Tutorial: Relativistic Pseudopotentials

Transcription

1 Relativistic Pseudopotentials Hirschegg 1/2006 Tutorial: Relativistic Pseudopotentials Hermann Stoll Institut für Theoretische Chemie Universität Stuttgart

2 Relativistic Pseudopotentials Hirschegg 1/ Relativistic effects in group 11 monohydrides Bond lengths (Angström) HF (nrel) DHF (rel) Experiment/Theory CuH AgH AuH ( 111 E)H M. Dolg in: Modern Methods and Algorithms of Quantum Chemistry, J. Grotendorst (ed.), NIC Series, Vol. 3, J ülich, 2000, p. 507.

3 Relativistic Pseudopotentials Hirschegg 1/2006 Outline Basic Ideas Model Potentials Shape-Consistent Potentials DFT-based Pseudopotentials Soft-Core Potentials and Separability Energy-Consistent Pseudopotentials Core-Polarization Potentials Conclusion

4 Relativistic Pseudopotentials Hirschegg 1/2006 Basic ideas Valence electrons responsible for chemical properties Frozen-core approximation Relativistic effects arise in core region Transfer of information from atom to molecule Note: analogy to DFT

5 Relativistic Pseudopotentials - Basic Ideas Hirschegg 1/2006 Valence-only Hamiltonian H P P = n v i [ 1 2 p 2 i + λ V P P λ (r iλ )] + n v i<j 1 r ij + λ<µ Q λ Q µ r λµ formally non-relativistic atomically parametrized V P P λ point-charge approximation for core interaction (Q λ = Z λ n c,λ )

6 Relativistic Pseudopotentials - Basic Ideas Hirschegg 1/2006 The zoo of pseudopotentials one-center/one-electron potentials (but: CPP) local or semi-local or non-local potentials full nodal structure or pseudo-orbital transformation MP, ECP soft-core or hard-core potentials 1-component or 2-component potentials (SO)

7 Relativistic Pseudopotentials Hirschegg 1/2006 Model potentials Search for AE ab-initio scheme where core contributions take one-center/one-electron form Hartree-Fock V cv = Z r + c (2J c K c ) (single core, non-relativistic) Assume non-overlapping cores cf. L. Seijo and Z. Barandiarán in: Relativistic Electronic Structure Theory, Part 2, P. Schwerdtfeger (ed.), Elsevier, Amsterdam, 2004, p. 417

8 Relativistic Pseudopotentials - Model potentials Hirschegg 1/2006 Add quasi-relativistic esc terms (Cowan-Griffin, Wood-Boring) V (nlj) rel = 1 2c 2 (ɛ nlj V ) 2 1 dv 4c 2 dr ( c 2 [ɛ nlj V ]) 1 ( d dr 2j + 1 ); 2r (single atom, IPA) Average out energy/orbital dependence of V (nlj) rel V rel = lj V (nlj) rel P lj ;

9 Relativistic Pseudopotentials - Model potentials Hirschegg 1/2006 Separate into scalar-relativistic and SO terms V av rel = l V SO rel = l 1 (l,l 1/2) (lv rel 2l (l,l+1/2) (V rel 2l (l + 1)V (l,l+1/2) rel )P l, V (l,l 1/2) rel )P l l spl, Shift core orbitals to high energies (not needed any more) V ls AIMP = c ( 2ɛ c ) φ c >< φ c

10 Relativistic Pseudopotentials - Model potentials Hirschegg 1/2006 Approximate the potentials for a given atomic state (frozen-core approximation) VAIMP av = Q r + C k r e α kr 2 + χ p lm l > A (l) pq < χ q lm l k l,m }{{} l,p,q }{{} Coulomb exchange, V av rel V SO AIMP = l ( k B (l) k r 2 (l) e β k r2 )P l l spl. Note: cv correlation neglected

11 Relativistic Pseudopotentials - Model potentials Hirschegg 1/2006 Basis sets Only valence orbitals need to be described explicitly (but: wiggles in the core region, tails at neighbouring atoms) large primitive sets, heavily contracted for high exponents Database Program MOLCAS

