Winterschool on Michael Dolg Actinide Chemistry Institut fur¨ Theoretische Chemie Helsinki Universit¨at zu K¨oln December 10-14, 2007 Relativistic Energy-consistent Ab Initio Pseudopotentials — New Developments for Actinides. ‘Stuttgart-(Dresden→Bonn→Cologne)‘ PPs: http://www.theochem.uni-stuttgart.de Valence orbital densities and core densities of Th 30+ 12+ 4+ Th Th Th 1.0 + 0.9 Σ 2 1 0.8 σ 5f 0.7 0.6 ThO 7sp e R 1 h =7 2 h =8 3 h =9 0.5 0.4 6d density (a.u.) 7s 0.3 0.2 0.1 4 h − 1=7 5 h − 1=8 0.0 0 1 2 3 4 5 6 7 8 radius (a.u.) Contents • Short introduction to energy-consistent relativistic pseudopotentials • What we currently can provide for lanthanides and actinides: Wood-Boring adjusted scalar-relativistic (one-component) large-core (Ln 4f-in-core, An 5f-in-core) and small-core (Ln 4f-in-valence, An 5f-in-valence) pseudopotentials, valence spin-orbit operators in the small core case and corresponding valence basis sets of different quality (VDZ to VQZ) and contraction schemes (segmented, generalized) based on ANOs. – some general considerations concerning the choice of the core: small-core vs. large-core PPs; f-in-core PPs (superconfiguration model (R.W.Field); feudal model (B.Bursten) ?) – calibration for atoms/molecules – examples for applications • What we would like to have for lanthanides and actinides: Multi-configuration Dirac-Hartree-Fock/Dirac-Coulomb-Breit adjusted relativistic (two- and one-component) small-core pseudopotentials and corresponding correlation- consistent valence basis sets. At present only available for heavy main group and all d-transition elements. • Conclusions and outlook Introduction: goals goal: relativistic and correlated quantum chemical investigation of systems with (one or several) heavy elements desireable: • suitable for ground and excited states (unambigous description in well-defined coupling scheme) • suitable for all types of bonding (van der Waals, covalent, ionic, metallic) and systems (atoms, molecules, clusters, polymers, solids) ↓ wavefunction-based (ab initio) methods with the best available relativistic Hamiltonian ! (The right answer for the right reason ... and a very high computational cost) also desireable: • reduction of computational effort • control of underlying approximations • tolerable loss of accuracy • possibility for systematic improvements ↓ approximate relativistic (ab initio/first principles) methods (A reasonable answer for good reasons ... and a moderate computational cost) Introduction: approximations in valence-only methods approximation: restriction of the explicit wavefunction-based quantum chemical ab initio treatment to the system of valence electrons ↓ frozen-core approximation and core-valence separation ↓ model potentials (MP), pseudopotentials (PP) and core-polarization potentials (CPP) ∗ ∗advantage: reduction of computational effort (simple inclusion of relativistic contributions) ∗disadvantage: reduction of accuracy critical: choice of the core (small, medium, large) Introduction: approximations in valence-only methods model potentials (Huzinaga, Klobukowski, Seijo, Barandiaran; Wahlgren, Gropen, Marian;...) valence orbitals (correct number of radial nodes) ↓ Ni AE/MP P(r) 4s vs. log10(r) pseudoorbital Ni PP(Ne-core) P(r) 4s vs. log10(r) transformation ↓ pseudopotentials ∗ (Hellmann; Preuß; Pitzer, Christiansen, ...; Durand, Barthelat, ...; Hay, Wadt; Stevens, ...; Stoll, Dolg, Schwerdtfeger, ...; ...) pseudo valence orbitals (reduced number of radial nodes) ∗advantage: computational savings with respect to one-particle basis set ∗disadvantage: formal transformation of (some) operators necessary, e.g., spin-orbit term, ... overestimation of correlation energies ? Introduction: construction of valence-only Hamiltonian methods of adjustment of ab initio pseudopotentials orbital adjustment: shape-consistent PPs reference data: all-electron valence orbitals and orbital energies (independent particle model) (Pitzer, Christiansen, ...; Durand, Barthelat, ...; Hay, Wadt; Stevens, ...) energy adjustment: energy-consistent PPs∗ reference data: all-electron total valence energies (quantum mechanical observables; independent particle model and beyond) (Stoll, Dolg, Schwerdtfeger, ...) ∗advantage: applicable for any quality of the wavefunction (SCF, MCSCF, CI, CC), e.g., adjustment in intermediate coupling scheme possible ∗disadvantage: relatively high computational effort problems with neutral or negative charged cores Introduction: approximations in quantum chemical first-principles methods ? exact solution or experiment rel. (4−k.) rel. e.g. DC,DCB rel. (2−c.) Hamiltonian ECP (MP,PP) e.g. DKH, CPD scalar−rel. (1−c.) e.g. CG VDZ VTZ VQZ ... "nrel." nrel. Min. DZ TZ QZ ... complete HF one−particle basis "HF" MP2 MCHF MP3 CISD MP4 CCSD .... DFT ? many−particle basisCCSD(T) CCSDT MRCI FCI .... Wanted: suitable compromise between accuracy and computational effort ! Note: ECPs lead to reductions in the sizes of the one- and many-electron basis sets ! Accuracy ECPs are an approximate relativistic quantum chemical scheme and preferably should be used for medium-sized to large systems (including also periodic ones). Nevertheless many results for atoms and diatomics have been produced in the last decades, however more accurate schemes are available for theses cases nowadays. • (highly) accurate atomic results are not the ultimate goal of ECP calculations ! → E. Eliav, U. Kaldor, I. Grant, C. Froese-Fischer, P. Indelicato, ... can do it much better at the AE level, at least they get the right answer for the right reason. → atoms provide reference data and good test cases, e.g., for the balanced description of “chemically relevant“ atomic configurations/states/levels. → An accuracy of 10−2 eV (80 cm−1) in excitation energies, ionization potentials, electron affinities usually sufficient. • (highly) accurate diatomic results are not the ultimate goal of ECP calculations ! → K. Faegri, K. Dyall, T. Saue, L. Visscher, ... could do it better at the AE level, at least if they apply large enough basis sets ! → diatomics nevertheless provide good test cases for the transferability of the ECPs from the atom to the molecule (calibration studies). −1 −2 −1 → An accuracy of 10 eV in De, 10 A˚ in Re and 10 cm in ωe should at least be achieved. Needed: “robust“ (small-core) and highly transferable ECPs based on the best available re- lativistic many-electron Hamiltonian (e.g. Dirac-Coulomb-Breit) and series of valence basis sets with the possiblity to approach systematically the basis set limit. MCDHF/DC+B-adjusted