Betahistine is a simple organic molecule (N-methyl-2-(pyridin-2-yl)ethanamine, C8H12N2) consisting of the pyridine ring and an aliphatic chain containing amine group (Fig. 1). Is structure is closely related to phenethylamine, amphetamine and/or histamine.

Fig. 1
figure 1

Structural formula and molecular structure of betahistine (PubChem CID 2366 (2004)). Color codes: C—cyan, N—blue, H—white

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Betahistine belongs to the drugs that amplify effects of neurotransmitters such as acetylcholine, histamine, norepinephrine, serotonin, and γ-aminobutyric acid (GABA). Betahistine primarily affects the histaminergic system: It acts as a partial agonist of the H1 histamine receptor, and as an antagonist of the H3 histamine receptor, it changes the generation of neuronal excitation, e.g., in the vestibular nuclei, thereby facilitating vestibular compensation. Betahistine can increase blood flow in the brain, especially in the inner ear, both cochlear and vestibular parts. Based on its pharmacological properties, betahistine is an effective medicament administrated in some diseases such as hearing loss, dizziness, vertigo, tinnitus and Menière's syndrome (Zamergrad et al. 2021; Murdin et al 2016; Wegner et al 2018; Kloos et al 2022). Biological effects of these species are encoded in their molecular and electronic structure. Nowadays, molecular properties can be effectively studied using high-quality ab initio calculations that include a part of the correlation energy.

Structural data have been retrieved from database (PubChem CID 2366 (2004)); it serves as an initial guess for the full geometry optimization in vacuo. For such a purpose the ab initio HF-MO-LCAO-SCF method was utilized using the ORCA package (Neese 2012, 2020, 2022). The basis set def2-TZVP consisting of 382 basis functions (N, C: {62,111/411/11/1}; H: {311/1}) was utilized (Weigend and Ahlrichs 2005). The stationary geometry has been confirmed by the complete vibrational analysis showing no imaginary frequencies.

Molecular properties were calculated at the Hartree–Fock SCF level, namely the energies of the HOMO (the highest occupied molecular orbital) and LUMO (the lowest unoccupied molecular orbital), the permanent dipole moment p and quadrupole moment Q and the dipole polarizability α (one-third trace of the polarizability tensor). The vertical ionization energy Ei and electron affinity Eeg were calculated upon the energy of the positively and/or negatively charged open-shell system in the UHF (unrestricted Hartree–Fock) variant (frozen optimum geometry for the neutral species). They allow expression of the chemical potential μ via the Mulliken electronegativity χM and the Pearson hardness ηP, i.e.,

$$\frac{\partial E}{{\partial N}} = - \mu = \chi_{M} = \, \left( {E_{i} {-}E_{{{\text{eg}}}} } \right)/2$$
(1)
$$\frac{1}{2}\frac{{\partial^{2} E}}{{\partial N^{2} }} = \eta_{P} = \, \left( {E_{i} + E_{{{\text{eg}}}} } \right)/2$$
(2)

These quantities can be viewed as an electronic gradient and electronic force constant with respect to the number of electrons N, respectively (Sen 1993). The hardness serves as a measure of the Lewis acidity/basicity (Pearson 1997). The molecular electrostatic potential is plotted as a 3D contour map on the isovalue surface of charge density (Politzer and Murray 2002; Politzer et al 1985). This indicates the basic and/or acidic sites suitable for electrophilic or nucleophilic interactions along the molecule. In the optimized geometry, full vibrational–rotational analysis was applied giving rise to the partition function and related thermodynamic functions: the internal energy U, entropy S and Gibbs energy G. A partial inclusion of the correlation energy by the 2nd-order perturbative configuration interaction using the Moller–Plesset partitioning (abbr. MP2) improves the calculated energetic properties: ionization energy, electron affinity, Mulliken electronegativity and the Pearson hardness.

