Theoretical study of intermolecular interaction energy for $\textrm{F}_{2}\cdots \textrm{F}_{2}$ complex

Abstract

The ab initio intermolecular pair potentials of $\textrm{F}_{2}$ dimer were calculated for five leading stable configurations, using the symmetry-adapted perturbation theory. We employ an improved Lennard–Jones potential to best fit the potential energy surface of each leading configuration. The molecular anisotropy is characterized through the expansion of the degrees of freedom of the analytical potential energy surface (PES) using the spherical harmonics. The resulting analytical PES is used to calculate the second virial coefficients and compared with the experimental values and other theoretical works to test the quality of the presented intermolecular potential. Finally, we performed the theoretical computation of viscosity and self-diffusion transport properties for the $\textrm{F}_{2} \cdots \textrm{F}_{2}$ system.

Appendices

Appendix A Second virial calculations

The value of $B_2(T)$ for a given temperature T is calculated through the following expansion (10):

$$\begin{aligned} B_{2}(T) = \left[ B_\textrm{cl}(T)+B_{I}^r(T)+B_{I}^{a,I}(T)+B_{I}^{a,\mu }(T)+\cdots \right] \end{aligned}$$

(10)

with the aid of Eq.’s (11)–(14) [35]:

$$\begin{aligned} B_\textrm{cl}(T)= & {} \dfrac{N_A}{4}\int ^{2\pi }_{0}\int ^{\pi }_{0}\sin {\theta _1}\int ^{\pi }_{0}\sin {\theta _2}\times \nonumber \\{} & {} \int ^{\infty }_{0}\left( 1-\exp \left( -\dfrac{V}{k_B T}\right) \right) r^{2}\textrm{drd}\theta _1\textrm{d}\theta _2\textrm{d}\phi \end{aligned}$$

(11)

$$\begin{aligned} B_{I}^r(T)= & {} \dfrac{N_{A}\hbar ^2}{96\mu k^{3}_{B}T^{3}} \int ^{2\pi }_{0}\int ^{\pi }_{0}\sin {\theta _1}\int ^{\pi }_{0}\sin {\theta _2}\times \nonumber \\{} & {} \int ^{\infty }_{0}\exp \left( -\dfrac{V}{k_B T}\right) \left( \frac{\partial V}{\partial r}\right) ^{2}r^{2}\mathrm drd\theta _1d\theta _2d\phi \end{aligned}$$

(12)

$$\begin{aligned} B_{I}^{a,I}(T)= & {} \dfrac{-N_{A}}{48k^{3}_{B}T^{3}}\int ^{2\pi }_{0}\int ^{\pi }_{0}\sin {\theta _1} \int ^{\pi }_{0}\sin {\theta _2}\int ^{\infty }_{0}\exp \left( -\dfrac{V}{k_B T}\right) \nonumber \\{} & {} \times \left[ \sum \limits _{i=1}^2\frac{\hbar ^2}{2I_{i}}\left( \cot {\theta _i}\frac{\partial V}{\partial \theta _i}+\frac{\partial ^2 V}{\partial \theta _i^2}+\frac{1}{\sin ^2{\theta _i}}\frac{\partial ^2 V}{\partial \phi ^2}\right) \right] \nonumber \\{} & {} \quad r^{2}\textrm{drd}\theta _1\textrm{d}\theta _2\textrm{d}\phi \end{aligned}$$

(13)

$$\begin{aligned} B_{I}^{a,\mu }(T)= & {} \dfrac{-N_{A}}{48k^{3}_{B}T^{3}}\int ^{2\pi }_{0}\int ^{\pi }_{0}\sin {\theta _1}\int ^{\pi }_{0}\sin {\theta _2}\int ^{\infty }_{0}\exp \left( -\dfrac{V}{k_B T}\right) \times \nonumber \\{} & {} \left[ \sum \limits _{i=1}^2\frac{\hbar ^2}{2\mu r^2}\left( \cot {\theta _i}\frac{\partial V}{\partial \theta _i}+\frac{\partial ^2 V}{\partial \theta _i^2}+\frac{1}{\sin ^2{\theta _i}}\frac{\partial ^2 V}{\partial \phi ^2}\right) \right] \nonumber \\{} & {} \quad r^{2}\textrm{drd}\theta _1\textrm{d}\theta _2\textrm{d}\phi \end{aligned}$$

(14)

in which $k_B$ and $N_A$ are Boltzmann’s and Avogadro’s constants, respectively.

