Pseudopotentials and Basis Sets. How to generate and test them
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1 Pseudopotentials and Basis Sets How to generate and test them
2 Pseudopotential idea Atomic Si Core electrons highly localized very depth energy are chemically inert 1s 2 2s 2 2p 6 3s 2 3p 2 Valence wave functions must be core valence orthogonal to the core wave functions
3 Cut-off radii Balance between softness and transferability controlled by R c R c SOFTNESS TRANSFERABILITY Larger R c : softer pseudo Si Shorter R c : harder pseudo
4 The atom program pseudopotential generation label xc flavor # valence shells # core orbitals n l Valence configuration Cutoff radii
5 Run the shell script (pg.sh) Procedure (I) Check contents of new directory (Si.tm2) Plot the pseudo-potentials/orbitals
6 Pseudo-wave function Procedure (II) Logarithmic derivative Radial Fourier-T Pseudopotential
7 Procedure (III)
8 Radial charge distribution: Compare all-electron with pseudo-charge
9 Pseudopotential testing (I) The all-electron (ae.sh) and pseudo-test (pt.sh) scripts: run script input pseudopotential to test
10 Pseudopotential testing (II) A transferable pseudo will reproduce the AE energy levels and wave functions in arbitrary environments
11 SUMMER SCHOOL ON COMPUTATIONAL MATERIALS SCIENCE Pseudopotential testing (III) Compute the energy of two different configurations Compute the difference in energy For the pseudopotential to be transferible: 3s2 3p2 (reference) 3s2 3p1 3d1 3s1 3p3 3s1 3p2 3d1 3s0 3p3 3d1
12 Large core-valence overlap Errors due to non-linearity of XC-potential
13 ~ V Non-linear core corrections Standard pseudopotential unscreening: Valence charge only V ps However xc V xc = V xc = V [ ρ ] + v ps scr [ ρ ]( r) V [ ρ ]( r) V [ ρ ]( r) v { V [ ρ + ρ ] V [ ρ ]} xc v c H [ ρ + ρ ]( r) V [ ρ ]( r) + V [ ρ ]( r) v c xc Core-correction v v xc v xc xc v c Keep core charge in pseudopotential generation V ps ps ( r) = V [ ρ + ρ ]( r) V [ ρ ]( r) V [ ρ + ρ ]( r) scr v c H v xc v c
14 New flag PCC input file
15 Pseudo-core & pseudo-valence charge
16 Smooth Fourier Transform The real-space grid required fineness depends on how you define the pseudopotential. The meshcutoff parameter can be determined from the Fourier Transform.
17 Basis generation
18 Key for f linear-scaling scaling: : LOCALITY Large system x2 Divide and Conquer W. Yang, Phys. Rev. Lett. 66, 1438 (1992)
19 Very efficient Atomic Orbitals Lack of systematic for convergence Main features: φ Ilm ( r )=R Il (r I )Y lm ( ˆ r I ) r = r R r I I 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 Size or number of functions Range of localization of these functions Shape or functional form used.
20 Basis Size (I) Depends on the required accuracy and available computational power Quick and dirty calculations Minimal basis set (single-ζ; SZ) Highly converged calculations Complete multiple-ζ + Polarization + Diffuse orbitals
21 Basis Size (II): improving One single radial function per angular momentum shell occupied in the free-atom. Single-ζ (minimal or SZ) Improving the quality? 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
22 Examples
23 Basis Size (III): Polarization Perturbative polarization Apply a small E field to the orbital we want to polarize E Atomic polarization Solve Schrödinger equation for higher angular momentum unbound in the free atom require short cut offs s s+p Si 3d orbitals E. Artacho et al, Phys. Stat. Sol. (b), 215, 809 (1999)
24 Basis Size (IV): Convergence Cohesion curves Bulk Si PW and NAO convergence J. Junquera et. al. PRB 64, (2001).
25 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 ) Accuracy and computational efficiency depend on the range of the atomic orbitals. Way to define all the cutoff radii in a balanced way.
26 Range (II): Energy Shift Easy approach to define the cutoff radii for the NAOs: 1 2r d dr 2 2 r l( l + 1) + 2 2r + V l ( r) φ l ( r) = ( ε l + δε l ) φl ( r) E. Artacho et al. Phys. Stat. Solidi (b) 215, 809 (1999) A single parameter for all cutoff radii Fireballs O. F. Sankey & D. J. Niklewski, Phys. Rev. B 40, 3979 (1989) BUT, a different cutoff radius for each orbital
27 Range (II): Convergence Bulk Si equal s, p orbitals radii J. Soler et al, J. Phys: Condens. Matter, 14, 2745 (2002)
28 Shape (I) The radial function shape is mainly determined by the pseudopotential. Extra parameters can be introduced to add flexibility: δq : extra charge per atomic specie. Confinement : imposed separately for each angular momentum shell.
29 Shape (II) Soft confinement (J. Junquera et al, Phys. Rev. B 64, (01) ) Shape of the optimal 3s orbital of Mg in MgO for different schemes Corresponding optimal confinement potential Better variational basis sets Removes the discontinuity of the derivative
30 PAO.Basis (I) # shells Add polarization 4s 3d 1 st ζ %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 Rc n=3 2 2 # n, l, Nzeta Rc nd ζ %endblock PAO.Basis
31 PAO.Basis (II): new generation charge %block PAO.Basis # Define Basis set Cu n=4 0 1 E V c r c n=4 1 1 E Polarization orbital n=3 2 1 E %endblock PAO.Basis
32 Procedure 1. Check the difference in energies involved in your problem 2. For semiquantitative results and general trends use SZ 3. 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». 4. Functional optimization of the basis
33 Pseudos & Basis repository 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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