Molecules zero-point energy

Molecule Zero point energy (kcal/mol) (kcal/mol)... 

Even at 0 K, molecules do not stand still. Quantum mechanically, this unexpected behavior can be explained by the existence of a so-called zero-point energy. Therefore, simplifying a molecule by thinking of it as a collection of balls and springs which mediate the forces acting between the atoms is not totally unrealistic, because one can easily imagine how such a mechanical model wobbles aroimd, once activated by an initial force. Consequently, the movement of each atom influences the motion of every other atom within the molecule, resulting in a com-... 

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. 

Fig. 4.9. DifiBoing zero-point energies ofprotium- and deuterium-substituted molecules as the cause of primary kinetic isotope effects.

Figure S-1. Form of a potential energy curve for diatomic molecule AB. VfrAa) is the potential energy, Tab is the intemuclear distance, is the equilibrium intemuclear distance, and D is the bond dissociation energy. (The zero point energy is neglected in the figure.)...

That is, the sum of the electronic energy and nuclear repulsion energy of the molecule at the specified nuclear configuration. This quantity is commonly referred to as the total energy. However, more complete and accurate energy predictions require a thermal or zero-point energy correction (see Chapter 4, p. 68). 

The thermochemistry section of the output also gives the zero-point energy for this system. The zero-point energy is a correction to the electronic energy of the molecule to account for the effects of molecular vibrations which persist even at 0 K. 

You ll need to run five calculations at each model chemistry oxygen atom, chlorine atom, O2, CIO and ozone (but don t forget that you can obtain lower-level energies from a higher-level calculation). Use the experimental geometries for the various molecules and the following scaled zero-point energy corrections ... 

The square of the wavefunction is finite beyond the classical turrfing points of the motion, and this is referred to as quantum-mechanical tunnelling. There is a further point worth noticing about the quantum-mechanical solutions. The harmonic oscillator is not allowed to have zero energy. The smallest allowed value of vibrational energy is h/2jt). k /fj. 0 + j) and this is called the zero point energy. Even at a temperature of OK, molecules have this residual energy. 

The molecular mechanics calculations discussed so far have been concerned with predictions of the possible equilibrium geometries of molecules in vacuo and at OK. Because of the classical treatment, there is no zero-point energy (which is a pure quantum-mechanical effect), and so the molecules are completely at rest at 0 K. There are therefore two problems that I have carefully avoided. First of all, I have not treated dynamical processes. Neither have I mentioned the effect of temperature, and for that matter, how do molecules know the temperature Secondly, very few scientists are interested in isolated molecules in the gas phase. Chemical reactions usually take place in solution and so we should ask how to tackle the solvent. We will pick up these problems in future chapters. 

The total energy in ab initio theory is given relative to the separated particles, i.e. bare nuclei and electrons. The experimental value for an atom is the sum of all the ionization potentials for a molecule there are additional contributions from the molecular bonds and associated zero-point energies. The experimental value for the total energy of H2O is —76.480 a.u., and the estimated contribution from relativistic effects is —0.045 a.u. Including a mass correction of 0.0028 a.u. (a non-Bom-Oppenheimer effect which accounts for the difference between finite and infinite nuclear masses) allows the experimental non-relativistic energy to be estimated at —76.438 0.003 a.u. ... 

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. 

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. 

We may fix our attention on the minimum of the potential-energy curve in Fig. 7 and ask how much higher the lowest vibrational level will lie. This energy gap between the potential minimum and the lowest vibrational level is equal to the vibrational energy of the molecule at the absolute zero of temperature and is known as the zero-point energy of... 

The quantum chemical methods introduced in part 2.2 calculate only individual molecules at the temperature of 0 K. The energies obtained in these cases represent the energies of the molecules directly in the minimum of the potential energy, i.e. the zero point energy which is evident at 0 K and the thermic energy of an ensemble of... 

By applying Eqs. (4) or (36) we can calculate the average vibrational energy of a molecule with N vibrational modes, and we do this with respect to the zero of the vibrational potentials, implying that we will include all zero-point energies ... 

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