Appendix B Reduced collision integrals

The reduced collision integrals used to calculate the viscosity and self-diffusion are calculated through the following formulas

$$\begin{aligned} \Omega ^{(i,j)*}=\frac{1}{(j+1)!}\int \limits _ 0^{\infty }e^{-\gamma ^2}(\gamma ^2)^{j+1}Q^{(i)*}\textrm{d}(\gamma ^2) \end{aligned}$$

(15)

where $\gamma ^2\equiv mv_0^2/4kT$ with $v_0$ being the initial relative speed of collision. $Q^{(i)*}$ is the reduced cross section given by

$$\begin{aligned} Q^{(i)*}=\frac{2}{R_0^2}\left[ 1-\frac{1}{2}\frac{1+(-1)^i}{1+i}\right] ^{-1} \int \limits _0^{\infty }\left( 1-\cos ^i{\chi }\right) b\textrm{d}b \end{aligned}$$

(16)

with

$$\begin{aligned} \chi =\pi -2b\int \limits _{r_m}^{\infty }\left\{ 1 -\frac{b^2}{r^2}-\frac{V(R,\theta _1,\theta _2,\phi )}{\frac{1}{4}mv_0^2}\right\} ^{-\frac{1}{2}}\frac{\mathrm{{d}}r}{r^2} \end{aligned}$$

(17)

where $r_m$ is the closest approach of molecules given as a function of b and $v_0$ through the equation

$$\begin{aligned} 1-\frac{b^2}{r_m^2}-\frac{V(r_m,\theta _1,\theta _2,\phi )}{\frac{1}{4}mv_0^2}=0 \end{aligned}$$

(18)

It is convenient to define adimensional variables such that

$$\begin{aligned} r^*= & {} r/R_0 \nonumber \\ b^*= & {} b/R_0 \nonumber \\ r_m^*= & {} r_m/R_0 \nonumber \\ (v_0^*)^2= & {} \frac{1}{4}\frac{mv_0^2}{D_e^\textrm{spher}} \end{aligned}$$

(19)

where $D_e^\textrm{spher}$ is the well depth of the isotropic term $V^\textrm{spher}_\textrm{avg}(R)$ calculated through the formula

$$\begin{aligned} V^\textrm{spher}_\textrm{avg}(R)=\int \limits _{-1}^{1}\int \limits _{-1}^{1}\int \limits _{0}^{2\pi }V(R,\theta _1,\theta _2,\phi )\textrm{d}\phi \textrm{d}(\cos {\theta _1})\textrm{d}(\cos {\theta _2}) \end{aligned}$$

(20)

This enables us to rewrite the integrals (16) and (17) in a more convenient way for numerical calculations. However, the lower limit of integration of Eq. (17) reveals a singularity of the function inside the integral that can be avoided by the substitution $\sin {\alpha }=r_m^*/r^*$ into integral in Eq. (17). The temperature can make the calculation of $\chi$ a tough task. For lower temperatures, some values of energy in Eq. (15) may cause the molecules to orbiting each other making the integral in Eq. (17) diverges. Actually, the problem becomes more evident when one tries to calculate the roots of equation (18) in which will appear two values of $r_m$ for such temperatures and energies. To circumvent this situation, one needs to avoid these regions in Eq. (17) as discussed in Hirschfelder et al. [36] work.

Finally, if one assumes that all relative orientations are equally probable, we have

$$\begin{aligned} \langle \Omega ^{(i,j)*}\rangle =\frac{1}{8\pi }\int \limits _{-1}^{1} \int \limits _{-1}^{1}\int \limits _{0}^{2\pi }\Omega ^{(i,j)*}\textrm{d}\phi \textrm{d}(\cos {\theta _1})\textrm{d}(\cos {\theta _2}) \end{aligned}$$

(21)

and

$$\begin{aligned}{} & {} f_{\eta }=1+\frac{3}{196}\left[ 8\frac{\langle \Omega ^{(2,3)*}\rangle }{\langle \Omega ^{(2,2)*}\rangle }-7\right] ^2 \nonumber \\{} & {} f_{D}=1+\frac{1}{8}\frac{\left[ 6\frac{\langle \Omega ^{(1,2)*}\rangle }{\langle \Omega ^{(1,1)*}\rangle }-5\right] ^2}{\left[ 2\frac{\langle \Omega ^{(2,2)*}\rangle }{\langle \Omega ^{(1,1)*}\rangle }+5\right] } \end{aligned}$$

(22)