One-electron energy per unit cell

With a given number of electrons in the solid the levels, doubly occupied, will be filled to a certain energy called the Fermi level, which corresponds to a specific value of k, kp. The total one electron energy per unit cell, /f/yV, is then obtained by integrating equation 13.13... 

The density N(E) of electronic states is defined such that N(E)dE is the number of one-electron states per unit cell and per spin, with energy between E and E + dE. It may be calculated from... 

The distances found between platinum centers in these molecules have been correlated with the resonating valence bond theory of metals introduced by Pauling. The experimentally characterized partially oxidized one-dimensional platinum complexes fit a correlation of bond number vs. metal-metal distances, and evidence is presented that Pt—Pt bond formation in the one-dimensional chains is resonance stabilized to produce equivalent Pt—Pt distances.297 The band structure of the Pt(CN)2- chain has also been studied by the extended Huckel method. From the band structure and the density of states it is possible to derive an expression for the total energy per unit cell as a function of partial oxidation of the polymer. The equilibrium Pt-Pt separation estimated from this calculation decreases to less than 3 A for a loss of 0.3 electrons per platinum.298... 

DFT-PW calculations [91] were performed on a SiC crystal for different SP meshes to investigate which one corresponds to the best convergence of the results for the total energy per unit cell. The experimental value a = 4.35 A was taken for the fee lattice constant of SiC, the electron-ion interaction was described by ultrasoft, Vandebildt-type pseudopotentials [93], the cutoff energy for the plane-wave basis set was taken to be 1000 eV. The generalized gradient corrections (GGA) DPT method was used (see Chap. 7). 

In [184] explicit expressions were presented for electron-correlation at the MP2 level and implemented for the total energy per unit cell and for the band structure of extended systems. Using MP2 for (the total energies of one deter-... 

If one examines the dependence of the different physical properties on the number of neighbor interactions explicitly taken into account, one finds that the position of the valence and conduction bands (or, if one recalls Koopmans theorem, the ionization potential and electron affinity needed for the interpretation of ESCA spectra), their widths (which have values between 6 and 10 eV), as weU as the gap and the charge on the carbon atom become practically constant after a few (5-7) neighbor interactions. On the other hand, this is not the case for the total energy per unit cell where one must account for a larger number of neighbors (the reasons for this were discussed in Section 1.3). For further details we refer the reader to the paper of Suhai. ... 

For the calculation of the correlation energy per unit cell in the ground state of a polymer (either conductor or an insulator) one can use any size-consistent method (perturbation theory /18/, coupled cluster expansion /19/, electron pair theories /20/, etc.). In the case of insulators one can Fourier transform the delocalized Bloch orbitals into site semilocalized Wannier functions (WF-s) and perform the excitations between Wannier functions belonging to near lying sites /4/. (For the generation of optimally localized Wannier functions see /21/.) This procedure is, however,... 

Here V1 is the lattice potential deriving from all sites except that labelled 0, and the fts are 7r-electron wavefunctions for the unpaired electrons in the TCNQ ions. If we have one ion per unit cell (hl = h2 == h0), the energy is given by (33),... 

The calculation of the ground-state energy of the Wigner electron crystal necessitates the self-consistent solution of the Slater-Kohn-Sham equations for the Bloch orbitals of a single fully occupied energy band, since there is one electron per unit cell and one is concerned with the spin-polarized state [45], This was accomplished by standard computational routines for energy band-... 

The Kronig-Penney solution illustrates that, for periodic systems, gaps can exist between bands of energy states. As for the case of a free electron gas, each band can hold 2N electrons where N is the number of wells present. In one dimension, this implies that if a well contains an odd number, one will have partially occupied bands. If one has an even number of electrons per well, one will have fully occupied energy bands. This distinction between odd and even numbers of electrons per cell is of fundamental importance. The Kronig-Penney model implies that crystals with an odd number of electrons per unit cell are always metallic whereas an even number of electrons per unit cell implies an... 

To illustrate how the geometric and electronic structures of a lattice are intimately related, let us now consider a dimerized infinite chain of hydrogen atoms. The unit cell of this system contains two hydrogen atoms and we must take into account two different resonance integrals, and one for the interaction of the two atoms within the unit cell and the other between two atoms in neighboring cells (Figure 15.3a). As there are now two Is orbitals per unit cell, we must first build the Bloch orbitals associated with each of them, evaluate the four matrix elements of the 2x2 secular determinant, and solve the secular equation to obtain two different energy values for each k vector. This leads to the expression ... 

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