Skip to main content
SpringerLink
Log in

A new method with improved phase-lag and stability properties for problems in quantum chemistry - an economical case

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

A new FINDIFF (Finite Difference) method with zeroing phase-lag and its derivatives up to order two, for initial or boundary value problems with periodical and/or oscillating solutions with an application on problems in Quantum Chemistry, is proposed in this paper. The method is considered to belong to the economical class of methods since it is of the highest possible algebraic order using the minimum number of function evaluations per step.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. M.M. Chawla, S.R. Sharma, Families of 5Th Order Nyström Methods for Y’’=F(X, Y) and Intervals of Periodicity. Computing 26(3), 247–256 (1981)

    Article  Google Scholar 

  2. J.M. Franco, M. Palacios, High-order P-stable multis\({\rm te}\)p Methods. J. Comput. Appl. Math. 30, 1–10 (1990)

    Article  Google Scholar 

  3. J.D. Lambert, Numerical Methods for Ordinary Differential Systems, The Initial Value Problem (John Wiley and Sons, 1991), pp. 104–107

    Google Scholar 

  4. E. Stiefel, D.G. Bettis, Stabilization of Cowell’s Method. Numer. Math. 13, 154–175 (1969)

    Article  Google Scholar 

  5. M.M. Chawla, S.R. Sharma, Intervals of Periodicity and Absolute Stability of Explicit Nyström Methods for Y’’=F(X, Y). Bit 21(4), 455–464 (1981)

    Article  Google Scholar 

  6. M.M. Chawla, Unconditionally Stable Noumerov-Type Methods for 2nd Order Differential-Equations. Bit 23(4), 541–542 (1983)

    Article  Google Scholar 

  7. http://www.burtleburtle.net/bob/math/multistep.html

  8. M.M. Chawla, P.S. Rao, A Noumerov-Type Method with Minimal Phase-Lag for the Integration of 2nd Order Periodic Initial-Value Problems. J. Comput. Appl. Math. 11(3), 277–281 (1984)

    Article  Google Scholar 

  9. M.M. Chawla, Numerov Made Explicit has Better Stability. Bit 24(1), 117–118 (1984)

    Article  Google Scholar 

  10. M.M. Chawla, P.S. Rao, High-Accuracy P-Stable Methods for Y” = F(T, Y). Ima J. Num. Anal. 6(2), 252-252(1986)

  11. T. Lyche, Chebyshevian multistep Methods for Ordinary Differential Eqations. Num. Math. 19, 65–75 (1972)

    Article  Google Scholar 

  12. T.E. Simos, P.S. Williams, A Finite Difference Method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 79, 189–205 (1997)

    Article  Google Scholar 

  13. R.M. Thomas, Phase properties of high order almost P-stable formulae. BIT 24, 225–238 (1984)

    Article  Google Scholar 

  14. J.D. Lambert, I.A. Watson, Symmetric multis\(\rm te\)p Methods for periodic initial values problems. J. Inst. Math. Appl. 18, 189–202 (1976)

    Article  Google Scholar 

  15. M.M. Chawla, A New Class of Explicit 2-S\(\rm te\)p 4Th Order Methods for Y’’ = F(T, Y) with Extended Intervals of Periodicity. J. Comput. Appl. Math. 14(3), 467–470 (1986)

    Article  Google Scholar 

  16. M.M. Chawla, B. Neta, Families of 2-S\(\rm te\)p 4Th-Order P-Stable Methods for 2Nd-Order Differential-Equations. J. Comput. Appl. Math. 15(2), 213–223 (1986)

    Article  Google Scholar 

  17. M.M. Chawla, P.S. Rao, A Noumerov-Type Method with Minimal Phase-Lag for the Integration of 2Nd-Order Periodic Initial-Value Problems .2. Explicit Method. J. Comput. Appl. Math. 15(3), 329–337 (1986)

    Article  Google Scholar 

  18. M.M. Chawla, P.S. Rao, B. Neta, 2-S\(\rm te\)p 4Th-Order P-Stable Methods with Phase-Lag of Order 6 for Y’’=F(T, Y). J. Comput. Appl. Math. 16(2), 233–236 (1986)

    Article  Google Scholar 

  19. M.M. Chawla, P.S. Rao, An Explicit 6Th-Order Method with Phase-Lag of Order 8 for Y’’=F(T, Y). J. Comput. Appl. Math. 17(3), 365–368 (1987)

    Article  Google Scholar 

  20. M.M. Chawla, M.A. Al-Zanaidi, Non-Dissipative Extended One-S\(\rm te\)p Methods for Oscillatory Problems. International Journal of Computer Mathematics 69(1–2), 85–100 (1998)

    Article  Google Scholar 

  21. M.M. Chawla, M.A. Al-Zanaidi, A Two-Stage Fourth-Order “Almost” P-Stable Method for Oscillatory Problems. J. Comput. Appl. Math. 89(1), 115–118 (1998)

  22. M.M. Chawla, M.A. Al-Zanaidi, S.S. Al-Ghonaim, Singly-Implicit Stabilized Extended One-S\(\rm te\)p Methods for Second-Order Initial-Value Problems with Oscillating Solutions. Math. Comput. Modell.29(2), 63–72 (1999)

    Article  Google Scholar 

  23. J.P. Coleman, Numerical-Methods for Y’’=F(X, Y) Via Rational-Approximations for the Cosine. Ima J. Num. Anal. 9(2), 145–165 (1989)

    Article  Google Scholar 

  24. J.P. Coleman, A.S. Booth, Analysis of A Family of Chebyshev Methods for Y’’ = F(X, Y). J. Comput. Appl. Math. 44(1), 95–114 (1992)

    Article  Google Scholar 

  25. J.P. Coleman, L. Gr, Ixaru, P-Stability and Exponential-Fitting Methods for Y’’=F(X, Y). Ima J. Num. Anal. 16(2), 179–199 (1996)

    Article  Google Scholar 

  26. J.P. Coleman, S.C. Duxbury, Mixed Collocation Methods for Y ’ ’ = F(X, Y). J. Comput. Appl. Math. 126(1–2), 47–75 (2000)

    Article  Google Scholar 

  27. LGr. Ixaru, S. Berceanu, Coleman Method Maximally Adapted to the Schrödinger-Equation. Comput. Phys. Commun. 44(1–2), 11–20 (1987)

    Article  Google Scholar 

  28. LGr. Ixaru, the Numerov Method and Singular Potentials. Journal of Computational Physics 72(1), 270–274 (1987)

    Article  CAS  Google Scholar 

  29. LGr. Ixaru, M. Rizea, Numerov Method Maximally Adapted to the Schrödinger-Equation. J. Comput. Phys. 73(2), 306–324 (1987)

