Table 1.

Closure expressions and excess chemical potential equations for various common closures and corrections.

Closure	Closure Relation
KH	$$\mathbf{g}(\mathbf{r})=\{\begin{array}{ll}exp({\mathbf{t}}^{\ast }(\mathbf{r}))\hfill & {\mathbf{t}}^{\ast }(\mathbf{r})<0\hfill \\ \mathbf{1}+{\mathbf{t}}^{\ast }(\mathbf{r})\hfill & {\mathbf{t}}^{\ast }(\mathbf{r})\ge 0\hfill \end{array}$$
HNC	g(r) = exp(t*(r))
PSE-n	$$\mathbf{g}(\mathbf{r})=\{\begin{array}{ll}exp({\mathbf{t}}^{\ast }(\mathbf{r}))\hfill & {\mathbf{t}}^{\ast }(\mathbf{r})<0\hfill \\ \sum _{i=0}^n\frac{{\mathbf{t}}^{\ast }{(\mathbf{r})}^i}{i!}\hfill & {\mathbf{t}}^{\ast }(\mathbf{r})\ge 0\hfill \end{array}$$
	t*(r) = −βu(r) + h(r) − c(r)
Closure	Excess Chemical Potential
KH	$$\mathrm{\Delta }{\mu }^{\text{KH}}=kT{\sum }_{\gamma }{\rho }_{\gamma }\int \frac{h_{\gamma }^2(\mathbf{r})}{2}\mathrm{\Theta }(-h_{\gamma }(\mathbf{r}))-c_{\gamma }(\mathbf{r})-\frac{h_{\gamma }(\mathbf{r})c_{\gamma }(\mathbf{r})}{2}\mathrm{d}\mathbf{r}$$
HNC	$$\mathrm{\Delta }{\mu }^{\text{HNC}}=kT{\sum }_{\gamma }{\rho }_{\gamma }\int \frac{h_{\gamma }^2(\mathbf{r})}{2}-c_{\gamma }(\mathbf{r})-\frac{h_{\gamma }(\mathbf{r})c_{\gamma }(\mathbf{r})}{2}\mathrm{d}\mathbf{r}$$
PSE-n	$$\mathrm{\Delta }{\mu }^{\text{PSE-}n}=kT{\sum }_{\gamma }{\rho }_{\gamma }\int \frac{h_{\gamma }^2(\mathbf{r})}{2}-c_{\gamma }(\mathbf{r})-\frac{h_{\gamma }(\mathbf{r})c_{\gamma }(\mathbf{r})}{2}-\frac{t_{\gamma }^{\ast }{(\mathbf{r})}^{n+1}}{(n+1)!}\mathrm{\Theta }(h_{\gamma }(\mathbf{r}))\mathrm{d}\mathbf{r}$$
Correction	Excess Chemical Potential
GF	$$\mathrm{\Delta }{\mu }^{\text{GF}}=kT{\sum }_{\gamma }{\rho }_{\gamma }\int -c_{\gamma }(\mathbf{r})-\frac{h_{\gamma }(\mathbf{r})c_{\gamma }(\mathbf{r})}{2}\mathrm{d}\mathbf{r}$$
UCT	ΔμUC = ΔμRISM + av + b
NgBT	$$\mathrm{\Delta }{\mu }^{\text{NgB}}=\mathrm{\Delta }{\mu }^{\text{RISM}}+\frac{kT{\rho }_{\mathrm{O}}}{2}(1-\gamma )\int c_{\mathrm{O}}^{\text{np}}(\mathbf{r})d\mathbf{r}$$
PC+/3D-RISM	$$\mathrm{\Delta }{\mu }^{\text{PC}+/3\mathrm{D}\text{-RISM}}=\mathrm{\Delta }{\mu }^{\text{RISM}}-\frac{1}{2}\mathit{kTv}\left(\frac{1}{{\chi }_{T}kT}+N_{\text{site}}{\rho }_{\text{Tot}}\right)$$