary but rather are dictated by the critical temperatures Tc of the solvents; namely, Tc,Ar ) 150.65 K and Tc,Xe ) 289.75 K. Thus for the Ar solutions the minimum T’s = Tc,Ar while for the Xe solution the minimum T = Tc,Xe. Thus the solutions are typically supercritical fluids. Subcritical solutions were avoided due to numerical difficulties. (For example, for states on or near the l-g phase boundary, the Percus-Yevick equation often had multiple solutions or otherwise converged poorly.) (ii) The Density Ranges. At the tops of Tables 3-5 are the ranges of the solvent density F and of the equivalent (but more transparent) solvent packing fraction (PF). (PF is defined in Table 3.) The PF range of 10-4 to 0.40 covers the entire density spectrum-ideal gas to dense fluid. For example, at T ) 300 K and PF ) 10-4 the Ar and Xe solvents are nearly ideal gases with pressures of, respectively, 0.186 bar and 0.115 bar. d. Thermodynamic State Dependence of τ for Three Model Solutions. We next look at how τ varies over the states in Tables 3-5. (i) Isobaric T-Dependence of τ. To illustrate the isobaric T-dependence of τ, consider as a typical example the Br2/Ar solution. For definiteness take PF ) 0.1. Then reading down the PF ) 0.1 column of Table 3: τ ) 90.33, 68.69, and 30.66 fs at, respectively, T ) 150, 300, and 1500 K, showing a very strong isobaric T-dependence for τ (as seen in Figure 3). (ii) Isothermal G-Dependence of τ. In contrast, τ is nearly isothermally F-independent. Consider again as an example the Br2/Ar solution at, for example, T ) 300 K. Reading across the T ) 300 K row shows that from PF ) 10-4 to 0.10 τ is fixed at 66.69 fs, and thus it remains constant to four significant figures as F increases by a factor of 1000. As F increases by another 4-fold so that PF ) 0.40, τ decreases only very slightly to 66.67 fs. Thus at T ) 300 K over a factor of 4000 density range τ changes by only 0.03%. Similar F-independent behavior is found at other T’s for the Br2/Ar system. Thus over the whole T-range of Table 3, the τ’s at PF ) 10-4 and 0.10 differ by only 0.00-0.03 fs. Also the percent changes in τ over the full range PF ) 10-4 to 0.40 are for all T’s < 0.35%. Specifically, these percent changes are 0.33% at T ) 150 K, 0.14% at T ) 200 K, 0.02% at T ) 400 K, 0.08% at T ) 600 K, 0.14% at T ) 800 K, and 0.23% at T ) 1500 K. From Table 3, the largest variations in τ occur for the smallest T’s and are probably due to enhanced error in our algorithm near Tc,Ar ) 150.65 K. (Percent changes up to PF ) 0.5 while still small are somewhat larger than those just given, probably again due to greater numerical error at high F.) The τ values in Tables 4 and 5 like those of Table 3 show isothermal F independence (Figure 2). Thus for I2/Xe solution of Table 4, for the full tabulated T range all τ’s at PF ) 0.10 differ negligibly from their values at PF ) 10-4. Also the maximum and minimum percent changes in τ over the range PF ) 10-4 to 0.40 are 0.24% (at 600 K) and 0.02% (at 280 K). Similarly, for the I2/Ar solution of Table 5, for all T’s the τ’s are virtually F-independent over the thousand-fold range PF )

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TABLE 3: Time τ (fs) Computed from Eq 3.5 for the Br2/Ar Solution As a Function of Temperature T (K) and Number Density G (cm-3) or Packing Fraction PF, where (PF) ) 1/6πGσVV3 with σVV as in Table 2a 10-22F (cm-3):

0.4474 × 10-3

0.04774

0.2387

0.4774

0.9549

1.432

1.910

PF:

10-4

0.01

0.05

0.10

0.20

0.30

0.40

90.31 87.87 83.55 79.82 73.68 68.79 66.69 58.47 52.68 48.31 42.03 37.64 34.36 30.69

90.32 87.87 83.55 79.82 73.69 68.79 66.69 58.47 52.68 48.31 42.03 37.64 34.36 30.68

90.32 87.88 83.56 79.83 73.69 68.80 66.69 58.47 52.68 48.30 42.02 37.63 34.35 30.67

90.33 87.89 83.57 79.84 73.69 68.80 66.69 58.47 52.67 48.29 42.01 37.62 34.34 30.66

90.32 87.88 83.55 79.83 73.68 68.79 66.68 58.45 52.65 48.27 41.98 37.60 34.32 30.64

90.20 87.78 83.48 79.77 73.65 68.77 66.66 58.44 52.64 48.26 41.96 37.58 34.29 30.62

90.01 87.63 83.38 79.71 73.63 68.77 66.67 58.46 52.66 48.27 41.97 37.58 34.30 30.62

T (K) τ (fs) 150 160 180 200 240 280 300 400 500 600 800 1000 1200 1500 a

Notice that for all T’s τ is nearly isothermally F-independent.