    Article  Google Scholar 

  30. LGr. Ixaru, H. De Meyer, G. Vanden Berghe, M. Van Daele, Expfit4 - A Fortran Program for the Numerical Solution of Systems of Nonlinear Second-Order Initial-Value Problems. Comput. Phys. Commun.100(1–2), 71–80 (1997)

    Article  CAS  Google Scholar 

  31. LGr. Ixaru, G. Vanden Berghe, H. De Meyer, M. Van Daele, Four-S\({\rm te}\)p Exponential-Fitted Methods for Nonlinear Physical Problems. Comput. Phys. Commun.100(1–2), 56–70 (1997)

    Article  CAS  Google Scholar 

  32. LGr. Ixaru, M. Rizea, Four S\({\rm te}\)p Methods for Y’’=F(X, Y).J. Comput. Appl. Math. 79(1), 87–99 (1997)

    Article  Google Scholar 

  33. M. Van Daele, G. Vanden Berghe, H. De Meyer, LGr. Ixaru, Exponential-Fitted Four-S\({\rm te}\)p Methods for Y ’ ’=F(X, Y). Int. J. Comput. Math.66(3–4), 299–309 (1998)

    Article  Google Scholar 

  34. LGr. Ixaru, B. Paternoster, A Conditionally P-Stable Fourth-Order Exponential-Fitting Method for Y ’ ’ = F(X, Y). J. Comput. Appl. Math. 106(1), 87–98 (1999)

    Article  Google Scholar 

  35. LGr. Ixaru, Numerical operations on oscillatory functions. Computers & Chemistry 25(1), 39–53 (2001)

    Article  CAS  Google Scholar 

  36. LGr. Ixaru, G. Vanden Berghe, H. De Meyer, Exponentially Fitted Variable Two-Step BDF Algorithm for First Order Odes. Comput. Phys. Commun.150(2), 116–128 (2003)

    Article  CAS  Google Scholar 

  37. Maxim A. Medvedev, T.E. Simos, A three-stages multistep teeming in phase Algorithm for computational problems in Chemistry. J. Math. Chem. 57(6), 1598–1617 (2019)

    Article  CAS  Google Scholar 

  38. Marina A. Medvedeva, T.E. Simos, An Accomplished Phase FD Process for DEs in Chemistry. J. Math. Chem. 57(10), 2208–2228 (2019)

    Article  CAS  Google Scholar 

  39. Yu.-Yu. MA, Chia-Liangh. LIN, T.E. Simos, An Integrated in Phase FD Procedure for DiffEqns in Chemical Problems. J. Math. Chem. 58(1), 6–28 (2020)

    Article  CAS  Google Scholar 

  40. Sheng Hao, T.E. Simos, A Phase Fitted FinDiff Process for DifEquns in Quantum Chemistry. J. Math. Chem. 58(2), 353–381 (2020)

    Article  CAS  Google Scholar 

  41. Xing Tong, T.E. Simos, A complete in phase FinitDiff procedure for DiffEquns in Chemistry. J. Math. Chem. 58(2), 407–438 (2020)

    Article  CAS  Google Scholar 

  42. Xiaoping Chen, T.E. Simos, A phase fitted FiniteDiffr process for DiffrntEqutns in Chemistry. J. Math. Chem. 58(6), 1059–1090 (2020)

    Article  CAS  Google Scholar 

  43. Z.H.E.N. ZHAO, J.U.N. LUO, C.H.I.A.-L.I.A.N.G. LIN, T.E. Simos, Full in Phase Finite Difference Algorithm for Differential Equations in Quantum Chemistry. J. Math. Chem. 58(6), 1197–1218 (2020)

    Article  CAS  Google Scholar 

  44. YU-YU MA, CHIA-LIANG LIN, T.E. Simos, A New Economical Method with Eliminated Phase-Lag and its Derivatives for Problems in Chemistry, J. Math. Chem., to appear

  45. Fei Hui, T.E. Simos, A new family of two stage symmetric two-Step Methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation. J. Math. Chem. 53(10), 2191–2213 (2015)

    Article  CAS  Google Scholar 

  46. L Gr. Ixaru, M. Rizea, Comparison of some four-Step Methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 38(3), 329–337 (1985)

    Article  CAS  Google Scholar 

  47. LGr. Ixaru, M. Micu, Topics in Theoretical Physics (Central Institute of Physics, Bucharest, 1978)

    Google Scholar 

  48. LGr. Ixaru, M. Rizea, A Numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies. Comput. Phys. Commun. 19, 23–27 (1980)

    Article  Google Scholar 

  49. J.R. Dormand, M.E.A. El-Mikkawy, P.J. Prince, Families of Runge-Kutta-Nyström formulae. IMA J. Numer. Anal. 7, 235–250 (1987)

    Article  Google Scholar 

  50. J.R. Dormand, P.J. Prince, A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)

    Article  Google Scholar 

  51. G.D. Quinlan, S. Tremaine, Symmetric Multistep Methods for the Numerical Integration of Planetary Orbits. Astronom. J. 100, 1694–1700 (1990)

    Article  Google Scholar 

  52. A.D. Raptis, A.C. Allison, Exponential-fitting Methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 14, 1–5 (1978)

    Article  Google Scholar 

  53. M.M. Chawla, P.S. Rao, An Noumerov-type Method with minimal phase-lag for the integration of second order periodic initial-value problems II Explicit Method. J. Comput. Appl. Math.15, 329–337 (1986)

    Article  Google Scholar 

  54. M.M. Chawla, P.S. Rao, An explicit sixth - order Method with phase-lag of order eight for \(y^{\prime \prime }=f(t, y)\). J. Comput. Appl. Math. 17, 363–368 (1987)

    Article  Google Scholar 

  55. M. Rizea, Exponential fitting Method for the time-dependent Schrödinger equation. J. Math. Chem. 48(1), 55–65 (2010)

    Article  CAS  Google Scholar 

  56. A. Konguetsof, Two-S\(\rm te\)p high order hybrid explicit Method for the numerical solution of the Schrödinger equation. J. Math. Chem. 48, 224–252 (2010)

    Article  CAS  Google Scholar 

  57. A.D. Raptis, J.R. Cash, A variable Step Method for the numerical integration of the one-dimensional Schrödinger equation. Comput. Phys. Commun. 36, 113–119 (1985)

