Calculation of excitation energies for heavy-element systems

Transcription

1 Calculation of excitation energies for heavy-element systems Stefan Knecht ETH Zürich, Laboratorium für Physikalische Chemie, Switzerland 1

2 Outline Part 1: Density Matrix Renormalization Group in a Nutshell (complement to Multiconfigurational methods for the f-elements ) How to choose/analyze active spaces for multiconfigurational methods? Insights from Quantum Information theory Part 2: Chemical bonding in actinides: U 2 Ground- and low-lying excited states of ThF + Part 3: Spectroscopy and conceptual chemistry: properties and excited states of UO 2 2+, OUN + and NUN 2

3 Acknowledgment Markus Reiher (ETH Zürich) Sebastian Keller (ETH Zürich) Stefano Battaglia (ETH Zürich) Örs Legeza (U Budapest) Pawel Tecmer (Mc Master U Hamilton) Luuk Visscher (VU Amsterdam) P. Bagus (U North Texas) Trond Saue (U Toulouse) Hans Jørgen Aa. Jensen (SDU Odense) Timo Fleig (U Toulouse) Morten N. Pedersen (SDU Odense) Andre Gomes (U Lille) 3

4 Part 2 4

5 SK, H. J. Aa. Jensen, and T. Saue, in preparation. 5

6 Consideration of 5f, 6d and 7s orbitals via trial calculations with different active spaces (26 orbitals, 12 electrons in total) Eight orbitals involved in the U-U bond three normal electron-pair bonds (7sσ g, 6dπ u ) two one-electron bonds (6dσ g, 6dδ g ) two weaker bonding orbitals (5fπ u, 5fδ g ) 7sσ g (2.00) 6dπ u (4.00) 6dσ g (0.97) 6dδ g (0.98) 5fπ u (0.63) 5fπ g (0.37) 10 electron quintuple bond Exchange stabilised ferromagnetic coupling of non-bonding 5f electrons Gagliardi and Roos, Nature 433:848, fδ g (0.63) 5fδ u (0.37) 5fφ u (1.00) 5fφ g (1.00)

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8 Bonding picture discussed in a scalar-relativistic framework so far but what about spin-orbit coupling? à does it change our view of the U-U bonding? à lowest-lying states of U 2? Ground state? 8

9 Our computational tool box in Dirac (diracprogram.org) Hartree Fock wave function analysis I wave function analysis II CI property module ^ z ^z ^ natural orbitals < l >, < s >, < j z > + occupation numbers KR MCSCF 4 component + 2 component + (1 component) projection analysis analysis modules Mulliken population 2D and 3D visualization à 2-Component CASSCF/RASSCF with variational SOC à Dyall.cv3z basis set 9

10 CAS(12,26) secondary anti bonding orbital energy [10 4 cm 1 ] s 6d (6d 6d ) 3/2 = 2081 cm 1 5/2 (5f 5/2 5f 7/2 ) = 6025 cm 1 GAS II CAS 7s: u 6d: u, g, u 5f: u, g, u, g 7s: g bonding SD 15 5f nonrelativistic scalar relativistic fully relativistic GAS I 6d:,, g u g 5f: g, u, g, u Valence orbital space of the U atom inactive Not feasible à DMRG! 10

11 6d: u, u 5f: u, g, u, GAS II g bonding Choice 1 Choice 2 6d: g, g CAS 5f: g, u, g, u GAS I Same active space as Roos and Gagliardi CAS(6,20) secondary anti bonding anti bonding 7s: u 6d: g bonding 7s: g 6d: u inactive SD GAS[(6,6)SD(6,20)] GAS II CAS GAS I secondary anti bonding 6d: u, u 5f: u, g, u, g bonding 6d: g, g 5f: g, u, g, u anti bonding 7s: u 6d: g bonding 7s: g 6d: u inactive SD 11

