  Understanding electronic properties of sub-stoichiometric phases of titanium
oxide such as Magn\'eli phase Ti4O7 is crucial in designing and modeling
resistive switching devices. Here we present our study on Magn\'eli phase Ti4O7
together with rutile TiO2 and Ti2O3 using density functional theory methods
with atomic-orbital-based self-interaction correction (ASIC). We predict a new
antiferromagnetic ground state in the low temperature phase (or LT phase), and
we explain energy difference with a competing antiferromagnetic state using a
Heisenberg model. The predicted energy ordering of these states in the LT phase
is calculated to be robust in a wide range of modeled isotropic strain. We have
also investigated the dependence of the electronic structures of the Ti-O
phases on stoichiometry. The splitting of titanium t2g orbitals is enhanced
with increasing oxygen deficiency as Ti-O is reduced. The electronic properties
of all these phases can be reasonably well described by applying ASIC with a
standard value for transition metal oxides of the empirical parameter {\alpha}
of 0.5 representing the magnitude of the applied self-interaction correction.