TABLE 4: Same as Table 3 Except for the I2/Xe Solution 10-22F (cm-3): PF

0.2771 × 10-3 10

-4

0.02771

0.1386

0.2771

0.5542

0.8313

1.108

0.01

0.05

0.10

0.20

0.30

0.40

115.31 112.03 99.12 89.94 82.95 72.83 65.69 60.30 54.20

115.32 112.04 99.13 89.94 82.95 72.83 65.69 60.29 54.19

115.38 112.09 99.15 89.95 82.95 72.82 65.67 60.26 54.16

115.33 112.05 99.16 89.98 82.99 72.84 65.69 60.27 54.16

115.30 112.07 99.30 90.14 83.15 72.98 65.80 60.37 54.23

T (K) 280 300 400 500 600 800 1000 1200 1500

115.32 112.03 99.11 89.94 82.95 72.83 65.70 60.30 54.20

τ (fs) 115.33 112.06 99.14 89.94 82.95 72.82 65.68 60.28 54.18

TABLE 5: Same as Table 3 Except for the I2/Ar Solution 10-22F (cm-3): PF:

0.4474 × 10-3 10

-4

0.04774

0.2387

0.4774

0.9549

1.432

1.910

0.01

0.05

0.10

0.20

0.30

0.40

88.70 84.94 78.71 73.71 71.56 63.09 57.08 52.51 45.90 41.26 37.77 33.83

88.70 84.95 78.72 73.72 71.56 63.09 57.08 52.51 45.91 41.27 37.77 33.83

88.70 84.95 78.72 73.73 71.57 63.10 57.09 52.52 45.91 41.27 37.77 33.83

88.63 84.89 78.69 73.71 71.56 63.11 57.11 52.54 45.93 41.28 37.78 33.84

88.41 84.72 78.60 73.67 71.54 63.13 57.14 52.58 45.96 41.31 37.81 33.86

88.17 84.56 78.54 73.67 71.55 63.20 57.23 52.66 46.04 41.38 37.87 33.91

T (K) τ (fs) 180 200 240 280 300 400 500 600 800 1000 1200 1500

88.70 84.95 78.71 73.71 71.56 63.09 57.07 52.51 45.90 41.26 37.77 33.83

10-4 to 0.10. Also the percent changes in τ over the full PF range 10-4 to 0.40 are typically ∼0.25% with the maximum and minimum changes being 0.60% (at 180 K) and 0.10% (at 300 K). However, it is not yet certain that this evidence has fundamental significance. The doubt arises because of the ˜˙2〉0]1/2. Namely, form of eq 3.5, which gives τ as τ ) [〈F˜2〉0/〈F ˜˙2〉0 accidently cancel if the F-dependencies of 〈F˜2〉0 and 〈F when forming τ, then the apparent isothermal F-independence

of τ in Tables 3-5 is artifactual. This is not impossible since ˜˙2〉 are the potential energy integrals for 〈F˜2〉 and 〈F 0

0

similar.11b-d Next we rule out such artifactual cancellations and, hence, clinch the fundamental significance of Tables 3-5. (The results of Table 3-5 would also be artifactual if 〈F˜2〉0 and ˜˙ 2〉0 were individually F-independent. This is, however, 〈F not the case.11b)

Liquid and Gas Phase VER Rates

J. Phys. Chem. A, Vol. 114, No. 16, 2010 5237 (ii) Fundamental Significance of Tables 3-5. Suppose ˜˙2〉0 actumomentarily that the F-dependencies of 〈F˜2〉0 and 〈F ally did fortuitously cancel, rendering the isothermal constancy of τ in Tables 3-5 artifactual. Since the potential energy ˜˙2〉0 used to form integrals for the approximations to 〈F˜2〉0 and 〈F τ1/2 are similar to the corresponding exact forms, the F-dependencies of the approximations should then also cancel, rendering τ1/2 isothermally F-independent. This prediction, however, conflicts with Table 6, thus ruling out the possibility of accidental cancellation, and hence verifying the fundamental significance of Tables 3-5. f. Tables 3-6 and the Rate Correlation Eq 1.1. The fluid solutions of Table 2 were chosen without bias, merely as representative simple VER systems. Thus Tables 3-6 and Figures 2 and 3 significantly support our hypothesis that as t f 0 C(t,F) is isothermally F-independent, and that, consequently, the molecular derivation of eq 1.1, given in section 2c, is sound. IV. Summary and Discussion

