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Algorithms for Total Energy and Forces in Condensed-Matter DFT codes. IPAM workshop “Density-Functional Theory Calculations for Modeling Materials and Bio-Molecular Properties and Functions” Oct. 31 – Nov. 5, 2005 P. Kratzer Fritz-Haber-Institut der MPG D-14195 Berlin-Dahlem, Germany.

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Algorithms for total energy and forces in condensed matter dft codes

Algorithms for Total Energy and Forces in Condensed-Matter DFT codes

IPAM workshop “Density-Functional Theory Calculations for Modeling Materials and Bio-Molecular Properties and Functions”

Oct. 31 – Nov. 5, 2005

P. Kratzer

Fritz-Haber-Institut der MPG

D-14195 Berlin-Dahlem, Germany

Dft basics
DFT basics DFT codes

Kohn & Hohenberg (1965)

For ground state properties, knowledge of the electronic density r(r) is sufficient. For any given external potential v0(r), the ground state energy

is the stationary point of a uniquely defined functional

Kohn & Sham (1966)

[ –2/2m + v0(r) + Veff[r] (r) ] Yj,k(r) = ej,kYj,k(r)

r(r) = j,k | Yj,k( r) |2

in daily practice:

Veff[r] (r) Veff(r(r)) (LDA)

Veff[r] (r) Veff(r(r), r(r) ) (GGA)

Outline DFT codes

  • flow chart of a typical DFT code

  • basis sets used to solve the Kohn-Sham equations

  • algorithms for calculating the KS wavefunctions and KS band energies

  • algorithms for charge self-consistency

  • algorithms for forces, structural optimization and molecular dynamics

initialize charge density DFT codes

initialize wavefunctions

construct new charge density

for all k determine wavefunctions spanning the occupied space

determine occupancies of states



static run

energy converged ?



relaxation run or molecular dynamics


forces converged ?


forces small ?

move ions




Dft methods for condensed matter systems
DFT methods for Condensed-Matter Systems DFT codes

  • All-electron methods

  • Pseudopotential / plane wave method: only valence electrons (which are involved in chemical bonding) are treated explicitly

1) ‘frozen core’ approximation

projector-augmented wave (PAW) method

2) fixed ‘pseudo-wavefunction’ approximation

Pseudopotentials and wavefunctions
Pseudopotentials and -wavefunctions DFT codes

  • idea: construct ‘pseudo-atom’ which has the valence states as its lowest electronic states

  • preserves scattering properties and total energy differences

  • removal of orbital nodes makes plane-wave expansion feasible

  • limitations: Can the pseudo-atomcorrectly describe the bonding in different environments ? → transferability

Basis sets used to represent wavefuntions
Basis sets used to represent wavefuntions DFT codes

  • All-electron: atomic orbitals + plane waves in interstitial region (matching condition)

  • All-electron: LMTO (atomic orbitals + spherical Bessel function tails, orthogonalized to neighboring atomic centers)

  • PAW: plane waves plus projectors on radial grid at atom centers (additive augmentation)

  • All-electron or pseudopotential: Gaussian orbitals

  • All-electron or pseudopotential: numerical atom-centered orbitals

  • pseudopotentials: plane waves






Eigenvalue problem pre conditioning
Eigenvalue problem: pre-conditioning DFT codes

  • spectral range of H: [Emin, Emax]

    in methods using plane-wave basis functions dominated by kinetic energy;

  • reducing the spectral range of H: pre-conditioning H → H’ = (L†)-1(H-E1)L-1 orH → H’’ = (L†L)-1(H-E1)C:= L†L ~ H-E1

  • diagonal pre-conditioner (Teter et al.)

Eigenvalue problem direct methods
Eigenvalue problem: ‘direct’ methods DFT codes

  • suitable for bulk systems or methods with atom-centered orbitals only

  • full diagonalization of the Hamiltonian matrix

  • Householder tri-diagonalization followed by

    • QL algorithm or

    • bracketing of selected eigenvalues by Sturmian sequence

      → all eigenvalues ej,k and eigenvectors Yj,k

  • practical up to a Hamiltonian matrix size of ~10,000 basis functions

Eigenvalue problem iterative methods
Eigenvalue problem: iterative methods DFT codes

  • Residual vector

  • Davidson / block Davidson methods(WIEN2k option runlapw -it)

    • iterative subspace (Krylov space)

    • e.g., spanned by joining the set of occupied states {Yj,k} with pre-conditioned sets of residues {C―1(H-E1) Yj,k}

    • lowest eigenvectors obtained by diagonalization in the subspace defines new set{Yj,k}

Eigenvalue problem variational approach
Eigenvalue problem: variational approach DFT codes

  • Diagonalization problem can be presented as a minimization problem for a quadratic form (the total energy) (1) (2)

  • typically applied in the context of very large basis sets (PP-PW)

  • molecules / insulators: only occupied subspace is required → Tr[H ] from eq. (1)

  • metals: → minimization of single residua required

Algorithms based on the variational principle for the total energy
Algorithms based on the variational principle for the total energy

  • Single-eigenvector methods: residuum minimization, e.g. by Pulay’s method

  • Methods propagating an eigenvector system {Ym}:(pre-conditioned) residuum is added to each Ym

    • Preserving the occupied subspace (= orthogonalization of residuum to all occupied states):

      • conjugate-gradient minimization

      • ‘line minimization’ of total energy

        Additional diagonalization / unitary rotation in the occupied subspace is needed ( for metals ) !

