# Calculated vibration spectrum of calcium hexahydroxodizincate dihydrate (qatranaite)

Andrei Postnikov, Anna Majtyka-Pilat, Dariusz Chrobak, Józef Deniszczyk

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Andrei Postnikov, Anna Majtyka-Pilat, Dariusz Chrobak, Józef Deniszczyk. Calculated vi-bration spectrum of calcium hexahydroxodizincate dihydrate (qatranaite). Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy [1994-..], 2024, 318 (25), pp.124414. 10.1016/j.saa.2024.124414. hal-05244991

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# Calculated vibration spectrum of calcium hexahydroxodizincate dihydrate (qatranaite)

Andrei Postnikov a,b,*, Anna Majtyka-Piłat b, Dariusz Chrobak b, Józef Deniszczyk b

<figure><img src="image_3.png" /></figure>

## GRAPHICAL ABSTRACT

<figure><img src="image_4.png" /></figure>

Zone-center phonons are calculated and analysed using projections of eigenvectors according to different criteria.

<figure><img src="image_5.png" /></figure>

Representative H-O-H scissoring mode (above) and Zn-O1-H1 bending mode (below)

## HIGHLIGHTS

* Projection of vibration eigenvectors according to different criteria is suggested.

* Contributions from different stretchings and bendings are naturally identified.

* Molecular-type vibrations make almost degenerate symmetry-constrained combinations.

* Vibrations throughout metal-oxygen cages make a dense quasi-continuous spectrum.

## ARTICLE INFO

**Keywords:**

Lattice vibrations

Raman spectroscopy

Density functional theory

## ABSTRACT

On the basis of first-principles electronic structure calculations, crystallographic parameters have been refined for calcium hydroxozincate (Qatranaite mineral), and the vibration properties (frequencies and eigenvectors) calculated. A detailed analysis of vibration modes is done, in the context of comparison with infrared and Raman spectra previously available. Special attention is paid to a *posteriori* symmetry analysis of vibration modes, discussing the latters' attribution to four irreducible representations of the *P*2₁/c space group, and to identifying stretchings and bendings of particular chemical bonds, pronounced in different vibrations. It turns out that high-frequency (>700 cm⁻¹) vibrations of hydroxyl groups bridging the Ca or Zn cations differ quite considerably for crystallographically distinct hydroxyl positions. It is shown that the vibrations involving hydroxyl groups and crystalline water typically come about in quadruplets at very close frequencies, whereby different irreducible representations reflect different combinations of similar “molecular” vibrations of four

<footnote>a LCP-A2MC, Université de Lorraine, Bd Arago 1, F-57078 Metz Cedex 3, France</footnote>

<footnote>b Institute of Materials Engineering, University of Silesia in Katowice, 75 Pułku Piechoty 1A, 41-500, Chorzów, Poland</footnote>

<footnote>* Corresponding author at: LCP-A2MC, Université de Lorraine, Bd Arago 1, F-57078 Metz Cedex 3, France.
*E-mail addresses:* andrei.postnikov@univ-lorraine.fr (A. Postnikov), jozef.deniszczyk@us.edu.pl (J. Deniszczyk).</footnote>
identical entities (of each hydroxyl or water) present in the unit cell. However, some vibrations show exceptions from this rule. In addition to interpretation of earlier experimental investigations, our study indicates that the low-frequency (< 700 cm⁻¹) vibrations within the cation–hydroxyl connected skeleton are of more “solid-state-like” character and cannot be reasonably interpreted in terms of “molecular” vibrations within ZnO₄ or CaO₆ units.

# 1. Introduction

The material of our present interest, CaZn₂(OH)₆·2H₂O, to be called in short calcium hydroxozincate (CHZ), got attention as an intermediate phase which forms in the process of (intentional) retarding of the setting of cement via adding zinc compounds. This attribution is probably due to Lieber and Gebauer [1]; research works centring at CHZ and relevant for building construction have been underway till now – see, e.g., Ref. [2,3] and references therein. Notably the use of CHZ for immobilising inorganic heavy metal sludge in cement-based treatment systems was mentioned in Ref. [2]. In a quite different context, CHZ compound became subject of extensive research due to its potential application as an active material in secondary Zn-pasted electrodes employed in batteries [4–7]. It turned out that the use of CHZ helps to improve charge/discharge reversibility in Zn-pasted battery electrodes, reducing dendrite deposit formation and resisting Zn-electrode shape change [8–10]. Furthermore, CHZ is mentioned in the literature as an efficient reversible negative electrode material for rechargeable zinc-air batteries [11,12]. Besides these electrochemical applications, CHZ is currently regarded as a potential material for protection of limestone and dolostone monuments because of its antifungal activity [13] and as a solid catalyst for biodiesel synthesis [14–16].

Liebau and Amel-Zadeh [17], presumably for the first time, prepared monocrystals and identified crystal structure as an array of corner-coupled tilted Ca-centered octahedra and Zn-centered tetrahedra. Tien-Chih Lin et al. [18] refined synthesis and structure characterisation, measured Fourier transform infrared (FT-IR) spectrum and assigned some vibration lines. Stahl and Jacobs [19] reported the crystal structure of CHZ (in good agreement with that of Ref. [17]), refined to include the hydrogen positions. Xavier et al. [20] described synthesis of CHZ microparticles (by microwave-hydrothermal method) and inspected their crystal structure, which turned to be nearly identical to that of Ref. [19].