Figure 2. Gaussian decay time τ as a function of packing fraction for the Ι2/Xe system of Table 4. The time τ is nearly independent of packing fraction over the range PF ) 10-4 (ideal gas) to PF ) 0.4 (dense fluid) for temperatures over the range 300-1500 K.

Figure 3. Gaussian decay time as a function of temperature for the Br2/Ar and Ι2(Xe) systems. Both systems show that τ decreases strongly with T as is needed for the increase of k(T,P) with T. All curves are plotted over the PF range of Figure 2 showing a PF independence consistent with that of Figure 2.

e. Comparison of “Exact” and Approximate Evaluations of τ. To do this, we compare τ with an approximation τ1/2 computed from the simplified form of eq 3.5 of eqs 2.15 and 2.16 of ref 11b. The results are given in Table 6. (i) Near G-Independence of τ Wersus the Strong G-Dependence of τ1/2. In Table 6, we compare τ and τ1/2 for the three model solutions over the range PF ) 10-4 to 0.40 and for representative temperatures from near Tc to 1200 K. Relevant to our work here is that Table 6 shows that τ1/2 always has a much stronger isothermal F-dependence than τ. Thus let pc(τ) and pc(τ1/2), be the percent changes in, respectively, τ and τ1/2 when PF is increased from 10-4 to 0.4. Then, for example, (i) pc(τ) ) 0.04% and pc(τ1/2) ) 8.23% for I2/Xe at T ) 300 K; (ii), pc(τ) ) 0.29% and pc(τ1/2) ) 8.32% for I2/Ar at 600 K; and (iii) pc(τ) ) 0.17% and pc(τ1/2) ) 10.85% at Br2/Ar at 1200 K. In summary, the above isothermal changes with density for τ1/2 are ∼10-100 times greater than those for τ.

The l-g rate correlation eq 1.1 is of such significance and generality that it may be properly called an empirical law of VER. Unsurprisingly, much effort has been expended on the molecular basis of this law. Virtually all of this effort has dealt with the development1,2,5 and application1-3 of the IBC model. As outlined in the Introduction this model gives a molecular interpretation and derivation of eq 1.1, along with a theoretical form for G. However, the IBC model has weaknesses6 as well as strengths. Thus here we proposed and tested a simple hypothesis that yields (without doubtful physical assumptions) eq 1.1. Namely, we proposed that to order t2 the faf C(t) ) C(t,F) of eq 1.5 is F-independent. The validations of this hypothesis in Figure 1 and Tables 3-6 lend it credence but, of course, do not prove it. For this reason, we next note the following. The emergence of eq 1.1 from time correlation function formulas for k(T,F)11b,f,g virtually requires that the time correlation function in question, in the case C(t,F), be F-independent over part of its time range. We will next argue that the F-independent part is the domain of shortest times, in accord with our hypothesis. To start, note that the time dependence of F˜(t) in eq 1.5 is determined by the relative motions of the solute and its nearby solvent molecules. Thus if C(t,F) is to be F-independent over a time domain, the corresponding motions must also be F-independent. However, motions can only be F-independent if they are unaffected by the intermolecular forces (so that each molecule moves like a free particle, independent of forces from its F-dependent “cage”); that is, if they are the system’s shortest time inertial motions, familiar from time-dependent solvation phenomena.14 In summary, our central hypothesis is not a mere guess. Rather, experimental fact, eq 1.1, virtually requires it. Additionally, further inference from eq 1.1 implies a plausible new basis for VER, inertial dynamics. This new basis has common features with but is sharply distinct from the IBC model. Like the IBC model it attributes VER to isolated binary solute-solvent interactions. However, the binary interactions in the new and IBC pictures are very different. Thus, as noted in the Introduction, in Herzfeld’s8 traditional IBC picture, the binary interactions are postulated to arise from rare high energy collisions. However, the importance or even the existence of these collisions has never

