APPENDIX 1
The Irving–Kirkwood pressure tensor is determined as follows:
$$\begin{gathered} \hat {p}({\mathbf{r}}) = kT\sum\limits_i {{{\rho }_{i}}({\mathbf{r}})} \hat {1} - \frac{1}{2}\sum\limits_{i > j} {\int {d{\mathbf{R}}} \frac{{{\mathbf{R}} \times {\mathbf{R}}}}{R}\Phi _{{ij}}^{'}(R)} \\ \times \,\,\int\limits_0^1 {d\eta \rho _{{ij}}^{{(2)}}({\mathbf{r}} - \eta {\mathbf{R}},} \,{\mathbf{r}} - \eta {\mathbf{R}} + {\mathbf{R}}), \\ \end{gathered} $$
where \(\hat {p}({\mathbf{r}})\) is the pressure tensor at point r, k is the Boltzmann constant, T is the temperature, \({{\rho }_{i}}({\mathbf{r}})\) is the one-particle distribution function (local density) of the i-type particles, \(\hat {1}\) is the unit tensor, R is the vector that connects two interacting particles located at distance R from each other and passes through point r (denotation \({\mathbf{R}} \times {\mathbf{R}}\) symbolizes the direct vector product, which represents a tensor), \(\Phi _{{ij}}^{'}(R)\) is the derivative of pair interaction potential \({{\Phi }_{{ij}}}(R)\) for the particles of the i and j types (i.e., the force of the interaction between these particles), and \(\rho _{{ij}}^{{(2)}}({\mathbf{r}} - \eta {\mathbf{R}},{\mathbf{r}} - \eta {\mathbf{R}} + {\mathbf{R}})\) is the two-particle distribution function for particles that are simultaneously located at points \({\mathbf{r}} - \eta {\mathbf{R}}\) and \({\mathbf{r}} - \eta {\mathbf{R}} + {\mathbf{R}}\) on opposite sides of a unit area with coordinate r (which is controlled by auxiliary variable η).
APPENDIX 2.
RESULT OF INTEGRATING EQ. (7)
$$\begin{gathered} {{I}_{{\varphi \varphi }}} = {{\sin }^{3}}{\kern 1pt} {{\theta }_{1}}{{\sin }^{3}}{\kern 1pt} {{\theta }_{2}}\ln \left( {\tan {\kern 1pt} \frac{{{{\theta }_{1}}}}{2}\tan {\kern 1pt} \frac{{{{\theta }_{2}}}}{2}} \right) \\ + \,\,3(\sin {\kern 1pt} {{\theta }_{2}} - \sin {\kern 1pt} {{\theta }_{1}}){{\sin }^{2}}{\kern 1pt} {{\theta }_{1}}{{\sin }^{2}}{\kern 1pt} {{\theta }_{2}}\sin {\kern 1pt} \Delta \\ + \,\,({{\sin }^{3}}{\kern 1pt} {{\theta }_{2}} - {{\sin }^{3}}{\kern 1pt} {{\theta }_{1}}) \\ \times \,\,\left( {\frac{1}{3}{\kern 1pt} {{{\sin }}^{3}}{\kern 1pt} \Delta - \sin {\kern 1pt} {{\theta }_{1}}\sin {\kern 1pt} {{\theta }_{2}}\sin {\kern 1pt} 2\Delta } \right) \\ - \,\,\frac{1}{5}{\kern 1pt} \sin {\kern 1pt} \Delta \left( {{{{\sin }}^{5}}{\kern 1pt} {{\theta }_{2}} - {{{\sin }}^{5}}{\kern 1pt} {{\theta }_{1}}} \right)\left( {1 - 4{\kern 1pt} {{{\cos }}^{2}}{\kern 1pt} \Delta } \right) \\ - \,\,(\cos {\kern 1pt} {{\theta }_{1}} + \cos {\kern 1pt} {{\theta }_{2}}) \\ \times \,\,\left( {\frac{3}{4}{\kern 1pt} \sin {\kern 1pt} 2{{\theta }_{1}}{\kern 1pt} \sin {\kern 1pt} 2{{\theta }_{2}}\cos {\kern 1pt} \Delta - {{{\cos }}^{3}}{\kern 1pt} \Delta } \right) \\ - \,\,({{\cos }^{3}}{\kern 1pt} {{\theta }_{1}} + {{\cos }^{3}}{\kern 1pt} {{\theta }_{2}}) \\ \times \,\,\left( {\cos {\kern 1pt} {{\theta }_{1}}{\kern 1pt} \cos {\kern 1pt} {{\theta }_{2}}\cos {\kern 1pt} 2\Delta - \frac{1}{3}{\kern 1pt} {{{\cos }}^{3}}{\kern 1pt} \Delta } \right) \\ + \,\,\frac{1}{5}({{\cos }^{5}}{\kern 1pt} {{\theta }_{1}} + {{\cos }^{5}}{\kern 1pt} {{\theta }_{2}})(1 - 4{\kern 1pt} {{\sin }^{2}}{\kern 1pt} \Delta )\cos {\kern 1pt} \Delta , \\ \end{gathered} $$
