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Spherical harmonics representation of the potential energy surface for the H2⋯H2 van der Waals complex

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We perform a study of the molecular anisotropy for the H2⋯H2 van der Waals system using a spherical harmonics expansion. We use six leading stable configurations to construct our analytical potential energy surface (PES) from ab initio calculations guided qualitatively by the symmetry-adapted perturbation theory (SAPT) analyses. We extrapolate the energies of the PES performed at the CCSD(T)/aug-cc-pVnZ (n = 2 and 3) levels to the complete basis set (CBS) limit. To best fit the shallow potential energy surface of each leading configuration with the intermolecular distance, it was employed an extended version of the Rydberg potential. To assess the quality of our extrapolated analytical PES, we calculate the second virial coefficients, which are in relatively good agreement with the experimental data. As a result, the spherical harmonics coefficients obtained might be of considerable relevance in spectroscopy and dynamics applications.

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This work was supported by a MCTI-PCI grant, Institutional Process Number 444327/2018-5, Individual Process Number 313555/2019-2.

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Correspondence to Alessandra F. Albernaz.

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Appendix 1: Hyperspherical harmonics expansion of the interaction potential

Applying the expansion of Eq. (4) to the LC’s of Fig. 2 one gets

$$ \begin{array}{@{}rcl@{}} \varepsilon^{H}&=&\varepsilon^{000}-\sqrt{5}\varepsilon^{202}+\sqrt{5}\varepsilon^{220}+\frac{5}{\sqrt{14}}\varepsilon^{222}+\frac{9}{4}\sqrt{\frac{5}{14}}\varepsilon^{224}+\frac{9}{4}\varepsilon^{404} \\ \varepsilon^{X}&=&\varepsilon^{000}-\sqrt{5}\varepsilon^{202}-\frac{\sqrt{5}}{2}\varepsilon^{220}-\frac{10}{\sqrt{14}}\varepsilon^{222}+\frac{3}{4}\sqrt{\frac{5}{14}}\varepsilon^{224}+\frac{9}{4}\varepsilon^{404} \\ \varepsilon^{T}&=&\varepsilon^{000}+\frac{\sqrt{5}}{2}\varepsilon^{202}-\frac{\sqrt{5}}{2}\varepsilon^{220}+\frac{5}{\sqrt{14}}\varepsilon^{222}-3\sqrt{\frac{5}{14}}\varepsilon^{224}+\frac{33}{8}\varepsilon^{404} \\ \varepsilon^{L}&=&\varepsilon^{000}+2\sqrt{5}\varepsilon^{202}+\sqrt{5}\varepsilon^{220}-\frac{10}{\sqrt{14}}\varepsilon^{222}+6\sqrt{\frac{5}{14}}\varepsilon^{224}+6\varepsilon^{404} \\ \varepsilon^{Z}&=&\varepsilon^{000}+\frac{\sqrt{5}}{2}\varepsilon^{202}+\sqrt{5}\varepsilon^{220}-\frac{5}{2\sqrt{14}}\varepsilon^{222}-\frac{39}{16}\sqrt{\frac{5}{14}}\varepsilon^{224}-\frac{39}{16}\varepsilon^{404} \\ \varepsilon^{S}&=&\varepsilon^{000}-\frac{\sqrt{5}}{4}\varepsilon^{202}-\frac{13\sqrt{5}}{32}\varepsilon^{220}-\frac{35}{8\sqrt{14}}\varepsilon^{222}-\frac{21}{64}\sqrt{\frac{5}{14}}\varepsilon^{224}-\frac{111}{64}\varepsilon^{404} \\ \end{array} $$

The respective moments for the analytical PES is obtained by the inversion of these relations given by

$$ \begin{array}{@{}rcl@{}} \varepsilon^{000}&=&\frac{1}{45}\left( 10\varepsilon^{H}-8\varepsilon^{X}+8\varepsilon^{T}+3\varepsilon^{L}+32\varepsilon^{S}\right) \\ \varepsilon^{202}&=&-\frac{1}{63\sqrt{5}}\left( 14\varepsilon^{H}+23\varepsilon^{X}-8\varepsilon^{T}-13\varepsilon^{L}-16\varepsilon^{S}\right) \\ \varepsilon^{220}&=&\frac{8}{45\sqrt{5}}\left( \varepsilon^{H}+\varepsilon^{X}-\varepsilon^{T}+3\varepsilon^{Z}-4\varepsilon^{S}\right) \\ \varepsilon^{222}&=&\frac{1}{45}\sqrt{\frac{2}{7}}\left( 13\varepsilon^{H}-23\varepsilon^{X}+8\varepsilon^{T}-2\varepsilon^{L}-12\varepsilon^{Z}+16\varepsilon^{S}\right) \\ \varepsilon^{224}&=&\frac{4}{45}\sqrt{\frac{2}{35}}\left( 6\varepsilon^{H}-9\varepsilon^{X}-6\varepsilon^{T}+5\varepsilon^{L}-12\varepsilon^{Z}+16\varepsilon^{S}\right) \\ \varepsilon^{404}&=&\frac{4}{315}\left( 9\varepsilon^{X}+6\varepsilon^{T}+\varepsilon^{L}-16\varepsilon^{S}\right) \end{array} $$

