The short-time intramolecular dynamics of solutes in liquids. II. Vibrational population relaxation

Journal of Chemical Physics

Volumn 107, Issue 8, 1997, Pages 3098-3120

  Goodyear, Grant ^a,b   Stratt, Richard M ^a  

^a BROWN UNIVERSITY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0042654594     PISSN: 00219606     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.474664     Document Type: Article

References (145)

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46. Gaussian and Exponential friction kernels have been explored in S. C. Tucker, J. Phys. Chem. 97, 1596 (1993), S. K. Reese, S. C. Tucker, and G. K. Schenter, J. Chem. Phys. 102, 104 (1995); G. D. Harp and B. J. Berne, Phys. Rev. A 2, 975 (1970); B. J. Berne and D. Forster, Annu. Rev. Phys. Chem. 22, 563 (1971); S. A. Adelman, R. Ravi, R. Muralidhar, and R. H. Stote, Adv. Chem. Phys. 84, 73 (1993).
47. Gaussian and Exponential friction kernels have been explored in S. C. Tucker, J. Phys. Chem. 97, 1596 (1993), S. K. Reese, S. C. Tucker, and G. K. Schenter, J. Chem. Phys. 102, 104 (1995); G. D. Harp and B. J. Berne, Phys. Rev. A 2, 975 (1970); B. J. Berne and D. Forster, Annu. Rev. Phys. Chem. 22, 563 (1971); S. A. Adelman, R. Ravi, R. Muralidhar, and R. H. Stote, Adv. Chem. Phys. 84, 73 (1993).
48. Gaussian and Exponential friction kernels have been explored in S. C. Tucker, J. Phys. Chem. 97, 1596 (1993), S. K. Reese, S. C. Tucker, and G. K. Schenter, J. Chem. Phys. 102, 104 (1995); G. D. Harp and B. J. Berne, Phys. Rev. A 2, 975 (1970); B. J. Berne and D. Forster, Annu. Rev. Phys. Chem. 22, 563 (1971); S. A. Adelman, R. Ravi, R. Muralidhar, and R. H. Stote, Adv. Chem. Phys. 84, 73 (1993).
49. Gaussian and Exponential friction kernels have been explored in S. C. Tucker, J. Phys. Chem. 97, 1596 (1993), S. K. Reese, S. C. Tucker, and G. K. Schenter, J. Chem. Phys. 102, 104 (1995); G. D. Harp and B. J. Berne, Phys. Rev. A 2, 975 (1970); B. J. Berne and D. Forster, Annu. Rev. Phys. Chem. 22, 563 (1971); S. A. Adelman, R. Ravi, R. Muralidhar, and R. H. Stote, Adv. Chem. Phys. 84, 73 (1993).
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68. B. M. Ladanyi and R. M. Stratt, J. Phys. Chem. 99, 2502 (1995); 100, 1266 (1996).
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70. 1 values reported in Ref. 40 should be replaced with the more accurate ones reported in the present paper.
71. R. E. Larsen, E. F. David, G. Goodyear, and R. M. Stratt, J. Chem. Phys. 107, 524 (1997).
72. (a) S. J. Schvaneveldt and R. F. Loring, J. Chem. Phys. 102, 2326 (1995);
73. (b) 104, 4736 (1996);
74. (c) J. Phys. Chem. 100, 10355 (1996).
75. P. Moore, A. Tokmakoff, T. Keyes, and M. D. Fayer, J. Chem. Phys. 103, 3325 (1995).
76. An alternative, liquid-theoretic INM approach to computing vibrational line shapes and vibrational friction kernels has been offered by Schvaneveldt and Loring, Ref. 43. Rather than deriving the INM friction and the INM dynamics more generally, and then deducing the line shape, the authors go after the INM line shape directly, observing that when they do so there is a quantity that seems to play the role of a friction.
