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Highly accurate numerical solution of Hartree-Fock equation with pseudospectral method for closed-shell atoms. (English) Zbl 1448.81472
Summary: The Hartree-Fock (HF) equation for atoms with closed (sub)shells is transformed with the pseudospectral (PS) method into a discrete eigenvalue equation for scaled orbitals on a finite radial grid. The Fock exchange operator and the Hartree potential are obtained from the respective Poisson equations also discretized using the PS representation. The numerical solution of the discrete HF equation for closed-(sub)shell atoms from He to No is robust, fast and gives extremely accurate results, with the accuracy superior to that of the previous HF calculations. A very moderate number of 33 to 71 radial grid points is sufficient to obtain total energies with 14 significant digits and occupied orbital energies with 12 to 14 digits in numerical calculations using the double precision (64-bit) of the floating-point format. The electron density at the nucleus is then determined with 13 significant digits and the Kato condition for the density and s orbitals is satisfied with the accuracy of 11 to 13 digits. The node structure of the exact HF orbitals is obtained and their asymptotic dependence, including the common exponential decay, is reproduced very accurately. The accuracy of the investigated quantities is further improved by performing the PS calculations in the quadruple precision (128-bit) floating-point arithmetic which provides the total energies with 25 significant digits while using only 80 to 130 grid points.
MSC:
81V45 Atomic physics
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
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[1] Hartree, DR, The wave mechanics of an atom with a non-Coulomb central field. part I. Theory and methods, Proc. Camb. Philos. Soc., 24, 89-110 (1928)
[2] Fock, V., Näherungsmethode zur lösung des quantenmechanischen mehrkörperproblems, Z. Phys., 61, 126-148 (1930) · JFM 56.1313.08
[3] Slater, JC, Note on Hartree’s method, Phys. Rev., 35, 210-211 (1930)
[4] Hartree, DR; Hartree, W., Self-consistent field, with exchange, for beryllium, Proc. R. Soc., A150, 9-33 (1935) · JFM 61.1574.01
[5] Johnson, WR, Atomic Structure Theory (2007), Berlin: Springer, Berlin
[6] Roothaan, CCJ, Self-consistent field theory for open shells of electronic systems, Rev. Mod. Phys., 32, 179-185 (1960) · Zbl 0092.23203
[7] Møller, C.; Plesset, MS, Note on an approximation treatment for many-electron systems, Phys. Rev., 46, 618-622 (1934) · JFM 60.0778.04
[8] Čížek, J., On the correlation problem in atomic and molecular systems. calculation of wavefunction components in Ursell-type expansion using quantum-field theoretical methods, J. Chem. Phys., 45, 4256-4266 (1966)
[9] Ma, A.; Drummond, ND; Towler, MD; Needs, RJ, All-electron quantum Monte Carlo calculations for the noble gas atoms He to Xe, Phys. Rev. E, 71, 066704 (2005)
[10] Plakhutin, BN; Gorelik, EV; Breslavskaya, NN, Koopmans’ theorem in the ROHF method: canonical form for the Hartree-Fock hamiltonian, J. Chem. Phys., 125, 204110 (2006)
[11] Plakhutin, BN; Davidson, ER, Combined open shell Hartree-Fock theory of atomic-molecular and nuclear systems, J. Math. Chem., 45, 859 (2009) · Zbl 1309.92088
[12] Froese Fischer, C., A multi-configuration Hartree-Fock program with improved stability, Comput. Phys. Commun., 4, 107-116 (1972)
[13] Froese Fischer, C., The Hartree-Fock Method for Atoms: A Numerical Approach (1977), New York: Wiley, New York
[14] Froese Fischer, C., A general multi-configuration Hartree-Fock program, Comput. Phys. Commun., 14, 145-153 (1978)
[15] Froese Fischer, C., General Hartree-Fock program, Comput. Phys. Commun., 43, 335-365 (1987)
[16] Froese Fischer, C.; Brage, T.; Jönsson, P., Computational Atomic Structure, An MCHF Approach (1997), Bristol: Institute of Physics Publishing, Bristol · Zbl 0997.00542
[17] Kobus, J.; Laaksonen, L.; Sundholm, D., A numerical Hartree-Fock program for diatomic molecules, Comput. Phys. Commun., 98, 346-358 (1996)
[18] Shiozaki, T.; Hirata, S., Grid-based numerical Hartree-Fock solutions of polyatomic molecules, Phys. Rev. A, 76, 040503(R) (2007)
[19] Artemyev, AN; Ludeña, EV; Karasiev, VV; Hernández, AJ, A finite B-spline basis set for accurate diatomic molecule calculations, J. Comput. Chem., 25, 368-374 (2004)
