## Departmental PhD Thesis Exam – Ioannis Anapolitanos

Everyone welcome.  Refreshments will be served in the Math
Lounge before the exam.
Monday, July 18, 2011, 3:10 p.m., in BA 6183, 40 St. George St.
PhD Candidate: Ioannis Anapolitanos
Thesis Title:  On van der Waals forces

Thesis 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. 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 of adding electrons). This condition
is well justified experimentally as can be seen from the table,
Atomic number Element Ionization energy Electron affinity
------------  ------- ----------------- -----------------
1             H       313.5             17.3
6             C       259.6             29
8             O       314.0             34
9             F       401.8             79.5
16            S       238.9             47
17            Cl      300.0             83.4
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 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 $| y_i - y_j |$, $i,j \in \{ 1, ..., M \}$, $i \ne j$,
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 $j$.

A copy of the thesis can be obtained by contacting  ioannis.anapolitanos@utoronto.ca.