### 3 Results and discussion

#### 3.Một General remarks on the crystal-chemistry of H_{3}PO_{2}

In the solid state, H_{3}PO_{2} exists as the tetrahedral HOP(O)H_{2} tautomer, where two H atoms are connected to P and the third H atom to O. Only the latter is *acidic*, whereas the atoms attached to P are *hydridic* and do not partake in hydrogen-bonding. The minor trigonal pyramidal HP(OH)_{2} tautomer exists for example in water or methanol solution according to the scheme.

It can be evidenced by the ready exchange of the hydridic Hs by D [10].

The P–H distances refined to ca. 1.28 Å (Table 3), which is by 0.07 Å (5%) shorter than the P–H distance in the H2PO2− (1.35 Å) ion as determined using neutron diffraction [11].

### Table 3:

T (K) | d (P–H) (Å) |
---|---|

100 | 1.250 (15), 1.277 (16) |

150 | 1.275 (15), 1.276 (15) |

175 | 1.276 (16), 1.283 (17) |

200 | 1.274 (19), 1.279 (19) |

225 | 2 × 1.273 (16) |

250 | 2 × 1.27 (2) |

275 | 2 × 1.281 (18) |

290 | 2 × 1.285 (17) |

#### 3.Hai Diffraction pattern

Before discussing the structure of H_{3}PO_{2}, let us analyze the diffraction patterns. Crystals of the HT phase can be indexed with an orthorhombic lattice and the systematic absences suggest the *P*2_{1}2_{1}Hai symmetry. On cooling, additional reflections appear at half-integer *l* values, which corresponds to a loss of translation symmetry, in particular to a doubling of the length of the *c* axis (Figure 1). These additional reflections are called *superstructure* reflections, in contrast to the *main* reflections, which are also produced by the HT phase.

The superstructure reflections are clearly observed at 200 K and absent at 225 K. Thus, the phase transition happens between these two temperatures.

The intensities of the superstrare generally weak and easily overlooked. This indicates rather subtle structural changes during the phase transition. With decreasing temperature, the relative intensity of the superstructure reflections increases, as seen in the Wilson plot in Figure 2(a). Accordingly, reliable intensity data of the superstructure reflections are available only for low-angle reflections (Figure 2(b)). As expected log (I) shows a nearly linear decrease with |s|2, owing to the atomic form factors and the thermal vibration. With increasing temperature and thus more thermal vibration, the slope becomes steeper. Interestingly, the effect is less pronounced for the superstructure reflections. This in turn means that the ratio Isuper:Ibasic increases with diffraction angle as shown in Figure 3(b). At low scattering angles, the superstructure reflections are weaker by more than two orders of magnitude; At higher scattering angles only by a factor of ca. 30.

#### 3.3 Refinement of the hydrogen bond

##### 3.3.Một The HT phase

The hydrogen bond in the HT phase is located on a twofold axis. In such a case, the hydrogen bond might either be symmetric with the H atom located on the rotation axis (O–H–O) or it might be statistically disordered ([O–H⋯O] ↔ [O⋯H–O]).

As mentioned in the introduction, conflicting information is found in the literature on this matter. 4 report of neutron diffraction measurements at −40 °C (233 K), slightly above the phase transition temperature, with indication of a splitting of the hydrogen position in Fourier maps. However, refinements carried out by 4 failed when using a split and therefore these authors attribute the apparent disorder to ‘series termination errors’. A symmetric hydrogen bond is likewise suggested by vibrational spectroscopy [12].

In this work, in all HT measurements, the maxima in difference Fourier maps were clearly split, suggesting an asymmetric bond (Figure 4). Moreover, refinements with a disorder model (H occupancy of 1/2) converged with and without distance restraints in place. The final refinements were performed *with* restraint of the O–H bond lengths, because free refinement led to unreasonably short O–H distances of ca. 0.7 Å.

Thus, with the caveat of an inherent difficulty of locating half-occupied H positions using X-ray data, we prefer a model with an asymmetric disordered hydrogen bond.

