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DFT in practice
Sergey V. Levchenko
Fritz-Haber-Institutder MPG, Berlin, DE
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Outline
• From fundamental theory to practical solutions
• General concepts:
- basis sets - integrals and grids, electrostatics, molecules and clusters versus periodic systems- scalar relativity- solution of the eigenvalue problem
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Equations to be solved
Hohenberg-Kohn theorems
� � → �� → Ψ ��,��,…,��; {�} → �[�],…Electron density
� � = ��Ψ∗ �,��,…,��; {�} Ψ �,��,…,��; {�} ���� …����
�
�� �� � − � �� � ��� − � = 0 → � � ,�,�…
Variational principle:
� � −?
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� = Ψ �� Ψ = −�
�∫Ψ∗ ��,��,… ∑
��
����
���� Ψ ��,��,… � � −
−∫Ψ∗ ��,��,… ∑��
������,� Ψ ��,��,… � � +
+∫Ψ∗ ��,��,… ∑�
������,� Ψ ��,��,… � � =
= −�
�∫
���� �,��
���������
��� − ∫� � ∑��
����� ��� + ∫
� �,��
����������′
Equations to be solved
with one- and two-particle density matrices:
�� �,�� = �Ψ∗ �,��,… Ψ �′,��,… ���� …����
� �,�� = �Ψ∗ �,�′,�� … Ψ �,�′,��,… ���� …����
Problem: kinetic energy and electron-electron interaction energy as a functional of �(�) -- ?
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Equations to be solved
Kohn-Sham approach:
�������� + ���� = −�
�∑ �� ∫��
∗ � ���� � ���� +
∫� � �(��)
����������� + ���[�]
where functions �� represent the density: � � = ∑ �� �� � ��
�
�� �� � − � ∫� � ��� − � = 0 →
�
���∗ �
� � − ∑ �� �� �� − 1� = 0 →
(Generalized) Kohn-Sham equations:
−��
�+ ���� � + ∫
� ��
�������� �� � + ∫���,�[{�}] �,�� �� �� ���� =
= ���� �
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Solving Kohn-Sham equations
−��
�+ ���� � + ∫
� ��
�������� �� � + ∫���,�[{�}] �,�� �� �� ���� =
= ���� �
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Initial guess: e.g. density
Calculate potential
Construct Hamiltonian
Obtain new eigenvalues, eigenvectors, and density
Converged?Update density / density matrix
(mixer, preconditioner)
Update potential
Yes
No
n0
V0
n j
nj+1= nj+f(Δn)
V j+1
Self-consistent field method
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When is the calculation finished?• Answer: When „property“ does not change
Convergence
Convergence threshold Very strictVery loose
Typical „properties“ to converge:
Change of the density n
Change of the sum of eigenvalues
Change of the total energy
Options differ between codes
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Initial guess: e.g. density
Calculate potential
Construct Hamiltonian
Obtain new eigenvalues, eigenvectors, and density
Converged?Update density / density matrix
(mixer, preconditioner)
Update potential
Yes
No
n0
V0
n j
nj+1= nj+f(Δn)
V j+1
Self-consistent field method (SCF)
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Importance of the initial guessElectronic structure exhibits multiple minima
• Qualitatively different behavior depending on guess
• Common issue for magnetic solids
1O2
3O2
Initial guess
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Importance of the initial guessElectronic structure exhibits multiple minima
• Qualitatively different behavior depending on guess
• Common issue for magnetic solids
1O2
3O2
How do we find „the right“ initial guess?
