start Technische Universitt Dresden Physikalische Chemie Gotthard Seifert Tight-binding Density Functional Theory DFTB an approximate Kohn-Sham DFT scheme Density Functional Theory Functional - electron density Total energy Ansatz Many particle problem (M electrons) Kohn-Sham-equations Approximation for V XC LDA (LDA Local Density Approximation) Gradient expansion GGA Methodology of approximate DFT Basic Concepts Local potential! Representation: Numerical on a grid Analytical with auxiliary functions (N nuclei) Many centre problem Ansatz Atomic Orbitals - LCAO Gauss type Orbitals - LCGTO Plane Waves - PW Muffin Tin Orbitals - LMTO Slater type Orbitals - LCSTO LCAO Ansatz Secular equations Hamilton matrix Overlap matrix LCAO method Practical and Computational aspects Basis sets, Approximations Basis functions Slater Type Orbitals - STO Gauss Type Orbitals GTO (cartesian Gaussians ) Atomic Orbitals - AO Atomic Orbitals AOs Analytical representation Linear combination of Slater type orbitals (STO) with Optimization of basis functions Confinement potential Example: Cu (r 0 =3.5,n 0 =4) Bonding behaviour (Linear combination of Cu-4s(A)-Cu-4s(B)) Variational behaviour (Band energies of Cu as function of r 0 ) Valence basis - basis function (AO) at A, B -core function at A, B V A - potential at A, B Core-Orthogonalization - orthogonalized basis function - non-orthogonalized basis function (AO) -core function at l Pseudopotentials V l PP I II Pseudopotentials for three centre (I) and crystal field (II) integrals Pseudopotential compensation (Example: Cu (fcc), i-neighbour shell) i 4s s s s s5s s5s s s minimal number of 3-centre integrals (numerical calculation) 2-centre integrals (analytical calc. Eschrig phys.stat.sol. b96, 329 (1979)) Optimization of the Potential V eff V j 0 potential of a neutral atom not free atom! Q = 0 for a neutral system Potential of atomic N and around N in N 2 (spherically averaged) Potential along the N-N axis in N 2 Matrix elements Example: N 2 molecule Neglect PP-terms Kohn-Sham energies in CO Neglect PP-terms SCF-DFT calculation (FPLO) Band Structure DFTB calculation Band Structure SCF-DFT calculation (FPLO) Band Structure DFTB calculation Heteronuclear Systems A - B Charge transfer A B not in real space!! q A, q B projection to basis functions on A and B but not Kohn-Sham energies in HF 11 11 11 2s F 2p F 1s H R eq Neglect PP-terms V 0 F, V 0 H ___ SCF Dipolmoment: DFTB 2.1 D exp. 1,8 D Cadmiumsulfide DFTB SCF-LCAO-DFT (FPLO) Density-Functional - Total energy electron density magnetization density Density fluctuations: Expansion of E DFT around n=n 0, =0 up to 2nd order Density-Functional based tight binding DF-TB Density-Functional total energy 2nd order approximation Cancellation of double counting terms E B /eV U(R jk ) E B - U(R jk ) Li 2 - dimer Short range repulsive energy U(R jk ) R/a B Approximations: Minimal (valence) basis in LCAO ansatz Neglect of pseudopotential terms in h 0 2-center representation! -Mulliken gross population at j 2nd order approximation in energy Approximation for magnetization density Hamiltonian: : Energy : Self Consistent Charge method SCC-DFTB Forces in DFTB Forces electronic contribution Forces contribution from repulsive energy U Practical Realization of DFTB Atomic DFT calculations Hamilton and Overlap matrix Solution of the secular problem Calculation of: Calculation of Energy and Forces Self consistent charge - SCC DFT calculations of reference molecules Repulsive energies end