    • Not preserving the occupied subspace:

      • Williams-Soler algorithm

      • Damped Joannopoulos algorithm

Conjugate gradient method
Conjugate-Gradient Method energy

  • It’s not always best to follow straight the gradient→ modified (conjugate) gradient

  • one-dimensional mimi-mization of the total energy (parameter f j )

Charge self consistency

lines of fixed energy r

Charge self-consistency

Two possible strategies:

  • separate loop in the hierarchy (WIEN2K, VASP, ..)

  • combined with iterative diagonalization loop (CASTEP, FHImd, …)

‘charge sloshing’

Two algorithms for self consistency

construct new charge density energy

and potential

construct new charge density

and potential

iterative diagonalization step

of H for fixed r

{Y(i-1)}→ {Y(i)}

|| Y(i) –Y(i-1) ||<d ?

(H-e1)Y<d ?

|| r(i) –r(i-1) ||=h ?

|| r(i) –r(i-1) ||=h ?

Two algorithms for self-consistency









Achieving charge self consistency
Achieving charge self-consistency energy

  • Residuum:

  • Pratt (single-step) mixing:

  • Multi-step mixing schemes:

    • Broyden mixing schemes: iterative update of Jacobian Jidea: find approximation to c during runtimeWIEN2K: mixer

    • Pulay’s residuum minimization

Total energy derivatives
Total-Energy derivatives energy

  • first derivatives

    • Pressure

    • stress

    • forces

      Formulas for direct implementation available !

  • second derivatives

    • force constant matrix, phonons

      Extra computational and/or implementation work needed !

Hellmann feynman theorem
Hellmann-Feynman theorem energy

  • Pulay forces vanish if the calculation has reached self-consistency and if basis set orthonormality persists independent of the atomic positions1st + 3rd term =

  • DFIBS=0 holds for pure plane-wave basis sets (methods 3,6), not for 1,2,3,5.

Combining dft with molecular dynamics

Born-Oppenheimer MD energy

Car-Parrinello MD

move ions

move ions

construct new charge density

and potential

construct new charge density

and potential

{Y(i-1)}→ {Y(i)}

{Y(i-1)}→ {Y(i)}

|| Y(i) –Y(i-1) ||=0 ?

|| Y(i) –Y(i-1) ||=0 ?

|| r(i) –r(i-1) ||=0 ?

|| r(i) –r(i-1) ||=0 ?

Forces converged?

Forces converged?

Combining DFT with Molecular Dynamics

Car parrinello molecular dynamics
Car-Parrinello Molecular Dynamics energy

  • treat nuclear and atomic coordinates on the same footing: generalized Lagrangian

  • equations of motion for the wavefunctions and coordinates

  • conserved quantity

  • in practical application: coupling to thermostat(s)

Schemes for damped wavefunction dynamics
Schemes for damped wavefunction dynamics energy

  • Second-order with dampingnumerical solution: integrate diagonal part (in the occupied subspace) analytically, remainder by finite-time step integration scheme (damped Joannopoulos), orthogonalize after advancing all wavefunctions

  • Dynamics modified to first order (Williams-Soler)

Comparison of algorithms pure plane waves
Comparison of Algorithms (pure plane-waves) energy

bulk semi-metal (MnAs), SFHIngx code

Summary energy

  • Algorithms for eigensystem calculations: preferred choice depends on basis set size.

  • Eigenvalue problem is coupled to charge-consistency problem, hence algorithms inspired by physics considerations.

  • Forces (in general: first derivatives) are most easily calculated in a plane-wave basis; other basis sets require the calculations of Pulay corrections.

Literature energy

  • G.K.H. Madsen et al., Phys. Rev. B 64, 195134 (2001) [WIEN2K].

  • W. E. Pickett, Comput. Phys. Rep. 9, 117(1989) [pseudopotential approach].

  • G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996) [comparison of algorithms].

  • M. Payne et al., Rev. Mod. Phys. 64, 1045 (1992) [iterative minimization].

  • R. Yu, D. Singh, and H. Krakauer, Phys. Rev. B 43, 6411 (1991) [forces in LAPW].

  • D. Singh, Phys. Rev. B 40, 5428(1989) [Davidson in LAPW].