The case of orbitals Trond Saue (Toulouse/Oslo) Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
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1 The case of orbitals Trond Saue (Toulouse/Oslo) Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
2 Some facts The lanthanide series: Orbital energies Numerical 4-component atomic Hartree-Fock calculations (GRASP). Average SCF over the 4f n 5d 1 6s 2 configuration. Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
3 More facts The lanthanide series: radial expectation values Numerical 4-component atomic Hartree-Fock calculations (GRASP). Average SCF over the 4f n 5d 1 6s 2 configuration. Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
4 Some facts The actinide series: Orbital energies Numerical 4-component atomic Hartree-Fock calculations (GRASP). Average SCF over the 5f n 6d 1 7s 2 configuration. Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
5 More facts The actinide series: radial expectation values Numerical 4-component atomic Hartree-Fock calculations (GRASP). Average SCF over the 5f n 6d 1 7s configuration. Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
6 The case Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
7 The case The Lu 4f orbitals are chemically active and mixes with the F(2p) orbitals. Hamiltonian: Method: ZORA DFT (PBE) Basis: QZ4P Slater-type orbitals (22s17p11d6f) Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
8 Do f orbitals participate in chemical bonding???? An answer requires: A leading question! that atomic f orbitals can be identified in the molecule that a clear disctinction between chemically active and inactive orbitals can be made Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
9 Probing covalency with X-ray spectroscopy E. I. Solomon, B. Hedman, K. O Hodgson, A. Dey, R. K. Szilagyi, Coor. Chem. Rev. 249 (2005) 97 Covalency: the number of pairs of electrons that a given atom shares with its neighbors [I. Langmuir, JACS 41 (1919) 868] Ligand K-edge XAS has been used to probe covalency in transition metal complexes [B. Hedman, K. O. Hodgson and E. Solomon, JACS 112 (1990) 1643; F. Neese, B. Hedman, K. O. Hodgson, and E. I. Solomon, Inorg. Chem. 38 (1999) 4854] More recently this approach has been extended to f element compounds Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
10 What is an orbital? R. S. Mulliken, Phys. Rev. 41(1932) 49; Science 157(1967) 13 Spectroscopic orbitals: e.g. canonical (HF) molecular orbitals [CMOs] Chemical orbitals: e.g. localized orbitals [LMOs] The HF energy of a closed-shell molecule is invariant to rotations amongst occupied orbitals. Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
11 Localization criteria Pipek-Mezey: minimize the number of centers spanning each molecular orbital occ nuclei G PM ( ) {ψ i } = Q i 2 A Boys: maximize orbital self-extension (or the distance between orbital charge centroids) occ G B (r1 {ψ i } = (ψ i ψ i r 2 ) 2 ) ψi ψ i i=1 Edmiston-Ruedenberg: maximize the electron self-repulsion of each molecular orbital occ ( ) G ER 1 {ψ i } = ψ i ψ i r 1 r 2 ψ iψ i i=1 Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137 i=1 A=1
12 From Lewis structures to bonding orbitals Carbon dioxide has double bonds, or? Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
13 Pipek-Mezey localized orbitals of CO 2 Hund suggested that two-center LMOs correspond to chemical bonds F. Hund, Z. Phys. 73 (1931) 565 d= d = d = d = Pipek-Mezey localization criterion: d i = ( nuclei A=1 ( Q i A ) 2 ) 1 For each C-O pair the localization procedure gives one σ-bond, two π-bonds and one lone pair! System r C O (Å) Bond energy (kcal/mol) CO CO H 2 CO Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
14 Resolution of the apparent octet rule violation T. A. Halgren, L. D. Brown, D. A. Kleier and W. N. Lipscomb, J. Am. Chem. Soc. 99(1977) 6793 Φ = (core)σo1 2 π2 O1x π2 O1y l O1 2 σ2 O2 π2 O2x π2 O2y l O2 2 Expansion of LMOs in the ground-state Slater determinant gives determinants that cancel (Pauli principle). The remaining determinants can be regrouped to MO structures, corresponding to the key valence-bond structures, in which the octet rule is not violated. Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
15 Back to our case LuF 3 Repeating the calculations of Dognon et al. we indeed find considerable mixing of lutetium 4f and ligand 2p orbitals Overlaps between fluorine and lutetium atomic orbitals are very small 1s 1/2,1/2 2s 1/2,1/2 2p 1/2,1/2 2p 3/2,1/2 2p 3/2,3/2 4f 5/2,1/ f 5/2,3/ f 5/2,5/ f 7/2,1/ f 7/2,3/ f 7/2,5/ f 7/2,7/ d 3/2,1/ d 3/2,3/ d 5/2,1/ d 5/2,3/ d 5/2,5/ s 1/2,1/ Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
16 Overlap driven vs. degeneracy driven covalency M. L. Neidig, D. L. Clark, R. L. Martin, Coor. Chem. Rev. 257 (2013) 394 Consider a two-center molecular orbital ψ = N (φ M + λφ L ) ; N = λSML + λ 2 Mixing coefficient λ (to lowest order) λ = H ML EM 0 E L 0 ; H ML S ML A large mixing coefficient may occur for large numerator: overlap driven covalency small denominator: near-degeneracy driven covalency It appears that bonding in LuF 3 is degeneracy driven....or? Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
17 Bonding analysis in localized orbitals Orbital energies obtained as expectation values of the Kohn Sham matrix Pipek-Mezey orbitals: Orb ε F1 F2 F3 Lu s p d f Boys orbitals: Orb ε F1 F2 F3 Lu s p d f No indication of Lu f covalency but...there is no unique definition of localized orbitals! Is there another way? Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
18 Our way Observations: In a closed-shell molecule one can freely rotate between the occupied orbitals without changing the total energy; this is the basis for orbital localization. If an open-shell is introduced, rotations between closed-shell and open-shell orbitals change the total energy. Strategy: Import 4f orbitals from an atomic calculation into the molecule and keep them frozen during initial optimization. Make a small fractional hole in the 4f shell (occ. = 14 ɛ) and see if the orbitals stay in this artificial open-shell in the ensuing optimization. Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
19 Results 14-ɛ E(au) Q(Lu) 4f 4f open 5d 6s (Energy : E au) The lutentium 4f orbitals do not participate in bonding! Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
20 What can we learn from X-ray spectroscopy? X-ray absorption spectroscopy probes spectroscopic (canonical), not chemical orbitals. So what about K-edge XAS? Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
21 We rest our case Sherlock Holmes and Doctor Watson took a tent and went on a camping trip. The first evening after a good meal and a bottle of wine they retired for the night. During the night Holmes woke Watson and said: I see millions of stars. What does that tell you, Watson? Watson replied: Astronomically it tells me that there are millions of stars and galaxies and billions of planets. Astrologically is shows that Saturn is in Leo. Theologically I can see God is all powerful and we are small and insignificant. Meteorlogically I believe we will have fine weather tomorrow. What does it tell you, Holmes? Holmes was silent a moment: Watson, you idiot!. It tells me that somebody has stolen our tent! Trond Saue (LCPQ, Toulouse) The case of orbitals ESQC / 137
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