In 2016, a natural mineral of the same chemical composition and the same space group as the synthetic compound previously studied has been discovered and named qatranaite [21]. Vapnik et al. [22] identified the mineral's crystal structure (including the hydrogen positions), measured its Raman spectrum and tentatively assigned some vibration lines. Majtyka-Piłat et al. [23] calculated electronic structure of CHZ from first principles within the density functional theory (DFT) and optimised the crystallographic parameters (excluding the positions of hydrogen atoms which were frozen), yielding a fair agreement (not entering here into discussing the details) with the results of the experimental works cited.

To the best of our knowledge, only a few studies of lattice vibrations in CHZ have been so far done. FT-IR spectroscopy results for synthetic CHZ have been reported by Lin et al. [18]. A micro-Raman spectra of nano-sized CHZ powders, in comparison with ZnO and Ca(OH)₂, were discussed by Yang et al. [10]. Raman spectrum of the natural mineral was published by Vapnik et al. [22]. All these three works suggested attribution of (groups of) modes, based on general experience with similar compounds. However, no calculation insight was so far available. The ambition of our present work is to offer an interpretation of vibration spectrum from first principles, analysing the IR or Raman activity of different modes. The vibration patterns of modes will be confronted with “common sense” assessment of their character, formulated in the experimental works cited. The calculation of vibrations is preceded by unconstrained optimisation of crystal structure, including (for the first time) the hydrogen positions.

The paper is organised as follows. Section 2 specifies the technical details of electronic structure calculation. Section 3 explains the analysis of vibration modes and different types of their eigenvectors' projection. Section 4 elucidates the crystal structure and its optimisation from first principles. Section 5 outlines the calculation results, in their comparison with experimental spectra. Section 6 summarises the findings.

# 2. Setup of electronic structure calculations

First-principles calculations of electronic structure, including forces on atoms and stress tensor, were performed using the (version 4.0 of) SIESTA method [24,25]. Exchange-correlation was treated within the generalised gradient approximation (GGA), in the formulation by Perdew-Burke-Ernzerhof [26,27]. The basis set of the “double-zeta with polarisation orbitals” quality was generated with the standard SIESTA procedure, using the PAO.EnergyShift parameter of 0.015 Ry, that resulted in the basis functions of the following maximal extension (in Bohr units): 7.999 (Ca), 6.195 (Zn), 4.037 (O), 4.951 (H). The MeshCutoff parameter that controls the fineness of real-space grid for sampling the charge density throughout the unit cell was chosen at 313 Ry (that yielded the grid step of about 0.09 Å); the effective k-space cutoff for Brillouin zone summation was taken at 31.7 Å, that corresponded to (10 × 6 × 11) divisions along the reciprocal lattice vectors. These parameters were subject to usual accuracy and convergence tests. After an optimisation of the ground-state structure (lattice parameters and atomic coordinates), the force constants have been calculated by applying six ±(x, y, z) trial Cartesian displacements of ±0.04 Bohr to each atom consecutively and registering the forces induced on all atoms in the unit cell. No symmetry analysis or symmetry-related simplification was done at this stage. The straightforward diagonalisation (within the Vibra tool being part of SIESTA package) yielded vibration frequencies and eigenvectors, which were then analysed as described in the following section.

# 3. Projection of vibration modes according to different criteria

If the calculation is done on a unit cell of the material in question (and not its replicated supercell), the hence resulting frequencies and eigenvectors would correspond to zone-centre ($\Gamma$) phonons, which can be probed in experiment by IR or Raman spectroscopy.

In case of hexahydroxodizincate dihydrate, the frozen phonon calculation at $\Gamma$ yields a dense spectrum of (42[atoms per unit cell]$\times$3 = 126) vibration modes, of which three are acoustic (hence pure rigid translations, yielding zero frequencies within the “numerical noise” tolerance level) and the rest are optical. Non-trivial vibration patterns within optical modes can be analysed in different ways. In principle, each vibration mode, on the basis of its eigenvector, can be visualised and directly inspected. This, however, may be of limited usefulness, because of mixed character of the majority of modes. Relevant information can be extracted, according to different criteria, by a projection technique which was earlier explained in Ref. [28] and implemented in post-processing tools developed for use with the SIESTA package [29] (and can be straightforwardly adapted to eigenvectors issued from any phonon calculation scheme). The projection onto different values of the phonon wave vector $q$, previously used on a number of occasions in order to select “genuine” zone-centre modes from a calculation done for
an enlarged supercell [30], or in order to reveal dispersion-like features in a non-perfectly-periodic alloy system [31],

$$I_N(\omega, \mathbf{q}) = \sum_i \left| \sum_{a \in \mathcal{N}} \mathbf{A}_i^a(\omega) \exp(i\mathbf{q}\mathbf{R}_a) \right|^2 \delta(\omega - \omega_i) \quad (1)$$

(a: atoms within an entity $\mathcal{N}$; i: vibration modes) are not relevant here, because our system of study is nominally periodic, and the calculation is done on a primitive unit cell, hence yields zone-centre phonon modes *a priori*. A natural decomposition of the vibration spectrum would be according to participation of different chemical species in the eigenvectors, e.g.,

$$I_N(\omega) = \sum_i \sum_{a \in \mathcal{N}} |\mathbf{A}_i^a(\omega)|^2 \delta(\omega - \omega_i), \quad (2)$$

which will open our discussion of the vibration properties in Section 5 below. Whereas the zone-centre vibration spectrum $\omega_i$ is discrete, an artificial broadening with a convenient function $\delta(\omega)$ is applied for better visibility of the spectral weights' distribution.

