Zero-point energy approximation

For these reactions of hydrogen, it is the isotope effect on the high frequency vibrational modes in the diatomic reactant and tri-atomic transition states which dominate in the calculation of the isotope effects using the TS model. Excitation into upper vibrational levels for these high frequency modes is negligible and the zero point energy approximation is appropriate (see Section 4.6.5.2 and Fig. 4.1). 

Low-temperature approximation. Zero-point-energy approximation. High-temperature approximation. 

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...

Figure B3.4.1. The potential surfaee for the eollinear D + H2 DH + H reaetion (this potential is the same as for H + H2 — H2 + H, but to make the produets and reaetants identifieation elearer the isotopieally substituted reaetion is used). The D + H2 reaetant arrangement and the DH + H produet arrangement are denoted. The eoordinates are r, the H2 distanee, and R, the distanee between the D and the H2 eentre of mass. Distanees are measured in angstroms the potential eontours shown are 4.7 eV-4.55 eV,.. ., -3.8 eV. (The potential energy is zero when the partieles are far from eaeh other. Only the first few eontours are shown.) For referenee, the zero-point energy for H2 is -4.47 eV, i.e. 0.27 eV above the H2 potential minimum (-4.74 eV) the room-temperature thennal kinetie energy is approximately 0.03 eV. The graph uses the aeeiirate Liu-Seigbalm-Triihlar-Horowitz (LSTH) potential surfaee [195].

How well do your results agree with the experimental value of about 3.4 kcal-mol i Since this observation is very approximate, we will not worry about zero-point energy corrections in this exercise. 

There is an evolution with time the older calculations correspond to isolated molecules in the gas phase without any corrections, the more recent ones include solvent effects, with different approximations, and also some corrections, like ZPE (zero-point energy correction). The contributions of some authors to the understanding of tautomerism have been significant. Some of their contributions are collected in Table II. 

MMl represents the mass and moment-of-inertia term that arises from the translational and rotational partition functions EXG, which may be approximated to unity at low temperatures, arises from excitation of vibrations, and finally ZPE is the vibrational zero-point-energy term. The relation between these terms and the isotopic enthalpy and entropy differences may be written... 

Applying the exchange approximation and neglecting the zero-point energy terms, we may safely limit the representation of the Hamiltonian Hsf ex within the following reduced base which accounts for the ground states of each mode and for the first (second) overtone of the fast (bending) mode ... 

W1/W2 theory and their variants would appear to represent a valuable addition to the computational chemist s toolbox, both for applications that require high-accuracy energetics for small molecules and as a potential source of parameterization data for more approximate methods. The extra cost of W2 theory (compared to W1 theory) does appear to translate into better results for heats of formation and electron affinities, but does not appear to be justified for ionization potentials and proton affinities, for which the W1 approach yields basically converged results. Explicit calculation of anharmonic zero-point energies (as opposed to scaling of harmonic ones) does lead to a further improvement in the quality of W2 heats of formation at the W1 level, the improvement is not sufficiently noticeable to justify the extra expense and difficulty. 

It has already been noted that the new quantum theory and the Schrodinger equation were introduced in 1926. This theory led to a solution for the hydrogen atom energy levels which agrees with Bohr theory. It also led to harmonic oscillator energy levels which differ from those of the older quantum mechanics by including a zero-point energy term. The developments of M. Born and J. R. Oppenheimer followed soon thereafter referred to as the Born-Oppenheimer approximation, these developments are the cornerstone of most modern considerations of isotope effects. 

Fig. 4.1 The zero point energy or low temperature approximation As temperature drops and u increases above u 4 the harmonic oscillator partition function Q (Harm. Osc.) is better and better approximated by the zero point energy term, exp(—u/2). For a typical CH stretching frequency, v = 3000 cm-1, u 4 at 1050 K and it is reasonable to use the ZPE approximation for that frequency at temperatures below 1000 k...

Fig. 10.1 Zero point energy diagrams, (a) An H or D atom attacking an H2 molecule. The TST isotope effect is negative (inverse, kn > kn) because there is no zero point isotope effect in the ground state, and tunneling is ignored in the TST approximation, (b) An H atom attacking either an H2 or D2 molecule. The isotope effect calculated in the TST approximation is positive (normal, kH > kn) because the zero point isotope effect in the ground state is larger than that in the transition state.

For historic and practical reasons hydrogen isotope effects are usually considered separately from heavy-atom isotope effects (i.e. 160/180, 160/170, etc.). The historic reason stems from the fact that prior to the mid-sixties analysis using the complete equation to describe isotope effects via computer calculations was impossible in most laboratories and it was necessary to employ various approximations. For H/D isotope effects the basic equation KIE = MMI x EXC x ZPE (see Equations 4.146 and 4.147) was often drastically simplified (with varying success) to KIE ZPE because of the dominant role of the zero point energy term. However that simplification is not possible when the relative contributions from MMI (mass moment of inertia) and EXC (excitation) become important, as they are for heavy atom isotope effects. This is because the isotope sensitive vibrational frequency differences are smaller for heavy atom than for H/D substitution. Presently... 

All approaches for the description of nonadiabatic dynamics discussed so far have used the simple quasi-classical approximation (16) to describe the dynamics of the nuclear degrees of freedom. As a consequence, these methods are in general not able to account for processes or observables for which quantum effects of the nuclear degrees of freedom are important. Such processes include nuclear tunneling, interference effects in wave-packet dynamics, and the conservation of zero-point energy. In contrast to quasi-classical approximations, semiclassical methods take into account the phase exp iSi/h) of a classical trajectory and are therefore capable—at least in principle—of describing quantum effects. 

For chemical purposes, substitution of total energy hypersurfaces by those based on the heat of formation seems more reasonable, with the difference given by the zero point energy corrections. However, their calculations from first principles can be rather cumbersome (12) and, moreover, for a given variation of some nuclear coordinates it usually can be assumed that the change in zero point energy is small compared to that of the total energy. On the other hand, se eral semiempirical quantum chemical procedures which are appropriately parametrized often yield satisfactory approximations for molecular heats of formation (10) and, therefore, AH hypersurfaces have become rather common. 

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