##### 3.3.Hai LT phase

In the LT phase, the hydrogen bond is ordered, *i.e.* there are distinct donor and acceptor O atoms. Figure 5 shows the difference Fourier maps obtained without the acidic H atom at the various measurement temperatures. The electron density becomes more diffuse close to the phase transition temperature, which anticipates the disordering of the hydrogen bond.

In an attempt to quantify the diffuseness of the electron density, refinements without restraining the O–H distance were performed. However, these led to unreasonably long O–H distances. While such long bonds might be due to strong hydrogen bonding [13], they are in contradiction with the observed maximum of the electron density. Refinements of the hydrogen atom as positionally disordered were inconclusive. Notably, the occupation ratio was independent of the temperature. A more detailed analysis of the dynamics and the geometry will require novel neutron diffraction experiments and/or charge-density refinements.

#### 3.4 Hydrogen bonding topology

The hydrogen bond geometries are compiled in Table 4. With an O⋯O distance of approximately 2.45 Å, the bonds are *strong* according to the classification of [15], as is expected for *resonance assisted hydrogen bonds* [16].

### Table 4:

T (K) | O–H⋯O | d (O–H) (Å) | d (H⋯O) (Å) | ∠(O–H⋯O) (°) | d (O⋯O) (Å) |
---|---|---|---|---|---|

100 | O2–H3⋯O1^{i} | 0.980 (17) | 1.475 (17) | 177 (2) | 2.4546 (11) |

150 | O2–H3⋯O1^{i} | 0.992 (16) | 1.461 (16) | 175 (2) | 2.4505 (11) |

175 | O2–H3⋯O1^{i} | 0.964 (19) | 1.488 (19) | 174 (3) | 2.4484 (13) |

200 | O2–H3⋯O1^{i} | 0.94 (2) | 1.51 (2) | 173 (3) | 2.4474 (15) |

225 | O1–H2⋯O1^{ii} | 0.85 (2) | 1.61 (2) | 169 (4) | 2.4473 (16) |

250 | O1–H2⋯O1^{ii} | 0.86 (2) | 1.60 (2) | 168 (5) | 2.444 (3) |

275 | O1–H2⋯O1^{ii} | 0.84 (2) | 1.62 (2) | 170 (4) | 2.4491 (19) |

290 | O1–H2⋯O1^{ii} | 0.84 (2) | 1.61 (2) | 172 (4) | 2.448 (2) |

On heating above the phase transition temperature, the acidic hydrogen becomes dynamically disordered in a 1:Một manner between two H_{3}PO_{2} molecules [Figure 6(b)]. The hydrogen bond strength does not change significantly at the phase transition temperature, with O⋯O distances still in the 2.45 Å range (Table 4).

Thus the symmetry relationship of the infinite chains in the HT and LT phases is of the *translationengleiche* type: The translation symmetry remains whereas the point groups of the rods are in a group/subgroup relationship of index 2 (i.e. every second point operation is lost on cooling).

The topology of a hydrogen-bonding network is conveniently represented by a graph (in the mathematical sense), where vertices represent ions or molecules and edges hydrogen bonds. In H_{3}PO_{2}, only inter-molecular hydrogen bonds exist and there is only one bond between two molecules. Therefore, the hydrogen-bonding graph is *simple*, which means that there are no loops (a vertex connected to itself) and at most one edge connecting two vertices.

Hydrogen bonds are directed (from *donor* to *acceptor*), and can be represented by directed edges, which results in a *directed* graph or, for short, *digraph*. The hydrogen-bonding topology of the LT phase of H_{3}PO_{2} is thus represented by the double-infinite digraph

(1)⋯→P→P→P→⋯

where *P* stands for a H_{3}PO_{2} molecule.

If the hydrogen bond is disordered in a 1:Một manner, as in the HT phase of H_{3}PO_{2}, it may be represented as a non-directed edge, resulting in the *undirected* graph

(2)⋯—P—P—P—⋯

This particular graph is the *double-infinite linear* graph.

Thus, from a topological point of view, the symmetry increase from the LT to the HT phase corresponds to the mapping of a digraph onto the corresponding undirected graph. In mathematics, such a mapping is said to ‘lose’ algebraic structure, namely the direction of the edges.

