The equilibrium structures of the A_XA_X, A_YA_Y, A_YB and adducts A_XA_XNa⁺, A_YA_YNa⁺, and A_YBNa⁺ are shown in Figure 1, Figure 2. The selected geometrical parameters and atomic charges are given in Table 1, Table 2. The dimers and adducts comprise two rings, each of the ring made up of five carbon and one oxygen atom inserted between. These rings are fixed jointly by the glycosidic bond, the bond which ensures the stability and stiffness of this molecule [7] [10] . The glycosidic bond bends the rings in opposite direction extricating rings from each other and thus reduces repulsive forces flanked by the two.

A_XA_X and A_XA_XNa⁺. In the A_XA_X dimer, glycosidic bond forms an angle C2-O12-C15 = 119˚ in Figure 1(a), and Table 1, whose respective length C2-O12 and C15-O12 are 1.438 Å and 1.400 Å correspondingly. In the A_XA_XNa⁺ complex the geometrical parameters are not changed drastically, the angle C2-O12-C15 is 118˚ the bond length C2-O12 is 1.426 Å and C15-O12 is 1.412 Å. That is the attachment of Na⁺ to the dimer does not alter significantly the shape of the glycosidic bond and this is because Na⁺ attaches at the distance far away from the bond. In the A_XA_X molecule hydroxyl group and amine group are in opposite sides to each other to reduce mutual repulsion of like charges. Amine groups are located in rather big distance between, 7.175 Å, thus the stable conformer of A_XA_X is formed. In the rings of A_XA_X the internal C-O-C groups exist, in the first ring

C4-O7-C5 angle is 114˚, its respective bond lengths are C4-O7 = 1.419 Å and C5-O7 = 1.433 Å, in the other ring the C15-O18-C16 angle is the same and the bond lengths are 1.423 Å for C15-O18 and 1.439 Å for C16-O18. That is the internal C-O-C fragments are alike in both rings of A_XA_X implying to have the same chemical environment before Na⁺ attachment.

In A_XA_XNa⁺ adduct, the position of the Na⁺ connection is ascribed to the negative charges of the N11, O1 and C4 (Table 1) and appropriate internuclear separations. The attachment of Na⁺ creates two new bonds Na⁺-O1 and Na⁺-N11. Compared to A_XA_X, in A_XA_XNa⁺ adduct the attached Na⁺ results in some modification to the original structure, the C4-O7-C5 = 117˚ in the first ring whose respective bond lengths are C4-O7 = 1.382 Å and C5-O7 = 1.448Å, and the next ring C15-O18-C16 = 116˚ whose respective bond lengths are 1.417 Å for C15-O18 and 1.445 Å for C16-O18. Furthermore in the adduct A_XA_XNa⁺, the internal hydrogen bond O8 - H29…O18 (1.853 Å) is formed which increases the adduct stability and stiffness.

A_YA_Y and A_YA_YNa⁺. In the dimer A_YA_Y the amine groups are separated by distance of 4.979 Å that is significantly less as compared to A_XA_X (7.175 Å). This may lead to decrease in the stability of the dimer A_YA_Y because of the repulsive force experienced between amine groups. Also this dimer has two hydrogen bonds O10-H32…O8 and O20-H43…O22 whose respective lengths are 1.865 and 1.820 Å. The formation of these internal hydrogen bonds favors the stability of the dimer. In the glycosidic bond of A_YA_Y, the angle C5-O9-C13 is equal to 117˚ and the bond lengths C5-O9 and C13-O9 are 1.438 Å and 1.398 Å correspondingly (Table 1) while in the A_YA_YNa⁺ complex, the similar angle C5-O9-C13 is 115˚ and the bond lengths are C5-O9 = 1.453 Å and C13-O9 = 1.406 Å. Thus akin to A_XA_XNa⁺ the attachment of Na⁺ to A_YA_Y does not bring essential change in the parameters of the glycosidic bond. At the same time contrary to A_XA_XNa⁺, the position of Na⁺ connection to A_YA_Y is close to the glycosidic bond; it attaches to two oxygen atoms (one of the glycosidic bond and another of the ring) and to the nitrogen atom connected with the second ring.

In A_YA_YNa⁺ adduct, the position of the Na⁺ attachment is credited to the negative charge −0.39 of the N12 in the dimer A_YA_Y and the appropriate internuclear separations and steric factors as well. Three new bonds are formed: Na⁺-O9, Na⁺-O18 and Na⁺-N12 with internuclear separations 2.335 Å, 2.253 Å, and 2.347 Å, respectively. Two hydrogen bonds O10-H32…O8 and O20-H43…O22 remain in adduct A_YA_YNa⁺, and their respective lengths do not change remarkably as compared to A_YA_Y.