Initial guess
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Random initialization• Traditional approach (for bandstructure
codes)
Initial guess
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Superposition of spheric atomicdensities
• Most common (for bandstructure codes)
Initial guess
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Extended Hückel Model [1]
Semi-empirical method:
- Linear combination of atomic orbitals
- Hamiltonian parameterized:
- Density from orbitals:
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
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Initial guess: e.g. density
Calculate potential
Construct Hamiltonian
Obtain new eigenvalues, eigenvectors, and density
Converged?Update density / density matrix
(mixer, preconditioner)
Update potential
Yes
No
n0
V0
n j
nj+1= nj+f(Δn)
V j+1
Self-consistent field method (SCF)
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Calculate new density from Kohn-Sham orbitals
Replace old density by new density
nnn jj =1
Example: H2, d=1.5Å, projected in 1 dimension
Density Update
Naive Mixing
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Calculate new density from Kohn-Sham orbitals
Replace old density by new density
nnn jj =1
Example: H2, d=1.5Å, projected in 1 dimension
Density Update
Naive Mixing
Typically does not work
- Best case: Osciallating, non-converging results- Worst case: Bistable, apparently converged
solution
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Calculate new density from Kohn-Sham orbitals
Replace old density by new density
Example: H2, d=1.5Å, projected in 1 dimension
Density Update
Linear Mixing
nnn jj = a1
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Calculate new density from Kohn-Sham orbitals
Replace old density by new density
Example: H2, d=1.5Å, projected in 1 dimension
Density Update
Linear Mixing
nnn jj = a1
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Calculate new density from Kohn-Sham orbitals
Replace old density by new density
Example: H2, d=1.5Å, projected in 1 dimension
Density Update
Linear Mixing
nnn jj = a1
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Calculate new density from Kohn-Sham orbitals
Replace old density by new density
Example: H2, d=1.5Å, projected in 1 dimension
Density Update
Linear Mixing
nnn jj = a1
system dependent
No clear recipe to choose ideal α
Guaranteed convergence
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[1] P. Pulay, Chem. Phys. Lett. 73 , 393 (1980).
Density Update
Define residual:
Change between input and output density
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)
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[1] P. Pulay, Chem. Phys. Lett. 73 , 393 (1980).
Density Update
Construct „optimal density“ minimizing residual norm, form new density
Learning from previous steps
Ideal case: convergence independent of a
Rules of thumb:
- a = 0.2 .. 0.6 for insulators
- a = 0.05 for metals
Pulay Mixing [1](Direct Inversion of Iterative Subspace)
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[1] P. Pulay, Chem. Phys. Lett. 73 , 393 (1980).
Density Update
Construct „optimal density“ minimizing residual norm, form new density
Learning from previous steps
Ideal case: convergence independent of a
Rules of thumb:
- a = 0.2 .. 0.6 for insulators
- a = 0.05 for metals
Pulay Mixing [1](Direct Inversion of Iterative Subspace)
Much faster than linear mixing
No guarantee for congerence
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Kerker (charge) preconditioning
Problem: Charge Sloshing
Charge „bounces“ back and forth
(Lack of) balance between long-range and short-range charge re-distribution
Typical for metals
Often encountered in „defective“ systems Localized defects (vacancies, interstitials)
Surfaces, Slabs, Thin Films
Density Update
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Charge preconditioning
Solution: Make α depend on ��(�)
Concept:
Treat density change in Fourier space:
Damp long-range oscillations:
Density Update
���� � = �� � + �Δ� � → ����� � = ��� � + �Δ�� �
�� � =1
�� � ��������
q0: Screening parameter
q >> q0: G ≈ a
Normal mixing
q << q0: G ≈ aq2/q02
Damping
� = ���
�� + ���
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Charge preconditioning
Density Update
8 layer Al slab, PBE calculation
no a priori known ideal choice for q0
Reasonable guess fromThomas-Fermi:
q0: Screening parameter� = ���
�� + ���
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A special problem: Metallic systems
Density Update
http://www.phys.ufl.edu/fermisurface/periodic_table.html
Metallic properties determined by Fermi surface
Fermi-surface: Collection of k-points at which e = eFermi
At eFermi occupation changes from 1 to 0
Problem: Frequent „level switching“ during SCF
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A special problem: Metallic systems
Density Update
Solution: Replace step function by a more smooth function
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]
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A special problem: Metallic systems
Density Update
Physical implications:
• States become fractionally occupied
• In DFT, well defined as ensemble average
• Broadening is not electron temperature (except: Fermi)
Numerical Implications
• Level switching is damped
• Much faster convergence
• Total energy now dependson σ, no longer variational
• Back-extrapolation possible
