# A new four-stages two-step phase fitted scheme for problems in quantum chemistry

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## Abstract

In this paper and for the first time in this research area we formulate a new multistage multistep full in phase method with meliorated properties. A theoretical, computational and numerical contemplation is also presented. The sufficiency of the new scheme is tried on using systems of coupled differential equations which represent quantum chemistry problems.

## Keywords

Phase-lag Derivative of the phase-lag Initial value problems Oscillating solution Symmetric Hybrid Multistep Schrödinger equation## Mathematics Subject Classification

65L05## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## Supplementary material

10910_2019_1018_MOESM1_ESM.pdf (78 kb)

## References

- 1.A.C. Allison, The numerical solution of coupled differential equations arising from the Schrödinger equation. J. Comput. Phys.
**6**, 378–391 (1970)Google Scholar - 2.K. 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)Google Scholar - 3.M. Liang, T.E. Simos, M. Liang, T.E. Simos, A new four stages symmetric two-step method with vanished phase-lag and its first derivative for the numerical integration of the Schrödinger equation. J. Math. Chem.
**54**(5), 1187–1211 (2016)Google Scholar - 4.X. Xi, T.E. Simos, A new high algebraic order four stages symmetric two-step method with vanished phase-lag and its first and second derivatives for the numerical solution of the Schrödinger equation and related problems. J. Math. Chem.
**54**(7), 1417–1439 (2016)Google Scholar - 5.F. Hui, T.E. Simos, Hybrid high algebraic order two-step method with vanished phase-lag and its first and second derivatives. MATCH Commun. Math. Comput. Chem.
**73**, 619–648 (2015)Google Scholar - 6.Z. 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)Google Scholar - 7.F. Hui, T.E. Simos, Four stages symmetric two-step P-Stable method with vanished phase-lag and its first, second, third and fourth derivatives. Appl. Comput. Math.
**15**(2), 220–238 (2016)Google Scholar - 8.W. Zhang, T.E. Simos, A high-order two-step phase-fitted method for the numerical solution of the Schrödinger equation. Mediterr. J. Math.
**13**(6), 5177–5194 (2016)Google Scholar - 9.L. Zhang, T.E. Simos, An efficient numerical method for the solution of the Schrödinger equation. Adv. Math. Phys. (2016). https://doi.org/10.1155/2016/8181927
- 10.M. Dong, T.E. Simos, A new high algebraic order efficient finite difference method for the solution of the Schrödinger equation. Filom. Filom.
**31**(15), 4999–5012 (2017)Google Scholar - 11.R. LIN, T.E. Simos, A two-step method with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation. Open Phys.
**14**, 628–642 (2016)Google Scholar - 12.H. 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)Google Scholar - 13.Z. 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)Google Scholar - 14.J. Ma, T.E. Simos, An efficient and computational effective method for second order problems. J. Math. Chem.
**55**, 1649–1668 (2017)Google Scholar - 15.L. Yang, T.E. Simos, An efficient and economical high order method for the numerical approximation of the Schrödinger equation. J. Math. Chem.
**55**(9), 1755–1778 (2017)Google Scholar - 16.V.N. Kovalnogov, R.V. Fedorov, V.M. Golovanov, B.M. Kostishko, T.E. Simos, A four stages numerical pair with optimal phase and stability properties. J. Math. Chem.
**56**(1), 81–102 (2018)Google Scholar - 17.K. Yan, T.E. Simos, A finite difference pair with improved phase and stability properties. J. Math. Chem.
**56**, 170–192 (2018)Google Scholar - 18.J. Fang, C. Liu, T.E. Simos, A hybric finite difference pair with maximum phase and stability properties. J. Math. Chem.
**56**, 423–448 (2018)Google Scholar - 19.J. Yao, T.E. Simos, New finite difference pair with optimized phase and stability properties. J. Math. Chem.
**56**(2), 449–476 (2018)Google Scholar - 20.J. Zheng, C. Liu, T.E. Simos, A new two-step finite difference pair with optimal properties. J. Math. Chem.
**56**(3), 770–798 (2018)Google Scholar - 21.X. Shi, T.E. Simos, New five-stages finite difference pair with optimized phase properties. J. Math. Chem.
**56**, 982–1010 (2018)Google Scholar - 22.C. Liu, T.E. Simos, A five-stages symmetric method with improved phase properties. J. Math. Chem.
