The Local Properties of the Electronic Structure

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1 The Local Properties of the Electronic Structure A Bridge between Theory and Experiment Valera Veryazov Valera.Veryazov@teokem.lu.se Department of Theoretical Chemistry Lund University Local properties p. 1/25

2 Atomic Valency in chemistry textbooks NaCl KClO 3 table salt Berthollet s salt Na +1 Cl 1 Na I Cl I K +1 (ClO 3 ) 1 K I Cl V O3 II The rule: Use stoichiometry and bond counting! Local properties p. 2/25

3 Drawbacks of arithmetic approach Stoichiometry: valency should change under bond breaking or bond formation. obviously fails for systems with bonds between same element: organic molecules, allotropes Bond counting: ignores bond lengths can beused only incovalent systems: crystalline NaCl: 6 neighbours, CsCl: 8. how count bondsin molecules withvanderwaals bonds, dimers, carbonyls? Local properties p. 3/25

4 Can quantum mechanics help? The expectation value of an operator: ˆΩ = ψ ˆΩψdτ = ψ ˆΩ ψ Examples of observables: energy, dipole moment. Atomic charge, valency are not observable! wemustrely onthe chemical sense toobtainthem as an interpretation of the results. Toanswer Can weuse NaCl insteadof KClO 3? one has to study reactions... Local properties p. 4/25

5 Outline methods to compute atomic valency introduction to density matrix formalism local properties of electronic structure practical examples Local properties p. 5/25

6 Methods to compute valency improved bond counting valency index geometry related counting of electrons atom-in-molecule density matrix analysis various atomic charges bondordersand covalency full valency Local properties p. 6/25

7 Valency index Buckingham potential: Φ 12 (r) = Aexp( Br) C/r 6 Usedas apair potential inionic systems. Valency index by I.D.Brown: 1. select trusted crystal structures with known atomic valencies 2. calibrate A, B and C toreproduce atomic valencies (instead of energy) 3. verify on unknown structures Result: (in 1985) about 10% of crystal structures in ICSD database have atoms with"strange" atomic valencies. Or, they have"strange" interatomic distances. Local properties p. 7/25

8 Atom-in-molecule Quantum chemistry can compute electron density as a function of coordinates. KClO 3 density computed byhartree-fock method: The problem: how to separate atoms and bonds? Local properties p. 8/25

9 Geometry based separation of atoms KClO 3 density minus individual ions density: K isalmost acation,oand Cl are toomixed inspace. Bader analysis: zero flux surfaces. Local properties p. 9/25

10 Electrostatic potential charges Wecan varypointcharges on atoms, inorder to reproduce (in least-squares mean) the computed density (in thesetof points, e.g. inagrid). For molecules with small amount of atoms, extra centers with atomic charge should be used. main problem: low transferability among similar molecules Local properties p. 10/25

11 Density matrix Density matrix is the result of any quantum chemical calculation: Hartree-Fock, DFT, multiconfigurational, etc. D µν = θ i C µi Cνi,where θ isanoccupation for orbital i. i We can construct population matrix P, as a product of density matrix and overlap matrix. P µν = (DS) µν Traceof P isthe number ofelectrons: µ P µµ = N Local properties p. 11/25

12 Population matrix Now wecansplit P intoatomic blocks (µ A) andassosiate N A withanatom. For non-atomic basis set we make a projection first. Local properties p. 12/25

13 Population analysis Density matrix and population matrix are invariant to unitarytransformation, but N A arenot. Mulliken charges (R. Mulliken, 1955) Löwdin charges (P-O. Löwdin, 1955) (orthogonalized by S 1/2 ) Natural Bond Order (NBO) (F. Weinhold, 1983) (orthogonalized with constrained occupations) Loprop charges (G.Karlström, 2004) (separate orthogonalization for occupied and virtuals). Local properties p. 13/25

14 Why do we need that many? If we take NaCl molecule, same method (HF), but different basis sets: from minimal to extended (independently). Mulliken charges vary from 0.58 to 0.96 Loprop charges vary from 0.85 to 0.89 Methyl ethylene sulfide: Mulliken NBO Loprop ESP C(H2) C(H) S C(H3) Local properties p. 14/25

15 Local properties from density matrix N A = P aa a A Q A = Z A N A W AB = P ab P ba a A,b B (K. Wiberg, 1968) C A = W AB B =A (D.R. Armstrong, 1973) Local properties p. 15/25

16 mixing ionic and covalent inionic case: Q A valency incovalent case: C A valency H 2 O Q A C A H O NaCl Q A C A Na Cl KClO 3 Q A C A K Cl O Local properties p. 16/25

17 Full atomic valency How to combine two opposite cases: covalent and ionic? A magic formula: ( ) V A = 1 2 C A + C 2 A +4Q2 A (V.Veryazov, 1991) V A Q A C A,if C A is small take fully transferred electrons and add half of covalent pairs. V A C A, if Q A issmall But, does it work better? Local properties p. 17/25

18 Sometime it does! H 2 O Q A C A V A H O NaCl Q A C A V A Na Cl KClO 3 Q A C A V A K Cl O Local properties p. 18/25

19 How to improve the definition? Problems: calculations have to be performed with good quality basis set But interpretation results are more stable with compact basis sets description of the covalent bond: high bond orders are less reliable description of delocalized metallic bond Local properties p. 19/25

20 How to treat discrepancies? Making a new formula for computed atomic valency wewanttomatch theexpected value. Butwhatif weget2.3,or 2.7 insteadof 2? is itanew valency state? is itadefectof calculation? is itadefectof interpretation? Local properties p. 20/25

21 Mixed valency Pb 2 O 3 crystal Spacegroup=P2 1 /a, Z=4 twokindsof Pb, both6-coordinated byo Interatomic distances: Å Å Dowehave Pb III or mixture Pb II and Pb IV? Full valency results: 2.32 and 3.94 Local properties p. 21/25

22 The importance of correct density What happens with NaCl molecule during dissociation? Hartree-Fock and DFT theory give ionic solution, so, full valency will never become 0. In multiconfigurational approach, e.g. CASSCF, full valency ofatoms goes from 1to0. Local properties p. 22/25

23 Non stoichiometric compounds HTSC materials: La 2 CuO 4+x,YBa 2 Cu 3 O 7 x Theformal valency of Cu ishigher than2, butisit Cu III oro I? Strong correlation effects influence the result of density calculations. Interpretation(charges, valency) varies dramatically Local properties p. 23/25

24 Surface relaxation How toestimate astability of asurface? Solution 1: compute all possible displacements, study saturation of broken bonds. Solution 2: compare valency in the bulk(unperturbed) and on the surface. Local properties p. 24/25

25 Conclusion: the local properties form an easy and convenient language to describe the electronic structure, and thus connect the theoretical description and observable reality. Thank you! Local properties p. 25/25

Chemical Bonding -- Lewis Theory (Chapter 9)

Chemical Bonding -- Lewis Theory (Chapter 9) Ionic Bonding 1. Ionic Bond Electrostatic attraction of positive (cation) and negative (anion) ions Neutral Atoms e - transfer (IE and EA) cation + anion Ionic