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Technische Universität 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 – AO’s  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) μνi 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 1σ1σ 1σ1σ 1π1π 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, μ=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 μν  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

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