**56**, 1313–1338 (2018)Google Scholar - 23.J. Yao, T.E. Simos, New five-stages two-step method with improved characteristics. J. Math. Chem.
**56**(6), 1567–1594 (2018)Google Scholar - 24.K. Yan, T.E. Simos, New Runge-Kutta type symmetric two-step method with optimized characteristics. J. Math. Chem.
**56**(8), 2454–2484 (2018)Google Scholar - 25.Z. Chen, C. Liu, T.E. Simos, New three-stages symmetric two step method with improved properties for second order initial/boundary value problems. J. Math. Chem.
**56**(9), 2591–2616 (2018)Google Scholar - 26.R. Hao, T.E. Simos, New Runge–Kutta type symmetric two step finite difference pair with improved properties for second order initial and/or boundary value problems. J. Math. Chem.
**56**(10), 3014–3044 (2018)Google Scholar - 27.G.-H. Qiu, C. Liu, T.E. Simos, A new multistep method with optimized characteristics for initial and/or boundary value problems. J. Math. Chem.
**57**(1), 119–148 (2019)Google Scholar - 28.T. Monovasilis, Z. Kalogiratou, T.E. Simos, Trigonometrical fitting conditions for two derivative Runge–Kutta methods. Numer. Algorithms
**79**, 787–800 (2018)Google Scholar - 29.G. Wang, T.E. Simos, New multiple stages two-step complete in phase algorithm with improved characteristics for second order initial/boundary value problems. J. Math. Chem.
**57**, 494–515 (2019)Google Scholar - 30.N. Yang, T.E. Simos, New four stages multistep in phase algorithm with best possible properties for second order problems. J. Math. Chem.
**57**(3), 895–917 (2019)Google Scholar - 31.F. Hui, T.E. Simos, New multistage two-step complete in phase scheme with improved properties for quantum chemistry problems. J. Math. Chem. online first
**(in press)**Google Scholar - 32.V.N. Kovalnogov, R.V. Fedorov, D.V. Suranov, T.E. Simos, Newmultiple stages scheme with improved properties for second order problems. J. Math. Chem.
**57**(1), 232–262 (2019)Google Scholar - 33.Z. Chen, C. Liu, C.-W. Hsu, T.E. Simos, A new multistage multistep full in phase algorithm with optimized charateristis for problems in chemistry. J. Math. Chem.
**(to appear)**Google Scholar - 34.V.N. Kovalnogov, R.V. Fedorov, A.A. Bondarenko, T.E. Simos, New hybrid two-step method with optimized phase and stability characteristics. J. Math. Chem.
**56**(8), 2302–2340 (2018)Google Scholar - 35.V.N. Kovalnogov, R.V. Fedorov, T.E. Simos, New hybrid symmetric two step scheme with optimized characteristics for second order problems. J. Math. Chem.
**56**(9), 2816–2844 (2018)Google Scholar - 36.C.J. Cramer,
*Essentials of Computational Chemistry*(Wiley, Chichester, 2004)Google Scholar - 37.F. Jensen,
*Introduction to Computational Chemistry*(Wiley, Chichester, 2007)Google Scholar - 38.A.R. Leach,
*Molecular Modelling—Principles and Applications*(Pearson, Essex, 2001)Google Scholar - 39.P. Atkins, R. Friedman,
*Molecular Quantum Mechanics*(Oxford University Press, Oxford, 2011)Google Scholar - 40.V.N. Kovalnogov, T.E. Simos, V.N. Kovalnogov, I.V. Shevchuk, Perspective of mathematical modeling and research of targeted formation of disperse phase clusters in working media for the next-generation power engineering technologies. AIP Conf. Proc.
**1863**, 560099 (2017)Google Scholar - 41.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 Conf. Proc.
**738**, 480004 (2016)Google Scholar - 42.V.N. Kovalnogov, R.V. Fedorov, T.V. Karpukhina, E.V. Tsvetova, Numerical analysis of the temperature stratification of the disperse flow. AIP Conf. Proc.
**1648**, 850033 (2015)Google Scholar - 43.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)Google Scholar - 44.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 - 45.S. Kottwitz,
*LaTeX Cookbook*(Packt Publishing Ltd., Birmingham, 2015), pp. 231–236Google Scholar - 46.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)Google Scholar - 47.Z.A. Anastassi, T.E. Simos, A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems. J. Comput. Appl. Math.
