Crystallographic and Optical Spectroscopic Study of Metal–Organic 2D Polymeric Crystals of Silver(I)– and Zinc(II)–Squarates

Abstract

Metal–organic framework materials, as innovative functional materials for nonlinear optical technologies, feature linear and nonlinear optical responses, such as a laser damage threshold, outstanding mechanical properties, thermal stability, and optical transparency. Their non-centrosymmetric crystal structure induces a higher-order nonlinear optical response, which guarantees technological applications. Zn^II– and Ag^I–squarate complexes are attractive templates for these purposes due to their good crystal growth, optical transparency, high thermal stability, etc. However, the space group type of the catena-((μ₂-squarato)-tetra-aqua-zinc(II)) complex ([Zn(C₄O₄)(H₂O)₄]) is debatable, (1) showing centro- and non-centrosymmetric monoclinic C2/c and Cc phases. The same is valid for the catena-((μ₃-squarato)-(μ₂-aqua)-silver(I)) complex (Ag₂C₄O₄), (2) exhibiting, so far, only a C2/c phase. This study is the first to report new crystallographic data on (1) and (2) re-determined at different temperatures (293(2) and 300(2)K) and the non-centrosymmetric Cc phase of (2), having different numbers of molecules per unit cell compared with the C2/c phase. There are high-resolution crystallographic measurements of single crystals, experimental electronic absorption, and vibrational spectroscopic data, together with ultra-high-resolution mass spectrometric ones. The experimental results are supported for theoretical optical and nonlinear optical properties obtained via high-accuracy static computational methods and molecular dynamics, using density functional theory as well as chemometrics.

1. Introduction

Nonlinear optical (NLO) crystals have found application as optical devices [1,2,3,4,5]. Second-harmonic generation (SHG) is a second-order nonlinear optical phenomenon, which is fundamentally important for developing advanced laser technologies. The major SHG process is defined as the conversion of a certain value of the light wavelength of its initial value or a two-times increase in the order of magnitude of the corresponding frequency.

The non-centrosymmetric crystal structure determined the generation of the SHG optical responses of the materials and their high laser damage threshold, which guarantees the corresponding practical applications of NLO crystals.

The Pockels effect of crystals has also been studied due to its applications in modulating functions in photonics [6]. As the SHG, the Pockels effect is observed in non-centrosymmetric crystals or centrosymmetric crystals, having strain-induced dielectric susceptibility. The experimental values of the SHG coefficients of a crystalline sample are comparable with the Pockels effect when comparing the Pockels coefficient βzzz (−ω; 0, +ω) with the SHG one [5].

Inorganic NLO materials feature an optical response and outstanding mechanical properties. So far, they have already been implemented in generating two monochromatic beams from a source in technologies, where a lack of tunable laser sources occurs.

However, the structural diversity of inorganics needs to be extended, due to the frequent need for an improvement in NLO performances. Organic NLO materials also provide prominent crystalline candidates for NLO technologies due to their flexible chemical structures and significant capability of chemical substitution. However, their practical applications have often been restricted due to optical instability and complex synthetic approaches to chemical modification.

However, organic–inorganic hybrid materials show a marked structural diversity of MOF (metal–organic framework) crystals, having 0D (zero)–3D (three) dimensions and high nonlinear optical performances, thus implementing the advantages of inorganic and organic materials. Particularly, their capability to relatively easily reduce dimensionality compared with organic crystals could be strengthened, thus enhancing their stability. The band gap of 0D–2D crystals increases and expands the wavelength range of the SHG response and laser damage threshold.

Therefore, the question of the symmetry of molecular crystals and ionic crystals is central to designing and developing linear optical and nonlinear optical technologies, respectively.

However, frequently, crystallographic reports show an inaccurate space group determination of molecular crystals (consider the details in [7]). Thus, relevant debates concerning the reliability of a determined space group type have begun to flourish [7,8]. The major reason for the latter fact is (i) the structural solution process variables of measurements, where reliability depends on the uncertainty of measurable parameters.

Therefore, first, there is the processing of data-blocks of the measured parameters in multiplication. There (ii) could be an observed spontaneous crystallization of crystals in different space groups [8,9]; furthermore, (iii) there are comparable parameters of structural solutions of a crystalline system in both centro- and non-centrosymmetric space groups [10]. The key aspects of the issue lie behind the aforementioned debates, accounting not only for the potential practical applications of crystalline materials but also for addressing fundamental questions, bridging the gap between molecular structure–crystal structure–molecular properties and mechanistic aspects in chemical reactions in condensed phases. Single-crystal X-ray diffraction (scXRD), as an analytical instrument, experimentally determines the electron density of atoms in molecules, thereby experimentally determining not only the geometry parameters but also the subtle electronic effects, if any.

In addition to the non-centrosymmetric crystal structure of the materials, the second basic prerequisite condition for determining molecular and ionic crystals as suitable NLO-phores, respectively, is detailed in [2,4].

Squarate-based MOFs evoke a great deal of interest, thus highlighting metal–squarate (M^II–C₄O₄)_n coordination nD polymers [10,11] due to their broad spectrum of technological applications involving catalysis; gas storage technologies, for instance, acetylene or hydrogen storage [12,13]; photonics and optoelectronics [14]; luminescent or light-emitting diodes [15]; materials with color-rendering properties [16]; electrochemistry, biomedicine, separation technologies [17]; environmental cleaning technologies for the treatment of, for instance, polluted ground and surface water with organics or with metal ions [18]; memory devices [19]; removal and purification methods; drug delivery; and more.

In particular, MOF application to nonlinear optics has been comprehensively discussed (2024) [20,21]. The tuning of hydrogen bond interactions is a key issue, amongst others, in crystallographic studies, including of squarete complexes, because squaric acid and its monoanion (HSq⁻) and dianion (Sq²⁻) are particularly attractive molecular templates for purposes of NLO materials research and crystal engineering, thus allowing for modulating the whole range of requested 0D–3D crystalline materials, such as, for example, complexes [M^II(C₄O₄)_x(C₄H₄N₂)(H₂O)_y] where M denotes transition metal ions (x = 1 or 4, y = 4 or 8) [16,22], showing several supramolecular motifs and metal-to-ligand coordination modes (Figure S1), thus yielding to unique structural topologies in its MOF coordination polymers. Both the HSq⁻ and Sq²⁻ ions readily coordinate and bridge a large number of transition metal ions and ions of lanthanides or actinides, thus showing a variety of structural dimensionalities and coordination modes of mononuclear and binuclear complexes [23,24,25].

Despite its polycentered coordination capability, anions of squaric acid also coordinate monodentately and bidentate, thus forming cheating complexes [16,26,27,28]. Non-centrosymmetric cubic (Pn3n) squarate crystals of {[ZnC₄O₄·2H₂O)·CH₃COOH]·H₂O} have been determined as well [29]. The crystallographic structural motif of discrete squarate anions has also been determined [30,31]. In addition, key aspects of the coordination capability of squaric acid anions include the formation of 1D polymers and crystals of mixed ligand complexes as well as the capability to stabilize doped crystals with tunable mixed metallic doped ones depending on the requested redox and magnetic properties of the new materials [32,33,34,35].