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TABLE 6: Density-Dependencies of Exact τ and Approximate τ1/2 (τ1/2’s in Brackets) Evaluations of Eq 3.5 at Representative Temperatures for the Model Solutionsa PF:

10-4

0.01

0.05

90.31 [80.22] 66.69 [60.88] 48.31 [45.05] 34.36 [32.53]

90.32 [80.06] 66.69 [60.78] 48.31 [44.97] 34.36 [32.46]

90.32 [79.38] 66.69 [60.34] 48.30 [44.63] 34.35 [32.19]

10-4

0.01

0.05

0.10

88.70 [79.56] 71.56 [65.26] 52.51 [48.78] 37.77 [35.55]

88.70 [79.44] 71.56 [65.17] 52.51 [48.70] 37.77 [35.48]

88.70 [78.94] 71.56 [64.78] 52.51 [48.38] 37.77 [35.22]

10-4

0.01

112.03 [99.43] 82.95 [75.63] 60.30 [56.11]

112.03 [99.24] 82.95 [75.51] 60.30 [56.02]

0.10

0.20

0.30

0.40

90.20 [74.94] 66.66 [56.50] 48.26 [41.58] 34.29 [30.00]

90.01 [72.72] 66.67 [54.56] 48.27 [40.10] 34.30 [29.00]

0.20

0.30

0.40

τ (fs) and τ1/2 (fs) for I2/Ar 88.70 [78.29] 71.57 [64.25] 52.52 [47.93] 37.77 [34.86]

88.63 [76.98] 71.56 [63.04] 52.54 [46.90] 37.78 [34.06]

88.41 [75.44] 71.54 [61.65] 52.58 [45.76] 37.81 [33.22]

88.17 [73.79] 71.55 [60.36] 52.66 [44.72] 37.87 [32.44]

0.05

0.10

0.20

0.30

0.40

112.04 [98.50] 82.95 [75.03] 60.29 [55.63]

τ (fs) and τ1/2(fs) for I2/Xe 112.06 [97.54] 82.95 [74.37] 60.28 [55.09]

112.09 [95.76] 82.95 [72.76] 60.26 [53.77]

112.05 [93.59] 82.99 [70.70] 60.27 [52.18]

112.07 [91.25] 83.15 [68.54] 60.37 [50.49]

T (K) 150 300 600 1200 PF:

τ (fs) and τ1/2 (fs) for Br2/Ar 90.33 90.32 [78.42] [76.85] 66.69 66.68 [59.76] [58.32] 48.29 48.27 [44.15] [42.98] 34.34 34.32 [31.82] [30.97]

T (K) 180 300 600 1200 PF: T (K) 300 600 1200 a

The τ’s but not the τ1/2’s are nearly F-independent at fixed T.

been rigorously justified. Moreover, Herzfeld’s mechanism conflicts with low temperature simulations.9 In contrast, for our picture the isolated binary interactions without doubt exist. They are merely the ordinary interactions between solute and solvent molecules when both are in the initial inertial phases of their trajectories. Relatedly, while Herzfeld unsymmetrically and arbitrarily selects a particular “first shell” solvent molecule to induce relaxation, in our picture all neighboring solvent molecules participate in VER. The interactions are, however, still binary since the solvent molecules move independently in the inertial regime. MD comparisons of our new inertial dynamics mechanism and the standard Herzfeld mechanism would be very helpful. Finally, we note that the importance of short time dynamics for VER has been noted by others.3h,15 Simpson and coworkers,3h in particular, have noted that this dynamics is expected to be F-independent. Appendix

and

F(ω) )

C(t) )

〈F˜(t)F˜〉0 〈F˜2〉0

(A.1)

(A.2)

(Equations A.1 and A.2 are eqs 1.5 and 2.1, with the latter’s F-dependencies suppressed.) In accord with focus of the main text, will be concerned here with the t f 0 and ω f ∞ limits of C(t) and F(ω) which we write as

lim C(t) ) lim C(t;τ,τ2,...)

(A.3)

lim F(ω) ) lim F(ω;τ,τ2,...)

