The Morse energy (solid line) is plotted together with the difference between the Morse and model energies (same units, cm−1, but much smaller scale, and plotted as line-points) for the electronic states (top plot) [α = 1.9Å−2] and (bottom plot) [β = 2.3Å−2].
The logarithm of Morse oscillator in Li2 Franck-Condon envelope for the transition , and weighted by the Lorentz distribution as the density of states, for three temperatures: (1) 10 K, (2) 100 K, and (3) 300 K.
The plots of Eqs. (32) and (33) . In panel 1, λ s = 0.016 eV, and in panels 2 and 3 λ s = 0.16 eV. In the first two panels, the temperature is T = 300 K. In the last panel, the temperature is T = 10 K. Note that the Δ energy scale differs in each panel as does the ln w s scale.
The two panels of this figure display ln (w) as a function of Δ for a temperature T = 100 K. In panel (a), line 1 displays the results of using Eq. (8) with λ s = 0.016 eV and ω i(f) = 500 (400) cm−1. Line 2 results from Eq. (11) and the same values of λ s and ω. Lines 3 and 4 display the results of Eqs. (8) and (11) with λ s = 0.16 eV. The plots in panel (b) repeat the sequence of panel (a), but use ω i(f) = 1300 (1200) cm−1.
The plots in these two panels are for T = 300 K and the same sequence of values of λ s and ω used in Fig. 4 .
Comparison of Morse and harmonic models for the inner sphere mode contribution to the rate constant. Panels (a) and (b) use the discrete summation of Morse and harmonic model Franck-Condon inner sphere factors together with a discrete summation of the Franck-Condon factors for the solvent modes. The plots in panels (a) and (b) are labeled “ML” for Morse model inner sphere/Lorentz density, and “HL” for Harmonic oscillator inner sphere/Lorentz density. Panels (c) and (d) use a combination of the discrete summations of inner shell Morse and harmonic model Franck-Condon factors and the Gaussian form for the solvent modes. The labels in these panels are “MG” for Morse model inner sphere/Gaussian solvent, and “HG” for harmonic oscillator inner sphere model/Gaussian solvent.
These plots show the percent difference between the predicted transition probabilities for the harmonnic versus the anharmonnic (Morse) models of the inner sphere mode. Panel (a) is the comparison of the two models shown in panels (a) and (b) of Fig. 6 . Panel (b) is a comparison of the two models shown in panels (c) and (d) of Fig. 6 . Only four effective masses are used: μ = 10, 20, 30, 40 gmw. The plots that start out in Δ with the lowest percent difference in each case are associated with the effective mass of 10 gmw. Successive plots that show increasing percent difference are associated with larger effective masses.
A surface plot of ln (w r, p ) for λ s = 0.16 eV, 5 ⩽ T ⩽ 300 (K), and 0 ⩽ Δ ⩽ 1.0 eV.
Plots of ln w versus T for 0.4 ⩽ Δ ⩽ 0.5. The topmost plot is that for Δ = 0.5 eV. Because of the symmetry of the transition probability (rate) surface in this region (see Fig. 8 ), the remaining plots appear almost as pairs. The apparent break at T = 25 K is allied with the transition from low temperature behavior seen in Fig. 8 . The temperature insensitivity at Δ = 0.5 eV is easier to see in this plot than it would be in an Arrhenius plot of ln (w) versus 1/T.
The first two panels list the (Lee-Koo et al. 51 ) parameters of the Morse potentials used to determine the Franck-Condon factors found here. To ensure consistency, the third panel lists parameters and conversions used in the analyses in this paper. Atomic units are not used. Units of length are listed and reported throughout in Ångström = 1.0 × 10−8 cm.
The Franck-Condon factors for the first six transitions in Li2 for the states and . The first value listed is the Lee-Koo et al., 51 result, the second is this work.
The mechanical harmonic oscillator model of the solvent: The effective oscillator frequency ω s and effective mass μ s are fixed. One selects either the oscillator displacement Δz s or a displacement energy λ s . The displacement energy is the Marcus repolarization energy, which converts to the displacement as a derived quantity. The value of H I, F comes from Yeganeh and Ratner. 26
Morse and related parameters for two models with equivalent harmonic frequencies . The calculations used 200 harmonic oscillator basis functions for each state. The quantities α and β are the orbital exponents for the harmonic oscillator basis functions in the lower and upper anharmonic (Morse) oscillator states. The common solvent parameters are those given in Table III . Two values of λ s are used: 0.016 and 0.16 eV. The respective solvent displacements for the mechanical solvent oscillator model are Δz = 0.6644 and Δz = 2.1012 Å. In all of these examples, a value of H i, f = 0.02 eV, used by Yeganeh and Ratner, 26 is also used here. Finally, the quantity rms-error(E) is the root-mean-square error of the variational calculation of the energy levels of against the Morse energies.
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