New Tools for Bonding Analysis

Nondynamical correlation (NDC, a.k.a., static correlation) refers to the situation where a single determinant is no longer adequate even as a zero-order description of the system. The familiar ‘gold standard’ CCSD(T) breaks down, DFT methods behave erratically, and a conventional MO picture is no longer adequate even for interpretation. High-order coupled-cluster (CC) methods are resilient but have very steep CPU time scalings, while multireference methods cannot be treated as ‘black boxes’. Hence it is very important to have diagnostics for the severity of this problem, so one knows whether the usual comp. chem. toolbox is adequate or one needs to go further. In a DFT context, we showed that the percent change in the DFT exchange energy between self-consistent DFT and HF densities was a good predictor for the importance of beyond-CCSD(T) correlation effects, albeit inferior to %TAE[(T)] for this purpose.
From a WFT perspective, one may extract information from the 1-particle reduced density matrix (1PDM), as done in the work of Matito and coworkers. For standard coupled cluster theory, 1PDM is nonhermitian at all levels. Stanton observed empirically that the asymmetry is greatest with ‘difficult’ molecules, and that it monotonically decays with improving CC level; this led us jointly to the DAD (density asymmetry diagnostic), which is unique in offering a direct gauge for residual deficiencies of the level of theory.
We propose4 several alternative diagnostics with this last property, but based on natural orbital occupations (i.e., eigenvalues of the 1DPM), which can be obtained for any correlated WFT, not just CC. The first, Δ𝐼𝑁𝐷[(T)], is the change in the Matito2 (I_ND ) ̅between CCSD and CCSD(T); the second, (I_ND^max [) ̅(T)], is more resilient to ‘dilution’ of small NDC-prone moieties by large chains or solvent clusters; while the third, rI[(T)]= Δ𝐼𝑁𝐷[(T)]/Δ𝐼T[(T)], appears to be a good energetic predictor as well (unlike traditional diagnostics like T1, D1, D2, M,…) Each of these can naturally be extended from CCSD(T) to higher-order methods, and asymptotically will vanish for an exact correlation treatment (i.e., full CI).

This contribution presents an Energy Decomposition Analysis (EDA) built from the Experimental Quantum Chemistry (EQC) framework. Departing from the original EQC partitioning, the EQC-EDA decomposes bond formation energies into orbital, charge-transfer, and electrostatic-reorganization terms, enabling more direct comparisons with conventional EDA methods. These terms can be further partitioned by fragment and into valence and core contributions, offering a complementary view of what drives chemical change. The decomposition is agnostic to the definition of atoms or fragments, and – unlike conventional EDA methods – its energy terms can in principle be estimated from experimental data. Electronegativity, defined as the average energy of valence electrons, is the interpretative backbone of EQC,3, 4 and we argue that it renders trends in some EQC-EDA terms qualitatively predictable from tabulated electronegativities and simple molecular-orbital arguments. After introducing the methodology, we show how an expression for the average electron energy within Density Functional Theory (DFT)5 provides a physically motivated constraint for rescaling Kohn–Sham energy levels. The resulting EQC-EDA, implemented in PySCF, is basis-set-insensitive and applicable across the periodic table. The approach cleanly distinguishes familiar bonding patterns such as ionic, covalent, and hydrogen bonding

E-mail address: luechow@pc.rwth-aachen.de
It is demonstrated that a topological analysis of the many-electron probability density |Ψ|2 (PDA) is useful for the analysis of chemical bonding. The stationary points of the probability density can be visualized in ordinary space. Additionally, basins, or attractor domains, are defined for the maxima of the probability density analogously to the 3D analysis of the electron density in the quantum theory of atoms in molecules by Bader and coworkers. The maxima of the probability density correspond to the locally most probable electron arrangements. They can be analyzed in many cases in terms of electron arrangements of the atoms. A local spin of the atom in the molecule can be defined easily and is discussed with respect to the covalent bond.
The saddle points of the probability density connect two basins of maxima and thus describe the electron exchange between atoms. A probability barrier is defined describing the electron delocalization via the saddle points.
The topological analysis of the probability density is based on the efficient sampling of |Ψ|2 with quantum Monte Carlo methods and is possible with arbitrary wave functions. The favorable scaling allows analysis of fairly large molecules.
New results regarding the local spin in molecules and their relation to chemical bonding are discussed as well the delocalization via probability barriers in aromatic molecules. Additionally, a comparison of first-row and second-row covalent bonding is presented from a probability density perspective.