77. D. W. Oxtoby, Mol. Phys. 34, 987 (1977); M. J. Fixman, J. Chem. Phys. 34, 369 (1961); R. J. Zwanzig, Chem. Phys. 34, 1931 (1961); R. J. Zwanzig, J. Chem. Phys. 36, 2227 (1962); P. S. Dardi and R. I. Cukier, ibid. 89, 4145 (1988); 95, 98 (1991).
78. D. W. Oxtoby, Mol. Phys. 34, 987 (1977); M. J. Fixman, J. Chem. Phys. 34, 369 (1961); R. J. Zwanzig, Chem. Phys. 34, 1931 (1961); R. J. Zwanzig, J. Chem. Phys. 36, 2227 (1962); P. S. Dardi and R. I. Cukier, ibid. 89, 4145 (1988); 95, 98 (1991).
79. D. W. Oxtoby, Mol. Phys. 34, 987 (1977); M. J. Fixman, J. Chem. Phys. 34, 369 (1961); R. J. Zwanzig, Chem. Phys. 34, 1931 (1961); R. J. Zwanzig, J. Chem. Phys. 36, 2227 (1962); P. S. Dardi and R. I. Cukier, ibid. 89, 4145 (1988); 95, 98 (1991).
80. D. W. Oxtoby, Mol. Phys. 34, 987 (1977); M. J. Fixman, J. Chem. Phys. 34, 369 (1961); R. J. Zwanzig, Chem. Phys. 34, 1931 (1961); R. J. Zwanzig, J. Chem. Phys. 36, 2227 (1962); P. S. Dardi and R. I. Cukier, ibid. 89, 4145 (1988); 95, 98 (1991).
81. D. W. Oxtoby, Mol. Phys. 34, 987 (1977); M. J. Fixman, J. Chem. Phys. 34, 369 (1961); R. J. Zwanzig, Chem. Phys. 34, 1931 (1961); R. J. Zwanzig, J. Chem. Phys. 36, 2227 (1962); P. S. Dardi and R. I. Cukier, ibid. 89, 4145 (1988); 95, 98 (1991).
82. D. W. Oxtoby, Mol. Phys. 34, 987 (1977); M. J. Fixman, J. Chem. Phys. 34, 369 (1961); R. J. Zwanzig, Chem. Phys. 34, 1931 (1961); R. J. Zwanzig, J. Chem. Phys. 36, 2227 (1962); P. S. Dardi and R. I. Cukier, ibid. 89, 4145 (1988); 95, 98 (1991).
83. M. E. Paige and C. B. Harris, Chem. Phys. 149, 37 (1990); J. Chem. Phys. 93, 3712 (1990); J. Chesnoy and J. J. Weis, ibid. 84, 5378 (1986).
84. M. E. Paige and C. B. Harris, Chem. Phys. 149, 37 (1990); J. Chem. Phys. 93, 3712 (1990); J. Chesnoy and J. J. Weis, ibid. 84, 5378 (1986).
85. M. E. Paige and C. B. Harris, Chem. Phys. 149, 37 (1990); J. Chem. Phys. 93, 3712 (1990); J. Chesnoy and J. J. Weis, ibid. 84, 5378 (1986).
86. P. A. Rejto and D. Chandler, J. Phys. Chem. 98, 12310 (1994).
87. ↔, correspond to 3 X 3 dyadic tensors. Bold-face symbols (R) denote macroscopic (extensive) vectors, while italicized bold-faced symbols (U) depict macroscopic square matrices.
88. As written, Eq. (2.1) implicitly assumes an atomic solution. The extension to a solution of rigid molecules requires nothing more complicated than extending the indices μ and ν to include Euler angles, and including the moment-of-inertia "masses" corresponding to those angular coordinates [see Refs. 37 and 39]. The size of the dynamical matrix changes to 5NX5N for linear molecules and 6NX6N for nonlinear molecules.
89. An exception is the mixture studies by L. J. Muller, D. Vanden Bout, and M. Berg, Ref. 17.
90. J. M. Ziman, Models of Disorder (Cambridge University Press, Cambridge, 1979), Sec. 9.3; E. N. Economou, Green's Functions in Quantum Physics (Springer, Berlin, 1990), Sec. 7.2.1.
