Pseudopotential. Meaning and role

Transcription

1 Pseudopotential. Meaning and role Jean-Pierre Flament Laboratoire de Physique des Lasers, Atomes et Molécules (PhLAM) Université de Lille-Sciences et technologies MSSC Torino, 2 to 7 september 2018

2 Overview What is an ECP and Why? requirements on ECP s How To make ECP s and implications Applications => Caveats => What cannot be done ECP databases

3 Motivation for ECP 1. experience (Mendeleev s table): valence electrons determine most of the structural, chemical and spectroscopic properties of molecules and crystals core electrons very little affected; just spectators screening the nuclei. 2. computations (ab initio, DFT) may be a considerable task as Z : large basis sets great number of 2-e integrals 3. try to get rid of the core electrons while keeping the accuracy of All Electron calculations. => find a cheap atomic potential transferable to molecules to mimic the effect of core electrons leading to good energies and Wave Functions (and densities). this is the role of Effective Core Potentials: pseudopotentials(pp) and Model Potentials (MP)

4 Requirements on ECP keep the properties of valence electrons: reduce the number of basis functions and gaussians (or PW) AE ε e, φ e ECP ε e, φ e 0.60 Iodine: 4s, 5s orbitals ε v, φ v ε v, φ v 0.40 rφ(r) 0.20 ε c, φ c F = ε c φ c φ c + c ε v φ v φ v + ε e φ e φ e v e { F φ v = F } ε c φ c φ c φ v c F MP φ v = F φ v = ε v φ v r φ v = φ v φ c φ c φ v c F PP φ v = {F c (ε c ε v ) φ c φ c } φ v F PP φ v = ε v φ v

5 the final form of F PP no approximation on F PP, φ v is a (arbitrary) pseudo-orbital: F PP φ v = { T Za r a } n c + V (φ c) (ε c ε v ) φ c φ c + V (φ v ) φ v = ε v φ v c=1 note ε v and V (φ v ) NOT V (φ v ). { } F PP φ v = T Za + W vc + V (φ v ) V (φ v r ) + V (φ v ) φ v = εv φ v a needs a 1-e operator to compute the red part => this is an other approximation: { } F PP φ v = T Z a eff + Va PP r (ra) + V (φ v ) φ v = ε v φ eff v, Za = Z a n c a Nuclear attraction + Coulomb (J c) = V elec Za + nc as r r a r a, hence Z eff a a.

6 the final form of F PP F PP φ v = { F PP φ v = { T Za r a T Z a eff r a } n c + V (φ c) (ε c ε v ) φ c φ c + V (φ v ) φ v = εv φ v c=1 + V PP a (ra) + V (φ v ) } φ v = ε v φ v φ v nodeless and smooth within the core; V PP a (r) = +l V l (r) lm lm = V L (r) + L [V l (r) V L (r)] P l (semi-local) l m= l l=0 L = l core max + 1. experience shows that V l (r) V L (r) l L as r a V l 0 faster then 1 r : V N kl l (r) = A kl r n kl e α kl r 2, n kl 2 k to determine V l (usually with numerical methods, not basis set expansions) solve the the AE problem: => ε v and φ v choose the pseudo-orbital(s) φ v (i) nodeless, (ii) close to φv r > rc, (iii) minimum undulation in core find the operator having ε v and φ v as eigensolutions (1 per l) (the inverse problem) => 3 commonly used inversion methods

7 norm-conserving PP L.R. Kahn, P. Baybutt, D.G. Truhlar, J. Chem. Phys., 65, 3826 (1976) φ v is a combination of the φ v s and φ c s ; the coefficients are obtained by minimization of a penalty function implementing the 3 conditions. Almost no human decision. solve the inverse { problem: from } F PP φ v = T Z a eff r + V PP (r) + V (φ v ) φ v = ε v φ v we have V PP (r) = ε v + Z a eff r {T + V (φ v )} φ v (r) φ v (r) then least-square fit the analytical formulas to the numerical V PP (r) and φ v (r). That s done. Problem: φ v is normalized: φ v = av φ v + av 2+a2 c φ v φ v even in the valence region where φ c 0. ac av 2+a2 c => TRANSFERABILITY? φ c so

8 norm-conserving PP example of Cl 2 E 27 kcal/mol; (E) 18 kcal/mol P. J. Hay, W. R. Wadt, and L. R. Kahn, J. Chem. Phys. 68, 3059 (1978) PP from L.R. Kahn, P. Baybutt, D.G. Truhlar, J. Chem. Phys., 65, 3826 (1976)

