4. Bond! Chemical Bond: Electronic Structure Methods at Work

This chapter plunges into applied quantum chemistry, with various examples, ranging from elementary notions, up to rather advanced tricks of know-how and non-routine procedures of control and analysis.

In the first section, the first-principles power of the ab initio techniques is illustrated by a simple example of geometry optimization, starting from random atoms, ending with a structure close to the experimental data, within various computational settings (HF, MP2, CCSD, DFT with different functionals). Besides assessing the performances of the different methods, in mutual respects and facing the experiment, we emphasize the fact that the experimental data are affected themselves by limitations, which should be judged with critical caution. The ab initio outputs offer inner consistency of datasets, sometimes superior to the available experimental information, in areas affected by instrumental margins. In general, the calculations can retrieve the experimental data only with semi-quantitative or qualitative accuracy, but this is yet sufficient for meaningful insight in underlying mechanisms, guidelines to the interpretation of experiment, and even predictive prospection in the quest of properties design.

The second section focuses on HF and DFT calculations on the water molecule example, revealing the relationship with ionization potentials, electronegativity, and chemical hardness (electrorigidity) and hinting at non-routine input controls, such as the fractional tuning of populations in DFT (with the ADF code) or orbital reordering trick in HF (with the GAMESS program).

Keeping the H₂O as play pool, the orbital shapes are discussed, first in the simple conjuncture of the Kohn–Sham outcome, followed by rather advanced technicalities in handling localized orbital bases, in a Valence Bond (VB) calculation, serving to extract a heuristic perspective on the hybridization scheme.

In a third section, the H₂ example forms the background for discussing the bond as spin-coupling phenomenology, constructing the Heisenberg-Dirac-van Vleck (HDvV) effective spin Hamiltonian. In continuation, other calculation procedures, such as Complete Active Space Self-Consistent Field (CASSCF) versus Broken-Symmetry (BS) approach, are illustrated, in a hands-on style, with specific input examples, interpreting the results in terms of the HDvV model parameters, mining for physical meaning in the depths of methodologies.

The final section presents the Valence Bond (VB) as a valuable paradigm, both as a calculation technique and as meaningful phenomenology. It is the right way to guide the calculations along the terms of customary chemical language, retrieving the directed bonds, hybrid orbitals, lone pairs, and Lewis structures, in standalone or resonating status. The VB calculations on the prototypic benzene example are put in clear relation with the larger frame of the CASSCF method, identifying the VB-type states in the full spectrum and equating them in an HDvV modeling. The exposition is closed with a tutorial showing nice graphic rules to write down a phenomenological VB modeling, in a given basis of resonance structures. The recall of VB concepts in the light of the modern computational scene carries both heuristic and methodological virtues, satisfying equally well the goals of didacticism or of exploratory research. A brief excursion is taken into the domain of molecular dynamics problems, emphasizing the virtues of the vibronic coupling paradigm (the account of mutual interaction of vibration modes of the nuclei with electron movement) in describing large classes of phenomena, from stereochemistry to reactivity. Particularly, the instability and metastability triggered in certain circumstances by the vibronic coupling determines phase transitions of technological interest, such as the information processing. The vibronic paradigm is a large frame including effects known as Jahn–Teller and pseudo Jahn–Teller type, determining distortion of molecules from formally higher possible symmetries. We show how the vibronic concepts can be adjusted to the actual computation methods, using the so-called Coupled Perturbed frames designed to perform derivatives of a self-consistent Hamiltonian, with respect to different parametric perturbations. The vibronic coupling can be regarded as interaction between spectral terms, e.g. ground state computed with a given method and excited states taken at the time dependent (TS) version of the chosen procedure. At the same time, the coupling can be equivalently and conveniently formulated as orbital promotions, proposing here the concept of vibronic orbitals, as tools of heuristic meaning and precise technical definition, in the course of a vibronic analysis. The vibronic perspective, performed on ab initio grounds, allows clear insight into hidden dynamic mechanisms. At the same time, the vibronic modeling can be qualitatively used to classify different phenomena, such as mixed valence. It can be proven also as a powerful model Hamiltonian strategy with the aim of accurate fitting of potential energy surfaces of different sorts, showing good interpolation and extrapolation features and a sound phenomenological meaning.

Finally, within the symmetry breaking chemical field theory, the intriguing electronegativity and chemical hardness density functional dependencies are here reversely considered by means of the anharmonic chemical field potential, so inducing the manifested density of chemical bond in the correct ontological order: from the quantum field/operators to observable/measurable chemical field.