    Article  CAS  Google Scholar 

  58. A.C. Allison, The numerical solution of coupled differential equations arising from the Schrödinger equation. J. Comput. Phys. 6, 378–391 (1970)

    Article  Google Scholar 

  59. R.B. Bernstein, A. Dalgarno, H. Massey, I.C. Percival, Thermal scattering of atoms by homonuclear diatomic molecules. Proc. Roy. Soc. Ser. A 274, 427–442 (1963)

    CAS  Google Scholar 

  60. R.B. Bernstein, Quantum mechanical (phase shift) analysis of differential elastic scattering of molecular beams. J. Chem. Phys. 33, 795–804 (1960)

    Article  CAS  Google Scholar 

  61. M. Rizea, V. Ledoux, M. Van Daele, G. Vanden Berghe, N. Carjan, Finite Difference approach for the two-dimensional Schrödinger equation with application to scission-neutron emission. Comput. Phys. Commun.179(7), 466–478 (2008)

    Article  CAS  Google Scholar 

  62. J.R. Dormand, P.J. Prince, A family of embedded Runge-Kutta formula. J. Comput. Appl. Math. 6, 19–26 (1980)

    Article  Google Scholar 

  63. Kenan Mu, T.E. Simos, A Runge-Kutta type implicit high algebraic order two-Step Method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of coupled differential equations arising from the Schrödinger equation. J. Math. Chem. 53, 1239–1256 (2015)

    Article  CAS  Google Scholar 

  64. LGr. Ixaru, M. Rizea, G. VandenBerghe, H. De Meyer, Weights of the Exponential Fitting Multistep Algorithms for First-Order Odes. J. Comput. Appl. Math. 132(1), 83–93 (2001)

    Article  Google Scholar 

  65. A.D. Raptis, J.R. Cash, Exponential and Bessel Fitting Methods for the Numerical-Solution of the Schrödinger-Equation. Comput. Phys. Commun.44(1–2), 95–103 (1987)

    Article  Google Scholar 

  66. C.D. Papageorgiou, A.D. Raptis, A Method for the Solution of the Schrödinger-Equation. Comput. Phys. Commun. 43(3), 325–328 (1987)

    Article  CAS  Google Scholar 

  67. Zhou Zhou, T.E. Simos, A new two stage symmetric two-Step Method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation. J. Math. Chem. 54, 442–465 (2016)

    Article  CAS  Google Scholar 

  68. A.D. Raptis, Exponential Multistep Methods for Ordinary Differential Equations. Bull. Greek Math. Soc. 25, 113–126 (1984)

    Google Scholar 

  69. Hang Ning, T.E. Simos, A low computational cost eight algebraic order hybrid Method with vanished phase-lag and its first, second, third and fourth derivatives for the approximate solution of the Schrödinger equation. J. Math. Chem. 53(6), 1295–1312 (2015)

    Article  CAS  Google Scholar 

  70. Zhiwei Wang, T.E. Simos, An economical eighth-order Method for the approximation of the solution of the Schrödinger equation. J. Math. Chem. 55, 717–733 (2017)

    Article  CAS  Google Scholar 

  71. Ke Yan, T.E. Simos, A Finite Difference Pair with Improved Phase and Stability Properties, J. Math. Chem., in press

  72. J.R. Cash, A.D. Raptis, A High-Order Method for the Numerical-Integration of the One-Dimensional Schrödinger-Equation. Comput. Phys. Commun. 33(4), 299–304 (1984)

    Article  CAS  Google Scholar 

  73. A.D. Raptis, Exponentially-Fitted Solutions of the Eigenvalue Shrödinger Equation with Automatic Error Control. Comput. Phys. Commun.28(4), 427–431 (1983)

    Article  Google Scholar 

  74. A.D. Raptis, 2-Step Methods for the Numerical-Solution of the Schrödinger-Equation. Comput. Phys. Commun. 28(4), 373–378 (1982)

    Google Scholar 

  75. A.D. Raptis, On the Numerical-Solution of the Schrödinger-Equation. Comput. Phys. Commun. 24(1), 1–4 (1981)

    Article  Google Scholar 

  76. A.D. Raptis, Exponential-Fitting Methods for the Numerical-Integration of the 4Th-Order Differential-Equation Y\(^{iv}\)+F.Y=G. Computing 24(2–3), 241–250 (1980)

    Article  Google Scholar 

  77. H. Van De Vyver, A Symplectic Exponentially Fitted Modified Runge-Kutta-Nyström Method for the Numerical Integration of Orbital Problems. New Astronomy 10(4), 261–269 (2005)

    Article  Google Scholar 

  78. H. Van De Vyver, On the Generation of P-Stable Exponentially Fitted Runge-Kutta-Nyström Methods By Exponentially Fitted Runge-Kutta Methods. J. Math. Chem. 188(2), 309–318 (2006)

    Article  Google Scholar 

  79. Marnix Van Daele, Guido Vanden Berghe, P-Stable Obrechkoff Methods of Arbitrary Order for Second-Order Differential Equations. Num. Algorith. 44(2), 115–131 (2007)

    Article  Google Scholar 

  80. M. Van DAELE, G. Vanden BERGHE, P-stable exponentially-fitted Obrechkoff Methods of arbitrary order for second-order differential equations. Num. Algorith. 46(4), 333–350 (2007)

    Article  Google Scholar 

  81. Yonglei Fang, Wu. Xinyuan, A Trigonometrically Fitted Explicit Numerov-Type Method for Second-Order Initial Value Problems with Oscillating Solutions. Appl. Num. Math. 58(3), 341–351 (2008)

    Article  Google Scholar 

  82. G. Vanden Berghe, M. Van Daele, Exponentially-fitted Obrechkoff Methods for second-order differential equations. Appl. Num. Math. 59(3–4), 815–829 (2009)

    Article  Google Scholar 

  83. D. Hollevoet, M. Van Daele, G. Vanden Berghe, the Optimal Exponentially-Fitted Numerov Method for Solving Two-Point Boundary Value Problems. J. Math. Chem. 230(1), 260–269 (2009)

    Article  Google Scholar 

  84. J.M. Franco, L. Rández, Explicit exponentially fitted two-Step hybrid Methods of high order for second-order oscillatory IVPs. Appl. Math. Comput. 273, 493–505 (2016)

    Article  Google Scholar 

  85. J.M. Franco, Y. Khiar, L. Rández, Two new embedded pairs of explicit Runge-Kutta Methods adapted to the numerical solution of oscillatory problems. Appl. Math. Comput. 252, 45–57 (2015)