All data calculated in vacuo have been recalculated by involvement of the solvent effect (water). For this purpose, the CPCM (Conductor-like Polarizable Continuum Model) has been utilized (Takano and Houk 2005). Prior to evaluation of molecular properties, the full geometry optimization has been done followed by the complete vibrational analysis. The calculated molecular properties of betahistine are listed in Table 1.

Table 1 Calculated molecular properties of betahistinea
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The internal energy and entropy are divided into contributions from vibrations, rotations and translation. The most important correction to the internal energy is the vibrational contribution of the value of 131 kcal mol−1 dominated by the zero-point energy. The overall entropy contribution S·Tø = 28 kcal mol−1 has a visible impact to the Gibbs energy.

A comparison of the solvent-free and solvent–water data shows: (i) The energies of HOMO and LUMO are not influenced by the solvent; (ii) total energies of the molecule and its ions are lower in the presence of the solvent; (iii) the solvation reduces the ionization energy and electron affinity; (iv) consequently the Pearson hardness is reduced; (v) the value of the dipole moment increases substantially in the solvent, whereas the isotropic quadrupole moment is insensitive; (vi) the polarizability increases owing to the solvation; (vii) thermodynamic functions are little sensitive to solvation.

The Pearson hardness reflects the resistance of the molecule against the electron transfer, and for the betahistine, it is 108 and 119 kcal mol−1 in vacuo by SCF and MP2 calculations, respectively. Upon solvation, these data are reduced to 53 and 74 kcal mol−1. They can be compared to the series of aminoacids, monoaminergic neurotransmitters and related drugs (Table 2).

Table 2 Calculated electronegativity and chemical hardness at MP2 levela
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A similarity of the species listed in Table 2 can be assessed by the cluster analysis (CA). CA is a classification method designed to group observations or variables into clusters based upon similarities between them using a numerical criterion (distance). Ward’s method defines the distance between 2 clusters in terms of the increase in the sum of squared deviations around the cluster means that would occur if the two clusters were joined (Statgraphics 2006).

Figure 2 shows that according to the applied Ward’s method and squared Euclidean norm as a distance criterion, three clusters are relevant. The target molecule No 1 (betahistine) has similar descriptors to No 2, 3, 4 and 5 (N-methylphenethylamine, phenethylamine and amphetamine, 1-phenethylamine), different from No 6, 7 and 8 (methamphetamine, ephedrine and octopamine) and very different from species No 9 and 10 (tyramine and histamine).

Fig. 2
figure 2

Cluster analysis (Wards method) showing similarity of species in Table 2 based upon the chemical potential (electronegativity), chemical hardness and polarizability calculated at MP2 level

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Simple QSAR calculations (Quantitative Structure–Activity Relationships) based upon increments gave the surface area S = 343 Å2, volume V = 514 Å3, molar refractivity R = 22.1 Å3, hydration energy  − 4.14 kcal mol−1 and octanol–water partition coefficient logP = 0.61 (Hyperchem 2008).

The molecular electrostatic potential is drawn in Fig. 3. This brings information about the active sites for the ligation; it can help in understanding the docking inside the receptor such as 3D orientation of the docked conformations of 1ETH with ligands—betahistine (Chaturvedi and Gupta 2021; Fossati et al 2001).

Fig. 3
figure 3

Molecular electrostatic potential and 3D mapped isosurface of charge density for the optimized structure of betahistine (contour 0.03 ea0−1; a0–bohr unit) calculated in vacuo at the SCF level

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On conclusions, ab initio MO-LCAO-SCF + MP2 calculations show that the molecule of betahistine, in comparison with some monoaminergic neurotransmitters, possesses a rather high ionization energy and consequently high electronegativity (84 kcal mol−1). Its hardness is rather low (119 kcal mol−1 in vacuo and 74 kcal mol−1 in water) and similar to analogous species like amphetamine. To this end, the calculated chemical potential and chemical hardness could be considered as new, non-additive collective molecular descriptors of the electronic structure of small bioactive molecules.