Cu (111), 5 layer slab, Gaussiansmearing, PBE calculation, 12x12x1 k-points
Pitfall: Adsorption of molecules
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Summary SCF
Ground state electron density determined iteratively
Initial guess needs initial thought, can change results qualiatively
Density update by– Linear mixing (slow)
– Pulay mixing
Convergence acceleration by – Preconditioner
– Broadening of states
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Representing KS states: Basis sets
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The tool for this workshop: FHI-aims (NAOs)
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Our choice: Numeric atom-centered basis functions
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Our choice: Numeric atom-centered basis functions
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Constructing a basis-set library
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Iterative selection of NAO basis functions
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Result: Hierarchical basis set library for all elements
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Accuracy: (H2O)2 hydrogen bond energy
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Basis-set superposition error (BSSE)
By construction, our NAO basis sets are close to completeness for very small BSSE for atomic dimers
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Basis-set superposition error (BSSE)
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Using numeric atom-centered basis functions
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Numeric atom-centered basis functions: Integration
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Overlapping atom-centered grids: “Partitioning of unity”
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Integration in practice: Large systems - small errors!
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Hartree potential (electrostatics): Overlapping multipoles
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Electrostatics: convergence with ����
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Periodic systems
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Structure optimization
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Global structure search
Born-Oppenheimer energy surface can contain several minima– Constitution isomery
– 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)
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Global structure search
Methods to find the global minimum:
o Basin Hopping
o Molecular dynamics: o Simulated annealing
o Minima Hopping
o Metadynamics
o Cluster expansion
o Genetic algorithm
o Diffusion methods
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Local structure optimization
Once we have a reasonable guess, find closest minimum
Different approaches possible:– Mapping of the potential energy surface
• Requires a lot of calculations
• Typically only for dynamics
52
dCC
aCCO
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Local structure optimization
Once we have a reasonable guess, find closest minimum
Different approaches possible:– Mapping of the whole potential energy surface
– Gradient free method: e.g., simplex
– Gradient-based methods
• Calculate gradient (a.k.a. „forces“)
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Total energy gradient
Search for minimum by following the gradient
��
��= �
���
��� +
���
���� � + � ��
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= �� ∑�� �����
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���� � ���
�
��=
��
��
�
��+
��
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��• First term vanishes
• Second term survives foratom-centered basis functions Pulay forces
(vanish when basis set approaches completeness)
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Total energy gradient
Search for minimum by following the gradient
Additional contributions from atom-centeredapproximations– Multipole expansion
– Relativistic corrections
– (Integration grids)
All straightforward but lengthy
Once we have the Force, how do we find the minimum?
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Geometry update – Steepest descent
Follow negative gradient to find minimum
Step-length α variable
Guaranteed but slow convergence
Oscillates near minimum
Not suitable for saddle points
Improved versions exists– Line minimization
(optimal step length)
– Conjugated gradient [1]
[1] M. Hestenes, E. Stiefel (1952). "Methods of Conjugate Gradients for Solving Linear Systems"
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Geometry update – (Quasi)Newton Methods
Approximate PES by quadratic function
Find minimum:
Newton: calculate exact HExpensive! Cheaper method needed
... Hessian
[1] J. Nocedal and S. J. Wright, “Numerical optimization” (Springer, 2006)
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−� ��
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Geometry update – (Quasi)Newton Methods
Approximate PES by quadratic function
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
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−� ��
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Geometry update – (Quasi)Newton Methods
Guess initial Hessian
Simple choice: Scaled unit matrix
Chemically motivated choice (e.g.: Lindh [1])
[1] R. Lindh et al. , Chem. Phys. Lett. 241 , 423 (1995).
