Applications of Topological Methods in Molecular Chemistry

Quantum Chemical Topology (QCT) is a branch of theoretical chemistry that uses the language of dynamical systems (e.g. attractor, basin, homeomorphism, gradient path/phase curve, separatrix, critical points) to partition chemical systems and characterise them via associated quantitative properties. This methodology can be applied to a variety of quantum mechanical functions, the oldest and most documented one being the electron density. We define and discuss the topological atom, and justify the name topology. Then we define the quantum atom without reference to the topological atom. Subsequently, it turns out that each topological atom is a quantum atom, a property that enables the construction of a topologically inspired force field called QCTFF. We briefly discuss the four primary energy contributions governing this force field under development, and how the machine learning method kriging captures the variation in these energies due to geometrical change. Finally, in a more philosophical style, we advocate falsification in the area of chemical interpretation by means of quantum mechanical tools, introducing the concept of a non-question.

Chemical graph theory (CGT) starts by defining matrices that represent the molecular graph then proceed to extract numbering-independent matrix invariants to be used as molecular descriptors in empirical quantitative structure to activity (or property) relationships (QSAR/QSPR). Two proposed matrix representations of molecular structure are presented in this chapter as alternatives to simple connectivity molecular graphs. Firstly, it is proposed to use a more “nuanced” connectivity matrix by weighing the “ones” entered in a CGT molecular graph matrix by the bond critical point electron densities associated with each bond path to yield what we term the “electron density-weighted adjacency/connectivity matrices (EDWAM/EDWCM)”. In a second approach, it is proposed to use the localization and delocalization indices of the quantum theory of atoms in molecules (QTAIM) to construct a richer representation of the molecular graph, a “fuzzy” graph, whereby an edge exists between any two atoms (measured by the delocalization index between them) whether they share a bond path or not. Such a fuzzy graph is represented by what we term “electron localization-delocalization matrix (LDM)”. We show that the LDM representations of a series of molecules provide a powerful tool for robust QSAR/QSPR modeling.

It is customary to conceive the interactions of all the constituents of a molecular system, i.e. electrons and nuclei, as Coulombic. However, in a more detailed analysis one may always find small but non-negligible non-Coulombic interactions in molecular systems originating from the finite size of nuclei, magnetic interactions, etc. While such small modifications of the Coulombic interactions do not seem to alter the nature of a molecular system in real world seriously, they are a serious obstacle for quantum chemical theories and methodologies which their formalism is strictly confined to the Coulombic interactions. Although the quantum theory of atoms in molecules (QTAIM) has been formulated originally for the Coulombic systems, some recent studies have demonstrated that most of its theoretical ingredients are not sensitive to the explicit form of the potential energy operator. However, the Coulombic interactions have been explicitly assumed in the mathematical procedure that is used to introduce the basin energy of an atom in a molecule. In this study it is demonstrated that the mathematical procedure may be extended to encompass the set of the homogeneous potential energy functions thus relegating adherence to the Coulombic interactions to introduce the energy of a real-space subsystem. On the other hand, this extension opens the door for seeking novel real-space subsystems, apart from atoms in molecules, in non-Coulombic systems. These novel real-space subsystems, quite different from the atoms in molecules, call for an extended formalism that goes beyond the orthodox QTAIM. Accordingly, based on a previous proposal the new formalism, which is not confined to the Coulombic systems nor to the atoms in molecules as the sole real-space subsystems, is termed the quantum theory of proper open subsystems (QTPOS) and its potential applications are detailed. The harmonic trap model, containing non-interacting fermions or bosons, is considered as an example for the QTPOS analysis. The QTPOS analysis of the bosonic systems is particularly quite unprecedented not attempted before.

Several potentially useful scalar and vector fields that have been scarcely or even never used to date in Quantum Chemical Topology are defined, computed, and analyzed for a few small molecules. The fields include the Ehrenfest force derived from the second order density matrix, which does not show many of the spurious features encountered when it is computed from the electronic stress tensor, the exchange-correlation (xc) potential, the potential acting on one electron in a molecule, and the additive and effective energy densities. The basic features of the topology of some of these fields are also explored and discussed, paying attention to their possible future interest.