**236**, 3880–3889 (2012)Google Scholar - 48.A.D. Raptis, T.E. Simos, A four-step phase-fitted method for the numerical integration of second order initial-value problem. BIT
**31**, 160–168 (1991)Google Scholar - 49.J.M. Franco, M. Palacios, High-order P-stable multistep methods. J. Comput. Appl. Math.
**30**, 1 (1990)Google Scholar - 50.J.D. Lambert,
*Numerical Methods for Ordinary Differential Systems, The Initial Value Problem*(Wiley, New York, 1991), pp. 104–107Google Scholar - 51.E. Stiefel, D.G. Bettis, Stabilization of Cowell’s method. Numer. Math.
**13**, 154–175 (1969)Google Scholar - 52.G.A. Panopoulos, Z.A. Anastassi, T.E. Simos, Two new optimized eight-step symmetric methods for the efficient solution of the Schrödinger equation and related problems. MATCH Commun. Math. Comput. Chem.
**60**(3), 773–785 (2008)Google Scholar - 53.G.A. Panopoulos, Z.A. Anastassi, T.E. Simos, Two optimized symmetric eight-step implicit methods for initial-value problems with oscillating solutions. J. Math. Chem.
**46**(2), 604–620 (2009)Google Scholar - 54.
- 55.T.E. Simos, P.S. Williams, Bessel and Neumann fitted methods for the numerical solution of the radial Schrödinger equation. Comput. Chem.
**21**, 175–179 (1977)Google Scholar - 56.T.E. Simos, J. Vigo-Aguiar, A dissipative exponentially-fitted method for the numerical solution of the Schrödinger equation and related problems. Comput. Phys. Commun.
**152**, 274–294 (2003)Google Scholar - 57.G. Psihoyios, T.E. Simos, A fourth algebraic order trigonometrically fitted predictor-corrector scheme for IVPs with oscillating solutions. J. Comput. Appl. Math.
**175**(1), 137–147 (2005)Google Scholar - 58.T. Lyche, Chebyshevian multistep methods for ordinary differential eqations. Numer. Math.
**19**, 65–75 (1972)Google Scholar - 59.R.M. Thomas, Phase properties of high order almost P-stable formulae. BIT
**24**, 225–238 (1984)Google Scholar - 60.J.D. Lambert, I.A. Watson, Symmetric multistep methods for periodic initial values problems. J. Inst. Math. Appl.
**18**, 189–202 (1976)Google Scholar - 61.A. Konguetsof, T.E. Simos, A generator of hybrid symmetric four-step methods for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math.
**158**(1), 93–106 (2003)Google Scholar - 62.Z. Kalogiratou, T. Monovasilis, T.E. Simos, Symplectic integrators for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math.
**158**(1), 83–92 (2003)Google Scholar - 63.Z. Kalogiratou, T.E. Simos, Newton–Cotes formulae for long-time integration. J. Comput. Appl. Math.
**158**(1), 75–82 (2003)Google Scholar - 64.G. Psihoyios, T.E. Simos, Trigonometrically fitted predictor–corrector methods for IVPs with oscillating solutions. J. Comput. Appl. Math.
**158**(1), 135–144 (2003)Google Scholar - 65.T.E. Simos, I.T. Famelis, C. Tsitouras, Zero dissipative, explicit Numerov-type methods for second order IVPs with oscillating solutions. Numer. Algorithms
**34**(1), 27–40 (2003)Google Scholar - 66.T.E. Simos, Dissipative trigonometrically-fitted methods for linear second-order IVPs with oscillating solution. Appl. Math. Lett.
**17**(5), 601–607 (2004)Google Scholar - 67.K. Tselios, T.E. Simos, Runge-Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics. J. Comput. Appl. Math.
**175**(1), 173–181 (2005)Google Scholar - 68.D.P. Sakas, T.E. Simos, Multiderivative methods of eighth algrebraic order with minimal phase-lag for the numerical solution of the radial Schrödinger equation. J. Comput. Appl. Math.
**175**(1), 161–172 (2005)Google Scholar - 69.G. Psihoyios, T.E. Simos, A fourth algebraic order trigonometrically fitted predictor-corrector scheme for IVPs with oscillating solutions. J. Comput. Appl. Math.
**175**(1), 137–147 (2005)Google Scholar - 70.Z.A. Anastassi, T.E. Simos, An optimized Runge–Kutta method for the solution of orbital problems. J. Comput. Appl. Math.
**175**(1), 1–9 (2005)Google Scholar - 71.T.E. Simos, Closed Newton–Cotes trigonometrically-fitted formulae of high order for long-time integration of orbital problems. Appl. Math. Lett.
