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Tight-binding Density Functional Theory DFTB
an approximate Kohn-Sham DFT scheme
Gotthard Seifert
Technische Universität Dresden Physikalische
Chemie
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Density Functional Theory
Many particle problem
(M electrons)
Functional
- electron density
Ansatz
Total energy
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Kohn-Sham-equations
Approximation for VXC ? LDA
(LDA Local Density Approximation)
Gradient expansion? GGA
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Methodology of approximate DFT Basic Concepts
Local potential!
Representation Numerical on a grid Analytical
with auxiliary functions
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Many centre problem
(N nuclei)
Ansatz
Atomic Orbitals - LCAO
Gauss type Orbitals - LCGTO
Slater type Orbitals - LCSTO
Plane Waves - PW
Muffin Tin Orbitals - LMTO
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LCAO method
LCAO Ansatz
Secular equations
Hamilton matrix
Overlap matrix
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Practical and Computational aspects Basis sets,
Approximations
Basis functions
Atomic Orbitals - AO
Gauss Type Orbitals GTO (cartesian Gaussians)
Slater Type Orbitals - STO
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• Atomic Orbitals AOs
• Analytical representation
• Linear combination of Slater type orbitals (STO)

with
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• Optimization of basis functions
• Confinement potential

Example Cu (r03.5,n04)

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• Bonding behaviour
• (Linear combination of Cu-4s(A)-Cu-4s(B))

• Variational behaviour
• (Band energies of Cu as function of r0)

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• Valence basis

-basis function (AO) at A, B

• -core function at A, B

• VA - potential at A, B

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• 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
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• Pseudopotential compensation
• (Example Cu (fcc), i-neighbour shell)

µ ? i
4s 4s 0 -0.0256 0.0108 -0.0148
4s 4s 1 -0.0033 0.0025 -0.0008
4s 4s 2 -0.0012 0.0014 0.0002
4s 4s 3 -0.0002 0.0001 -0.0001
4s 5s 0 -0.0597 0.0471 -0.0126
4s 5s 1 -0.0118 0.0107 -0.0011
5s 5s 0 -0.2120 0.2073 -0.0053
5s 5s 1 -0.0472 0.0465 -0.0007
minimal number of 3-centre integrals (numerical
calculation) 2-centre integrals (analytical
calc. Eschrig phys.stat.sol. b96, 329 (1979))
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• Optimization of the Potential
• Veff

Q 0 for a neutral system
Vj0 potential of a neutral atom not free atom!
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Potential of atomic N and around N in
N2 (spherically averaged)
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• Matrix elements

Example N2 molecule
Neglect PP-terms
Potential along the N-N axis in N2
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Kohn-Sham energies in CO
Neglect PP-terms
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Band Structure
SCF-DFT calculation (FPLO)
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Band Structure
DFTB calculation
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Band Structure
SCF-DFT calculation (FPLO)
DFTB calculation
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Heteronuclear Systems A - B
Charge transfer A B not in real space!! qA,
qB projection to basis functions on A and B
but not
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Kohn-Sham energies in HF
1p
1sH
2pF
1s
2sF
1s
- - - Neglect PP-terms V0F, V0H
Req.
___ SCF
Dipolmoment DFTB 2.1 D exp. 1,8 D
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Cadmiumsulfide
DFTB SCF-LCAO-DFT (FPLO)
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Density-Functional - Total energy
electron density
magnetization density
Density fluctuations
Expansion of EDFT around nn0, µ0 up to 2nd order
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Density-Functional total energy 2nd order
approximation
Density-Functional based tight binding DF-TB
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Cancellation of double counting terms
U(Rjk)
EB/eV
R/aB
Li2 - dimer
EB - U(Rjk)
Short range repulsive energy
U(Rjk)
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Approximations
Minimal (valence) basis in LCAO ansatz
Neglect of pseudopotential terms in h0µ?
?2-center representation!
-Mulliken gross population at j
2nd order approximation in energy
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Approximation for magnetization density
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Energy
Hamiltonian
Self Consistent Charge method SCC-DFTB
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Forces in DFTB
Forces electronic contribution
Forces contribution from repulsive energy U
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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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