Characterization and Application of Nanomaterials (2023) Volume 6 Issue 1 doi:10.24294/can.v6i1.1852 1 Original Research Article Construction of semiclassical interatomic B–B pair potential to char- acterize all-boron nanomaterials Levan Chkhartishvili Engineering Physics Department, Georgian Technical University, Tbilisi 0160, Georgia. E-mail: levanchkhartish- vili@gtu.ge ABSTRACT The semiclassical boron–boron interatomic pair potential is constructed in an integral form allowing its converting into the analytical one. It is an ab initio B–B potential free of any semiempirical adjusting parameters, which would serve as an effective tool for the theoretical characterization of all-boron and boron-rich nanomaterials. Keywords: Interatomic Potential; Semiclassical Approach; Ground State Parameters; Nanomaterial; Boron 1. Introduction Currently, the prospective wide technological applications of boro- phenes and boron-rich nanomaterials in general are of special research interests due to their variable interatomic bonding mechanism and relat- ed unique complex of physical and chemical properties (see some of re- cent reviews[1–6]). Among them, the small all-boron clusters Bn (contain- ing 푛 < 20 atoms) preferring (quasi) planar structures play an important role in characterization of borophenes and other 2D boron nanomaterials as can serve for their building blocks[7]. Boron nanoclusters’ ground state parameters—bond lengths, specif- ic (per atom) binding energy, atomic vibration frequencies, etc.—can be estimated on the basis of interatomic B–B pair potentials depended on a few rigorously chosen semiempirical parameters—see the paper and also the review of Chkhartishvili[8,9] which summarize results of similar at- tempts. Same approach has been found useful for the characterization of the relative stability of small (quasi) planar boron clusters, including the most abundant species B11, B12, and B13 in different charge states[10,11]. As is known, to study molecular properties quantitatively, e.g., de- termining the spectroscopic data or performing the collision calculations, potential curves of diatomic molecules are useful in analytical form. An- alytical pair interatomic potential curves are also needed to deduce the polyatomic molecular curves. In view of this, in the present work, we demonstrate how it is possible based on semiclassical approach to con- struct the B–B interatomic pair potential in an integral form, which is reducible to the analytical one. Paper is organized as follows. After this introduction, there is given a short review on interatomic potentials in general. Then charge density and potential distributions in boron atom are described semiclassically. They are used to construct the B–B inter- action potential energy curve in an integral form. And finally, based on discussion of obtained results some conclusions are drawn. ARTICLE INFO Received: 10 December 2022 Accepted: 6 February 2023 Available online: 17 February 2023 COPYRIGHT Copyright © 2023 by author(s). Characterization and Application of Nanomaterials is published by En- Press Publisher. This work is li- censed under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0). https://creativecommons.org/licenses /by-nc/4.0/ 2 2. Interatomic potentials When developing the numerical physical model of a material, the ab initio approaches, such as the DFT (Density Functional Theory) or QC (Quantum Chemistry), providing the best accura- cy on its electronic properties is limited to frag- ments up to thousands of atoms. To larger systems, it should be employed the computations by classi- cal MD (Molecular Dynamics), MC (Monte Carlo) or FE (Finite Element) methods, which are much faster but less accurate and assume that constitu- ent atoms are solid spheres influenced by interac- tions and in a good approximation follow the clas- sical equations of motion. Then behavior of materials in the elastic, electric or magnetic force fields depends on type and energy of interatomic bonding and the general understanding of their properties can be based on interaction potentials between atoms or ions[12]. The (long-range) interatomic forces, i.e., gradients of interatomic potentials prescribed as functions of atomic coordinates, play key role in capillarity and wetting phenomena as well[13]. The choice of forms or these potentials depends on the particular problem features. Recent Special Issue of the journal Molecules: “Intermolecular forces: From atoms and molecules to nanostructures”[14] has been devoted the relationship between forces act- ing between the atomic system particles and its properties across different scales: from molecules, simple aggregates or small clusters to nanostruc- tures and other types of condensed matter at the mesoscale. The parameters of introduced types of interatomic potentials can be derived both from experiments or quantum mechanics. To understand molecules chemical properties and also for their force fields high-quality empiri- cal parameterization, the obtaining of accurate conformational energetics is of significance. Mol- ecule conformational energy includes strain ener- gy coming from bond distortions, valence angles, torsion strains, etc., and enables to describe devia- tions of actual molecular geometry from the ideal one. As for intermolecular forces, they control most of the properties of the materials, such as their existence in solid, liquid or gaseous states, relative stability and chemical reactivity. The ab initio potentials are found in DFT or electron gas approximations, in which various contributions in the multi-atomic system energy— Coulomb, kinetic, exchange, correlation, etc.