Structural motifs and the symmetry of squarate-based MOFs are often difficult to control via tuning the intermolecular/interionic interactions of species. At this point, it should be highlighted that HSq⁻ and Sq²⁻ ligands rapidly, in mild synthetic conditions and in high-yield synthetic yields, produce chemical substituted derivatives [36,37]. The correlation between molecular and crystalline spectroscopic properties is of particular importance in designing new functional NLO materials because of both experimental and theoretical spectroscopic tools that are used to determine the molecular and crystalline properties in the designed materials in the initial synthetic stages. Both the synthesis of ligands and coordination compounds as well as the crystallization process could lead to difficulties, including those associated with the correlation between the properties of complexes in solution with those resulting in the solid-state crystalline phase [38]. The latter statement is particularly valid to Zn^II- and Ag^I- complexes, which often show stable solvate species in solution, thus dissolving crystalline compounds [39].

Despite this, as can be seen in the case of [Zn(H₂O)₆²⁺] counterion-containing complexes, the coordination species are stable in both the crystalline state and solution [40,41].

It is, perhaps, time for the advantages of Zn^II- and Ag^I-metal ion-containing complexes such as MOF NLO-phores to be proposed. If the optical properties of such compounds are examined, particularly, then their important characteristic qualities should be highlighted, such as the broad optical transparency window [42] and significant thermal stability, among others [43,44,45,46,47,48,49]. Innovations (2024) [21] have shown that Zn^II-ion-based MOFs with potential applications as NLO technologies and tunable NLO responses are characterized by high thermal stability (T = 600 °C) and a third-harmonic-generation response (3ω) of 2.9 × 10⁻¹² esu (λ = 1500 nm). However, the presence of conjugated organic ligands causes a relatively restricted transparency window λ_max > 600 nm.

Solid-state thermo-gravimetric analysis (TGV) of transition metal ion-containing squarates and their mixed ligand derivatives, including the results from the analysis of [Zn(HC₄O₄)₂(OH₂)₄], shows high thermal stability in corresponding squarate moieties up to T = 400 °C [43,44,45,46,47,48] and depending on the type of mixed ligands in the inner coordination sphere. The thermal decomposition path of the latter complex is characterized by four stages: [Zn(HC₄O₄)₂(OH₂)₄] → [Zn(HC₄O₄)₂(OH₂)₂] (T = 161 °C) → [Zn(HC₄O₄)₂] (T = 210 °C) → [Zn(HC₄O₄)] (T = 310 °C) → [ZnO] (T = 400 °C) [43].

For the purposes of this study, it should be underlined that a vast amount of work has been performed in determining the molecular and crystal structure of (1) because it is obtained not only through the rapid interaction and excellent crystal growth of high-quality single crystals (see in [49]), but, also, both the squarates and hydrogen squarates frequently show pressure-induced changes in their lattice structure from monoclinic to tetragonal and a change in their asymmetric-to-centrosymmetric crystalline structures, particularly highlighting hydrogen squarate compounds [50]. Particularly, squarate crystals of Zn(II) ions show crystal growth of large dimensions, 5–7 mm [49]. It should be mentioned that the crystal structure of the hydrogen squarate complex [Zn(HC₄O₄)(H₂O)₄] is a centrosymmetric triclinic one with space group type P-1 (Table 1 [43]).

However, the space group type of Zn(II) ion-containing squarate crystal catena-((μ-squarato)-tetra-aqua-zinc(II)) ([Zn(C₄O₄)(H₂O)₄]) studied in this work is debatable, as well. Research efforts so far have shown that it crystallizes both centro- and non-centrosymmetrically monoclinic space group types C2/c [51,52] and Cc [9].

Since the crystallographic refinement conditions only show (h,k,l) reflections with h + k = 2n and (h,k = 0,l) reflections with l = 2n, there are Cc or C2/c groups [51,52,53,54,55,56]. Then, the final solution and description of the crystal structure of the discussed complex must be made in non-centrosymmetric group type Cc [9]. There is, therefore, support for this view, looking at the obtained low R₁ parameter of the structural solution showing R₁ = 0.023 (see Table 1 and data in [9]).

Table 1. Crystallographic refinement data on catena-((μ-hydrogen squarato)-tetra-aqua-zinc(II)) and catena-((μ₂-squarato)-tetra-aqua-zinc(II)); (1)_1 and (1)_2 denote data of a single crystal of complex catena-((μ₂-squarato)-tetra-aqua-zinc(II)) measured in duplication at different temperatures, while (1)_3 denotes data on the complex measured from different single crystal of the sample.

Complex	[Zn(HC4O4)(H2O)4]	[Zn(C4O4)(H2O)4]
CCDC	929462	-	-
Ref.	[43]	[51]	[9]
Empirical formula	C8H10O12Zn	ZnC4O4,4H2O	ZnC4O4,4H2O
Moiety formula	C8H10O12Zn	‘C8 O16 Zn2’	‘C8 O16 Zn2’
Formula mass	-	249.48	-
Crystal system	Triclinic	Monoclinic	Monoclinic
Space Group	P-1	C2/c	Cc
a [Å]	5.0919(5)	8.986(2)	9.012(2)
b [Å]	7.3113(7)	13.333(2)	13.336(3)
c [Å]	8.7536(7)	6.694(3)	6.746(2)
α [o]	66.440(6)	90.00	90.00
β [o]	77.254(7)	99.67(2)	99.33(2)
γ [o]	75.480(7)	90.00	90.00
V [Å3]	286.50(5)	790.6(3)	800.0
Z	1	4	-
ρ [g·cm−1]	2.107	2.096	2.07
F000	184	-	-
Μ (Mo-K) [mm−1]	2.216	31.9	-
T [K]	100(2)	120(2)	-
θ range	4.68–29.66	40–90	-
Refl. collected	3003	19653	2639
Unique refl.	1308	4140	-
R1[2σ(I)]	0.0513	0.023	-
R1 (all data)	0.0534	0.024	0.039
wR2	0.1325	0.024	0.042
GooF	1.092	-	-
Diff. peak/hole [e/Å3]	1.464/−1.792	0.79/−1.13	-
Complex	[Zn(C4O4)(H2O)4]
CCDC	1917571	1917547	1565990
Single crystal	(1)_1	(1)_2	(1)_3
Ref.	This work [55]	This work [56]	This work
Empirical formula	ZnC4O4,4H2O	ZnC4O4,4H2O	ZnC4O4,4H2O
Moiety formula	‘C8 O16 Zn2’	‘C8 O16 Zn2’	C4 O8 Zn
Formula mass	482.82	482.82	241.41
Crystal system	Monoclinic	Monoclinic	Monoclinic
Space Group	C2/c	C2/c	C2/c
a [Å]	9.003(3)	9.003(3)	8.982(3)
b [Å]	13.295(5)	13.295(5)	13.315(5)
c [Å]	6.746(3)	6.746(3)	6.734(3)
α [o]	90.00	90.00	90.00
β [o]	99.244(17)	99.244(17)	99.327(15)
γ [o]	90.00	90.00	90.00
V [Å3]	797.0(5)	797.0(5)	794.7(5)
Z	2	2	4
ρ [g·cm−1]	2.012	2.012	2.018
F000	472	472	472
μ(Mo-K) [mm−1]	3.095	3.095	3.104
T [K]	293(2)	300(2)	293(2)
θ range	4.33–25.29	4.33–25.11	2.76–25.12
Refl. collected	1180	1157	1200
Unique refl.	643	641	586
R1[2σ(I)]	0.0796	0.0581	0.0608
R1 (all data)	0.0906	0.1562	0.0660
wR2	0.1937	0.1532	0.0721
GooF	1.013	1.383	1.128
Diff. peak/hole [e/Å3]	1.414/−1.477	0.674/−1.059	1.835/−1.344