streching
bending
torsion
k parameterized [1]
Update as search progresses [2]
[2] J. Nocedal and S. J. Wright, “Numerical optimization” (Springer, 2006)
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Geometry update – (Quasi)Newton Methods
Initial Hessian is critical for performance
May even influence result
(combination with convergence thresholds)
Courtesy of Elisabeth Wruss
Diagonal Hessian Model Hessian (Fischer)
Lower in energy
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Challenges of Quasi-Newton Methods
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
• Enforce upper limit for ΔR
• Evaluate quality of quadratic model
• Adjust ΔRmax based on q
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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
Quasi-Newton method de-facto standard– Requires approximation and update of Hessian
– Step control by line search or trust radius method
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Vibrations in the harmonic approximation
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Why calculating vibrations?
Vibrations give important information about the system:– Classification of stationary points (minimum / saddle point)
– If saddle-point: Provides search direction
– Thermodynamic data
• Zero-point energy
• Partition sum
• Finite-temperature effects
– Connection to experiment:
• Infrared intensities: derivative of dipole moment
• Raman intensities: derivative of polarizabilty
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How good is the harmonic approximatino?
Morse-potential:
Morse:
Harmonic oscillator:
Harmonic osc.
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How good is the harmonic approximation?
Morse-potential:
Morse:
Harmonic oscillator:
Harmonic osc.
Morse:
Harmonic osc.
Reasonable modelfor small displacements(~10% of bond length)
Maximum displacement:
About 0.1Å at roomtemperature
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Vibrations
Expand Energy in Taylor series:
Hessian H
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Interpretation of results:
• Single negative ω: Transition state
Calculating Vibrations
Solve Newton‘s equations of motion:– Exponential ansatz:
– Leads to generalized eigenvalue problem:
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From vibrations to thermodynamics
Partition function
Free energy for finite temperature
Hessian from geometry optimization is not sufficent– Analytic second derivative using perturbation theory [1]
– Numerical differentiation
• Computationally very expensive
• Contains parameter (displacement), needs to be checked carefully
• Often single displacement is not sensible to sample all vibrations
[1] S. Baroni et al. , Rev. Mod. Phys. 73 , 515 (2001).
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Conclusions
Often calculated in harmonic approximation
Yield information about stability of geometry
Required for temperature effects
Anharmonic effects via molecular dynamics
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Relativity
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Simple approximation to scalar relativity
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Simple approximation to scalar relativity
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Fixing ZORA
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Atomic ZORA and scaled ZORA in practice
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Computational scaling: two subproblems
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CPU time versus system size
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CPU time versus system size
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Parallel eigenvalue solvers - the problem
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Parallel eigenvalue solvers - the problem
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A massively-parallel dense eigensolver: ELPA
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The key improvement: two-step trigonalization
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The key improvement: Two-step trigonalization
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ELPA: Performance
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ELPA: Performance (Cray XC30, 2013)
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Generalized Kohn-Sham DFT: Exact exchange
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Resolution of identity (RI)
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Hybrid functionals -- scaling with system size
Periodic GaAs, HSE06 hybrid functional
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Hybrid functionals -- CPU scaling
Periodic GaAs, HSE06 hybrid functional
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Summary
![Page 91: DFT in practice - Theory Department - Fritz Haber Instituteth.fhi-berlin.mpg.de/sitesub/meetings/dft-workshop-201… · · 2016-05-02DFT in practice Sergey V. Levchenko Fritz-Haber-Institut](https://reader031.vdocument.in/reader031/viewer/2022022011/5b0d82ef7f8b9a02508dcf35/html5/thumbnails/91.jpg)
Our approach to all-electron simulations: FHI-aims
Volker Blum(Duke Univ.)
Igor Ying Zhang (FHI)
Karsten Reuter (TU Munich)
Patrick Rinke (Aalto Univ., Finland)
Ville Havu (Aalto Univ., Finland)