## Thermal destruction of the initial structures

Table 1 shows the temperature limits of structural stability for all selected configurations of clusters. As noted above, in cubic polyhedra, an increase in the number of square and hexagonal faces occurs simultaneously with a decrease in the number of pentagons. We will estimate the dependence of stability on the number of squares, the number of which in such polyhedra varies from 0 to 6 [7, 12].

First, we note that for water polyhedra with the number of molecules N = 20, the temperature limits of structural stability are completely consistent with the stabilization energies at zero temperature [7]. The most stable configuration is still configuration no. 3 {444} with the largest number of square faces (Fig. 1). The configuration of regular dodecahedron no. 1, formed only by pentagonal faces, is the least stable of the three. For water polyhedra with the number of molecules N = 24, the agreement of the results is also good. The most stable configuration under the influence of temperature is still configuration no. 7 {608} with the maximum number of square faces. Unlike the results at 0 K, configuration no. 4 {0,12,2} is not the least stable. But the advantage of configuration no. 7 over all others is still very strong.

The following is a description of the main types of defects that destroy the initial polyhedral shape of the clusters. Although the clusters under study are empty, thermally induced rearrangements rarely immediately lead to a collapse of the initial structure with the appearance of distant jumper bonds and the movement of some molecules to the central part of the cluster. Structural changes begin with the appearance of surface defects. Thermal fluctuations

Table 1

The temperature limits of structural stability (K) of water polyhedra nos. 1–7 for ten realizations of the heating process

No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

1 | 110 | 150 | 170 | 130 | 110 | 110 | 150 |

2 | 110 | 130 | 150 | 110 | 130 | 110 | 130 |

3 | 130 | 130 | 150 | 130 | 130 | 90 | 130 |

4 | 130 | 150 | 130 | 130 | 90 | 130 | 170 |

5 | 130 | 90 | 150 | 110 | 90 | 110 | 150 |

6 | 110 | 150 | 170 | 130 | 90 | 110 | 170 |

7 | 130 | 130 | 150 | 130 | 90 | 130 | 150 |

8 | 130 | 130 | 170 | 130 | 110 | 90 | 170 |

9 | 130 | 130 | 170 | 130 | 130 | 130 | 190 |

10 | 130 | 150 | 150 | 110 | 130 | 110 | 170 |

Average | 124 | 134 | 156 | 124 | 110 | 112 | 158 |

often lead to temporary structural damage, after which the original structure is restored. We analyzed only fatal defects that lead to further destruction of the original structure. These defects can be divided into three main types.

The most common surface defect is the appearance of a bridge inside a pentagonal or hexagonal face (Fig. 2a). Adjacent square and triangle most often appear instead of a pentagon (sqtr-defect). Such a transformation is also observed in the simulation of an isolated cyclic pentamer [18]. In our case, it is common for a triangle to be open; one of the H-bonds is broken. At the site of such a defect, the surface of the polyhedron becomes concave. There are two main reasons for this transformation. This can be caused by intense thermal fluctuations of the free O–H group, especially if such groups are located nearby. Under the influence of temperature or the repulsive force from the neighboring free hydrogen atom, one of these atoms breaks the existing one and creates a new surface H-bond. In this case, the neighboring molecule forms a bridge bond inside the pentagonal face. In addition, the cause of such a bridge may be the deformation vibrations of the pentagonal face itself. This bond occurs when the pentagonal face becomes very narrow and elongated.

Here, it is important to emphasize that the formation of the sqtr-defect leads to the appearance of 4-coordinated molecules (Fig. 2a), although it causes strong deviations from the tetrahedral coordination of the H-bond network. The transformation of the pentagonal face into square one at the initial stage of the structure change is the most typical defect. In polyhedra with a large number of hexagons, a similar narrowing of the hexagonal ring to the pentagonal one often occurs.