12 Relativistic Pseudopotentials - Model potentials Hirschegg 1/2006 Benchmark studies XH 4, XH 2, XO (X = Ge, Sn, Pb) (14-ve-MP, SCF results) R e : 1 pm Re rel : 10% D e : few kcal XH (X = Cl, Br, I) (7-,17-ve-MP, SO, corr. included) R e : 1 pm ω e : 1% D e : 0.1 ev

13 Relativistic Pseudopotentials - Model potentials Hirschegg 1/2006 Current/future developments Accounting for core-valence correlation (CPP) Explicit treatment of relativistic effects (DKH) Model potentials for deformed atoms/ions in crystal surroundings

14 Relativistic Pseudopotentials Hirschegg 1/2006 Shape-consistent pseudopotentials Search for potentials yielding radially nodeless pseudo-valence orbitals reduction of computational effort removal of cv orthogonality Start from AE reference calculation (HF, DHF) Transform valence orbitals/spinors: pseudo-orbital transformation φ lj (r) φ φ lj (r) for r r c lj (r) = f lj (r) for r < r c Apply normalization and continuity conditions for f lj (r)

15 Relativistic Pseudopotentials - Shape-consistent potentials Hirschegg 1/2006 Valence spinors vs. pseudo-valence spinors of I AE MCDHF/DC vs. PP(7,MCDHF) 0.8 5s 1/ p 1/2 5p 3/2 P(a.u.) log 10 (r[a.u.]) M. Dolg in: Relativistic Electronic Structure Theory, Part 1, P. Schwerdtfeger (ed.), Elsevier, Amsterdam, 2002, p. 793

16 Relativistic Pseudopotentials - Shape-consistent potentials Hirschegg 1/2006 5s valence and pseudo valence orbitals of I 1.0 P 5s (r) 0.5 SKBJ CEP HW 0.0 MWB SDF AE, WB log 10 (r[a.u.]) M. Dolg in: Relativistic Electronic Structure Theory, Part 1, P. Schwerdtfeger (ed.), Elsevier, Amsterdam, 2002, p. 793

17 Relativistic Pseudopotentials - Shape-consistent potentials Hirschegg 1/2006 5p valence and pseudo valence orbitals of I 0.8 P 5p (r) 0.4 SKBJ CEP HW 0.0 MWB SDF AE, WB log 10 (r[a.u.]) M. Dolg in: Relativistic Electronic Structure Theory, Part 1, P. Schwerdtfeger (ed.), Elsevier, Amsterdam, 2002, p. 793

18 Relativistic Pseudopotentials - Shape-consistent potentials Hirschegg 1/ PP and AE orbitals for E114, 7s 2 7p 2 7s 1/2 7p 1/2 7p 3/ P(r) r (a.u.) B. Metz and H. Stoll, unpublished

19 Relativistic Pseudopotentials - Shape-consistent potentials Hirschegg 1/2006 Invert radial Fock equation for given ɛ lj, φ lj (r) to solve for V P P ( 1 2 d 2 l(l + 1) + dr2 2r 2 + V P P lj (r) + W v lj[{ φ l j }]) φ lj (r) = ɛ lj φlj (r) locality assumption V P P = lj V P P lj (r) P lj Note: Relativistic effects are implicitly included in V P P when using relativistic reference data Note: Only one orbital is fitted for each (l, j) combination

20 Relativistic Pseudopotentials - Shape-consistent potentials Hirschegg 1/2006 Analytical fit of V P P V P P = Q r + lj ( k A lj,k r n lj,k 2 e α lj,kr 2 )P lj Separate into V P P av, V P P SO Note: No 1/r 3 dependence in V P P SO Variants Analytical fit after V P P av Neglecting l dependence of V P P av Truncating V P P SO, VSO P P separation for l L for l L