A discrimination of vibration modes can be done based on their compatibility of one or another irreducible representation (irreps) of the space group. The symmetry analysis can reasonably be done in advance, in order to identify the symmetry coordinates (i.e., combinations of Cartesian displacements of different atoms) which will make a basis of each irrep. Then, harvesting of force constants and subsequent diagonalisation can be done for each irrep independently. Otherwise, the symmetry analysis can be done *a posteriori*, by projecting the phonon eigenvectors $A_i^a$ onto symmetry coordinates $S_i^{av}$ describing the combined movement of equivalent (symmetry-related) atoms *a* of the (sub)species $\mathcal{N}$ within the irreducible representation $v$:

$$I_N^v(\omega) = \sum_i \left| \sum_{a \in \mathcal{N}} A_i^a S_i^{av} \right|^2 \delta(\omega - \omega_i), \quad (3)$$

as was suggested in Ref. [28]. The scalar product concerns the Cartesian components of the eigenvector and of the symmetry coordinates. These latter can be consulted e.g. at the web resources of the Bilbao Crystallographic Center [32,33], in the Solid State Theory Applications (SAM) section [34].

A yet different way of analysis will be an identification of stretching or bendings of particular (groups of) bonds. A direct inspection of vibration patterns might be complicated by a mixed character of modes. Here again, a projection technique can be useful. A (taken to square) stretching contribution for a bond between atoms *a* and *β* situated along the normal direction $\mathbf{n}^{a\beta} = (\mathbf{R}^\beta - \mathbf{R}^\gamma)/R^{a\beta}$ will be

$$I_{\alpha-\beta}(\omega) = \sum_i \left[ \left( \frac{\mathbf{A}_i^\beta}{\sqrt{m_\beta}} - \frac{\mathbf{A}_i^\alpha}{\sqrt{m_\alpha}} \right) \mathbf{n}^{a\beta} \right]^2 \delta(\omega - \omega_i). \quad (4)$$

The summation can then be done over all the bonds of interest representing the $(\alpha-\beta)$ species. The division by square roots of masses was needed in order to express the “genuine” atom displacements from the vibration eigenvectors. In previous examples this was not necessary because the contributions from different species came out separately, and respective normalisations of $I_N$ were not related.

The “weights” of bending of $\alpha-\beta-\gamma$ bonds (the $\beta$ atom in the middle) can be expressed as (taken to square) increments of the bending angle $\theta$, or specifically (see Appendix)

$$I_{\beta<\gamma}^{\alpha}(\omega) = \sum_i \left[ \left( \frac{\mathbf{A}_i^\alpha}{\sqrt{m_\alpha}} - \frac{\mathbf{A}_i^\beta}{\sqrt{m_\beta}} \right) \frac{\mathbf{n}^{\beta\alpha} \cot \theta - \mathbf{n}^{\beta\gamma} \csc \theta}{R^{\alpha\beta}} + \left( \frac{\mathbf{A}_i^\gamma}{\sqrt{m_\gamma}} - \frac{\mathbf{A}_i^\alpha}{\sqrt{m_\alpha}} \right) \frac{\mathbf{n}^{\beta\gamma} \cot \theta - \mathbf{n}^{\alpha\beta} \csc \theta}{R^{\gamma\beta}} \right]^2 \delta(\omega - \omega_i). \quad (5)$$

In the following, we present the calculated vibration spectrum and discuss the “nature” of its different parts, under an angle of a comparison with available experimental data.

<table><caption>**Table 1**
Lattice parameters and bond lengths according to experimental works and the present *SiESTA* calculation. (Hydrogen bonds are indicated by dots).</caption><thead><tr><th>Property</th><th>Exp. [19]<br>(synthetic)</th><th>Exp. [22]<br>(mineral)</th><th>Calc.<br>(present)</th></tr></thead><tbody><tr><td>a (Å)</td><td>6.372</td><td>6.389</td><td>6.355</td></tr><tr><td>b (Å)</td><td>10.940</td><td>10.969</td><td>10.856</td></tr><tr><td>c (Å)</td><td>5.749</td><td>5.759</td><td>5.844</td></tr><tr><td>β (°)</td><td>101.94°</td><td>101.95°</td><td>106.5°</td></tr><tr><td>Bond lengths (Å)</td><td></td><td></td><td></td></tr><tr><td>Ca—O2</td><td>2.365</td><td>2.371</td><td>2.400</td></tr><tr><td>Ca—O3</td><td>2.329</td><td>2.335</td><td>2.405</td></tr><tr><td>Ca—O4</td><td>2.358</td><td>2.366</td><td>2.398</td></tr><tr><td>Zn—O1</td><td>1.965</td><td>1.972</td><td>2.000</td></tr><tr><td>Zn—O1</td><td>1.988</td><td>1.990</td><td>2.030</td></tr><tr><td>Zn—O2</td><td>1.948</td><td>1.955</td><td>2.003</td></tr><tr><td>Zn—O3</td><td>1.936</td><td>1.942</td><td>1.990</td></tr><tr><td>O1—H1</td><td>0.85</td><td>0.899</td><td>1.044</td></tr><tr><td>O2—H2</td><td>0.68</td><td>0.899</td><td>0.997</td></tr><tr><td>O3—H3</td><td>0.68</td><td>0.900</td><td>1.033</td></tr><tr><td>O4—H5</td><td>0.84</td><td>0.899</td><td>1.084</td></tr><tr><td>O4—H5</td><td>0.84</td><td>0.899</td><td>1.084</td></tr><tr><td>O1···(H4)-O4</td><td>2.704</td><td>2.71</td><td>2.587</td></tr><tr><td>O2···(H1)-O1</td><td>2.795</td><td>2.80</td><td>2.610</td></tr><tr><td>O3···(H5)-O4</td><td>2.698</td><td>2.70</td><td>2.536</td></tr><tr><td>O4···(H3)-O3</td><td>2.911</td><td>2.91</td><td>2.643</td></tr></tbody></table>

