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Relativistic calculations of $C_6$ and $C_8$ coefficients for strontium dimers

The electric dipole and quadrupole polarizabilities of the $5s5p~^3\!P_1^o$ state and the $C_6$ and $C_8$ coefficients for the $^1\!S_0 +\, ^1\!S_0$ and $^1\!S_0 +\, ^3\!P_1^o$ dimers of strontium are calculated using a high-precision relativistic approach that combines configuration interaction and linearized coupled-cluster methods. Our recommended values of the long range dispersion coefficients for the $0_u$ and $1_u$ energy levels are $C_6(0_u)=3771(32)$ a.u. and $C_6(1_u)= 4001(33)$ a.u., respectively. They are in good agreement with recent results from experimental photoassociation data. We also calculate $C_8$ coefficients for Sr dimers, which are needed for precise determination of long-range interaction potential. We confirm the experimental value for the magic wavelength, where the Stark shift on the $^1\!S_0$-$^3\!P_1^o$ transition vanishes. The accuracy of calculations is analyzed and uncertainties are assigned to all quantities reported in this work.