Table 1. Crystallographic refinement data on catena-((μ-hydrogen squarato)-tetra-aqua-zinc(II)) and catena-((μ₂-squarato)-tetra-aqua-zinc(II)); (1)_1 and (1)_2 denote data of a single crystal of complex catena-((μ₂-squarato)-tetra-aqua-zinc(II)) measured in duplication at different temperatures, while (1)_3 denotes data on the complex measured from different single crystal of the sample.

Complex	[Zn(HC4O4)(H2O)4]	[Zn(C4O4)(H2O)4]
CCDC	929462	-	-
Ref.	[43]	[51]	[9]
Empirical formula	C8H10O12Zn	ZnC4O4,4H2O	ZnC4O4,4H2O
Moiety formula	C8H10O12Zn	‘C8 O16 Zn2’	‘C8 O16 Zn2’
Formula mass	-	249.48	-
Crystal system	Triclinic	Monoclinic	Monoclinic
Space Group	P-1	C2/c	Cc
a [Å]	5.0919(5)	8.986(2)	9.012(2)
b [Å]	7.3113(7)	13.333(2)	13.336(3)
c [Å]	8.7536(7)	6.694(3)	6.746(2)
α [o]	66.440(6)	90.00	90.00
β [o]	77.254(7)	99.67(2)	99.33(2)
γ [o]	75.480(7)	90.00	90.00
V [Å3]	286.50(5)	790.6(3)	800.0
Z	1	4	-
ρ [g·cm−1]	2.107	2.096	2.07
F000	184	-	-
Μ (Mo-K) [mm−1]	2.216	31.9	-
T [K]	100(2)	120(2)	-
θ range	4.68–29.66	40–90	-
Refl. collected	3003	19653	2639
Unique refl.	1308	4140	-
R1[2σ(I)]	0.0513	0.023	-
R1 (all data)	0.0534	0.024	0.039
wR2	0.1325	0.024	0.042
GooF	1.092	-	-
Diff. peak/hole [e/Å3]	1.464/−1.792	0.79/−1.13	-
Complex	[Zn(C4O4)(H2O)4]
CCDC	1917571	1917547	1565990
Single crystal	(1)_1	(1)_2	(1)_3
Ref.	This work [55]	This work [56]	This work
Empirical formula	ZnC4O4,4H2O	ZnC4O4,4H2O	ZnC4O4,4H2O
Moiety formula	‘C8 O16 Zn2’	‘C8 O16 Zn2’	C4 O8 Zn
Formula mass	482.82	482.82	241.41
Crystal system	Monoclinic	Monoclinic	Monoclinic
Space Group	C2/c	C2/c	C2/c
a [Å]	9.003(3)	9.003(3)	8.982(3)
b [Å]	13.295(5)	13.295(5)	13.315(5)
c [Å]	6.746(3)	6.746(3)	6.734(3)
α [o]	90.00	90.00	90.00
β [o]	99.244(17)	99.244(17)	99.327(15)
γ [o]	90.00	90.00	90.00
V [Å3]	797.0(5)	797.0(5)	794.7(5)
Z	2	2	4
ρ [g·cm−1]	2.012	2.012	2.018
F000	472	472	472
μ(Mo-K) [mm−1]	3.095	3.095	3.104
T [K]	293(2)	300(2)	293(2)
θ range	4.33–25.29	4.33–25.11	2.76–25.12
Refl. collected	1180	1157	1200
Unique refl.	643	641	586
R1[2σ(I)]	0.0796	0.0581	0.0608
R1 (all data)	0.0906	0.1562	0.0660
wR2	0.1937	0.1532	0.0721
GooF	1.013	1.383	1.128
Diff. peak/hole [e/Å3]	1.414/−1.477	0.674/−1.059	1.835/−1.344

Therefore, we have not found an objective reason to reject the latter view, looking closer at the available experimental facts. In an effort to obtain the non-centrosymmetric phase of the complex, in this study, we re-determined the crystal structure of catena-((μ₂-squarato)-tetra-aqua-zinc(II)) (1) in triplicate at different temperatures, T = 293(2) and 300(2)K, thus obtaining a centrosymmetric phase with the C2/c space group type (see Table 1, CCDC (Cambridge Crystallographic Data Centre) 1917571, 1917547, and 1565990). The new results support the view of [51], crystallographically determining the same analyte at T = 120(2)K. To the question, however, “Does the solution of the crystal structure should be in monoclinic noncentrosymmetric space group type Cc?”, this work provides an answer by solving the structure of the complex CCDC 1917547 ((1)_2) into the Cc space group type. We found an increasing R₁ parameter from R₁ = 0.058 (C2/c) to 0.0782 (Cc), in addition to the results from symmetry tests suggesting the former space group type (see below).

The results from [51] and new data on complex (1) do not reject, however, the view that the discussed compound could crystallize into non-centrosymmetric space group type Cc. Due to further results in this study, it is possible to admit that in certain experimental crystallization conditions, complex (1) should produce non-centrosymmetric crystals. The latter position could be held looking at new crystallographic results reported first, herein, of catena-((μ₃-hydrogen squarato)-(μ₂-aqua)-silver(I)) (2) (CCDC 2387639 and 2387641). As the crystal structure of complex (1), the Ag(I) coordination compound crystallizes in monoclinic space group type C2/c (CCDC 771415 [53,54]), measured at T = 199(2) K. The new data on the same complex at T = 300(2) K (CCDC 2387641,

Table 2. Crystallographic refinement data on catena-((μ₃-squarato)-(μ₂-aqua)-silver(I)).

Complex	[Ag(C4O4)O] n	[Ag(C4O4)O] n	[Ag(C4O4)O] n
CCDC	771415	2387639	2387641
Single crystal	(2)_1	(2)_2	(2)_3
Refs.	[53,54]	This work	This work
Empirical formula	C4AgO5	C4AgO5	C4AgO5
Moiety formula	C4AgO5	C4AgO5	C4AgO5
Formula mass	235.91	459.81	235.91
Crystal system	Monoclinic	Monoclinic	Monoclinic
Space Group	C2/c	Cc	C2/c
a [Å]	13.572(9)	13.491(11)	13.594(6)
b [Å]	8.229(6)	8.233(11)	8.251(3)
c [Å]	11.108(7)	11.038(13)	11.107(4)
α [o]	90.00	90.00	90.00
β [o]	118.142(17)	117.94(5)	118.094(11)
γ [o]	90.00	90.00	90.00
V [Å3]	1093.9(12)	1083(2)	1099.0(7)
Z	8	4	8
ρ [g·cm−1]	2.865	2.820	2.852
F000	888	864	888
μ(Mo-K) [mm−1]	3.561	3.666	3.617
T [K]	199(2)	300(2)	300(2)
θ range	3.00–25.07	3.01–24.94	3.00–25.24
Refl. collected	3173	1555	1655
Unique refl.	951	1034	963
R1[2σ(I)]	0.0558	0.1544	0.2323
R1 (all data)	0.0587	0.1781	0.0944
wR2	0.1819	0.3626	0.1044
GooF	0.952	2.026	1.003
Diff. peak/hole [e/Å3]	2.591/−1.161	3.374/−2.879	3.344/−1.724

) agree with previously reported data.

