Abstract
The auxiliary functions provide efficient computation of integrals arising at the self-consistent field level for molecules using Slater-type bases. This applies both in relativistic and non-relativistic electronic structure theory. The relativistic molecular auxiliary functions derived in our previous paper (Bağcı and Hoggan, Phys Rev E 91:023303, 2015) are discussed here in detail. Two solution methods are proposed in the present study. The ill-conditioned binomial series representation formulae are first replaced by a convergent series representation for incomplete beta functions. They are then improved by inserting an extra parameter used to extend the domain of convergence. Highly accurate results can be achieved for integrals by the procedures discussed in the present study which place no restrictions on quantum numbers in all ranges of orbital parameters. The difficulty of obtaining analytical relations associated with using non-integer Slater-type orbitals which are non-analytic in the sense of complex analysis at \(r=0\) is, therefore, eliminated.
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References
Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications, New York
Agmon S (1982) Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of n-body schrödinger operations. Princeton University Press, Princeton
Appell P (1925) Sur les fonctions hypergéométriques de plusieurs variables, les polynômes d’Hermite et autres fonctions sphériques dans l’hyperespace. Mémorial des sciences mathématiques, Gauthier-Villars
Bağcı A (2017) Notes on mathematical difficulties arising in relativistic SCF approximation. arXiv:1603.02307 [physics.chem-ph]
Bağcı A, Hoggan PE (2014) Performance of numerical approximation on the calculation of overlap integrals with noninteger Slater-type orbitals. Phys Rev E 89(7):053307. https://doi.org/10.1103/PhysRevE.89.0533070
Bağcı A, Hoggan PE (2015a) Benchmark values for molecular two-electron integrals arising from the Dirac equation. Phys Rev E 91(2):023303. https://doi.org/10.1103/PhysRevE.91.023303
Bağcı A, Hoggan PE (2015b) Benchmark values for molecular three-center integrals arising in the Dirac equation. Phys Rev E 92(4):043301. https://doi.org/10.1103/PhysRevE.92.043301
Bağcı A, Hoggan PE (2016) Solution of the Dirac equation using the Rayleigh-Ritz method: flexible basis coupling large and small components. Results for one-electron systems. Phys Rev E 94(1):013302. https://doi.org/10.1103/PhysRevE.94.013302
Barnett MP, Coulson CA (1951) The evaluation of integrals occurring in the theory of molecular structure. Parts I&II. Philos Trans R Soc Lond A Math Phys Eng Sci 243(864):221–249. https://doi.org/10.1098/rsta.1951.0003
Bouferguene A, Weatherford CA, Jones HW (1999) Addition theorem of Slater-type orbitals: application to \({{\rm H}}_{2}^{+}\) in a strong magnetic field. Phys Rev E 59(2):2412–2423. https://doi.org/10.1103/PhysRevE.59.2412
Colavecchia FD, Gasaneo G (2004) f1: a code to compute Appell’s F1 hypergeometric function. Comput Phys Commun 157(1):32–38. https://doi.org/10.1016/S0010-4655(03)00490-9
Colavecchia FD, Gasaneo G, Miraglia JE (2001) Numerical evaluation of Appell’s F1 hypergeometric function. Comput Phys Commun 138(1):29–43. https://doi.org/10.1016/S0010-4655(01)00186-2
Condon EU, Shortley GH (1935) The theory of atomic spectra. Cambridge University Press, Cambridge
Coulson CA (1942) Two-centre integrals occurring in the theory of molecular structure. Math Proc Camb Philos Soc 38(2):210–223. https://doi.org/10.1017/S0305004100021873
Ema I, López R, Fernández JJ, Ramírez G, Rico JF (2008) Auxiliary functions for molecular integrals with Slater-type orbitals. II. Gauss transform methods. Int J Quantum Chem 108(1):25–39. https://doi.org/10.1002/qua.21409
Fernández JJ, López R, Ema I, Ramírez G, Fernández RJ (2006) Auxiliary functions for molecular integrals with Slater-type orbitals. I. Translation methods. Int J Quantum Chem 106(9):1986–1997. https://doi.org/10.1002/qua.21002
Gorder RAV (2017) On the utility of the homotopy analysis method for non-analytic and global solutions to nonlinear differential equations. Numer Algorithm 76(1):151–162. https://doi.org/10.1007/s11075-016-0248-y
Grant IP (2007) Relativistic quantum theory of atoms and molecules. Springer, New York
Guseinov II (1970) Analytical evaluation of two-centre Coulomb, hybrid and one-electron integrals for Slater-type orbitals. J Phys B Atom Mol Phys 3(11):1399–1412. https://doi.org/10.1088/0022-3700/3/11/001
