Algorithms for total energy and forces in condensed matter dft codes
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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

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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

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

Outline

  • 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


Algorithms for total energy and forces in condensed matter dft codes

initialize charge density

initialize wavefunctions

construct new charge density

for all k determine wavefunctions spanning the occupied space

determine occupancies of states

no

yes

static run

energy converged ?

STOP

yes

relaxation run or molecular dynamics

no

forces converged ?

yes

forces small ?

move ions

STOP

yes

no


Dft methods for condensed matter systems

DFT methods for Condensed-Matter Systems

  • 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

  • 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

  • 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

LCAOs

LCAOs

LCAOs

LCAOs

PWs


Implementations basis set sizes

Implementations, basis set sizes


Eigenvalue problem pre conditioning

Eigenvalue problem: pre-conditioning

  • 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

  • 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

  • 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

  • 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

  • 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 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

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

No

No

Yes

Yes

No

STOP

No

STOP


Achieving charge self consistency

Achieving charge self-consistency

  • 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

  • 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

  • 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.


Forces in lapw

Forces in LAPW


Combining dft with molecular dynamics

Born-Oppenheimer MD

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

  • 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

  • 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)

bulk semi-metal (MnAs), SFHIngx code


Summary

Summary

  • 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

Literature

  • 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].


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