Density of states factor

The density of states factors of (2irv) for SDW and (irv) for CDW have been removed. The excitonic response functions (X ) correspond to coupling of electrons and holes in different chains in even (+) and odd (-) combinations. [For more details see ref. 12.]... 

With this formulation of the problem the calculation of quenching rate constants is reduced to evaluating various matrix elements of the type y of Franck-Condon factors for the complex, and of the density of states factor p. 

The Franck-Condon factors are weighted by the density-of-states factor if the fragment is treated as a rigid rotor-harmonic oscillator, /> (e) is given by... 

Figure 12 Optical intensities (in inverse hartrees) as a function of fi) (in hartrees) for the He atom. Top panel) The exact KS and LDA spectra. (Lower panel) The TDDFT corrected spectra. LDA/ALDA results are from Ref. 247 but unshifted. The exact calculations are from Ref. 248, multiplied by the density of states factor (see text), and the experimental results are from Ref. 249.

In a more recent development, it has been shown that the "equally likely" hypothesis of Edwards is not essential for the definition of a temperature-like quantity and the much weaker condition of factorizability of distributions (the density of states factorization discussed in the previous section, e.g.) is sufficient [5-7]. The necessary conditions for being able to define a temperature-like variable and a statistical ensemble based on this variable are (I) the existence of a physical quantity that is conserved by the natural dynamics of the system (in thermal systems energy is conserved but in dissipative granular media, it is not) and (II) the frequency of finding different states with the same value of the conserved quantity is factorizable The latter condition implies that if one... 

Under the assumption that the matrix elements can be treated as constants, they can be factored out of the integral. This is a good approximation for most crystals. By comparison with equation Al.3.84. it is possible to define a fiinction similar to the density of states. In this case, since both valence and conduction band states are included, the fiinction is called the joint density of states ... 

Table B2.2.1 Continuum wavefiinction nonnalization, density of states and cross section factors.

Figure B3.3.5. Energy distributions. The probability density is proportional to the product of the density of states and the Boltzmaim factor.

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

In a canonical ensemble the probability f canon(T E) of visiting a point in phase space wi an energy E is proportional to the Boltzmarm factor, = exp(—E/lcgT), multiplied by tl density of states, (E), where the number of states between E and E + dE is given 1 n E)6E. Thus ... 

The density of states increases rapidly with energy but the Boltzmann factor decrease exponentially, meaning that Pcanon(T, E) is bell-shaped, with values that can vary by mar orders of magnitude as the energy changes. In the multicanonical method the simulatic... 

Here r is the radius vector from the origin to a point R in the crystal, t is the electron-pair-bond function in the region near R, Pfc is the momentum vector corresponding to the three quantum numbers k (the density of states being calculated in the usual way), h is Planck s constant, and G is the normalizing factor. 

Although energy conservation constraints dictate which VP channels are open, it is the nature of the intermolecular interactions, the density of states and the coupling strengths between the states that ultimately dictate the nature of the dynamics and the onset of IVR. These factors are dependent on the particular combinations of rare gas atom and dihalogen molecule species constituting the complex. For example, Cline et al. showed that, in contrast to He Bra, Av = 2 VP in the He Cla and Ne Cla complexes proceeds via a direct... 

NIS provides an absolute measurement of the so-called normal mode composition factors that characterize the extent of involvement of the resonant nucleus in a given normal mode. On the basis of the analysis of experimental NIS data, one can therefore construct a partial vibrational density of states (PVDOS) that can be... 

Here, L(v) is a lineshape function that integrates to unity, v is the frequency,/ is the Lamb-Mossbauer factor, and the desired side bands have an area fraction / that is proportional to which hence determines the relative peak heights in a NIS spectrum. More details are provided in Appendix 2 (Part III, 3 of CD-ROM). An equivalent and often more suggestive display of the NIS spectrum is the PVDOS approach, which describes the NIS signal in terms of the partial vibrational density of states ... 

Here, 6 is the Dirac delta function, U is the potential energy function, and q represents the 3N coordinates. In this expression, the integral is performed over the entire configuration space - each coordinate runs over the volume of the simulation box, and the delta function selects only those configurations of energy S. The N term factors out the identical configurations which differ only by particle permutation. It is worth noting that the density of states is an implicit function of N and V,... 

In principle, valence band XPS spectra reveal all the electronic states involved in bonding, and are one of the few ways of extracting an experimental band structure. In practice, however, their analysis has been limited to a qualitative comparison with the calculated density of states. When appropriate correction factors are applied, it is possible to fit these valence band spectra to component peaks that represent the atomic orbital contributions, in analogy to the projected density of states. This type of fitting procedure requires an appreciation of the restraints that must be applied to limit the number of component peaks, their breadth and splitting, and their line-shapes. 

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