Guseinov II (2007) Expansion formulae for two-center integer and noninteger n STO charge densities and their use in evaluation of multi-center integrals. J Math Chem 42(3):415–422. https://doi.org/10.1007/s10910-006-9111-z
Guseinov II (2009) Use of auxiliary functions \(Q_{ns}^{q}\) and \(G_{-ns}^{q}\) in evaluation of multicenter integrals over integer and noninteger n-Slater type orbitals arising in Hartree-Fock-Roothaan equations for molecules. J Math Chem 45(4):974–980. https://doi.org/10.1007/s10910-008-9431-2
Guseinov II, Ertürk M (2012) Use of noninteger n-generalized exponential type orbitals with hyperbolic cosine in atomic calculations. Int J Quantum Chem 112(6):1559–1565. https://doi.org/10.1002/qua.23133
Guseinov II, Mamedov BA (2002a) On the calculation of arbitrary multielectron molecular integrals over Slater-type orbitals using recurrence relations for overlap integrals. III. Auxiliary functions \(Q_{nn^{\prime }}^{q}\) and \(G_{-nn^{\prime }}^{q}\). Int J Quantum Chem 86(5):440–449. https://doi.org/10.1002/qua.10045
Guseinov II, Mamedov BA (2002b) Evaluation of overlap integrals with integer and noninteger n Slater-type orbitals using auxiliary functions. Mol Model Annu 8(9):272–276. https://doi.org/10.1007/s00894-002-0098-5
Guseinov II, Mamedov BA (2002c) Computation of multicenter nuclear-attraction integrals of integer and noninteger n Slater orbitals using auxiliary functions. J Theoret Comput Chem 1(1):17–24. https://doi.org/10.1142/S0219633602000130
Guseinov II, Mamedov BA (2005) Fast evaluation of molecular auxiliary functions \(A_{\alpha }\) and \(B_{n}\) by analytical relations. J Math Chem 38(1):21–26. https://doi.org/10.1007/s10910-005-4527-4
Guseinov II, Mamedov BA, Kara M, Orbay M (2001) On the computation of molecular auxiliary functions \(A_{n}\) and \(B_{n}\). Pramana 56(5):691–696. https://doi.org/10.1007/s12043-001-0093-x
Guseinov II, Mamedov BA, Sünel N (2002) Computation of molecular integrals over Slater-type orbitals. X. Calculation of overlap integrals with integer and noninteger n Slater orbitals using complete orthonormal sets of exponential functions. J Mol Struct THEOCHEM 593(1):71–77. https://doi.org/10.1016/S0166-1280(02)00074-X
Harris FE, Michels HH (1967) The evaluation of molecular integrals for slater-type orbitals. In: Prigogine I (ed) Adv Chem Phys, vol 13. Wiley, Hoboken, pp 205–266. https://doi.org/10.1002/9780470140154.ch8
Harris FE (2002) Analytic evaluation of two-center STO electron repulsion integrals via ellipsoidal expansion. Int J Quantum Chem 88(6):701–734. https://doi.org/10.1002/qua.10181
Harris FE (2003) Comment: “On the computation of molecular auxiliary functions \(A_{n}\) and \(B_{n}\)”. Pramana 61(4):C779–C780
Harris FE (2004) Efficient evaluation of the molecular auxiliary function \(B_{n}\) by downward recursion. Int J Quantum Chem 100(2):142–145. https://doi.org/10.1002/qua.10812
Harris FE, Michels HH (1965) Multicenter integrals in quantum mechanics. I. Expansion of slater-type orbitals about a new origin. J Chem Phys 43(10):S165–S169. https://doi.org/10.1063/1.1701480
Harris FE, Michels HH (1966) Multicenter integrals in quantum mechanics. II. Evaluation of electron-repulsion integrals for slater-type orbitals. J Chem Phys 45(1):116–123. https://doi.org/10.1063/1.1727293
Hoggan PE (2011) Slater-type orbital basis sets: reliable and rapid solution of the Schrödinger equation for accurate molecular properties. In: Popelier P (ed) Solving the Schrödinger equation. Imperial College Press, Covent Garden. https://doi.org/10.1142/9781848167254_0007
Kato T (1957) On the eigenfunctions of many-particle systems in quantum mechanics. Commun Pure Appl Math 10(2):151–177. https://doi.org/10.1002/cpa.3160100201
Koborov VI, Hilico L, Karr JPh (2013) Calculation of the relativistic Bethe logarithm in the two-center problem. Phys Rev A 87(6):062506. https://doi.org/10.1103/PhysRevA.87.062506
Koga T, Kanayama K (1997) Noninteger principal quantum numbers increase the efficiency of Slater-type basis sets: singly charged cations and anions. J Phys B Atom Mol Opt Phys 30(7):1623–1631. https://doi.org/10.1088/0953-4075/30/7/004
Koga T, Kanayama K, Thakkar AJ (1997) Noninteger principal quantum numbers increase the efficiency of Slater-type basis sets. Int J Quantum Chem 62(1):1–11. https://doi.org/10.1002/(SICI)1097-461X(1997)62:1<1::AID-QUA1>3.0.CO;2-#
Koga T, Shimazaki T, Satoh T (2000) Noninteger principal quantum numbers increase the efficiency of Slater-type basis sets: double-zeta approximation. J Mol Struct THEOCHEM 496(1):95–100. https://doi.org/10.1016/S0166-1280(99)00176-1