—are additively taken into account. In this case, in cal- culating the interaction potentials for heteroatomic systems, it is more correct to use a separate com- bination of these types of contributions. Recent alternative to both empirical and ab initio interatomic potentials are ML (Machine Learning) potentials. ML is becoming a method of choice for modeling complex chemical processes and materials providing a surrogate model trained on a reference dataset that can be used to establish a relationship between molecular structure and chemical properties. ML trained on quantum me- chanical calculations is a powerful tool for model- ing the PES (Potential Energy Surface). A critical factor for the automated discovery of robust inter- atomic potentials by ML is the quality and diver- sity of the training dataset. The course to atomistic simulations[15] pro- vides classification of popular interatomic poten- tials; their characterization in terms of accuracy, transferability and computational speed; potential cut-off procedures; short review of potentials used in MD and MC methods; derivation of the force from pair potential and force fields in materials; relationship between pair potential and elastic constants; limitations of pair potentials; and de- scription of pair potentials versus many-atomic potentials. The NIST’s (National Institute of Standards and Technology) IPR (Interatomic Potentials Re- pository)[16] serves for a source of interatomic po- tentials and force fields, in which there are pre- sented all the possible classes of potentials and materials (metals, semiconductors, oxides and carbon-containing systems). Interatomic and also fictional potentials are provided both for elemen- tary and non-elementary (alloys and compounds) materials. It is specially noted that: (i) multi- component potentials may not be applicable to the full composition range; (ii) coarse-grained poten- 3 tials reduce the simulation complexity by repre- senting molecular or alloy compositions with a single particle type; and (iii) fictional potentials purposefully fitted to target properties are not able to represent real materials accurately. 2.1 Empirical potentials Interactions between atoms can be modeled using different two- or many-atomic potentials. The total interaction potential can be written as a series of terms depending on the position of one, two, three, … atoms at a definite moment of time: 푈�����(푟⃗�, 푟⃗�, 푟⃗�, … ) = ∑ 푈�(푟⃗�)� + ∑ 푈�(푟⃗�, 푟⃗�)�� + ∑ 푈�(푟⃗�, 푟⃗�, 푟⃗�)��� + ⋯. Here 푟⃗�, 푟⃗�, 푟⃗�, … are the radius-vectors of atoms and 푈�, 푈�, 푈�, …—one-, two-, three-atomic, … potentials, respectively. The term 푈� is an atom’s potential in self- consistent or external force field. As for the term 푈� , in the simplest ionic bonding model, it is expressed by the potential energy of Coulomb or electrostatic interaction between a pair of oppositely charged ions: 푈�(푟) = − 퐴 푟⁄ , where 퐴 > 0 is the constant de- pendent on effective point charges of interacting atomic ions and 푟 = |푟⃗� − 푟⃗�| is the distance sepa- rating them. The Madelung or electrostatic energy of an ionic crystal is the sum of potential energies of interaction between constituent ions. A nega- tive overall energy implies attraction and then system’s stability. Electrostatic attraction between cations and anions increases as they approach each other, until at some distance their electron clouds begin to overlap leading to repulsion. These two forces of opposite signs are balanced at equilibrium separa- tion. The repulsive energy typically is formulated as 푈�(푟) = 퐵 푟�⁄ , where 퐵 > 0 and 푛 > 0 are the empirical constants. Bond length and binding en- ergy can be determined by minimizing the total potential energy: 푈(푟) = 푈�(푟) + 푈�(푟) = 퐵 푟�⁄ − 퐴 푟⁄ . Alternatively, the repulsion can be expressed by the Born–Mayer potential: 푈���(푟) = 퐶 exp(− 푟 푎⁄ ) with 퐶 > 0 and 푎 > 0 for constants. Usually, the interatomic potentials obtained empirically are approximated by model potentials without separation of terms with clear physical sense. These are Lennard–Jones 푈��(푟) = 퐵 푟��⁄ − 퐴 푟�⁄ , Morse 푈�(푟) = 퐷(exp(−2훼(푟 − 푎) − exp(−훼(푟 − 푎)) , Buckingham 푈�(푟) = 퐶 exp(− 푟 푎⁄ ) − 퐴 푟�⁄ , … potentials including two more positive parameters: 퐷 and 훼. Sometimes, the Lennard–Jones potential, which is frequently used for modeling rare gas crystals bounded by van der Waals forces, is pre- sented as 푈��(푟) = 4퐷((푟� 푟⁄ )�� − (푟� 푟⁄ )�) , where 푟�, called as van der Waals diameter, is the value of 푟, when 푈��(푟) function is zero, and 퐷 is the potential well depth. The origin of such intera- tomic interaction can be imagined as interplay between van der Waals attraction and Pauli repul- sion related to zero point fluctuations of electrons leading to induced dipole forces and their short- range interaction due to exclusion principle, re- spectively. An important special form of 푈��(푟) is the Mie–Lennard–Jones potential: 푈����(푟) = (퐷 (푏 − 푎)⁄ )�푎(푟��� 푟⁄ )� − 푏(푟��� 푟⁄ )�� , where 퐷 and 푟��� denote depth and coordinate of the potential minimum, 푎 and 푏 are the numerical parameters, 1 < 푎 < 푏, characterizing long-range action and rigidity of the potential. To construct adequate theoretical model for a material, it is necessary to use the empirical po- tentials that correctly take into account all the available types of interactions in a wide range of interatomic distances. In the simplest case, the pair interatomic potential can be chosen in the form of 푈(푟) = 퐷(푋(푟)� − 2푋(푟)), where 푋(푟) is some function of distance 푟 . The first term takes into account the repulsion, while the sec- ond—the attraction of atoms. The expediency of choosing this approximation is due to following reasons. It, firstly, quite adequately describes the pair interaction energy dependence on the intera- tomic distance and, secondly, allows one to obtain