91. J. M. Ziman, Models of Disorder (Cambridge University Press, Cambridge, 1979), Sec. 9.3; E. N. Economou, Green's Functions in Quantum Physics (Springer, Berlin, 1990), Sec. 7.2.1.
92. A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 149, 374 (1983); 153, 445(E) (1984); E. Pollak, J. Chem. Phys. 85, 865 (1986).
93. A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 149, 374 (1983); 153, 445(E) (1984); E. Pollak, J. Chem. Phys. 85, 865 (1986).
94. A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 149, 374 (1983); 153, 445(E) (1984); E. Pollak, J. Chem. Phys. 85, 865 (1986).
95. In Eq. (2.14) [and in Figs. 4, 9, and 10], we have omitted a term proportional to δ(ω) that results from the particular formulation of the friction given in Eq. (2.13). See Ref. 41. Moreover, we have neglected the unstable imaginary modes in performing the transform. However, these imaginary modes make such a small contribution to the associated friction spectrum (here only 1%) that we feel justified in neglecting these modes.
96. B. M. Ladanyi and S. Klein, J. Chem. Phys. 105, 1552 (1996).
97. μ.
98. Recall, however, that our rigid-rotor/harmonic-oscillator Hamiltonian neglects direct intramolecular rotational-vibrational coupling.
99. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, Oxford, 1989), Chap. 4.
100. Without some multiple time scale method an MD simulation would be much harder because of the disparity between the vibrational and translational time scales in the problem.
101. We have treated the tagged coordinate as a one-dimensional variable that can range over all positive and negative values. For vibrational relaxation involving rotation, though, x probably should be treated as a radial coordinate, and thus the integrals should contain the appropriate Jacobian factors.
102. B. J. Berne, M. E. Tuckerman, J. E. Straub, and A. L. R. Bug, Ref. 24.
103. This development essentially echoes that of Levine, Shapiro, and Pollak, Ref. 28.
104. In its most literal form, this approximation suggests that the strongly interacting solute atoms are somehow far enough apart that the solvent structures around each solute atom do not perturb each other. The resulting friction kernel is reasonable, however (see Ref. 61, or Fig. 5).
105. J. E. Straub, M. Borkovec, and B. J. Berne, J. Chem. Phys. 89, 4833 (1988). Both this reference and Ref. 61 detail some of the possible pitfalls in using a single-particle friction to treat the friction on a diatomic molecule.
106. α〉 = 3, corresponding to the three atomic translational degrees of freedom.
107. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, Oxford, 1989), Chap. 1.
108. In fact, the INM friction kernel has no explicit dependence on the oscillator frequency whatsoever. There is, however, an implicit dependence inasmuch as the frequency determines the distribution of initial x values.
109. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, Oxford, 1989), Chaps. 3, 5.
110. The solute enters as an edge defect in the initial lattice.
111. eg.
112. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge University Press, Cambridge, 1992), Chap. 11.
113. A couple of technical points are in order. When the solute is allowed to translate we discard the three translational modes in each snapshot since they would not contribute in the thermodynamic limit. A frozen solute breaks translational symmetry, so translational modes do not appear. In performing the calculation it is also useful to keep in mind that the diagonalizations are the slow step in this procedure; the simulation itself is extremely rapid by comparison.
114. LJ). In the averaged spectra we bin the spikes, approximating the delta functions with prelimit rectangles.
115. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University Press, Cambridge, 1992), Sec. 15.5.
116. jω)] leads to faster convergence.
117. j), and the sum of the indices [(i-j)+j] is always i. Other than that minor slip, though, the transcription of Gregory's formula is correct, as can be verified in F. B. Hildebrand, Introduction to Numerical Analysis. 2nd ed. (Dover, New York, 1987), pp. 202-203.