Appendix 2: Second virial expansion

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


$$ \begin{array}{@{}rcl@{}} B_{cl}(T) &=& \frac{-N_{A}}{4}{\int}^{2\pi}_{0}{\int}^{\pi}_{0}\sin{\theta_{1}}{\int}^{\pi}_{0}\sin{\theta_{2}}\times \\ &&{\int}^{\infty}_{0}\left( 1-\exp \left( -\frac{V}{k_{B} T}\right)\right)r^{2}drd\theta_{1}d\theta_{2}d\phi \end{array} $$
$$ \begin{array}{@{}rcl@{}} {B_{I}^{r}}(T) = \frac{N_{A}\hbar^{2}}{48\mu {k^{3}_{B}}T^{3}}&&{\int}^{2\pi}_{0}{\int}^{\pi}_{0}\sin{\theta_{1}}{\int}^{\pi}_{0}\sin{\theta_{2}}\times \\ &&{\int}^{\infty}_{0}\exp\left( -\frac{V}{k_{B} T}\right)\left( \frac{\partial V}{\partial r}\right)^{2}r^{2}drd\theta_{1}d\theta_{2}d\phi \end{array} $$
$$ \begin{array}{@{}rcl@{}} &&B_{I}^{a,I}(T)= \frac{-N_{A}}{48{k^{3}_{B}}T^{3}}{\int}^{2\pi}_{0}{\int}^{\pi}_{0}\sin{\theta_{1}}{\int}^{\pi}_{0}\sin{\theta_{2}}{\int}^{\infty}_{0}\exp\left( -\frac{V}{k_{B} T}\right)\times \\ &&\left[\sum\limits_{i=1}^{2}\frac{\hbar^{2}}{2I_{i}}\left( \cot{\theta_{i}}\frac{\partial V}{\partial\theta_{i}}+\frac{\partial^{2} V}{{\partial\theta_{i}^{2}}}+\frac{1}{\sin^{2}{\theta_{i}}}\frac{\partial^{2} V}{\partial\phi^{2}}\right)\right]r^{2}drd\theta_{1}d\theta_{2}d\phi \\ \end{array} $$
$$ \begin{array}{@{}rcl@{}} &&B_{I}^{a,\mu}(T)= \frac{-N_{A}}{48{k^{3}_{B}}T^{3}}{\int}^{2\pi}_{0}{\int}^{\pi}_{0}\sin{\theta_{1}}{\int}^{\pi}_{0}\sin{\theta_{2}}{\int}^{\infty}_{0}\exp\left( -\frac{V}{k_{B} T}\right)\times \\ &&\left[\sum\limits_{i=1}^{2}\frac{\hbar^{2}}{2\mu r^{2}}\left( \cot{\theta_{i}}\frac{\partial V}{\partial\theta_{i}}+\frac{\partial^{2} V}{{\partial\theta_{i}^{2}}}+\frac{1}{\sin^{2}{\theta_{i}}}\frac{\partial^{2} V}{\partial\phi^{2}}\right)\right]r^{2}drd\theta_{1}d\theta_{2}d\phi \\ \end{array} $$
$$ \begin{array}{@{}rcl@{}} B_{II}^{r}(T) &=& \frac{-N_{A}\hbar^{4}}{1920\mu^{2}{k^{4}_{B}}T^{4}}{\int}^{2\pi}_{0}{\int}^{\pi}_{0}\sin{\theta_{1}}{\int}^{\pi}_{0}\sin{\theta_{2}}{\int}^{\infty}_{0}\exp\left( -\frac{V}{k_{B} T}\right)\times \\ &&\left[\left( \frac{\partial^{2} V}{\partial r^{2}}\right)^{2}+\frac{2}{r^{2}}\left( \frac{\partial V}{\partial r}\right)^{2}+\frac{10}{9k_{B} T}\frac{1}{r}\left( \frac{\partial V}{\partial r}\right)^{3}\right. \\ &-&\left.\frac{5}{36{k^{2}_{B}} T^{2}}\left( \frac{\partial V}{\partial r}\right)^{4}\right]r^{2}drd\theta_{1}d\theta_{2}d\phi \\ \end{array} $$

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Barreto, P.R.P., Cruz, A.C.P.S., Euclides, H.O. et al. Spherical harmonics representation of the potential energy surface for the H2⋯H2 van der Waals complex. J Mol Model 26, 277 (2020).

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