(A.4)

tf0

tf0

and

ωf∞

In this Appendix we prove the wing function theorem eq 2.22. To start we define the normalized faf C(t) and its frequency spectrum F(ω) by

∫°∞ cos ωtC(t) dt

ωf∞

Notice in eqs A.3 and A.4 we use the detailed notation of eqs 2.14 and 2.15. This is done so that the nature of the simplifications of C(t) and F(ω) which occur as t f 0 and ω f ∞ are transparent. Before going on, we emphasize that in eqs A.3 and A.4 limtf0 C(t) does not denote C(0) ) 1 and limωf∞ F(ω) does not denote F(∞) ) 0. Rather, these and other similar limits appearing

Liquid and Gas Phase VER Rates

J. Phys. Chem. A, Vol. 114, No. 16, 2010 5239

here denote the asymptotic functional forms which C(t) and F(ω) tend to as t f 0 and ω f ∞. For example, for the simplest case of a Gaussian faf and its spectrum, from eqs 3.2 and 3.3 the asymptotic functional forms are merely the functions themselves. That is,

( )

lim CG(t) ) exp tf0

1 t2 2 τ2

and

lim FG(ω) )

ωf∞

( ) ( πτ2 2

1/2

1 exp - ω2τ2 2

)

Equations A.10 and A.11 suggest that no universal wing function type is possible. Fortunately, this is not so, since analysis13 shows that the long-range faf’s are unphysical. Thus, only Gaussian-like wing functions occur. Consequently, we will henceforth focus solely on short-range fafs, for brevity to simply be called faf’s, defined as those which conform to eq A.8. Next we discard the imprecise eq A.6. To start, we introduce finite times tcut that conform to tcut J t* but that are otherwise arbitrary. Then the ω f ∞ limit of eq A.2 may be written as

(A.5)

lim F(ω) ) lim lim

ωf∞ tcutf∞

ωf∞

These precise interpretations of limits have been emphasized since they are essential to our later contour integration analysis. To start, we temporarily split C(t)at a time t ) t* into short time “head” and long time “tail” parts CH(t) and CT(t) defined by13

CH(t) ≡

{

C(t) for t e t* 0 for t > t*

and

CT(t) ≡

{

0 for t e t* C(t) for t > t*

(A.6)

∫0t

cut

(A.12)

C(t) cos ωt dt

(Equation A.12 holds only for short-range faf’s. For long-range faf’s it must be replaced by limωf∞ F[ω] ) limωf∞ ∫0∞C(t) cos ωt dt. Because of this replacement, the results of this Appendix are invalid for long-range faf’s.) To continue, we transform eq A.12 into a complex time integration17 over the contour of Figure 4, C ≡ C1 + C2 + C3, where 0 < χ < π and where tcut f ∞. We thus introduce the complex time z ) t + ir, the complex faf C(z), and the complex integrand

I(z) ≡ C(z) exp(iωz)

(The sharp cutoff at t* is, of course, not rigorous, so the discussion of eqs A.6-A.11 is only qualitative. However, from eq A.12 on, our analysis is rigorous.) From eq A.6 C(t) ) CH(t) + CT(t) and thus from eq A.2 F(ω) ) FH(ω) + FT(ω) where

(A.13)

We then consider the contour integral

∫C I(z) dz ) ∫C I(z) dz + ∫C I(z) dz + ∫C I(z) dz 1

2

3

(A.14) FH(T)(ω) ≡

∫0

∞

CH(T)(t) cos ω dt

(A.7)

Next two distinct faf types can occur.13 These have, respectively, short and long-range CT(t)’s. These distinct types yield wing functions with, respectively, Gaussian-like16 and slowly decaying (for example exponential-like) forms. The two types are defined precisely by the conditions13

lim FT(ω) , lim FH(ω)

ωf∞

ωf∞

for short range CT(t)’s

From eq A.12 and Figure 4

lim F(ω) ) lim Re

ωf∞

ωf∞

∫C I(z) dz

(A.15)

1

yielding from eq A.14

lim F(ω) ) lim Re[

ωf∞

ωf∞

∫C I(z) dz - ∫C I(z) dz - ∫C I(z) dz] 2

3

(A.16)

(A.8)

We will show that ∫CI(z) dz and ∫C2I(z) dz vanish so that eq A.16 reduces to

and

lim FT(ω) . lim FH(ω)

ωf∞

ωf∞

for long range CT(t)’s

lim F(ω) ) - lim Re

(A.9) Given eqs A.6-A.9 the wing functions for the two types have the distinct Fourier integral representations

ωf∞

ωf∞

and ∞ CT(t) cos ωt dt ∫ t* ωf∞

ωf∞

for long range CT(t)’s and slowly decaying wings (A.11)

(A.17)

and

C(z) f 0 on the contour C2

lim F(ω) ) lim FT(ω) ) lim

ωf∞

3

C(z) is an analytic on and inside the contour C (A.18a)

∫0t* CH(t) cos ωt dt ωf∞

for short range CT(t)’s and Gaussian-like wings (A.10)

∫C I(z) dz

The vanishings just noted follow from two properties of C(z), namely,

lim F(ω) ) lim FH(ω) ) lim

ωf∞

ωf∞

(A.18b)

Given eqs A.18a, the vanishings of ∫C and ∫C2 follow from, respectively, the Cauchy integral theorem and (an adaptation of) the standard proof of Jordan’s lemma.

