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Orbital HP-clouds for solving Schrödinger equation in quantum mechanics. (English) Zbl 1173.81301
Summary: Solving Schrödinger equation in quantum mechanics presents a challenging task in numerical methods due to the high order behavior and high dimension characteristics in the wave functions, in addition to the highly coupled nature between wave functions. This work introduces orbital and polynomial enrichment functions to the partition of unity for solution of Schrödinger equation under the framework of HP-Clouds. An intrinsic enrichment of orbital function and extrinsic enrichment of monomial functions are proposed. Due to the employment of higher order basis functions, a higher order stabilized conforming nodal integration is developed. The proposed methods are implemented using the density functional theory for solution of Schrödinger equation. Analysis of several single and multi-electron/nucleus structures demonstrates the effectiveness of the proposed method.

MSC:
81-08 Computational methods for problems pertaining to quantum theory
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[1] Alemany, M.M.G.; Jain, M.; Kronik, L.; Chelikowsky, J.R., Real-space pseudopotential method for computing the electronic properties of periodic systems, Phys. rev. B, 69, 075101-075106, (2004)
[2] Atluri, S.N.; Zhu, T., The meshless local petrov – galerkin (MLPG) approach for solving problems in elasto-statics, Comput. mech., 25, 169-179, (2000) · Zbl 0976.74078
[3] Babuška, I.; Ihlenburg, F.; Paik, E.; Sauter, S., A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution, Comput. methods appl. mech. engrg., 128, 325-359, (1995) · Zbl 0863.73055
[4] Babuška, I.; Melenk, J.M., The partition of unity finite element method, Int. J. numer. methods engrg., 40, 727-758, (1997) · Zbl 0949.65117
[5] Babuška, I.; Banerjee, U.; Osborn, J.E., Survey of meshless and generalized finite element methods: a unified approach, Acta numer., 12, 1-125, (2003) · Zbl 1048.65105
[6] Babuška, I.; Zhang, Z., The partition of unity method for the elastically supported beam, Comput. methods appl. mech. engrg., 152, 1-18, (1998) · Zbl 0959.74078
[7] Beissel, S.; Belytschko, T., Nodal integration of the element-free Galerkin method, Comput. methods appl. mech. engrg., 139, 49-74, (1996) · Zbl 0918.73329
[8] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Comput. methods appl. mech. engrg., special issue on meshless methods, 139, 3-47, (1996) · Zbl 0891.73075
[9] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Int. J. numer. methods engrg., 37, 229-256, (1994) · Zbl 0796.73077
[10] Belytschko, T.; Lu, Y.Y.; Gu, L., Crack propagation by element-free Galerkin methods, Engrg. fract. mech., 51, 295-315, (1995)
[11] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods for static and dynamic fracture, Int. J. solids struct., 32, 2547-2570, (1995) · Zbl 0918.73268
[12] Bonet, J.; Kulasegaram, S., Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulation, Int. J. numer. methods engrg., 47, 1189-1214, (1999) · Zbl 0964.76071
[13] Bransden, B.H.; Joachain, C.J., Quantum mechanics, (2000), Prentice-Hall Englewood Cliffs, NJ
[14] Chelikowsky, J.R.; Troullier, N.; Saad, Y., Finite-difference-pseudopotential method: electronic structure calculations without a basis, Phys. rev. lett., 72, 1240-1243, (1994)
[15] Chelikowsky, J.R.; Troullier, N.; Wu, K.; Saad, Y., Higher-order finite-difference pseudopotential method: an application to diatomic molecules, Phys. rev. B, 50, 11355-11364, (1994)
[16] Chen, J.S.; Pan, C.; Wu, C.T.; Liu, W.K., Reproducing kernel particle methods for large deformation analysis of nonlinear structures, Comput. methods appl. mech. engrg., 139, 195-227, (1996) · Zbl 0918.73330
[17] Chen, J.S.; Wu, C.T.; Yoon, S.; You, Y., A stabilized conforming nodal integration for Galerkin meshfree methods, Int. J. numer. methods engrg., 50, 435-466, (2001) · Zbl 1011.74081
[18] Chen, J.S.; Wu, C.T.; Yoon, S.; You, Y., Nonlinear version of stabilized conforming nodal integration for Galerkin meshfree methods, Int. J. numer. methods engrg., 53, 2587-2615, (2002) · Zbl 1098.74732
[19] De, S.; Bathe, K.J., The method of finite spheres, Comput. mech., 25, 329-345, (2000) · Zbl 0952.65091
[20] Dolbow, J.; Belytschko, T., Numerical integration of Galerkin weak form in meshfree methods, Comput. mech., 23, 219-230, (1999) · Zbl 0963.74076
[21] Duarte, C.A.M.; Babuška, I.; Oden, J.T., Generalized finite element methods for three dimensional structural mechanics problems, Comput. struct., 77, 215-232, (2000)
[22] Duarte, C.A.M.; Oden, J.T., An h-p adaptive method using clouds, Comput. methods appl. mech. engrg., 139, 237-262, (1996) · Zbl 0918.73328
[23] Duarte, C.A.M.; Oden, J.T., Hp clouds – an hp meshless method, Numer. methods partial diff. eq., 12, 673-705, (1996) · Zbl 0869.65069
[24] Garcia, O.; Fancello, E.A.; Barcellos, C.S.; Duarte, C.A.M., Hp-clouds in mindlin’s thick plate model, Int. J. numer. methods engrg., 47, 1381-1400, (2000) · Zbl 0987.74067
[25] Gingold, R.A.; Monaghan, J.J., Smoothed particle hydrodynamics: theory and application to non-spherical stars, Monthly notices R. astron. soc., 181, 375-389, (1977) · Zbl 0421.76032
[26] Hehre, W.J.; Radom, L.; Schleyer, P.; Pople, J.A., Ab initio molecular orbital theory, (1986), Wiley New York
[27] Heinemann, D.; Fricke, B.; Kolb, D., Solution of the hartree – fock – slater equations for diatomic molecules by the finite-element method, Phys. rev. A, 38, 4994-5001, (1988)
[28] Hermansson, B.; Yevick, D., Finite-element approach to band-structure analysis, Phys. rev. B, 33, 7241-7242, (1986)
[29] Ihm, J.; Zunger, A.; Cohen, M.L., Momentum-space formalism for the total energy of solids, J. phys. C, 12, 4409-4422, (1979)
[30] Kohn, W.; Sham, L.J., Self-consistent equations including exchange and correlation, Phys. rev., 140, A1133-A1138, (1965)
[31] Lancaster, P.; Salkauskas, K., Surfaces generated by moving least square method, Math. comput., 37, 141-158, (1981) · Zbl 0469.41005
[32] Liszka, T.J.; Duarte, C.A.M.; Tworzydlo, W.W., Hp-meshless cloud method, Comput. methods appl. mech. engrg., 139, 263-288, (1996) · Zbl 0893.73077
[33] Liu, W.K.; Jun, S.; Zhang, Y.F., Reproducing kernel particle method, Int. J. numer. methods fluids, 20, 1081-1106, (1995) · Zbl 0881.76072
[34] J.M. Melenk, On generalized finite element methods. Ph.D. dissertation, University of Maryland, College Park, MD 1995.