<table><caption>**Table 2**
Optimised coordinates of atoms after GGA *SiESTA* calculation. Lattice parameters are specified in Table 1.</caption><thead><tr><th>Site</th><th>x</th><th>y</th><th>z</th><th>Site</th><th>x</th><th>y</th><th>z</th></tr></thead><tbody><tr><td>Ca</td><td>0</td><td>0</td><td>0</td><td>H1</td><td>0.515</td><td>0.599</td><td>0.261</td></tr><tr><td>Zn</td><td>0.533</td><td>0.670</td><td>0.659</td><td>H2</td><td>0.360</td><td>0.543</td><td>0.913</td></tr><tr><td>O1</td><td>0.438</td><td>0.678</td><td>0.297</td><td>H3</td><td>0.918</td><td>0.634</td><td>0.912</td></tr><tr><td>O2</td><td>0.369</td><td>0.529</td><td>0.747</td><td>H4</td><td>0.185</td><td>0.661</td><td>0.223</td></tr><tr><td>O3</td><td>0.854</td><td>0.643</td><td>0.729</td><td>H5</td><td>0.952</td><td>0.737</td><td>0.213</td></tr><tr><td>O4</td><td>0.018</td><td>0.646</td><td>0.198</td><td></td><td></td><td></td><td></td></tr></tbody></table>

## 4. Crystal structure

The definitions of the crystal structure reasonably agree in the experimental works cited; however, some differences in the convention can be pointed out. Ref. [17] cites an acute crystallographic angle $\beta = 78.02^{\circ}$, whereas later works give an obtuse one, $\beta \approx 101.9^{\circ}$, i.e., complementary to $\pi$; consequently, the *x* coordinates in Ref. [17] correspond to $(1-x)$ in later works. Definitions of O1 and O3 sites in Ref. [19] and subsequent works are reversed with respect to Ref. [17]. Sites H41 and H42 (hydrogen atoms of the water molecule) become H4 and H5 respectively in Ref. [22,23]. The coordinates optimised in Ref. [23] are compared with experimental ones, presumably those of Ref. [22] (cf. Table 2 of Ref. [23]); in fact these cited numbers differ from the original ones in the Table 3 of Ref. [22] by a $(0 \frac{1}{2} \frac{1}{2})$ internal translation (that connects the two equivalent Ca positions), with occasional errors of up to 0.01 slipped into some coordinates. Anyway, there is no controversy as for the space group $(P2_1/c; Nr 14)$ and the Wyckoff positions. Occupied are $4e$ (Zn; four distinct O; five distinct H sites) and $2a$ (Ca), that makes $2 \times [CaZn_2(OH)_6 \cdot 2H_2O]$ per unit cell.

In our present calculation, we departed from the coordinates as stipulated in the convention of Ref. [23], hence with $(0 \frac{1}{2} \frac{1}{2})$ translation; the structure was then fully relaxed, so some inaccuracies of initial coordinates were not relevant.

The results of our *SiESTA* optimisation are outlined in Tables 1 (lattice parameters and interatomic distances) and 2 (coordinates of Wyckoff positions). Bond lengths from experimental works [19,22] are reproduced in Table 1 for the sake of convenient discussion; the (less enlightening) atom coordinates can be consulted in the works cited.

The crystal structure, otherwise depicted in Fig. 1 of Ref. [23] and in Fig. 5 of Ref. [22], is shown in our Fig. 1 in such a way as to underline
<figure><img src="image_6.png" /><figcaption>Fig. 1. Crystal structure of qatranaite (as optimised in first-principles calculation), with monoclinic unit cell shown and inequivalent atomic positions marked. Two Ca atoms (large blue spheres) and four Zn atoms (grey spheres) of the unit cell are shown with full coordination by hydroxyl and water molecules; this results in seven OH groups being shown twice. The corresponding doubled pairs are labelled by different colours; the unique (not repeated) sites are labelled in black. Hydrogen bonds are drawn by thin lines. See text for discussion. The figure is generated with the VESTA software [35,36].</figcaption></figure>

the full coordination and connectivity of every metal atom. As a result, seven hydroxyl groups are lattice-translated and shown in duplicate; for better perception, such pairs are labelled in distinct colours. The counting of atoms in Fig. 1 will then lead to the nominal chemical formula as follows:

$$2 \times [\text{Ca(OH)}_4(\text{H}_2\text{O})_2] + 3 \times [\text{Zn(OH)}_3] + \text{Zn(OH)}_2 - 7 \times (\text{OH}) \\ \rightarrow 2 \times [\text{CaZn}_4(\text{OH})_6 \cdot 2\text{H}_2\text{O}].$$

From Ref. [19] on, the notation of distinct O and H sites in the literature is consistent: O1–H1 are hydroxyl groups at the shared corners of two Zn-centered tetrahedra; O2–H2 and O3–H3 are (distinct) hydroxyl groups at common corners of Zn-centered tetrahedra and Ca-centered octahedra; H4–O4–H5 are water molecules at two opposite (not further connected) vertices of Ca-centered octahedra. Further on, the hydrogen bonds O1···H4, O2···H1, O3···H5 and O4···H3 are formed in the lattice. (The latter two do not come about in Fig. 1 because this would demand to show more translated structure elements, but they can be seen in Fig. 5 of Ref. [22]). H2 do not participate in hydrogen bonds.