the widely used analytical formulas for interaction potential energies and forces between systems of atoms, in particular, above mentioned Lennard– Jones and Morse potentials. The use of the Len- nard–Jones potential leads to simple and easy-to- calculate explicit relations that describe the physi- cal properties, such as the potential energy of sol- 4 id structure, liquid surface tension and solid sur- face energies, and heat of sublimation. Its disad- vantage is the power-law dependence of the repul- sion energy, which gives too fast fall as the dis- tance increases. Morse potential, which takes into account the repulsive forces “softness”, is de- prived of this shortcoming and preferable to be used in modeling interactions in nanosystems, as well as quantum mechanical calculations since it allows one to obtain Schrodinger equation solu- tions in an explicit form. In the vicinity of mini- mum, any pair interaction potential is represented as a parabolic dependence 푈(푟) = −퐷 + 푘(푟 − 푟���)� 2⁄ , where 푘 is the bond stiffness factor. In various solid state physics applications, it turns out to be useful the Mie–Lennard–Jones potential, whose parameters for most of elements of the Periodic Table were estimated by Magome- dov[17]. The potential well depth 퐷 and equilibri- um distance 푟��� values were calculated from heat of sublimation and the lattice constant or molar volume. To estimate the pair interaction potential of different atoms, it can be used the Lorentz– Berthelot empirical combination rules: 푈��� → 푈�� = �푈�푈� and 푟��� → 푟�� = (푟� + 푟�) 2⁄ , where indices A and B denote the interacting at- oms. Three-atomic term 푈� frequently is ex- pressed by the Tersoff potential: 푈�(푟⃗�, 푟⃗�, 푟⃗�) = (1 2⁄ ) � 푓��푟��� ��� �푎��푓��푟��� + 푏��푓��푟���� where 푟�� = |푟⃗� − 푟⃗�| is the distance separating 푖 and 푗 atoms, 푓�(푟) = −훼 exp(−휆�푟) is the attrac- tive three-atomic potential and 푓�(푟) = 휌 exp(−휆�푟) is the repulsive two-atomic term (here parameters 훼, 휆� , 휌 and 휆� are the positive constants), while 푓�(푟) is a smooth cutoff function. The three-atomic contributions arise due to the bond-order parameters 푎�� dependent on bonds length and angle between them. The parameters 푏�� limiting the repulsive interactions range can be chosen as functions only of bonds length. The most recent comprehensive review of empirical interatomic potentials designed to re- produce materials elastic properties, defect ener- gies, bond formation and breaking, redox reac- tions, etc., has been done in the study of Muser et al.[18]. There are described the most popular two- atomic potentials such as: embedded-atom model potential for metals, bond-order potential for co- valently bonded semiconductors, polarizable po- tentials including ionic systems of atoms, and quantum-Drude oscillator model potential mim- icking multi-atomic dispersion. Emphasis is laid on the constraints ensue from the functional form of a potential. The review highlights potentials with simple functional forms allowing the analyti- cal treatment. Below, some of interesting exam- ples of practical applications of the empirical po- tentials are given. Periodic Table’s carbon subgroup ele- ments—C, Si, Ge, Sn and Pb—are of interests because of occurring of the covalent-to-metallic bond-type transition. The available experimental data were used[19] to obtain the parameters of their pair interatomic potentials represented in the Mie– Lennard–Jones form. All the different approaches suggested for the self-consistent determination have same disadvantage: it is unclear whether it is correct or not to use the parameters obtained for a free atoms pair for double covalent bond in a crys- tal. To answer this question, it was investigated the evolution of the potential depth 퐷 value exper- imentally determined from the crystal properties during the covalent-to-metallic bond-type transi- tion and demonstrated that 퐷 of a covalent bond determined from the bulk modulus is approxi- mately twice the value which follows from the crystal atomization energy. A conclusion was drawn about the covalent bonding nature: such bond between in a crystal is double and these two bonds differ in potential depth value. Each of the generalized valence electrons realizes strong and weak bonds with own and alien ions, respectively. A double covalent bond under elastic, i.e., re- versible, deformation of the crystal is about twice stronger than in the case of its sublimation, i.e., its destruction, because two valence electrons cannot depart from each other without breaking the weaker one. Bonds of both types are active at elastic deformation, but only weak bonds break at 5 plastic one, i.e., irreversible deformation. This explains the covalent crystals high brittleness along with their high strength. Recently, based on paired covalent bond model, it has been deter- mined[20] that the causes of both the appearance of surface cracks on a semiconductor crystal at tem- perature 푇 below its brittle-to-plastic transition point 푇��� , 푇 < 푇��� , and such transition at 푇 > 푇���. At small deformations of a covalent crystal, it is energetically preferable to create a surface by irreversible rupture, than by reversible stretching. The brittle–plastic transition in elemental covalent crystals would accompanied by the surface cova- lent bonds metallization. It is shown that the tran- sition temperature under static loads has an upper limit: 푇��� 푇�⁄ < 0.45 , where 푇� is the melting temperature. Thus, an analytical (i.e., without computer simulation) method was suggested for calculating the brittle–plastic transition tempera- ture for elemental covalent crystals. To overcome some drawbacks of previous determinations of four parameters characterizing the Mie–Lennard–Jones potential