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16
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85034872253
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(b) Compare numerical results in Figs. 10–12 in the second of these papers, to obtain this maximum rate
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17
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85034881043
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(c) An explicit expression of the variable proportional to X can be found as Eq. (20) in the second of these papers, or in the E(t) in Ref. 11 below.
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27
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0012916750
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These authors consider a dipole and the effect of the reaction field on [formula omitted] In this case, [formula omitted] differs somewhat from that given by Eq. (1.1). For example, the [formula omitted] in this equation is replaced by [formula omitted] where [formula omitted] is related to the polarizability α assumed for the solute molecule with volume [formula omitted] by the Clausius‐Mossoti relation [formula omitted] Other model geometries and charge distributions (e.g., a pair of separated charges) will lead to still other results, and we plan to discuss this topic elsewhere. Examples of free energy calculations for systems with separated charge distributions are given in, and in references cited therein, (c) However, different authors, e.g., Ref. 4 vs Ref. 11, have made different choices for [formula omitted]” in the case of multirelaxation time solvents. This point is discussed further in Ref. 24 below
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(1965)
J. Chem. Phys.
, vol.43
, pp. 58-1261
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Marcus, R.A.1
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38
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84950852405
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In the particular case that the “vibrational” motion of the solute is some internal rotation, it may also become diffusive during the course of reaching the transition state. In order for the adiabatic treatment mentioned above to be applicable to this case, too (i.e., fast internal motions), the relaxation time of such a diffusive motion would have to be appreciably shorter than that of the diffusive solvent orientational fluctuations.
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42
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84950793365
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In this extension one solves a pair of coupled reaction‐diffusion equations for the system on the two electronic‐state surfaces. The distribution function given later becomes a two‐dimensional vector and the reaction rate k(X) becomes a matrix. Otherwise, the development is similar to the present though mathematically more complex.
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43
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85034875804
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See, for example, Ref. 17 and references cited therein, or Ref. 12.
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46
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84950791466
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When [formula omitted] Eq. (4.10) should read [formula omitted]
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48
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85034875871
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To show this one need consider only the behavior near [formula omitted] since a possible pole may arise there. At sufficiently small [formula omitted] can be written as [formula omitted] since the other eigenvalues of H are nonzero. Then a(s) becomes [formula omitted] In this case [formula omitted] becomes [formula omitted] which is independent of s. In two of the terms in brackets in Eq. (5.23), one finds a cancellation upon introducing the above result for [formula omitted] and a(s), and the bracketted term then becomes [formula omitted] which again is independent of s. The remaining factor [formula omitted] has no singularity at [formula omitted] as long as [formula omitted] is nonzero.
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50
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85034873769
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When the eigenfunction of [formula omitted] is written as [formula omitted] with the eigenvalue of [formula omitted] for [formula omitted] with [formula omitted] replaced by [formula omitted] is given by [formula omitted]
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52
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85034870281
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Considering that k can be scaled by [formula omitted] and both H and h by [formula omitted] we see that [formula omitted] is composed of [formula omitted] [formula omitted] and [formula omitted] where 〈g|k in Eq. (7.8) was replaced by [formula omitted] anticipating the procedure shown in Eq. (7.14). These three quantities can be calculated by performing the t, X, and y integrations, appearing in Eqs. (7.10) and (7.14). For these integrations, where we use Eq. (3.2) for [formula omitted] Eqs. (3.6) and (4.1) for [formula omitted] and Eq. (5.1) for [formula omitted] we can change the tX, and Y variables, respectively, to scaled ones [formula omitted] [formula omitted] and [formula omitted] all of which are dimensionless. Then, the first of the three quantities mentioned above depends only on [formula omitted] and [formula omitted] which characterize the shift of the origin and the curvature of [formula omitted] relative to [formula omitted] while the last two ones of these depend also on [formula omitted] since [formula omitted] From Eqs. (3.7) and (4.8) we can use the pair [formula omitted] and [formula omitted] instead of the pair [formula omitted] and [formula omitted] Therefore, [formula omitted] depends only on [formula omitted] [formula omitted] and [formula omitted] The same property can be verified also for [formula omitted] given by Eq. (7.9). Thus, we see that both [formula omitted] and [formula omitted] given by Eq. (7.7) depend only on [formula omitted] [formula omitted] and [formula omitted]
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55
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85034870833
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At [formula omitted] (Fig. 1) there is no vibrational barrier and so the tunneling referred to is for [formula omitted]
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56
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85034880157
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To calculate [formula omitted] denned by Eq. (1.1), for use in Fig. 9, we need [formula omitted] [formula omitted] and [formula omitted] However, no data is available on [formula omitted] which is the square of the refractive index, at the highest temperature point (333 K) and the lowest three temperature points (227, 209, and 184 K) in the In [formula omitted] vs [formula omitted] plot of this figure for 1‐propanol. The [formula omitted] was set at 2.2 at these temperatures therein, since it is known to be 2.21 at 313 K (Ref. 20), 2.24 at 293 K (Ref. 20), and 2.11 at 133 K (Ref. 35). For 1‐hexanol, it was set at 2.2 at the lowest three temperature points (274, 248, and 233 K), since it is known to be 2.15 at 333 K, 2.17 at 313 K, and 2.19 at 293 K (Ref. 20).
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60
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85034875259
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Apart from a factor of 2: The intersection of the two surfaces serves as a boundary in the present treatment (cf. Sec. VIII).
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