21 Relativistic Pseudopotentials - Shape-consistent potentials Hirschegg 1/2006 Databases Christiansen, Ermler, Pitzer, Ross... (ECP) pac/reps.html Hay, Wadt Stevens, Basch, Krauss (CEP) Titov et al. (GRECP) Programs COLUMBUS, GAUSSIAN, GAMESS, MOLPRO, TURBOMOLE,... CRYSTAL Note: Library basis sets often rather small, re-optimization possible

22 Relativistic Pseudopotentials - Shape-consistent potentials Hirschegg 1/2006 How to improve transferability Choose reasonably small core Example: Ce 4f, 5d, 6s ( chemical choice): not recommended relaxation of 5sp shell with varying 4f occupation 1-3 ev fc errors 4f, 5spd, 6s (medium core): use as f-in-core V P P relaxation of 4spd shell with varying 4f occupation ev fc errors elimination of 5spd nodes in 4f region 4spdf, 5spd, 6s (small core): general-purpose V P P 10 3 ev fc errors

23 Relativistic Pseudopotentials - Shape-consistent potentials Hirschegg 1/2006 Ce orbital energies s 5d 6s 5d f 4f ε (a.u.) p 5p s 5s -2.0 nrel(hf) rel(dhf) M. Dolg in: Relativistic Electronic Structure Theory, Part 1, P. Schwerdtfeger (ed.), Elsevier, Amsterdam, 2002, p. 793

24 Relativistic Pseudopotentials - Shape-consistent potentials Hirschegg 1/2006 density (a.u.) Valence orbital densities and core densities of Ce Ce 30+ Ce 12+ Ce 4+ 4f 5d R e CeO 4fφ 1 6spσ 1 3 Φ radius (a.u.) 6s M. Dolg in: Relativistic Electronic Structure Theory, Part 1, P. Schwerdtfeger (ed.), Elsevier, Amsterdam, 2002, p. 793

25 Relativistic Pseudopotentials - Shape-consistent potentials Hirschegg 1/2006 How to improve transferability (cont d) Avoid occupation of core-like orbitals no problem for single atoms (local ansatz) but too weak Pauli repulsion in molecules (core other-atom) XF (X = group 1 atom, 1-ve-PP) R e -10 pm use 9-ve-PP

26 Relativistic Pseudopotentials Hirschegg 1/2006 DFT-based pseudopotentials Replace HF/DHF modelling of cv interaction by KS/DKS ( K c ) V xc [ρ c + ρ v ] V xc [ρ v ] c (potential) < ψ v c ( K c ) ψ v > E xc [ρ c + ρ v ] E xc [ρ v ] E xc [ρ c ] ; (energy) Caution: Non-linearity of V xc, E xc.

27 Relativistic Pseudopotentials - DFT-based potentials Hirschegg 1/2006 Possible solutions Neglect overlap of ρ c and ρ v Do not include xc effects in the MPs or exclude them a-posteriori with ECPs NLCC (simplified representation of ρ c ) LDA, GGA, hybrid GGA, OEP,... based PPs

28 Relativistic Pseudopotentials Hirschegg 1/2006 Soft-core pseudopotentials and Separability Computational effort depends on valence basis set used pseudo-orbital transformation sufficient for GTOs smooth/near-constant PP desirable for plane waves no strong Pauli repulsion no strong attraction

29 Relativistic Pseudopotentials - Soft-core / Separability Hirschegg 1/2006 How to achieve smoothness Impose additional conditions on pseudo-orbital transformation Minimize high-kinetic-energy Fourier components of φ lj Increase r c Relax norm-conservation (Vanderbilt)

30 Relativistic Pseudopotentials - Soft-core / Separability Hirschegg 1/2006 Semi-local vs. non-local potentials V P P av = V loc (r) + l V l(r)p l }{{} special problems for matrix elements V P P av = V loc (r) + l,m l,p,q χ p lm l > A (l) pq < χ q lm l }{{} overlap integrals with auxiliary basis set

31 Relativistic Pseudopotentials - Soft-core / Separability Hirschegg 1/2006 How to achieve a separable form χ lml > = V l (r) φ lml > V P P av = V loc (r) + l,m l χ lml > (Kleinman, Bylander) 1 < χ lml φ lml > < χ lm l V P P = V loc (r) + or more generally χ nlml > = V l (r) φ nlml > B nlml,n l m l n,l,m l,n,l,m l = < χ nlml φ n l m l > χ nlml > B 1 nlm,n l m < χ n l m l Further generalization (change of metric) leads to ultra-soft PPs

32 Relativistic Pseudopotentials Hirschegg 1/2006 Energy-consistent potentials Which information are the PP derived from? atomic data single-reference states (but: low-lying excited states?) DHF/DKS (but: jj-coupling?)