12 Ω = 4 g Ω = 4 u Ω = 5 g Ω = 5 u Ω = 6 g Ω = 6 u Ω = 7 g Ω = 7 u Ω = 8 g Ω = 8 u Ω = 9 g Relative energy in au Ω = Σ + Λ Term symbol: 2S+1 Λ Ω(g/u) CAS(6,20) Proposed(ground(state( (Gagliardi(and(Roos,(Nature(433,(848((2005))( R(U U) in Angström 12

13 Ω = 4 g Ω = 4 u Ω = 5 g Ω = 5 u Ω = 6 g Ω = 6 u Ω = 7 g Ω = 7 u Ω = 8 g Ω = 8 u Ω = 9 g Relative energy in au Ω = Σ + Λ Term symbol: 2S+1 Λ Ω(g/u) CAS(6,20) Proposed(ground(state( (Gagliardi(and(Roos,(Nature(433,(848((2005))( R(U U) in Angström 13

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15 Example N 2 : 15

16 Example N 2 : CAS(6,6) Total σ π Effective bond order R(N N) in Angström 16

17 Classification of bonding and anti-bonding orbitals 17

18 Classification of bonding and anti-bonding orbitals 18

19 Effective bond-order generalized: 19

20 Effective bond orders for the lowest two Ω states: à non-integer Σ and Λ values à Qualitative agreement between CAS and RAS data 20

21 Effective bond orders for the lowest two Ω states: à non-integer Σ and Λ values à Qualitative agreement between CAS and RAS data 21

22 4.1 4 Ω = 9 g CAS Ω = 9 g RAS Ω = 8 g CAS Ω = 8 g RAS Effective bond order R(U U) in Angström 22

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24 Summary: Our 2c-SO CASSCF calculations also predict close-lying states of different Ω BUT a different ground state separated by ~ 0.5 ev with an EBO of about 3.8 which suggests a quadruple bond! Comparison of electronic configurations: 24

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26 Ground- and low-lying excited states of ThF + M. Denis, M. N. Pedersen, H. J. Aa. Jensen, A. Gomes, M. K. Nayak, SK, and T. Fleig, submitted to New J. Phys (2014) 26

27 Ground- and low-lying excited states of ThF + hot candidate in search of the eedm (Yale-Harvard collaboration, group of E. Cornell (JILA)) à P,T violation: new physics beyond the standard model two close-lying 3 Δ (7s 1 6d 1, science state ) and 1 Σ (7s 2 ) states à theoretical data (single-reference CC + pert. SO from MRCI) inconclusive on ground state assignment B. J. Barker, I. O. Antonov, M. C. Heaven, K. A. Peterson, JCP 136, (2012) 27

28 Computational approach 4c-/2c-Hamiltonian (variational SOC) dyall.cv3z/dyall.cv4z basis set (Th); aug-cc-pv(t/q)z (F) MRCI and IH-FSCC-SD IH-FSCC is a multi-reference CC approach à simultaneous account for dynamic/static correlation à practically intruder-state free à size-consistent (GS); size-extensive (@EXC) à cluster operators S (h,p) for each sector (holes,particles) à model space P = P m + P i ; correlation space Q 28

29 Active spaces for IHFSCC and MRCI Our choice for ThF + : ThF 3+ (0h,0p) à ThF 2+ (0h,1p) à ThF + (0h,2p) 29

30 Ground- and low-lying excited states of ThF + QZ basis CBS extrapolation 30

31 Ground- and low-lying excited states of ThF + QZ basis CBS extrapolation 31

32 Spectroscopic constants for the Ω=1 and Ω=0 + states 32

33 Summary Differential correlation effects (à Fermi hole in triplet state) play an important role Large basis sets required Molecular-mean-field SOC and Gaunt contribution nonnegligible 3 Δ state seems to be lower by ~ 320 cm -1 compared to 1 Σ Suggestion for experimental re-investigation 33

34 Backup slides