Re: orbitals, flowers, quantum puzzlement;
Cyberkatru wrote:
For isolated atoms, rho is spherically symmetric, giving symmetric
shapes. For molecules, rho is in fact a function of the coordinates of
all nuclei involved, and there is no longer any reason to have more
symmetry than the symmetry of the configuration of nuclei has, which is
very little and often none.
Hmm. OK. But what determines the relative positions of the nuclei which are
each in and of themselves, basically spherically symmetric?
You can understand this already classically. If you have 6 spherical
argon atoms that interact with each other by a Lennard-Jones pair
potential, what you get will be something unsymmetric. There are
several local minima in the potential energy surface, each determining
a metastable state. The ground state is the global minimizer.
The quantum version is more or less similar (in the Born-Oppenheimer
approximation, which suffices for most of chemistry) although the
classical view must be taken with a grain of salt. In particular, the
potential energy gets quantum modifications. There may also be
peculiarities due to surface crossing, where the Born-Oppenheimer
approximation breaks down.
The shape of molecules is therefore mainly determined by the geometry
of the positions of the nuclei. In equilibrium, these arrange
themselves such that the potential energy, i.e., the smallest
eigenvalue of the Hamiltonian operator for the electrons is minimal
among all other positions (or at least a local minimum from which a
deeper lying state is very difficult to reach).
Arnold Neumaier
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