Long-range interactions between two helium atoms

The dispersion coefficients $C_n$ (with n up to 10) for the long-range interaction between two helium atoms are calculated using variation method in Hylleraas coordinates. For the any combined system He($nS$)-He($n^{'}S$) among the ground state He$(1\,^1\!S)$ and the lowest excited $S$ states He$(2\,^1\!S)$ and He$(2\,^3\!S)$, significant improvements are made upon previous calculation and our results provide definitive values for these coefficients. For the He$(1\,^1\!S)$-He$(2\,^3\!P)$ system, we also assess coefficients $C_6^{\pm}$ and $C_6^0$ using a semi-empirical method based on tabulated oscillator strengths and available photoionization cross sections. The results from the two methods agree to within 1.5\%. In addition, we have calculated dispersion coefficients $C_ {n\leq10}$ for the four systems He$(1\,^1\!S)$-He$(2\,^1\!P)$, He$(2\,^1\!S)$-He$(2\,^3\!P)$, He$(2\,^3\!S)$-He$(2\,^3\!P)$, and He$(2\,^3\!P)$-He$(2\,^3\!P)$.