  We present a catalog of 1.4 million photometrically-selected quasar
candidates in the southern hemisphere over the $\sim 5000\,{\rm deg^2}$ Dark
Energy Survey (DES) wide survey area. We combine optical photometry from the
DES second data release (DR2) with available near-infrared (NIR) and the
all-sky unWISE mid-infrared photometry in the selection. We build models of
quasars, galaxies, and stars with multivariate Skew-t distributions in the
multi-dimensional space of relative fluxes as functions of redshift (or color
for stars) and magnitude. Our selection algorithm assigns probabilities for
quasars, galaxies, and stars, and simultaneously calculates photometric
redshifts (photo-$z$) for quasar and galaxy candidates. Benchmarking on
spectroscopically confirmed objects, we successfully classify (with photometry)
94.7% of quasars, 99.3% of galaxies, and 96.3% of stars when all IR bands (NIR
$YJHK$ and WISE $W1W2$) are available. The classification and photo-$z$
regression success rates decrease when fewer bands are available. Our quasar
(galaxy) photo-$z$ quality, defined as the fraction of objects with the
difference between the photo-$z$ $z_p$ and the spectroscopic redshift $z_s$,
$|\Delta z| = |z_s - z_p|/(1 + z_s)\le 0.1$, is 92.2% (98.1%) when all IR bands
are available, decreasing to 72.2% (90.0%) using optical DES data only. Our
photometric quasar catalog achieves estimated completeness of 89% and purity of
79% at $r<21.5$ (0.68 million quasar candidates), with reduced completeness and
purity at $21.5<r\lesssim 24$. Among the 1.4 million quasar candidates, 87,857
have existing spectra and 84,978 (96.7%) of them are spectroscopically
confirmed quasars. Finally, we provide quasar, galaxy, and star probabilities
for all (0.69 billion) photometric sources in the DES DR2 coadded photometric
catalog.
