DFT in practice - th.fhi-berlin.mpg.de

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DFT in practice Sergey V. Levchenko Fritz-Haber-Institut der MPG, Berlin, DE
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Transcript of DFT in practice - th.fhi-berlin.mpg.de

Outline
• General concepts:
- basis sets - integrals and grids, electrostatics, molecules and clusters versus periodic systems - scalar relativity - solution of the eigenvalue problem
Equations to be solved
, = Ψ∗ ,,… Ψ ′,,… …
, = Ψ∗ ,′, … Ψ ,′,,… …
Problem: kinetic energy and electron-electron interaction energy as a functional of () -- ?
Equations to be solved

Converged? Update density / density matrix
(mixer, preconditioner)
Update potential
V j+1
Self-consistent field method
When is the calculation finished? • Answer: When „property“ does not change
Convergence
Typical „properties“ to converge:
Change of the sum of eigenvalues
Change of the total energy
Options differ between codes
Initial guess: e.g. density
Converged? Update density / density matrix
(mixer, preconditioner)
Update potential
Importance of the initial guess Electronic structure exhibits multiple minima
• Qualitatively different behavior depending on guess
• Common issue for magnetic solids
1O2
3O2
Importance of the initial guess Electronic structure exhibits multiple minima
• Qualitatively different behavior depending on guess
• Common issue for magnetic solids
1O2
3O2
Initial guess
codes)
Initial guess
- Hamiltonian parameterized:
Initial guess
[1] R. Hoffmann, J Chem. Phys (1963), 1397; [2] R. S. Mulliken, J. Chem. Phys. (1946) 497; [3] M. Wolfsberg and L. Helmholtz, J. Chem. Phys (1952), 837
Initial guess: e.g. density
Converged? Update density / density matrix
(mixer, preconditioner)
Update potential
nnn jj =1
Density Update
Naive Mixing
nnn jj =1
Density Update
Naive Mixing
solution
Example: H2, d=1.5Å, projected in 1 dimension
Density Update
Linear Mixing
Example: H2, d=1.5Å, projected in 1 dimension
Density Update
Linear Mixing
Example: H2, d=1.5Å, projected in 1 dimension
Density Update
Linear Mixing
Example: H2, d=1.5Å, projected in 1 dimension
Density Update
Linear Mixing
Guaranteed convergence
Density Update
Define residual:
Predict residual of next step from previous steps
Idea: obtain b minimizing norm ∫
Pitfall: Only use limited number of previous steps
Pulay Mixing [1] (Direct Inversion of Iterative Subspace)
[1] P. Pulay, Chem. Phys. Lett. 73 , 393 (1980).
Density Update
Learning from previous steps
Rules of thumb:
- a = 0.05 for metals
[1] P. Pulay, Chem. Phys. Lett. 73 , 393 (1980).
Density Update
Learning from previous steps
Rules of thumb:
- a = 0.05 for metals
Much faster than linear mixing
No guarantee for congerence
(Lack of) balance between long-range and short-range charge re-distribution
Typical for metals
Surfaces, Slabs, Thin Films
Concept:
Damp long-range oscillations:
Damping
no a priori known ideal choice for q0
Reasonable guess from Thomas-Fermi:
Density Update
Fermi-surface: Collection of k-points at which e = eFermi
At eFermi occupation changes from 1 to 0
Problem: Frequent „level switching“ during SCF
A special problem: Metallic systems
Density Update
Methfessel-Paxton [3]
[1] N. Mermin, Phys. Rev.137 , A1441 (1965).[2] C.-L. Fu, K.-H. Ho, Phys. Rev. B 28 , 5480 (1983). [3] M. Methfessel, A. Paxton, Phys. Rev. B 40 , 3616 (1989).
Gaussian [2]
Fermi [1]
Density Update
Physical implications:
Numerical Implications
• Back-extrapolation possible
Cu (111), 5 layer slab, Gaussian smearing, PBE calculation, 12x12x1 k- points
Pitfall: Adsorption of molecules
Initial guess needs initial thought, can change results qualiatively
Density update by – Linear mixing (slow)
– Pulay mixing
The tool for this workshop: FHI-aims (NAOs)
Our choice: Numeric atom-centered basis functions
Our choice: Numeric atom-centered basis functions
Constructing a basis-set library
Result: Hierarchical basis set library for all elements
Accuracy: (H2O)2 hydrogen bond energy
Basis-set superposition error (BSSE)
By construction, our NAO basis sets are close to completeness for very small BSSE for atomic dimers
Basis-set superposition error (BSSE)
Overlapping atom-centered grids: “Partitioning of unity”
Integration in practice: Large systems - small errors!
Hartree potential (electrostatics): Overlapping multipoles
Electrostatics: convergence with
– Configuration isomery
– Conformation isomery
– Polymorphism
50
• System in equilibrium is given by ensemble average over all minima
• Often dominated by global minimum (but watch out for tautomers)
Global structure search
o Basin Hopping
o Minima Hopping
• Requires a lot of calculations
• Typically only for dynamics
Different approaches possible: – Mapping of the whole potential energy surface
– Gradient free method: e.g., simplex
– Gradient-based methods

