Revision of Quantum Chemistry Valence Formulas based on Density Matrix Analysis

Abstract

Valence is a fundamental yet ambiguously defined concept in chemistry, playing a central role in the description of molecular structure, bonding, and reactivity. This thesis examines the quantum mechanical Veryazov valence formula (referred to as the original formula), which determines atomic valence based on density matrix analysis. The aim of this work is to assess the robustness of this valence formula with respect to different computational methods and basis sets and to investigate the consequences of modifications of the original formula.
As a first step, the original formula was applied to a diverse database comprising molecular and crystalline systems. This analysis revealed that the behavior of the formula depends strongly on the system.
For a first group of molecules, consisting mainly of small and chemically simple systems, the valence values obtained with the original formula agree with expected valence values and show only minor sensitivity to methods and basis sets. For these systems, valence can be regarded as a well-defined and stable property within the framework of the original formula. This group therefore serves as a reference set for further analysis.
In contrast, for a second group of larger and chemically more complex systems, including heavier elements, hypervalent species, ionic compounds, and transition-metal molecules, the valence values obtained show a more pronounced dependence on the method and basis sets. In these cases, valence emerges as a less well-defined property. Expected valence values for this second group are therefore used for comparative analysis rather than as reference data.
Further investigations were focused primarily on the reference group of molecules with well-defined valence. Based on this reference set, four approaches were undertaken in order to modify the original formula and evaluate the resulting consequences on the valence: (i) variations of the ionic and covalent contributions using alternative charge analyses and modified weighting functions in the covalent term, (ii) fitted valence formulas with parameters optimized for selected groups of molecules, (iii) targeted adaptations of the original formula for classes of molecules sharing common chemical features such as hypervalency or lone pairs, and (iv) modifications of the density matrix, including the orbital occupation scheme.
Among the investigated modifications, changes to the treatment of the covalent contribution have the strongest impact on the resulting valence values, whereas modifications of the density matrix itself do not lead to increased stability.

Popular Abstract

For many people, chemistry appears almost like a form of magic. This perception probably arises because chemists describe and explain phenomena that can be very difficult to observe directly. Central concepts of chemistry are based on entities that are invisible to the naked eye and often even to microscopes, yet these concepts are used to predict and control the behavior of matter.
According to chemical theory, all matter is composed of atoms. Atoms consist of nuclei and electrons and combine to form molecules through chemical bonds. These bonds are commonly understood as shared electrons that occupy the region between atoms and hold them together. Although most molecules themselves are too small to be seen directly, they are used to explain a wide range of observable phenomena.
Today, there is overwhelming experimental evidence for the existence of atoms, molecules, and electrons, and many theories based on them have been repeatedly confirmed. Nevertheless, the challenge of observing these fundamental entities remains, particularly when it comes to describing and quantifying chemical bonding.
Not all atoms form the same number of bonds. While noble gases typically form none, carbon, for example, usually forms four bonds, oxygen two, and hydrogen one. The number of bonds an atom forms within a molecule is referred to as its valence. Valence is a key concept for understanding molecular structure, reactivity, and many other chemical properties. For simple molecules, valence can often be assigned reliably using chemical sense, drawing on established bonding patterns, empirical knowledge, and straightforward counting of bonds. However, because chemical bonds are not directly observable, especially in large or chemically complex systems, valence cannot, in general, be determined by inspection alone.
Instead, valence must be calculated. This requires mathematical formulas that can be applied to molecules in a general and reliable way. Such calculations are naturally based on quantum mechanics, which describes the behavior of electrons, the particles responsible for chemical bonding. The already existing quantum mechanical approach, on which this work is based, describes bonds as consisting of ionic and covalent contributions, representing two extreme types of bonding in which electrons are either shared equally (covalent) or transferred (ionic) between atoms.
While these approaches work well in certain contexts, they fail in others. The goal of this work is to better understand an existing quantum mechanical valence formula, to identify its robustness to different computational methods, and to explore how modifying this formula changes and influences the obtained values. By doing so, this work aims to contribute to a deeper understanding of chemical bonding and to help make chemistry appear less like magic and more like what it truly is: a systematic and powerful framework for understanding the world around us.

• author: Hoffmann, Judith
• supervisor: Valera Veryazov
• organization: Department of Chemistry
• course: KEMR10 20252
• year: 2026
• type: H2 - Master's Degree (Two Years)
• subject: Chemistry
• keywords: valence, theoretical chemistry, computational chemistry, Veryazov valence formula
• language: English
• id: 9221367
• date added to LUP: 2026-02-11 11:38:55
• date last changed: 2026-02-11 11:38:55