**22**(10), 1616–1621 (2009)Google Scholar - 72.S. Stavroyiannis, T.E. Simos, Optimization as a function of the phase-lag order of nonlinear explicit two-step P-stable method for linear periodic IVPs. Appl. Math. Lett.
**59**(10), 2467–2474 (2009)Google Scholar - 73.T.E. Simos, Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation. Acta Appl. Math.
**110**(3), 1331–1352 (2010)Google Scholar - 74.T.E. Simos, New stable closed Newton–Cotes trigonometrically fitted formulae for long-time integration. Abstr. Appl. Anal.
**2012**, 15. https://doi.org/10.1155/2012/182536 - 75.T.E. Simos, Optimizing a hybrid two-step method for the numerical solution of the Schrödinger equation and related problems with respect to phase-lag. J. Appl. Math.
**2012**, 17 (2012). https://doi.org/10.1155/2012/420387 Google Scholar - 76.I. Alolyan, T.E. Simos, A high algebraic order multistage explicit four-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives for the numerical solution of the Schrödinger equation. J. Math. Chem.
**53**(8), 1915–1942 (2015)Google Scholar - 77.I. Alolyan, T.E. Simos, Efficient low computational cost hybrid explicit four-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical integration of the Schrödinger equation. J. Math. Chem.
**53**(8), 1808–1834 (2015)Google Scholar - 78.I. Alolyan, T.E. Simos, A high algebraic order predictor–corrector explicit method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the Schrödinger equation and related problems. J. Math. Chem.
**53**(7), 1495–1522 (2015)Google Scholar - 79.I. Alolyan, T.E. Simos, A family of explicit linear six-step methods with vanished phase-lag and its first derivative. J. Math. Chem.
**52**(8), 2087–2118 (2014)Google Scholar - 80.T.E. Simos, An explicit four-step method with vanished phase-lag and its first and second derivatives. J. Math. Chem.
**52**(3), 833–855 (2014)Google Scholar - 81.I. Alolyan, T.E. Simos, A Runge–Kutta type four-step method with vanished phase-lag and its first and second derivatives for each level for the numerical integration of the Schrödinger equation. J. Math. Chem.
**52**(3), 917–947 (2014)Google Scholar - 82.I. Alolyan, T.E. Simos, A predictor–corrector explicit four-step method with vanished phase-lag and its first, second and third derivatives for the numerical integration of the Schrödinger equation. J. Math. Chem.
**53**(2), 685–717 (2015)Google Scholar - 83.I. Alolyan, T.E. Simos, A hybrid type four-step method with vanished phase-lag and its first, second and third derivatives for each level for the numerical integration of the Schrödinger equation. J. Math. Chem.
**52**(9), 2334–2379 (2014)Google Scholar - 84.G.A. Panopoulos, T.E. Simos, A new optimized symmetric 8-step semi-embedded predictor–corrector method for the numerical solution of the radial Schrödinger equation and related orbital problems. J. Math. Chem.
**51**(7), 1914–1937 (2013)Google Scholar - 85.T.E. Simos, New high order multiderivative explicit four-step methods with vanished phase-lag and its derivatives for the approximate solution of the Schrödinger equation. Part I: construction and theoretical analysis. J. Math. Chem.
**51**(1), 194–226 (2013)Google Scholar - 86.T.E. Simos, High order closed Newton–Cotes exponentially and trigonometrically fitted formulae as multilayer symplectic integrators and their application to the radial Schrödinger equation. J. Math. Chem.
**50**(5), 1224–1261 (2012)Google Scholar - 87.D.F. Papadopoulos, T.E. Simos, A modified Runge–Kutta–Nyström method by using phase lag properties for the numerical solution of orbital problems. Appl. Math. Inf. Sci.
**7**(2), 433–437 (2013)Google Scholar - 88.T. Monovasilis, Z. Kalogiratou, T.E. Simos, Exponentially fitted symplectic Runge–Kutta–Nyström methods. Appl. Math. Inf. Sci.
**7**(1), 81–85 (2013)Google Scholar - 89.G.A. Panopoulos, T.E. Simos, An optimized symmetric 8-step semi-embedded Predictor–Corrector method for IVPs with oscillating solutions. Appl. Math. Inf. Sci.
**7**(1), 73–80 (2013)Google Scholar - 90.D.F. Papadopoulos, T.E. Simos, The use of phase lag and amplification error derivatives for the construction of a modified Runge–Kutta–Nyström method. Abstract Appl. Anal.