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Next we validate eqs A.18. To start, we prove that C(z) is an entire function. (That is, C[z] is analytic for all finite |z|.) To show this, we show that the Maclaurin series for C(t)

t2 t4 C(t) ) 1 - 〈F˙˜ 2〉0 /〈F˜2〉0 + 〈F¨˜ 2〉0 /〈F˜2〉0 + · · · 2! 4! (A.19) converges for all t. Then from complex variable theory, the radius of convergence of the complex Maclaurin series for C(z) is unbounded, and hence, C(z) is entire. Equivalently, eq A.19 continued to the complex plane

z2 z4 C(z) ) 1 - 〈F˙˜ 2〉0 /〈F˜2〉0 + 〈F¨˜ 2〉0 /〈F˜2〉0 + · · · 2! 4! (A.20)

∞ exp[i(χ + ωR cos χ)] ∫ 0 ωf∞

lim F(ω) ) Re[ lim

ωf∞

×

exp(-ωR sin χ)C[R exp(iχ)] dR] (A.22) where we have let tcut f ∞. We further define the new variable X by

X ) ωR sin χ

(A.23)

where X > 0 since for 0 < χ < π sin χ > 0. Changing the integration variable in eq A.22 from R to X yields

[

∞ (ω sin χ)-1 × ∫ 0 ωf∞

lim F(ω) ) Re lim

ωf∞

[ ω sinX χ exp(iχ)] dX]

exp[i(χ + X cos χ)]exp(-X)C converges for all finite z. The convergence of C(t) for -∞ < t < ∞ follows from Newton’s equation of motion, which implies that for all t that the time-dependence of F˜ is exactly generated by the Liouville operator power series

1 + iLt +

(iL)2t2 + · · · ≡ exp(iLt) 2!

(A.21)

Consequently, deriving eq A.19 from eq A.21 and eq A.1 written as C(t) ) 〈F˜2〉0-1〈[exp(iLt)F˜]F˜〉0 yields that eq A.19 converges for all t, implying that C(z) is entire and that, hence, eq A.18a holds. Moreover, since C(z) is entire, it is continuous near the real axis. Thus keeping χ > 0, but letting it become arbitrarily small, on C2 the faf C(z) can be made arbitrarily close to C(tcut). Then since C(tcut) f 0 eq A.18b follows. In summary, we have validated eqs A.18a and thus justified eq A.17. Next, to analyze eq A.17, we use the polar form of z, which on C3 is from Figure 4. z ) R exp(iχ). Then ∫C3I(z) dz in eq A.17 reduces to the one-dimensional integral, -∫0tcutexp(iχ)I(R exp[iχ]) dR. Then eqs A.13 and A.17 yield

(A.24) In eq A.24 the integrand is negligible for X . 1, since then the factor exp(-X) is negligible while the factor Re[(exp(i(χ + X cos χ))C[ ]] is bounded. Thus one may cut off the integral in eq A.24 at a finite upper limit X† . 1, yielding with arbitrarily small error

[

lim F(ω) ) Re lim

ωf∞

ωf∞

∫0X (ω sin χ)-1 †

×

[ ω sinX χ exp(iχ)] dX]

exp[i(χ + X cos χ)] exp(-X)C

(A.25) Next since X in eq A.25 is bounded by X†, over the whole integration range the argument of C[ ], (X)/(ω sin χ) exp(iχ), approaches zero as ω f ∞. Thus in eq A.25 we may replace C[ ] by another valid even faf C0[ ], which approaches C(z) as |z| f 0 but which is simpler than C(z) at larger z. Then eq A.25 reduces to

Figure 4. Contour C ) C1 + C2 + C3 used in the derivation of eq A.28 from eq A.13.

