Molecular orbital localization and geodesic paths between resonance structures
Molecular orbital theory provides a rigorous quantum-mechanical description of electronic structure, but its most common form — using delocalized canonical orbitals — often appears far removed from the intuitive bonding pictures used in chemistry. Concepts such as single bonds, lone pairs, and resonance structures arise from localized electron pairs, while canonical molecular orbitals are typically spread over the entire molecule. Bridging this gap between formal theory and chemical intuition is one of the key motivations for orbital localization methods.
The Foster-Boys localization procedure is one of the most widely used approaches to obtain spatially compact molecular orbitals. By applying a unitary transformation to the canonical orbitals, it produces localized orbitals that resemble Lewis-type bonding patterns without altering the underlying wavefunction or any physical observables. Interestingly, for molecules that can be described by multiple resonance structures, the localization procedure may yield different but equally valid sets of localized orbitals, each corresponding to a distinct bonding picture.
This situation raises a deeper theoretical question: if these orbital sets describe the same electronic state, how are they connected mathematically? Since orbital representations of a fixed wavefunction are related by unitary transformations, one can view different localized solutions as points in a space of possible orbital bases. Exploring the shortest path — a geodesic — between such representations provides a geometric perspective on orbital transformations and offers new insight into the relationship between resonance concepts and quantum-mechanical descriptions.
This project therefore combines electronic structure theory, chemical bonding concepts, and elements of linear algebra and geometry. By studying orbital localization and the transformations between alternative localized solutions, you will connect abstract mathematical structures with familiar chemical ideas in a rigorous and exploratory way.
Figure 1: (left) Delocalized molecular orbitals of benzene. (right) Localized molecular orbital of benzene after Foster-Boys localization procedure. This is one of the resonance structures of benzene.
Project outline
- Perform an electronic structure calculation and obtain canonical molecular orbitals.
- Apply the Foster-Boys localization procedure and analyze the spatial character of the localized orbitals.
- Identify molecules with multiple resonance-like bonding patterns and obtain alternative localized orbital solutions.
- Express the transformation between two localized orbital sets as a unitary matrix.
- Investigate the concept of the shortest (geodesic) path between these orbital representations and discuss its mathematical and chemical interpretation.
Expected outcome
A report describing the localization procedure, visualizations and interpretations of the localized orbitals, and a comparison of different resonance-like solutions. The report should include a discussion of unitary transformations between orbital sets and a qualitative or quantitative analysis of the shortest transformation path, highlighting both the mathematical structure and the limits of interpreting this path as a physical process.