1. 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 Berlin-Dahlem, Germany

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

3. 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

4. relaxation run or molecular dynamics
initialize charge density
initialize wavefunctions
construct new charge density
for all k determine wavefunctions spanning the occupied space
determine occupancies of states no
energy converged ?
yes
static run
STOP
yes
relaxation run or molecular dynamics no
forces converged ?
yes
forces small ?
move ions
STOP no
yes

5. 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

6. 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-atom correctly describe the bonding in different environments ? → transferability

7. 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

8. Implementations, basis set sizes
(examples)
bulk compound
surface,
oligo-peptide 1
WIEN2K
~200
~20,000 2
TB-LMTO
~50
~1000 3
CP-PAW, VASP, abinit
5x103…5x105 4
Gaussian,
Quickstep, …
50-500
103…104 5
Dmol3 6
CPMD, abinit, sfhingx, FHImd
104…106

9. 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 or H → H’’ = (L†L)-1(H-E1) C:= L†L ~ H-E1
diagonal pre-conditioner (Teter et al.)

10. 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

11. 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}

12. 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

13. 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

14. 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 )

15. Charge self-consistency
Two possible strategies:
separate loop in the hierarchy (WIEN2K, VASP, ..)
combined with iterative diagonalization loop (CASTEP, FHImd, …)
‘charge sloshing’
lines of fixed r

16. Two algorithms for self-consistency
construct new charge density
and potential
|| r(i) –r(i-1) ||=h ?
(H-e1)Y<d ?
iterative diagonalization step
of H for fixed r
construct new charge density
and potential
|| r(i) –r(i-1) ||=h ?
|| Y(i) –Y(i-1) ||<d ?
{Y(i-1)}→ {Y(i)} No No
Yes
Yes No
STOP No
STOP

17. Achieving charge self-consistency
Residuum:
Pratt (single-step) mixing:
Multi-step mixing schemes:
Broyden mixing schemes: iterative update of Jacobian J idea: find approximation to c during runtime WIEN2K: mixer
Pulay’s residuum minimization

18. 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 !

19. Hellmann-Feynman theorem
Pulay forces vanish if the calculation has reached self-consistency and if basis set orthonormality persists independent of the atomic positions 1st + 3rd term =
DFIBS=0 holds for pure plane-wave basis sets (methods 3,6), not for 1,2,3,5.

20. Forces in LAPW

21. Combining DFT with Molecular Dynamics
Born-Oppenheimer MD
Car-Parrinello MD
construct new charge density
and potential
|| r(i) –r(i-1) ||=0 ?
|| Y(i) –Y(i-1) ||=0 ?
{Y(i-1)}→ {Y(i)}
move ions
Forces converged?
construct new charge density
and potential
|| r(i) –r(i-1) ||=0 ?
|| Y(i) –Y(i-1) ||=0 ?
{Y(i-1)}→ {Y(i)}
move ions
Forces converged?

22. 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)

23. Schemes for damped wavefunction dynamics
Second-order with damping numerical 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)

24. Comparison of Algorithms (pure plane-waves)
bulk semi-metal (MnAs), SFHIngx code

25. 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.

26. Literature
G.K.H. Madsen et al., Phys. Rev. B 64, (2001) [WIEN2K].
W. E. Pickett, Comput. Phys. Rep. 9, 117(1989) [pseudopotential approach].
G. Kresse and J. Furthmüller, Phys. Rev. B 54, (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].