One can note good agreement of our present GGA calculation with experimental works in what regards the lattice parameters; the unit cell volume is 387.7 Å³, to be compared with 392.0 Å³ of Ref. [19] and 394.85 Å³ of Ref. [22], i.e., within 2% of experimental estimation. The crystallographic angle, 4.5% off, may seem problematic, but such things happen in anisotropic systems, as errors in optimised interatomic distances occur unevenly in different directions. The Ca–O and Zn–O bond lengths are accurate (slightly overestimated) within 2% of experimental values. Such agreement can be considered acceptable for a state-of-art DFT calculation not employing special effort to push for ultimate chemical accuracy, e.g., ad hoc selection of special tuning of exchange–correlation potentials, or of dispersion interaction schemes. A systematic overestimation of O–H bond lengths may be seen as a drawback, probably attributable to non-inclusion of dispersion interactions. A similar overestimation of O–H mean distance is visible in simulations of liquid water by Fernández-Serra et al. [37], done with SiESTA method, like in our present study. O···O hydrogen-mediated “bonds” (cf. the four concluding lines of Table 1) come out somehow shortened, as compared to experiment. As a result, the protons are placed more “half-way” in between the two oxygen atoms than the experimental measurement suggests. It is not our priority in the present work to strive for ultimate accuracy in O–H bond length determination, the more so that this is a special issue of modern computational efforts, see e.g. Ref. [38,39]. Tolerating the “standard” DFT approach, however, we can already anticipate a red shift in our predicted O–H stretching frequencies, by force of a quite general observation, markedly illustrated in Fig. 2 of Ref. [39] for O–H bonds, that dilating the bond weakens the force constant and hence the corresponding frequencies.

<figure><img src="image_7.png" /><figcaption>Fig. 2. Density of modes (a discrete spectrum, artificially broadened for better visibility) weighted according to participation of different species in the corresponding vibration eigenvector. Three acoustic modes with zero frequency are removed. The inset zooms onto the low-frequency part of the spectrum.</figcaption></figure>

## 5. Phonon calculation results

The symmetry analysis within the $P2_1/c$ space group tells us that the vibrations of atoms in (2a) Wyckoff positions (Ca, in the present case) make three $A_u$ and three $B_u$ vibration modes, all being IR active. Each group of symmetry-related atoms in (4e) positions (i.e., all Zn, O, and H) contributes to all $A_g$, $A_u$, $B_g$ and $B_u$ irreps, with three vibration modes in each. $A_g$ and $B_g$ modes are Raman active and not IR active. Given, in total, 10 distinct (4e) positions and one (2a), this amounts to the following reduction of the $\Gamma$ representation (that in the basis of individual atomic displacements) and counting of vibration modes: $\Gamma = 30A_g \oplus 30B_g \oplus 33A_u \oplus 33B_u$, i.e., 126 modes in total.

The whole zone-centre vibration spectrum, resolved into contributions from different chemical species according to Eq. (2), is shown in
Bending and stretching of O-H bonds

<figure><img src="image_8.png" /><figcaption>**Fig. 3.** Projections of vibration eigenvectors onto different (O–H) stretchings and (Cation–O–H) bendings. Scales of individual plots are arbitrarily adjusted for convenience. See text for discussion.</figcaption></figure>

**Fig. 2.** The vibrations of calcium, zinc and oxygen (in what regards the latter's bonds with Ca and Zn) are confined within the frequency range 100 to 600 cm⁻¹. This part of the spectrum is shown in the inset. The frequency range from 700 to 3500 cm⁻¹ is covered by (inequally spaced) vibrations involving O and H. In order to “understand” the nature of different vibration modes, their immediate one-by-one inspection can be done; for a preliminary overfly, projections onto different stretchings and bendings, calculated along Eqs. (4) and (5) respectively and shown in Fig. 3, may be instructive. Fig. 4 depicts projections, calculated along Eq. (3), of vibration spectrum onto different irreps, resolved moreover into site-specific atom-related contributions.

5.1. Vibrations involving hydroxyl groups and water molecules

We start our discussion with high-frequency part (from 700 cm⁻¹ upwards). A general observation is that, since each species (Wyckoff position) is present in quadruple, every given O–H vibration comes about in four symmetry-distinct combinations of its “molecular” manifestations, according to four irreps of the space group. This neatly splits the general spectrum into four (two Raman active and two IR active) irrep-resolved constituents. Some of vibrations which come about in four panels of Fig. 4 are nearly degenerate (e.g., all H1-related modes); the other species-resolved contributions exhibit a similar distribution of weights throughout the frequency range in different irreps, whereas their individual lines are markedly shifted. We discuss them group by group, in the context of suggested experimental assignment.

The fastest vibration (at ~3450 cm⁻¹) in each irrep is stretching of O2–H2 hydroxyl groups (those bridging Zn to Ca), as is evidenced by Fig. 3. A snapshot of one such mode is shown in Fig. 5d. Lin et al. [18] assign measured “sharp bands at 3495 and 3615 cm⁻¹ ... to ... the O–H stretching vibrations”, not discriminating Zn–(OH)–Ca from Zn–(OH)–Zn bridges. Vibrations detected at even higher frequencies (from 3770 cm⁻¹ on) in the IR spectrum of Lin et al. were attributed by these authors “to O–H stretching vibrations due to discrete noninteracting OH groups on the surface”, obviously missing in our calculation. Vapnik

et al. [22] ascribe their Raman line at 3624 cm⁻¹ to O2–H2 vibration and that at 3497 cm⁻¹ – to O1–H1 ... O2 and O3–H3 ... O4 vibrations.

