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Model of diatomic homonuclear molecule scattering by atom or barriers. (English) Zbl 1392.81214
Vishnevskiy, Vladimir M. (ed.) et al., Distributed computer and communication networks. 19th international conference, DCCN 2016, Moscow, Russia, November 21–25, 2016. Revised selected papers. Cham: Springer (ISBN 978-3-319-51916-6/pbk; 978-3-319-51917-3/ebook). Communications in Computer and Information Science 678, 511-524 (2016).
Summary: The mathematical model of quantum tunnelling of diatomic homonuclear molecules through repulsive barriers or scattering by an atom is formulated in the s-wave approximation. The 2D boundary-value problem (BVP) in polar coordinates is reduced to a 1D BVP for a set of second-order ODEs by means of Kantorovich expansion over the set of parametric basis functions. The algorithm for calculating the asymptotic form of the parametric basis functions and effective potentials of the ODEs at large values of the parameter (hyperradial variable) is presented. The solution is sought by matching the numerical solution in one of the subintervals with the analytical solution in the adjacent one. The efficiency of the algorithm is confirmed by comparing the calculated solutions with those of the parametric eigenvalue problem obtained by applying the finite element method in the entire domain of definition at large values of the parameter. The applicability of algorithms and software are demonstrated by the example of benchmark calculations of discrete energy spectrum of the trimer $$\mathrm{Be}_3$$ in collinear configuration.
For the entire collection see [Zbl 1386.68012].
##### MSC:
 81U05 $$2$$-body potential quantum scattering theory 81V55 Molecular physics 81V45 Atomic physics 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
##### Software:
KANTBP; KANTBP 2.0
Full Text:
##### References:
 [1] 1.Chuluunbaatar, O., Gusev, A.A., Vinitsky, S.I., Abrashkevich, A.G.: ODPEVP: a program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem. Comput. Phys. Commun. 181, 1358-1375 (2009) · Zbl 1198.15002 [2] 2.Chuluunbaatar, O., Gusev, A.A., Abrashkevich, A.G., et al.: KANTBP: a program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach. Comput. Phys. Commun. 177, 649-675 (2007) · Zbl 1196.81283 [3] 3.Chuluunbaatar, O., Gusev, A.A., Vinitsky, S.I., Abrashkevich, A.G.: KANTBP 2.0: new version of a program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach. Comput. Phys. Commun. 179, 685-693 (2008) · Zbl 1197.81008 [4] 4.Chuluunbaatar, O., Gusev, A.A., Vinitsky, S.I., Abrashkevich, A.G.: KANTBP 3.0: new version of a program for computing energy levels, reflection and transmission matrices, and corresponding wave functions in the coupled-channel adiabatic approach. Comput. Phys. Commun. 185, 3341-3343 (2014) · Zbl 1360.81333 [5] 5.Efimov, V.N.: Weakly-bound states of three resonantly-interacting particles. Soviet J. Nucl. Phys. 12, 589-595 (1971) [6] 6.Efimov, V.: Energy levels of three resonantly interacting particles. Nucl. Phys. A 210, 157-188 (1973) [7] 7.Efimov, V.: Few-body physics: Giant trimers true to scale. Nat. Phys. 5, 533-534 (2009) [8] 8.Fonseca, A.C., Redish, E.F., Shanley, P.E.: Efimov effect in an analytically solvable model. Nucl. Phys. A 320, 273-288 (1979) [9] 9.Gusev, A.A., Chuluunbaatar, O., Vinitsky, S.I., Abrashkevich, A.G.: POTHEA: a program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined 2D elliptic partial differential equation. Comput. Phys. Commun. 185, 2636-2654 (2014) · Zbl 1360.35053 [10] 10.Gusev, A.A., Chuluunbaatar, O., Vinitsky, S.I., Abrashkevich, A.G.: Description of a program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the coupled parametric self-adjoined elliptic differential equations. Bull. Peoples’ Friendsh. Univ. Russ. Ser. Math. Inf. Sci. Phys. 2, 336-341 (2014) · Zbl 1360.35053 [11] 11.Gusev, A.A., Hai, L.L.: Algorithm for solving the two-dimensional boundary value problem for model of quantum tunneling of a diatomic molecule through repulsive barriers. Bull. Peoples’ Friendship Univ. Russia. Ser. Math. Inf. Sci. Phys. 1, 15-36 (2015) [12] 12.Gusev, A.A., Hai, L.L., Chuluunbaatar, O., Vinitsky, S.I.: Programm KANTBP 4M for solving boundary problems for a system of ordinary differential equations of the second order (2015). [13] 13.Koput, J.: The ground-state potential energy function of a beryllium dimer determined using the single-reference coupled-cluster approach. Chem. Phys. 13, 20311-20317 (2011) [14] 14.Krassovitskiy, P.M., Pen’kov, F.M.: Contribution of resonance tunneling of molecule to physical observables. J. Phys. B 47, 225210 (2014) [15] 15.Merritt, J.M., Bondybey, V.E., Heaven, M.C.: Beryllium dimercaught in the act of bonding. Science 324, 1548-1551 (2009) [16] 16.Mitin, A.V.: Ab initio calculations of weakly bonded He [17] 17.Patkowski, K., Špirko, V., Szalewicz, K.: On the elusive twelfth vibrational state of beryllium dimer. Science 326, 1382-1384 (2009) [18] 18.Pijper, E., Fasolino, A.: Quantum surface diffusion of vibrationally excited molecular dimers. J. Chem. Phys. 126, 014708 (2007) [19] 19.Vinitsky, S., Gusev, A., Chuluunbaatar, O., Hai, L., Góźdź, A., Derbov, V., Krassovitskiy, P.: Symbolic-numeric algorithm for solving the problem of quantum tunneling of a diatomic molecule through repulsive barriers. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2014. LNCS, vol. 8660, pp. 472-490. Springer, Heidelberg (2014). doi: · Zbl 1353.81007 [20] 20.Wang, J., Wang, G., Zhao, J.: Density functional study of beryllium clusters, with gradient correction. J. Phys.: Condens. Matter 13, L753-L758 (2001) [21] 21.Zaccanti, M., Deissler, B., D’Errico, C., et al.: Observation of an Efimov spectrum in an atomic system. Nat. Phys. 5, 586-591 (2009)
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