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Please use this identifier to cite or link to this item: http://hdl.handle.net/1807/32060

Title: Mathematical Theory of van der Waals forces
Authors: Anapolitanos, Ioannis
Advisor: Sigal, Israel Michael
Department: Mathematics
Keywords: van der Waals forces
Issue Date: 19-Jan-2012
Abstract: The van der Waals forces, which are forces between neutral atoms and molecules, play an important role in physics (e.g. in phase transitions), chemistry (e.g. in chemical reactions) and biology (e.g. in determining properties of DNA). These forces are of quantum nature and it is long being conjectured and experimentally verified that they have universal behaviour at large separations: they are attractive and decay as the inverse sixth power of the pairwise distance between the atoms or molecules. In this thesis we prove the van der Waals law under the technical condition that ionization energies (energies of removing electrons) of atoms are larger than electron affinities (energies released when adding electrons). This condition is well justified experimentally as can be seen from the table, \newline \begin{tabular}{|c|c|c|c|} \hline Atomic number & Element & Ionization energy (kcal/mol)& Electron affinity (kcal/mol) \\ \hline 1 & H & 313.5 & 17.3 \\ \hline 6 & C & 259.6 & 29 \\ \hline 8 & O & 314.0 & 34 \\ \hline 9 & F & 401.8 & 79.5 \\ \hline 16 & S & 238.9 & 47 \\ \hline 17 & Cl & 300.0 & 83.4 \\ \hline \end{tabular} \newline where we give ionization energies and electron affinities for a small sample of atoms, and is obvious from heuristic considerations (the attraction of an electron to a positive ion is much stronger than to a neutral atom), however it is not proved so far rigorously. We verify this condition for systems of hydrogen atoms. With an informal definition of the cohesive energy $W(y),\ y=(y_1,...,y_M)$ between $M$ atoms as the difference between the lowest (ground state) energy, $E(y)$, of the system of the atoms with their nuclei fixed at the positions $y_1,...,y_M$ and the sum, $\sum_{j=1}^M E_j$, of lowest (ground state) energies of the non-interacting atoms, we show that for $|y_i-y_j|,\ i,j \in \{1,...,M\}, i \neq j,$ large enough, $$W(y)=-\sum_{i<j}^{1,M} \frac{\sigma_{ij}}{|y_i-y_j|^6}+O(\sum_{i<j}^{1,M} \frac{1}{|y_i-y_j|^7})$$ where $\sigma_{ij}$ are in principle computable positive constants depending on the nature of the atoms $i$ and $j$.
URI: http://hdl.handle.net/1807/32060
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