However, as the later-tabulated parameters reveal, we obtained a non-centrosymmetric monoclinic phase of complex (2), showing the Cc space group type. The application of symmetry tests [7] (see below) shows a lack of additional symmetry operations. In addition, the crystallographic solution in centrosummetric space group type C2/c shows an increasing R₁ parameter up to R₁ = 0.1746. The same is valid for a structural solution in the I2/c space group type. The latter new results, together with those reported previously, clearly show that, despite the research effort, so far, knowledge of the crystallization behavior of even simple squatates and hydrogen squarate crystals of Zn(II)- and Ag(I)-ions does not imply knowledge of the preferred monoclinic space group type, looking at C2/c and Cc, and depending on the experimental conditions of the synthesis and crystal growth. The similarity of the crystallization preference of the two Zn(II) and Ag(I)-ions and common space group types of their coordination species with squaric acid anions suggests that there is a practical relationship between the coordination capability of the ions and the squarate ligands. The current study in the later context contributes, crucially, to our further understanding of the issue, which is closely related not only to the fundamental issue of chemical crystallography associated with the development of robust tests for the unambiguous determination of crystal space group types [7] but also to the practical fields of NLO-phore research on MOF-based new materials, due to the advantages of squarate crystals of Zn(II)- and Ag(I)-ions.

Further arguments and results shedding light on the latter issue should be particularly valid and important because of Ag(I) and Zn(II)-complexes, which often crystallize into monoclinic C-type space groups, showing possible or pseudo new space group type Cc, whose refinements are often unstable [53]. In other words, complexes of Zn(II) and An(I) provide prospective templates for the design of new MOFs NLO-phores, due to their tend to produce non-centrosymmetrical crystals.

The purpose of this study is twofold. It provides the theoretical and experimental optical properties of crystals (1) and (2) in condensed phases using crystallographic data on both the centro- and non-centrosymmetric space group types. It also provides experimental and theoretical data on single-crystal X-ray diffraction, Fourier-transform infrared spectroscopy, and high-accuracy static and molecular dynamics results from molecular and crystalline structures and the optical and NLO- properties of coordination species (1) and (2).

Since, the molecular design, synthesis, crystallographic analysis, and spectroscopic studies—both theoretical and experimental—are the primary focuses of MOF materials research, as well as being a strategic step in constructing new and specific supramolecular arrangements of crystals, then an in-depth comprehension of the correlation among molecular and crystal structures, and the physical–optical properties of materials, contribute crucially to the development of the MOF-based materials research field [57,58,59,60,61,62,63].

2. Materials and Methods

Complexes (1) and (2) were isolated by mixing ZnCl₂.2H₂O (0.1773 g, Sigma-Aldrich, St. Louis, MI, USA) or AgNO₃ (0.1690 g) inorganic salts with 25 mL squaric acid (0.115 g, Sigma-Aldrich, St. Louis, MI, USA) in solvent mixture CH₃OH:H₂O = 1:2 at T = 200 °C with a yield 70% of (1) and 38% of (2). Analytical calculations for complex [Zn(C₄O₄)·4H₂O] (1): C, 19.26; H, 3.23; O, 51.30 showed: (1) C, 19.28; H, 3.20%. Analytical calculations for complex [C₄O₅Ag] (2): C, 20.37; O, 33.91 showed: (2) C, 20.33%. The single crystals of the analytes were obtained using slow evaporation approach to ambient conditions for 10 days. Consider detail in [53].

ScXRD (h,k,l) reflections were collected via Bruker Smart X2S diffractometer, Mo Ka source irradiation, and ω scan mode. Figure 1 depicts their crystal structures using PLATON plots [64]. We used absorption correction approach based on multiple scanned reflections. The crystallographic solutions were achieved via direct method and SHELXS-97 [65,66,67]. The structural refinement was carried out using the F² approach [65,66,67]. There are anisotropic displacement parameters for all heavy atoms. The refinement data are shown in Table 1 and Table 2.

The monopole and multipole electron density refinement was carried out by means of XD2016 and MoPro v16 program packages [68,69,70,71,72], thus employing the Hansen–Coppens methodology. The experimental structural factors were further processed by WinGX 2014 [73] towards data quality. WTANAL and DRK plot analyses of the structure factors were carried out. In addition, residual analyses and THMA were performed, thus evaluating the thermal motion on the basis of the experimentally measured U_ij values [74,75].

High-performance liquid chromatography electrospray ionization (HPLC–ESI-tandem) MS/MS data were obtained via TSQ 7000 instrument (Thermo Fisher Inc., Rockville, MD, USA), using composition of mobile phase 0.1% (v/v) H₂O solution of HCOOH, 0.1% (v/v) HCOOH in CH₃CN; 20.0% (v/v) HCOOH in CH₃CN:CH₃OH mixture 1:1; and 10.0% KOH/NaCO₃ in CH₃OH. A triple-quadruple mass spectrometer (TSQ 7000 Thermo Electron, Dreieich, Germany), having ESI 2 source, was used for ESI-MS data. There following conditions were set: capillary temperature of 275.00 and −63.17 °C; sheath gas of 50.00 psi; source voltage of 5 kV; and capillary voltage of 33.95 V. Samples of (1) and (2) were dissolved in CH₃CN (1 mg·mL⁻¹) and were injected into ion source using an auto-sampler (Surveyor) with a CH₃CN flow (0.2 mL·min⁻¹). The data processing was carried out using Excalibur 1.4 software within mass range m/z 100–600. LTQ Orbitrap XL (Thermo Fisher Inc.) mass spectrometer was utilized, as well. The chromatographic analysis was carried out via Gynkotek (Germering, Germany) HPLC instrument, Kromasil 100 C18 column (250 × 20 mm, 7 μm; Eka Chemicals, Bohus, Sweden), and a UV detector at 250 nm. The mobile phase was CH₃CN:H₂O (90:10, v/v), with a flow rate of 4 mL·min⁻¹. The HPLC was made via Phenomenex (Torrance, CA, USA) RP-18 column (Jupiter 300 150 × 2 mm, 3 μm). We used Shimadzu UFLC XR (Kyoto, Japan) instrument, auto-sampler, PDA, and an on-line degasser and column thermostat, as well Phenomenex Luna Phenyl-Hexyl column (150 × 3 mm i.d., 3 μm particle size). The mobile phase was 0.02% (v/v) TFA in H₂O (solvent A) and CH₃CN:CH₃OH 75:25 (v/v; solvent B).