    Article  Google Scholar 

  86. J.M. Franco, I. Gomez, L. Rández, Optimization of explicit two-Step hybrid Methods for solving orbital and oscillatory problems. Comput. Phys. Commun.185(10), 2527–2537 (2014)

    Article  CAS  Google Scholar 

  87. J.M. Franco, I. Gomez, Trigonometrically fitted nonlinear two-Step Methods for solving second order oscillatory IVPs. Appl. Math. Comput. 232, 643–657 (2014)

    Article  Google Scholar 

  88. J.M. Franco, I. Gomez, Symplectic explicit Methods of Runge-Kutta-Nyström type for solving perturbed oscillators. J. Comput. Appl. Math. 260, 482–493 (2014)

    Article  Google Scholar 

  89. J.M. Franco, I. Gomez, Some procedures for the construction of high-order exponentially fitted Runge-Kutta-Nyström Methods of explicit type. Comput. Phys. Commun. 184(4), 1310–1321 (2013)

    Article  CAS  Google Scholar 

  90. M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, On some new low storage implementations of time advancing Runge-Kutta Methods. J. Comput. Appl. Math. 236(15), 3665–3675 (2012)

    Article  Google Scholar 

  91. M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, Symmetric and symplectic exponentially fitted Runge-Kutta Methods of high order. Comput. Phys. Commun. 181(12), 2044–2056 (2010)

    Article  CAS  Google Scholar 

  92. M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, On high order symmetric and symplectic trigonometrically fitted Runge-Kutta Methods with an even number of stages. BIT Num. Math. 50(1), 3–21 (2010)

    Article  Google Scholar 

  93. J.M. Franco, I. Gomez, Accuracy and linear Stability of RKN Methods for solving second-order stiff problems. Appl. Num. Math. 59(5), 959–975 (2009)

    Article  Google Scholar 

  94. M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, Sixth-order symmetric and symplectic exponentially fitted Runge-Kutta Methods of the Gauss type. J. Comput. Appl. Math. 223(1), 387–398 (2009)

    Article  Google Scholar 

  95. M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, Structure preservation of exponentially fitted Runge-Kutta Methods. J. Comput. Appl. Math. 218(2), 421–434 (2008)

    Article  Google Scholar 

  96. M. Calvo, J.M. Franco, J.I. Montijano, L. Rández, Sixth-order symmetric and symplectic exponentially fitted modified Runge-Kutta Methods of Gauss type. Comput. Phys. Commun.178(10), 732–744 (2008)

    Article  CAS  Google Scholar 

  97. J.M. Franco, Exponentially fitted symplectic integrators of RKN type for solving oscillatory problems. Comput. Phys. Commun. 177(6), 479–492 (2007)

    Article  CAS  Google Scholar 

  98. J.M. Franco, New Methods for oscillatory systems based on ARKN Methods. Appl. Num. Math. 56(8), 1040–1053 (2006)

    Article  Google Scholar 

  99. J.M. Franco, Runge-Kutta-Nyström Methods adapted to the numerical integration of perturbed oscillators. Comput. Phys. Commun. 147, 770–787 (2002)

    Article  CAS  Google Scholar 

  100. J.M. Franco, Stability of explicit ARKN Methods for perturbed oscillators. J. Comput. Appl. Math. 173, 389–396 (2005)

    Article  Google Scholar 

  101. X.Y. Wu, X. You, J.Y. Li, Note on derivation of order conditions for ARKN Methods for perturbed oscillators. Comput. Phys. Commun. 180, 1545–1549 (2009)

    Article  CAS  Google Scholar 

  102. A. Tocino, J. Vigo-Aguiar, Symplectic conditions for exponential fitting Runge-Kutta-Nyström Methods. Math. Comput. Modell.42, 873–876 (2005)

    Article  Google Scholar 

  103. L. Brugnano, F. Iavernaro, D. Trigiante, Hamiltonian boundary value Methods (energy preserving discrete line integral Methods). JNAIAM J, Numer. Anal. Ind. Appl. Math. 5, 17–37 (2010)

    Google Scholar 

  104. F. Iavernaro, D. Trigiante, High-order symmetric schemes for the energy conservation of polynomial Hamiltonian problems. JNAIAM J, Numer. Anal. Ind. Appl. Math. 4, 87–101 (2009)

    Google Scholar 

  105. A. Konguetsof, A generator of families of two-Step numerical Methods with free parameters and minimal phase-lag. J. Math. Chem. 55(9), 1808–1832 (2017)

    Article  CAS  Google Scholar 

  106. A. Konguetsof, A hybrid Method with phase-lag and derivatives equal to zero for the numerical integration of the Schrödinger equation. J. Math. Chem. 49(7), 1330–1356 (2011)

    Article  CAS  Google Scholar 

  107. Hans Van de Vyver, A phase-fitted and amplification-fitted explicit two-Step hybrid Method for second-order periodic initial value problems. Int.J. Modern Phys. C 17(5), 663–675 (2006)

    Article  Google Scholar 

  108. Hans Van de Vyver, An explicit Numerov-type Method for second-order differential equations with oscillating solutions. Comput. Math. Appl. 53(9), 1339–1348 (2007)

    Article  Google Scholar 

  109. Yonglei Fang, Wu. Xinyuan, A trigonometrically fitted explicit hybrid Method for the numerical integration of orbital problems. Appl. Math. Comput. 189(1), 178–185 (2007)

    Article  Google Scholar 

  110. Beny Neta, P-stable high-order super-implicit and Obrechkoff Methods for periodic initial value problems. Comput. Math. Appl. 54(1), 117–126 (2007)

    Article  Google Scholar 

  111. Hans Van de Vyver, Phase-fitted and amplification-fitted two-Step hybrid Methods for y ’ ’ = f (x, y). J. Comput. Appl. Math. 209(1), 33–53 (2007)

    Article  Google Scholar 

  112. Hans Van de Vyver, Efficient one-Step Methods for the Schrödinger equation. MATCH-Commun. Math. Comput. Chem. 60(3), 711–732 (2008)

    Google Scholar 

  113. J. Martín-Vaquero, J. Vigo-Aguiar, Exponential fitted Gauss, Radau and Lobatto Methods of low order. Num. Algorith. 48(4), 327–346 (2008)

    Article  Google Scholar 

  114. A. Konguetsof, A new two-Step hybrid Method for the numerical solution of the Schrödinger equation. J. Math. Chem. 47(2), 871–890 (2010)

    Article  CAS  Google Scholar 

  115. A. Fatheah, Hendi, P-Stable Higher Derivative Methods with Minimal Phase-Lag for Solving Second Order Differential Equations. J. Appl. Math. 2011, 407151 (2011)

