Density of states convolution

Figure 1 Valence ADC[3] ionizadon spectra (full lines) of the (a) ethane, (b) n-butane, (c) n-hexane and (d) n-octane compounds in their all-staggered confoimadon (6-31G results). Hie dotted curves represent the partial contribution to the convoluted density of states arising from sateilims with Fn < 0.40.

Direct correlations between the results of MO calculations and those of experiments (on electronic structure) have until recently been plagued by a number of problems related to the experimental state of the art The data obtained from visible and ultraviolet absorption spectra arise from transitions between occupied and unoccupied levels, and hence represent a convoluted density of states. In addition, valence electronic transitions in solids in the range of 10-40 eV are accessible experimentally only under limited and difficult conditions deeper spectral regions require varied experimental techniques to probe them. 

The excess density of states figures straightforwardly into the canonical partition function. Substituting the convolution in (1.4) into (1.3) and making the substitution S" = S — S", it follows that... 

However, UPS and XPS do not both image the density of states in entirely the same way. In XPS, the photoelectrons originating from the valence band leave the sample with kinetic energies over 1 keV. In UPS, the exciting energy is on the order of 21 eV, and the kinetic energy of the electrons is low, say between 5 and 16 eV. This means that the final state of the photoelectron is within the unoccupied part of the density of states of the metal. As a result, the UPS spectrum represents a convolution of the densities of occupied and unoccupied states, which is sometimes called the "Joint Density of States."... 

Furthermore, if the magnitude of the tunneling matrix, M, does not change appreciably in the interval of interest, the tunneling current may be viewed as simply the convolution of the density of states of the respective electrodes. 

Figure 9 shows the photoemission valence band spectrum of Th metal. A comparison of the high resolution XPS valence band spectra of Th (resolving two distinct peaks at 1.8 and 0.6 eV below Ep) with a calculated total (s-d) density of states, (convoluted for broadening effects as lifetime and instrumental effects) gives a nearly complete agreement. The two peaks are attributed to 6d states. 

The most recent calculations, however, of the photoemission final state multiplet intensity for the 5 f initial state show also an intensity distribution different from the measured one. This may be partially corrected by accounting for the spectrometer transmission and the varying energy resolution of 0.12, 0.17, 0.17 and 1,3 eV for 21.2, 40.8, 48.4, and 1253.6 eV excitation. However, the UPS spectra are additionally distorted by a much stronger contribution of secondary electrons and the 5 f emission is superimposed upon the (6d7s) conduction electron density of states, background intensity of which was not considered in the calculated spectrum In the calculations, furthermore, in order to account for the excitation of electron-hole pairs, and in order to simulate instrumental resolution, the multiplet lines were broadened by a convolution with Doniach-Sunjic line shapes (for the first effect) and Gaussian profiles (for the second effect). The same parameters as in the case of the calculations for lanthanide metals were used for the asymmetry and the halfwidths ... 

The observed second harmonic signal is the double convolution of the oscillator density of states function Ds(e) with a thermal broadening function F(eVQ+E-h 0 ( Fig.5a ) and a modulation broadening function G(E) ( Fig.Sb ). 

It can be shown formally that the total density of states for two independent degrees of freedom with the densities N (E) and N2(E) is obtained as a convolution of the densities. 

Whilst the tunneling current is a convolution of the tip and sample density of states, it is the energy spectra of the sample states that we try to determine. For positive biases on the sample, where electrons are injected from the tip into unoccupied sample states, the tunnel current will be dominated by electrons close to the Fermi energy of the tip. Under these conditions the density of states of the tip can be taken to be constant and the structure of the spectrum corresponds largely to the spectrum of... 

Finally, if a system has been factored into parts a and jS, the total density of states may be obtained by convolution, viz. 

Similarly, convoluting the individual densities of states, an equation for s oscillators is found, viz. 

Experimental information on the valence levels comes essentially from photoemission XPS and UPS measure densities of states (DOSs) convoluted with absorption cross sections, and these DOS values can be compared with those computed from VEH valence-band structures [195]. This has now been done for several CPs and the agreement is good. It would be more instructive to compare the actual band structure to angle-resolved (ARUPS) measurements, but this has never been done. What comes nearest is an ARUPS study of a series of long alkanes taken as models for polyethylene, a nonconjugated polymer [196]. 