A broad systematisation of vibration frequencies within the hydroxyl ion in different substances, in dependence on the metal-oxygen bond length and other aspects of bonding, was proposed in Ref. [40]. The fundamental frequency of the free OH⁻ ion, 3555.6 cm⁻¹, is often hardened by the presence of metal ion to which oxygen is bonded, the effect being the stronger the shorter the metal-oxygen bond is; this is in line with elevated frequencies of hydroxyl stretching vibrations at the surface. However, a softening of O–H vibration occurs “when hydrogen is bonded” (see Ref. [41]). Out of three distinct hydroxyl groups in the structure, O2–H2 is singled out in that its hydrogen is not involved in hydrogen bond, so its stretching vibration is closest to the fundamental hydroxyl one. Still, since all oxygen atoms participate in hydrogen bonds (see Table 1), the stretching vibrations are softened.

At considerably lower frequencies, ~2840 cm⁻¹, we find the stretching vibrations within the O3–H3 hydroxyl group, the other one bridging Zn to Ca. These vibrations resonate with symmetric stretching of water molecules. The stretching of O1–H1 bonds – those within hydroxyls of the Zn–(OH)–Zn bridges – occurs at yet lower frequency, 2600–2700 cm⁻¹, and mixes with a solo movement of H4 atoms from the water molecule towards their respective nearest O1. In fact it looks like “symmetric stretching” (see Fig. 5c) within the H1–O1 ... H4 groups. The immediately following modes (on going downwards in frequency), those at 2470–2500 cm⁻¹, represent similar O1–H1 stretching, now organised “antisymmetrically” with respect to hydrogen from a neighbouring water molecule (see Fig. 5b), to produce a chained O4–H4 ... O1–H1 (oxygens agains hydrogens) movement.

It is interesting that these stretching vibrations of hydroxyl groups, otherwise represented by peaks in Fig. 3, occur within “similar” fragments assembled by hydrogen bonds into different patterns, i.e., –H5···O3–H3···O4– and –H4···O1–H1···O2–. The effect of hydrogen bond cooperativity, i.e., frequency shift due to hydrogen-bonded concationation of different chemical fragments, has been systematised by Maes and Smets [42] over a big amount of published data.
<figure><img src="image_9.png" /><figcaption>**Fig. 4.** Density of modes projected onto four irreducible representations of the *P*2<sub>1</sub>/<i>c</i> space group according to Eq. (3), resolved into contributions from different hydrogen sites. Scales of individual plots are adjusted for convenience; the frequencies of the strongest peaks are indicated. See text for discussion.</figcaption></figure>

The isolated peak representing almost exclusively the vibrations of H5, seen near 2000 cm⁻¹ in four irreps, namely (see Fig. 4) at 1862 cm⁻¹ (*A*<sub>g</sub>), 1892 cm⁻¹ (*B*<sub>u</sub>), 2015 cm⁻¹ (*A*<sub>u</sub>) and 2154 cm⁻¹ (*B*<sub>g</sub>), corresponds to stretching of O4–H5 bond “against” the H5···O3 hydrogen bond. No peaks in this region (1800–2500 cm⁻¹) are detected in the Raman spectrum [22], whereas in the IR spectrum [18], four features (2285, 2080, 1830 and 1707 cm⁻¹) are accompanied by question marks, without interpretation. Not being able to simulate the IR transmission spectrum directly, we leave the possible specific identification of these features to future studies.

The following (going downwards) mode (1640–1660 cm⁻¹), with contributions of H4 and H5, is a neat bending (scissoring) within water molecules. This is made visible in bending projection of vibrations (Fig. 3), calculated along Eq. (5). An exemplary snapshot of one such mode (that with *A*<sub>g</sub> symmetry) is shown in Fig. 5a. Lin et al. [18] attribute “a band at 1630–1600 cm⁻¹” to H–O–H bending; Vapnik et al. [22] confirm that “the very weak band at 1612 cm⁻¹ originated from bending vibrations of H₂O”. Therefore for this mode one

can assert an unambiguous attribution and perfect agreement with experiment.

The next dense group of modes (from 1500 cm⁻¹ downwards) describes waggings of different hydroxyl groups in their different combinations, some combined with waggings (as a whole) of water molecules. The discussion concerning the corresponding part of spectrum in the experimental works is scarce. Lin et al. [18] attributed the band at 1070 cm⁻¹ “to the Zn–O–H bending vibration” and the band at ~940 cm⁻¹ “to bridging O–H bending vibration”. Vapnik et al. [22] assigned the 1065 cm⁻¹ line in their Raman spectrum to “vZn–O–H” vibrations and that at 990 cm⁻¹ to “vZn–O–Zn” vibrations. In our analysis, the “Cation–O–H bending” projections of vibration modes (shown in “bending” panels of Fig. 3) include in fact the contributions from both “along the Cation–O–Cation bridge” waggings and “around the bridge” twistings of any given O–H bond. It is convenient to discuss them separately for each hydroxyl species, whose selected vibrations are depicted as snapshots in Fig. 6. The four modes (at 1309, 1291, 1282 and 1274 cm⁻¹, prominent in four different irreps of Fig. 4 and merged in the major peak in the “Zn–O1–H1 bending” panel of Fig. 3)
<figure><img src="image_10.png" /><figcaption>**Fig. 5.** Snapshots of selected vibration modes representing scissoring of water molecules (a), “antisymmetric” (b) and “symmetric” (c) stretchings within the H4···O1–H1 group, and (d) O2–H2 stretching. The fragment’s cut and orientation correspond to those in Fig. 1.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 6.** Snapshots of several selected O–H bending modes predominantly representing around-the-bridge twistings (a,c,e) and along-the-bridge waggings (b,d,f) of O2H2 (a,b), O3H3 (c,d) and O1H1 (e,f) hydroxyl groups. The fragment’s cut and orientation correspond to those in Fig. 1.</figcaption></figure>

have more or less clean character of O1–H1 wagging in-plane with the Zn–O1–Zn bridge. The modes at 1129, 1123, 1117 and 1108 cm⁻¹, also spread over four different irreps and merged into a minor peak in the same figure, represent fairly net twisting motions of O1–H1 around the Zn–O1–Zn bridge.

