Ground state zero-point energy

The actual situation involving barrierless discharge is not altogether clear when finite temperatures obtain, for then transitions to some excited vibrational states in the product species could be barrierless, while transition to the ground-state, zero-point, energy level may not be. 

The bond-stretch model provides an upper limit for kinetic isotope effects that arise solely from ground state zero-point energy effects. Observations that deviate from this model imply a nonclassical effect. Provided that potential artifacts are controlled, the observation of KlEs that disobey the bond-stretch predictions calls into question the basic theory. 

The ground state zero point heats of formation are known for all species in 5), as are the excitation energies X A2 A Ax for methyl and a Ax b Bx for methylene.Taking the singlet-triplet splitting in CH2 to be 9.5 kcal/mol, one obtains AE = -94 kcal/mol for reaction (5), so reaction (1) is clearly favored over (4). In fact, using the appropriate states, reaction (4) is slightly endothermic (by about 0.5 eV). 

The constant of integration is zero at zero temperature all the modes go to the unique non-degenerate ground state corresponding to the zero point energy. For this state S log(g) = log(l) = 0, a confmnation of the Third Law of Thennodynamics for the photon gas. 

The second correction is much larger. The residual energy that the molecule ion has in the ground state above the T),- at the e(]nilibrintn bond length is the zero point energy. /PH. 

To obtain the G2 value of Eq we add five corrections to the starting energy, [MP4/6-31 lG(d,p)] and then add the zero point energy to obtain the ground-state energy from the energy at the bottom of the potential well. In Pople s notation these additive terms are... 

The vibrational enthalpy consists of two parts, the first is a sum of hv/2 contributions, this is the zero-point energies. The second part depends on temperature, and is a contribution from molecules which are not in the vibrational ground state. This contribution goes toward zero as the temperature goes to zero when all molecules are in the ground state. Note also that the sum over vibrational frequencies runs over 3Ai — 6 for the reactant(s), but only 3A1 — 7 for the TS. At the TS, one of the normal vibrations has been transformed into the reaction coordinate, which formally has an imaginary frequency. 

Passing to the Boson operators by aid of Table II, and after neglecting the zero-point-energy of the fast mode, we obtain a quantum representation we shall name I, in which the effective Hamiltonians of the slow mode corresponding respectively to the ground and first excited states of the fast mode are... 

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 ... 

The allyl anion ground-state conformation is Czv at 6-31G HF and C2 at MP2. The energy difference, however, is only 0.2 kcalmol-1 and when the zero-point energy (ZPE)... 

Note that the condition n = 0 is not allowed since that would imply tp = 0, everywhere, which is not square integrable as required by the Born condition. The ground-state, or zero-point energy... 

It is interesting to contrast the rate ratio for reactions 10.1 and 10.4 where either H or D atoms react with H2 with that for reactions 10.1 and 10.7 where common H atoms react with either H2 or D2 (compare Figs. 10.1a and b). In the first case, (kH,HH/kD,HH), there is a ZPE difference in the transition state but not the ground state consequently the high temperature KIE is inverse. In the second (kH,HH/kH,DD), however, there are zero point energy differences in both the transition and ground states. We expect the vibrational force constants to be smaller in the more loosely bound transition as compared to the ground state. The isotope effects scale with the force constant differences. Consequently RT[ln(kH,HH/kH,DD)] =... 

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.

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