118. j), and the sum of the indices [(i-j)+j] is always i. Other than that minor slip, though, the transcription of Gregory's formula is correct, as can be verified in F. B. Hildebrand, Introduction to Numerical Analysis. 2nd ed. (Dover, New York, 1987), pp. 202-203.
119. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University Press, Cambridge, 1992), Sec. 12.3.
120. D. A. McQuarrie, Statistical Mechanics (Harper & Row, New York, 1976), pp. 498-499.
121. P. J. Davis and P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975), pp. 147-150.
122. Although Wan and Stratt's derivation differs from that of Wu and Loring [Ref. 38, the method used by Schvaneveldt and Loring, Ref. 43(a)], the resulting equation for the density of states is nevertheless the same.
123. We define the first solvation shell as those atoms whose distance from a solute site lies within the first minimum in the solvent/solute-site radial distribution function. Following Ref. 39, it is straightforward to project out the portion of the friction spectrum that arises from motions of the first solvation shell, and we find that for our model this projected spectrum captures 99.998% of the total spectrum.
124. However, neither the averaged friction spectrum nor its projections show any dependence on system size. G. Goodyear, doctoral thesis, Brown University, 1997.
125. S. R. Nagel, A. Rahman, and G. S. Grest, Phys. Rev. Lett. 47, 1665 (1981); S. R. Nagel, G. S. Grest, and A. Rahman, ibid. 53, 368 (1984); J. Canisius and J. L. van Hemmen, J. Phys. C 18, 4873 (1985); M. L. Williams and H J. Maris, Phys. Rev. B 31, 4508 (1985);
126. S. R. Nagel, A. Rahman, and G. S. Grest, Phys. Rev. Lett. 47, 1665 (1981); S. R. Nagel, G. S. Grest, and A. Rahman, ibid. 53, 368 (1984); J. Canisius and J. L. van Hemmen, J. Phys. C 18, 4873 (1985); M. L. Williams and H J. Maris, Phys. Rev. B 31, 4508 (1985);
127. S. R. Nagel, A. Rahman, and G. S. Grest, Phys. Rev. Lett. 47, 1665 (1981); S. R. Nagel, G. S. Grest, and A. Rahman, ibid. 53, 368 (1984); J. Canisius and J. L. van Hemmen, J. Phys. C 18, 4873 (1985); M. L. Williams and H J. Maris, Phys. Rev. B 31, 4508 (1985);
128. S. R. Nagel, A. Rahman, and G. S. Grest, Phys. Rev. Lett. 47, 1665 (1981); S. R. Nagel, G. S. Grest, and A. Rahman, ibid. 53, 368 (1984); J. Canisius and J. L. van Hemmen, J. Phys. C 18, 4873 (1985); M. L. Williams and H J. Maris, Phys. Rev. B 31, 4508 (1985);
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130. B. B. Laird and H. Schober, Phys. Rev. Lett. 66, 636 (1991); H. Schober and B. B. Laird, Phys. Rev. B 44, 6746 (1991); U. Buchenau, Philos. Mag. B 65, 303 (1992); H. R. Schober, C. Oligshleger, and B. B. Laird, J. Non-Cryst. Solids 156-158, 965 (1993); S. D. Bembenek and B. B. Laird, J. Chem. Phys. 104, 5199 (1996); S. D. Bembenek and B. B. Laird, Phys. Rev. Lett. 74, 936 (1995).
131. B. B. Laird and H. Schober, Phys. Rev. Lett. 66, 636 (1991); H. Schober and B. B. Laird, Phys. Rev. B 44, 6746 (1991); U. Buchenau, Philos. Mag. B 65, 303 (1992); H. R. Schober, C. Oligshleger, and B. B. Laird, J. Non-Cryst. Solids 156-158, 965 (1993); S. D. Bembenek and B. B. Laird, J. Chem. Phys. 104, 5199 (1996); S. D. Bembenek and B. B. Laird, Phys. Rev. Lett. 74, 936 (1995).
