Multiconfigurational methods for the f-elements

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1 Multiconfigurational methods for the f-elements Andy Kerridge Winter School in Theoretical f-element Chemistry Helsinki, Finland, December 5 th -8 th, 204

2 Overview CASPT2: reference weights and intruder states Uranyl (revisited) Spin-orbit coupling: RASSI 6d-5f transitions in uranium chlorides What next: beyond CASSCF Literature examples

3 The Reference Weight The first order approximation to the exact wavefunction is given by Normalising gives To a good approximation Ψ = Ψ 0 + Ψ Ψ Ψ = Ψ 0 Ψ 0 + Ψ 0 Ψ = + S, S = Ψ Ψ Ψ = ω Ψ 0 + ω Ψ ω = ω = + S + α N/2 + Ψ Ψ 0 + Ψ Ψ Where N is the number of electrons and α is determined empirically, α~0.05

4 Intruder States Configurations in V SD can have zeroth order energies close to (or below) E 0 and can make significant contributions to the corrected wavefunction. Such intruder states, if weakly interacting, can be handled via level shifting H 0 H 0 + ε This shift increases the magnitude of the energy denominator, reducing the effect of the intruder. The corrected energy is given by E 2 E 2 ε ω Strongly interacting intruder states must be handled via an increase in the active space Other perturbational products are available! e.g. NEVPT2

5 Uranyl (revisited) u + g g E CASSCF = 42 nm u CASPT2 ε E (nm) ω GS ω ES First absorption band in Cs 2 UO 2 Cl 4 : 498 nm Denning, Inorg. Chem. 26:72 (987)

Spin-Orbit Coupling: RASSI The RAS State Interaction (RASSI) procedure can be used to calculate coupling and interactions between CASSCF/RASSCF states An effective SO

6 Spin-Orbit Coupling: RASSI The RAS State Interaction (RASSI) procedure can be used to calculate coupling and interactions between CASSCF/RASSCF states An effective SO Hamiltonian can be constructed using atomic mean field integrals Matrix Elements are given by the Wigner-Eckart theorem, and are only non-zero between states with S S 2 = 0,, M M 2 = 0,

7 6d-5f Transitions in Uranium Chlorides 5f 2 : 49 SOF states 5f 6d : 70 SOF states 2 SOC states in total UCl 6 2- S I D G [UCl 5 THF] - P F H UCl 4 THF 5f 2

8 6d-5f Transitions in Uranium Chlorides 5f 2 : 49 SOF states 5f 6d : 70 SOF states 2 SOC states in total UCl 2-6 S I D S I D G [UCl 5 THF] - G P P F F H UCl 4 THF H 5f 2 O h CF: 49 states

9 6d-5f Transitions in Uranium Chlorides 5f 2 : 49 SOF states 5f 6d : 70 SOF states 2 SOC states in total UCl 2-6 S S I D S 0 I 6 D 2 I D G G 4 P 2 G P P [UCl 5 THF] - P P 0 F F F 4 F F 2 H UCl 4 THF H H 6 H 5 H 4 5f 2 O h CF: 49 states SOC: 9 states

Theory Exp. Theory.4.68.40.7.4.79 2.94 2.82 2.94 2.90.0.00 P 2.

10 6d-5f Transitions in Uranium Chlorides 5f 2 : 49 SOF states 5f 6d : 70 SOF states 2 SOC states in total CASPT2(2,) ANO(TZP) UCl 6 2- [UCl 5 THF] - UCl 6 2- [UCl 5 THF] - UCl 4 THF UCl 4 THF F 2 F G 4 Exp. Theory Exp. Theory Exp. Theory P ~ Hashem, et al., RSC Adv. :450 (20)