33 Relativistic Pseudopotentials - Energy-consistent potentials Hirschegg 1/2006 Alternative: atomic valence-energy spectra historically: single-valence-electron ions experimental reference data recent approach: neutral atoms, near-neutral ions ab-initio reference data AE-MCDHF

34 Relativistic Pseudopotentials - Energy-consistent potentials Hirschegg 1/2006 Parametrization of V P P V P P = Q r + lj ( k A lj,k r n lj,k 2 e α lj,kr 2 )P lj i w i (E AE i E P P i ) 2 = Min. target accuracy 10 2 ev for all i, with k 3

35 Relativistic Pseudopotentials - Energy-consistent potentials Hirschegg 1/2006 Features radially nodeless orbitals (semi-local ansatz) pseudo-orbitals in core region controlled by PP ansatz pseudo-orbitals in valence region correct for V P P = Q/r Remaining approximations frozen core (no cv correlation) limited transferability of pseudo-orbital derived properties (v correlation) differential errors 1mH

36 Relativistic Pseudopotentials - Energy-consistent potentials Hirschegg 1/2006 Database scalar-relativistic PP, SO-PP, CPP for various core sizes + valence basis sets cc-pvnz basis sets under development (K. Peterson) Programs COLUMBUS, GAUSSIAN, GAMESS, MOLCAS, MOLPRO, TURBOMOLE,...

37 Relativistic Pseudopotentials - Energy-consistent potentials Hirschegg 1/2006 Benchmark studies R e 0.2 pm ω e 1 cm 1 D e 0.2 kcal/mol Examples group 18 polarizabilities α e 1-2% spectroscopic constants of group 11 diatomics

38 Relativistic Pseudopotentials - Energy-consistent potentials Hirschegg 1/2006 Hg, atomic excitations (CCSD(T), aug-cc-pwcv5z) 4 MDF MWB MWB (readjusted) 3 2 Deviation from experiment (mh) P 6s 6p 1P 6s 6p 3S 6s 7s 1S 6s 7s 3P 6s 7p 1D 6s 6d 3D 6s 6d 2S 6s 2P 6p 2D 6d 1S L. Thiel et al., J. Chem. Phys. 119 (2003) 9008; D. Figgen and H. Stoll, in preparation

39 Relativistic Pseudopotentials - Energy-consistent potentials Hirschegg 1/2006 Hg, 6s 1 6p 1 excitation errors a PP b AE d CCSD CCSD(T) FSCCSD FSCCSD c FSCCSD FSCCSD e exp. 3 P P P P P P a) in mh b) PP calculations by Figgen et al. (in preparation), 20 electrons correlated, aug-cc-pwcvqz c) correction for 4f correlation d) all-electron calculations by Kaldor et al. (J. Phys. B 33 (2000) 667), 20 electrons correlated e) 34 electrons correlated

40 Relativistic Pseudopotentials - Energy-consistent potentials Hirschegg 1/2006 K.A. Peterson et al., J. Chem. Phys. 119 (2003) 11099, 11113

41 Relativistic Pseudopotentials - Energy-consistent potentials Hirschegg 1/2006 replacements SCF/MP2 results for AuH 0.06 r e (Å) EC-PP, nrel. EC-PP, rel. all PP 19 PP 11 P. Schwerdtfeger et al., J. Chem. Phys. 113 (2000) 7110

