  The rate constants for the formation, destruction, and collisional excitation
of SH$^+$ are calculated from quantum mechanical approaches using two new
SH$_2^+$ potential energy surfaces (PESs) of $^4A''$ and $^2A''$ electronic
symmetry. The PESs were developed to describe all adiabatic states correlating
to the SH$^+$ ($^3\Sigma^-$) + H($^2S$) channel. The formation of SH$^+$
through the S$^+$ + H$_2$ reaction is endothermic by $\approx$ 9860 K, and
requires at least two vibrational quanta on the H$_2$ molecule to yield
significant reactivity. Quasi-classical calculations of the total formation
rate constant for H$_2$($v=2$) are in very good agreement with the quantum
results above 100K. Further quasi-classical calculations are then performed for
$v=3$, 4, and 5 to cover all vibrationally excited H$_2$ levels significantly
populated in dense photodissociation regions (PDR). The new calculated
formation and destruction rate constants are two to six times larger than the
previous ones and have been introduced in the Meudon PDR code to simulate the
physical and illuminating conditions in the Orion bar prototypical PDR. New
astrochemical models based on the new molecular data produce four times larger
SH$^+$ column densities, in agreement with those inferred from recent ALMA
observations of the Orion bar.