Liquid and Gas Phase VER Rates

[

∫

X†

lim F(ω) ) Re lim

ωf∞

0

ωf∞

J. Phys. Chem. A, Vol. 114, No. 16, 2010 5241

(ω sin χ)-1 exp[i(χ + X cos χ)] ×

exp(-X)C0

[ ω sinX χ exp(iχ)] dX]

(A.26)

Then reversing the steps which led from eq A.22 to eq A.25, eq A.26 becomes

lim F(ω) ) Re[ lim

ωf∞

ωf∞

∫0∞ exp[i(χ + ωR cos χ)]

×

exp(-ωR sin χ)C0[R exp(iχ)] dR] (A.27) Finally, we return to the real axis by applying to eq A.27 the inverse of the steps leading from eq A.12 to eq A.22. This yields ∞ C0(t) cos ωt dt ∫ 0 ωf∞

lim F(ω) ) lim

ωf∞

(A.28)

Equations A.27 and A.28 are identical to, respectively, eqs A.22 and A.12 if in the latter one lets C(t) f C0(t). This substitution is crucial since it shows that limωf∞ F(ω) is determined by C0(t), which must coincide with C(t) only at short times. This is a great simplification since while C(t), as seen in eq A.3, depends on all parameters in eq 2.12, C0(t) may be taken to depend on only the order t2 parameter τ2 ) τ. That is,

C0(t) ) C0(t;τ)

(A.29)

Next we define, as in eq 2.19

F0(ω) )

∫0∞ cos ωtC0(t;τ) dt

(A.30)

Equations A.29 and A.30 show that F0(ω) ) F0(ω;τ). That is, in contrast to F(ω) ) F(ω;τ,τ4,...), which depends on all the parameters of eq 2.12, F0(ω) depends only on τ. Finally, from eqs A.28 and A.30, limωf∞ F(ω) ) limωf∞ F0(ω). Thus we have proven

lim F(ω) ) lim F(ω;τ,τ4,...) ) lim F0(ω;τ)

ωf∞

ωf∞

ωf∞

(A.31) Equation A.31 is the wing function theorem eq 2.22. (One more comment is needed. Order t2 agreement of a function with C[t] is not sufficient to guarantee that this function is on acceptable asymptotic faf C0[t]; namely, one in eq A.28 yields a physical wing function. Analysis shows that an acceptable C0[t] must also share the mathematical properties of C[t]. For example like C[z], C0[z] must be entire.)

