Total energy calculations

A general discussion of the method we are using for the evaluation of the total energy - the Local Density Functional - is presented in full detail in another paper in this volume (Ref. 12). In this Section we summarize the details of its application to our particular cases of GaAs and Ge. The starting point is the ionic 

The pseudopotentials of Ga and As are plotted in Fig. 2.1 in reciprocal space. Fig. Al.l in Appendix shows their variation in direct space, and compares them with the full atomic potentials. For Germanium we choose a potential which is an average of those for Ga and As. [The main advantage of this choice (over the Ge pseudopotential given in Ref. 15 and previously used in Ref. 9) is that the predicted equilibrium lattice constant is close to the experimental value (-1.6 % error). This is crucial for all lattice dynamical calculations. The convergence properties with number of plane waves are expected to be similar to those of GaAs.] 

Their variation in reciprocal spac is given by the formula 

The alternative units of q displayed at the bottom assume a=5.65 A. 

All the calculations are performed in momentum space and (unless otherwise stated) plane waves with kinetic energy up to 9.15 Ry are included in the expansions of the wave functions. Only those with kinetic energy S 2.55 Ry are dealt with exact, the remaining ones are treated by Lowdin perturbation theory up to second order. This corresponds to approximately 21 + 125 waves when working with the two-atoms cells, 43 + 240 when working with the doubled (four atoms) ui it cells, 85 + 500 on quadrupled cells, etc. Two to five special k-points are used for Brillouin zone integration (corresponding to (222) in the notation of 

but the principal goal there is to clarify the nature of elasticity sufficiently to allow the introduction of approximate models that can be used when the distortions are not uniform. 

The term Fco, is the observed cohesive energy it will not interest us here. The constant Cq is called the ladial force constant. Its value is about forty to fifty electron volts for most semiconductors and will be tabulated later. In this chapter, we focus on angular distortions and add this empirical radial interaction where it is needed. 

It is possible, by using the LCAO approach or a pseudopotential approach, to make a calculation of energy bands for the distorted lattice. There are still two atoms per primitive cell, so no serious difficulty is encountered. The sum of the 

Any energy band can be written as a Fourier series in the Brillouin Zone in the form 

Notice how this is done, first, for a one-dimensional case for which the translations T arc integral multiples of the atomic spacing a. If the special point k = n/(2a) is chosen, the two terms sum to zero (note 

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Feibelman P J 1987 Force and total-energy calculations for a spatially compact adsorbate on an extended, metallic crystal surface Phys. Rev. B 35 2626... 

Payne M C, Teter M P, Allan D C, Arias T A and Joanopoulos J D 1992 Iterative minimization techniques for ab /M/o total energy calculations molecular dynamics and conjugate gradient Rev. Mod. Phys. 64 1045... 

Fig. 11.38 Lag ejfects in ab initio molecular dynamics. (Figure redrawn from Payne MC, M P Teter, D C Allan, R A Arias and D ] Joannopoidos 1992. Iterative Minimisaticm Techniques for Ab Initio Total-Energy Calculations Molecular Dynamics and Conjugate Gradients. Reviews of Modern Physics 64 1045-1097.)...

J. Ihm. Total energy calculations in solid state physics. Rep Prog Phys 57 105, 1988. 

The entries in the remaining sections of the table perform the finol higher-level total energy calculation at a... 

Our best understanding to date of this qV relation is that it has the same origin in the DFT-LDA equations as the Harris energy approximation. " The concept that certain fragments retain their identity in total energy calculations that was demonstrated by Harris... 

Experimentally it is found that the Fe-Co and Fe-Ni alloys undergo a structural transformation from the bee structure to the hep or fee structures, respectively, with increasing number of valence electrons, while the Fe-Cu alloy is unstable at most concentrations. In addition to this some of the alloy phases show a partial ordering of the constituting atoms. One may wonder if this structural behaviour can be simply understood from a filling of essentially common bands or if the alloying implies a modification of the electronic structure and as a consequence also the structural stability. In this paper we try to answer this question and reproduce the observed structural behaviour by means of accurate alloy theory and total energy calcul ions. 