Each edge is marked with a label, which is called the *voltage*, which indicates the translational component needed to move the molecule into the unit cell when crossing unit-cell boundaries. In the graph above, the 1‾00 label thus means that the molecule in the unit cell is connected via a hydrogen bond to a molecule outside the unit cell. This molecule is translated back into the unit cell by a translation of −**a**. In analogy to the infinite graphs, the symmetry increase is again characterized by a loss of algebraic structure, *viz.* the directionality of the edges.

#### 3.5 Space group symmetry

Our experiments confirm the *P*2_{1}2_{1}Hai space group symmetry of the HT phase described by 4. The chains of hydrogen-bonded H_{3}PO_{2} molecules are arranged in a checkerboard pattern as shown in Figure 7(a). The chains alternately appear in two orientations with respect to [001], which is the direction of the twofold axes.

As has been mentioned above, on transition from LT to HT, directionality in the [100] direction is lost. In return, given a HT chain, the corresponding LT chain can adopt one out of two orientations with respect to the *a*-axis.

There is in principle an infinity of ways of distributing the two orientations, which can lead to either loss of translational symmetry, point group symmetry, or both with respect to the HT phase.

In the actual crystals, the orthorhombic 222 point group is retained and the translation symmetry is reduced. This corresponds to a *klassengleiche* symmetry reduction, i.e. the crystal class stays unchanged. In particular, the *c* cell parameter is doubled, because the chains related by a **c** translation in the HT phase possess opposite orientation of the hydrogen bonding in the LT phase [Figure 7(b)].

Note that whereas the point symmetry of the overall structure is retained on cooling (*klassengleiche* transition), the point symmetry of the individual H_{3}PO_{2} chains is reduced while the translation lattice is retained (*translationengleiche* transition). However, the lost twofold rotation of the chains is retained as a 2_{1} screw rotation of the overall crystal structure, resulting in an overall *P*2_{1}2_{1}2_{1} symmetry.

The ordering of the hydrogen bonding network in the LT phase is accompanied by a very subtle tilting of the H_{3}PO_{2} molecules with respect to the HT phase owing to the loss of the twofold rotation (Figure 7). It can be quantified by the angle of the O–O segment of the H_{3}PO_{2} molecule to the (001) plane. As expected, the deviation from the HT symmetry is more pronounced the lower the temperature [100 K: 3.11°; 150 K: 2.83°; 175 K: 2.54°; 200 K: 1.92°; ≥225 K: 0°]. The larger deviation from the HT symmetry causes higher-intensity superstructure reflections.

#### 3.6 Twinning

Both H_{3}PO_{2} phases are enantiomorphous. Their 222 point group is a merohedry, *i.e.* the symmetry is a subgroup of the *mmm* point symmetry of the lattice. This means that in principle the crystals could be twinned by merohedry, here by inversion. Unfortunately, the present data is not of sufficient quality to confirm or preclude such twinning. Even though the Flack parameter [18] refines to zero within the standard uncertainty, the latter is too large for any definite statement (ca. 0.2–0.3). This also means that the absolute structure could not be determined.

From a crystal-chemical point of view, such a twinning is unlikely. The hydrogen-bonded H_{3}PO_{2} chains are enantiomorphous (Figure 6) and all chains in the crystal are of the same orientation (both phases crystallize in Sohnke space groups). Twinning by inversion would require a structural transition from one chain enantiomorph to the other.

Since the phase transition from HT to LT is of the *klassengleiche* type, i.e. the point symmetry is retained, there are no additional orientation states in the LT phase. Thus, the phase transition does not lead to twinning on cooling. On the other hand, due to loss of translation symmetry, there are two domain states of the LT phase, which are related by a **c** translation of the HT phase. An association of such domains is known as *antiphase* domains [19] and difficult to detect with routine diffraction methods. Notably, in the diffraction patterns of LT H_{3}PO_{2} we did not observe broadening of reflections, which would be indicative of small domain sizes.

#### 3.7 Evolution of lattice parameters

Figure 8 gives the evolution of the *a*, *b* and *c* cell parameters with temperature. There is no distinct point of discontinuity at the phase transition temperature. Whereas the *b* and *c* cell parameters increase with temperature, the *a* cell parameter is virtually constant.