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WIREs Comput Mol Sci
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Principal interacting orbital: A chemically intuitive method for deciphering bonding interaction

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Abstract Modular bonding picture is a central part of chemical understanding as exemplified by the wide usage of Lewis structure as a chemical language to describe molecular electronic structures. A chemically intuitive bonding picture requires the descriptors to be localized and transferable. Molecular orbitals directly derived from quantum calculations are too delocalized to interpret in this regard, hence many efforts have been devoted to the development of bonding analysis methods. After a brief overview of various bonding analysis methods, this review outlines the framework of principal interacting orbital (PIO) analysis and presents its role in recovering intuitive chemical concepts and producing modular bonding pictures. The PIO analysis identifies the most important fragment orbitals that are involved in fragment interactions and characterizes a one‐to‐one orbital interaction pattern that underlies a unique set of bonding and anti‐bonding orbitals derived from pairwise orbital interactions between two fragments. By making use of the principal component analysis (PCA), PIO analysis can effectively combine complex delocalized interaction patterns involved in numerous molecular orbitals into a handful of localized PIOs to provide easily interpretable results with intimate connection to localized chemical concepts, while maintaining necessary delocalized features when dealing with systems that involve mutlicentered interactions such as cluster compounds and various reactions. Such adaptability guarantees the robustness of PIO analysis with respect to change of substituents and the transferability of obtained PIOs across different systems. This article is categorized under: Structure and Mechanism > Molecular Structures
Principal interacting spin orbital (PISO) analysis on O2 between the two O atoms. (a) Four most contributing PISMOs. Total contribution of the 4 PISMOs: 98.7%. Isovalue 0.07 (e/bohr3)1/2. (b) MO diagram for O2
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Principal interacting spin orbital (PISO) analysis on B2H4 radical between the two BH2 groups. (a) Three most contributing PISO pairs. Total contribution: 95.7%. Isovalue 0.05 (e/bohr3)1/2. (b) Spin density of the radical. Isovalue 0.0025 (e/bohr3)
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Principal interacting orbital (PIO) analysis of [Rh@B18], the Rh center is chosen as the first fragment and the B18 as the second fragment. The label S+(s) stands for in‐phase combination of S orbitals of the two B9 units which matches the s orbital of the Rh center, and other labels are similar
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Frontier orbitals of a [B8] ring. The frontier orbitals of other [Bn] rings are similar
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Principal interacting orbital (PIO) analysis on [Co©B8] cluster between the Co center and the [B8] ring. Isovalue: 0.05 (e/bohr3)1/2. Orbitals are computed according to calculations presented in Ref.
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Principal interacting orbital (PIO) analysis on [Pd2Sn18]4− cluster between two [PdSn9] moieties. Total contribution: 67.0%. Isovalue: 0.05 (e/bohr3)1/2. (Reprinted with permission from Reference . Copyright 2018 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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Principal interacting orbital (PIO) analysis on [Pd3Ge18R6]2− cluster between two [Ge9R3] moieties. (a) Top two PIO pairs between two [Ge9R3] moieties. (b) A frontier orbital diagram of the [Ge9R3]+ moiety. (c) A schematic plot showing the electronic structure of the [Pd3Ge18R6]2‐ cluster. Total contribution: 61.5%. Isovalue: 0.05 (e/bohr3)1/2. Adapted with permission from Ref. . Copyright (2019) American Chemical Society
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Principal interacting orbital (PIO) analysis on two dicupraannulene compounds between two C4H4 ligands (including neighboring Li atoms) as well as their resonance structures. (a) Top two PIO pairs between two C4H4 ligands in Li4Cu2(C4H4)2. Total contribution: 91.9%. (b) Two resonance structures of Li4Cu2(C4H4)2 accounting for the inter‐ligand delocalization. (c) Top two PIO pairs between two C4H4 ligands in Li2Cu2(C4H4)2. Total contribution: 59.7%. (d) The only resonance structure of Li2Cu2(C4H4)2 without inter‐ligand delocalization. Isovalue for all: 0.05 (e/bohr3)1/2. Reproduced from Ref. with permission from The Royal Society of Chemistry
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Principal interacting orbital (PIO) analysis on Li4Pd(C4H4)2 between two C4H4 ligands (including neighboring Li atoms) as well as its resonance structures. Total contribution of 2 PIO pairs: 91.8%. Isovalue: 0.05 (e/bohr3)1/2. Reproduced from Ref. with permission from The Royal Society of Chemistry
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Principal interacting orbital (PIO) analysis on [Fe(CO)3Ac] between the [Fe(CO)3] moiety and Ac atom. Total contribution: 94.1%. Isovalue: 0.05 (e/bohr3)1/2. Orbitals are computed according to calculations presented in Ref. . Copyright 2019 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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Principal interacting orbital (PIO )analysis on [Fe(CO)3Sc] between the [Fe(CO)3] moiety and Sc atom. Total contribution: 94.5%. Isovalue: 0.05 (e/bohr3)1/2. (Reprinted with permission from Reference . Copyright 2019 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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Principal interacting orbital (PIO) analysis on the triple bond in [Os2Cl8]2− with the two [OsCl4] moieties as two fragments. Total contribution: 96.3%. Isovalue: 0.10 (e/bohr3)1/2. Orbitals are computed according to calculations presented in Ref.
