Zero-point energy, definition

Fig. 28. Schematic of potential energy surfaces of the vinoxy radical system. All energies are in eV, include zero-point energy, and are relative to CH2CHO (X2A//). Calculated energies are compared with experimentally-determined values in parentheses. Transition states 1—5 are labelled, along with the rate constant definitions from RRKM calculations. The solid potential curves to the left of vinoxy retain Cs symmetry. The avoided crossing (dotted lines) which forms TS5 arises when Cs symmetry is broken by out-of-plane motion. (From Osborn et al.67)...

Fig.1 Calculated free energy diagram for hydrogen evolution at a potential U = 0 V relative to the standard hydrogen electrode at pH = 0. The free energy of H+ + e is by definition the same as that of j - i at standard conditions. The free energy of H atoms bound to different catalysts is then found by calculating the free energy with respect to molecular hydrogen including zero-point energies and entropy terms (reprinted from Ref 83 with permission).

The first term on the right side is exactly common to all steps and the second term as well practically, provided that the rate and in consequence dkT In IBT are referred to unit area of catalyst and consists of one or a few adjacent, physically identical sites of adsorption as exemplified in Section II,D,2,a the third term must be small because of a narrow phase space in which the critical system on a definite is confined. The critical system, e.g., of recombination of hydrogen adatoms, contributes through the third term only 1.94 eu (cal/deg mole) at 25° as calculated from vibrational frequencies of their five normal modes (35). The latter term vanishes exactly in case, where the critical system rests practically on the zero point energy level. The fourth term vanishes, if a is practically unoccupied. If occupied, the term is of common value to every step, provided that is common to them. The fourth term has been calculated for tr of recombination of hydrogen adatoms consisting of two adjacent adsorption sites a a of the adatoms each situated right above a metal atom of adjacent metal atoms 3.52 A apart on (llO)-lattice plane of nickel, i.e., = [2a],... 

Energy characteristics of atoms also define, to a large extent, the strengths of their bonds in molecules, polyatomic ions and radicals. The work required to disrupt a chemical bond, e.g. to separate chemically bonded atoms from the equilibrium distance to a practically infinite one (in the ground state) is called bond energy Eb). In case of the A2 and AX molecules, Eb is equal to the dissociation energy of the molecule (De) which can be determined by thermochemical, calorimetric, kinetic, mass-spectroscopic and molecular spectroscopic techniques. By definition, De characterizes atoms in molecules at the equilibrium state with zero-point energy,... 

The energy gap between the bottom of the well and the v = 0 ground vibrational state is called the zero-point energy, which can be approximated in a computational study, after calculating the harmonic vibrational frequencies. We discuss this further in Section 3.5. We also note from Figure 2.14 two different definitions for the bond dissociation energy De, derived from the very bottom of the well (and therefore a theoretical observation only) and Dq, derived from the ground vibrational state of the well (and therefore an experimental observation). 

Having given the definition of isotope effects on chemical shifts, isotope effects can be further subdivided into intrinsic or equilibrium. In the latter case, isotope substitution many lead to a change in the chemical equilibrium due to a difference in zero-point energies as illustrated in Figure 6.3 (for an explanation, see later). 

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