  Potential Energy Surface for the ($\mathrm{\bar{p}} - \mathrm{He}^{2+} -
\mathrm{He}$) system is calculated in the framework of the restricted (singlet
spin state) Hartree-Fock method with subsequent account of the electronic
correlations within the second order perturbation method (MP2). The geometry of
heavy particles is described in variables of distance $r$ from nucleus $a$ to
antiproton, distance $R$ from the center of mass of the $(\bar{p} - a)$ pair to
$\mathrm{He}$ atom containing nucleus $b$, and an angle $\theta$ between
$\mathbf{r}$ and $\mathbf{R}$. The potential $V(R,r,\cos\theta)$ of the
interaction between $\mathrm{He}$ atom and a $\bar{p} - a$ subsystem involves a
total energy of two electrons in the field of three heavy particles and Coulomb
interactions of the $b$ nucleus with antiproton and the $a$ nucleus. The
expansion of this potential in terms of Legendre polynomials
$\mathrm{P}_k(\cos\theta)$ is obtained. Matrices of the multipole terms
$V^k(r,R)$ ($k=0,1,2$) are obtained in the basis of antiprotonic helium ion
states. The results are compared with the model potential that was used earlier
in the calculations of collisional Stark transitions of
$(\mathrm{\bar{p}He}^{2+})$ ion.Total cross sections of collisional Stark
transitions obtained with the PES potentials exceed the model results by 15 -
20\% at $E \leq 12$ K.
