The Aufbau principle in the Theory of Operators (Elias Polak, UniGe)

12.04.2022 14:00

A century ago the so–called Aufbau principle (German: ”building up”) was
postulated by Niels Bohr as a model for the characterization of the electron
configuration in atoms. It was based on recent experimental radiation results
and on the new scientific domain of Quantum Mechanics. The aim was to
present a reasonable physical description of the ground state of an atomic sys-
tem by ordering the stationary states upwards in the energy levels. Over the
decades, its etymology became vague and in other languages one can now find
this principle under a different name. Today, the eigenfunctions with their
eigenvalues of differential operators correspond to these stationary states at
specific energy levels. The Aufbau principle suggests that one should use the
eigenstates that minimize a lower dimensional eigenvalue problem in order
to construct the ground state solution, which is equivalent to the minimum
of the expectation value of the operator describing the higher dimensional
quantum system. This interpretation provides a well defined mathematical
formulation in the view of unbounded operators acting on subdomains of
Hilbert spaces. It is based on Ritz’s method, which was introduced to find
the solution of a variational problem by developing the solution in a lin-
ear combination of a given sequence of functions. However, without a proof
for the exact quantum chemical operator, approximations in computational
methods as well as chemical models show cases of violation of the princi-
ple. Nevertheless, modern numerical techniques in computational chemistry
can benefit from this analysis in order to provide robust and physical sound
results.

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Room 1-05, Séminaire d'analyse numérique

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