Steric repulsion is one of the most frequently invoked concepts in chemistry, yet one of the least rigorously characterized: selectivity, conformational preferences, and reactivity are routinely rationalized by “steric clashes” depicted in a qualitative, often sketchy manner. The challenge is to convert its quantum-mechanical origin (the Pauli exclusion principle), a wavefunction-level constraint, into a real-space, chemically interpretable quantity revealing where exclusion operates and how much it costs. We introduce SELF (Steric Exclusion Localization Function), a new real-space descriptor that localizes and quantifies steric repulsion in three dimensions with atomic resolution, from a single routine quantum-mechanical calculation. SELF maps the local kinetic energy excess arising from same-spin exclusion between interacting electron distributions, delivering 3D isosurfaces, integrated repulsion scores (which correlate with EDA/SAPT exchange-repulsion energies), and per-atom decompositions. It applies equally to intermolecular contacts and, unlike most energy decomposition schemes, to intramolecular repulsion within a single structure, while naturally capturing the anisotropy of repulsion around atoms that distance-based steric models miss. Implemented in IGMPlot software, SELF adds a quantitative, energy-based measure of the steric repulsion that real-space indicators such as sign(λ₂)ρ identify but do not quantify energetically. Selected applications, from sterically controlled organocatalysis to π-hole interaction, will illustrate its scope. SELF bridges quantum theory and experimental practice, offering a practical tool for rational design, mechanistic analysis, and teaching.

Classical chemical concepts—such as the chemical bond, oxidation state, aromaticity, donor–acceptor interactions—are inherently heuristic. While they lack precise, mathematically defined formulations, they remain indispensable for describing, rationalizing, and predicting chemical behavior. This duality represents a fundamental challenge: bridging the conceptual language of chemistry with the rigorous formalism of quantum chemistry.

In this work, we discuss the use of effective fragment orbitals (EFOs) and their occupations as a powerful and versatile tool to address this gap. The EFOs generalize Mayer’s effective atomic orbitals (eff-AOs) to molecular fragments, providing a direct orbital-level representation of fragment-based electron density from the molecular wavefunction, without any additional fragment reference calculations. By analyzing the EFOs and their associated occupation numbers, we obtain a rigorous yet chemically intuitive picture of electronic structure.

We will show that key chemical concepts can be quantitatively extracted from this framework, including oxidation states via spin-resolved EFO populations, donor–acceptor interactions, inductive and mesomeric effects as distinct orbital interaction patterns induced by substituents, mechanistic rationalization of reaction pathways, and electronic structure features relevant to spin-crossover properties. By translating qualitative chemical concepts into well-defined orbital descriptors, EFOs offer a unified language that connects chemical intuition with first-principles quantum chemistry.

Valence bond (VB) theory, as one of modern chemical bonding theories, offers intuitive insights into various chemical problems. However, ab initio VB theory has long faced challenges in computational efficiency and accuracy due to the use of nonorthogonal orbitals. This talk will focus on the recent advances in algorithms and methods of ab initio VB theory. By implementing the low-rank algorithms, such as the resolution of the identity (RI) and the chain of spheres for exchange (COSX), the computational cost of valence bond self-consistent field (VBSCF) method is significantly reduced, and thereby enhancing its efficiency and extending its applicability. Furthermore, I will present post-VBSCF methods, including second-order valence bond perturbation theory (VBPT2) and density functional valence bond (DFVB). These methods incorporate dynamic correlation into VB calculations, substantially improving computational accuracy for strongly correlated systems.

As is well known, there are significant differences in the structures and energetics even of relatively small carbon compounds and their silicon analogues, as is to be expected from the so-called first-row anomaly. It is shown that spin-coupled generalized valence bond (SCGVB) theory, in combination with other approaches, provides useful insights into the nature and causes of such chemical bonding differences for Si and C. A limitation that arises in the quantitative analysis of hydridization is likely to be an issue also for other modern VB treatments.

Today, computational quantum chemistry calculations can provide near-quantitative answers to questions about the strength of covalent interactions, like chemical bonds, and non-covalent interactions like hydrogen bonds, with only modest compute requirements. This is a triumph of the quantum chemistry community, through development of state-of-the-art density functional theory (DFT) over the past 60 years.

Yet the results of those calculations are like numerical experiments. They tell you the value of the quantity: for instance that the water-water hydrogen bond is 20 kJ/mol, or the H-H bond in hydrogen is over 400 kJ/mol. They do not tell you the driving force that causes the interaction.

To fill this gap, it is desirable to have quantitative ways to associate contributions to binding with physical and chemical driving forces. This is the task of energy decomposition analysis (EDA), and although EDAs cannot be considered completely mature, they are increasingly used for applications such as understanding non-covalent interactions, chemical bonds, and even aspects of chemical catalysis and reactivity.
I will describe key elements that define the Absolutely Localized MO (ALMO) EDA, which attempts to decompose an interaction energy such as a bond or hydrogen bond into underlying physical driving forces, such as Pauli repulsions, spin-pairing, permanent and induced electrostatics, dispersion, and charge-transfer. I will discuss what this EDA reveals about the old yet and yet not fully resolved question of the driving force behind conventional covalent bonding, as well as some unconventional bonds.