    Google Scholar 

  116. Hans Van de Vyver, Comparison of some special optimized fourth-order Runge-Kutta Methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 166(2), 109–122 (2005)

    Article  CAS  Google Scholar 

  117. Zhongcheng Wang, Deying Zhao, Yongming Dai, Wu. Dongmei, An improved trigonometrically fitted P-stable Obrechkoff Method for periodic initial-value problems. Proc. Royal Soc. A-Math. Phys. Eng. Sci. 461(2058), 1639–1658 (2005)

    Article  Google Scholar 

  118. M. Van Daele, G. Vanden Berghe, H. De Meyer, Properties and Implementation of R-Adams Methods Based On Mixed-Type Interpolation. Comput. Math. Appl. 30(10), 37–54 (1995)

    Article  Google Scholar 

  119. J. Vigo-Aguiar, L.M. Quintales, A parallel ODE solver adapted to oscillatory problems. J. Super Comput. 19(2), 163–171 (2001)

    Article  Google Scholar 

  120. Zhongcheng Wang, Trigonometrically-fitted Method with the Fourier frequency spectrum for undamped Duffing equation. Comput. Phys. Commun. 174(2), 109–118 (2006)

    Article  CAS  Google Scholar 

  121. Zhongcheng Wang, Trigonometrically-fitted Method for a periodic initial value problem with two frequencies. Comput. Phys. Commun. 175(4), 241–249 (2006)

    Article  CAS  Google Scholar 

  122. J. Vigo-Aguiar, Josè M. Ferrandiz, A general procedure for the adaptation of multistep Algorithms to the integration of oscillatory problems. SIAM J. Num. Anal. 35(4), 1684–1708 (1998)

    Article  Google Scholar 

  123. J. Vigo-Aguiar, H. Ramos, Dissipative Chebyshev exponential-fitted Methods for numerical solution of second-order differential equations. J. Comput. Appl. Math. 158(1), 187–211 (2003)

    Article  Google Scholar 

  124. J. Vigo-Aguiar, S. Natesan, A parallel boundary value technique for singularly perturbed two-point boundary value problems. J. Super Comput. 27(2), 195–206 (2004)

    Article  Google Scholar 

  125. Chen Tang, Haiqing Yan, Hao Zhang, WenRun Li, The various order explicit multistep exponential fitting for systems of ordinary differential equations. J. Comput. Appl.Math. 169(1), 171–182 (2004)

    Article  Google Scholar 

  126. Chen Tang, Haiqing Yan, Hao Zhang, Zhanqing Chen, Ming Liu, Guimin Zhang, The arbitrary order implicit multistep schemes of exponential fitting and their applications. J. Comput. Appl. Math. 173(1), 155–168 (2005)

    Article  Google Scholar 

  127. Hans Van de Vyver, Frequency evaluation for exponentially fitted Runge-Kutta Methods. J. Comput. Appl. Math. 184(2), 442–463 (2005)

    Article  Google Scholar 

  128. J.P. Coleman, L. Gr, Ixaru, Truncation Errors in exponential fitting for oscillatory problems. SIAM J. Num. Anal. 44(4), 1441–1465 (2006)

    Article  Google Scholar 

  129. J. Martín-Vaquero, J. Vigo-Aguiar, Adapted BDF Algorithms: Higher-order Methods and their Stability. J. Sci. Comput. 32(2), 287–313 (2007)

    Article  Google Scholar 

  130. J. Vigo-Aguiar, J. Martín-Vaquero, H. Ramos, Exponential fitting BDF-Runge-Kutta Algorithms. Comput. Phys. Commun. 178(1), 15–34 (2008)

    Article  CAS  Google Scholar 

  131. Beatrice Paternoster, Present state-of-the-art in exponential fitting. A contribution dedicated to Liviu Ixaru on his 70th birthday. Comput. Phys. Commun. 183(12), 2499–2512 (2012)

    Article  CAS  Google Scholar 

  132. Zhongcheng Wang, Obrechkoff one-Step Method fitted with Fourier spectrum for undamped Duffing equation. Comput. Phys. Commun. 175(11–12), 692–699 (2006)

    Article  CAS  Google Scholar 

  133. Chunfeng Wang, Zhongcheng Wang, A P-stable eighteenth-order six-Step Method for periodic initial value problems. Int. J. Modern Phys. C 18(3), 419–431 (2007)

    Article  Google Scholar 

  134. Jiaqi Chen, Zhongcheng Wang, Hezhu Shao, Hailing Hao, Highly-accurate ground state energies of the He atom and the He-like ions by Hartree SCF calculation with Obrechkoff Method. Comput. Phys. Communi. 179(7), 486–491 (2008)

    Article  CAS  Google Scholar 

  135. Hezhu Shao, Zhongcheng Wang, Arbitrarily precise numerical solutions of the one-dimensional Schrödinger equation. Comput. Phys. Commun. 180(1), 1–7 (2009)

    Article  CAS  Google Scholar 

  136. Hezhu Shao, Zhongcheng Wang, Numerical solutions of the time-dependent Schrödinger equation: Reduction of the Error due to space discretization. Phys. Rev. E 79(5), 056705 (2009)

    Article  CAS  Google Scholar 

  137. Zhongcheng Wang, Hezhu Shao, A new kind of discretization scheme for solving a two-dimensional time-independent Schrödinger equation. Comput. Phys. Commun. 180(6), 842–849 (2009)

    Article  CAS  Google Scholar 

  138. T.E. Simos, Exponentially fitted Runge-Kutta Methods for the numerical solution of the Schrödinger equation and related problems. Comput. Mater. Sci. 18, 315–332 (2000)

    Article  Google Scholar 

  139. C.J. Cramer, Essentials of Computational Chemistry (Wiley, Chichester, 2004)

    Google Scholar 

  140. F. Jensen, Introduction to Computational Chemistry (Wiley, Chichester, 2007)

    Google Scholar 

  141. A.R. Leach, Molecular Modelling - Principles and Applications (Pearson, Essex, 2001)

    Google Scholar 

  142. P. Atkins, R. Friedman, Molecular Quantum Mechanics (Oxford Univ. Press, Oxford, 2011).

    Google Scholar 

  143. V.N. Kovalnogov, R.V. Fedorov, D.A. Generalov, Y.A. Khakhalev, A.N. Zolotov, Numerical research of turbulent boundary layer based on the fractal dimension of pressure fluctuations. AIP Conference Proceedings 738, 480004 (2016)

