  Spin properties of two interacting electrons in a quantum dot (QD) embedded
in a nanowire with controlled aspect ratio and longitudinal magnetic fields are
investigated by using a configuration interaction (CI) method and exact
diagonalization (ED) techniques. The developed CI theory based on a
three-dimensional (3D) parabolic model provides explicit formulations of the
Coulomb matrix elements and allows for straightforward and efficient numerical
implementation. Our studies reveal fruitful features of spin singlet-triplet
transitions of two electrons confined in a nanowire quantum dot (NWQD), as a
consequence of the competing effects of geometry-controlled kinetic energy
quantization, the various Coulomb interactions, and spin Zeeman energies. The
developed theory is further employed to study the spin phase diagram of two
quantum-confined electrons in the regime of "cross over" dimensionality, from
quasi-two-dimensional (disk-like) QDs to finite one-dimensional (rod-like) QDs.
