Intrinsic bond orbitals

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.[1][4]

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.[4]

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.[4] 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.[1]

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 ${\displaystyle n_{A}(i')}$ of orbital ${\displaystyle i'}$ on atom ${\displaystyle A}$. The localization is performed in the spirit of the Pipek-Mezey localization scheme, maximizing a localization functional ${\displaystyle L}$.[1]

${\displaystyle L=\sum _{i}^{occ}\sum _{A}^{atom}[n_{A}(i')]^{p}}$

with ${\displaystyle p=4}$ or ${\displaystyle 2}$. While the choice of the exponent ${\displaystyle p}$ does not affect the resulting IBOs in most cases, the choice of ${\displaystyle p=4}$ localizes the orbitals in aromatic systems unlike ${\displaystyle p=2}$.[1]

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.

${\displaystyle |i'\rangle =\sum _{i}^{occ}|i\rangle U_{ii'}}$

The unitary matrix ${\displaystyle U_{ii'}}$, which produces the localized IBOs upon matrix multiplication with set of occupied MOs ${\displaystyle |i\rangle }$, 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.[1]

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.[1]

In 2015, Knizia and Klein introduced the analysis of electron flow in reactions with IBO as a non-empirical and straight-forward method of evaluating curly arrow mechanisms. Since IBOs are exact representations of Kohn-Sham wavefunctions, they can provide physical conformation for curly arrows mechanisms based on first-principles. IBOs usually represent chemical bonds and lone pairs, this method allows for elucidation of bond rearrangements in terms of the elemental steps and their sequence. By calculating the root mean square deviations of the partial charge distributions compared to the initial charge distribution, IBOs taking active part in a reaction can be distinguished from those that remain unchanged along the IRC.[2]

Knizia and Klein demonstrate the versatility of this method in their original report, first presenting a simple SN₂-type self-exchange reaction of H₃CCl and Cl^–, followed by the migration of π bonds in a substitution reaction (SN₂) and π- to σ-bond transformations in a Claisen rearrangement. Electron flow can be easily followed by observing the migration of an IBO and bond types are easily distinguished based on the geometries of the IBOs. The value of IBO analysis along the IRC[2] especially shows for complex reactions, such as a cyclopropanation reaction with only one transition state and without intermediates, reported by Haven et al.[16] Calculations by Knizia and Klein yielded a precise curly arrow mechanism for this reaction.[2]

• Localized molecular orbitals