Intrinsic bond orbitals (IBO) are
localized molecular orbitals Localized molecular orbitals are molecular orbitals which are concentrated in a limited spatial region of a molecule, such as a specific bond or lone pair on a specific atom. They can be used to relate molecular orbital calculations to simple bondin ...
giving exact and non-empirical representations of
wave functions
In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi, respec ...
. 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
The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier in 1937. Wannier functions are the localized molecular orbitals of crystalline systems.
The Wannier functions f ...
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
Density functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
(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
In linear algebra, orthogonalization is the process of finding a Set (mathematics), set of orthogonal vectors that span (linear algebra), span a particular linear subspace, subspace. Formally, starting with a linearly independent set of vectors ...
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
of orbital
on atom
. The localization is performed in the spirit of the
Pipek-Mezey localization scheme, maximizing a localization functional
.
:
with
or
. While the choice of the exponent
does not affect the resulting IBOs in most cases, the choice of
localizes the orbitals in aromatic systems unlike
.
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
In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two fermion ...
wavefunctions towards
unitary rotations.
:
The unitary matrix
, which produces the localized IBOs upon matrix multiplication with set of occupied MOs
, 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
In chemistry, pi bonds (π bonds) are covalent chemical bonds, in each of which two lobes of an orbital on one atom overlap with two lobes of an orbital on another atom, and in which this overlap occurs laterally. Each of these atomic orbitals ...
, but also aromatic systems and non-trivial bonds. The differentiation of σ and π bonds in
acrylic acid
Acrylic acid (IUPAC: prop-2-enoic acid) is an organic compound with the formula CH2=CHCOOH. It is the simplest unsaturated carboxylic acid, consisting of a vinyl group connected directly to a carboxylic acid terminus. This colorless liquid has ...
is possible based on IBO geometries, as are the identification of the IBOs corresponding to the oxygen lone pairs.
Benzene
Benzene is an Organic compound, organic chemical compound with the Chemical formula#Molecular formula, molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar hexagonal Ring (chemistry), ring with one hyd ...
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
A three-center two-electron (3c–2e) bond is an electron-deficient chemical bond where three atoms share two electrons. The combination of three atomic orbitals form three molecular orbitals: one bonding, one ''non''-bonding, and one ''anti''- ...
.

Transition metal compounds
IBO analysis was used to explain the stability of electron rich
gold-carbene complexes, mimicking reactive intermediates in gold catalysis. While these complexes are sometimes depicted with a Au-C double bond, representing the sigma donation of the carbene and π backbonding of Au, IBO analysis points towards a minimal amount of π-backbonding with the respective orbital mainly localized on Au. The σ-donating carbene orbital is likewise strongly polarized towards C. Stabilization of the compound thus occurs through strong donation of the aromatic carbene substituents into the carbene carbon p-orbitals, outcompeting the Au-π-backdonation. IBO analysis was thus able to negate the double bond character of the gold-carbene complexes and provided deep insight into the electronic structure of Cy
3P-Au-C(4-OMe-C
6H
4)
2) (Cy = cyclohexyl).
The π-backbonding character was again evaluated for gold-
vinylidene
Methylidenecarbene (systematically named λ2-ethene and dihydrido-1κ2''H''-dicarbon(''C''—''C'')) is an organic compound with the chemical formula (also written [] or ). It is a metastability, metastable proton tautomer of acetylene, which on ...
complexes, as another common type of gold catalysis intermediates. IBO analysis revealed significantly stronger π-backbonding for the gold-vinylidenes compared to the gold-carbenes. This was attributed to the geometric inability of aromatic vinylidene substituents to compete with Au for π-interactions since the respective orbitals are perpendicular to each other.
Knizia and Klein similarly employed IBO for the analysis of
3NO">e(CO)3NOsup>–. The even polarization of IBOs between Fe and N points towards a covalently bonded NO ligand. The double bond occurs via two d-p π-interactions and results in a formal Fe
0 center. Confirmed by further calculations, IBO proved as a fast and straightforward method to interpret bonding in this case. Making use of the low computational cost, a
Cloke-Wilson rearrangement catalyzed by
3NO">e(CO)3NOsup>– was investigated by constructing the IBOs for every stationary point along the IRC. It was found that one of the Fe-NO π bonds takes active part in catalysis by electron transfer to and from the substrate, explaining the unique catalytic activity of
3NO">e(CO)3NOsup>– compared to the isoelectronic
4">e(CO)4sup>2–.
Apart from the above mentioned compounds, the IBO method has been employed to investigate various other transition metal complexes, such as gold-diarylallenylidenes
or diplatinum diboranyl complexes, proving as a valuable tool to gain insight into the extent and nature of bonding.
Main group compounds

IBO analysis has been employed in main group chemistry to elucidate oftentimes non-trivial electronic structure. The bonding of phosphaaluminirenes was, for example, investigated showing a 3-center-2π-electron bond of the AlCP cycle. Further application was found for confirming the
distonic nature of a phosphorus containing radical cation reported by Chen et al. (see figure). While the IAO charge analysis yielded a positive charge on the chelated P, IBOs showed the localization of the unpaired electron on the other P atom, confirming the spatial separation of
radical site and charge.
Another example is the elucidation of the electronic structure of the hexamethylbenzene dication. Three π bonding IBOs were found between the basal C
5Me
5 plane and the apical C, reminiscent of
Cp* coordination complexes. The three π bonds are thereby polarized towards the apical C, which in turn coordinates to a CH
3+ cation with its lone pair. IBO analysis therefore revealed the
Lewis-acidic and Lewis-basic character of the apical C.