The infrared (IR) spectra were collected via Bomem Michelson 100 FTIR spectrometer (4000–400 cm⁻¹, ±1 cm⁻¹ resolution, and 200 scans), with Specac wire-grid polarizer. We used GRAMS AI 7 software (Thermo Fisher Scientific Inc.) for data processing.

The UV–VIS–NIR spectra were measured via Evolution 300 spectrometer (Thermo Fisher Scientific Inc.) within a range of 200–1100 nm.

2.1. Theory/Computations

Consider detail in [8]. Crystallographic data on (1) and (2) were utilized for input atomic coordinates and quantum chemical computations. The UV-VIS spectra were calculated using time-dependent density functional theory (TD-DFT) and the Tamm–Dancoff approximation [76,77,78], thus also showing excellent predictive capability of MOFs [19].

2.2. Chemometrics

We used R4Cal Open Office STATISTICs 7.0 software. The t- and F-tests were utilized to evaluate the equality of variables. Analysis of variance (ANOVA) was performed [79,80,81,82,83] together with non-parametric Kolmogorov–Smirnov [84], Wilcoxon–Mann–Whitney [85], and Mood’s median tests [86].

3. Results

3.1. Analysis of Chromatograms and Mass Spectra in Solution

As detailed in [39,40,41], the proper experimental and theoretical design of MOF-based materials research is actually in successful operation only when there is reliable assignment of the coordination species in both the solution and in the solid crystalline state. Chromatographic and mass spectrometric data on complex (2) show that at the retention time, RT = 14.34 min, MS peaks are observed at m/z 145/147 of the Ag^I-hydride complex of type [Ag^IH(H₃O⁺)(H₂O)]. As can be seen, the oxidation state of the metal ion is maintained (+1).

Conversely, Zn^II complexes of simple O-containing organic ligands often produce solvate complexes of the zinc metal ions in different oxidation states (consider detail in [40,41]).

3.2. Crystallographic Data

The crystal structures of complex [Zn(C₄O₄)(H₂O)₄] (1) at T = 120 K [51], reported previously and herein (CCDC 1917571 [55], 1917547 [56], and 1565990 (this work), Table 1 and Figure 1), at T = 293(2) and 300(2) K are the same. The four complexes are monoclinic and have the C2/c space group, thus lacking the discrepancy between the three reported independent structural solutions of the three new crystals (1)_1, (1)_2, and (1)_3 and ADDSYM test data [7,57]. A pro argument for the latter statement is reached testing the structural solution of the crystallographic data-block of the structure in the same complex, CCDC 1917547, solved as non-centrosymmetric monoclinic space group type Cc (Supporting Information). As Figure S3 reveals, the ADDSYM test shows missing or additional symmetry operations and suggests solution in the C2/c space group type. Despite the fact that the structural solution in the Cc space group type shows a low R₁ parameter (R₁ = 0.0782, Supporting Information), the determination of the crystal structure is performed using the ADDSYM criterion, thus yielding R₁ = 0.0581 (Table 1) in the C2/c space group type.

The Zn(II)-ion in complex (1) exhibits the distorted octahedral geometry of metal chromophore Zn^IIO₆ and r(Zn–O) bonds, with lengths of 2.068, 2.072, and 2.129 Å (Figure 1). The corresponding bond angles ∠(O–Zn–O) vary as follows: 88.08, 88.54, 88.68, 91.46, and 94.42°, respectively. The squarate dianion shows the μ_1,3-bonding type and acts as a bridging ligand. The interplanar angle between squarate p-aromatic systems is 5.02°, while the interplane distance is 3.559 Å (Figure 2).

The comparative analysis with compound [Zn(HC₄O₄)₂(OH₂)₄] [43] shows that, again, the Oh*-geometry of the Zn^IIO₆ metal chromophore connected to four O-centers of the solvent water molecules and two monodentately bonded hydrogen squarate anions HC₄O₄- arranged mutually in a trans-configuration is distorted. The molecular chains of the crystal are connected with [Zn(OH₂)₄]²⁺ sub-structural units.

Some studies [53,54] briefly touched on the coordination capability of Ag(I) with a squarate dianion, showing that the complex catena-((μ₃-squarato)-(μ₂-aqua)-silver(I)) (2) (CCDC 771415, T = 199(2)K) forms a stable square pyramidal geometry for the Ag^IO₅ metal chromophore, thus exhibiting bond lengths r(Ag–O) = 2.317, 2.352, 2.498, and 2.512 Å of the equatorial Ag–O bonds as well as 2.613 Å of the axial Ag–O bond. Consider the data on crystal (2)_1 in Table 2. This study reports on the determination of the crystals in complex (1) obtained due to independent synthesis (crystal (2)_3, CCDC 2387641) at a different temperature, T = 300(2), again showing good crystal growth and a low R₁ parameter (R₁ = 0.0944), despite the fact that it is determined at a higher temperature. The crystallographic refinement data show good agreement with the results from other works [53,54], again showing centrosymmetric space group type C2/c. The fact that the ADDSYM test was conducted in PLATON 21098 software [64] confirms the same space group type could be claimed to be free of errors in the structural analysis.

However, the purpose of this study is not to produce only crystallographic data on compounds (1) and (2) at different temperatures but merely to discuss the effect of the independent measurements in the multiplication of one and the same coordination compounds on the crystallographic parameters because, routinely, as mentioned, they are used to predict the NLO properties of crystals among other ones. Thus, variation in the experimental geometry parameters affect the energetics of the molecular crystals. Since the theoretical quantum chemical data involve the high-accuracy determination of the energy parameter, among other molecular properties, up to six decimal signs, as can be expected, the variation in the initial atomic coordinates of the molecules in the computed crystals should affect the theoretical parameters, as well. The later issue seemed of importance due to the fact that subtle electronic effects could be accounted for with significant errors only as a result of the variation in the crystallographic inputs of the theoretical computations. This study tackles the latter topics following this route in its explanation of chemometrics. As the ANOVA data in Table S1 show, the two datasheets of crystallographic variables for crystals (2)_1 and (2)_3 are not statistically significantly different. Therefore, they also belong to complex (2), with the C2/c space group type. Within the framework of the multiplication of measurements, the geometry parameters should be tackled with the corresponding standard deviations, thus yielding a = 13.58375 ± 0.01534, b = 8.24045 ± 0.01534, c = 11.10805 ± 9.19239 × 10⁻⁴ Å; β = 118.11814 ± 0.03398°, V = 1096.491 ± 3.64726 ų and r = 2.8585 ± 0.00919 g·cm⁻¹. As can be seen within the framework of multiple crystallographic re-determinations, the accuracy of the input atomic coordinates affects the second decimal sign, and, thus, the properties of such crystals could be reliably predicted with relatively low-cost computational approaches. As can be expected, depending on the number of theoretical computations, both the energy parameters should account for the uncertainty in the input crystallographic atomic coordinates.