    Article  Google Scholar 

  144. V.N. Kovalnogov, R.V. Fedorov, T.V. Karpukhina, E.V. Tsvetova, Numerical analysis of the temperature stratification of the disperse flow. AIP Conference Proceedings 1648, 850033 (2015)

    Article  Google Scholar 

  145. N. Kovalnogov, E. Nadyseva, O. Shakhov, V. Kovalnogov, Control of turbulent transfer in the boundary layer through applied periodic effects. Izvestiya Vysshikh Uchebnykh Zavedenii Aviatsionaya Tekhnika (1), 49–53 (1998)

  146. V.N. Kovalnogov, R.V. Fedorov, D.A. Generalov, Modeling and development of cooling technology of turbine engine blades. Int. Rev. Mech. Eng. 9(4), 331–335 (2015)

    Google Scholar 

  147. Stefan Kottwitz, LaTeX Cookbook, Pages 231-236, Packt Publishing Ltd., Birmingham B3 2PB, UK (2015)

Download references

Author information

Authors and Affiliations

  1. Zhejiang Fashion Institute of Technology, 495 Fenghua Road, Zhejiang, 315211, Ningbo, China

    Xingyuan Li

  2. Huzhou University, Huzhou, 313000, China

    Chia-Liang Lin

  3. South Ural State University, 76, Lenin Aven., Chelyabinsk, 454080, Russia

    T. E. Simos

  4. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

    T. E. Simos

  5. Data Recovery Key Laboratory of Sichun Province, Neijing Normal Univ., Neijiang, China

    T. E. Simos

  6. 10 Konitsis Street, Amphithea - Paleon Phaleron, GR-175 64, Athens, Greece

    T. E. Simos

Authors
  1. Xingyuan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chia-Liang Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. E. Simos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. E. Simos.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest or other ethical conflicts concerning this paper

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

T.E. Simos: Highly Cited Researcher (2001-2013 List, 2017 List and 2018 List), Active Member of the European Academy of Sciences and Arts. Active Member of the European Academy of Sciences. Corresponding Member of European Academy of Arts, Sciences and Humanities.

