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Theoretical Foundations of the Bond-Orbital Projection Formalism 2/33 Updated: 2016-02-25 WORKSHOP Institut de Química Computacional i Catàlisi AA newnew perspectiveperspective onon quantifyingquantifying electronelectron localizationlocalization andand delocalizationdelocalization inin molecularmolecular systemssystems TheoreticalTheoretical foundationsfoundations ofof thethe bond-orbitalbond-orbital projectionprojection formalismformalism dr Dariusz Szczepanik GIRONA JAGIELLONIAN UNIVERSITY January 21st, 2016 Department of Theoretical Chemistry Theoretical foundations of the bond-orbital projection formalism 2/33 Presentation plan 1. Electron density, atoms in molecules, chemical bonds. 2. Localization and delocalization components of the electron density 3. Electron delocalization between atoms. 4. Electron delocalization between bonds. 5. The effectiveness of bond conjugation as an aromaticity criterion. Theoretical foundations of the bond-orbital projection formalism 3/33 Electron density, atoms in molecules, chemical bonds I Hohenberg–Kohn theorem: the ground-state electron density (ED) uniquely determines the potential and thus all physicochemical properties of the molecular system. What about traditional concepts such as atom and chemical bond? Theoretical foundations of the bond-orbital projection formalism 4/33 Electron density, atoms in molecules, chemical bonds Partitioning of the electron density into atomic contributions (charges): ● Bader's charges (QTAIM approach), ● Hirshfeld's charges, ● Politzer's charges, ● Voronoi charges, ● Coulson's charges, ● Mulliken's charges, ● Löwdin's charges, ● Weinhold's (natural) charges, ● Merz-Kollman's charges, ● Breneman's charges, ● Szigeti charges, and many others... A difficult choice! Theoretical foundations of the bond-orbital projection formalism 5/33 Electron density, atoms in molecules, chemical bonds Delocalization of atomic charges – chemical bonding analyses: ● Bond critical points (QTAIM by Bader), ● Coulson's, Wiberg's, Mayer's and Gopinathan-Jug's bond orders, ● Localized molecular orbitals (Boys, Edmiston-Ruedenberg, Pipek-Mezey schemes) ● Natural bond orbitals (NBO), ● Natural orbitals for chemical valence (NOCV), ● Localized orbitals of bond orders (LOBO), ● Iterative double-atom partitioning by orthogonal projectors (IDAP), ● Electron localization function (ELF), ● Localized orbital locator (LOL), ● Single exponential decay detector (SEDD), ● Reduced density gradient (RDG), and many others... A difficult choice! Theoretical foundations of the bond-orbital projection formalism 6/33 Electron density, atoms in molecules, chemical bonds Delocalization of chemical bonds – a multicenter electron-sharing analyses: ● Scanning for multicenter bonding within the framework of the NBO analysis, ● Adaptive natural density partitioning (AdNDP) analysis, ● Multicenter delocalization descriptors (MCI,DI,ESI), ● Bridgeman-Empson's three-center bonding analysis, ● Electron density of delocalized bonds approach (EDDB), and others... Is it possible to probe electron localization and delocalization within one theoretical paradigm? Theoretical foundations of the bond-orbital projection formalism 7/33 Localization and delocalization components of the electron density .. electrons electrons electrons ED(r) EDLA(r) EDLB(r) EDDB(r) EDDA(r) ED(r) – electron density, EDLA(r) – density of electrons localized on atoms, EDDA(r) – density of electrons delocalized between atoms, ! EDLB(r) – electron density of localized bonds, EDDB(r) – electron density of delocalized bonds, Theoretical foundations of the bond-orbital projection formalism 8/33 Electron delocalization between atoms (EDDA) Electron density layer representing the population of electrons delocalized between atoms, EDDA(r), is crucial for the whole ED-decomposition procedure. It involves the Hilbert-space partitioning scheme and is defined through the following steps: Definition of the matrix 1.Transformation of non-orthogonal atomic orbitals (AOs) into the represe- ntation of natural atomic orbitals (NAOs). 2.Solving the eigenproblems of a set of Jug's matrices, representing all possible bonds (interactions) in a molecule, to obtain two-center bond- order orbitals (2cBO) and their occupations. 