In addition to the new crystallographic data in complex (2) obtained in different experimental conditions, the current study reports new the non-centrosymmetric phase of complex (2), crystallizing into monoclinic space group type Cc (CCDC 2387639, crystal (2)_2). The application of the ADDSYM test used for PLATON software shows, in this case, a lack of missing or additional symmetry operations (Figure S4).

At this point, if one is asked to criticize anything so far, there are perhaps only two objections worth mentioning, which this study shall briefly address. The first objection is that the current study claims that there is a new non-centrosymmetric crystalline phase of the same complex (2), but not for the new non-centrosymmetric complex of Ag(I) ions with a squarate anionic ligand. The geometry parameters of crystals (2)_1 or (2)_3 of the centrosymmetric phases are not statistically significantly different from the experimental parameters of the non-centrosymmetric phase (2)_2 (Table S2). Therefore, despite the fact that there are different crystallographic structural data reported from independent crystals obtained in independent synthetic experimental conditions, the experimentally determined unit cell parameters are not statistically distinguishable mutually, and, thus, the corresponding data belong to different crystallographic symmetric phases of the same complex (2) but not to different complexes.

In order to satisfy readers immediately on all possible objections, next, a second objection is addressed, relating to the structural solution of crystal (2)_2 in centrosymmetric space group system C2/c. The CIF data on the latter solution are provided, herein, as well. As can be seen, they show a higher R₁ parameter (R₁ = 0.1746) compared with the solution of the complex in a non-centrosymmetric phase Cc. The same is valid to solve the discussed structure in the I2/c space group type. Thus, in order to arrive at an empirically evident knowledge of the true space group type of crystalline phase (2)_2 in complex (2) (CCDC 2387639), we used two arguments: (i) from the ADDSYM test of symmetry and (ii) the R1 parameter, which convinced me. Thus, I am able to convince the reader that the latter phase is a non-centrosymmetric, rather than a centrosymmetric, phase. In an attempt to completely grasp the discussed issue, I highlight that the structural solution of (2)_2 in the Cc space group type was not performed on the basis of the statement above, that when there are two solutions of a crystalline phase, the non-centrosymmetric one should be carried out [9]. Rather, as the statistical data show, the statistical and chemometric performances of the crystallographic solution into the Cc space group type are better than those of the centrosymmetric solution. Therefore, the final structural solution of (2)_2 was based on objective criteria in statistical and chemometric tests.

The crystal structure consists of a square pyramidal Ag^IO₅ metal chromophore, showing bond lengths r(Ag–O) = 2.545, 2.522, 2.333, and 2.410 Å of the equatorial Ag–O bonds as well as 2.426 Å of the axial Ag–O bond. The ∠(O–Ag–O) angles are 88.3(5), 89.9(8), 87.7(6), and 94.8(8)°, respectively. In contrast to centrosymmetric phase (2)_1 containing Z = 8, the crystal structure of non-centrosymmetric phase Cc shows Z = 4. The two O-centers act as bridging ligands between two Ag^I-ions, showing r(Ag–O) = 2.426 and 2.668 Å. The angles ∠(Ag–O–Ag) are 91.4(0) and 98.2(3)°. There is contact r(Ag^…Ag) = 3.587 Å. The complex shows high thermal stability up to 480 K. The thermoanalytical data on the Ag^I-complex of squarate dianions detailing the decomposition reactions were examined. The following decomposition reaction has been proposed: Ag₂C₄O_4(s) + CO_(g) → 2Ag_(s) + CO₂ + [C₄O₃] at T = 515–535 K [87,88,89,90,91]. Despite the fact that, as mentioned, the Ag^I complexes show the significant stability of their (+1) oxidation state of the metal center, stable Ag²⁺ species at T = 485–500 K have been proposed, as well [88]. The following reaction has been proposed: Ag₂C₂O₄ → Ag⁰ + {Ag²⁺(C₄O₄)²⁻}.

3.3. Electronic Optical Properties in Solution

Looking at the electronic absorption properties of complexes (1) and (2) in solution, they shall be described as virtually close to each other, due to determination of the mass spectrometric data stable solvate complexes of the metal ions (see Section 3.1). The observed experimentally electronic absorption bands within the ultraviolet (UV) range of the electromagnetic spectrum (Figure 3), therefore, should be assigned to the electronic transition within the framework of the squarate anions or ligands. The experimental data are explained well enough via theoretical quantum chemical computations. As can be seen, the employment of SOS/DQ or EOM approaches and the LANL2DZ basis set to crystallographic data on the complex species or to their optimized geometries (below) yields an accuracy of λ_max of 3.65 nm (see also Figure S5 and Table S3). The observed UV bands, as expected, are assigned to the n→p transitions of the squarate dianion. The obtained theoretical value λ_max = 268.77 nm (f = 0.3538) agrees excellently with the experimental data in CH₃OH or H₂O reported in this study, as well as data from other authors [89].

Apart from the advantages of complex (2), highlighted above, as a prospective new MOF-based NLO-phore, such as its non-centrosymmetric crystalline phase and high temperature stability up to T = 535 K, the optical transparence of the crystals within 270–1100 nm should be added.

3.4. Vibrational Properties in Crystalline State

The theoretical vibrational modes were obtained using optimized geometries of (1) and (2). Table S4 summarizes the energetics of the optimized species, using the atomic coordinates shown in Table S5. The obtained vibration modes in the ground states are tabulated, as well (Table S6). Figure S6, in the latter context, illustrates the theoretical IR spectra together with the selected visualized vibrational motions of the species. The IR bands at 1804–1750, 1620, and 1530 cm⁻¹ belong to the ν_C=O, ν_C=C, and ν_C=C+ν_C=O modes of the squarate ligand. Bands within 1460–1100 cm⁻¹ belong to the ν_C-C mode, while the band at 727 cm⁻¹ belongs to the ring breathing mode. The 650 and 317 cm⁻¹ bands belong to the δ_ring and δ_CO modes. The shown experimental and theoretical data on M062X/LANL2DZ agree well with the results from comprehensive analysis of the vibrational properties of squarate complex species of alkali metal ions [90], as well. A Raman mode has been assigned at 385 cm⁻¹ (λ_ex = 514.5 nm) o ν_Zn–O stretching vibration of the Zn^II-squarate complex [91].

