HF-DFT is included as an automatic calculation scheme in Q-Chem 6.0.
The input file requires only the "DC_DFT" keyword in addition to the basic DFT calculation.
The gradient is also available.
The following example code is water calculation using HF-DFT.
$molecule
0 1
o -1.07602 -1.24264 0.65991
h -0.49714 -0.99588 -0.47841
h -0.08653 -1.26594 1.06125
$end
$rem
jobtype sp
method pbe
dc_dft true
basis def2-qzvppd
$end