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Pseudopotentials in DFT and VMC

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Pseudopotentials in DFT and VMC Lubos Mitas North Carolina State [email protected] Urbana, July 2007Ahhh, pseudopotentials ...Difficult subject ... why ?- highly technical, difficult - often the most complicated parts of the codes- “only” an auxiliary concept, not really fundamental :-(but - it saves (b)millions of hours of computer time- enables to do calculations/predictions which otherwise are impossible- forces you to learn/understand electronic structure a lot deeperLub os_Mita [email protected] su.e d uOutline of this talk- Total energy as a function of Z (atomic number)- Core vs valence: energy and length scales- Idea of “pseudo-ion”: effective potential in the core + valence electrons - Helps in DFT and basis set methods: smaller basis, less states- Helps in QMC: smaller total energies, significant gain in efficiency- PP (ECP in quantum chem.) norm-conserving construction- Evaluation of PP terms in VMC/QMC- Existing tables, accuracy, errors to watch- Ideas on many-body construction of PP Lub os_Mita [email protected] su.e d uDensity Functional Theory : self-consitent loop Hartree-Fock: min self-consitent loopLub os_Mita [email protected] su.e d uEtot=∫Ftot[r]d rHFr1,r2,...=Det[{irj}]r=∑ioccupi2r[TkinVextVeff ,DFT]i=EiiReminder: one-particle electronic structure methods, DFT and HFEHF= HF∣H∣HF =[TkinVextVeff ,HF{j}]i=EiiVeff ,HF=VCoulExch[{i}]Eigenvalues/energies of the one-particle levels [ in a.u. ~ 27 eV ] Deep core qualitatively: n is the principal q. number Valence different, strongly modified by e-e interactions ~ a few eVs One-particle eigenvalues for carbon C (Z=6) and copper Cu (Z=29) 2p E_1s= - 0.5 Valence 4s E_4s = - 0.2 2s E_2s= - 0.7 3d E_3d = - 0.5 1s E_1s= - 11.0 3p E_3p = - 3.5 3s E_3s = - 5.0 Core 2p E_2p = -35. 2s E_2s = - 41. 1s E_1s = - 392. Note: different energy scales in core vs. valenceLub os_Mita [email protected] su.e d uAtom with nuclear charge Z:energies of one-aprticle statesEn=−Z2/2n2,Lub os_Mita [email protected] su.e d uLength scales of core vs valence electrons: carbon atom 1s state bonding region 2p 2s r_coreLub os_Mita [email protected] su.e d u Core vs valence electrons: copper atom, semicore states 3s,3p 1s,2s 3s bonding region 4s 3d r_coreLub os_Mita [email protected] su.e d uCore vs valence in bonds (C2): isosurfaces of molecular orbitals valence (hybridized, bond1s core states (unchanged) formation) _ +Difficulties from cores for DFT (HF, etc):large/huge basis and/or combined basis necessaryLub os_Mita [email protected] su.e d u Clearly difficult to describe both core and valence: - core states/electrons are highly localized and have large energies: require very accurate description: nuclear cusp -> - requires very localized description and basis - valence states have small energies, affected significantly due to bonding; states at or above Fermi level in solids can be even completely delocalized like a free-particle wave, very smooth - calls for very smooth basis, plane wave almost idealcorer≈exp−ZrDifficulties from cores for DFT (and HF):large/huge basis and/or combined basis necessary IILub os_Mita [email protected] su.e d u The best all-electron basis sets for solids based on combination of two types of basis (FLAPW method): - in spheres around atoms localized states: numerical radial meshes - between spheres plane waves - matching/continuity of orbitals on sphere surfaces very complicated - perfectly working approach within DFT but could be expensive - development took a long time and was rather slow - ultimately, cores are inefficient if you are interested in valence properties (think about a heavy atom, most of the states are in core) - can we get rid of the core electrons completely ?Idea of core – valence partitioningLub os_Mita [email protected] su.e d u - core states/electrons appear to be rigid and do not affect valence electronic structure (bonds, excitations, band gaps, conductivity) much due to the different energy and length scales - get rid of the core states/electrons and keep only the valence ones - represent the core by an effective operator (cannot be a simple potential, must be angular momentum dependent because of different number of core states in s, p, d angular momentum channels) - valence electrons feel a pseudopotential operator (instead of core e-)Vps−ion=∑lvlr∑m∣lm lm∣=∑l=0lmax[vlr−vloc]∑m∣lm lm∣vlocrDictionary and notationsLub os_Mita [email protected] su.e d u In condensed matter physics: pseudopotentials or PPs In quantum chemistry : effective core potentials or ECPs - radial pseudopotential function for a given l-symmetry channel - outside the core will be just - Z_eff/r = - (Z-Z_core)/r projection operator onto a given |lm> state -> nonlocal!!! - number of different occupied channels -> number of nonlocal projection operatorsvlr∣lm lm∣Vps−ion=∑lvlr∑m∣lm lm∣=∑l=0lmax[vlr−vloc]∑m∣lm lm∣vlocrvlocrlmaxNonlocality ? What does it mean ?Lub os_Mita [email protected] su.e d u Remember the self-consistent loop/one-particle eigenfunction eq. ? In the simplest atomic case the nonlocality means that each symmetry channel has different ionic (pseudo)potential s p etc[Tkin Vext=ion Veff ,HF{j}]i=Eii[Tkin vs Veff ,HF{j}]s=Ess[Tkin vp Veff ,HF{j}]p=EppOne-particle construction of PP:norm-conservationLub os_Mita [email protected] su.e d u How to construct such an operator so as to represent the action of the core electrons as closely as possible to the true atom The most important is the charge: beyond certain r_core radius the valence states should look identical like if the core electrons were present - outside the r_core the charge will then be the same as pseudo-charge,


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