Bonding Analysis and Reactivity

Variational Hirshfeld partitioning is a powerful framework for constructing approximate models of the electronic density of molecules and materials. While its attractive features have been derived from its mathematical definition,1 this talk explores the reverse direction: what is the minimal set of properties, or axioms, that uniquely define the variational Hirshfeld framework?

We show that four intuitive axioms are sufficient: locality, additivity, coordinate invariance, and normalization. Beyond clarifying the mathematical foundation of variational Hirshfeld partitioning, these axioms carry two significant implications. First, within a broad family of partitioning schemes, every alternative method must violate at least one of these four basic axioms. Second, the axioms constrain the set of admissible model densities for atomic fragments. These results establish theoretical guardrails for density partitioning and offer clear guidance for designing future atoms-in-molecules methods.

A fundamental understanding of chemical bonding through quantum chemistry is a cornerstone of the rational design and development of modern catalytic systems. Advanced quantum-mechanical techniques and bonding descriptors provide a rigorous framework for elucidating the electronic rearrangements, orbital interactions, and potential-energy surfaces that govern transition states and overall reactivity. In particular, the integration of state-of-the-art computational methods, such as density functional theory (DFT) and emerging foundation-level machine-learning potentials (MLPs), offers unprecedented opportunities to map complex reaction spaces and accelerate the discovery of efficient and sustainable catalysts.

In homogeneous catalysis, quantum chemical topology (QCT), including the electron localization function and bonding evolution theory, has successfully bridged the gap between traditional electron-pushing formalisms and energy-based descriptors. This is illustrated, for example, by metal–ligand cooperation in challenging processes such as N–H bond activation by ruthenium(II) pincer complexes.

Such electronic-structure insight is also instrumental in the design of catalytic systems based on earth-abundant transition metals. We have developed hemilability-controlled manganese and cobalt complexes supported by triazole-based P–N ligands that exhibit high efficiency in dehydrogenative Si–O bond-forming reactions. These transformations proceed through non-classical mechanisms involving frustrated Lewis pairs (FLPs), whose design was guided by the analysis of chemical-bonding indices.

Nuclear quantum effects (NQEs) are relevant to a wide range of chemical phenomena, including photochemistry, proton-coupled electron-transfer reactions, and the description of common intermolecular hydrogen bonds. The latter may require a quantum-mechanical treatment of atomic nuclei, particularly protons. Several approaches have been developed for this purpose, many of which involve extensive potential-energy-surface calculations followed by quantum or semiclassical dynamics. However, the computational cost of these methods scales unfavourably with system size and with the number of nuclei treated quantum mechanically.

Multicomponent methods provide an alternative by using wavefunction theory or density functional theory (DFT) to treat selected nuclei and electrons quantum mechanically and on an equal footing. Among these approaches, notable progress has been made within the nuclear–electronic orbital (NEO) formalism. The NEO framework offers a promising route for systematically incorporating NQEs, although its practical application has traditionally been limited by the substantial computational cost of existing algorithms.

Considerable effort has therefore been devoted to developing scalable implementations, achieving speed-ups of one order of magnitude or more with almost no loss of accuracy. Moreover, NEO provides a unique framework for quantifying the influence of NQEs on the electronic density and on associated chemical-bonding descriptors, with potential applications ranging from studies of chemical reactivity to the training of machine-learning models.

Here, we present recent developments in the NEO framework within our group, currently available in Molpro. Particular emphasis will be placed on evaluating how these methods describe the influence of NQEs on intra- and intermolecular interactions. Overall, our aim is to make multicomponent simulations more efficient and accessible, both in terms of computational cost and the chemical information that can be extracted from them, thereby extending the day-to-day quantum-chemistry toolbox.

The use of electronic-structure theory to obtain total molecular properties has been standard practice for several decades. As atoms form the central building blocks of all chemistry, considerable interest has also been devoted to decomposing the total molecular energy into atomic contributions. Yet, molecular properties and chemical reactivity are not determined by atoms alone: molecules only materialize when their constituent atoms interact by forming bonds, and chemical reactions are driven by bond breaking and formation. Knowledge of the contributions of individual bonds to such properties could therefore provide insight into the influence of their local environments and help rationalize experimental observations.
We present our recent efforts to develop new bond-centered decomposition schemes aimed at decomposing any arbitrary molecular property into local contributions associated with individual bonds. Starting from Eriksen’s atom-centered decomposition scheme, which relies on intrinsic atomic orbitals and localized molecular orbitals, the atomic contributions are redistributed into a quantum-chemical atom- and atom-pair-based decomposition. These contributions are subsequently further decomposed into conceptual bond-based components using an atom-to-bond mapping based on bond orders. Multiple approaches are explored and assessed through their application to a series of compounds containing a variety of functional groups.