The wagging/rocking/twisting of the O3–H3 groups occupy about the same range of frequencies; however, different modes are more shifted apart and considerably contaminated by admixture with H4 and H5 vibrations. The general trend is that the softest modes of this family, namely 885 cm⁻¹ (*B*<sub>g</sub>) and 896 cm⁻¹ (*A*<sub>u</sub>), which are markedly split off
<figure><img src="image_12.png" /><figcaption>Fig. 7. The same as in Fig. 4, zoomed onto the low-frequency region, plotted with smaller artificial broadening.</figcaption></figure>

from the rest, but also 1025 cm⁻<fnref n="1" /> (*A*<sub>g</sub>) and 1036 cm⁻<fnref n="1" /> (*B*<sub>u</sub>) which fall within a dense set of lines, represent “rotational” rocking/twisting of H3 around the O3. The “mid-band” vibration lines, i.e., 1098 cm⁻<fnref n="1" /> (*B*<sub>u</sub>), 1161 cm⁻<fnref n="1" /> (*A*<sub>g</sub>), 1250 cm⁻<fnref n="1" /> (*B*<sub>g</sub>), 1260 cm⁻<fnref n="1" /> (*A*<sub>u</sub>), reveal predominantly “along the bridge” wagging character of H3 vibration. Finally, the hardest split-off modes (1426 and 1428 cm⁻<fnref n="1" />) regain in part a rotational rocking/twisting character. It should be noted that this gradual drift of vibrations’ polarisation from ~longitudinal to ~transversal and back, with respect to the Ca–O3–Zn bridge direction, on passing the 800–1500 cm⁻<fnref n="1" /> range of frequencies, is implied but not clearly marked.

The O2–H2 group, markedly different from the other hydroxyl species in what regards the frequency and the “linewidth”, exhibits very neat twisting “around” the Ca–O2–Zn bridge at 742 cm⁻<fnref n="1" />, as a vibration mode of *A*<sub>g</sub> symmetry, and at 747 cm⁻<fnref n="1" /> within the *B*<sub>g</sub> symmetry.<sup>1</sup> It is noteworthy that no counterparts of such twisting movement come about in *A*<sub>u</sub> or *B*<sub>u</sub> irreps. The immediately “neighboring” vibrations of H2, at 798 cm⁻<fnref n="1" /> (*B*<sub>g</sub>), 802 cm⁻<fnref n="1" /> (*A*<sub>g</sub>), 855 cm⁻<fnref n="1" /> (*A*<sub>u</sub>) and 856 cm⁻<fnref n="1" /> (*B*<sub>u</sub>), indicated in Fig. 4 and grouped into two distinct peaks in the “(Cation)–O2–H2 bending” panel of Fig. 3, represent, indeed, neat in-plane waggings “along the bridge”.

5.2. Vibrations involving cation-oxygen cages and chains

The vibration spectrum at lower frequencies (up to 600 cm⁻<fnref n="1" />) is quite dense and more difficult to make sense of. Its decomposition into four irreps (Fig. 7) does not seem much enlightening, but in fact it bears features of “solid state” vibration spectrum, i.e. a quasi continuous one, differently from a “molecular” character of high-frequency O–H-related vibrations. Obviously, the vibrations with different wave vectors **q** ought to propagate along the chains of corner-sharing cation-centred polyhedra, resonating at different frequencies. It might have been interesting to explore the corresponding ω(**q**) dispersions, not immediately available in experiment but accessible in a calculation, should it be done on sufficiently large supercell, using the **q**-sensitive projection technique of Eq. (1) a *posteriori*. The projections according to different irreps just select some of these lattice waves which happen to be, in their different ways, commensurate with the lattice translation vectors.

A different view on the same bunch of information, under an angle of cation-oxygen stretchings and bendings (see Fig. 8), permits to establish some classification. The stretchings within the Ca-centered octahedra occur predominantly in the range 200 to 400 cm⁻<fnref n="1" />, whereby the Ca–O2 vibrations tend to contribute more at the softer end of this interval, and Ca–O3 – at the harder end. The stretchings within the Zn-centered tetrahedra are grouped around 500 cm⁻<fnref n="1" />, whereby the Zn–O3 vibration is more “selective” (in the sense of the frequency

<footnote n="1">¹ Both these symmetries being Raman active, we point out that Yang et al. [10] report marked isolated peaks near this frequency in both CHZ (709 cm⁻¹) and Ca(OH)₂ (720 cm⁻¹) – but not in ZnO – not offering any interpretation.</footnote>
<figure><img src="image_13.png" /><figcaption>**Fig. 8.** Projections of vibration eigenvectors onto different (cation–O) stretchings and (cation1–O–cation2) bendings. Scales of individual plots are arbitrarily adjusted for convenience. See text for discussion.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 9.** Snapshots of two vibration modes, of $B_u$ symmetry (a) and $A_g$ symmetry (b), representing deformations of cation-centred polyhedra. See text for discussion.</figcaption></figure>

range involved) than Zn–O2 or, especially, Zn–O1.<fnref n="2" /> The bendings of cation–O–cation bridges occur throughout the frequency range 100 to 550 cm⁻¹.