It is shown that the quantum mechanical theory of static magnetic properties can be reformulated in terms of electronic current densities induced by an external magnetic field and permanent magnetic dipole moments at the nuclei. Theoretical relationships are reported to evaluate magnetizability, nuclear magnetic shielding and nuclear spin-spin coupling via the equations of classical electromagnetism, assuming that the current density is evaluated by quantum mechanical methods. Emphasis is placed on the advantage of the proposed formulation, as an alternative to procedures based on perturbation theory, as regards interpretation of response allowing for the ideas of current density tensor and current susceptibility vector. Visualisation of the electronic interaction with a magnetic field and intramolecular perturbations, e.g., nuclear magnetic dipoles, is made possible via current density maps, nuclear shielding density maps and plots of nuclear spin-spin coupling density. Topological analysis of the quantum mechanical current density in terms of Gomes stagnation graphs is shown to yield fundamental information for understanding magnetic response. Examples are given for a few archetypal molecules. A topological definition of delocalized electron currents is proposed.

In this work, the Fukui function will be analyzed using the framework of the topological analysis. First, the Fukui function will be introduced as part of the Density Functional Theory of Chemical Reactivity, and its chemical interpretation will be discussed. Then, some applications showing the importance of the topological analysis will be presented. The applications cover from acids and basis of Lewis, substituted benzenes and as an orientation predictor for the most favorable interaction between clusters (used as building blocks) to form larger structures.

Two types of the main topological properties of potential energy surfaces are compared, where the first types are related to the chemical processes, conformational changes and chemical reactions along the potential energy surface, and where the second types are describing the presence, interrelations, structural variability, and shape variations of identifiable chemical species associated with the potential surface. Some new relations are obtained when the families of topologically equivalent reaction paths representing reaction mechanisms at some energy bound, and the algebraic structure of the fundamental group of reaction mechanisms for a given collection of atoms (that is, for a given stoichiometry) are constrained by the collection of “catchment regions” of the potential surface, representing chemical species. These relations, providing additional detail when they are compared to the more traditional, unconstrained cases, are phrased in terms of potential energy surface level set relations and the originally integer, but “unquantized” continuous variables of the Universal Molecule model.

Chemical structure and bonding are key features and concepts in chemical systems which are used in deriving structure–property relationships, and hence in predicting physical and chemical properties of compounds. Even though the contemporary high standards in determination using theoretical methods and experimental techniques, questions of chemical bonds as well as their evolution along a reaction pathway are still highly controversial. We present a conceptionally approach to dissect chemical structure and reactivity (bond formation and breaking processes) in the nucleation and formation of Ag on AgVO₃ provoked in this crystal by the electron-beam irradiation, and glycolic acid decomposition using concepts from quantum chemical topology. The electronic activity that drives the structure and the molecular mechanism of the reaction was identified, fully characterized, and associated with specific chemical events, bond forming/breaking processes.

Various Kekulé–structure–based models, aimed at describing π-electron conjugation in polycyclic aromatic compounds are briefly described. Our main concern are benzenoid hydrocarbons, π-electron systems in which the Kekulé–structure–based approaches are expected to yield the best results. Although there are numerous examples in which reasonings based on Kekulé structures render correct results, there exist cases in which significant violations are encountered. Perylene, its homologs, and derivatives are characteristic representatives of such “anomalous” conjugated systems. Violations from the predictions of the Kekulé–structure–based models are verified by means of a variety of Kekulé–structure–independent theoretical methods.

The concept of aromaticity is elusive; it is not directly observable. Somewhat surprisingly, given the fuzzy character of this concept, there exist a number of very simple mathematical rules that can account for the aromaticity of a large number of organic and inorganic molecules. Among them we can mention Hückel’s, Baird’s, Wade-Mingos’, and Hirsch’s rules. In this chapter we summarize recent advances carried out in our group in the study of these aromaticity rules.

A simple Hückel Hamiltonian is used and modified to describe localized states, where the electron pairs are confined to bonds between two atoms, or to lone pairs. The electronic delocalization can be considered either as a mixture of these localized states, or through a standard Hückel calculation. The two Hückel-Lewis methods described here attempt to find the coefficients of the mixture, based on energy or overlap consistence with the standard Hückel results. After the description of the two methods, test examples are used to show advantages and drawbacks of the different approaches. In any case, the results are compared to the NBO-NRT approach which is used on the electronic density obtained from standard DFT hybrids calculations such as B3LYP/6-31+G(d). This chapter ends with an introduction to the HuLiS program in which the two methods are implemented.

The present chapter shows first that the topological Hückel Hamiltonian provides an analytical expression of both the singly occupied Molecular Orbitals and the spin density distribution of mono- and poly-radical conjugated hydrocarbons. It permits a new derivation of the Ovchinnikov’s rule (first established from a magnetic model Hamiltonian), which predicts the preferred ground state spin multiplicity from the topology of the molecule. From the Hubbard simplified representation of the bi-electronic Hamiltonian one obtains directly, without any matrix diagonalization, a reasonable evaluation of the singlet-triplet energy difference. For singlet di-radicals the method enables one to predict whether the Ms = 0 single-determinant solution is subject to a spin-symmetry breaking. The spin polarization of the closed shells, which is a different phenomenon, of bi-electronic origin, increases the value of the magnetic coupling in these systems, contrasts the spin densities between negative and positive values and spatially extends the spin distribution. Numerical Unrestricted Density Functional Theory calculations illustrate the relevance of the predictions of the topological model.

