  The first-forbidden transition of nuclear $\beta$ decay is expected to play
crucial roles in many aspects in nuclear physics, nuclear astrophysics and
particle physics such as the stellar $\beta$-decay rates and the reactor
anti-neutrino spectra. In this work we develop the projected shell model (PSM)
for description of first-forbidden transition of nuclear $\beta$ decay for the
first time. Detailed theoretical framework and logics are provided, and 35
dominant first-forbidden transitions that are expected to be important for the
reactor anti-neutrino spectra problems are calculated and compared
systematically with the data to test the new development of the PSM. The
corresponding experimental Log$f_0 t$ values are described reasonably, and the
quenching factors of nuclear matrix elements are found to affect the Log$f_0 t$
values as well as the related shape factors, which may be helpful for better
understanding of the reactor anti-neutrino spectra problems.
