Atoms and molecules in soft confinement potentials

ABSTRACT

We present a detailed non-relativistic study of the atoms H, He, C and K and the molecule CH${}_4$ in the centre of a spherical soft confinement potential of the form $V_N(r)=(r/r_0{)}^N$ with stiffness parameter N and confinement radius $r_0$. The soft confinement potential approaches the hard-wall limit as $N\to \mathrm{\infty }$, giving a more detailed picture of spherical confinement. The confined hydrogen atom is considered as a base model: it is treated numerically to obtain ground- and excited-state energies and nodal positions of the eigenstates to study the convergence towards the hard-wall limit. We also derive some important analytical relations. The use of Gaussian basis sets is analysed. We find that, for increasing stiffness parameter N, the convergence towards the basis-set limit becomes problematic. As an application, we report dipole polarisabilities for different values of N and $r_0$ of hydrogen. For helium, we determine electron correlation effects with varying N and $r_0$, and discuss the virial theorem for both soft and hard confinements in the limit $r_0\to 0$. For carbon, a change in the orbital population from 2s${}^2$2p${}^2$ to 2s${}^0$2$p^4$ is observed with decreasing $r_0$, while, for potassium, we observe a change from the ${}^2$S to ${}^2$D ground state at small $r_0$ values. For CH${}_4$, we show that the one-particle density becomes more spherical with increasing confinement. A possible application of soft confinement to atoms and molecules under high pressure is discussed

Prof. Jürgen Gauss observing Schrödinger's cat under quantum confinement.

GRAPHICAL ABSTRACT

PACS CLASSIFICATIONS:

• 37.30.+i
• 32.10.-f
• 31.15.ap
• 31.70.Dk

Acknowledgments

This work is in memory of Michael Wormit and Boris Pavlov, who both passed away at an early stage of this project in 2015. The authors thank Jonas Wiebke, Kyle Beloy, Dirk Andrae and Mathieu Lewin for fruitful discussions. The authors are grateful to the Centre for Advanced Study at the Norwegian Academy of Science and Letters, Oslo, Norway, where much of this work was carried out under the project ‘Molecules in Extreme Environments’ during the academic year 2017–2018.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

PS is indebted to the Royal Society of New Zealand for financial support in terms of a Marsden Fund (14-MAU-034). DS acknowledges financial support from Magnus Ehrnrooth Foundation, Swedish Cultural Foundation in Finland, and the Academy of Finland (275845 and 314821). TH acknowledges support by the Norwegian Research Council (Norges Forskningsråd) through the CoE Hylleraas Centre for Quantum Molecular Sciences Grant No. 262695. LFP is grateful for the support from the Slovak Research and Development Agency (Agentúra na Podporu Výskumu a Vývoja) (APVV-15-0105) and the Scientific Grant Agency of the Slovak Republic (Vedecká Grantová Agentúra MŠVVaŠ SR a SAV) (1/0777/19).