Natural Bond−Bond Polarizability: A Hü ckel-Like Electronic Delocalization Index H. E. Zimmerman † and F. Weinhold* Department of Chemistry and Theoretical Chemistry Institute, University of Wisconsin, Madison, Wisconsin 53706, United States ABSTRACT: We show how the bond−bond polarizability index, as originally introduced by Coulson and Longuet−Higgins in the Hü ckel-theoretic context, can be generalized in the natural bond orbital (NBO) framework to ab initio molecular orbital and density functional theory levels. We demonstrate that such a “natural bond−bond polarizability” (NBBP) index provides a flexible and quantitative descriptor for a broad spectrum of delocalization effects ranging from strong π aromaticity to weak intra- and intermolecular hyperconjugative phenomena. Illustrative applications are presented for representative delocalization effects in saturated and unsaturated species, chemical reactions, and hydrogen-bonding interactions. ■ INTRODUCTION Coulson and Longuet−Higgins 1 originally introduced the concept of bond−bond polarizability Π b;b′ in their classic molecular orbital (MO) treatment of π-electron systems. If π b = 2 −1/2 [p r + p s ] is a normalized π bond between p-orbitals on atoms r, s, and π b′ =2 −1/2 [p t + p u ] similarly between atoms t, u, the bond−bond polarizability can be written as ∑∑ ε ε Π =Π = + + − ′ cc cc cc cc 2 ( )( ) /( ) b;b rs;tu j(occ) k(vir) rj sk rk sj tj uk tk uj k j (1) where c rj and c rk , respectively, denote the LCAO coefficients of p r in an occupied (φ j ) or virtual (φ k ) π MO and ε j and ε k are the corresponding orbital energies. In the Hü ckel framework, Π b;b′ can be expressed as β β β Π =Π =∂ ∂ = ∂ ∂ ∂ =Π ′ P E / 1 2 / b;b rs;tu rs tu 2 rs tu tu;rs (2) where P rs is the r−s π bond order (off-diagonal density matrix element) and β tu the off-diagonal Hü ckel matrix element between p t and p u . Equation 2 identifies the physical significance of Π b;b′ , which measures how the r−s bond order is affected by changes in the off-diagonal Hü ckel matrix element β tu (due, e.g., to changes in t−u interatomic distance or other perturbations 2 ). Colloquially speaking, Π b;b′ predicts how loudly bond b “squeals” if bond b′ is “pinched”. Equations 1 and 2 were obtained with the usual Hü ckel π- electron orthogonality assumption, ⟨p r | p s ⟩ = δ rs . Chirgwin and Coulson 3 developed a generalized expression that includes overlap corrections, and McWeeny 4 discussed possible self- consistent generalizations for such Hü ckel-type formulas (cf. Greenwood and Hayward 5 ). In the intermediate neglect of differential overlap (INDO) approximation, Pople and Santry 6 developed well-known approximations for nuclear spin coupling constants that incorporate the related “atom−atom polarizability” index (Π rr;ss ). However, little use has been made of Π rs;tu in the framework of modern ab initio molecular orbital (MO) and density functional theory (DFT). The arguments leading to eqs 1, 2 can be applied virtually without modification in the framework of natural bond orbital (NBO) analysis of ab initio wave functions. 7 Each localized bond NBO b is expressed in terms of natural hybrid orbitals (NHOs) h r , h s and associated polarization coefficients a r , a s , = = + b b ah ah rs r r s s (3) that satisfy Hü ckel-like orthonormality relations δ ⟨ | ⟩= hh r s rs (4a) ⟨ |⟩= + = bb a a 1 r 2 s 2 (4b) In this case, the β rs parameter can be identified as the r−s Fock or Kohn−Sham matrix element between NHOs h r and h s β = =⟨ | | ⟩ hF h F () rs rs r op s (5) but the formulas are otherwise unmodified. Whereas the original Hü ckel treatment reflects only topological (graph- theoretic connectivity) aspects of carbon planar π networks, eq 5 incorporates the quantitative dependencies on geometry, electronegativity differences, and other chemically significant variables. Equations 1 and 2 can be applied to describe σ and π bond−bond polarizability in arbitrary molecular systems for which ab initio MO/DFT densities and corresponding NBOs have been obtained. We refer to the Π rs;tu defined by eqs 1−5 as Special Issue: Howard Zimmerman Memorial Issue Received: July 31, 2012 Published: August 22, 2012 Article pubs.acs.org/joc © 2012 American Chemical Society 1844 dx.doi.org/10.1021/jo301620k | J. Org. Chem. 2013, 78, 1844−1850