Efficient Electronic Energy Functionals for Tight-Binding
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of new electronic energy functionals. The purpose of this paper is to explain how electronic energy functionals other than the Hohenberg-Kohn (H-K) functional can be constructed and applied to localized bases, here loosely called tight-binding. This Sec. outlines the difficulties of using plane wave basis sets and of the self-consistent minimization of the H-K functional. There follow Secs. describing the construction of generalized functionals with convenient variational parameters, the calculation of energy differences and forces from the generalized functionals, the construction of approximate electron densities and potentials together with their optimization, and a brief discussion of the advantages of atomic exchange and correlation over the Local Density Approximation (LDA). The reason it is necessary to use localized electronic bases for big systems is to keep small the number of non-zero matrix elements in each row and column of the electronic Hamiltonian. Atomic-like orbitals for the electrons only extend over a few shells of neighbors, so the calculation of Hamiltonian matrix elements between localized orbitals involves only a few atoms, and there are at most a few hundred non-zero elements in each row or column of the Hamiltonian matrix. With localized bases the effects of distant parts of the system arise from the way the localized orbitals combine to form extended states. Unlike localized orbitals, each plane wave overlaps every atom in the system, so calculation of the Hamiltonian requires the Fourier transform of the potential, a process whose length increases faster than the number of inequivalent atoms. Worse, the number of non-zero matrix elements in each row or column of the Hamiltonian grows as the number of inequivalent atoms making any attempt to diagonalize the Hamiltonian grow as the cube of the number of non-zero elements. The use of localized bases is further justified by a remarkable property of wave equations such as the electronic Schroedinger equation, that the density of states, weighted by its intensity on some localized function, is exponentially insensitive to distant parts of the system [1 ]. For the Mat. Res. Soc. Symp. Proc. Vol. 491 ©1998 Materials Research Society
present discussion, this property tells us that the energy it takes to displace an atom or the force on an atom, is insensitive to distant parts of the system. This means that plane wave bases with their equal weighting of all parts of the system are ill suited to the calculation of quantities such as energy differences and forces due to local disturbances, which depend only on nearby parts of the system. The energy differences and forces are just the quantities which should be calculated, not total energies as is often done. The simplest way to see this is that as the number of inequivalent atoms grows, so does the total energy, while the energy differences and forces do not. Consequently, the total energies have to be calculated ever more accurately in order to keep constant the accuracy of the energy differences
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