[20] Morrison, JC; Boyd, S.; Marsano, L.; Bialecki, B.; Ericsson, T.; Santos, JP, Numerical methods for solving the Hartree-Fock equations of diatomic molecules I, Comput. Phys. Commun., 5, 959-985 (2009) · Zbl 1364.65263
[21] Saito, SL, Hartree-Fock-Roothaan energies and expectation values for the neutral atoms He to Uuo: the B-spline expansion method, At. Data Nucl. Data Tables, 95, 836 (2009)
[22] Froese Fischer, C., A B-spline Hartree-Fock program, Comput. Phys. Commun., 182, 1315-1326 (2011) · Zbl 1262.81265
[23] Zatsarinny, O.; Froese Fischer, C., A B-spline Dirac-Hartree-Fock program, Comput. Phys. Commun., 202, 287-303 (2016) · Zbl 1348.81030
[24] Hu, SL; Zhao, ZX; Shi, TY, B-spline one-center method for molecular Hartree-Fock calculations, Int. J. Quant. Chem., 114, 441-448 (2014)
[25] Ozaki, T.; Toyoda, M., Accurate finite element method for atomic calculations based on density functional theory and Hartree-Fock method, Comput. Phys. Commun., 182, 1245-1252 (2011) · Zbl 1262.81267
[26] Yanai, T.; Fann, GI; Gan, Z.; Harrison, RJ, Multiresolution quantum chemistry in multiwavelet bases: Hartree-Fock exchange, J. Chem. Phys., 121, 6680-6688 (2004)
[27] Hachmann, J.; Galek, PTA; Yanai, T.; Chan, GKL; Handy, NC, The nodes of Hartree-Fock wavefunctions and their orbitals, Chem. Phys. Lett., 392, 55-61 (2004)
[28] Anderson, JB, Quantum chemistry by random walk. \(H^2P, \text{H}^+_\[3D_{3h} \]^1A_1^{\prime }, \text{ H }_2^3\varSigma^+_u, \text{ H }_4^1\varSigma^+_g, Be^1S\), J. Chem. Phys., 65, 4121 (1976)
[29] Nakatsuji, H., Scaled Schrödinger equation and the exactwave function, Phys. Rev. Lett., 93, 030403 (2004)
[30] Nakatsuji, H., General method of solving the Schrödinger equation of atoms and molecules, Phys. Rev. A, 72, 062110 (2005)
[31] Nakatsuji, H.; Nakashima, H.; Kurokawa, Y.; Ishikawa, A., Solving the Schrödinger equation of atoms and molecules without analytical integration based on the free iterative-complement-interaction wave function, Phys. Rev. Lett., 99, 240402 (2007)
[32] Kurokawa, Y.; Nakashima, H.; Nakatsuji, H., Free iterative-complement-interaction calculations of the hydrogen molecule, Phys. Rev. A, 72, 062502 (2005)
[33] Nakatsuji, H.; Nakashima, H.; Kurokawa, YI, Solving the Schrödinger equation of atoms and molecules with the free-complement chemical-formula theory: first-row atoms and small molecules, J. Chem. Phys., 149, 114106 (2018)
[34] Nakatsuji, H., Full configuration-interaction calculations with the simplest iterative complement method: merit of the inverse Hamiltonian, Phys. Rev. A, 84, 062507 (2011)
[35] Wang, J.; Chu, SI; Laughlin, C., Multiphoton detachment of \(\text{ H }^-\). II. Intensity-dependent photodetachment rates and threshold behavior-complex-scaling generalized pseudospectral method, Phys. Rev. A, 50, 3208-3215 (1994)
[36] Boyd, JP; Rangan, C.; Bucksbaum, PH, Pseudospectral methods on a semi-infinite interval with application to the hydrogen atom: a comparison of the mapped Fourier-sine method with Laguerre series and rational Chebyshev expansions, J. Comp. Phys., 188, 56-74 (2003) · Zbl 1028.65086
[37] Lo, JQW; Shizgal, BD, Pseudospectral methods of solution of the Schrödinger equation, J. Math. Chem., 44, 787-801 (2008) · Zbl 1217.81058
[38] Shizgal, BD, Pseudospectral method of solution of the Schrödinger equation with nonclassical polynomials; the Morse and Pöschl-Teller (SUSY) potentials, Comput. Theor. Chem., 1084, 51-58 (2016)
[39] Morrison, CL; Shizgal, B., Pseudospectral solution of the Schrödinger equation for the Rosen-Morse and Eckart potentials, J. Math. Chem., 57, 1035 (2019) · Zbl 1414.81093
[40] Kowari, K., A numerical analysis of motion in symmetric double-well harmonic potentials using pseudospectral methods, Chem. Phys. Lett. (2019)
[41] Roy, AK; Chu, SI, Density-functional calculations on singly and doubly excited Rydberg states of many-electron atoms, Phys. Rev. A, 65, 052508 (2002)
[42] Cinal, M.; Holas, A., Noniterative accurate algorithm for the exact exchange potential of density-functional theory, Phys. Rev. A, 76, 042510 (2007)
[43] Friesner, RA, Solution of self-consistent field electronic structure equations by a pseudospectral method, Chem. Phys. Lett., 116, 39-43 (1985)
[44] Friesner, RA, Solution of the Hartree-Fock equations by a pseudospectral method: application to diatomic molecules, J. Chem. Phys., 85, 1462-1468 (1986)