as applied to crystals, a different parameterization method was introduced by Magomedov[21]. It is based on the best agreement of the crystal thermoelastic prop- erties calculated values from experimental data such as: crystal sublimation energy at zero tem- perature, thermal expansion coefficient and iso- thermal elasticity modulus at room temperature, and pressure–volume dependence according to the equation of state’s room temperature isotherm curve. Method verification for iron Fe and gold Au showed good results, as well as its application for accurate calculation of some refractory metals (such as niobium Nb, tantalum Ta, molybdenum Mo and tungsten W) Debye temperature and sub- limation and surface energies. Disordered Au–Fe substitution alloys were studied by Magomedov[22]. Namely, based on Mie–Lennard–Jones-type inter- atomic potential parameters, Au and Fe fcc (face- centered cubic) and bcc (body-centered cubic) structures were analytically determined, and the composition dependences of the Au–Fe alloys properties at the fcc–bcc structural phase transi- tion were found. Then, the key parameters of acti- vation processes were calculated[23] in various structures of iron. These are: Gibbs energy, en- thalpy, entropy, and volume both for processes of formation of electrically neutral vacancies and self-diffusion of atoms. A beryllium–tungsten Be–W interatomic po- tential was derived[24] using a formalism originat- ed from the Pauling bond order concept. It found to be suitable for simulation of plasma–wall inter- actions (Be surface with W atom and vice versa) taking place in fusion reactor. Obtained interac- tion energies are qualitatively similar to that from ab initio, namely, DFT calculations showing that diffusion of Be into bulk W is not energetically favorable and the opposite is true for the reversed system. This Be–W potential can reasonably de- scribe BexWy molecules with 푥, 푦 = 1, 2, 3, 4 and intermetallic phases Be2W and Be12W as well. A correlation between binding energy of an individual atom in metal lattice and its macro- scopic parameters like Debye temperature, Young’s modulus and sound speed was consid- ered in the study of Erokhin and Kalashnikov[25]. Based on the Lennard–Jones potential, the anhar- monic zero-point oscillations of a crystal were analyzed[26] within the framework of the diatomic model. It is shown that their amplitude cannot exceed the limiting value which is a certain part of the equilibrium interatomic distance. The com- pression of a crystal decreases the zero oscilla- tions amplitude, while the tension increases it. It was found that crystal melting point depends on the de Boer parameter. The processes of melting and solidification of gold nanoclusters consisting of 43–1,055 atoms were studied[27] using the MC method and the Gupta multiatomic potential. It was shown that the temperature-dependences of the specific internal energy potential part and av- erage first coordination number have pronounced hysteresis. The work previously carried out in the con- text of quantitative crystal engineering involving the analysis of intermolecular interactions such as carbon (tetrel), pnicogen, chalcogen, and halogen bonding using experimental charge density meth- odology has been reviewed by Thomas et al.[28]. The focus was to extract electron density distribu- tion in the intermolecular space and to obtain 6 guidelines to evaluate the strength and directional- ity of such interactions towards the design of mo- lecular crystals with desired properties. In this formalism, the atomic electron density is divided into three terms: spherical core and spherical and asperical valence electron densities. It was demonstrated power and limits of X-ray diffrac- tion experimental analysis using the CDMM (Charge Density Multipole Modeling) approach. Using the continuum approximation for in- teracting atoms and the Mie–Grüneisen Theory, a simple equation of state for a monatomic crystal was constructed[29], which describes the phase diagram even in the vicinity of the critical point. The use of the Lennard–Jones formula for a pair potential made it possible to analytically find ex- pressions for the critical volume, pressure, and temperature. Based on investigation[30] of interactions of a rigid sphere with another rigid sphere and half- space using the Lennard–Jones potential, by their integrating over the surfaces and volumes, respec- tively, the analytical forms of surface tractions and total force between two rigid spheres were obtained. First of them can be used for the de- scription of adhesive contact between rigid and elastic bodies. The FVT (Free Volume Theory) extended to explicitly include the hard-sphere character of colloidal depletants into the free vol- ume fraction expression was used[31] as a basis for comprehensive calculations performed to predict the phase behavior of large spherical colloids mixed with small spherical ones acting as de- pletant. The parameters of the Mie–Lennard–Jones pair potential for nearest bond-forming fullerenes or interfullerene interaction in fcc-fullerites were determined[32] from the data on correlation re- vealed between fullerene mass and corresponding fullerite properties. The experimentally observed fact that under the same conditions C70 fullerite is more stable than C60 was explained[33] by calcula- tion equilibrium bond energy between C60–C60 and C70–C70 fullerene pairs using the potential energy curve for interaction between two identical hollow spherical molecules, which is given with formula including the Lennard–Jones potential parameters. It has been emphasized[34] that pair-wise in- teratomic potentials presentable in analytical form serve for powerful theoretical tools to model vari- ous