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Principal interacting orbital (PIO) analysis on the quadruple bond in [Re2Cl8]2− with the two [ReCl4] moieties as two fragments and comparison with two canonical MOs. Note that there are two canonical MOs both having significant metal–metal σ‐bonding component and hence are more difficult to interpret. Total contribution of 4 PIMOs: 96.0%. Isovalue: 0.10 (e/bohr3)1/2 for PIO and 0.05 (e/bohr3)1/2 for CMO. Orbitals are computed according to calculations presented in Ref.
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Principal interacting orbital (PIO) analysis on [Fe(CO)3B] between the [Fe(CO)3] moiety and B atom. Total contribution: 98.3%. Isovalue: 0.05 (e/bohr3)1/2. PBI, PIO‐based bond index. (This figure is a derivative of Reference by Springer Nature, used under CC BY 4.0.)
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Dominant principal interacting orbital (PIO) pairs of C2H4 versus Cl in Zeise's salt. Note that PIMO is not available because of the incomplete fragmentation. Total contribution: 68.4%. Isovalue: 0.10 (e/bohr3)1/2. PBI, PIO‐based bond index
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Principal interacting orbital (PIO) analysis on the Zeise's salt [PtCl3(C2H4)] with PtCl3 being one fragment and the C2H4 being the other. Total contribution: 91.7%. Isovalue: 0.10 (e/bohr3)1/2. PBI, PIO‐based bond index. PIOs are adapted from Ref.
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Lewis structure of the Zeise's salt [PtCl3(C2H4)], and the Dewar–Chatt–Duncason model describing the interaction between the Pt center and the C2H4 ligand
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Scree plot of the principal interacting orbital (PIO)‐based bond index (PBI) values derived from PIO analysis on aniline with the phenyl and amino groups as two fragments, respectively
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Principal interacting orbital (PIO) analysis on aniline between the phenyl and amino groups showing the electron‐donating effect of amino group on benzene. Total contribution: 94.2%. Isovalue: 0.10 (e/bohr3)1/2. PBI, PIO‐based bond index. PIOs are adapted from ref.
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Electron‐donating effect of amino group on benzene
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The evolution of principal interacting orbital (PIO)‐based bond index (PBI) and contribution of top two PIO pairs against intrinsic reaction coordinate (IRC) in the Diels–Alder reactions between butadiene and ethene. (Reprinted with permission from Reference . Copyright 2018 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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Schematic diagram showing the relationship between fragment molecular orbital and principal interacting orbital. Isovalue: 0.05 (e/bohr3)1/2. (Reprinted with permission from Reference . Copyright 2018 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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Principal interacting orbital (PIO) analysis on the Diels–Alder reaction between hexadecaoctaene and ethene. Total contribution: 95.5%. Isovalue: 0.05 (e/bohr3)1/2. PBI, PIO‐based bond index. PIOs are adapted from Ref.
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Mutual delocalization in Diels–Alder reactions
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Principal interacting orbital (PIO) analysis on the H─Cl bonding interactions in hydrogen chloride. Isovalue: 0.10 (e/bohr3)1/2. FMO, fragment molecular orbital
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Charge decomposition analysis on the H─Cl σ‐bonding interactions in hydrogen chloride. All structures presented in this review are optimized using PBE0/def2TZVP and all orbitals are computed based on the same level of theory. Isovalue: 0.10 (e/bohr3)1/2. CMO, canonical molecular orbital; FMO fragment molecular orbital
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A flow chart demonstrating how principal interacting orbitals (PIOs) are obtained. CDA, charge decomposition analysis; FMO fragment molecular orbital; MO, molecular orbital; NAO, natural atomic orbital; NBO, natural bond orbital; PCA, principal component analysis. Figure reproduced from Ref.
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Overview of common bonding analysis tools
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Orbital interaction patterns that can arise in open‐shell systems and comparison with ordinary closed‐shell two‐orbital interaction scheme
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