The Energy Zero

The formulae given in Table 4.1 for the molecular partition functions enable us to write the partition function ratio qheavy/qiight or q2/qi where, by the usual convention, the subscript 2 refers to the heavy isotopomer and 1 refers to the light isotopomer if heavy and light are appropriate designations. Then, a ratio of such partition function ratios enables one to evaluate the isotope effect on a gas phase equilibrium constant, as pointed out above. Before continuing, it is appropriate to 

2 In the equations in this part of the text the frequency v (cycles per second) has units of sec-1. 

The units of Planck s constant h and the Boltzmann constant k are J s and J K—, respectively. 

Chemists, however, are fond of expressing frequency in units of reciprocal wave length or wave 

1010 cm s and X is the wave length. Therefore should one choose to express frequency in units 

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Lorentzian shape centered at the original energy of the adsorbate. The choice of the energy zero is the same as in the subsequent two figures, but is irrelevant in this case. 

The term N pb 4> ) accounts for the excess charges transferred from the reference electrode at the electrostatic potential < ef (which we have aheady defined as the energy zero) to the electrode. Thus, with (5.11) this term can be rewritten as e(< e /< )> where we have introduced the total excess charge at the electrode... 

First, we describe the various system parameters, primarily adapted from Newns (1969). From the energy dispersion relation (2.32), the bulk states are distributed through a band, centered at a, and with width Wb = 4 / . The Fermi level Ef is taken to be at the center of this band, and is chosen to be the energy zero (so that Ef = a = 0, for all systems). The position of /, relative to the vacuum level, is given by the work function (j>, whence the isolated H adatom level, relative to Ef is... 

Figure 5 Density of states of Ni V clusters with N — 5, 6, and 7, calculated by the tight binding method sp (dashed lines) and d (continuous lines). Positive and negative values correspond to up and down J, spins, respectively. The Fermi level is at the energy zero. Adapted with permission from Ref. 45.

The material of the previous subsections may now be assembled into a useful result. If the energy zero for equation (18) is defined to be Uh then a simplification takes place we may write, remembering ah was defined to be the antibonding orbital, simply... 

I-ig. d. Benz> 1 MO s ami excitation processes for the six and eight electron species. Blackened portions and numbers indicate LCAO MO electron densities. Energies in the center column are in units of the absolute value of fj with, an isolated p-orbitul taken as the energy zero,... 

Because the energy zero is arbitrary, the calculated energy is relative. It is meaningful only to compare energies calculated for different configurations of chemically identical systems. 

Potential-energy diagram for molecular iodine. The energy zero has been arbitrarily set at the minimum of the ground-state potential. 

The potential energy function C/(ri---rjv) expresses the energy of an assembly of N atoms or ions as a function of the nuclear coordinates ri - rjv. The Bom-Oppenheimer approximation is, of comse, implicit in the use of such fimctions but there is no explicit inclusion of the effects of the electronic stractme of the system such effects are subsumed into the potential function. The energy zero for such functions is normally taken to be that of the component atoms (or ions) at rest at infinity, that is, the self energies (electron-nuclear) of the atoms (or ions) are not included in U. 

Figure 4. Orbital resolved rf-band of Pts for bare [Xbuik- s]-Pt3 (left) and oxygen-chemisorbed [Xbuik- s]-Pt3-02 (right) surfaces. The spectra are numbered after the extended molecule Xs-Pt3 (see Fig. 1) used in the surface description. The corresponding bulk surfaces Xbun are 3, 8, Pt (red) 11, 12, Pt (blue) and 13, 14, Co3Pt (chocolate). The orbital resolved rf-band of the standalone Pt3 trimer (green, left) is shown as reference. In all panels, the Fermi level is the energy zero. All molecules are oriented such that the Pts cluster is in the X-Y plane.

Figure 6. Partial DOS projected onto the O atoms (red) and ci-band of Pt atoms (blue) for oxygen-chemisorbed [Xbuik-Xs]-Pt3-02 surfaces. The right panels show an expanded view of the O2 spectra (red) using as reference the spectrum of free O2 (green, labels are given in the top panel). A c> antibonding is indexed according to whether it originates from or p-orbitals. The bulk surfaces are 6, Co3Pt, 7, Co, 8, Pt, 9, Ni, and 10, Fe. The ci-band of bulk Pt—left panel, shaded area—is shown as a reference. The Fermi level is the energy zero in all cases.

Fig. 4. Total DOS of bulk VO2 for the monoclinic (left diagram) and tetragonal rutile (right diagram) lattice geometry. The energy zero coincides with the energy of the highest occupied state. The DOSs are given in states per unit volume and per eV.

J=0), then H (298.15 K)-H (0 K) would be 0.254 kcal mol less for "normal" than for "equilibrium" Hg. This would change the difference between AjH (0 K) and a H (298.15 K) for all species involving hydrogen (8, 10). No such change would occur if we chose the lowest level (v=0, J=0) as the energy zero for ortho-H. "Equilibrium" Hg is the form which parallels most substances, i.e., those maintaining equilibrium among all rotational levels (9). 

In summary, because the inner shell energy levels shift in the direction of higher binding energy by an unknown quantity, only qualitative information is obtained on the relative position of the electron distributions between metal and compound. However, provided the shift is approximately the same for 3p and 3d, it is possible to deduce from our results an energy diagram for each oxide (cf. Fig. 12, where the energy zero is the bottom of the conduction band). 

Figure 1.5. Comparison of the energies of icosahedral (I), decahedral (D), and close-packed (C) LJ/v clusters. The energy zero is mi, a function fitted to the energies of the first...

Fig. 11. Calculated density of states for a Si l 11 2 X 1 surface, (after Schliiter et al. [149]). The energy zero is taken at the bulk valence band edge. dout are the states associated with atoms moving upwards out of the surface plane and those from inward-moving atoms.

Energies have always to be measured relative to some chosen initial state. So far we have chosen zero separation as this reference level. For many purposes, however, it is more convenient to choose infinite separation as the energy zero so that Figure 2.4 takes the form shown in Figure 2.6. The curve now... 

Figure 2.7 Influence of repulsive forces on the free energy of interaction of surfaces as a function of separation, taking the energy at infinite separation as the energy zero. Work has to be done on the system to bring the surfaces together.

Figure 1.2 The energy levels available to an electron in a hydrogen atom. The energies are given hy —AfrP, and each level is n-fold degenerate. The lowest energy correspond to n = 1. The energy zero is taken at n = cc, when the electron is removed from the atom...

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