TOTAL ENERGY CALCULATIONS OF ALLOYS LOCALLY SELF-CONSISTENT GREEN S FUNCTION METHOD... 

LA. Abrikosov, A.V. Ruban, B. Johansson, and H.L. Skriver, Total energy calculations of random alloys Connolly-Wiliams and CPA methods, in Stability of Materials , Series E Applied Sciences, A. Gonis, P.E.A. Turchi, and J. Kudrnovsky, ed., Kluwer Academic Publishers, the Netherlands (1996). 

In the present work, we report on a new semi-empirical theoretical approach which allows us to perform spin and symmetry unconstrained total energy calculations for clusters of transition metal atoms in a co .putationally efficient way. Our approach is based on the Tight Binding Molecular Dynamics (TBMD) method. 

K.M. Ho and B.N. Harmon, First-principles total energy calculations applied to displacive transformations. 

In practice it is only necessary to do the total energy calculation for the block with some convenient termination at the origin of x and to calculate the excess energy rex(O) for this case. The true excess energy can then be obtained by adding the excess energy of the correct terminating piece. An example should make the procedure clear. 

The precursor of such atomistic studies is a description of atomic interactions or, generally, knowledge of the dependence of the total energy of the system on the positions of the atoms. In principle, this is available in ab-initio total energy calculations based on the loc density functional theory (see, for example, Pettifor and Cottrell 1992). However, for extended defects, such as dislocations and interfaces, such calculations are only feasible when the number of atoms included into the calculation is well below one hundred. Hence, only very special cases can be treated in this framework and, indeed, the bulk of the dislocation and interfacial... 

TOTAL ENERGY CALCULATIONS IN THE TIGHT-BINDING APPROXIMATION... 

In crystals with the LI2 structure (the fcc-based ordered structure), there exist three independent elastic constants-in the contracted notation, Cn, C12 and 044. A set of three independent ab initio total-energy calculations (i.e. total energy as a function of strain) is required to determine these elastic constants. We have determined the bulk modulus, Cii, and C44 from distortion energies associated with uniform hydrostatic pressure, uniaxial strain and pure shear strain, respectively. The shear moduli for the 001 plane along the [100] direction and for the 110 plane along the [110] direction, are G ooi = G44 and G no = (Cu — G12), respectively. The shear anisotropy factor, A = provides a measure of the degree of anisotropy of the electronic charge... 

Use of the Born-Oppenheimer approximation is implicit for any many-body problem involving electrons and nuclei as it allows us to separate electronic and nuclear coordinates in many-body wave function. Because of the large difference between electronic and ionic masses, the nuclei can be treated as an adiabatic background for instantaneous motion of electrons. So with this adiabatic approximation the many-body problem is reduced to the solution of the dynamics of the electrons in some frozen-in configuration of the nuclei. However, the total energy calculations are still impossible without making further simplifications and approximations. 

In concluding, we point out an essential role of vibrational spectra In theoretical studies. Total energy calculations yield quantities of much Interest, like equilibrium geometries and binding energies, which are not accessible In a direct experimental way. Only the vibrational quantities can be meaningfully compared with experiment and provide a way to assess the adequacy of these calculations. 

Kresse G, Furthmuller J. 1996a. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput Mater Sci 6 15-50. 

Fig. 3. Contour plot of the energy surface for H+ in a (110) plane through the atoms in Si. The zero of energy is arbitrarily chosen at T. The black dots represent Si atoms at their unrelaxed positions the relaxations (which are different for different H positions) are not shown but are taken into account in the total-energy calculations. The contour interval is 0.1 eV (Reprinted with permission from the American Physical Society, Van de Walle, era/., 1989.)... 

LaFemina, J.P. Total energy calculations of semiconductor surface reconstructions. Surf. Sci. Rep. 1992,16, 133-260. 

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