3.5. Nonlinear Optical Properties

In this sub-section, the author would like to add that the claim regarding the prospective application of complex (2) to linear optical and nonlinear optical technologies is based not only on the fact that it is among the rare cases of Ag^I complexes of squarates, which crystallizes non-centrosymmetrically, as they are at about 15% of the reported crystals of Ag^I-ion and at about 13% of complexes of Zn^II- and Ag^I-ions with the discussed ligand crystallized into the non-centrosymmetric space group type (Table S7). Conversely, it shows significant thermal stability and optical transparency within 200–1100 nm but also looking at the theoretical linear optical and nonlinear optical properties. The comparative analysis involves data on complex (2) and the squarate dianion. We obtained the following results from polarizability and hyperpolarizability tensor components [(C₄O₄)²⁻] (polar. 85.1708865, −0.000005, 85.1471524, −0.0001605, 0.0005283, 23.2082861; hyper polar. −0.0098898, 0.1864671, −0.0071, 0.275977, 0.0786893, −0.1005592, 0.0792977, −0.0001254, 0.0049768, 0.0518928); (2) (polar. 259.6438627, −5.2250367, 168.1356433, 1.5268616, −0.1398028, 62.4900084; hyper polar. −8.6881902, −27.0819917, 6.9304239, −0.4725642, −4.9880504, 3.8684769, 1.0596874, −0.8279703, −2.8137591, −3.4648956). As can be seen, there is an increasing α_xx tensor component at a magnitude order of 3.04-times and 879.43-times the β_xxx tensor component in the complex compared with the free squarate dianionic ligand. Readers who give the mater attention concentrating the latter data and results from KDP—inorganic material with marked NLO properties, which is already used in the field of nonlinear optical technologies, as highlighted above—shall clearly see that the α_xx value of complex (2) has a larger value at a magnitude order of 7.00-times compared with KDP computed at the same level of theory (KDP (polar. 37.0776932, −0.000518, 32.6220945, −0.0037454, −3.1567931, 32.8203542; hyper polar. −234.6417743, 0.0909451, −38.4587409, −0.1151177, −0.0412643, −41.1818089, 0.0247814, −53.9553874, −0.026954, −0.0198434).

Table 3. Electric dipole moment (μ); dipole polarizability, α (esu units = cm³, SI units = C²m²J⁻¹), and first dipole hyperpolarizability, β, where || and _|_ denote parallel and perpendicular values with respect to z axis, as well as tensor-components x, y, and z (esu units = statvolt⁻¹·cm⁴, SI units = C³·m³·J⁻²) data on complex (2).

Electric dipole moment
	[a.u.]	Debye	10−30 SI
μtot	0.317187 × 10−1	0.806208 × 10−1	0.268922
μx	0.000000	0.000000	0.000000
μy	0.000000	0.000000	0.000000
μz	0.317187 × 10−1	0.806208 × 10−1	0.268922
Dipole polarizability
α(0,0)	[au]	[10−24 esu]	[10−40 SI]
αiso	0.425867 × 103	0.631070 × 102	0.702160 × 102
αaniso	0.986272 × 103	0.146150 × 103	0.162614 × 103
αxx	0.633700 × 103	0.939047 × 102	0.104483 × 103
αyx	0.103717 × 103	0.153692 × 102	0.171006 × 102
αyy	0.154964 × 103	0.229634 × 102	0.255502 × 102
αzx	−0.500754 × 103	−0.742041 × 102	−0.825632 × 102
αzy	−0.495223 × 102	−0.733845 × 101	−0.816513 × 101
αzz	0.488936 × 103	0.724529 × 102	0.806147 × 102
α(−ω;ω) ω = 455.6 nm	[au]	10−24 esu	10−40 SI
αiso	0.153553 × 103	0.227542 × 102	0.253175 × 102
αaniso	0.587493 × 103	0.870575 × 102	0.968645 × 102
αxx	0.327207 × 103	0.484871 × 102	0.539492 × 102
αyx	0.227125 × 102	0.336565 × 101	0.374479 × 101
αyy	−0.970495 × 102	−0.143813 × 102	−0.160013 × 102
αzx	−0.241757 × 103	−0.358247 × 102	−0.398603 × 102
αzy	−0.815340 × 102	−0.120821 × 102	−0.134431 × 102
αzz	0.230502 × 103	0.341568 × 102	0.380046 × 102
First dipole hyperpolarizability
β(0;0,0)	[a.u.]	[10−30 esu]	[10−50 SI]
β|| (z)	0.450178 × 101	0.388919 × 10−1	0.144343 × 10−1
β_|_(z)	0.150059 × 101	0.129640 × 10−1	0.481145 × 10−2
βx	−0.369368 × 102	−0.319105	−0.118433
βy	−0.124310 × 102	−0.107394	−0.398583 × 10−1
βz	0.225089 × 102	0.194459	0.721717 × 10−1
β||	0.900112 × 101	0.777627 × 10−1	0.288609 × 10−1
βxxx	−0.819435 × 101	−0.707928 × 10−1	−0.262740 × 10−1
βxxy	−0.923337 × 101	−0.797692 × 10−1	−0.296055 × 10−1
βyxy	−0.164386 × 101	−0.142017 × 10−1	−0.527081 × 10−2
βyyy	−0.358041D × 101	−0.309320 × 10−1	−0.114801 × 10−1
βxxz	0.736136	0.635964 × 10−2	0.236032 × 10−2
βyxz	−0.210114 × 101	−0.181522 × 10−1	−0.673702 × 10−2
βyyz	−0.191303 × 101	−0.165271 × 10−1	−0.613386 × 10−2
βzxz	−0.247405 × 101	−0.213738 × 10−1	−0.793269 × 10−2
βzyz	0.867012 × 101	0.749031 × 10−1	0.277995 × 10−1
βzzz	0.867986 × 101	0.749873 × 10−1	0.278308 × 10−1
β(−ω;ω,0) ω = 455.6 nm
	[a.u.]	[10−30 esu]	[10−50 SI]
β|| (z)	−0.106836 × 104	−0.922982 × 101	−0.342556 × 101
β_|_(z)	−0.999635 × 102	−0.863607	−0.320519
βx	0.408269 × 104	0.352712 × 102	0.130906 × 102
βy	−0.271454 × 104	−0.234515 × 102	−0.870381 × 101
βz	−0.534181 × 104	−0.461491 × 102	−0.171278 × 102
β||	0.145013 × 104	0.125280 × 102	0.464965 × 101
βxxx	0.396853 × 103	0.342850 × 101	0.127246 × 101
βyxx	0.621301 × 102	0.536756	0.199212
βyyx	−0.322483 × 103	−0.278600 × 101	−0.103400 × 101
βzxx	−0.327435 × 103	−0.282878 × 101	−0.104987 × 101
βzyx	−0.113067 × 103	−0.976813	−0.362535
βzzx	0.279499 × 103	0.241465 × 101	0.896175
βxxy	−0.425822 × 103	−0.367877 × 101	−0.136534 × 101
βyxy	0.705009 × 103	0.609073 × 101	0.226051 × 101
βyyy	−0.423235 × 103	−0.365642 × 101	−0.135704 × 101
βzxy	0.496533 × 103	0.428966 × 101	0.159207 × 101
βzyy	−0.962529 × 103	−0.831550 × 101	−0.308622 × 101
βzzy	−0.752474 × 103	−0.650079 × 101	−0.241270 × 101
βxxz	−0.843863 × 103	−0.729032 × 101	−0.270573 × 101
βyxz	0.220077 × 103	0.190130 × 101	0.705647
βyyz	0.322371 × 103	0.278504 × 101	0.103364 × 101
βzxz	0.762547 × 103	0.658781 × 101	0.244500 × 101
βzyz	−0.195401 × 103	−0.168811 × 101	−0.626526
βzzz	−0.746798 × 103	−0.645175 × 101	−0.239450 × 101

summarizes the frequency-dependent dipole polarizability and first dipole hyperpolarizability. Table S8 shows data on KDP. The β(−ω; ω; 0)_‖(z) (ω = 455.6 nm) of (2) is 3.64-times larger than the value of KDP, while β_zzz(−ω; ω; 0) of the complex is 2.255-times larger compared with the KDP one.