The second type of fatal defect is the occurrence of an inverse O–H group, i.e. when the "dangling" hydrogen atom becomes directed towards the cluster center (Fig. 2b). This type of defect was observed quite often (in about three cases out of ten), but basically in clusters with 20 molecules. Such a rearrangement is accompanied by a local flattening of the polyhedral surface or a small deepening. Subsequently, this often leads to the appearance of a distant bridge and the destruction of the polyhedral shape. Stability over 1 ps for a structure with a reverse O–H group seems to be quite unexpected. This is primarily because the most stable proton configurations of water polyhedra were selected for the study.

Another, rarer type of surface defect is the topological transformation according to the scheme: pentagon and hexagon ◊ two squares and pentagon (Fig. 2c). Such a spontaneous structural transformation was observed, in particular, in cluster no. 3 {444}. In this case, a noticeable distortion of the tetrahedral coordination is compensated by the appearance of two 4-coordinated molecules. The resulting polyhedron {643} is no longer cubic, but its surface is convex. Having a fairly even polyhedral shape, it surpasses regular polyhedra in the number of H-bonds. It is not difficult to make sure that for an even number of molecules *N* in the initial cubic polyhedron, the number of emerging 4-coordinated molecules *k* is also even. This is because the doubled number of H-bonds is determined by the expression \(2n=4k+3\left(N-k\right)=3N+k\). Each pair of 4-coordinated molecules increases the number of H-bonds by one (Fig. 1), since in cubic polyhedra the number of bonds *n*0 is one and a half times greater than the number of vertices \(2{n}_{0}=3N\). All other types of fatal defects appeared extremely rarely. Such rare defects include, for example, an inversion with the simultaneous appearance of a distant jumper.

The spontaneous appearance of surface defects and their stability for some time means a small difference in the energies of these configurations from the initial ones. Recall that for the study, the lowest energy proton configurations were selected from a huge set of configurations that satisfy the Bernal–Fowler ice rules. Fig. 3 shows configurations of the two lower energy levels of cluster no. 1 (regular pentagonal dodecahedron) following the discrete SWEB model (106 and 1541 configurations) [8]. The total number of defect-free configurations in this cluster is 30026 [5]. The calculations were performed using the Amoeba force field. The minimum energy EMIN = -41.303 kJ/mol. The energy of geometrically optimized structure containing the sqtr-defect ESQTR = -41.248 kJ/mol (Fig. 3, circle). This is the energy of the most stable sqtr-defect of those that spontaneously occurred during the simulation. It is noteworthy that it is only slightly inferior to the energy of the lowest energy defect-free configuration of regular water dodecahedron. The energy of the geometrically optimized structure of non-cubic polyhedron with two 4-coordinated molecules ENC = -41.180 (Fig. 3, rhombus), which is also quite surprising, since this is only one of the possible polyhedra. Special calculations were performed for the defect in the form of an inverse O–H group. All O–H groups were sequentially inverted for all 106 configurations of the lower energy level in the SWEB model (1060 structures; the number of dangling hydrogens is N/2 [5]). The minimum energy of the configuration with the inverse O–H group is noticeably lower: EOH = -41,008 (Fig. 3, triangle). However, on the set of all defect-free configurations of a regular dodecahedron, such a configuration should still be considered among the most stable. Cartesian coordinates of all atoms of the three defect configurations can be found in the Supporting Information. Coordinates of all seven defect-free configurations of clusters nos. 1–7 can be found in ref. 7.

As the temperature increases, the distinction between the water clusters with the same number of molecules begins to disappear. This mainly occurs in the transition region from solid-like to liquid-like states of clusters. However, one important point needs to be made here. The states of the clusters at low temperatures are not equilibrium, because there are significantly more stable non-polyhedral structures. For example, the global minimum energy structure of a water cluster with 20 molecules represents three fused pentagonal prisms [19, 20]. A spontaneous transition to this structure, as well as to other very stable structures (finite ice nanotubes), is extremely unlikely due to the high barriers between adjacent local minima on the potential energy surface. When water polyhedra are heated, these structures practically do not manifest themselves in any way, and the transition from solid-like to liquid-like state occurs without them. It is extremely difficult to calculate the actual thermodynamic properties of such clusters [21, 22]. This is not the purpose of our work. We only evaluate the difference in stability of various polyhedral configurations at different temperatures including the region of melting-like transition.