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Review
. 2006 Aug;106(8):3140-69.
doi: 10.1021/cr050308e.

Multidimensional tunneling, recrossing, and the transmission coefficient for enzymatic reactions

Jingzhi Pu  1 , Jiali Gao, Donald G Truhlar
Affiliations
Review

Multidimensional tunneling, recrossing, and the transmission coefficient for enzymatic reactions

Jingzhi Pu et al. Chem Rev. 2006 Aug.
No abstract available

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Figures

Figure 1
Figure 1
Schematic tunneling paths (green and red) for an H transfer reaction as functions of two isoinertial rectilinear coordinates. (For example, for a triatomic reaction D – X + A → D + X –A, where D and A are donor and acceptor atoms, x would be the mass-scaled distance of A to DX, and y would be the mass-scaled distance of D to X.) The black curves are potential energy surface contours plotted in a mass-weighted coordinate. The figure shows a two-dimensional cut through the (3N − 1)-dimensional space. The minimum energy path (MEP) is depicted as a blue curve that connects the reactant (R) and product (P) regions. In a one-dimensional tunneling model, the reaction path curvature is ignored, and the tunneling path is the MEP. When the reaction path is moderately curved, the dominant tunneling path (depicted in red and called a small-curvature (SC) tunneling path) corner-cuts the MEP on its concave side. Although the tunneling path does not follow the MEP (and hence is not perfectly adiabatic), the effective potential along this kind of path is adiabatic. In the limit of reaction paths with large curvature, the optimal tunneling paths (depicted in green and called large-curvature (LC) tunneling paths) are straight lines connecting the reactant and product valleys; the effective potential for these tunneling paths is nonadiabatic. For a symmetric reaction, the distance between the donor and acceptor atoms is approximately constant along LC tunneling paths. A brown arrow is used to depict the direction of increasing the donor–acceptor distance.
Figure 2
Figure 2
Schematic trajectory for an H transfer reaction as a function of the H-to-donor and H-to-acceptor distances. The black curves are potential energy surface contours. Keep in mind that the figure shows a two-dimensional cut through the (3N − 1)-dimensional space. The minimum energy path (MEP) is depicted as a blue curve that connects the reactant (R) and product (P) regions. Three possible transition state dividing surfaces are shown. The magenta curve represents the projection of an example trajectory into this 2D cut; only the portion of the trajectory from the reactant to slightly past the dynamical bottleneck is shown, but we assume that the remainder of the trajectory proceeds to products without recrossing any of the three dividing surfaces. The conventional transition state dividing surface (DS1 in green) is orthogonal to the MEP at the saddle point, and it is crossed twice in the forward direction by the example trajectory; therefore, is has a transmission coefficient less than unity. Displacing the dividing surface to DS2 (also in green) also gives a dividing surface that is crossed twice in the forward direction. DS3 (in red) is a variationally improved transition state that is not recrossed, yielding a unity recrossing transmission. (The canonical variational transition state is defined to minimize recrossing for the canonical ensemble, not for a single trajectory, as used here for illustrative purposes only.) Note that DS3 is rotated as compared to DS2. Since the reaction coordinate is the degree of freedom normal to the dividing surface, rotating the dividing surface corresponds to rotating the reaction coordinate, that is, choosing a different reaction coordinate. Although the dividing surfaces are shown as hyperplanes in this diagram (in a 2D diagram, a hyperplane is a straight line; in a 3D world, a hyperplane is a 2D plane; in the 3N-dimensional coordinate space, a hyperplane has dimension 3N − 1), general dividing surfaces can be nonplanar, and general reaction coordinates can be curved. For example, a dividing surface defined by a linear transformation of Cartesian coordinates would be nonplanar (nonstraight in this picture) because the axes are nonlinear functions of Cartesians, a dividing surface defined as a difference of bond stretches would curved in a Cartesian coordinate system, and an energy gap reaction coordinate would be curved in almost any coordinate system. Notice that the dividing surface defined by z = 0 where z is defined in eq 9 would be a straight line at an angle of 45° in this figure (and mass-weighting the two distances would change this angle); for comparison, we note that DS1, DS2, and DS3 are at angles of 11, 31, and 56°, respectively.
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