11 What Next? Beyond CASSCF Split-CASSCF Define a small principle space (A) and a larger extended space (B). Use Lowdin s partioning method to map eigenvalue problem onto A. No full-ci Quasi-Complete Active Space (QCAS) Divide CAS into any number of subspaces, each of which is a CAS Generalised Active Space Self Consistent Field (GASSCF) Generalisation of RASSCF. Arbitrary number of active spaces. Accumulated minimum and maximum occupations numbers define the configurations in the CI expansion. Occupation Restricted Multiple Active Spaces (ORMAS) Restrict electron occupation of a series of subspaces Density Matrix Renormalisation Group CASSCF (DMRG-CASSCF) Employs the DMRG algorithm as an alternative to exact diagonalisation of the Hamiltonian.

u ) two one-electron bonds (6d g, 6d g ) two weaker bonding orbitals (5f u, 5f g ) 7s g (2.00) 6d u (4.00) 6d g (0.97) 6d g (0.98) 5f u (0.

12 Examples: The Uranium Dimer Gagliardi and Roos, Nature 4:848, 2005 Consideration of 5f, 6d and 7s orbitals via trial calculations with different active spaces (26 orbitals, 2 electrons in total) Eight orbitals involved in the U-U bond three normal electron-pair bonds (7s g, 6d u ) two one-electron bonds (6d g, 6d g ) two weaker bonding orbitals (5f u, 5f g ) 7s g (2.00) 6d u (4.00) 6d g (0.97) 6d g (0.98) 5f u (0.6) 5f g (0.7) 0 electron quintuple bond Exchange stabilised ferromagnetic coupling of nonbonding 5f electrons 5f g (0.6) 5f u (0.7) 5f u (.00) 5f g (.00)

Sulfur-bridge organodysprosium SMM τ = τ 0 e U eff/k B T U eff = cm - m J = 5/2 m J = /2 = cm - thermally

13 Examples: Dy Single-Molecule Magnet Tuna et al., Angew. Chem. Int. Ed. 5:6976 (202) SOC electronic structure required to understand high barrier to magnetization reversal in Sulfur-bridge organodysprosium SMM τ = τ 0 e U eff/k B T U eff = cm - m J = 5/2 m J = /2 = cm - thermally activated relaxation High magnetic anisotropy Easy axes ~ perpendicular to Dy 2 S 2 plane Ising Hamiltonian for magnetic interaction g x = g y = g z = 9.6 J = 2.2 cm

14 5f non-bonding U(5f)-O(2p) U(6d)-O(2p) Examples: Uranyl, UO 2 2+ Pierloot and Van Besien, JCP 2:20409 (2005) Strong covalent U-O interactions No -, -interactions CASSCF(2,2) for GS CASSCF(2,6) for ES Six two electron, three centre bonding orbitals two U-O triple bonds Pseudocore 6p orbital destabilises u bond Many excited states with similar energies UO 2 2+ : g g : 529 nm

15 5f non-bonding U(5f)-O(2p) U(6d)-O(2p) Examples: Uranyl, UO 2 2+ Pierloot and Van Besien, JCP 2:20409 (2005) Strong covalent U-O interactions No -, -interactions CASSCF(2,2) for GS CASSCF(2,6) for ES Six two electron, three centre bonding orbitals two U-O triple bonds Pseudocore 6p orbital destabilises u bond Many excited states with similar energies UO 2 2+ : g g : 529 nm [UO 2 Cl 4 ] 2- : g g : 499 nm

16 5f non-bonding U(5f)-O(2p) U(6d)-O(2p) Examples: Uranyl, UO 2 2+ Pierloot and Van Besien, JCP 2:20409 (2005) Strong covalent U-O interactions No -, -interactions CASSCF(2,2) for GS CASSCF(2,6) for ES Six two electron, three centre bonding orbitals two U-O triple bonds Pseudocore 6p orbital destabilises u bond Many excited states with similar energies UO 2 2+ : g g : 529 nm [UO 2 Cl 4 ] 2- : g g : 499 nm Cs 2 UO 2 Cl 4 : 498 nm

Calculation of excitation energies for heavy-element systems

Calculation of excitation energies for heavy-element systems Stefan Knecht ETH Zürich, Laboratorium für Physikalische Chemie, Switzerland http://www.reiher.ethz.ch/people/knechste stefan.knecht@phys.chem.ethz.ch