**2013**, 11. https://doi.org/10.1155/2013/910624 - 91.I. Alolyan, Z.A. Anastassi, T.E. Simos, A new family of symmetric linear four-step methods for the efficient integration of the Schrödinger equation and related oscillatory problems. Appl. Math. Comput
**218**(9), 5370–5382 (2012)Google Scholar - 92.I. Alolyan, T.E. Simos, A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation. Comput. Math. Appl.
**62**(10), 3756–3774 (2011)Google Scholar - 93.C. Tsitouras, I.T. Famelis, T.E. Simos, On modified Runge–Kutta trees and methods. Comput. Math. Appl.
**62**(4), 2101–2111 (2011)Google Scholar - 94.C. Tsitouras, I.T. Famelis, T.E. Simos, Phase-fitted Runge–Kutta pairs of orders 8(7). J. Comput. Appl. Math.
**321**, 226–231 (2017)Google Scholar - 95.M.A. Medvedev, T.E. Simos, C. Tsitouras, Trigonometric-fitted hybrid four-step methods of sixth order for solving \(y^{\prime \prime }(x)=f(x, y)\). Math. Methods Appl. Sci.
**42**(2), 710–716 (2019)Google Scholar - 96.Z. Kalogiratou, T. Monovasilis, T.E. Simos, New fifth order two-derivative Runge–Kutta methods with constant and frequency dependent coefficients. Math. Methods Appl. Sci. early view
**(in press)**Google Scholar - 97.M.A. Medvedev, T.E. Simos, C. Tsitouras, Hybrid, phase-fitted, four-step methods of seventh order for solving x”(t) = f (t, x). Math. Methods Appl. Sci. early view
**(in press)**Google Scholar - 98.T.E. Simos, C. Tsitouras, Evolutionary generation of high order, explicit two step methods for second order linear IVPs. Math. Methods Appl. Sci.
**40**, 6276–6284 (2017)Google Scholar - 99.T.E. Simos, C. Tsitouras, A new family of 7 stages, eighth-order explicit Numerov-type methods. Math. Methods Appl. Sci.
**40**, 7867–7878 (2017)Google Scholar - 100.D.B. Berg, T.E. Simos, C. Tsitouras, Trigonometric fitted, eighth-order explicit Numerov-type methods. Math. Methods Appl. Sci.
**41**, 1845–1854 (2018)Google Scholar - 101.T.E. Simos, C. Tsitouras, Fitted modifications of classical Runge–Kutta pairs of orders 5(4). Math. Methods Appl. Sci.
**41**(12), 4549–4559 (2018)Google Scholar - 102.C. Tsitouras, T.E. Simos, Trigonometric fitted explicit Numerov type method with vanishing phase-lag and its first and second derivatives. Mediterr. J. Math.
**15**(4), 168 (2018). https://doi.org/10.1007/s00009-018-1216-7 Google Scholar - 103.M.A. Medvedev, T.E. Simos, C. Tsitouras, Fitted modifications of Runge–Kutta pairs of orders 6(5). Math. Methods Appl. Sci.
**41**(16), 6184–6194 (2018)Google Scholar - 104.M.A. Medvedev, T.E. Simos, C. Tsitouras, Explicit, two stage, sixth order, hybrid four–step methods for solving \(y^{\prime \prime }(x)=f(x,y)\). Math. Methods Appl. Sci.
**41**(16), 6997–7006 (2018)Google Scholar - 105.T.E. Simos, C. Tsitouras, I.T. Famelis, Explicit Numerov type methods with constant coefficients: a review. Appl. Comput. Math.
**16**(2), 89–113 (2017)Google Scholar - 106.T.E. Simos, C. Tsitouras, High phase-lag order, four-step methods for solving \(y^{\prime \prime }=f(x, y)\). Appl. Comput. Math.
**17**(3), 307–316 (2018)Google Scholar - 107.A.A. Kosti, Z.A. Anastassi, T.E. Simos, Construction of an optimized explicit Runge–Kutta–Nyström method for the numerical solution of oscillatory initial value problems. Comput. Math. Appl.
**61**(11), 3381–3390 (2011)Google Scholar - 108.Z. Kalogiratou, T. Monovasilis, T.E. Simos, New modified Runge–Kutta–Nystrom methods for the numerical integration of the Schrödinger equation. Comput. Math. Appl.
