The short-time intramolecular dynamics of solutes in liquids. I. An instantaneous-normal-mode theory for friction

Journal of Chemical Physics

Volumn 105, Issue 22, 1996, Pages 10050-10071

  Goodyear, Grant ^a   Stratt, Richard M ^a  

^a BROWN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001237035     PISSN: 00219606     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.472835     Document Type: Article

References (163)

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115. We assume appropriate mass-weighting here (see Ref. 73).
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117. In principle the tagged coordinate, x, could well be vectorial in character. For notational simplicity, however, we shall assume x is a scalar in this development.
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119. This step is not a trivial as it may appear. Within a site-site pair-potential representation of the potential energy, each solute-solvent pair potential will involve both intramolecular and center-of-mass coordinates.
120. If orientational degrees of freedom are included, then the "masses" will not only contain instantaneous values of the bath coordinates (see Ref. 73), but may also depend on the time-dependent tagged coordinate. This situation can arise because the bath coordinates will often include the nontagged intramolecular solute coordinates: if x were the interatomic distance in a diatomic, then because of centrifugal distortion, the bath would contain angular momentum terms with an x-dependent moment of inertia. Since the effects are small in practice, we shall ignore this last complication for the purpose of this paper.
121. 0.
122. l(ω)], and the "R" and "I" subscripts denote the real and imaginary parts of the transform, respectively.
123. R(ω) has a delta function contribution at zero frequency - but the J(ω)'s are identical.
124. eq=1.25σ and spatially fixed. Note that only configurational information, not dynamics, is used in these INM calculations.
125. LJ the (Lennard-Jones) unit of time and the solvent mass, respectively]. Note also that the INM spectrum of couplings becomes complex if there are any imaginary modes in the INM spectrum. However, these imaginary modes make such a small contribution (here only 1%) to the associated influence spectrum [Eqs. (41)-(43)] that we feel justified in plotting only the real portion in this and subsequent spectrum-of-couplings figures.
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129. B.-C. Xu and R. M. Stratt, J. Chem. Phys. 92, 1923 (1990).
130. An implicit assumption here is that the kinetic energy depends only on velocities, not positions. This restriction is not fundamental, but working beyond it requires specifying the particulars of the system.
131. Insofar as the real-frequency modes are concerned, the initial configuration distribution (the one corresponding to the true potential surface) is just a nonequilibrium starting point from which the set of harmonic oscillators must relax according to their own dictates. The imaginary modes lead to unphysical behavior at long times simply because they have no stable equilibrium to reach.
132. D. A. McQuarrie, Statistical Mechanics (Harper and Row, New York, 1976), p. 502.
133. max, that will capture the entire spectrum to within an arbitrarily small tolerance.
134. 2)> [as opposed to the frequency-weighted spectrum, Eqs. (41) and (42)] will be nonzero at λ=0 as well.
135. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 231.
136. -2=∞.
137. 2 [see Ref. 73].
138. 4 snapshots were employed to reduce the noise further.
139. Usually, one needs only the (configuration-averaged) friction kernel and the potential of mean force to compute correlation functions from the GLE (see Ref. 39). If detailed solute dynamics are required, however, then the GLE must be solved as a stochastic differential equation, with the fluctuating force treated as a stochastic variable. See, for example, J. D. Doll and D. R. Dion, J. Chem. Phys. 65, 3762 (1976);
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141. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, Oxford, 1989), Chap. 4.
142. x exp(-st)f(t).
143. Recall also that ℱ is independent of the initial velocity of the tagged coordinate.
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145. R(ω) in the numerator. Contour integration proves that the added term vanishes. See P. J. Davis and P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975), pp. 147-150.
146. x(ω) is just the full density of states D(ω).
147. usp(ω)/2, and these are the quantities plotted as the single-particle J(ω)'s in Fig. 5.
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151. See also Deutch and Silbey, Ref. 45.
152. -(1/2)ω2 f t 2-1].
153. The influence spectrum is, of course, invariant to changing the signs of the mode coefficients.
154. Since the fluctuating force vanishes at time 0, we cannot literally integrate Eq. (83), with A the fluctuating force, in order to obtain the GLE friction. If one could do so the INM friction would satisfy the ordinary fluctuation-dissipation relation, Eq. (2). [See Sees. III C and IV C.] Nonetheless, our derivation does indeed show that Eq. (85), the analogue of Eq. (84), is correct.
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