Zero-point energy separation

It will be convenient to measure the E f relative to the molecular zero points and to account for any change of zero point energy separately. 

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 discussion above has been directed principally to thermally induced spin transitions, but other physical perturbations can either initiate or modify a spin transition. The effect of a change in the external pressure has been widely studied and is treated in detail in Chap. 22. The normal effect of an increase in pressure is to stabilise the low spin state, i.e. to increase the transition temperature. This can be understood in terms of the volume reduction which accompanies the high spin—dow spin change, arising primarily from the shorter metal-donor atom distances in the low spin form. An increase in pressure effectively increases the separation between the zero point energies of the low spin and high spin states by the work term PAV. The application of pressure can in fact induce a transition in a HS system for which a thermal transition does not occur. This applies in complex systems, e.g. in [Fe (phen)2Cl2] [158] and also in the simple binary compounds iron(II) oxide [159] and iron(II) sulfide [160]. Transitions such as those in these simple binary systems can be expected in minerals of iron and other first transition series metals in the deep mantle and core of the earth. 

According to two simultaneous hypotheses15,16, the principal component of the 7 of simple metals (from lithium to aluminum) originates in the shift in the zero-point energy of the plasma models of the system when one perfect crystal is broken into two separate crystals. Some simplifications and arbitrary assumptions lead finally to the relation... 

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... 

At low temperatures and light frequencies, the separative effect per unit shift (the terms in brackets in eq. 11.44 and 11.45) approaches (figure 11.9) and the Helmholtz free energy difference approaches the differences in zero-point energies. At high T (low frequencies), the separative effect per unit shift approaches zero and the total separative effect sis ")/approaches 1, so that no isotopic fractionation is observed. 

The right-hand side can be separated into five parts. The first part is the enthalpy at 0 K, the second represents the zero point energy, the third is the Debye energy term, the fourth is an approximation for the Cp — C correction while the last part arises from the difference in electronic specific heats. 

As for graphite, its zero-point energy, ZPE = R6 + jR0 , is most conveniently deduced from Debye s theory [197,198] by separating the lattice vibrations into two approximately independent parts, with Debye temperatures (in plane) and 6j (perpendicular). A balanced evaluation gives ZPE 3.68 kcal/mol [199]. 

In a crystalline antiferromagnet the moment on each ion is less than it would be on the free ion. There are two separate phenomena involved here. One is the zero-point energy of the spin waves, which reduces the moment on each ion by a factor (Anderson 1952, Ziman 1952)... 

Two separate curves of potential energy versus rxy may be constructed, one for the >XH vibration in its ground state (with zero point energy only), and another for the excited state when a quantum of energy has been absorbed to excite the vXH vibration. 

LATTICE ENERGY OF CRYSTAL. The decrease in energy accompanying the process of bringing the tons, when separated front each other by an infinite distance, to the positions they occupy in the stable lattice It is made up of contributions from the electrostatic forces between the ions, from the repulsive forces associated with the ovedap of electron shells, front the van der Waals forces, and from the zero-point energy. 

We often think of there being two separate quantum features of nonclassical oscillators, the change in energy levels in quantal units hv (or Tko) and the finite zero-point energy... 

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