Fig. 9 depicts the snapshots of two “typical” vibration modes with wavevectors close to 500 cm⁻¹. As could be anticipated, the hydroxyl groups and water molecules vibrate as rigid entities. The deformation of cation-centred polyhedra is of complex character, propagating throughout the lattice via vertice-connected chains. The mode in Fig. 9a exhibits a bending deformation of O3–Ca–O3 diagonals, combined with Ca wiggling between the two H₂O molecules, along the O4–Ca–O4 line (in fact, in the centre-of-masses reference it is the octahedral cage that moves around immobile Ca atoms). The mode in Fig. 9b exhibits a scissoring of O2–Ca–O3 angles, not affecting the Ca–O4 bonds. The Zn-centered tetrahedra are distorted in all cases in different ways which are difficult to characterise in simple terms.

The experimental spectra contain a number of pronounced lines in this spectral range; their discussion is done in terms of vibrations within cation-centred polyhedra. Lin et al. [18] attribute the IR band at 564 cm⁻¹ as being due “to the Zn–O vibration in the skeleton of ZnO₄”; the bands detected at 780, 707, 490 and 458 cm⁻¹ are left

without assignment. Vapnik et al. [22] report a number of intense lines in the low-frequency part of their Raman spectra and vaguely attribute those at 252, 297, 344, 440, 449 and 479 cm⁻¹ “to the vibrations in Zn(OH)₄<fnref n="2" />⁻ tetrahedra”, with “possible contribution of Ca–O vibrations within Ca(OH)₄(OH₂)₂<fnref n="2" />⁻ octahedra”. An inclusion of Raman intensities in a calculation would facilitate an unambiguous interpretation of the experimentally measured lines.

## 6. Conclusions

Summarising, we present an analysis, based on first-principles calculations, of vibration spectrum of calcium hexahydroxodizincate dihydrate, and revise the preliminary attribution, done within earlier experimental works, of some marked features in IR and Raman spectra of this material. Our analysis, beyond direct inspection of selected vibration modes, includes the latters' classification according to different irreps of the space group, and the highlighting contributions from specific stretching and bending atomic movements. The projection technique for vibration eigenvectors, applied *a posteriori*, is flexible and might be found useful for elucidating complicated vibration patterns. Concerning specific numerical results of the present work, we think that identification of stretching and bending frequencies of different hydroxyl groups, hitherto discussed “all together” in experimental works, which however vary quite considerably depending on the hydroxyl's position in the structure, is a novel and potentially interesting result.

<footnote n="2">² The peaks at 252 cm⁻¹ and 356 cm⁻¹ in the micro-Raman spectra of CHZ by Yang et al. [10] are close to those reported in the same work for Ca(OH)₂, whereas the peak at 428 cm⁻¹ falls near a prominent one in the spectrum of ZnO, hereby reinforcing our conclusion that Zn–O stretching vibrations occur in this frequency domain.</footnote>
### CRediT authorship contribution statement

**Andrei Postnikov:** Writing – original draft, Visualization, Super-vision, Methodology, Investigation. **Anna Majtyka-Piłat:** Writing – review & editing, Formal analysis, Data curation. **Dariusz Chrobak:** Writing – review & editing, Formal analysis. **Józef Deniszczyk:** Writing – review & editing, Supervision, Formal analysis, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Data availability

Data will be made available on request.

### Acknowledgement

A.P. thanks the mesocenter of calculation EXPLOR at the Université de Lorraine (project 2019CPMXX0918) where the calculations have been done.

### Appendix. Contribution of bond bendings

A triplet of atoms situated at $R_1$, $R_2$, $R_3$ defines a bond angle $\theta = 2\angle_{3}$ between vectors $r_{23} = R_3 - R_2$ and $r_{21} = R_1 - R_2$; the normal to the plane of three atoms is $N = r_{23} \times r_{21}$; the vectors in the plane applied to atoms 1 and 3 so as to open (increase) the bond angle are, respectively,

$$ 
\begin{aligned}
T_1 &= N \times r_{21} = r_{21}(r_{21}r_{23}) - r_{23}r_{21}^2, \\
T_3 &= r_{23} \times N = r_{23}(r_{21}r_{23}) - r_{21}r_{23}^2,
\end{aligned}
$$

with the respective norms

$$ T_1 = r_{21} \sqrt{r_{23}^2 r_{21}^2 - (r_{21} r_{23})^2} = r_{21}^2 r_{23} \sin \theta \quad \text{and} \quad T_3 = r_{21} r_{23} \sin \theta, $$

hence normalised to unity:

$$ t_1 = \frac{T_1}{T_1} = \frac{r_{21}}{r_{21}} \cot \theta - \frac{r_{23}}{r_{23}} \csc \theta \quad \text{and} $$

$$ t_3 = \frac{T_3}{T_3} = \frac{r_{23}}{r_{23}} \cot \theta - \frac{r_{21}}{r_{21}} \csc \theta. $$

The relative displacements of peripheric atoms against the middle one, in terms of phonon eigenvectors $A^a$ and masses $m_a$, are

$$ d_{12} = \left( \frac{A^1}{\sqrt{m_1}} - \frac{A^2}{\sqrt{m_2}} \right), \quad d_{32} = \left( \frac{A^3}{\sqrt{m_3}} - \frac{A^2}{\sqrt{m_2}} \right), $$

and the increment of bond angle due to these displacements is

$$ \Delta\theta = \frac{(d_{12} t_1)}{r_{12}} + \frac{(d_{32} t_3)}{r_{32}}. $$

Depending on the phase of the vibration, this increment can be positive or negative. Taking it to square reveals the contribution of the bond bending under discussion in the vibration pattern in question, hence Eq. (5).

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