A bond path being a line of maximum electron density linking attractors of two atoms is often applied in various studies as a criterion of the existence of numerous interactions such as for example hydrogen, halogen or pnicogen bond. It covers cases of atom-atom energetically stabilized links, from weak van der Waals interactions, through stronger Lewis acid–Lewis base interactions up to covalent bonds. The location of bond paths also allows interpreting mechanisms of interactions and, in general, of chemical reactions. The Quantum Theory of Atoms in Molecules (QTAIM) results are mainly presented here; however they are supported by other approaches as, for example, the Natural Bond Orbitals (NBO) method or the σ-hole concept. The most important orbital-orbital interactions determined from the NBO method and characterizing different types of interactions are presented. The analysis of the distribution of the electron charge density is also performed here for numerous systems; this is shown that the regions of the concentration and depletion of the electron density coincide with the regions of the negative and positive regions of the electrostatic potential. The role of the analysis of the laplacian of the electron density is shown on the basis of numerous interactions.

In this chapter. we will show how Bader’s atoms-in-molecules theory enables to unravel the main physicochemical factors that drive the formation of halogen bonds, which are intriguing and fascinating noncovalent interactions at work as well as in crystals, biological and chemical systems, and which have found numerous applications in, among other fields, drug design and supramolecular chemistry. In particular, the use of Pendás and coworkers’ interacting quantum atoms scheme will cast the light on the nature of such interactions (more or less electrostatic, more or less covalent) and will provide useful hints to account for the existence or absence of energy minima in the corresponding potential energy surface. Importantly, such a rationalizing approach can be carried out whatever the system and also possesses predictive power.

The main characteristics of beryllium bonds formed by the interaction of different Lewis bases with BeX₂ (X = H, F) moieties have been analyzed by means of the Charge Displacement (CD) function. This analysis is systematically compared with that provided by other approaches based on the topology of the electron density, namely the quantum theory of atoms in molecules (QTAIM) and the electron localization function (ELF). The CD scheme provides a quantitative description of the charge transfer that gives rise to the formation of beryllium bonds. For systems of suitable symmetry, its decomposition into symmetry contributions allows to easily identify the mechanisms involved in the charge transfer process, as well as to quantify possible back-donations. The CD function analysis also provides a clear quantitative description of cooperativity between the beryllium and halogen bonds in ternary F₂Be:FCl:N-base (N-base = NH₃, NHCH₂, NCH) complexes, confirming the trends obtained by the QTAIM and ELF methods. The different viewpoints each of these methodologies provide are clearly complementary, the CD being the only one that permits to quantify the charge transfer from the Lewis base to the Lewis acid.

The electronic structure of nitrite group (–ONO) has been studied for 21 inorganic and organic nitrites using topological analysis of Electron Localisation Function (ELF) for the DFT(B2PLYP)/aug-cc-pVTZ and DFT(B3LYP)/aug-cc-pVTZ optimised geometrical structures. The N–O bonds exhibit populations smaller than 2e, thus including the N⁺O⁻, N⁻O⁺ Lewis-type structures in the description of electron density delocalisation is of great importance. The main focus of the ELF analysis was formally single N–O bond in the nitrite group (–O–NO). The results have yielded four different types of local topology: (a) single local maximum V(N,O) with the disynaptic bonding basin, (b) two local maxima V(N), V(O) with monosynaptic non-bonding basins, (c) single local maximum V(N) with monosynaptic non-bonding basin, (d) absence of the local maxima in the N–O bond. Analysis of relationships between basin population values, calculated for the V(N,O), V(N) and V(O) basins, and the N–O bond length, has shown overall trends that can be qualitatively described by the catastrophe theory.

This chapter aims to present QTAIM and ELF topological analyzes in the framework of two-component relativistic computations. Attention is focused on spin-orbit coupling (SOC) effects on the chemical bond in systems containing heavy atoms. The emblematic At₂ and uranyl species have been studied as a relevant test set. The presented methodology appears particularly suitable for evidencing relativistic effects on bonding schemes. The influence of SOC was found to depend, not only of the involved heavy atoms, but also of the bond nature. Furthermore, the robustness of QTAIM and ELF for analyzing wave functions built from spinors has been verified.