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 [ –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)

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

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

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 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

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 (examples) bulk compound surface, oligo-peptide 1 WIEN2K ~200 ~20,000 2 TB-LMTO ~50 ~1000 3 CP-PAW, VASP, abinit 100..200 5x103…5x105 4 Gaussian, Quickstep, … 50-500 103…104 5 Dmol3 6 CPMD, abinit, sfhingx, FHImd 100..500 104…106

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

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

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

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

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 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.

Forces in LAPW

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?

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 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)

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

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 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].