nanosystems: nanoparticles, nanotubes, full- erenes, AFM (Atomic Force Microscope) probes, etc. In particular, using so-called equilibrium MD simulations the effect of electrostatic interactions influence on heat transfer mechanism and interfa- cial properties was investigated[35] for the hexago- nal boron nitride−water system Kapitza resistance in nanoscale planar (nanosheet) and cylindrical (nanotube) geometries. A water molecule was imaged by the simple point charge model due to its reliability, precision, and relatively low com- putational cost. An optimized Tersoff potential was used to model the h-BN sheet–tube interac- tions. And the pairwise interactions between at- oms or ions were described by adding Coulomb and Lennard–Jones potentials. The optimum structure of some materials at the nanoscale, in- cluding boron-based clusters, was modeled[36] by employing standard and ab initio MD simulations. The used instantaneous forces on atoms were cal- culated from Lennard–Jones, van der Waals, Cou- lomb, etc. potentials. 2.2 Ab initio potentials The ZRPM (Zero-Range Potential Model) treats[37,38] the atomic and molecular potentials as short-range potential-wells with a shallow energy level near the continuum spectrum boundary. It is a schematic description of these potentials for cases (e.g., negative atomic ion), in which the in- ternal structure details are not too significant. The basic idea of the approach is to replace the wave equation solution inside the well by a boundary condition at its center. The single potential well model useful for atoms can be directly generalized for the case of molecules, when there are several potential wells. Dolgonosov[39] has proposed an atomic elec- tron gas model together with the generalized charges theory and the subsequent development of the interatomic interactions theory for the ab initio description of covalent bonding and van der 7 Waals forces, as well as complex molecules ad- sorption on homogeneous surfaces. In the studies of Shukla and Eliasson[40–42], it was pointed out a short-range attractive force be- tween two ions screened by degenerate electron gas in an unmagnetized plasma. At quantum scale, due to that force it can arise ordered ion structures such as ion clusters or Coulomb ion lattices, as well as the phase separations in dense quantum plasmas, e.g., from solid to liquid–vapor. Corre- sponding electric potential is attributed to the quantum statistical pressure and the quantum Bohm potential, as well as the electron exchange and electron correlations due to electron-1/2 spin. Bonding in the excited alkali dimers involv- ing resonant ionic, covalent and steric interactions was studied[43] by ab initio calculation for case of second, third, fourth and fifth 1Σu +-states of lithi- um diatomic molecule Li2. In particular, the corre- sponding potential energy curves were obtained and applied for high resolution laser spectroscopy. Usually, conformational profiles are obtained with DFT methods, using of which is time- consuming when the molecules are relatively large or there are many molecules of interest. Wang et al.[44] compared several possible alterna- tives to this traditional approach, including a neu- ral network potential. It was found that a sequen- tial geometry optimization with the semiempirical method and single-point energy high-level DFT calculation can provide satisfactory conforma- tional energy profiles hundreds of times faster. The concept of electronegativity 휒 reformu- lated within DFT has acquired a central place in chemical reactivity due to its special relationship with the chemical potential 휇:휇 = −휒, and defini- tion of so-called local electronegativity viewed as the functional variation of the system energy with respect to the electronic density in a given poten- tial environment. On the one hand, the chemical hardness concept realization within the conceptual DFT is approached with perspective of electro- negativity and hardness equalization of atoms in molecules. On the other hand, the maximum hardness principle presents a relation with the chemical stability of the hardness concept. In light of these concepts and inverse relation between hardness and polarizability, the minimum polar- izability principle has been proposed by Kaya and Putz[45]. Additionally, this review includes appli- cations of the chemical hardness concept. Eluci- dating the quantum nature of the chemical bond is fundamental to establishing the directed chemical synthesis of new compounds with predefined properties and reactivity aiming at specific inter- actions. The structures, intermolecular interac- tions, and energy of some energetic materials crystal models were comparatively predicted based on MD quantum chemical simulations[46]. Detecting the intermolecular interactions would provide fundamental insights for the energetic materials crystal engineering. Electronic structure of charged defects in crystals often is calculated in the supercell models, which include the jellium counter charges to maintain system’s overall neutrality. However, the correction related to these artificially charges be- comes paramount for low-dimensional crystals, where they may induce the spurious vacuum states. A corresponding self-consistent potential correction scheme was presented in the study of Silva et al.[47]. A problem of identification of pa- rameters of the Mie–Lennard–Jones- and Morse- type pairwise interatomic potentials from the in- teraction between metal atom and graphene layer was considered by Rekhviashvili et al.