Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Parameter Estimation of Potentials which are Solutions of some Second-Order Ordinary Differential Equation

Discontinuity, Nonlinearity, and Complexity 12(1) (2023) 207--229 | DOI:10.5890/DNC.2023.03.015

S. A. Surulere, T. Malange, M. Y. Shatalov, A. C. Mkolesia

Download Full Text PDF



The parameter estimation of interatomic potentials are considered as a solution of some second order ordinary differential equation. The developed method for the multiple goal function approach can be applied to a wide range of transcendental nonlinear problems. The parameter estimation of several interatomic potentials in classical functional forms are estimated using the objective least squares function method. Potentials such as Lennard-Jones, Classical Rydberg, Classical, Generalized Morse and Biswas-Hamann potential were each considered. Two new interatomic potentials, Modified Generalized Morse and Modified Lennard-Jones potential are proposed in this text. Numerical estimates were obtained using gold atom for numerical simulation in MathCad\textsuperscript{\textregistered} software while the potential energy curves are constructed and reconstructed in Mathematica\textsuperscript{\textregistered}. The estimated parameters gave good fit to experimental data sets of gold atom as the error plots of the potential energy curves are small. The values obtained for the goal function also shows that the approximated parameters values are good approximations.


  1. [1]  Lim, T.C. (2004), Relationship and discrepancies among typical interatomic potential functions, Chinese Physics Letters, 21(11), 2167-2170.
  2. [2]  Jones, J.E. (1924), On the determination of molecular fields. II. from the equation of state of a gas, in Proc of Royal Society of London A: Mathematical, Physical and Engineering Sciences, 106, 463-477.
  3. [3]  Kaplan, I.G. (2006), Intermolecular Interactions: Physical Picture, Computational Methods and Model Potentials, John Wiley \& Sons.
  4. [4]  Lim, T.C. (2005), A functionally flexible interatomic energy function based on classical potentials, Chemical Physics, 320, 54-58.
  5. [5]  Bart{o}k-P{a}rtay, A. (2010), The Gaussian Approximation Potential: an Interatomic Potential Derived from first Principles Quantum Mechanics, Springer Science \& Business Media.
  6. [6]  Rydberg, R. (1932), Graphische darstellung einiger bandenspektroskopischer ergebnisse, Zeitschrift f{\"u}r, Physik A Hadrons and Nuclei, 73(5), 376-385.
  7. [7]  Lim, T.C.(2004), Connection among classical interatomic potential functions, Journal of Mathematical Chemistry, 36(3), 261-269.
  8. [8]  Lim, T.C. (2011), Application of Extended-Rydberg parameters in general Morse potential functions, Journal of Mathematical Chemistry, 49, 1086-1091.
  9. [9]  Morse, P.M. (1929), Diatomic molecules according to the wave mechanics. II. Vibrational levels, Physical Review, 34(1), 57-64.
  10. [10]  Lim, T.C. (2004), Relationship between Morse and Murrell-Mottram potentials at long range, Journal of Mathematical Chemistry, 30(2), 139-145.
  11. [11]  Girifalco, L.A. and Weizer, V.G. (1959), Application of the Morse potential function to cubic metals, Physical Review, 114(3), 687.
  12. [12]  Wilhelm, E. and Battino, R. (1971), Estimation of Lennard-Jones (6-12) pair potential parameters from gas solubility data, The Journal of Chemical Physics, 55(8), 4012-4017.
  13. [13]  Nguyen, V.H., Trinh, T.H., and Nguyen, B.D. (2015), Calculation of Morse potential parameters of bcc crystals and application to anharmonic interatomic effective potential, local force constant, VNU Journal of Science: Mathematics-Physics, 31(3), 25-30.
  14. [14]  Taseli, H. (1998), Exact solutions for vibrational levels of the Morse potential, Journal of Physics A: Mathematical and General, 31(2), 779-788.
  15. [15]  Lincoln, R.C., Koliwad, K.M., and Ghate, P.B. (1967), Morse-potential evaluation of second-and third-order elastic constants of some cubic metals, Physical Review, 157(3), 463-466.
  16. [16]  Konowalow, D. and Guberman, S. (1968), Estimation of Morse potential parameters from critical constants and acentric factor, Industrial $\&$ Engineering Chemistry Fundamentals, 7(4), 622-625.
  17. [17]  Pamuk, H. and Halicio\v{g}lu, T. (1976), Evaluation of Morse parameters for metals, Physica Status Solidi (a), 37(2), 695-699.
  18. [18]  Kikawa, C. (2013), Methods to Solve Transcendental Least-Squares Problems and Their Statistical Inferences, Ph. D. thesis, Tshwane University of Technology, South Africa.
  19. [19]  Kikawa, C.R., Shatalov, M.Y., and Kloppers, P.H. (2015), A method for computing initial approximations for a 3-parameter exponential function, Physical Science International Journal, 6, 203-208.
  20. [20]  Malange, T.N. (2016), Alternative Parameter Estimation Methods of the Classical Rydberg Interatomic Potential, Master's thesis, Tshwane University of Technology, South Africa.
  21. [21]  Surulere, S.A., Shatalov, M.Y., Mkolesia, A.C., and Fedotov, I. (2020), A modern approach for the identification of the classical and modified generalized Morse potential, Nanoscience $\&$ Nanotechnology - Asia, 10(2), 142-151.
  22. [22]  Mkolesia, A.C., Kikawa, C.R., and Shatalov, M.Y. (2016), Estimation of the Rayleigh distribution parameter, Transylvanian Review, 24(8), 1158-1163.
  23. [23]  Olsson, P.A. (2010), Transverse resonant properties of strained gold nanowires, Journal of Applied Physics, 108(3), 1-10, 034318.
  24. [24]  Surulere, S.A., Shatalov, M.Y., Mkolesia, A.C., and Adeniji, A.A. (2020), A comparative investigation of complex conjugate eigenvalues of generalized Morse and classical Lennard-Jones potential for metal atoms, Nanoscience $\&$ Nanotechnology - Asia, 10(3), 356-363.
  25. [25]  Erko{\c{c}}, {\c{S}}. (1997), Empirical many-body potential energy functions used in computer simulations of condensed matter properties, Physics Reports, 278(2), 79-105.
  26. [26]  Kozlov, E.V., Popov, L.E., and Starostenkov, M.D. (1972), Calculation of the Morse potential for solid gold, Russian Physics Journal, 15(3), 395-396.
  27. [27]  Surulere, S.A., Shatalov, M.Y., Mkolesia, A.C., and Ehigie, J.O. (2020), The integral-differential and integral approach for the estimation of the classical Lennard-Jones and Biswas-Hamann potentials, International Journal of Mathematical Modelling and Numerical Optimisation, 10(3), 239-254.
  28. [28]  Surulere, S.A., Shatalov, M.Y., Mkolesia, A.C., Malange, T.N., and Adeniji, A.A. (2020), The integral-differential and integral approach for the exact solution of the hybrid functional forms for Morse potential, International Journal of Applied Mathematics, 50(2), 242-250.