A_YB and A_YBNa⁺. The equilibrium structures of the A_YB complex and the A_YBNa⁺ adduct are shown in Figure 2; and the selected geometrical parameters and atomic charges are given in Table 2. In the glycosidic bond of A_YB connecting two rings the angle C2-O12-C15 is equal to 118˚, and bond lengths are C2-O12 = 1.438 Å and O12-C15 = 1.408 Å. In the A_YBNa⁺, the C2-O12-C18 angle is 119˚ and the respective bond lengths C2-O12 and O12-C15 are 1.457 Å and 1.378 Å, thus slight change in the C-O-C parameters are noticed when Na⁺ is attached. In A_YBNa⁺, three bonds are formed between Na⁺ and negatively charged oxygen atoms O18, O21, O22. In A_YBNa⁺ adduct the amino groups are located far apart at the distance of 7.244 Å between two nitrogen atoms to ebb electron-electron repulsion. Whereas in A_YB molecule there is no internal hydrogen bond, and in its adduct the hydrogen bond N23-H47…O10 is formed; this connotes that attachment of Na⁺ stabilizes the species.

In the calculated vibrational spectra there were no imaginary frequencies revealing the equilibrium geometrical structures of the species. The theoretical spectra of the dimer molecules and adducts with Na⁺ ion are presented in Figures 3-5. The general feature of all these spectra is an existence of three regions of vibrational modes, which are approximately (i) below 1700 cm⁻¹, (ii) about 3000 cm⁻¹, and (iii) above 3300 cm⁻¹. The assignment of the most intensive peaks is considered below.

A_XA_X and A_XA_XNa⁺. The vibrational spectra of A_XA_X and A_XA_XNa⁺ adducts are shown in Figure 3. As is seen, majority of bands are located at a low frequency region 100 - 1690 cm⁻¹ which correspond to O-H rocking, NH₂ scissoring, and ring bending vibrations. For the dimer A_XA_X the strongest peak seen at 1067 cm⁻¹ is ascribed to bending of the first ring, and the subsequent peak at 1352 cm⁻¹ is assigned to the bending of the second ring. An intricate vibration mode at 717 cm⁻¹ relates to the bending of different fragments of the molecule. The middle region ranges from 2900 - 3100 cm⁻¹ in both A_XA_X and A_XA_XNa⁺ adduct; the most intensive peak relates to C-H stretching at about 3034 cm⁻¹. The higher frequency region ranges between 3400 - 3750 cm⁻¹ for both A_XA_X and the A_XA_XNa⁺ adduct where very intense O-H stretching modes are seen at about 3500 cm⁻¹ and 3474 cm⁻¹ for A_XA_X and A_XA_XNa⁺, respectively. Also the peaks at 3581 cm⁻¹ (A_XA_X) and 3444 cm⁻¹ (A_XA_XNa⁺) observed are assigned to the combination of O-H and N-H stretching modes.

A_YA_Y and A_YA_YNa⁺. The vibrational spectra of A_YA_Y and A_YA_YNa⁺ are shown in Figure 4. Majority of the vibrational bands manifests at the first frequency region. The most intensive vibrational peak is seen at 1089 cm⁻¹ associated to bending of the ring, other peaks are seen also at 186 cm⁻¹ relating to O-H rocking and ring bending, at 800 cm⁻¹ NH₂ wagging, O-H rocking and a vibration band at ~1670 cm⁻¹ assigned to the NH₂ scissoring. The similar bands are seen in A_YA_YNa⁺ adduct. In the middle region at about 3000 cm⁻¹ C-H stretching vibrational modes are seen for both A_YA_Y and A_YA_YNa⁺ adduct; and in the higher frequency region O-H and NH₂ stretching vibrations may be observed between 3400 - 3750 cm⁻¹ in both cases.

A_YB and A_YBNa⁺. In the IR spectra of A_YB and A_YBNa⁺ the most intensive bands are seen in low and high frequency regions (Figure 5). In the spectrum of A_YB molecule, the peak at 1741 cm⁻¹ is assigned to a combination of C=C bond stretching and N-H rocking, the next band at 1035 cm⁻¹ corresponds to the combination of C-O stretching and ring bending. Other peaks are ascribed to O-H rocking, N-H scissoring, and knotty vibration mode of the ring. In the middle region the C-H stretching vibration is observed at 3019 cm⁻¹ for A_YB and at 3061 cm⁻¹ for the adduct. Lastly in the third region, N-H, and O-H stretching vibration modes of the A_YB are

seen between 3391 - 3668 cm⁻¹ while for A_YBNa⁺ adduct the frequency 3359 cm⁻¹ is assigned to the stretching vibration of NH…O (hydrogen bond).