[48] using the continuum approximation. The potential pa- rameters were calculated by the DFT method from the equilibrium adsorption energy and distances in the adatom–graphene system. The advantage of this method is that the empirical combining rules are not used. The Mie–Lennard–Jones potential was found to be the most suitable for describing such interaction. The problem with the Morse po- tential is that the exponential function cannot equally correctly describe the attractive and repul- sive forces. 2.3 Machine learning potentials In ML schemes, the potentials of interaction between constituent atoms 푖 and 푗 and crystal mo- lar energy can be represented in forms of 푈���푟��� = ∑ 퐴� 푟�� �⁄� and 푢 = ∑ ∑ ∑ 퐴� 푟�� �⁄���� , respectively. Here 퐴� is the constant, summation 8 is done over unit cells number l = 0, 1, 2, … and index 푖 is assumed to refer to the atom in the cell with l = 0. In these lattice sums, the terms with indices k = 0, 1, 2 describe so-called long-range interactions, because corresponding volume inte- grals over whole space diverge. One has to calcu- late them only inside a finite sphere. The well- known Ewald method developed for the Coulomb part, k = 1, can be generalized to terms with k = 0, 1, 2, 3 as well. Below, a few recent examples of ML potentials development are given. A machine learning scheme[49] for an unbi- ased and accurate representation of interatomic potentials is a combination of an artificial neural network and a simple approach for reconstruction of pair interatomic (e.g., Al–Al, He–He and Xe– Xe) potentials in elementary crystals providing accuracy comparable with ab initio ones, but at a small computational cost. This method can be ap- plied to structures of real systems of atoms by MD simulations. A highly automated approach to da- taset construction and then building a potential for elementary aluminum, called as ANI-Al, was pre- sented in the research of Smith et al.[50]. In this active scheme, the ML potential under develop- ment is used to drive non-equilibrium MD simula- tions with time-varying applied temperatures. Whenever a configuration is reached for which the ML uncertainty is large, new QC data is col- lected. The ML model is periodically retrained on all the available QC data. The final ANI-Al poten- tial makes accurate predictions of radial distribu- tion function in melt, liquid-solid coexistence curve, and crystal properties such as defect ener- gies and barriers. It was performed a 1.3 × 106 atom shock simulation and shown that corre- sponding force predictions agree well with DFT calculations. In the study of Mortazavi et al.[51], it was shown that ML interatomic potentials trained over short ab initio MD trajectories enable ab initio multiscale modeling, in which DFT is hierarchi- cally bridged to efficiently simulate macroscopic structures. It was demonstrated that such approach can efficiently predict the lattice thermal conduc- tivity of graphene and borophene pristine phases, as well as graphene–borophene interfaces. Thus, ML interatomic potentials enable ab initio mul- tiscale modeling via hierarchical employment of DFT/MD/FE simulations for computational de- sign of novel nanostructures. The structural prop- erties of amorphous boron nitride a-BN doped with varying amount of carbon C were modeled[52] by generating versatile force fields using ab initio and designing realistic disordered BN:C ML sim- ulations. The recent review[53] highlights develop- ments in the use of ML to evaluate chemical properties such as partial atomic charges, dipole moments, spin and electron densities, and chemi- cal bonding, as well as to obtain a reduced QC description. There is overviewed several neural network architectures, their predictive capabilities, generality and transferability, and illustrated their applicability to various chemical properties. It is emphasized that ML molecular representations resemble QC analogues demonstrating the ability of the models to capture the underlying physics. It is also discussed how ML models can describe non-local quantum effects. The observed trends demonstrate that this field is evolving towards physics-based models augmented by ML. 3. Charge density and potential distributions in boron atom Atom is a bounded system of electrically in- teracting electric charges—positive nucleus and negative electrons. So, to calculate potential ener- gy of interaction between two atoms in fully theo- retical manner, one needs their detailed electronic structure, including the explicit expressions for electric charge density and electric field potential distributions in interacting atoms. Recently, it has been demonstrated[54,55] that electronic structure of any bounded system of at- oms—molecules, clusters or even condensed mat- ter—with a good accuracy can be calculated with- in the semiclassical approximation expressing electric SCF (Self Consistent Field) affecting atomic electrons by Coulomb-like (pseudo) poten- tials. Based on electric charge density and electric field potential radial distributions in constituent atoms obtained in this way, one can construct semiclassical interatomic pair potentials needed 9 for calculating the important physical characteris- tics of any bounded system of atoms. In such type semiclassical approximation, for ground-state electronic configuration 1s22s22p1, the radial wave functions 푅�(푞�(푟)) of five elec- trons, k = 1, 2, 3, 4, 5 (1) of an isolated electrically neutral boron B atom with nuclei centered at the origin, 푟⃗ = 0, are 푅��푞�(푟)� = �� 푍� 푟� � � 2exp �− 푞�(푟) 2 � (2) 푅��푞�(푟)� = �� 푍� 푟� � � 2 exp �− 푞�(푟) 2 � (3) 푅��푞�(푟)� = �� 푍� 푟� � � 1 2√2 �2 − 푞�(푟)� exp �− 푞�(푟) 2 � (4) 푅��푞�(푟)� = �� 푍� 푟� � � 1 2√2 �2 − 푞�(푟)� exp �− 푞�(푟) 2 � (5) and 푅��푞�(푟)� = �� 푍� 푟� � � 1 2√6 푞�(푟) exp �− 푞�(푟) 2 � (6) Here the variables 푞�(푟) = 2푍� 푛� 푟 푟� (7) stand for radial wave functions’ arguments. The constant 푟� = ℏ� 푒�푚 ≈ 0.53 Å (8) is the Bohr radius, 푛� = 푛� = 1 (9) 푛� = 푛� = 2 (10) and 푛� = 2 (11) are the electron orbitals’ principal quantum num- bers, while the parameters 푍� equal to 푍�� ≡ 푍� = 푍� ≈ 4.69 (12) 푍�� ≡ 푍� = 푍� ≈ 2.76 (13) and 푍� ≈ 1.48 (14) respectively. These are the nucleus effective charge numbers, respectively, for 1s2, 2s2, and 2p1 electrons bounded in boron atom. They could be considered for known numerical quantities. On the one hand, in the Coulomb-like intra- atomic field the classical orbit radii 푟� of electrons or their characteristic displacements from the nu- cleus could be found from the relations: 푟� 푟� = 푛� � 푍� (15) On the other hand, the nuclear charge radii of boron stable isotopes 10B and 11B with charge number of 푍 = 5 approximately equal to[56] 푅� ≈ 2.4 × 10�� Å (16) 푅� 푟� ≈ 2.2 × 10�� (17) and, consequently, in the boron atom the nucleus electric charge radius is negligible in comparison with electrons’ characteristic displacements from it: 푅� 푟� ≪ 1 (18) These relations meet a key assumption of the semiclassical method used that atomic nucleus could be considered as point electric charge. The squared semiclassical radial wave func- tions, 푅�� � (푟) ≡ 푅� �(푟) = 푅� �(푟) = 4푍�� � 푟� � exp �− 2푍��푟 푟� � 10 (19) 푅�� � (푟) ≡ 푅� �(푟) = 푅� �(푟) = 푍�� � 8푟� � �2 − 푍��푟 푟� � � exp �− 푍��푟 푟� � = � 푍�� � 2푟� � − 푍�� � 푟 2푟� � + 푍�� � 푟� 8푟� � � exp �− 푍��푟 푟� � (20) and 푅� �(푟) = 푍� � 24푟� � � 푍�푟 푟� � � exp �− 푍�푟 푟� � = 푍� �푟� 24푟� � exp �− 푍�푟 푟� � (21) determine the semiclassical electrical charge den- sity distribution in the isolated boron atom: 휌�(푟⃗) = 5푒훿(푟⃗) − 푒 4휋 � 푅� �(|푟⃗|) ��� ��� = 푒 �5훿(푟⃗) − 2푅�� � (푟) + 2푅�� � (푟) + 푅� �(푟) 4휋 � (22) Here, the first term containing Dirac delta- function 훿(푟⃗) stands for positive charge density of point-like nucleus with charge number of 푍 = 5 and the second term is negative charge density of electron cloud. From the corresponding semiclassical elec- trical charge density radial distribution. 휌�(푟⃗) = 푒 �5훿(푟⃗) − 2푍�� � 휋푟� � exp �− 2푍�푟 푟� � − � 푍�� � 4휋푟� � − 푍�� � 푟 4휋푟� � + 푍�� � 푟� 16휋푟� �� exp �− 푍��푟 푟� � − 푍� �푟� 96휋푟� � exp �− 푍�푟 푟� �� (23) the Poisson’s equation determines that of the elec- trical field potential, 휑�(푟) = 푒 �� 2 푟 + 2푍�� 푟� � exp �− 2푍��푟 푟� � + � 2 푟 + 3푍�� 2푟� + 푍�� � 푟 2푟� � + 푍�� � 푟� 4푟� � � exp �− 푍��푟 푟� � + � 1 푟 + 3푍� 4푟� + 푍� �푟 4푟� � + 푍� �푟� 24푟� �� exp �− 푍�푟 푟� �� (24) in the boron atom centered at the origin: 푟⃗ = 0. 4. Potential energy of B–B interac- tion In general, an atom electrically interacting with another one affects its electric charge density and, consequently, electric field potential distribu- tions. However, in the vicinity of interatomic chemical bond equilibrium length such redistribu- tions are relatively small: both in finite and infi- nite bounded systems of atoms electric charge density and electric field potential distributions can be approximated by the simple superposition of corresponding distributions in constituent at- oms in their isolated states, when they are local- ized in structure sites. Assuming electric charge density and related electric field potential redistributions in electrical- ly interacting pair of boron atoms displaced at vector 푎⃗ (Figure 1) to be negligible, the semiclas- sical potential energy of such interaction can be found as volume integral: 푈���(푎) = � 푑�푟⃗ 휑�(푟)휌�(푟⃗ − 푎⃗) (25) Figure 1. To calculation of B–B pair potential. 11 The trivial integration over the polar angle 0 ≤ 휙 ≤ 2휋 yields the following result: 푈���(푎) = 5푒� �� 2 푎 + 2푍�� 푟� � exp �− 2푍��푎 푟� � + � 2 푎 + 3푍�� 2푟� + 푍�� � 푎 2푟� � + 푍�� � 푎� 4푟� � � exp �− 푍��푎 푟� � + � 1 푎 + 3푍� 4푟� + 푍� �푎 4푟� � + 푍� �푎� 24푟� �� exp �− 푍�푎 푟� �� − 푒� � 푑푟 푟� ∞ � �� 2 푟 + 2푍�� 푟� � exp �− 2푍��푟 푟� � + � 2 푟 + 3푍�� 2푟� + 푍�� � 푟 2푟� � + 푍�� � 푟� 4푟� � � exp �− 푍��푟 푟� � + � 1 푟 + 3푍� 4푟� + 푍� �푟 4푟� � + 푍� �푟� 24푟� �� exp �− 푍�푟 푟� �� × � 푑휃 sin 휃 � 4푍�� � 푟� � exp �− 2푍�√푟� − 2푎푟 cos 휃 + 푎� 푟� � � � + � 푍�� � 2푟� � − 푍�� � √푟� − 2푎푟 cos 휃 + 푎� 2푟� � + 푍�� � (푟� − 2푎푟 cos 휃 + 푎�) 8푟� � � exp �− 푍��√푟� − 2푎푟 cos 휃 + 푎� 푟� � + 푍� �(푟� − 2푎푟 cos 휃 + 푎�) 48푟� � exp �− 푍�√푟� − 2푎푟 cos 휃 + 푎� 푟� �� (26) As for integration over the azimuthal angle 0 ≤ 휃 ≤ 휋 , it can be conducted via two successive trans- formations into new integration variables 푥 and 푡: cos 휃 = 푥 (27) and 푟� − 2푎푟푥 + 푎� = 푡� (28) finally leading to the form 푈���(푎) = 5푒� �� 2 푎 + 2푍�� 푟� � exp �− 2푍��푎 푟� � + � 2 푎 + 3푍�� 2푟� + 푍�� � 푎 2푟� � + 푍�� � 푎� 4푟� � � exp �− 푍��푎 푟� � + � 1 푎 + 3푍� 4푟� + 푍� �푎 4푟� � + 푍� �푎� 24푟� �� exp �− 푍�푎 푟� �� − 푒� 푟�푎 � 푑푟 푟 ∞ � �� 2 푟 + 2푍�� 푟� � exp �− 2푍��푟 푟� � + � 2 푟 + 3푍�� 2푟� + 푍�� � 푟 2푟� � + 푍�� � 푟� 4푟� � � exp �− 푍��푟 푟� � + � 1 푟 + 3푍� 4푟� + 푍� �푟 4푟� � + 푍� �푟� 24푟� �� exp �− 푍�푟 푟� �� × �퐽��(푟) + 퐽��(푟) + 퐽�(푟)� (29) where 퐽��(푟) = 4푍�� � 푟� � � 푑푡 푡 exp �− 2푍��푡 푟� � ��� |���| = 푍�� ��1 + 2푍��|푟 − 푎| 푟� � exp �− 2푍��|푟 − 푎| 푟� � − �1 + 2푍��(푟 + 푎) 푟� � exp �− 2푍��(푟 + 푎) 푟� �� (30) 12 퐽��(푟) = � 푑푡 푡 � 푍�� � 2푟� � − 푍�� � 푡 2푟� � + 푍�� � 푡� 8푟� � � exp �− 푍��푡 푟� � = ��� |���| = 푍�� 4 ��1 + 푍��|푟 − 푎| 푟� − 푍�� � |푟 − 푎|� 2푟� � + 푍�� � |푟 − 푎|� 2푟� � � exp �− 푍��|푟 − 푎| 푟� � − �1 + 푍��(푟 + 푎) 푟� − 푍�� � (푟 + 푎)� 2푟� � + 푍�� � (푟 + 푎)� 2푟� � � exp �− 푍��(푟 + 푎) 푟� �� (31) and 퐽�(푟) = 푍� �푡� 48푟� � � 푑푡 푡 exp �− 푍�푡 푟� � ��� |���| = 푍� 8 ��1 + 푍�|푟 − 푎| 푟� + 푍� �|푟 − 푎|� 2푟� � + 푍� �|푟 − 푎|� 6푟� � � exp �− 푍�|푟 − 푎| 푟� � − �1 + 푍�(푟 + 푎) 푟� + 푍� �(푟 + 푎)� 2푟� � + 푍� �(푟 + 푎)� 6푟� � � exp �− 푍�(푟 + 푎) 푟� �� (32) are the definite radial, 0 ≤ 푟 ≤ ∞, integrals. Thus, semiclassical approximation has al- lowed us to express the boron–boron pair poten- tial 푈��� = 푈���(푎) (33) in dependence on interatomic distance 푎 by a ra- dial integral. Schematic view of an interatomic pair poten- tial 푈 as a function of the interatomic distance 푟 is shown in Figure 2. Here 푟� is the equilibrium in- teratomic distance, i.e., bond length in corre- sponding diatomic molecule, and 푈� is its binding energy (not corrected for relative atomic vibra- tions zero-point energy). Figure 2. Schematic of interatomic pair potential. The integration over the radius is possible to conduct in elementary functions, although it will give the result in a cumbersome analytical form. Therefore, it seems that for the practical applica- tions of the constructed B–B potential often it will be necessary to convert it in a numerical form. It is clear that the same goal can be achieved by the direct numerical integration. 