Zero-point energy illustration

Figure A3.13.il. Illustration of the time evolution of redueed two-dimensional probability densities I I and I I for the exeitation of CHD between 50 and 70 fs (see [154] for further details). The full eurve is a eut of tire potential energy surfaee at the momentary absorbed energy eorresponding to 3000 em during the entire time interval shown here (as6000 em, if zero point energy is ineluded). The dashed eurves show the energy uneertainty of the time-dependent wave paeket, approximately 500 em Left-hand side exeitation along the v-axis (see figure A3.13.5). The vertieal axis in the two-dimensional eontour line representations is...

Primary isotope effects tend to be dominated by the difference in zero-point energies, as illustrated in Figure 15.2. Because the reaction coordinate is the breaking bond, and because there is little or no ZPVE associated with this mode in the TS structure, the full difference in reactant ZPVEs enters into the difference in zero-point-including potential energy barriers. 

Fig. 6.1.1 An illustration of the barrier height Eo- The zero-point energy levels of the activated complex and the reactants are indicated by solid lines. Note that the zero-point energy in the activated complex comes from vibrational degrees of freedom orthogonal to the reaction coordinate. In the classical barrier height Ec 1, vibrational zero-point energies are not included.

Fig. 8.2.1 An illustration of Eq. (8.6). The zero-point energy levels for an exothermic reaction are indicated by solid lines and the average internal energies of the products and the reactants (relative to the zero-point levels) are given by dashed lines.

Indeed, the Debye approximation is more appropriate in any problem where the modes of lowest frequency are important, as they are in thermal properties at low temperatures. In cases were all modes are important, such as in the evaluation of the total zero-point energy, the simpler Einstein model may be preferable. Notice that even within the Debye approximation the frequencies are concentrated near the highest frequency, called the Debye frequency. This is illustrated in Fig. 9-7. 

Cartoon of a C-L potential energy curve, illustrating the bond stretch-anharmonicity explanation of the inductive and steric effects of deuterium and tritium. The anharmonicity and zero-point energies are greatly exaggerated. 

Figure 6.2 Triatomic model of H-transfer illustrating changes in zero-point energies of normal vibrations between the initial and transition states.

The expected changes in the zero-point energies of the H transferred are illustrated schematically in Fig. 6.6 for the degenerate case, as proposed by Westhei-mer [45]. The antisymmetric stretch in the initial state exhibits quite different ZPEs for H and for D as the force constants are large. This vibration becomes imaginary in the transition state, which is assumed here to be located in the minimum of q2, i.e. the ZPE of the antisymmetric stretch is lost in the transition state. The ZPE of the symmetric stretch in the transition state is small and exhibits little isotope dependence. We note that ZPE is built up in the bending vibration in the... 

When isotopic fractionation takes place, for example through a strenghtening of the H-bond in the intermediate leading to reduced zero-point energies [50], the factor will be larger than 2, leading to an increase of as illustrated in Fig. 

The fit of the experimental data to Eq. (6.48) is very satisfactory, as illustrated in Eig. 6.22(a), where the solid lines were recalculated here using the Bell-Limbach model, with the parameters included in Table 6.4. This result also means that there is no substantial decrease in the zero-point energies of the two protons in the cis-intermediate states as compared to the initial and final trans-states, as this would increase the HD/DD isotope effect beyond the value of 2 as was illustrated in Fig. 6.14(c). 

Figure 1.2 illustrates transitions of the three types mentioned for a diatomic molecule. As the figure shows, rotational intervals tend to increase as the rotational quantum number J increases, whereas vibrational intervals tend to decrease as the vibrational quantum number v increases. The dashed line below each electronic level indicates the zero-point energy that must exist even at a temperature of absolute zero as a result of Heisenberg s uncertainty principle ... 

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