[35] Melenk, J.M.; Babuška, I., The partition of unity finite element method: basic theory and applications, Comput. methods appl. mech. engrg., 139, 289-314, (1996) · Zbl 0881.65099
[36] Melenk, J.M.; Babuška, I., Approximation with harmonic and generalized harmonic polynomial in the partition of unity method, Comput. assisted mech. engrg. sci., 4, 607-632, (1997) · Zbl 0951.65128
[37] Mendonca, P.T.; Barcellos, C.S.; Duarte, C.A.M., Investigations on the hp-cloud method by solving Timoshenko beam problems, Comput. mech., 25, 286-295, (2000) · Zbl 0987.74070
[38] Möes, N.; Dolbow, J.; Belytschko, T., A finite element method for crack growth without remeshing, Int. J. numer. methods engrg., 46, 131-150, (1999) · Zbl 0955.74066
[39] Nayroles, B.; Touzot, G.; Villon, P., Generalizing the finite element method: diffuse approximation and diffuse elements, Comput. mech., 10, 307-318, (1992) · Zbl 0764.65068
[40] Oden, J.T.; Duarte, C.A.M.; Zienkiewicz, O.C., A new cloud-based hp finite element method, Comput. methods appl. mech. engrg., 153, 117-126, (1998) · Zbl 0956.74062
[41] Oñate, E.; Idelsohn, S.; Zienkiewicz, O.C.; Taylor, R.L., A finite point method in computational mechanics: applications to convective transport and fluid, Int. J. numer. methods engrg., 39, 3839-3866, (1996) · Zbl 0884.76068
[42] Parr, R.G.; Yang, W., Density functional theory of atoms and molecules, (1989), Oxford University Press New York
[43] Pask, J.E.; Sterne, P.A., Real-space formulation of the electrostatic potential and total energy of solids, Phys. rev. B, 71, 113101-113104, (2005)
[44] Pickett, W.E., Pseudopotential methods in condensed matter applications, Comput. phys. rep., 9, 115-197, (1989)
[45] P.W. Randles, L.D. Libersky, A.G. Petschek, On neighbors, derivatives, and viscosity in particle codes, in: Proceeding of ECCM Conference, Munich, Germany, 31 August-3 September, 1999.
[46] Sukumar, N.; Moran, B.; Belytschko, T., The natural element method in solid mechanics, Int. J. numer. methods engrg., 43, 839-887, (1998) · Zbl 0940.74078
[47] Sulsky, D.; Chen, Z.; Schreyer, H.L., A particle method for history-dependent materials, Comput. methods appl. mech. engrg., 118, 179-196, (1994) · Zbl 0851.73078
[48] Szabo, A.; Ostlund, N.S., Modern quantum chemistry: introduction to advanced electronic structure theory, (1989), McGraw-Hill New York
[49] Tsuchida, E.; Tsukada, M., Electronic-structure calculations based on the finite-element method, Phys. rev. B, 52, 5573-5578, (1995)
[50] Tsuchida, E.; Tsukada, M., Adaptive finite-element method for electronic-structure calculations, Phys. rev. B, 54, 7602-7605, (1996)
[51] Tsuchida, E.; Tsukada, M., Large-scale electronic-structure calculations based on the adaptive finite-element method, J. phys. soc. Japan, 67, 3844-3858, (1998)
[52] Wang, D.; Chen, J.S.; Sun, L., Homogenization of magnetostrictive particle-filled elastomers using an interface-enriched reproducing kernel particle method, J. finite element anal. des., 39/8, 765-782, (2003)
[53] White, S.R.; Wilkins, J.W.; Teter, M.P., Finite-element method for electronic structure, Phys. rev. B, 39, 5819-5833, (1989)
[54] Yamakawa, S.; Hyodo, S., Electronic state calculation of hydrogen in metal clusters based on Gaussian-FEM mixed basis function, J. alloys compounds, 231, 356-357, (2003)
[55] Yamakawa, S.; Hyodo, S., Gaussian finite-element mixed-basis method for electronic structure calculations, Phys. rev. B, 71, 035113-035121, (2005)
[56] You, Y.; Chen, J.S.; Lu, H., Filter, reproducing kernel, and adaptive meshfree methods, Comput. mech., 31, 316-326, (2003) · Zbl 1038.74681
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