5. Discussion and conclusion Further work aims to convert the obtained in integral form semiclassical B–B potential function into analytical and/or numerical forms, which will allow to determine the B–B bond’s parameters such as bond length, dissociation energy, frequen- cy of relative atomic vibrations, etc. They should be compared with currently available data given below. To the best our knowledge for the first time the fully theoretical pair potential energy curves for low-lying energy states of diboron molecule B2 were constructed[57] by so-called complete- active-space SCF method at the multi-reference CI (Configuration Interaction) level of theory. The potential curves of ground and some of low-lying excited electronic states of B2 neutral molecule and B2 + positive ion were also obtained[58] by us- ing another CI approach. The B2 + cation ground state showed a rather shallow potential curve with a bond length of 2.13 Å and vibration quantum of 0.052 eV, when compared with B2 neutral ground state with that of 1.59 Å and 0.131 eV, respective- ly. In result of bonding electron loss, the B2 + mo- lecular ion ground-state dissociation energy of 1.94 eV was found to be significantly smaller than that of B2: 3.06 eV. Later, same authors by exten- sive multi-reference CI calculations were con- structed[59] B2 and B2 + potential curves yielding 13 the following sets of parameters 1.59 Å, 2.75 eV, 0.131 eV and 2.12 Å, 1.90 eV, 0.052 eV, respec- tively. And according to the one more multi- reference CI study[60], the ground-state curve pa- rameters of B2 such as bond length, dissociation energy and two atoms relative vibration quantum are of 1.60–1.61 Å, 2.70–2.78 eV and 0.128– 0.129 eV, respectively. First principles quadratic CI method was used[61] to calculate the equilibri- um potential energy curves of ground and low- lying excited electronic states of B2 and B2 +. The corresponding analytical potentials were con- structed by the fitting calculation results to the Murrell–Sorbie potential energy function. Curve parameters obtained for B2 and B2 + were, respec- tively, 1.62 Å, 3.14 eV, 0.125 eV and 2.18 Å, 1.69 eV, 0.052 eV. A non-SCF DFT based construction of non- orthogonal TB (Tight-Binding) matrix elements for B–B, N–N, B–N, B–H and N–H systems with- in the framework of the LCAO (Linear Combina- tion of Atomic Orbitals) formalism was present- ed[62] using the LDA (Local Density Approxima- tion). Despite the simplicity of the scheme con- sidering only two-center Hamiltonian integrals and overlap matrix elements, the method has been proven to be sufficiently accurate and transferable to all scale B–B(N,H) structures from small clus- ters and molecules to crystalline solids and solid surfaces. The calculation of forces from these TB potentials is straightforward and allows an appli- cation of the method to MD simulations of struc- ture formation in complex BNH systems. For the B2 molecule ground-state interatomic potential, the energy curve was also construct- ed[63,64] within a quasi-classical approach. The ob- tained in such way curve’s parameters are as fol- lows: equilibrium bond length of 1.78 Å, dissocia- tion energy of 2.80 eV, and vibration quantum of 0.130 eV. Most of NIST’s IPR potentials are presented in efficient “universal” shifted Lennard–Jones model for all KIM API supported species devel- oped by Elliott and Akerson in 2015[65]. Cohesive energy graphs generated for each elemental crys- tals supported by the model show the cohesive energy versus volume-per-atom for four mono- atomic cubic phases: sc (simple cubic), bcc, fcc, and diamond-like. The curve with the lowest min- imum is the ground state of the crystal, if stable. Point is that the crystal structure is enforced in these calculations, so the phase may not be stable. In particular, cohesive energy graphs are available for elemental boron, i.e., B–B, and some boron- containing systems: B–N, B–Hf, B–Zr and B–C– N. The local nature of different types of boron– boron bonds—B∙B, B–B, B=B and B≡B—from the topological analysis of ELF (Electron Locali- zation Function) perspective was investigated[66] in number of boron-containing molecules. As for the experimental parameters of B–B pair interatomic potential curve, they are available in the reference book[67]: 1.59 Å, 3.09 eV, 0.130 eV. Currently, for neutral diboron B2 molecule in ground state configuration (σ1s)2(σ*1s)2(σ2s)2 (σ*2s)2(π2p)2 bond length and bond energy are estimated[68] as 0.159 nm and 3.00 eV, respective- ly. In similar way, one can construct different semiclassical interatomic pair potentials to charac- terize the nanomaterials containing not only boron, but some other chemical elements as well. For example, results obtained on the basis of semi- classical boron–nitrogen, i.e., B–N, potential for pristine and doped hexagonal boron nitride h-BN nanotubes prospective for toxic gas sensors can be compared with DFT ones on their electrical sensi- tivity toward ethyl benzene C8H10 and phosphine PH3 molecules[69,70]. 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