  We have carried out calculations towards the goal of reducing the inaccuracy
of the Sr optical atomic clock to 1$\times10^{-17}$ and below. We calculated
a.c. polarizabilities of the $5s^2 ^1S_0$ and $5s5p ^3P_0^o$ clock states that
are important for reducing the uncertainty of blackbody radiation-induced
frequency shifts for the $^1S_0 - ^3P_0^o$ clock transition. We determined four
low-lying even-parity states whose total contribution to the static
polarizability of the $^3P_0^o$ clock state is at the level of 90%. We show
that if the contribution of these states is experimentally known with 0.1%
accuracy, the same accuracy can be achieved for the total polarizability of the
$^3P_0^o$ state. The corresponding uncertainty for the blackbody shift at a
fixed room temperature will be below 1$\times10^{-17}$. The calculations are
confirmed by a number of experimental measurements on various Sr properties.
