1. Molecular Structure in Topology and Quantum Mechanics
- Figure 1: Molecular structures with nuclei as attractors in maps of the gradient vector field of their charge densities (for the plane) (Bader 1990, p. 30).
In general: the molecular graph is the network of bond paths linking pairs of neighbouring nuclear attractors. An atom, free or bound, is defined as the union of an attractor and its basin. Atoms, bonds, and structure are topological consequences of a measurable molecular charge distribution. In a next step, it is necessary to demonstrate that the topological atom and its properties have a basis in quantum mechanics. Topological atoms and bonds have a meaning in real three-dimensional space. But this structure is not reflected in the properties of the abstract infinite-dimensional Hilbert-space of the molecular state function. The state function y contains all of the information that can be known about a nuclear quantum system. From an operational point of view, there is too much and redundant information in the state function because of the indistinguishability of the electrons or because of the symmetry of their interactions. Some of it is unnecessary as a result of the two-body nature of the Coulombic interaction. Thus, there is a reduction of information in passing from the state function in the infinite-dimensional Hilbert space to the charge distribution function in the real three-dimensional space. But, on the other hand, we get a description of the molecular structure in the observable and measurable space.
2. Molecular Symmetry in Group Theory
Figure 2: Orbital symmetry of Kekulé's ring structure of benzene (Mainzer 1996, p. 498).
The fourth valence electron corresponds to the pz orbital, which is above and below the plane with its two dumb-bells, each perpendicular in the nodes of the carbon atom. The pz orbitals overlap with their respective neighbors and form a p-bond. Figure 2b shows a p-orbital of benzene. In contrast to the s-bond, the p-bond is weak, so that the p-electrons can be easily influenced by extremal forces, and thus determine many of the spectroscopic characteristics of benzene. s and p orbitals of benzene can be distinguished by their symmetry behavior in a reflection on the xy plane. While s orbitals fs do not change their sign during the reflection z ®-z and are therefore symmetric, antisymmetry occurs with the p orbitals fp:
- fs(x,y, -z) = fs(x,y,z)
fp(x,y, -z) = -fp(x,y,z)
3. Molecular Complexity in Dynamical Systems Theory
- Figure 3: Supramolecular cluster in a ball-and-stick representation as an example of a complex near-equilibrium system (Müller 1995, p.2293)
Molecular cavities can be used as containers for other chemicals or even for medicaments which need to be transported within the human organism. An iron-storage protein that occurs in many higher organisms is ferritin. It is an unusual host-guest system consisting of an organic host (an aprotein) and a variable inorganic guest (an iron core). Depending on the external demand, iron can either be removed from this system or incorporated into it. Complex chemical aggregates like polyoxometalates are frequently discovered to be based upon regular convex polyhedra, such as Platonic solids. But their collective electronic and/or magnetic properties cannot be deduced from the known properties of these building blocks. According to the catch-phrase "from molecules to materials" supramolecular chemistry applies the "blue-prints" of conservative self-organization to build up complex materials on the nanometer scale with novel catalytic, electronic, electrochemical, optical, magnetic and photochemical properties. Multi-property materials are extremely interesting.
4. Molecular Symmetry and Complexity as Technological and Philosophical Perspectives of Research
Nombre: Edymar Gonzalez A