[45] Ringnalda, MN; Belhadj, M.; Friesner, RA, Pseudospectral Hartree-Fock theory: applications and algorithmic improvements, J. Chem. Phys., 93, 3397-3407 (1990)
[46] Won, Y.; Lee, JG; Ringnalda, N.; Friesner, RA, Pseudospectral Hartree-Fock gradient calculations, J. Chem. Phys., 94, 8152-8157 (1991)
[47] Heyl, JS; Thirumalai, A., Pseudospectral methods for atoms in strong magnetic fields, Mon. Not. R. Astron. Soc., 407, 590-598 (2010)
[48] Thirumalai, A.; Heyl, JS, Two-dimensional pseudospectral Hartree-Fock method for low-Z atoms in intense magnetic fields, Phys. Rev. A, 89, 052522 (2014)
[49] Bunge, CF; Barrientos, JA; Bunge, AV; Cogordan, JA, Hartree-Fock and Roothaan-Hartree-Fock energies for the ground states of He through Xe, Phys. Rev. A, 46, 3691-3696 (1992)
[50] Tatewaki, H.; Koga, YT; Thakkar, AJ, Numerical Hartree-Fock energies of low-lying excited states of neutral atoms with \(Z\le 18\), J. Chem. Phys., 101, 4945 (1994)
[51] Bunge, CF; Barrientos, JA; Bunge, AV, Roothaan-Hartree-Fock ground-state atomic wave functions: slater-type orbital expansions and expectation values for Z=2-54, At. Data Nucl. Data Tables, 53, 113-162 (1993)
[52] Koga, T., Hyperbolic cosine functions applied to atomic Roothaan-Hartree-Fock wave functions, J. Mol. Struct. (Theochem), 451, 1-10 (1998)
[53] Cinal, M., Direct mapping between exchange potentials of Hartree-Fock and Kohn-Sham schemes as origin of orbital proximity, J. Chem. Phys., 132, 014101 (2010)
[54] Cinal, M.; Holas, A., Exact exchange potential evaluated from occupied Kohn-Sham and Hartree-Fock solutions, Phys. Rev. A, 83, 062515 (2011)
[55] Cinal, M., Depurated inversion method for orbital-specific exchange potentials, Int. J. Quant. Chem., 120, e26101 (2020)
[56] The formula (6) was misprinted in the previous paper by the present author [53]
[57] Boyd, JP, Chebyshev and Fourier Spectral Methods (2000), Mineola: Dover Publications, Mineola
[58] Shizgal, B., Spectral Methods in Chemistry and Physics (2015), New York: Springer, New York · Zbl 1327.81007
[59] D. Funaro, Fortran Routines for Spectral Methods. Pubblicazioni—Istituto di analisi numerica del Consiglio Nazionale delle Ricerche, No. 891, Pavia (1993). Available on-line http://morespace.unimore.it/danielefunaro/routines/. Accessed 30 Dec 2019
[60] Note that there is a misprint in the work by Wang et al. [35] where the factor \(P_N(x_i)/P_N(x_j)\) is missing in the expression \(g^{\prime \prime }_j(x_i)\) with \(i \ne j\)
[61] Legendre polynomial. Wolfram Functions Site, http://functions.wolfram.com/05.03.23.0010.01. Accessed 30 Dec 2019
[62] Becke, AD, A multicenter numerical integration scheme for polyatomic molecules, J. Chem. Phys., 88, 2547-2553 (1988)
[63] Pulay, P., Convergence acceleration of iterative sequences. The case of scf iteration, Chem. Phys. Lett., 73, 393-398 (1980)
[64] Intel Math Kernel Library. https://software.intel.com/content/www/us/en/develop/tools/math-kernel-library.html. Accessed 22 May 2020
[65] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J.D. Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide, 3rd edn. (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999). http://www.netlib.org/lapack/lug/
[66] Koga, T.; Watanabe, S.; Kanayama, K.; Yasuda, R.; Thakkar, AJ, Improved Roothaan-Hartree-Fock wave functions for atoms and ions with \(N\le 54\), J. Chem. Phys., 103, 3000-3005 (1995)
[67] Mendez, MPA; Mitnik, DM; Miraglia, JE, Depurated inversion method for orbital-specific exchange potentials, Int. J. Quant. Chem., 116, 1882-1890 (2016)
[68] Takeda, N.; Hatano, Y.; Yamamoto, S.; Tatewaki, H., Counting nodal surfaces in molecular orbitals: elimination of artificial nodes, Comput. Theor. Chem., 1045, 99-112 (2014)
[69] Handy, NC; Marron, MT; Silverstone, HJ, Long-range behavior of Hartree-Fock orbitals, Phys. Rev., 180, 45-48 (1969)
[70] Handler, GS; Smith, DW; Silverstone, HJ, Asymptotic behavior of atomic Hartree-Fock orbitals, J. Chem. Phys., 73, 3936-3938 (1980)
[71] Ishida, T.; Ohno, K., On the asymptotic behavior of Hartree-Fock orbitals, Theor. Chim. Acta., 81, 355-364 (1992)
[72] Nogueira, F.; Castro, A.; Marques, M.; Fiolhais, C.; Nogueira, F.; Marques, M., A tutorial on density functional theory, A primer in density functional theory, 224 (2003), Berlin: Springer, Berlin
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