The results from the latter tables are in accordance with what the author was taught, i.e., that the metal–organic framework materials based on Ag^I-or Zn^II complexes of squarate ligands produce comparable linear optical and nonlinear optical responses in marked inorganic NLO materials, such as the used standard KDP one.

Looking at classical theory describing the light-scattering effects in matter arising due to charge oscillations and current distribution induced by the interaction with incident electromagnetic waves, the dominant contribution comes from the oscillating effects of the electric dipole, which is linearly related to the electric field strength E_β and second-rank polar tensor α_αβ, called polarizability [92] (Equation (S1)). μ_α denotes the dipole moment, while μ_α⁰ means a permanent dipole moment. β_αβγ denotes third-rank tensor or hyperpolarizability. In the cases of centrosymmetric molecules, μ_α⁰ and β_αβγ vanished.

The average value of the mean molecular polarizability as a function of temperature is given in Equation (S2). It depends on the electronic, vibrational, and rotational states, assuming that polarizability in the electronic ground state α_el could be separated from the corresponding ro-vibrational contributions [93]. The Z denotes the rotation vibration partition function. It is given by Equation (S3). The Z can be approximated to Z = Z_vibZ_rot, and, thus, the functional relation of polarizability with temperature can be accurately predicted in computational methods, as well. The function Z_vibr(T) = f(T) shows a nonlinear relationship within T = 0–500 K [93]. The results from computations of complex (2), as shown in

Table 4. Vibrational average of electric-field properties of electric dipole of optimized asymmetric unit cell content of complex (2) at T = 0 and 298 K.

T [K]	Level of Theory	X [Debye]	Y [Debye]	Z [Debye]
0	STO-3G	−	−1.4453	0.0007
298		−2.7078	−1.4347	0.0007
0	LANL2DZ	−11.2328	−0.6061	0.0003
298		−11.2407	−0.6120	0.0003

and

Table 5. Thermodynamic properties and partition functions of optimized asymmetric unit cell content of complex (2) at various temperatures [K].

	Harmonic Value		SPT Anharmonic Value
T [K]	298.15
	LANL2DZ	STO-3G	LANL2DZ	STO-3G
Qvib	0.13987 × 10−16	0.26471 × 10−17	0.18714 × 10−16	0.35544 × 10−17
QZvib	0.11518 × 102	0.36366 × 102	0.12179 × 102	0.39512 × 102
Sp.Heat(V) J/(mol K)	0.85862 × 102	0.10169 × 103	0.86655 × 102	0.10246 × 103
Sp.Heat(P) J/(mol K)	0.94177 × 102	0.11000 × 103	0.94969 × 102	0.11077 × 103
T [K]	500.00
Qvib	0.25010 × 10−8	0.30844 × 10−8	0.31466 × 10−8	0.39466 × 10−8
QZvib	0.12056 × 103	0.79637 × 103	0.13182 × 10−3	0.89811 × 103
Sp.Heat(V) J/(mol K)	0.12028 × 103	0.13490 × 103	0.12111 × 103	0.13576 × 103
Sp.Heat(P) J/(mol K)	0.12860 × 103	0.14322 × 103	0.12943 × 103	0.14408 × 103
T [K]	1000.00
Qvib	0.13402	0.11370 × 101	0.16589	0.14477 × 101
QZvib	0.29425 × 105	0.57772 × 106	0.33955 × 105	0.69061 × 106
Sp.Heat(V) J/(mol K)	0.15865 × 103	0.17292 × 103	0.15915 × 103	0.17351 × 103
Sp.Heat(P) J/(mol K)	0.16697 × 103	0.18123 × 103	0.16746 × 103	0.18183 × 103
T [K]	1500.00
Qvib	0.76657 × 103	0.16686 × 105	0.95443 × 103	0.21493 × 105
QZvib	0.27899 × 107	0.10625 × 109	0.33148 × 107	0.13122 × 109
Sp.Heat(V) J/(mol K)	0.17088 × 103	0.18601 × 103	0.17116 × 103	0.18636 × 103
Sp.Heat(P) J/(mol K)	0.17919 × 103	0.19432 × 103	0.17947 × 103	0.19467 × 103
T [K]	2000.00
Qvib	0.25484 × 106	0.10363 × 108	0.31918 × 106	0.13476 × 108
QZvib	0.11941 × 109	0.73870 × 1010	0.14440 × 109	0.93079 × 1010
Sp.Heat(V) J/(mol K)	0.17586 × 103	0.19152 × 103	0.17603 × 103	0.19174 × 103
Sp.Heat(P) J/(mol K)	0.18417 × 103	0.19984 × 103	0.18434 × 103	0.20006 × 103

, agree well with the data on the latter reference. The thermally averaged microscopic second hyperpolarizability (Γ) is given by Equation (S4) [94]. There are two separate contributions of (i) temperature-independent second hyperpolarizability, due to distortion of the electronic structure due to the effect of the applied electric field, and (ii) a temperature-dependent first hyperpolarizability, obtained as a result of the orientational effect of the applied electric field on the permanent dipole moment. Equations (S5) and (S6) further detail the mutual relationships (consider [94]). Therefore, a linear relationship between the averaged microscopic second hyperpolarizability and 1/T is established [94].

4. Conclusions

So, a reasonable conclusion from this study might be that it deals with new, first reported in the literature, non-centrosymmetric phases of the catena-((μ₃-squarato)-(μ₂-aqua)-silver(I)) complex of Ag^I ions and squarate ligand crystallizing into the monoclinic Cc space group type. It shows high thermal stability up to T = 535 K, good crystal growth, and optical transparency within 264–1100 nm, and, thus, it appears to be a prominent candidate for metal–organic framework-based linear optical and nonlinear optical crystalline materials. In addition, in condensed phases, the complex shows the high stability of the Ag^I-oxidation state of the metal center. There is a long-standing opinion that the non-centrosymmetric crystal structure is a precondition—in addition to the outstanding mechanical properties of the crystals—for the generation of a higher-order NLO response of crystals and their tunable laser damage threshold, used in technological applications of the designed crystals to nonlinear optical technologies. Perhaps there may be someone who would prefer to highlight the novelty of this study also reporting data on crystallographic re-determination in the multiplication of catena-((μ₂-squarato)-tetra-aqua-zinc(II)) in different experimental conditions toward temperature, thus discussing the effect of the uncertainty of crystallographic variables on the theoretical optical properties of the materials together with their energetics, which the current study also provided. Furthermore, this study tackled, from the perspective of statistical test data, the multiplication measurements of complex catena-((μ₃-squarato)-(μ₂-aqua)-silver(I)), in addition to detailed correlation between the molecular crystal structure and optical properties, both experimentally and theoretically. However, the author arrived at the present conclusion in order to underline the prospective applications of the studied complexes as innovative metal–organic framework-based materials to many interdisciplinary branches of technology and industry, rather than to focus the reader’s attention on their effect on the fundamental science connected with the further in-depth understanding of the coordination chemistry of transition metal ions or chemical crystallography, among others.