42 g replacements Relativistic Pseudopotentials - Energy-consistent potentials Hirschegg 1/2006 SCF/MP2 results for AuH D e (ev) EC-PP, nrel. EC-PP, rel. all PP 19 PP 11 P. Schwerdtfeger et al., J. Chem. Phys. 113 (2000) 7110

43 Relativistic Pseudopotentials - Energy-consistent potentials Hirschegg 1/2006 SCF/MP2 results for AuH rag replacements 100 ω e (ev) EC-PP, nrel. EC-PP, rel. all PP 19 PP P. Schwerdtfeger et al., J. Chem. Phys. 113 (2000) 7110

44 Relativistic Pseudopotentials - Energy-consistent potentials Hirschegg 1/2006 SCF/MP2 results for AuH g replacements 0.2 µ e (Debye) all PP EC-PP, nrel. EC-PP, rel. PP P. Schwerdtfeger et al., J. Chem. Phys. 113 (2000) 7110

45 Relativistic Pseudopotentials - Energy-consistent potentials Hirschegg 1/2006 D. Figgen et al., Chem Phys. 311 (2005) 227

46 Relativistic Pseudopotentials - Energy-consistent potentials Hirschegg 1/2006 K.A. Peterson and C. Puzzarini, Theor. Chem. Acc. 114 (2005) 283

47 *) $" %& +, -. ( (! "# ' $ $. /. $. /. Relativistic Pseudopotentials - Energy-consistent potentials Hirschegg 1/2006 I.S. Lim et al., J. Chem. Phys. 122 (2005) ; ibid., 124 (2006)

48 Relativistic Pseudopotentials Hirschegg 1/2006 Core-polarization potentials cv correlation How to recover static and dynamic core-polarization? non-frozen-core effects for large, easily polarizable cores

49 Relativistic Pseudopotentials - Core-polarization potentials Hirschegg 1/2006 CPP formalism (W. Meyer) V CP P = 1 2 f λ = i α D,λf 2 λ λ r λi rλi 3 g λ (r λi ) r λµ Q µ r 3 g λ (r λµ ) µ( λ) λµ g λ (r) = (1 exp( δ λ r 2 )) m adjustment to IP exp IP PP for 1-ve ions Program : MOLPRO

50 Relativistic Pseudopotentials - Core-polarization potentials Hirschegg 1/ Calibration study of the iodine dimer I 2 convergence of the bond length bond length (Angstroem) SCF CCSD(T) CCSD(T)+SO experiment maximum angular quantum number in the basis set M.Dolg, Mol. Phys. 88 (1996) 1645

51 Relativistic Pseudopotentials - Core-polarization potentials Hirschegg 1/ Calibration study of the iodine dimer I 2 convergence of the binding energy binding energy (ev) SCF CCSD(T) CCSD(T)+SO experiment maximum angular quantum number in the basis set M.Dolg, Mol. Phys. 88 (1996) 1645

52 Relativistic Pseudopotentials - Core-polarization potentials Hirschegg 1/ Calibration study of the iodine dimer I 2 convergence of the vibrational constant 240 vibrational constant (cm 1 ) SCF CCSD(T) CCSD(T)+SO experiment maximum angular quantum number in the basis set M.Dolg, Mol. Phys. 88 (1996) 1645

53 Relativistic Pseudopotentials Hirschegg 1/2006 Concluding remarks PP provide an accurate description of valence properties (comparable to AE ab-initio treatment) PP enable an implicit treatment of relativistic effects in a formally non-relativistic framework (direct/indirect effects for core and valence system) PP/CPP efficiently account for core-valence correlation effects Caution: difficulties with core-sensitive properties (EFG, NMR)

54 Relativistic Pseudopotentials Hirschegg 1/2006 Recommended reading P. Pyykkö and H. Stoll in: R.S.C. Spec. Period. Rep. Chemical Modelling, Applications and Theory, Vol. 1, 2000, p. 239 M. Dolg in: Relativistic Electronic Structure Theory, Part 1 (Fundamentals), P. Schwerdtfeger (ed.), Elsevier, Amsterdam, 2002, p. 793