$$\begin{gathered} 5{{I}_{{zz}}} = - \frac{1}{2}\sin {\kern 1pt} {{\theta }_{1}}\sin {\kern 1pt} {{\theta }_{2}}({{\sin }^{2}}{\kern 1pt} {{\theta }_{1}}\cos {\kern 1pt} {{\theta }_{2}} \\ + \,\,{{\sin }^{2}}{\kern 1pt} {{\theta }_{2}}\cos {\kern 1pt} {{\theta }_{1}}) - \sin {\kern 1pt} {{\theta }_{1}}\sin {\kern 1pt} {{\theta }_{2}} \\ \times \,\,\left( {3\sin {\kern 1pt} {{\theta }_{1}}\sin {\kern 1pt} {{\theta }_{2}}\cos {\kern 1pt} \Delta - 3{\kern 1pt} {{{\sin }}^{2}}{\kern 1pt} \Delta \frac{{}}{{}}} \right. \\ \left. { - \,\,\frac{1}{2}{\kern 1pt} {{{\sin }}^{2}}{\kern 1pt} {{\theta }_{1}}{{{\sin }}^{2}}{\kern 1pt} {{\theta }_{2}}} \right)\ln \left( {\tan {\kern 1pt} \frac{{{{\theta }_{1}}}}{2}\tan {\kern 1pt} \frac{{{{\theta }_{2}}}}{2}} \right) \\ + \,\,(\sin {\kern 1pt} {{\theta }_{2}} - \sin {\kern 1pt} {{\theta }_{1}})({{\sin }^{3}}{\kern 1pt} \Delta - 3{\kern 1pt} \sin {\kern 1pt} {{\theta }_{1}}\sin {\kern 1pt} {{\theta }_{2}}\sin {\kern 1pt} 2\Delta \\ + \,\,3{\kern 1pt} \sin {\kern 1pt} {{\theta }_{1}}\sin {\kern 1pt} {{\theta }_{2}}\sin {\kern 1pt} \Delta ) + ({{\sin }^{3}}{\kern 1pt} {{\theta }_{2}} - {{\sin }^{3}}{\kern 1pt} {{\theta }_{1}}) \\ \times \,\,\left( {\frac{1}{2}{\kern 1pt} \cos {\kern 1pt} \Delta \sin {\kern 1pt} 2\Delta - \frac{1}{3}{\kern 1pt} {{{\sin }}^{3}}{\kern 1pt} \Delta } \right) - (\cos {\kern 1pt} {{\theta }_{1}} + \cos {\kern 1pt} {{\theta }_{2}}) \\ \times \,\,(3{\kern 1pt} \sin {\kern 1pt} {{\theta }_{1}}\sin {\kern 1pt} {{\theta }_{2}}\cos {\kern 1pt} 2\Delta - {{\cos }^{3}}{\kern 1pt} \Delta ) \\ - \,\,({{\cos }^{3}}{\kern 1pt} {{\theta }_{1}} + {{\cos }^{3}}{\kern 1pt} {{\theta }_{2}})\left( {\frac{1}{3}{\kern 1pt} {{{\cos }}^{3}}{\kern 1pt} \Delta - \frac{1}{2}{\kern 1pt} \sin {\kern 1pt} \Delta \sin {\kern 1pt} 2\Delta } \right), \\ \end{gathered} $$
\({{I}_{{rr}}} = 5{{I}_{{zz}}} - {{I}_{{\varphi \varphi }}}\) [Eq. (16)],
$$\begin{gathered} {{I}_{{r\varphi }}} = - 3{{\sin }^{2}}{{\theta }_{1}}{{\sin }^{2}}{{\theta }_{2}}\sin \Delta \ln \left( {\tan \frac{{{{\theta }_{1}}}}{2}\tan \frac{{{{\theta }_{2}}}}{2}} \right) \\ + \,\,(\sin {{\theta }_{2}} - \sin {{\theta }_{1}})\sin {{\theta }_{1}}\sin {{\theta }_{2}} \\ \times \,\,[3(\sin {{\theta }_{1}}\sin {{\theta }_{2}}\cos \Delta - {{\sin }^{2}}\Delta ) - \sin {{\theta }_{1}}\sin {{\theta }_{2}}] \\ + \,\,({{\sin }^{3}}{{\theta }_{2}} - {{\sin }^{3}}{{\theta }_{1}})({{\sin }^{2}}\Delta \cos \Delta \\ - \,\,\sin {{\theta }_{1}}\sin {{\theta }_{2}}\cos 2\Delta ) + \frac{1}{5}({{\sin }^{5}}{{\theta }_{2}} - {{\sin }^{5}}{{\theta }_{1}}) \\ \times \,\,(1 - 4{{\sin }^{2}}\Delta )\cos \Delta - 3(\cos {{\theta }_{1}} + \cos {{\theta }_{2}}){{\sin }^{2}}{{\theta }_{1}} \\ \times \,\,{{\sin }^{2}}{{\theta }_{2}}\sin \Delta + ({{\cos }^{3}}{{\theta }_{1}} + {{\cos }^{3}}{{\theta }_{2}})\sin \Delta \cos \Delta \cos \theta \\ + \,\,\frac{1}{5}({{\cos }^{5}}{{\theta }_{1}} + {{\cos }^{5}}{{\theta }_{2}})\sin \Delta (1 - 4{{\cos }^{2}}\Delta ), \\ \end{gathered} $$
where \(\Delta \equiv {{\theta }_{2}} - {{\theta }_{1}}.\)
