Elstner_DFTB - Introduction to DFTB Marcus Elstner July 28,...

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Introduction to DFTB Marcus Elstner July 28, 2006
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I. Non-selfconsistent solution of the KS equations DFT can treat up to 100 atoms in routine applications, sometimes even more and about several ps in MD simulations. Very often, one would like to go to larger systems, therefore approximations to DFT are required. To get started, consider a case, where you know the ground state density ρ 0 already to sufficient accuracy. In this case, one can omit the self consistent solution of the KS equations and get the orbitals immediately through: ± - 1 2 2 + v eff [ ρ 0 ] ² φ i = ± i φ i ( ρ 0 stands for a proper chosen input density in the following). This saves a factor of 5 -10 already, however, it is the starting point for further approxi- mations.
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Consider a minimal basis set consisting of atomic orbitals, i.e. η μ = 2 s, 2 p x , 2 p y , 2 p z for first row elements (we omit the core states in the following, since they are in a good approximation chemically inactive) and η μ = 1 s for H. With the basis set expansion φ i = X μ c i μ η μ and the Hamiltonian ˆ H [ ρ 0 ] = ˆ T + v eff [ ρ 0 ] we find: X μ c i μ ˆ H [ ρ 0 ] | η μ > = ± i X μ c i μ | η μ > (1) Multiplication with < η ν | X μ c i μ < η ν | ˆ H [ ρ 0 ] | η μ > = ± i X μ c i μ < η ν | η μ > (2)
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or in matrix notation HC = SC ± (3) This means, we just have to solve the eigenvalue equation once, i.e. we have to diagonalize the Hamilton matrix H μν = < η ν | ˆ H [ ρ 0 ] | η μ > . Note, that our basis set is non-orthogonal , i.e. the overlap matrix S μν = < η ν | η μ > appears in the eigenvalue equations. In empirical schemes, the basis functions are taken to be orthogonal, i.e. S μν = δ μν . Background is the so called L¨owdin orthogonalization. Introducing orthonor- mal orbitals means multiplying with S - 1 / 2 and inserting a ’1’: S - 1 / 2 HS - 1 / 2 S 1 / 2 C = S - 1 / 2 S 1 / 2 S 1 / 2 C ± to get the orthonormal equations ( C 0 = S 1 / 2 C ): H 0 C 0 = C 0 ± Introducing orthonormal orbitals means effectively changing the Hamiltonian. And this is convenient, since in empirical schemes the Hamitonmatrix is com- pletely fitted to empirical data, e.g. for Carbon to the solid state band- structures of several crystal structures (diamond, graphite, bcc etc.), or, in
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uckel theory, to properties of Hydrocarbons. Diagonalization leads to the one-particle energies ± i , i.e. to the so called electronic energy: E elec = X i ± i If we compare this to the total energy in DFT, E [ ρ ] = occ X i ± i - 1 2 Z ρ 0 ( r ) ρ 0 ( r 0 ) | r - r 0 | drdr 0 + E xc [ ρ 0 ] - Z v xc ( r ) ρ 0 ( r ) dr + 1 2 X αβ Z α Z β R αβ (4) it is obvious, that a big part of energy is missing, the so called double- counting and core-core repulsion terms in DFT. First of all, it is interesting to note, that the double counting terms depend on the input/reference density ρ 0 only. The XC parts are hard to evaluate, however, in GGA we can say that they decay exponentially due to the exponential decay of the density-overlap.
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Elstner_DFTB - Introduction to DFTB Marcus Elstner July 28,...

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