References and Notes (1) Chesnoy, J.; Gale, G. M. Annu. ReV. Phys. Fr. 1984, 9, 893. (2) For ultrasound studies of the l-g correlation see, for example: (a) Herzfeld, K. F.; Litowitz, T. A. Absorption and Dispersion of Ultrasonic WaVes; Academic Press: New York, 1959, especially section 95. (b) Madigosky, W. A.; Litowitz, T. A. J. Chem. Phys. 1961, 34, 489 (especially Figure 3). (3) For representative laser studies of the l-g correlation see: (a) Brueck, S. R. J.; Deutsch, T. F.; Osgood, R. M., Jr. Chem. Phys. Lett. 1977, 51, 339. (b) Gale, G. M.; Delalande, C. Chem. Phys. 1978, 34, 205. (c) Protz, R.; Maier, M. Chem. Phys. Lett. 1979, 64, 27. (d) Chesnoy, J.; Ricard, D. Chem. Phys. 1982, 67, 347. (e) Chatelet, M.; Kieffer, J.; Oksengorn, B. Chem. Phys. 1983, 79, 413. (f) Knutson, J. T.; Weitz, E. Chem. Phys. Lett. 1984, 104, 71. (g) Lupo, D. W.; Lucas, D. J. Phys. Chem. 1986, 90, 5105. (h) Andrew, J. J.; Harriss, A. P.; McDermott, D. C.; Williams, H. T.; Madden, P. A.; Simpson, C. J. S. M. Chem. Phys. 1989, 139, 369 (especially sections 4 and 5). (i) Zittel, P. F. J. Phys. Chem. 1991, 95, 6802. (j) Schwarzer, D.; Troe, J.; Zereka, M. J. Chem. Phys. 1997, 107, 8380. (k) Dlott, D. D. Chem. Phys. 2001, 266, 149. (4) The longest measured T1’s J 1 s. For example, see: (a) Brueck, S. R. J.; Osgood, R. M., Jr. Chem. Phys. Lett. 1976, 39, 568. (b) Chandler, D. W.; Ewing, G. E. J. Chem. Phys. 1980, 73, 4904. The shortest measured T1’s are j1 ps. For example, see: (c) Gale, G. M.; Gallot, G.; Hoche, F.; Lascoux, N.; Bratos, S.; Leicknam, J.-Cl Phys. ReV. Lett. 1999, 82, 1068. (d) Laenen, R.; Gale, G. M.; Lascoux, N. J. Phys. Chem. A 1999, 103, 10708. (e) Deak, J. C.; Rhea, S. T.; Iwaki, L. K.; Dlott, D. D. J. Phys. Chem. A 2000, 104, 4866. (5) Davis, P. K.; Oppenheim, I. J. Chem. Phys. 1999, 57, 505. (6) For example, see: (a) Fixman, M. J. Chem. Phys. 1961, 34, 309. (b) Zwanzig, R. J. Chem. Phys. 1961, 34, 1931. (c) Dardi, P. S.; Cukier, R. I. J. Chem. Phys. 1988, 89, 4145. (7) Reference 1, pp 915-916 and 944-945. (8) (a) Reference 2a, p 95. (b) Herzfeld, K. F. J. Chem. Phys. 1962, 36, 3305. (9) For example, see: Buckingham, M. R.; Williams, H. T.; Pennington, R. S.; Simpson, C. J. S. M.; Maricq, M. M. Chem. Phys. 1985, 98, 179 (section 4.3). (10) Chesnoy, J.; Weis, J. J. J. Chem. Phys. 1986, 84, 5378. (11) For our work see: (a) Adelman, S. A.; Stote, R. H. J. Chem. Phys. 1988, 88, 4397. Stote, R. H.; Adelman, S. A. J. Chem. Phys. 1988, 88, 4415. (b) Adelman, S. A.; Muralidhar, R.; Stote, R. H. J. Chem. Phys. 1991, 95, 2738. (c) Adelman, S. A.; Ravi, R.; Muralidhar, R.; Stote, R. H. AdV. Chem. Phys. 1993, 84, 73. (d) Adelman, S. A.; Stote, R. H.; Muralidhar, R. J. Chem. Phys. 1993, 99, 1320. (e) Adelman, S. A.; Muralidhar, R.; Stote, R. H. J. Chem. Phys. 1993, 99, 1333. (f) Miller, D. W.; Adelman, S. A. J. Chem. Phys. 2002, 117, 2672. (g) Miller, D. W.; Adelman, S. A. J. Chem. Phys. 2002, 117, 2688. (12) For the work of others, for example see: (a) Berne, B. J.; Tuckerman, M. E.; Straub, J. E.; Bug, A. L. R. J. Chem. Phys. 1990, 93, 5084. (b) Whitnell, R. M.; Wilson, K. R.; Hynes, J. T. J. Chem. Phys. 1992, 96, 5354. (c) Ferrario, M.; Klein, M. L.; McDonald, I. R. Chem. Phys. Lett. 1993, 213, 537. (d) Egorov, S. A.; Skinner, J. L. J. Chem. Phys. 1996, 105, 7047. (e) Poulsen, J.; Nymand, T. M.; Keiding, S. R. Chem. Phys. Lett. 2001, 43, 581. (f) Sibert, E. L., III; Roy, R. J. Chem. Phys. 2002, 116, 237. (g) Chorny, I.; Viecelli, J.; Benjamin, I. J. Chem. Phys. 2002, 116, 8904. (13) Adelman, S. A. J. Phys. Chem. B 2009, 113, 5528. (14) For example, see: Fleming, G. R.; Cho, M. Annu. ReV. Phys. Chem. 1996, 47, 109, and the many references therein. (15) For example see: (a) Biswas, R.; Bhattacharyya, S.; Bagchi, B. J. Chem. Phys. 1998, 108, 4963. (b) Larsen, R. E.; Stratt, R. M. J. Chem. Phys. 1999, 110, 1036. (c) Bagchi, B.; Biswas, R. AdV. Chem. Phys. 1999, 109, 207. (d) Deng, Y. Q.; Ladanyi, B. M.; Stratt, R. M. J. Chem. Phys. 2002, 117, 10752. (16) Equation A.10 type wing functions derive from CH(t), which simulations (like those of ref 11f) show to be Gaussian-like; hence, our term Gaussian-like wing functions. (17) Complex variable theorems used here are proven in: Arkfen, G. B.; Weber, H. J. Mathematical Methods for Physicists; Academic Press: New York, 1995; Chapter 6.

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