**60**(6), 1639–1647 (2010)Google Scholar - 109.T. Monovasilis, Z. Kalogiratou, T.E. Simos, A family of trigonometrically fitted partitioned Runge–Kutta symplectic methods. Appl. Math. Comput.
**209**(1), 91–96 (2009)Google Scholar - 110.T. Monovasilis, Z. Kalogiratou, T.E. Simos, Construction of exponentially fitted symplectic Runge–Kutta–Nyström methods from partitioned Runge–Kutta methods. Mediterr. J. Math.
**13**(4), 2271–2285 (2016)Google Scholar - 111.T. Monovasilis, Z. Kalogiratou, H. Ramos, T.E. Simos, Modified two-step hybrid methods for the numerical integration of oscillatory problems. Math. Methods Appl. Sci.
**40**(4), 5286–5294 (2017)Google Scholar - 112.T.E. SIMOS, Multistage symmetric two-step P-stable method with vanished phase-lag and its first, second and third derivatives. Appl. Comput. Math.
**14**(3), 296–315 (2015)Google Scholar - 113.Z. Kalogiratou, T. Monovasilis, H. Ramos, T.E. Simos, A new approach on the construction of trigonometrically fitted two step hybrid methods. J. Comput. Appl. Math.
**303**, 146–155 (2016)Google Scholar - 114.H. Ramos, Z. Kalogiratou, T. Monovasilis, T.E. Simos, An optimized two-step hybrid block method for solving general second order initial-value problems. Numer. Algorithms
**72**, 1089–1102 (2016)Google Scholar - 115.T.E. Simos, High order closed Newton–Cotes trigonometrically-fitted formulae for the numerical solution of the Schrödinger equation. Appl. Math. Comput.
**209**(1), 137–151 (2009)Google Scholar - 116.A. Konguetsof, T.E. Simos, An exponentially-fitted and trigonometrically-fitted method for the numerical solution of periodic initial-value problems. Comput. Math. Appl.
**45**(1–3), 547–554 (2003)Google Scholar - 117.T.E. Simos, A new explicit hybrid four-step method with vanished phase-lag and its derivatives. J. Math. Chem.
**52**(7), 1690–1716 (2014)Google Scholar - 118.T.E. Simos, On the explicit four-step methods with vanished phase-lag and its first derivative. Appl. Math. Inf. Sci.
**8**(2), 447–458 (2014)Google Scholar - 119.G.A. Panopoulos, T.E. Simos, A new optimized symmetric embedded predictor–corrector method (EPCM) for initial-value problems with oscillatory solutions. Appl. Math. Inf. Sci.
**8**(2), 703–713 (2014)Google Scholar - 120.G.A. Panopoulos, T.E. Simos, An eight-step semi-embedded predictor-corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown. J. Comput. Appl. Math.
**290**, 1–15 (2015)Google Scholar - 121.F. 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)Google Scholar - 122.L.G. 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)Google Scholar - 123.L.G. Ixaru, M. Micu,
*Topics in Theoretical Physics*(Central Institute of Physics, Bucharest, 1978)Google Scholar - 124.L.G. 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)Google Scholar - 125.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)Google Scholar - 126.J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math.
**6**, 19–26 (1980)Google Scholar - 127.G.D. Quinlan, S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J.
**100**, 1694–1700 (1990)Google Scholar - 128.A.D. Raptis, A.C. Allison, Exponential-fitting methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun.
**14**, 1–5 (1978)Google Scholar - 129.M.M. Chawla, P.S. Rao, An Noumerov-typ 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)Google Scholar - 130.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)Google Scholar - 131.T.E. Simos, A new Numerov-type method for the numerical solution of the Schrödinger equation. J. Math. Chem.
**46**, 981–1007 (2009)Google Scholar - 132.A. Konguetsof, Two-step high order hybrid explicit method for the numerical solution of the Schrödinger equation. J. Math. Chem.
**48**, 224–252 (2010)Google Scholar - 133.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)Google Scholar - 134.R.B. Bernstein, A. Dalgarno, H. Massey, I.C. Percival, Thermal scattering of atoms by homonuclear diatomic molecules. Proc. R. Soc. Ser. A
**274**, 427–442 (1963)Google Scholar - 135.R.B. Bernstein, Quantum mechanical (phase shift) analysis of differential elastic scattering of molecular beams. J. Chem. Phys.
**33**, 795–804 (1960)Google Scholar - 136.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)Google Scholar - 137.J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formula. J. Comput. Appl. Math.
**6**, 19–26 (1980)Google Scholar

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