Optical absorption and recombination processes involve two or more particles and so may include correlation effects. Electron-hole pairs form excitons in a crystal, with the result that the absorption and emission spectra are not described by the one-particle density of states distributions. Although excitons can exist in an amorphous material (see Chapter 3), they are not detected in the optical spectra and the absorption is described by the convolution of the one-particle densities of valence and conduction band states. The correlation effects in... 

Lastly, it is generally assumed that 0.5eV is the best possible resolution for solid state XPS measurements and the experimental resolution function is reasonably well reproduced by a Gaussian of full width f at half maximum of 0.7eV. A final "theoretical XPS spectrum" is obtained after correction of the basic density of states function by cross-section effects and convolution by the experimental resolution function (16) ... 

It has been emphasized that STM is sensitive to topography convoluted with the electronic density of states. Spectroscopic characterization of surface states by STM is a challening field of research to be intensified for a better understanding of the chemical reactivity of interfaces. There are still fundamental effects which could be clarified definitively by direct observation. The characterization of transport properties, as demonstrated in Sec. 6, is complementary to STM and STS, and the combination of several techniques should provide a comprehensive description of charge transfer at electrodes. 

Both photoemission and inverse photoemission require reasonable sample conductivity and their application to hard insulators such as MgO and AI2O3 is problematic. Both techniques also involve the complication that inelastic electron energy loss processes become convoluted with electron emission or decay. This may give rise to spectral features in regions where none are expected fi om the density of states [24,25] and care must always be taken to exclude these features before considering assignment to surface states. 

Figure 12. The density of states (DS) on the potential energy (e) space for an M7 cluster, (a) DS in configuration space f2g(s) calculated for the Lennard-Jones potential, increasing with 8. (b) DS in momentum space (1p(E — s), decreasing with 8. (c) A product 3(e) = Qq(e)Qp(E — e) giving the total density of state at the total energy E after the convoluting integral over 8. 3(8) has a single sharp peak. (Reproduced from Ref. 11 with permission.)...

FlP(E — e— jPaijPa) Density of states in momentum space that is a convolution of the components orthogonal and parallel to the direction of pa. E e—jPa is the kinetic energy allowed in the orthogonal directions. 

We wish to compare the valence band density of states (DOS) of f.c.c. and h.c.p. metals with and without stacking faults. We therefore adopt a mixture of the f.c.c. and h.c.p. structures as a representative of the stacking fault structure of either of these structures. To calculate the DOS we summed up the squares of the coefficients of molecular orbital wave functions and convoluted the summed squares with the Gaussian of full width 0.5 eV at half maximum. For these DOS calculations we chose the metals Mg, Ti, Co, Cu and Zn. The model clusters employed here for both the f.c.c. and the h.c.p. structures were made of 13 atoms i.e., a central atom and 12 equidistant neighbor atoms. These structures are shown in Fig. 1. We reproduced the typical electronic structures in bulk materials by extracting the molecular orbitals localized only on the central atom from all the molecular orbitals which contributed - those localized on ligand atoms as well as on the central atom. To perform calculations we take the symmetry of the cluster as C3, and the number... 

Fig. 10.17 (a) INS spectrum of /raw-polyacetylene, (b) calculated density-of-states convoluted with a Gaussian lineshape and the instrument resolution function and (c) as (b) including the effects of the Debye-Waller factor and phonon wings. Reproduced from [29] with permission of Elsevier. 

For the calculation using Variflex, the number of a variational transition q uantum s tates, N ej, w as given b y t he v ariationally d etermined minimum in Nej (R), as a function of the bond length along the reaction coordinate R, which was calculated by the method developed by Wardlaw-Marcus [6, 7] and Klippenstein [8]. The basis of their methods involves a separation of modes into conserved and transitional modes. With this separation, one can evaluate the number of states by Monte Carlo integration for the convolution of the sum of vibrational quantum states for the conserved modes with the classical phase space density of states for the transitional modes. 

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