Level densities

Forst W 1971 Methods for calculating energy-level densities Chem. Rev. 71 339-56... 

Although Dirac s equation does not directly admit of a completely self-consistent single-particle interpretation, such an interpretation is physically acceptable and of practical use, provided the potential varies little over distances of the order of the Compton wavelength (h/mc) of the particle in question. It allows, for instance, first-order relativistic corrections to the spectrum of the hydrogen atom and to the core-level densities of many-electron atoms. The latter aspect is of special chemical importance. The required calculations are invariably numerical in nature and this eliminates the need to investigate central-field solutions in the same detail as for Schrodinger s equation. A brief outline suffices. 

The energy level density is not important in determining the magnitude of the isotope effect at high pressure. At the low pressure limit, again for thermal activation,... 

The calculations of Feibelman and Hamann have expressly addressed the surface electronic perturbation by sulfur as well as by Cl and The sulfur-induced total charge density vanishes beyond the immediately adjacent substrate atom site. However, the Fermi-level density of states, which is not screened, and which governs the ability of the surface to respond to the presence of other species, is substantially reduced by the sulfur even at nonadjacent sites. Finally, the results for several impurities indicate a correlation between the electronegativity of the impurity and its relative perturbation of the Fermi-level density of states, a result which could be very relevant to the poisoning of H2 and CO chemisorption by S,C1, and as discussed above. 

Fig. S-S8. Electron levels of dehydrated redox particles, H ld + bh /h = H,d, adsorbed on an interface of metal electrodes D = state density (electron level density) 6 = adsorption coverage shVi - most probable vacant electron level of adsorbed protons (oxidants) eH(d = most probable occupied electron level of adsorbed hydrogen atoms (reductants) RO.d = adsorbed redox particles.

In the state of Fermi level pinning, the Fermi level at the interface is at the surface state level both where the level density is high and where the electron level is in the state of degeneracy similar to an allowed band level for electrons in metals. The Fermi level pinning is thus regarded as quasi-metallization of the interface of semiconductor electrodes, making semiconductor electrodes behave like metal electrodes at which all the change of electrode potential occurs in the compact layer. 

D. Zero-Point Energy Problem and Level Density... 

Due to the large-level density of the lower-lying adiabatic electronic state, the chances of a back transfer of the adiabatic population are quite small for a multidimensional molecular system. To a good approximation, one may therefore assume that subsequent to an electronic transition a random walker will stay on the lower adiabatic potential-energy surface [175]. This observation suggests a physically appealing computational scheme to calculate the time evolution of the system for longer times. First, the initial decay of the adiabatic population is calculated within the QCL approach up to a time to, when the... 

To determine the optimal value of quantum correction y, several criteria have been proposed, all of which are based on the idea that an appropriate classical theory should correctly reproduce long-time hmits of the electronic populations. (Since the populations are proportional to the mean energy of the corresponding electronic oscillator, this condition also conserves the ZPE of this oscillator.) Employing phase-space theory, it has been shown that this requirement leads to the condition that the state-specihc level densities... 

It is interesting to note that the latter criterion imphes that the ground-state level density completely dominates the total level density— that is, that No E) N E). Hence the assumption (98) of complete decay into the adiabatic ground state is equivalent to the criterion that the classical and quantum total level densities should be equivalent. Furthermore, it is clear that this criterion determines an upper limit of 7. This is because larger values of the quantum correction would result in ground-state population larger than one (or negative excited-state populations). 

It is instmctive to first consider how the classical approximation of the total level density, Nc E), depends on the amount of electronic ZPE included. Let us begin with a one-mode problem—that is, Model 1 Va, for which Nc (E) can be evaluated analytically for high enough energies [226]. One obtains... 

Figure 20. Total integral level density N E) as obtained for Model IVa. The mapping calculation (y = 1, upper line) is seen to match the quantum staircase function almost perfectly, while the mean-field trajectory results (y = 0, lower line) underestimate the correct level density considerably.

Although the classical mapping formulation yields the correct quantum-mechanical level density in the special case of a one-mode spin-boson model, the classical approximation deteriorates for mulhdimensional problems, since the classical oscillators may transfer their ZPE. As a hrst example. Fig. 21a compares Nc E) as obtained for Model I in the limiting cases y = 0 and 1 (thin solid lines) to the exact quantum-mechanical density N E) (thick line). The classical level density is seen to be either much higher (for y = 1) or much lower (for y = 0) than the quantum result. Since the integral level density can be... 

Figure 21. (a) Total integral level density N E) and (b) normalized state-specific level density... 

Figure 22 shows the same quantities for the intramolecular electron-transfer Model IVb. Similar to what occurs in the pyrazine model, the classical level density obtained with y = 1 overestimates the total and state-specific level density while for y = 0 the classical level densities are too small. Employing a ZPE correction of y = 0.8 results in a very good agreement with the total quantum mechanical level density, while the criterion to reproduce the state-specific level density results in a ZPE correction of y = 0.6. 

Employing the alternative criterion that requires the agreement of the classical and the quantum-mechanical total level density, we have also calculated the time-dependent observables for the quantum correction y =... 

Having determined the appropriate value of the quantum correction from the comparison of classical and quantum level densities, it is interesting to study the accuracy of the simple approximation (99). Extracting from Fig. 19 the longtime limits of the adiabatic ground-state populations as Pq j = 0, oo) = 0.75 and Pq j = 1, oo) = 1.25, the difference of the two populations yields Ky2 Ti) = 0.5, just as predicted by Eq. (99). Furthermore, we may employ the approximation to estimate the optimal quantum correction. Assuming that Pq oo) = 1, we obtain y = 0.5, which is in qualitative agreement with the results obtained above. ... 

In the case of Model II, neither the state-specihc nor the total quantum-mechanical level densities are available. To determine the optimal value of the ZPE correction, therefore criterion (98) was applied, which yielded y = 0.6. The mapping results thus obtained (panels D and G) are seen to reproduce the quantum result almost quantitatively. It should be noted that this ZPE adjustment ensures that the adiabatic population probabilities remain within [0, 1] and at the same time also yields the best agreement with the quantum diabatic populations. 

We note that the integral over the energetically allowed phase space— that is, the classical level density (97)—was found in Fig. 20 to be in excellent agreement with the quantum-mechanical level density. This finding indicates that there is a valid correspondence between the quantum-mechanical two-state system and its classical mapping representation. A similar conclusion was drawn in a recent smdy of a mapped two-state problem, which focused on the Lyapunov exponents and the energy level statistics of the system [124, 235]. 

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