3.Projection of the 2cBO metric onto the subspace of occupied MOs. Theoretical foundations of the bond-orbital projection formalism 9/33 Electron delocalization between atoms (EDDA) 1. The occupancy-weighted symmetric orthogonalization (OWSO) of atomic orbitals (AOs) to natural atomic orbitals (NAOs): A). Transformation from cartesian to pure AOs. B). Partitioning and symmetrization of intra-atomic blocks of the overlap and the Coulson's density matrix (D). C). Löwdin orthogonalization of the intra-atomic blocks of the DM. D). Solving the eigenproblems for intra-atomic blocks of the DM. E). Division of eigenfunctions into the (pre-orthogonalized) natural minimal basis (NMB) and the complementary natural Rydberg's basis (NRB). F). Interatomic Gramm-Schmidt orthogonalization of NRB to NMB. G). Repeating the B,C and D steps, but only for the NRB 'orbitals'. H). A separate interatomic occupancy-weighted orthogonalization of both subsets, I). Re-orthogonalization of both subspaces by repeating the B,C and D steps. Features of NAOs: – The effective dimensionality of the AO space is reduced to that of the formal NMB subspace. – For isolated atoms NAOs coincide with natural orbitals (NO). – NAOs mostly retain a well-localized one-center character. – NAOs are intrinsically stable toward basis set extensions. Theoretical foundations of the bond-orbital projection formalism 10/33 Electron delocalization between atoms (EDDA) 2. Construction of a set of two-center bond orbitals. – electron density in the NAO basis reads – the Jug's matrix is a (α,β)-diatomic block matrix of type: = – the eigenvectors of represent the two-center bond orbitals of three different types: bonding >0, non-bonding =0, and antibonding <0 , and both subsets of 2cBOs form a paired-orbital basis. – for duodemponent density matrices, the Wiberg's bond-order (covalency) index reads: Theoretical foundations of the bond-orbital projection formalism 11/33 Electron delocalization between atoms (EDDA) 2. Construction of a set of two-center bond orbitals. – bonding 2cBOs CCbond Acetylene – antibonding 2cBOs For typical molecular systems with well defined Lewis-like electronic structures the highest-occupied 2cBOs form a set of nearly orthogonal bond orbitals and, consequen- tly, twice the sum of Wiberg's indices between all pairs of covalently bonded atoms approximates the population of electrons delicalized between all atoms (EPDA). Theoretical foundations of the bond-orbital projection formalism 12/33 Electron delocalization between atoms (EDDA) 3. Projection of 2cBOs onto the subspace of occupied MOs Obviously, for accurate calculations as well as in the case of large molecular systems with non-typical bonds and weak interactions, twice the sum of Wiberg's covalencies n sometimes exceeds the exact EPDA due to nonorthogonal all = overcounting. In such situations we have two choices: ( 2 ) I. Restore orthogonality of the highest-occupied 2cBOs within the iterative double-atom partitioning procedure using orthogonal projectors or, more familiar, by transformation to the subset of bonding NBOs. II. Remove the nonorthogonal electron overcounting by the following projection cascade: NAO → MO(occupied) → 2cBO(bonding) → MO(occupied) → NAO , which is fully equivalent to the following orthogonal similarity transformation: Theoretical foundations of the bond-orbital projection formalism 13/33 Electron delocalization between atoms (EDDA) 3. Projection of 2cBOs onto the subspace of occupied MOs 8 25% 14% 6 π-electrons 4 2 Number of of Number 0 4.800 4.311 6.000 4.889 6.857 4.903 7.500 4.768 - + 2+ C5H5 C6H6 C7H7 C8H8 Projections through the subspace of occupied MOs remove π-electron overcounting...… a little too much! So where is the problem? Theoretical foundations of the bond-orbital projection formalism 14/33 Electron delocalization between atoms (EDDA) 3. Projection of 2cBOs onto the subspace of occupied MOs Due to nonorthogonalities both bonding and antibonding 2cBOs are linear combinations of MOocc and MOvir. Therefore, the projection cascade MUST involve both 2cBO subspaces. ! c o m p l e m e n t a r y + NAO → MO(occupied) → 2cBO(all) → MO(occupied) → NAO Theoretical foundations of the bond-orbital projection formalism 15/33 Electron delocalization between atoms (EDDA) 3. Projection of 2cBOs onto the subspace of occupied MOs 25% 8 14% 6 π-electrons 4 2 Number of of Number 4.800 4.311 6.000 5.760 4.889 6.000 6.857 4.903 5.877 7.500 4.768 0 5.625 - + 2+ C5H5 C6H6 C7H7 C8H8 NAO → MO(occupied) → 2cBO(all) → MO(occupied) → NAO Theoretical foundations of the bond-orbital projection formalism 16/33 Electron delocalization between atoms (EDDA) ED(r) EDLA(r) EDDA(r) 30.0 2.3 27.7 (1.7) Theoretical foundations of the bond-orbital projection formalism 17/33 Electron delocalization between atoms (EDDA) ED(r) 30.0 30.0 30.0 EDDA(r) 29.3 24.2 23.4 EDLA(r) 0.7 5.8 6.6 Theoretical foundations of the bond-orbital projection formalism 18/33 Electron delocalization between bonds The EDDA component of the electron density can be further partitioned: EDDA(r) = EDLB(r) + EDDB(r) EDLB(r) – electron density of localized (two-center) bonds, EPLB EDDB(r) – electron density of delocalized (multi-center) bonds, EPDB 0 – localized 2cBO 1 – delocalized 2cBO Theoretical foundations of the bond-orbital projection formalism 19/33 Electron delocalization between bonds Construction
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