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Pseudopotentials and Basis Sets SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 How to generate and test them

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1s 2 2s 2 2p 6 3s 2 3p 2 core valence Valence wave functions must be orthogonal to the core wave functions Atomic SiCore electrons… highly localized very depth energy … are chemically inert SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Pseudopotential idea

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Si SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Cut-off radii TRANSFERABILITY SOFTNESS RcRc Shorter R c : harder pseudo Larger R c : softer pseudo Balance between softness and transferability controlled by R c

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The “atom” program SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 “pseudopotential generation” label # core orbitals # valence shells xc flavor Valence configuration Cutoff radii n l

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Procedure (I) SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Run the shell script (pg.sh) Check contents of new directory (Si.tm2) Plot the pseudo-potentials/orbitals

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Procedure (II) SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Pseudo-wave function Pseudopotential Logarithmic derivative Radial Fourier-T

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Procedure (III) SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005

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Radial charge distribution: SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Compare all-electron with pseudo-charge

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Pseudopotential testing (I) SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 The all-electron (ae.sh) and pseudo-test (pt.sh) scripts: run scriptinputpseudopotential to test

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SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Pseudopotential testing (II) A transferable pseudo will reproduce the AE energy levels and wave functions in arbitrary environments

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Compute the energy of two different configurations Compute the difference in energy For the pseudopotential to be transferible: SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Pseudopotential testing (III) → 3s 2 3p 2 (reference) → 3s 2 3p 1 3d 1 → 3s 1 3p 3 → 3s 1 3p 2 3d 1 → 3s 0 3p 3 3d 1

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Large core-valence overlap SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Errors due to non-linearity of XC-potential

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Non-linear core corrections SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Standard pseudopotential unscreening: Valence charge only However… Core-correction Keep core charge in pseudopotential generation

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PCC input file SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 New flag

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Pseudo-core & pseudo-valence charge SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005

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Smooth Fourier Transform SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 The real-space grid required fineness depends on how you define the pseudopotential. The meshcutoff parameter can be determined from the Fourier Transform.

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Basis generation SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005

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Key for linear-scaling: LOCALITY W. Yang, Phys. Rev. Lett. 66, 1438 (1992) “Divide and Conquer” x2 Large system SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005

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Atomic Orbitals Very efficient Lack of systematic for convergence Main features: Numerical Atomic Orbitals (NAOs): Numerical solution of the KS Hamiltonian for the isolated pseudoatom with the same approximations (xc, pseudos) as for the condensed system SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 – Size or number of functions – Range of localization of these functions – Shape or functional form used.

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SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Basis Size (I) Highly converged calculations Complete multiple- Polarization + Diffuse orbitals Quick and dirty calculations Minimal basis set (single- ; SZ) Depends on the required accuracy and available computational power

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Improving the quality? SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Basis Size (II): improving One single radial function per angular momentum shell occupied in the free-atom. Single- (minimal or SZ) Radial flexibilization: Add more than one radial function within the same angular momentum shell Multiple-ζ Angular flexibilization: Add shells of different angular momentum Polarization

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SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Examples

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E. Artacho et al, Phys. Stat. Sol. (b), 215, 809 (1999) Atomic polarization Perturbative polarization Apply a small E field to the orbital we want to polarize E s s+p unbound in the free atom require short cut offs Si 3d orbitals SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Basis Size (III): Polarization Solve Schrödinger equation for higher angular momentum

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Bulk Si Cohesion curves PW and NAO convergence SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Basis Size (IV): Convergence J. Junquera et. al. PRB 64, (2001).

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Range (I): How to get sparsity for O(n) Neglecting interactions below a tolerance or beyond some scope of neighbours numerical instablilities for high tolerances. Strictly localized atomic orbitals (zero beyond a given cutoff radius, r c ) SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Accuracy and computational efficiency depend on the range of the atomic orbitals. Way to define all the cutoff radii in a balanced way.

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Fireballs O. F. Sankey & D. J. Niklewski, Phys. Rev. B 40, 3979 (1989) E. Artacho et al. Phys. Stat. Solidi (b) 215, 809 (1999) SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Range (II): Energy Shift Easy approach to define the cutoff radii for the NAOs: A single parameter for all cutoff radii… …BUT, a different cutoff radius for each orbital

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J. Soler et al, J. Phys: Condens. Matter, 14, 2745 (2002) equal s, p orbitals radii SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Range (II): Convergence Bulk Si

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Shape (I) SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Q : extra charge per atomic specie. Confinement : imposed separately for each angular momentum shell. The radial function shape is mainly determined by the pseudopotential. Extra parameters can be introduced to add flexibility:

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Shape (II) Shape of the optimal 3s orbital of Mg in MgO for different schemes Corresponding optimal confinement potential Soft confinement (J. Junquera et al, Phys. Rev. B 64, (01) ) Better variational basis sets Removes the discontinuity of the derivative SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005

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PAO.Basis (I) %block PAO.Basis # Define Basis set Cu 2 # Species label, number of l-shells n=4 0 2 P 1 # n, l, Nzeta, Polarization, NzetaPol n=3 2 2 # n, l, Nzeta %endblock PAO.Basis SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, s 3d Rc 1 st 2 nd # shells Add polarization

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PAO.Basis (II): new generation %block PAO.Basis # Define Basis set Cu n=4 0 1 E n=4 1 1 E n=3 2 1 E %endblock PAO.Basis SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 charge rcrc VcVc Polarization orbital

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SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005Procedure Check the difference in energies involved in your problem For semiquantitative results and general trends use SZ Improve the basis: Automatic DZP (Split Valence & Perturbative Polarization): High quality for most systems Good valence between well converged results & computational cost ‘Standard’ Rule of thumb in Quantum Chemistry: « a basis should always be doubled before being polarized » Functional optimization of the basis

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Pseudos & Basis repository SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE University of Illinois at Urbana-Champaign, June 13-23, 2005 Pseudopotentials and basis sets available in the SIESTA web page: Uploaded by users input files to generate them Plots of the radial functions Documentation of the tests done Author’s contact information The PAO is pseudopotential-dependent. Check also in the user’s mailing list.

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