Appendix

Appendix

$$\begin{aligned}LTE_{CL}= & {} LTE_{NM142S4SPD1} = LTE_{NM142S4SPD2} \approx \\\approx & {} h^{16} \, \ell _{0} = h^{16} \, \left[ {\frac{25387\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) \upsilon \left( x \right) \, EXPR_{1}}{128296396800}}\right. \\&+{\frac{2491\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) \upsilon \left( x \right) \left( {\frac{\mathrm{d}^{5}}{\mathrm{d}{x}^{5}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) }{9542707200}}\\&+ {\frac{16271\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) \upsilon \left( x \right) \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) ^{2}{\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) }{15395567616}}\\&+ {\frac{169441\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) }{144333446400}}\\&+ {\frac{371\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) {\frac{\mathrm{d}^{10}}{\mathrm{d}{x}^{10}}}PTN \left( x \right) }{104969779200}} \end{aligned}$$
$$\begin{aligned}&+ {\frac{1007\, \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) {\frac{\mathrm{d}^{9}}{\mathrm{d}{x}^{9}}}PTN \left( x \right) }{88820582400}}\\&+ {\frac{53\, \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) \upsilon \left( x \right) {\frac{\mathrm{d}^{10}}{\mathrm{d}{x}^{10}}}PTN \left( x \right) }{29606860800}}\\&+ {\frac{53\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) \upsilon \left( x \right) {\frac{\mathrm{d}^{11}}{\mathrm{d}{x}^{11}}}PTN \left( x \right) }{85530931200}}\\&+ {\frac{39167\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{4}\upsilon \left( x \right) {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) }{288666892800}}\\&+ {\frac{522739\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{2}\upsilon \left( x \right) \left( {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) \right) ^{2}}{1154667571200}}\\&+ {\frac{8533\, \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) \upsilon \left( x \right) \left( {\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) \right) ^{2}}{44410291200}}\\&+ {\frac{4823\, \left( {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) \right) ^{2}\upsilon \left( x \right) {\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) }{22205145600}}\\&+ {\frac{624181\, \left( PTN \left( x \right) \right) ^{2}\upsilon \left( x \right) \left( {\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) \right) ^{2}}{4041336499200}}\\&+ {\frac{53\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) \right) ^{2}}{159045120}} \end{aligned}$$
$$\begin{aligned} &+ {\frac{1641569\, \left( PTN \left( x \right) \right) ^{2}\upsilon \left( x \right) \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) ^{3}}{8082672998400}}\\&+ {\frac{23479\, \left( PTN \left( x \right) \right) ^{3}\upsilon \left( x \right) {\frac{\mathrm{d}^{8}}{\mathrm{d}{x}^{8}}}PTN \left( x \right) }{4041336499200}}\\&+ {\frac{1219\,PTN \left( x \right) \upsilon \left( x \right) {\frac{\mathrm{d}^{12}}{\mathrm{d}{x}^{12}}}PTN \left( x \right) }{8082672998400}}\\&+ {\frac{48707\, \left( PTN \left( x \right) \right) ^{3}\upsilon \left( x \right) \left( {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) \right) ^{2}}{505167062400}}\\&+ {\frac{4717\, \left( PTN \left( x \right) \right) ^{2}\upsilon \left( x \right) {\frac{\mathrm{d}^{10}}{\mathrm{d}{x}^{10}}}PTN \left( x \right) }{2694224332800}}\\&+ {\frac{16589\, \left( PTN \left( x \right) \right) ^{4}\upsilon \left( x \right) \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) ^{2}}{769778380800}}\\&+ {\frac{53\, \left( PTN \left( x \right) \right) ^{6}\upsilon \left( x \right) {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) }{128296396800}}\\&+ {\frac{110399\, \left( PTN \left( x \right) \right) ^{4}\upsilon \left( x \right) {\frac{\mathrm{d}^{6}}{\mathrm{d}{x}^{6}}}PTN \left( x \right) }{16165345996800}}\\&+ {\frac{53\, \left( PTN \left( x \right) \right) ^{5}\upsilon \left( x \right) {\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) }{18041680800}}\\&+ {\frac{2491\,PTN \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) {\frac{\mathrm{d}^{11}}{\mathrm{d}{x}^{11}}}PTN \left( x \right) }{4041336499200}}\\&+ {\frac{901\, \left( PTN \left( x \right) \right) ^{3} \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) {\frac{\mathrm{d}^{7}}{\mathrm{d}{x}^{7}}}PTN \left( x \right) }{126291765600}}\\&+ {\frac{2334809\,PTN \left( x \right) \upsilon \left( x \right) \left( {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) \right) \, EXPR_{2}}{2020668249600}}\\&+ {\frac{53\, \left( {\frac{\mathrm{d}^{14}}{\mathrm{d}{x}^{14}}}PTN \left( x \right) \right) \upsilon \left( x \right) }{32330691993600}}\\&+ {\frac{53\, \left( {\frac{\mathrm{d}^{13}}{\mathrm{d}{x}^{13}}}PTN \left( x \right) \right) {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) }{2309335142400}}\\&+ {\frac{53\, \left( PTN \left( x \right) \right) ^{8}\upsilon \left( x \right) }{32330691993600}}\end{aligned}$$
$$\begin{aligned}&+ {\frac{53\, \left( PTN \left( x \right) \right) ^{6} \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) }{577333785600}}\\&+ {\frac{371\, \left( PTN \left( x \right) \right) ^{3} \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{3}}{28866689280}}\\&+ {\frac{1007\, \left( PTN \left( x \right) \right) ^{5}\upsilon \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{2}}{577333785600}}\\&+ {\frac{2173\, \left( PTN \left( x \right) \right) ^{2}\upsilon \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{4}}{57733378560}}\\&+ {\frac{53\, \left( {\frac{\mathrm{d}^{5}}{\mathrm{d}{x}^{5}}}PTN \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) {\frac{\mathrm{d}^{6}}{\mathrm{d}{x}^{6}}}PTN \left( x \right) }{1009324800}}\\&+ {\frac{2491\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{2}\upsilon \left( x \right) {\frac{\mathrm{d}^{8}}{\mathrm{d}{x}^{8}}}PTN \left( x \right) }{104969779200}}\\&+ {\frac{371\, \left( {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) {\frac{\mathrm{d}^{8}}{\mathrm{d}{x}^{8}}}PTN \left( x \right) }{14803430400}}\\&+ {\frac{689\, \left( {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) \right) \upsilon \left( x \right) {\frac{\mathrm{d}^{9}}{\mathrm{d}{x}^{9}}}PTN \left( x \right) }{177641164800}}\\&+ {\frac{53\, \left( {\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) \right) \upsilon \left( x \right) {\frac{\mathrm{d}^{8}}{\mathrm{d}{x}^{8}}}PTN \left( x \right) }{8074598400}}\\&+ {\frac{1007\, \left( {\frac{\mathrm{d}^{5}}{\mathrm{d}{x}^{5}}}PTN \left( x \right) \right) \upsilon \left( x \right) {\frac{\mathrm{d}^{7}}{\mathrm{d}{x}^{7}}}PTN \left( x \right) }{113044377600}}\\&+ {\frac{7897\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{3}\upsilon \left( x \right) {\frac{\mathrm{d}^{5}}{\mathrm{d}{x}^{5}}}PTN \left( x \right) }{52484889600}}\\&+ {\frac{265\, \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) ^{2}\upsilon \left( x \right) {\frac{\mathrm{d}^{6}}{\mathrm{d}{x}^{6}}}PTN \left( x \right) }{2368548864}}\\&+ {\frac{53\, \left( {\frac{\mathrm{d}^{6}}{\mathrm{d}{x}^{6}}}PTN \left( x \right) \right) ^{2}\upsilon \left( x \right) }{10766131200}}\\&+ {\frac{53\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{5}{\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) }{2624244480}} \end{aligned}$$
$$\begin{aligned}&+ {\frac{371\, \left( {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) \right) ^{3}{\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) }{1850428800}}\\&+ {\frac{53\, \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) ^{4}\upsilon \left( x \right) }{526344192}}\\&+ {\frac{31747\, \left( PTN \left( x \right) \right) ^{3}\upsilon \left( x \right) \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) }{202066824960}}\\&+ {\frac{3551\, \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) {\frac{\mathrm{d}^{5}}{\mathrm{d}{x}^{5}}}PTN \left( x \right) }{9868953600}}\\&+ {\frac{35351\, \left( PTN \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \, EXPR_{3}}{192444595200}}\\&+ {\frac{53\, \left( {\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) {\frac{\mathrm{d}^{7}}{\mathrm{d}{x}^{7}}}PTN \left( x \right) }{1284595200}}\\&+ {\frac{265\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{3} \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) {\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) }{1049697792}}\\&+ {\frac{26977\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) {\frac{\mathrm{d}^{7}}{\mathrm{d}{x}^{7}}}PTN \left( x \right) }{367394227200}}\\&+ {\frac{168169\, \left( PTN \left( x \right) \right) ^{3}\upsilon \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{5}}{\mathrm{d}{x}^{5}}}PTN \left( x \right) }{2020668249600}}\\&+ {\frac{54007\,PTN \left( x \right) \upsilon \left( x \right) \left( {\frac{\mathrm{d}^{5}}{\mathrm{d}{x}^{5}}}PTN \left( x \right) \right) ^{2}}{734788454400}}\\&+ {\frac{1961\, \left( PTN \left( x \right) \right) ^{4} \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) {\frac{\mathrm{d}^{5}}{\mathrm{d}{x}^{5}}}PTN \left( x \right) }{384889190400}}\\&+ {\frac{53\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) ^{3}}{106913664}}\\&+ {\frac{1643\, \left( PTN \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) {\frac{\mathrm{d}^{9}}{\mathrm{d}{x}^{9}}}PTN \left( x \right) }{449037388800}} \end{aligned}$$
$$\begin{aligned}&+ {\frac{371\, \left( PTN \left( x \right) \right) ^{5} \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) }{288666892800}}\\&+ {\frac{53\, \left( PTN \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) }{198806400}}\\&+ {\frac{5671\,PTN \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) \right) ^{2}{\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) }{9020840400}}\\&+ {\frac{8851\,PTN \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{2}{\frac{\mathrm{d}^{5}}{\mathrm{d}{x}^{5}}}PTN \left( x \right) }{32074099200}}\\&+ {\frac{61427\,PTN \left( x \right) \upsilon \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{2}{\frac{\mathrm{d}^{6}}{\mathrm{d}{x}^{6}}}PTN \left( x \right) }{310872038400}}\\&+ {\frac{2809\, \left( PTN \left( x \right) \right) ^{3} \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) }{28866689280}}\\&+ {\frac{2173\, \left( PTN \left( x \right) \right) ^{3} \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) }{36083361600}}\\&+ {\frac{11183\, \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \, EXPR_{4}}{11102572800}}\\&+ {\frac{67363\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \, EXPR_{5}}{192444595200}}\\&+ {\frac{622697\, \left( PTN \left( x \right) \right) ^{2}\upsilon \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{2}{\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) }{2020668249600}}\\&+ {\frac{793781\,PTN \left( x \right) \upsilon \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{3}{\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) }{2020668249600}} \end{aligned}$$
$$\begin{aligned} &+ {\frac{63017\, \left( PTN \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{2}{\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) }{288666892800}}\\&+ {\frac{11819\,PTN \left( x \right) \upsilon \left( x \right) \left( {\frac{\mathrm{d}^{6}}{\mathrm{d}{x}^{6}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) }{91848556800}}\\&+ {\frac{20087\,PTN \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{3}{\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) }{72166723200}}\\&+ {\frac{2491\,PTN \left( x \right) \upsilon \left( x \right) \left( {\frac{\mathrm{d}^{9}}{\mathrm{d}{x}^{9}}}PTN \left( x \right) \right) {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) }{168389020800}}\\&+ {\frac{170713\,PTN \left( x \right) \upsilon \left( x \right) \left( {\frac{\mathrm{d}^{8}}{\mathrm{d}{x}^{8}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) }{4041336499200}}\\&+ {\frac{346037\,PTN \left( x \right) \upsilon \left( x \right) \left( {\frac{\mathrm{d}^{7}}{\mathrm{d}{x}^{7}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) }{4041336499200}}\\&+ {\frac{26129\,PTN \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}^{6}}{\mathrm{d}{x}^{6}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) }{144333446400}}\\&+ {\frac{583\,PTN \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}^{5}}{\mathrm{d}{x}^{5}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) }{2385676800}}\\&+ {\frac{65243\,PTN \left( x \right) \upsilon \left( x \right) \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) ^{2}{\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) }{91848556800}}\\&+ {\frac{23797\,PTN \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}^{8}}{\mathrm{d}{x}^{8}}}PTN \left( x \right) \right) {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) }{673556083200}}\\&+ {\frac{97997\,PTN \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}^{7}}{\mathrm{d}{x}^{7}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) }{1010334124800}}\\&+ {\frac{14893\, \left( PTN \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) ^{2}}{52484889600}}\end{aligned}$$
$$\begin{aligned}&+ {\frac{33443\, \left( PTN \left( x \right) \right) ^{3}\upsilon \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{2}{\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) }{288666892800}}\\&+ {\frac{371\, \left( PTN \left( x \right) \right) ^{4} \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) }{28866689280}}\\&+ {\frac{38213\, \left( PTN \left( x \right) \right) ^{4}\upsilon \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) }{1154667571200}}\\&+ {\frac{176543\, \left( PTN \left( x \right) \right) ^{2}\upsilon \left( x \right) \left( {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{5}}{\mathrm{d}{x}^{5}}}PTN \left( x \right) }{673556083200}}\\&+ {\frac{1277989\, \left( PTN \left( x \right) \right) ^{2}\upsilon \left( x \right) \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{6}}{\mathrm{d}{x}^{6}}}PTN \left( x \right) }{8082672998400}}\\&+ {\frac{87821\, \left( PTN \left( x \right) \right) ^{2}\upsilon \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{7}}{\mathrm{d}{x}^{7}}}PTN \left( x \right) }{1347112166400}}\\&+ {\frac{232511\,PTN \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) ^{2}{\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) }{288666892800}}\\&+ {\frac{6307\,PTN \left( x \right) \upsilon \left( x \right) \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) \left( {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) \right) ^{2}}{7401715200}}\\&+ {\frac{270883\,PTN \left( x \right) \upsilon \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) ^{2}}{367394227200}} \end{aligned}$$
$$\begin{aligned} &+ {\frac{5353\, \left( PTN \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{6}}{\mathrm{d}{x}^{6}}}PTN \left( x \right) }{64148198400}}\\&+ {\frac{298867\, \left( PTN \left( x \right) \right) ^{2}\upsilon \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) \, EXPR_{6}}{310872038400}}\\&+ {\frac{438787\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) ^{2}\upsilon \left( x \right) \left( {\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) }{577333785600}}\\&+ {\frac{65773\,PTN \left( x \right) \upsilon \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) \, EXPR_{7}}{80826729984}}\\&+ {\frac{29839\, \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) \upsilon \left( x \right) \left( {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{5}}{\mathrm{d}{x}^{5}}}PTN \left( x \right) }{88820582400}}\\&+ {\frac{148771\,PTN \left( x \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) \, EXPR_{8}}{144333446400}}\\&+ {\frac{164353\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) \left( {\frac{\mathrm{d}}{\mathrm{d}x}}\upsilon \left( x \right) \right) \left( {\frac{\mathrm{d}^{5}}{\mathrm{d}{x}^{5}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) }{288666892800}}\\&+ \left. {\frac{12137\, \left( {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) \right) \upsilon \left( x \right) \left( {\frac{\mathrm{d}^{7}}{\mathrm{d}{x}^{7}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) }{107768973312}} \right] \end{aligned}$$

