Third and fourth virial coefficients of hard hyperspheres of arbitrary dimensionality

We derive exact expressions for the third virial coefficient $B_3$ and two of the three terms $B_4\mathrm{(⧠)\hspace{0.17em}}\mathrm{and}{\mathrm{\hspace{0.17em}B}}_4\mathrm{(⧅)}$ contributing to the fourth virial coefficient $B_4$ for an assembly of hard hyperspheres of arbitrary dimensionality d. A semiempirical formula, accurate for all d, is proposed for the remaining term $B_4\mathrm{(⊠).}$ This formula is suggested by the large‐d asymptotic behavior of $B_3$ and $B_4\mathrm{(⧠).}$ We find that $B_4$ is positive for ${\mathrm{d<d}}_4$ and negative for ${\mathrm{d>d}}_4,$ with $d_4$ estimated to be $d_4\text{≈7.8,}$ whereas $B_4{\mathrm{(⧠)+B}}_4\mathrm{(⧅)}$ changes sign for d = 9.073