9 Shape-consistent PP (1) P.A. Christiansen, Y. S. Lee, K.S. Pitzer, J. Chem. Phys., 71, 4445 (1979) (2) P. Durand, J.-C. Barthelat, Theor. Chim. Acta, 38, 283 (1975) rφ(r) Carbon: 1s,2s orbitals; 2s pseudo-orbital choose φ v (r) = φ v (r) r l+2 4 a i r i i=0 r r c r r c φ v φ v = 1 continuity of φ v (r) and it s first 3 derivatives at rc r choose r c (conditions on d2 rφv dr 2 ). find V PP (1) as previously (2) by solving the F PP equation ( => ε v ) and minimizing the deviation of ε v from εv

10 Energy-adjusted PP M. Dolg, H. Stoll, H. Preuss, J. Chem. Phys., 86, 866 (1987) from the previous strategies forget all but the idea of limiting the calculations to valence electrons, the analytical form of V PP (limited to n kl = 0) instead use the total hamiltonian H PP = nv { } T i Z eff + V r PP (i) + nv 1 i i r i>j ij fit V PP to reproduce total energy differences AE and PP calculations on ion A +q with 1 valence electron (several l, 2 L) => A nl, α nl SEFIT AE and PP calculations on A and A + : several 2S+1 L states => refine the previous A nl MEFIT can be applied to any partition of electrons into core and valence not restricted to having a single (ε v, φ v ) for each l value better spectrum

11 applicability As can be guessed from the foregoing discussion the e-e interaction V (φ) as well as the kinetic energy operator T are general and ECP s can be determined for HF and DFT, classical (Schrödinger) and relativistic (various approximations) calculations. HF based ECP s are usually used also in DFT calculations. I know few DFT based ECP s N. Trouillier, J.L. Martins, Phys.Rev., 43, 1993, (1991) PW, L. Andzelm, E. Radzio, D.R. Salahub, J. Chem. Phys., 83, 4573 (1985) opium I. Grinberg, N. J. Ramer, A. M. Rappe, Phys. Rev. B62, 2311 (2000) For solid state PW (?). Search for ghost states

12 application to molecules and crystals calculations use H ECP = { n v T i i a Z eff a r ai + Va ECP (i) } + n v i>j 1 r ij Use an ECP basis set of the same quality as you would use for the valence in AE calculations suppress core electrons => underlying assumption: in the molecule the core orbitals are localized on each atom remain frozen in their atomic state => not too much polarized do not participate to the physics/chemistry under study The last two items raise the question of the core/valence partition i.e. small core or large core? small core more or less means Dolg s PP.

13 polarizable core elements at the right of Mendeleev s table have little polarizable cores: (e.g. Al -> Ar) ns np in valence may be sufficient elements at the left of Mendeleev s table have polarizable cores: (e.g. Na, Mg) at least (n-1)p ns np in valence. => to avoid the (n-1)p shell use Core Polarization Potentials 1 V CPP (r) = 1 α c fc f c 2 c fc = rc r 3 c C(r c, ρ c) c c Rcc R 3 cc Z c C(r c, ρ c) = (1 e (rc /ρc )2 ) nc n c = 1, 2 BUT only in MOLPRO (?); there are few of them... search the literature 1 W. Müller,J. Flesch, W. Meyer J. Chem. Phys. 80, 3297 (1984)

14 core orbitals participating to the physics/chemistry? Mainly transition elements often the outermost core orbitals have a non negligible overlap with valence ones => they may appear in bonding MO s => excitation energies with contraction/expansion of the outer core => large (differential) correlation effects recommended (n-1)s (n-1)p (n-1)d ns np (e.g. Fe: 3s3p3d4s4p) Lanthanides and actinides have f open-shells (n-2)f (n-1)s (n-1)p (n-1)d ns np for some of them, f orbitals may be put in-core 2, for chemical studies. (V PP then depends on the oxidation state -number of electrons in the F shell-) 2 A. Moritz, M. Dolg, Theor. Chem. Acc. 121, 297 (2008)

15 What you CAN NOT do with ECP Obvioulsy computing transitions or properties implying the core electrons! NEXAFS, ISEELS: excitation of core electrons by X-ray or Electron Energy Loss NMR, ( ESR) and all observables implying short range operators Ln, δ(r n) r 3 n

16 Where to find ECP s many ECP s and related basis sets in many program formats 1 many ECP s and related basis sets formatted for molpro, gaussian, turbomole, crystal 1 relativistic ECP s and related basis sets formatted for gaussian, columbus ~ pchristi/reps.html 1 Notes on pseudopotentials generation. A good HOW-TO with references to PP databases for solid-state. ~ giannozz/atom/doc.pdf search the literature for CPP s: look at the paper cited in the database in any case: have a look ar the original article (references are always given) A comprehensive article: L. Seijo, Z. Barandiarán, Computational chemistry reviews of current trends (world scientific, Singapur), 4, (1999) 1 accessible through a clickable periodic table