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2. Tight-binding Density Functional Theory DFTB an approximate Kohn-Sham DFT scheme Gotthard Seifert Technische Universität Dresden Physikalische Chemie

3. Density Functional Theory Many particle problem (M electrons) Functional - electron density Ansatz Total energy

4. Kohn-Sham-equations Approximation for VXC LDA (LDA – Local Density Approximation) Gradient expansion GGA

5. Methodology of approximate DFT Basic Concepts Local potential! Representation: Numerical on a grid Analytical with auxiliary functions

6. Many centre problem (N nuclei) Ansatz Atomic Orbitals - LCAO Gauss type Orbitals - LCGTO Slater type Orbitals - LCSTO Plane Waves - PW Muffin Tin Orbitals - LMTO

7. LCAO method LCAO Ansatz Secular equations Hamilton matrix Overlap matrix

8. Practical and Computational aspects Basis sets, Approximations… Basis functions Atomic Orbitals - AO Gauss Type Orbitals – GTO (cartesian Gaussians) Slater Type Orbitals - STO

9. Atomic Orbitals – AO’s • Analytical representation • Linear combination of Slater type orbitals (STO) with

10. Optimization of basis functions • Confinement potential Example: Cu (r0=3.5,n0=4)

11. Bonding” behaviour • (Linear combination of Cu-4s(A)-Cu-4s(B)) • Variational behaviour • (Band energies of Cu as function of r0)

12. Valence basis -basis function (AO) at A, B -core function at A, B VA - potential at A, B

13. Core-Orthogonalization - orthogonalized basis function -non-orthogonalized basis function (AO) • -core function at l • Pseudopotentials VlPP I II Pseudopotentials for three centre (I) and crystal field (II) integrals

14. Pseudopotential compensation • (Example: Cu (fcc), i-neighbour shell) minimal number of 3-centre integrals (numerical calculation) 2-centre integrals (analytical calc.– Eschrigphys.stat.sol. b96, 329 (1979))

15. Optimization of the Potential Veff Q = 0 – for a neutral system Vj0 – potential of a „neutral“ atomnotfree atom!

16. Potential ofatomic N and around N in N2 (spherically averaged)

17. Matrix elements Example: N2 molecule Neglect PP-terms Potential along the N-N axis in N2

18. Kohn-Sham energies in CO Neglect PP-terms

19. Band Structure SCF-DFT calculation (FPLO)

20. Band Structure DFTB calculation

21. Band Structure SCF-DFT calculation (FPLO) DFTB calculation

22. Heteronuclear Systems A - B Charge transfer A B not in real space!! qA, qB projection to basis functions on Aand B but not

23. Kohn-Sham energies in HF 1π 1sH 2pF 1σ 2sF 1σ - - -Neglect PP-terms V0F, V0H Req. ___ SCF Dipolmoment:DFTB – 2.1 D exp. 1,8 D

24. Cadmiumsulfide — DFTB — SCF-LCAO-DFT (FPLO)

26. Density-Functional - Total energy electron density magnetization density Density fluctuations: Expansion of EDFT around n=n0, μ=0 up to 2nd order

27. Density-Functional total energy 2nd order approximation Density-Functional based „tight binding“ DF-TB

28. Cancellation of „double counting terms“ U(Rjk) EB/eV R/aB Li2 - dimer EB - U(Rjk) Short range repulsive energy U(Rjk)

29. Approximations: Minimal(valence) basis in LCAO ansatz Neglect of pseudopotential terms in h0μν2-center representation! -Mulliken gross population at j 2nd order approximation in energy

30. Approximation for magnetization density

31. Energy: Hamiltonian:: Self Consistent Charge method SCC-DFTB

32. Forces in DFTB Forces – electronic contribution Forces – contribution from repulsive energy U

33. Practical Realization of DFTB DFT calculations of reference molecules Atomic DFT calculations Repulsive energies Hamilton and Overlap matrix Self consistent charge - SCC Solution of the secular problem Calculation of: Calculation of Energy and Forces

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