The Density of States

The density of states for a one-dimensional system diverges as 0. This divergence of D E) is not a serious issue as the integral of the density of states remains finite. In tliree dimensions, it is straightforward to show that... 

One can detennine the total number of electrons in the system by integrating the density of states up to the highest occupied energy level. The energy of the highest occupied state is called the Eermi level or Eermi energy, E ... 

Another usefiil quantity is defining the electronic structure of a solid is the electronic density of states. In general the density of states can be defined as... 

Figure Al.3.15. Density of states for silieon (bottom panel) as ealeulated from empirieal pseudopotential [25], The top panel represents the photoemission speetra as measured by x-ray photoemission speetroseopy [30], The density of states is a measure of the photoemission speetra.

By examining the spatial eharaeter of the wavefiinetions, it is possible to attribute atomie eharaeteristies to the density of states speetnun. For example, the lowest states, 8 to 12 eV below the top of the valenee band, are s-like and arise from the atomie 3s states. From 4 to 6 eV below the top of the valenee band are states that are also s-like, but ehange eharaeter very rapidly toward the valenee band maximum. The states residing within 4 eV of the top of the valenee band are p and arise from the 3p states. 

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 ... 

Note that if we identify the sum over 8-fimctions with the density of states, then equation (A1.6.88) is just Femii s Golden Rule, which we employed in section A 1.6.1. This is consistent with the interpretation of the absorption spectmm as the transition rate from state to state n. 

The only modification of equation (Al.6.90) for spontaneous Raman scattering is the multiplication by the density of states of the cavity, equation (Al.6.24). leading to a prefactor of the fonn cojCOg. ... 

Thus many aspects of statistical mechanics involve techniques appropriate to systems with large N. In this respect, even the non-interacting systems are instructive and lead to non-trivial calculations. The degeneracy fiinction that is considered in this subsection is an essential ingredient of the fonnal and general methods of statistical mechanics. The degeneracy fiinction is often referred to as the density of states. 

Then F( ) = S(/ -t d )- 2( ), and the density of states D E) = dS/d/ . A system containing a large number of particles N, or an indefinite number of particles but with a macroscopic size volume V, normally has the number of states S, which approaches asymptotically to... 

The complete thennodynainics of a system can now be obtained as follows. Let die isolated system withAi particles, which occupies a volume V and has an energy E within a small uncertainty E, be modelled by a microscopic Flamiltonian Ti. First, find the density of states p( ) from the Flamiltonian. Next, obtain the entropy as S(E, V, N) = log V E) or, alternatively, by either of the other two equivalent expressions... 

The canonical distribution corresponds to the probability density for the system to be in a specific microstate with energy E- H, from it one can also obtain the probability P( ) that the system has an energy between E and E + AE i the density of states D E) is known. This is because, classically. 

For a spin-zero particle in a cubic box, the density of states is... 

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The coimection between statistical mechanics and thennodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a hannonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

The RRKM rate constant is often expressed as an average classical flux tlirough the transition state [18,19 and 20]. To show that this is the case, first recall that the density of states p( ) for the reactant may be expressed as... 

Sinee there is one state per unit of sueh volume, E) is also the iiumber of states with energy less than or equal to E, and is ealled the integrated density of states. The iiumber of states g E),AE with energy between E and E + E, the density of states, is the derivative of 4i ... 

Figure B3.2.8. Comparison of the photoemission speetnim for the eetineite (Na Se) and the density of states ealeulated by tire AFC ELAPW-/cp method...

Pisani [169] has used the density of states from periodic FIP (see B3.2.2.4) slab calculations to describe the host in which the cluster is embedded, where the applications have been primarily to ionic crystals such as LiE. The original calculation to derive the external Coulomb and exchange fields is usually done on a finite cluster and at a low level of ab initio theory (typically minimum basis set FIP, one electron only per atom treated explicitly). 

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

Variation of the density of states, D(E), for the simple ID lattice, shown with the corresponding energy igram. 

Ihi. slope of the energy versus k curve the flatter the band the greater the density of states at that energy. 

The origins of the Finnis-Sinclair potential [Finnis and Sinclair 1984] lie in the density of states and the moments theorem. Recall that the density of states D(E) (see Section 3.8.5) describes the distribution of electronic states in the system. D(E) gives the number of states between E and E - - 8E. Such a distribution can be described in terms of its moments. The moments are usually defined relative to the energy of the atomic orbital from which the molecular orbitals are formed. The mth moment, fi", is given by ... 

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... 

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