132. B. B. Laird and H. Schober, Phys. Rev. Lett. 66, 636 (1991); H. Schober and B. B. Laird, Phys. Rev. B 44, 6746 (1991); U. Buchenau, Philos. Mag. B 65, 303 (1992); H. R. Schober, C. Oligshleger, and B. B. Laird, J. Non-Cryst. Solids 156-158, 965 (1993); S. D. Bembenek and B. B. Laird, J. Chem. Phys. 104, 5199 (1996); S. D. Bembenek and B. B. Laird, Phys. Rev. Lett. 74, 936 (1995).
133. B. B. Laird and H. Schober, Phys. Rev. Lett. 66, 636 (1991); H. Schober and B. B. Laird, Phys. Rev. B 44, 6746 (1991); U. Buchenau, Philos. Mag. B 65, 303 (1992); H. R. Schober, C. Oligshleger, and B. B. Laird, J. Non-Cryst. Solids 156-158, 965 (1993); S. D. Bembenek and B. B. Laird, J. Chem. Phys. 104, 5199 (1996); S. D. Bembenek and B. B. Laird, Phys. Rev. Lett. 74, 936 (1995).
134. B. B. Laird and H. Schober, Phys. Rev. Lett. 66, 636 (1991); H. Schober and B. B. Laird, Phys. Rev. B 44, 6746 (1991); U. Buchenau, Philos. Mag. B 65, 303 (1992); H. R. Schober, C. Oligshleger, and B. B. Laird, J. Non-Cryst. Solids 156-158, 965 (1993); S. D. Bembenek and B. B. Laird, J. Chem. Phys. 104, 5199 (1996); S. D. Bembenek and B. B. Laird, Phys. Rev. Lett. 74, 936 (1995).
135. B. B. Laird and H. Schober, Phys. Rev. Lett. 66, 636 (1991); H. Schober and B. B. Laird, Phys. Rev. B 44, 6746 (1991); U. Buchenau, Philos. Mag. B 65, 303 (1992); H. R. Schober, C. Oligshleger, and B. B. Laird, J. Non-Cryst. Solids 156-158, 965 (1993); S. D. Bembenek and B. B. Laird, J. Chem. Phys. 104, 5199 (1996); S. D. Bembenek and B. B. Laird, Phys. Rev. Lett. 74, 936 (1995).
136. 1/2.
137. -1 for both chloroform and carbon tetrachloride solvents, corresponding to about a 13% change in density over the temperature range studied.
138. The cavity distribution function y(r) measures the ratio of the radial distribution function, g(r), to its gas-phase value. The two distributions are equivalent, of course, but solution-phase IBC theories generally have R* in the core, where y(r) is numerically more convenient because it divides out the contribution due to the bare-potential's steeply rising wall.
139. R. M. Stratt, Int. J. Thermophys 18, 899 (1997).
140. ̂= 1) case here.
141. R. E. Larsen (unpublished).
142. E. Cortés, B. J. West, and K. Lindenberg, J. Chem. Phys. 82, 2708 (1985).
143. Recent advances in formulating quantum mechanical instantaneous normal modes may very well allow us to handle quantum mechanical situations. J. Cao and G. A. Voth, J. Chem. Phys. 101, 6184 (1994); C. Chakravarty and R. Ramaswamy (preprint); S. A. Corcelli and J. D. Doll, Chem. Phys. Lett. 263, 671 (1996).
144. Recent advances in formulating quantum mechanical instantaneous normal modes may very well allow us to handle quantum mechanical situations. J. Cao and G. A. Voth, J. Chem. Phys. 101, 6184 (1994); C. Chakravarty and R. Ramaswamy (preprint); S. A. Corcelli and J. D. Doll, Chem. Phys. Lett. 263, 671 (1996).
145. Recent advances in formulating quantum mechanical instantaneous normal modes may very well allow us to handle quantum mechanical situations. J. Cao and G. A. Voth, J. Chem. Phys. 101, 6184 (1994); C. Chakravarty and R. Ramaswamy (preprint); S. A. Corcelli and J. D. Doll, Chem. Phys. Lett. 263, 671 (1996).