Intrinsic bond orbitals

Intrinsic bond orbitals (IBO) are localized molecular orbitals giving exact and non-empirical representations of wave functions. They are obtained by unitary transformation and form an orthogonal set of orbitals localized on a minimal number of atoms. IBOs present an intuitive and unbiased interpretation of chemical bonding with naturally arising Lewis structures. For this reason IBOs have been successfully employed for the elucidation of molecular structures and electron flow along the intrinsic reaction coordinate (IRC). IBOs have also found application as Wannier functions in the study of solids.

Theory The IBO method entails molecular wave-functions calculated using self-consistent field (SCF) methods such as Kohn-Sham density functional theory (DFT) which are expressed as linear combinations of localized molecular orbitals. In order to arrive at IBOs, intrinsic atomic orbitals (IAOs) are first calculated as representations of a molecular wave function for which each IAO can be assigned to a specific atom. This allows for a chemically intuitive orbital picture as opposed to the commonly used large and diffuse basis sets for the construction of more complex molecular wavefunctions. IAOs are constructed from tabulated free-atom AOs of standard basis-sets under consideration of the molecular environment. This yields polarized atomic orbitals that resemble the free-atom AOs as much as possible, before orthonormalization of the polarized AOs results in the set of IAOs. IAOs are thus a minimal basis for a given molecule in which atomic contributions can be distinctly assigned. The sum of all IAOs spans exactly over the molecular orbitals which renders them an exact representation of the wavefunction. Since IAOs are associated with a specific atom, they can provide atom specific properties such as the partial charge. Compared to other charges, such as the Mulliken charge, the IAO charges are independent of the employed basis set. IBOs are constructed as a linear combination over IAOs with the condition of minimizing the number of atoms over which the orbital charge is spread. Each IBO can thereby be divided into the contributions of the atoms as the electronic occupation n A ( i ′ ) {\displaystyle n_{A}(i')} of orbital i ′ {\displaystyle i'} on atom A {\displaystyle A} . The localization is performed in the spirit of the Pipek-Mezey localization scheme, maximizing a localization functional L {\displaystyle L} .

L = ∑ i o c c ∑ A a t o m [ n A ( i ′ ) ] p {\displaystyle L=\sum _{i}^{occ}\sum _{A}^{atom}[n_{A}(i')]^{p}}

with p = 4 {\displaystyle p=4} or 2 {\displaystyle 2} . While the choice of the exponent p {\displaystyle p} does not affect the resulting IBOs in most cases, the choice of p = 4 {\displaystyle p=4} localizes the orbitals in aromatic systems unlike p = 2 {\displaystyle p=2} . The process of IBO construction is performed by unitary tranfomation of canonical MOs, which ensures that the IBOs remain an exact and physically accurate representation of the molecular wavefunction due to the invariance of Slater determinant wavefunctions towards unitary rotations.

| i ′ ⟩ = ∑ i o c c | i ⟩ U i i ′ {\displaystyle |i'\rangle =\sum _{i}^{occ}|i\rangle U_{ii'}}

The unitary matrix U i i ′ {\displaystyle U_{ii'}} , which produces the localized IBOs upon matrix multiplication with set of occupied MOs | i ⟩ {\displaystyle |i\rangle } , is thereby chosen to effectively minimize spread of IBOs over the atoms of a molecule. The product is a set of localized IBOs, closely resembling the chemically intuitive shapes of molecular orbitals, allowing for distinction of bond types, atomic contributions and polarization.

Application in structure and bonding In his original paper introducing IBOs, Knizia showed the versatility of his method for describing not only classical bonding situations, such as the σ and π bond, but also aromatic systems and non-trivial bonds. The differentiation of σ and π bonds in acrylic acid is possible based on IBO geometries, as are the identification of the IBOs corresponding to the oxygen lone pairs. Benzene provided an example of a delocalized aromatic system to test the IBO method. Apart from the C-C and C-H σ-bonds, the six electron π-system is expressed as three delocalized IBOs. Representation of non-Lewis bonding was demonstrated on diborane B2H6, with one IBO stretching over B-H-B, corresponding to the 3-center-2-electron bond.

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