Concluding this section on the spectra of the dimer molecules and the adducts with Na⁺ ion, some similarities may be noted which are apparently due to the same functional groups of atoms, NH₂, CH, OH, COC, and other fragments, and the structural resemblance of the species as well. In the AANa⁺, and ABNa⁺ adducts, the Na⁺ ion only participates in low frequency vibrations, less than 395 cm^-1 of low IR intensity. Worth to mention here that the findings of this study are in accordance with the theoretical spectra of the glucosamine molecules A, AA, and AAA [7] , and ANa⁺ and BNa⁺ adducts [20] as well with the experimental IR spectra of chitosan [26] [27] .

The reactions of the sodium ion attachment to the dimer molecules were considered:

The energies of the reactions D_rE were calculated as the difference between the total energies of the product and reactants:

The enthalpies of the reactions ∆_rH°(0) were obtained using ∆_rE and the zero-point vibration energy corrections ∆ZPVE:

where h is the Plank’s constant, c is the speed of light in the free space, , and are the sums of the vibration frequencies of the product and reactants, respectively.

The enthalpies of the reactions (1)-(3) have been calculated under different basis sets, 6-31G(d), 6-31G(d, p), 6-311G(d), 6-311G(d, p), 6-31++G(d, p) and 6-311++G(d, p); the results are presented in Figure 5. For all three reactions, the similarity of the plots is observed. The general trend observed reveals the increase of the enthalpies of the reactions with the basis set extension. The difference between the lowest and highest values of ∆_rH°(0) for each reaction is about 50 kJ∙mol⁻¹, thereafter the uncertainties for the enthalpies of the reactions may be estimated as half of this difference, that is ~25 kJ∙mol⁻¹. The values energies ∆_rE, ∆ZPVE, and enthalpies ∆_rH°(0) of the reactions obtained with the most extended basis set 6-311++G(d, p) are given in Table 3. The results indicate the association processes are highly exothermic, that implies a strong interaction between the cation and dimer molecules and high energetic stability of the corresponding adducts formed. The data obtained here are close to the enthalpies of attachment of sodium ion to monomer molecules, −196 kJ∙mol⁻¹ (A_X + Na⁺ = A_XNa⁺), −227 kJ∙mol⁻¹ (A_Y + Na⁺ = A_YNa⁺), and −229 kJ∙mol⁻¹ (B + Na⁺ = BNa⁺) [20] . Apparently likeness of these results relates to the similarity of the interacting moieties of the monomer or dimer molecules with Na⁺ ion.

The Gibbs free energies ∆_rG°(T) of reactions (1)-(3) were calculated using the following equation:

where ∆_rH°(T) is the enthalpy of the reaction at temperature T, ∆_rS°(T) is the change in entropy of the reaction. The thermodynamic functions of the A_XA_X, A_YA_Y, A_YB, A_XA_XNa⁺, A_YA_YNa⁺ and A_YBNa⁺, have been computed using the OpenThermo software [28] , those for Na⁺ gaseous ions are taken from [29] . The geometrical parameters and vibrational frequencies needed for the thermodynamic functions calculations have been obtained under 6-31G(d) basis set. The thermodynamic functions of the species are given in Table 4 for the temperature

^aΦ°(T) = −[H°(T) − H°(0) − TS°(T)]/T is the reduced Gibbs free energy.

range between 298 K and 1000 K. To find the enthalpies ∆_rH°(T), the enthalpy increments H°(T) − H°(0) were used:

where the values of ∆_rH°(0) were obtained using Equation (5) and the values of ∆_rE calculated with 6-311++G(d, p) basis set.

The plot of the Gibbs free energies of reactions (1), (2) and (3) are presented in Figure 6. The values of ∆_rG°(T) are negative in a broad temperature range, this indicates that the attachment processes are spontaneous; ∆_rG°(T) become positive at temperatures higher than 1700 K for the reaction (1), and 2000 K for (2) and (3). For the reverse process it is evident that their corresponding adducts are stable with respect of the detachment of Na⁺ ion in a broad temperature range (Figure 7).