Applications of IBO for cluster compounds have included zirconium doped boron clusters. IBO analysis showed, that the unusual stability of the neutral ZrB
12 cluster stems from several multicenter σ bonds. The B-B σ bonding orbitals extend to the central Zr atom, forming the mulicenter bonds. This example displays the method's aptitude to analyze cluster compounds and multicenter bonding.
Valence virtual intrinsic bond orbitals
Although IBOs typically describe occupied orbitals, the description of unoccupied orbitals can likewise be of value for interpreting chemical interactions. Valence virtual IBOs (vvIBOs) were introduced with the investigation of high valent formal Ni(IV) complexes. The bonding and antibonding manifold of the compound were described using IBOs and vvIBOs respectively. Compared to the widely used
HOMO/LUMOs which are often spread over the whole molecule and can be difficult to interpret, vvIBOs allow for more direct interpretation of chemical interactions with unoccupied orbitals.
Electron flow along the IRC
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.
Knizia and Klein demonstrate the versatility of this method in their original report, first presenting a simple
SN2-type self-exchange reaction of H
3CCl and Cl
–, followed by the migration of π bonds in a substitution reaction (S
N2) and π- to σ-bond transformations in a
Claisen rearrangement
The Claisen rearrangement is a powerful carbon–carbon chemical bond, bond-forming chemical reaction discovered by Rainer Ludwig Claisen. The heating of an allyl Vinyl group, vinyl ether will initiate a Sigmatropic reaction, ,3sigmatropic r ...
. 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
especially shows for complex reactions, such as a
cyclopropanation
In organic chemistry, cyclopropanation refers to any chemical process which generates cyclopropane () Ring (chemistry), rings. It is an important process in modern chemistry as many useful compounds bear this motif; for example pyrethroid insectic ...
reaction with only one transition state and without intermediates, reported by Haven et al. Calculations by Knizia and Klein yielded a precise curly arrow mechanism for this reaction.
Closed-shell systems
Examples of IBO analysis along the IRC included the investigation of C-H bond activation by gold-vinylidene complexes. Through this method, it is possible to discern between concerted and stepwise reactions. The previously thought single step C-H activation reaction was in this case revealed to consist of three distinct phases: i) hydride transfer, ii) C-C bond formation and iii) sigma to pi rearrangement of the lone pair coordinated to Au.

Other reports of IBO analysis along the IRC include the elucidation and confirmation of a previously proposed mechanism for a
,3 sigmatropic rearrangement of a Au(I)-vinyl species or the epoxidation of alkene by
peracids.
For the latter, the textbook four curly arrow mechanism was found to be physically inaccurate. Instead, seven changing IBOs were found, yielding an ideal mechanism featuring seven curly arrows.
The combination of IBO analysis with other computational methods, such as
natural bond orbital (NBO) analysis for a Ti-catalyzed pyrrole synthesis or natural localized molecular orbital (NLMO) analysis for an intramolecular
cycloaddition
In organic chemistry, a cycloaddition is a chemical reaction in which "two or more Unsaturated hydrocarbon, unsaturated molecules (or parts of the same molecule) combine with the formation of a cyclic adduct in which there is a net reduction of th ...
of a phosphaalkene to an arene has likewise led to insightful results regarding the specifics of the reaction mechanisms.

Open-shell systems
Klein and Knizia furthermore introduced the first examples of IBOs used for analysis of
open-shell systems during
proton-coupled electron transfer (PCET) and
hydrogen atom transfer (HAT). The differentiation between pCET, a separate but concerted electron and proton transfer, and HAT, the transfer of a hydrogen atom, were shown for two well-studied model systems of enzymatic Fe-oxo active sites. IBOs along the IRC were calculated for the
alpha and beta spin manifold respectively. While the IBO of the alpha spin electron travelled together with the proton to take part in the formation of a new H-O bond in case of HAT, the electron was transferred to the Fe-center separated from the transferred proton for PCET. The successful application of the IBO method for these two examples of open-shell systems was suggested to pave the way for broader applications to similar problems.
[{{Cite journal, last1=Klein, first1=Johannes E. M. N., last2=Knizia, first2=Gerald, date=2018-09-10, title=cPCET versus HAT: A Direct Theoretical Method for Distinguishing X-H Bond-Activation Mechanisms, journal=Angewandte Chemie, language=en, volume=130, issue=37, pages=12089–12093, doi=10.1002/ange.201805511, bibcode=2018AngCh.13012089K , doi-access=free, pmc=6175160]
See also
*
Localized molecular orbitals Localized molecular orbitals are molecular orbitals which are concentrated in a limited spatial region of a molecule, such as a specific bond or lone pair on a specific atom. They can be used to relate molecular orbital calculations to simple bondin ...
References
Electronic structure methods