PHYSICAL REVIEW B 101, 214303 (2020) Editors’ Suggestion Nonexistence of the decahedral Si 20 H 20 cage: Levinthal’s paradox revisited Deb Sankar De , 1 Bastian Schaefer, 1 Bernd von Issendorff, 2 and Stefan Goedecker 1 1 Department of Physics, Universität Basel, Klingelbergstr. 82, 4056 Basel, Switzerland 2 Department of Physics, Universität Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany (Received 18 January 2020; accepted 23 April 2020; published 1 June 2020) The decahedral cage is the theoretically established ground state of the hydrogen saturated Si 20 H 20 fullerene. However it has never been observed experimentally. Based on an extensive exploration of the potential energy surface and by constructing theoretical reaction pathways from possible initial structures to the ground state of Si 20 H 20 , we show that there is no driving force towards the global minimum. There exists a huge number of intermediate structures that consist mainly of collapsed cages. Visiting all these intermediate states to find the ground state is not possible on experimentally relevant time scales. In this way the ground state becomes kinetically inaccessible. We contrast the features of the potential energy landscape of Si 20 H 20 with that of C 60 which spontaneously forms by condensation. DOI: 10.1103/PhysRevB.101.214303 I. INTRODUCTION Condensed matter systems can adopt a huge number of structures. This comes from the fact that the potential en- ergy function has an equally large number of local minima, each of which corresponds to a metastable structure. Bulk materials can for instance be found in a huge number of amorphous structures, clusters in a very large number of isomers, and biomolecules in an astronomically large number of conformers. It is at present not fully understood which structure out of this huge number of theoretically possible structures can really be found in nature. This problem has extensively been discussed in the context of protein folding. The number of possible conformers grows exponentially with the number of residues. However only one configuration out of this exponentially large number of possible conformations, the configuration with the lowest free energy, is formed by a folding mechanism in living organisms and can perform its re- quired biological role. Based on a simple estimate of the time required to jump from one intermediate structure into another one, Levinthal [1] argued that the folding, i.e., the time to arrive at its correct ground state, should require a time longer than the age of the universe, even if only a small fraction of all possible configurations has to be visited as intermediates along the reaction pathway describing the folding. Since it is however experimentally well established that proteins do fold on a quite short time scale, these arguments are known as the Levinthal paradox. The Levinthal paradox was resolved by the folding funnel hypothesis [2] which shows that on a funnel like potential energy surface the system can fall into the ground state at the bottom of the funnel along a reaction pathway that contains only a modest number of intermedi- ate states. This is possible because the reaction pathway is embedded in some low-dimensional manifold of the funnel, whereas the entire funnel is a high-dimensional object in con- figuration space that consequently contains an exponentially large number of local minima. In order to establish such a short reaction pathway, it is however necessary to have a strong driving force towards the global minimum. Such a driving force gives rise to a reaction pathway whose downhill barriers are systematically lower than the uphill barriers. By definition the downhill barrier is the barrier that the system has to overcome when it jumps from one intermediate state into another one that is lower in energy, whereas the uphill barrier is the one that has to be overcome when it jumps into a higher energy state. In this way a strong directionality or driving force towards the global minimum is imposed since, according to standard transition state theory, it is more likely to cross low barriers than high barriers. It is widely believed that those proteins that have such a strong directionality were selected during the biological evolution of life because they can rapidly fold into a well defined functional structure. A fundamental question that we want to answer is whether a nanosystem with a well defined ground state will necessarily have such a funnel-like structure that will ensure that it will form quasiautomatically by some kind of physical self assembly process on a short time scale or whether, on the contrary, there exist systems whose ground state is virtually not accessible by a short directional reaction pathway on a reasonable time scale. As an example we will study the hydrogen saturated Si 20 H 20 fullerene. Nanosciences requires building blocks on the nanome- ter scale with a large variety of properties. Carbon-based nanostructures such as fullerenes, nanotubes, nanosheets, and carbon based polymers are common building blocks [3–5]. Some similar structures do exist for Si as well, namely lin- ear polysilanes, silicon nanosheets, and nanotubes [6,7], but silicon fullerenes have not been found so far. While carbon atoms can readily adjust their valence states to participate in single, double, and triple bonds, silicon strongly prefers sp 3 hybridization and single bonds. Therefore, although C 20 is the lowest stable fullerene structure, quantum chemical calculations show the Si 20 fullerene to be highly unstable [8]. The stability of a cage structure can be enhanced by var- ious modifications. One theoretically proposed possibility is 2469-9950/2020/101(21)/214303(8) 214303-1 ©2020 American Physical Society