where

$$\begin{aligned} \upsilon \left( x \right)= & {} \upsilon _{n}, \\ EXPR_{1}= & {} \left( {\frac{\mathrm{d}^{6}}{\mathrm{d}{x}^{6}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) , \\ EXPR_{2}= & {} \left( {\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) \right) {\frac{\mathrm{d}}{\mathrm{d}x}}PTN \left( x \right) , \\ EXPR_{3}= & {} \left( {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{5}}{\mathrm{d}{x}^{5}}}PTN \left( x \right) , \\ EXPR_{4}= & {} \left( {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) , \\ EXPR_{5}= & {} \left( {\frac{\mathrm{d}^{6}}{\mathrm{d}{x}^{6}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) , \\ EXPR_{6}= & {} \left( {\frac{\mathrm{d}^{3}}{\mathrm{d}{x}^{3}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) , \\ EXPR_{7}= & {} \left( {\frac{\mathrm{d}^{5}}{\mathrm{d}{x}^{5}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) , \\ EXPR_{8}= & {} \left( {\frac{\mathrm{d}^{4}}{\mathrm{d}{x}^{4}}}PTN \left( x \right) \right) {\frac{\mathrm{d}^{2}}{\mathrm{d}{x}^{2}}}PTN \left( x \right) \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, X., Lin, CL. & Simos, T.E. A new method with improved phase-lag and stability properties for problems in quantum chemistry - an economical case. J Math Chem 59, 1571–1602 (2021). https://doi.org/10.1007/s10910-021-01245-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-021-01245-3

Keywords

Mathematics Subject Classification

Search

Navigation