small-core PPs for actinides: Hamiltonian of reference all-electron calculations All-electron Hamiltonoperator: n n N ZλZµ Hˆ = h(i)ˆ + ˆg(i, j) + r Xi Xi<j λ<µX λµ Dirac operator (correct to O(c−∞): 2 hD(i) = c~αi~pi +(βi − I4)c + Vλ(riλ) , Xλ Coulomb operator: 1 gC(i, j) = rij Coulomb-Gaunt operator (Lorentz or Feynman gauge), not even correct to O(c−2): 1 ~αi~αj gCG(i, j) = − rij rij Coulomb-Breit operator (Coulomb gauge), correct to O(c−2): 1 1 (~αi~rij)(~αj~rij) gCB(i, j) = − [~αi~αj + 2 ] rij 2rij rij Considering the frequency of the exchanged photon (Coulomb gauge): 1 eiωrij/c eiωrij/c − 1 g(i,j,ω) = − ~α ~α − (~α ∇ )(~α ∇ ) i j i rij j rij 2 2 rij rij ω rij/c MCDHF/DC+B-adjusted small-core PPs for actinides: Hamiltonian of reference all-electron calculations In the low frequency limit ωrij << 1 this reduces to the Breit interaction (magnetic inter- action and retardation). Energy conservation requires ω = ǫs − ǫr = ǫq − ǫp. Dirac matrices: ~0 ~σ I 0 ~α = and β = 2 ~σ ~0 0 −I2 Pauli matrices: 0 1 0 −i 1 0 σ = σ = σ = x 1 0 y i 0 z 0 −1 DCB Hamiltonian: one of the most accurate many-electron Hamiltonians (O(c−2)) ! Suitable for the generation of atomic reference data for relativistic valence-only schemes ! Beyond the DCB Hamiltonian: The lowest order Feynman diagrams: electron-electron interaction via exchange of a single virtual photon, vacuum polarization and self-energy correction (cf. book written by K.G.Dyall and K. Faegri). Suitable implementation and importance of the latter for actinides ? MCDHF calculations for U: various contributions U AE DHF results (in cm−1): result ----- deviation ---- result -- deviation -- result ----- deviation ---- result -- deviation -- 5f 6d 7s 7p HFDB HFDB HFD+B HFD MCDHF/DC+B MCDHF/DC+B(w) 5f 6d 7s 7p HFDB HFDB HFD+B HFD MCDHF/DC+B MCDHF/DC+B(w) Fermi Point Fermi Fermi Fermi Fermi nucleus Fermi Point Fermi Fermi Fermi Fermi nucleus A=238 A=238 A=238 A=238 A=238 A=238 A=238 A=238 A=238 A=238 (*) (*) (*) (*) (**) (**) (*) (*) (*) (*) (**) (**) 5 99459 252 4 1126 99462.63 26.79 2 3 1 15120 28 -1 -675 2 3 59251 119 -1 -611 59248.83 -31.22 42 49572 223 3 709 2 2 2 4640 -85 -1 -779 4638.54 -24.64 4 1 1 30790 154 2 670 30792.34 17.68 2 2 1 1 23857 2 -1 -764 4 1 1 47895 222 2 706 47896.84 11.76 2 2 1 49992 -10 -1 -729 49990.25 -30.11 4 1 69159 224 2 735 69161.06 13.55 2 1 2 1 17449 -129 -1 -897 17447.06 -34.13 4 2 15780 76 2 627 15782.04 20.13 2 1 2 47433 -156 -1 -862 47431.36 -28.98 4 1 1 30712 138 2 648 30713.75 13.03 2 21 76494.09 -43.27 4 1 54593 126 1 677 54594.01 15.82 2 1 1 1 74925.44 -42.91 4 1 80130.98 5.36 1 4 1 38781 -49 -2 -1549 38779.68 -53.48 321 13124 97 0 78 1 4 85256 54 -2 -1477 85254.51 -59.25 3 2 54576 177 0 138 54575.18 -6.55 1 3 2 31450 -176 -2 -1673 31448.66 -52.56 311117200 75 0 14 1 3 1 1 52488 -78 -2 -1655 312 0 0 0 0 0.00 0.00 1 3 1 79451 -89 -2 -1616 79449.36 -58.37 3 1 1 42328 63 0 44 42327.62 -5.05 1 2 2 80779 -251 -2 -1769 80777.54 -57.39 3 2 1 7516 -40 0 -93 7516.23 -8.28 1 2 2 1 49776 -224 -2 -1810 3 2 36289 -68 0 -62 36288.95 -3.63 1 2 1 1 10350.39 -72.16 3 11 61569.42 -16.62 1 1 2 1 16100.09 -72.68 31 1 70265.01 -16.63 4 1 31966.53 -89.31 4 2 81605.91 -83.19 (*) Mosyagin, Petrov, Titov, Tupitsyn, in: Progress in Theoretical Chemistry and Physics, Vol.