(vanish when basis set approaches completeness)
Total energy gradient
– Relativistic corrections
– (Integration grids)
All straightforward but lengthy

Step-length α variable
(optimal step length)
– Conjugated gradient [1]
[1] M. Hestenes, E. Stiefel (1952). "Methods of Conjugate Gradients for Solving Linear Systems"
Geometry update – (Quasi)Newton Methods
Find minimum:
... Hessian
[1] J. Nocedal and S. J. Wright, “Numerical optimization” (Springer, 2006)
Δ ≈ +
Find minimum: – Newton: calculate exact H
– Quasi-Newton: approximate H
• Update as search progresses [1]
[1] J. Nocedal and S. J. Wright, “Numerical optimization” (Springer, 2006)
... Hessian
Δ ≈ +
Chemically motivated choice (e.g.: Lindh [1])
[1] R. Lindh et al. , Chem. Phys. Lett. 241 , 423 (1995).
streching
bending
torsion
Update as search progresses [2]
[2] J. Nocedal and S. J. Wright, “Numerical optimization” (Springer, 2006)
Geometry update – (Quasi)Newton Methods
May even influence result
(combination with convergence thresholds)
Courtesy of Elisabeth Wruss
Lower in energy
Soft degrees of freedom can cause large ΔR
Step control needed: – Line search method: If new point is worse than old, interpolate
– Trust radius method
Conclusions
(Global optimization: PES feature-rich, methods to find global minima exist)
Local geometry optimization: Follow gradient – Hellman-Feynman from moving potentials
– Pulay from moving basis functions
– + additional terms
Vibrations in the harmonic approximation
Why calculating vibrations?
– If saddle-point: Provides search direction
– Thermodynamic data
• Zero-point energy
• Partition sum
• Finite-temperature effects
• Raman intensities: derivative of polarizabilty
How good is the harmonic approximatino?
Morse-potential:
Morse:
Morse-potential:
Morse:
Maximum displacement:
Vibrations
Hessian H
Calculating Vibrations
– Leads to generalized eigenvalue problem:
~ /

Hessian from geometry optimization is not sufficent – Analytic second derivative using perturbation theory [1]
– Numerical differentiation
• Often single displacement is not sensible to sample all vibrations
[1] S. Baroni et al. , Rev. Mod. Phys. 73 , 515 (2001).
Conclusions
Yield information about stability of geometry
Required for temperature effects
Relativity
Fixing ZORA
Computational scaling: two subproblems
ELPA: Performance
Resolution of identity (RI)
Periodic GaAs, HSE06 hybrid functional
Hybrid functionals -- CPU scaling
Summary
Volker Blum (Duke Univ.)
Igor Ying Zhang (FHI)
Karsten Reuter (TU Munich)