Wiberg Bond Order

About

The Wiberg bond order (WBO) is a numerical value that quantifies the strength and nature of a bond between two atoms in a molecule. It is derived from Natural Bond Orbital (NBO) analysis, which is a computational method used to describe bonding in terms of localized atomic orbitals. The Wiberg bond order provides insight into the bonding character, such as whether a bond is single, double, triple, or if there is significant delocalization or partial bonding between atoms.

Key Aspects of Wiberg Bond Order:

1. Definition:

   • The Wiberg bond order is derived from the density matrix in the basis of atomic orbitals and is related to the amount of electron density shared between two atoms.

   • Mathematically, it is calculated from the elements of the density matrix ($P$) and the overlap matrix ($S$): $W_{AB} = \sum_{\mu \in A, \nu \in B} P_{\mu\nu} S_{\mu\nu}$ where is the density matrix element between atomic orbitals $\mu$ and $\nu$, and $S_{\mu\nu}$ is the overlap matrix element. The summation runs over atomic orbitals associated with atoms $A$ and $B$.

2. Interpretation:

   • A Wiberg bond order of approximately 1 suggests a single bond, around 2 suggests a double bond, and around 3 suggests a triple bond between two atoms.

   • Values less than 1 indicate weaker or partial bonds, such as those in delocalized systems or bonds between atoms with significant differences in electronegativity.

   • Non-integer values may also occur in molecules where electron density is delocalized, such as in conjugated systems (e.g., aromatic compounds like benzene).

3. Comparison to Other Bond Orders:

   • The Wiberg bond order is somewhat analogous to Mulliken bond order, but the Wiberg method tends to give more accurate and chemically meaningful results because it accounts for the overlap of atomic orbitals better.

   • It is also related to, but not the same as, bond orders derived from other methods such as Covalent bond indices or Bader charge analysis.

4. Applications:

   • Bond Strength: The Wiberg bond order gives insight into the strength of bonds. Higher bond orders typically correlate with stronger bonds.

   • Bond Length: There is often a correlation between bond order and bond length; bonds with higher Wiberg bond orders are usually shorter.

   • Resonance Structures: It can identify delocalization effects in molecules, where a bond order is fractional due to resonance (e.g., 1.5 in the case of bonds in aromatic rings like benzene).

   • Non-Covalent Interactions: Wiberg bond orders can also be used to assess non-covalent interactions, such as hydrogen bonding or van der Waals forces, though these typically have very low bond order values.

5. Limitations:

   • The value of the Wiberg bond order depends on the quality of the computational method and the basis set used. Different levels of theory can give slightly different bond orders.

   • It is not always applicable to very weak or non-covalent interactions because it focuses on covalent bonding descriptions.

Summary:

The Wiberg bond order is a powerful tool in computational chemistry that helps quantify the strength and character of bonds between atoms. By providing insights into the bonding patterns within molecules, it helps in understanding molecular structure, stability, and reactivity.

Method

The Fukui Indices are calculated at the GFN2-xTB level of theory with xTB 6.6.0

Find

The Wiberg Bond Order is found under the Bond section in the property tree.