Extended bond orbitals

Bond orbitals are constructed ft om s/r hybrids for the simple covalent tetrahedral structure energies are written in terms of a eovalent energy V2 and a polar energy K3. There are matrix elements between bond orbitals that broaden the electron levels into bands. In a preliminary study of the bands for perfect crystals, the energies for all bands at k = 0 arc written in terms of matrix elements from the Solid State Tabic. For calculation of other properties, a Bond Orbital Approximation eliminates the need to find the bands themselves and permits the description of bonds in imperfect and noncrystalline solids. Errors in the Bond Orbital Approximation can be corrected by using perturbation theory to construct extended bond orbitals. Two major trends in covalent bonds over the periodic table, polarity and metallicity, arc both defined in terms of parameters from the Solid State Table. This representation of the electronic structure extends to covalent planar and filamentary structures. 

D The Bond Orbital Approximation and Extended Bond Orbitals ... 

We define extended bond orbitals to be bond orbitals corrected in perturbation theory for those matrix elements that couple bonding and antibonding orbitals. This is done directly by using Eq. (1-16). Then corrections are added to every bond orbital 6> these arise from the interaction, 14, with neighboring antibond orbitals la),... 

The matrix elements between extended bond orbitals on different sites will not be evaluated until Chapter 6 for now, examine the matrix elements between extended bonding and antibonding orbitals on neighboring sites to affirm that they are of second order in <61 //1 a>, rather than first order as in Fig. 3-4. As long as they arc of second order, their neglect (implicit in the Bond Orbital Approximation for extended orbitals) is justified. 

Consider then the matrix clement H A), with the central site for the different from the central site for the /1>. If the two sites are sufficiently far apart that the corresponding b H a is zero, then the corona of one extended orbital will not overlap the central site of the other only corona-corona terms will enter the matrix element and they will be of order K b /( i )- If the sites arc close enough that <61 //1 a> = Kba is nonzero, then the corona of each will overlap the central cell of the other, giving two terms, Fba/(fib - A.) and // h>Fba/( a - fib)- The sum of these just cancels the direct i ,a contribution, leaving only contributions of order Fab/(cb — extended bond orbitals. 

To obtain an idea of what variation from material to material is expected, we may evaluate such matrix elements by ignoring the corona of the extended bond orbitals and in fact by taking only the matrix element for a bond orbital and an antibonding orbital at the same bond site. Using Eqs. (3-13) and (3-16),... 

Comparison of interbond matrix elements (eV) obtained in different ways. The first entry in each list is based upon the parameters of the Solid State Table. The second entry is the matrix clement based upon the Chadi-Cohen parameters (1975). The third entry is the value for extended bond orbitals obtained by Sokel (1976) by using perturbation theory and the Chadi-Cohen parameters. The fourth entry is the value obtained by Pantclides and Harrison by fitting the observed bands. 

Eq. (9-24) gives the effective charge within the Bond Orbital Approximation. It would be of interest to calculate corrections in terms of extended bond orbitals, but this has not been done to date. 

Extended bond orbitals were introduced on page 83 of the text, but few of the corresponding corrections to the properties were calculated. Since publication corrections have been made to the total energy of semiconductors to obtain cohesion, heats of solution, and corrections to the dielectric properties/ There have also been studies of Coulomb effects in semiconductors and insulators, including self-consistency and the "many-body" enhancement of the gap, in the same spirit as the analyses in this text. 

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