Schematic representation of the semiclassical instanton approach. A one-dimensional case is shown (which is equivalent to the WKB approach). Potential barrier V(s) along the reaction coordinate s is illustrated in the right column. The left column corresponds to the inverted barrier −V(s), on which ℏβ-periodic trajectories (instantons) are sought. Rows (a), (b), and (c) illustrate instanton trajectories (left figure) and their corresponding tunneling paths (right figure) at inverse temperatures β1, β2, and β3, respectively, such that β1 > β2 > β3. The inverse temperature β3 lies in the vicinity of the crossover temperature.
Instanton trajectory in D2C(CH)3CH3 at 352 K determined on the EVB PES of Sec. III. The trajectory is scaled five times to be visible, i.e., Fourier coefficients C j in Eq. (4.7), j > 1 are multiplied by the factor of 5. The red lines stand for tunnel paths of H(D) atoms, the blue lines stand for tunnel paths of C atoms. (a) front view, (b) top view.
The energy E inst and the abbreviated action W inst of ℏβ-periodic classical trajectories on the inverted PES of D2C(CH)3CH3. The vertical dashed line and the arrow indicate the crossover temperature T c = 1/κBβ c , defined as W inst (β c ) = 0. The horizontal dashed line indicates the classical potential barrier height V 0. Temperature T is in K; E inst and W inst are in atomic units.
The difference between and its saddle point value for the isotopologue D2C(CH)3CH3 at different temperatures β = 1/κB T. The frequencies are 32 stable vibrational frequencies at the transition state configuration. The units are kcal/mol for F vib and Kelvins for T.
The effective abbreviated action (left axis, solid circles), and the instanton abbreviated action W inst (β) (right axis, open circles) as a function of the inverse temperature β for (a) D2C(CH)3CH3 and (b) H2C(CH)3CD3. The root of defines the new crossover temperature . The original instanton crossover temperature, T c , is shown for reference. Atomic units are used for and W, and Kelvins for T.
Potential barrier along the reaction coordinate s. Solid line represents the effective barrier formed by one-dimensional instanton trajectories, and dashed line represents effective one-dimensional barrier which includes zero-point energy ΔE zp contributions of the orthogonal degrees of freedom. Only the harmonic parts of the barriers are shown in figure for simplicity.
H/D kinetic isotope effect as a function of temperature for the1,5 sigmatropic rearrangement reaction of cis-1,3-pentadiene. Solid circles represent experimental results;57 open triangles-transition state theory (TST) results with no tunneling;53 open circles-CVT/SCT theory;54 and open squares-QI theory.55 The solid line represents semiclassical instanton results of the present paper with from Eq. (5.4). Contributions of factors from Eqs. (5.5) to (5.7) to the semiclassical KIE: dashed line corresponds to the single TST factor (5.5); dotted-dashed line corresponds to the product of the TST (5.5) and the tunneling (5.6) factors; solid line is the product of all three factors (5.5)–(5.7). Dotted line represents the present semiclassical results for the case when the parameters of the original PES, β c and , are used instead of the effective parameters and
12C/13C kinetic isotope effect as a function of temperature in cis-1,3-pentadiene. The notation is the same as in Figure 7: solid line represents the semiclassical instanton results, dashed line represents TST results, and dotted-dashed line corresponds to the TST results with tunneling under the effective harmonic barrier.
Calculated vibrational frequencies of isotopologues of C5H8 in cm−1.
Experimental and theoretical kinetic isotope effects in cis-1,3-pentadiene for different temperatures.
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