Kotani M, Amemiya A, Ishiguro E, Kimura T (1963) Table of molecular integrals. Maruzen Company Ltd., Tokyo
Lesiuk M, Moszynski R (2014a) Reexamination of the calculation of two-center, two-electron integrals over Slater-type orbitals. II. Neumann expansion of the exchange integrals. Phys Rev E 90(6):063319. https://doi.org/10.1103/PhysRevE.90.063319
Lesiuk M, Moszynski R (2014b) Reexamination of the calculation of two-center, two-electron integrals over Slater-type orbitals. I. Coulomb and hybrid integrals. Phys Rev E 90(6):063318. https://doi.org/10.1103/PhysRevE.90.063318
Liao SJ (2004) Beyond perturbation: introduction to the homotopy analysis method. Chapman&Hall/CRC, Boca Raton
Liu CS (2010) The essence of the generalized Newton binomial theorem. Commun Nonlinear Sci Numer Simul 15(10):2766–2768. https://doi.org/10.1016/j.cnsns.2009.11.004
Löwdin PO (1956) Quantum theory of cohesive properties of solids. Adv Phys 5(17):1–171. https://doi.org/10.1080/00018735600101155
Mekelleche SM, Baba-Ahmed A (2000) Unified analytical treatment of one-electron two-center integrals with noninteger n Slater-type orbitals. Theoret Chem Acc 103(6):63–468. https://doi.org/10.1007/s002149900084
Mulliken RS, Rieke CA, Orloff D, Orloff H (1949) Formulas and numerical tables for overlap integrals. J Chem Phys 17(12):1248–1267. https://doi.org/10.1063/1.1747150
Pachucki K, Puchalski M, Remiddi E (2004) Recursion relations for the generic Hylleraas three-electron integral. Phys Rev A 70(6):032502. https://doi.org/10.1103/PhysRevA.70.032502
Parr RG, Hubert WJ (1957) Why not use slater orbitals of nonintegral principal quantum number? J Chem Phys 26(2):424–424. https://doi.org/10.1063/1.1743314
Roothaan CCJ (1951) A study of two-center integrals useful in calculations on molecular structure I. J Chem Phys 19(12):1445–1458. https://doi.org/10.1063/1.1748100
Roothaan CCJ (1956) Study of two-center integrals useful in calculations on molecular structure. IV. The auxiliary functions \(C\alpha \beta \gamma \epsilon (\rho a,\rho b)\) for \(\alpha \ge 0\). J Chem Phys 24(5):947–960. https://doi.org/10.1063/1.1742721
Rüdenberg K (1951) A study of two-center integrals useful in calculations on molecular structure. II. The two-center exchange integrals. J Chem Phys 19(12):1459–1477. https://doi.org/10.1063/1.1748101
Slater JC (1930) Atomic shielding constants. Phys Rev 36(1):57–64. https://doi.org/10.1103/PhysRev.36.57
Steinborn EO, Rüedenberg K (1973) Rotation and translation of regular and irregular solid spherical harmonics. Adv Quantum Chem 7(1):1–81. https://doi.org/10.1016/S0065-3276(08)60558-4
Temme NM (1994) Computational aspects of incomplete gamma functions with large complex parameters. In: Zahar RVM (ed) Approximation and computation: a festschrift in honor of Walter Gautschi: proceedings of the Purdue conference, December 2–5, 1993. Birkhäuser, Boston, pp 551–562.https://doi.org/10.1007/978-1-4684-7415-2_37
Wang X (2012) Recursion formulas for appell functions. Integral Transf Spec Funct 23(6):421–433. https://doi.org/10.1080/10652469.2011.596483
Weniger EJ (2002) Addition theorems as three-dimensional Taylor expansions. II. B functions and other exponentially decaying functions. Int J Quantum Chem 90(1):92–104. https://doi.org/10.1002/qua.948
Weniger EJ (2008) On the analyticity of Laguerre series. J Phys A Math Theoret 41(42):425207. https://doi.org/10.1088/1751-8113/41/42/425207
Willock DJ (2009) Appendix 9: the atomic orbitals of hydrogen. In: Molecular symmetry. Wiley, Chichester. https://doi.org/10.1002/9780470747414.app9
Acknowledgements
A.B. would like to thank the Department of Physics, Faculty of Arts and Sciences, Pamukkale University for providing working facilities. He is also particularly grateful to Prof. Dr. Muzaffer Adak and Prof. Dr. Mestan Kalay for fruitful discussions.
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This contribution is the written, peer-reviewed version of a paper presented at the International Conference “The Quantum World of Molecules: from Orbitals to Spin Networks”, held at the Accademia Nazionale dei Lincei in Rome on 27–28 April, 2017.
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Bağcı, A., Hoggan, P.E. Analytical evaluation of relativistic molecular integrals. I. Auxiliary functions. Rend. Fis. Acc. Lincei 29, 191–197 (2018). https://doi.org/10.1007/s12210-018-0669-8
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DOI: https://doi.org/10.1007/s12210-018-0669-8