  Aluminum oxide (alumina, Al$_{2}$O$_{3}$) is a promising candidate as a
primary dust condensate in the atmospheres of oxygen-rich evolved stars.
Therefore, alumina \textit{seed} particles might trigger the onset of stellar
dust formation and of stellar mass loss in the wind. However, the formation of
alumina dust grains is not well understood.} {To shed light on the initial
steps of cosmic dust formation (i.e. nucleation) in oxygen-rich environments by
a quantum-chemical bottom-up approach.} {Starting with an elemental gas-phase
composition, we construct a detailed chemical-kinetic network describing the
formation and destruction of aluminium-bearing molecules and dust-forming
(Al$_{2}$O$_{3}$)$_n$ clusters up to the size of dimers ($n$=2) coagulating to
tetramers ($n=$4). Intermediary species include the prevalent gas-phase
molecules AlO and AlOH, and Al$_x$O$_y$ clusters with $x=$1$-$5, $y=$1$-$6. The
resulting extensive network is applied to two model stars, representing a
semi-regular variable and a Mira-type, and to different circumstellar gas
trajectories including a non-pulsating outflow and a pulsating model. The
growth of larger-sized (Al$_{2}$O$_{3}$)$_n$ clusters with $n=$4$-$10 is
described by the temperature-dependent Gibbs free energies of the most
favourable structures (i.e. the global minima clusters) as derived from global
optimisation techniques and calculated by density functional theory. We provide
energies, bond characteristics, electrostatic properties and vibrational
spectra of the clusters as a function of size $n$ and compare these to corundum
corresponding to the crystalline bulk limit ($n\rightarrow \infty$). {The
circumstellar aluminium gas-phase chemistry in oxygen-rich giants is primarily
controlled by AlOH and AlO, which are tightly coupled by the reactions
AlO+H$_2$, AlO+H$_2$O, and their reverse. Models of ...}
