# High Order Split Operators for the Time-Dependent Wavepacket method of Triatomic Reactive Scattering in Hyperspherical Coordinates

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory: Coordinates System, Hamiltonian and Split Operators

#### 2.1. Theory: Mass Scaled Jacobi Coordinate and Initial WavePacket

#### 2.2. Hyperspherical Coordinate for Triatomic Reactive Scattering

#### 2.3. Split Operators

#### 2.3.1. Second Order Split Operator

#### 2.3.2. High Order Split Operator

#### 2.4. Split Operator in the APH Coordinate

## 3. Results and Discussion

#### 3.1. H + H ${}_{2}$ Reaction

#### 3.2. O + O ${}_{2}$ Reaction

#### 3.3. F + HD→ HF + D Reaction

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Kosloff, R. Time-dependent quantum-mechanical methods for molecular dynamics. J. Phys. Chem.
**1988**, 92, 2087–2100. [Google Scholar] [CrossRef] - Baumert, T.; Engel, V.; Meier, C.; Gerber, G. High laser field effects in multiphoton ionization of Na
_{2}: Experiment and quantum calculations. Chem. Phys. Lett.**1992**, 200, 488–494. [Google Scholar] [CrossRef] - Heitz, M.C.; Durand, G.; Spiegelman, F.; Meier, C. Time-resolved photoelectron spectra as probe of excited state dynamics: A full quantum study of the Na
_{2}F cluster. J. Chem. Phys.**2003**, 118, 1282. [Google Scholar] [CrossRef] - Sun, Z.; Lou, N. Autler-Townes Splitting in the Multiphoton Resonance Ionization Spectrum of Molecules Produced by Ultrashort Laser Pulses. Phys. Rev. Lett.
**2003**, 91, 023002. [Google Scholar] [CrossRef] [PubMed] - Nyman, G.; Yu, H.G. Quantum theory of bimolecular chemical reactions. Rep. Prog. Phys.
**2000**, 63, 1001–1059. [Google Scholar] [CrossRef] - Crawford, J.; Parker, G.A. State-to-state three-atom time-dependent reactive scattering in hyperspherical coordinates. J. Chem. Phys.
**2013**, 138, 054313. [Google Scholar] [CrossRef] [PubMed] - Althorpe, S.C. Quantum wavepacket method for state-to-state reactive cross sections. J. Chem. Phys.
**2001**, 114, 1601. [Google Scholar] [CrossRef] - Sun, Z.; Xu, X.; Lee, S.Y.; Zhang, D.H. A Reactant-Coordinate-Based Time-Dependent Wave Packet Method for Triatomic State-to-State Reaction Dynamics: Application to the H + O
_{2}Reaction. J. Phys. Chem. A**2009**, 113, 4145–4154. [Google Scholar] [CrossRef] [PubMed] - Pack, R.T.; Packer, G.A. Quantum reactive scattering in three dimensions using hyperspherical (APH) coordinates. Theory. J. Chem. Phys.
**1987**, 87, 3888. [Google Scholar] [CrossRef] - Zhao, H.L.; Hu, X.X.; Sun, Z.G. Quantum wavepacket method for state-to-state reactive cross sections in hyperspherical coordinates. J. Chem. Phys.
**2018**, 149, 174103. [Google Scholar] [CrossRef] - Zhao, H.L.; Umer, U.; Hu, X.X.; Xie, D.Q.; Sun, Z.G. An interaction-asymptotic region decomposition method for general state-to-state reactive scatterings. J. Chem. Phys.
**2019**, 150, 134105. [Google Scholar] [CrossRef] [PubMed] - Kosloff, D.; Kosloff, R. A Fourier method solution for the time-dependent Schrödinger equation as a tool in molecular dynamics. J. Comput. Phys.
**1983**, 52, 35–53. [Google Scholar] [CrossRef] - Kosloff, R.; Kosloff, D. Absorbing boundaries for wave propagation problems. J. Comput. Phys.
**1986**, 63, 363–376. [Google Scholar] [CrossRef] - Leforestier, C.L.; Bisseling, R.H.; Cerjan, C.; Feit, M.D.; Friesner, R.; Guldberg, A.; Hammerich, A.; Jolicard, G.; Karrlein, W.; Meyer, H.D.; et al. A comparison of different propagation schemes for the time-dependent Schrödinger equation. J. Comput. Phys.
**1991**, 94, 59–80. [Google Scholar] [CrossRef] - Fattal, E.; Kosloff, R. Phase space approach for optimizing grid representations: The mapped Fourier method. Phys. Rev. E
**1996**, 53, 1217. [Google Scholar] [CrossRef] - Kokoouline, V.; Dulieu, O.; Kosloff, R.; Masnou-Seeuws, F. Mapped Fourier methods for long-range molecules: Application to perturbations in the Rb
_{2}(0π+) photoassociation spectrum. J. Chem. Phys.**1999**, 110, 9865. [Google Scholar] [CrossRef] - Harris, D.O.; Engerholm, G.G.; Gwinn, W.D. Calculation of Matrix Elements for One-Dimensional Quantum-Mechanical Problems and the Application to Anharmonic Oscillators. J. Chem. Phys.
**1965**, 43, 1515. [Google Scholar] [CrossRef] - Light, J.C.; Carrington, T., Jr. Discrete-variable representations and their utilization. Adv. Chem. Phys.
**2000**, 114, 263–310. [Google Scholar] - Ba<i>c</i>˘i<i>c</i>´, Z.; Light, J.C. Theoretical methods for rovibrational states of floppy molecules. Annu. Rev. Phys. Chem.
**1989**, 40, 469–498. [Google Scholar] - Light, J.C.; Hamilton, I.P.; Lill, J.V. Generalized discrete variable approximation in quantum mechanics. J. Chem. Phys.
**1985**, 82, 1400. [Google Scholar] [CrossRef] - Tal-Ezer, H.; Kosloff, R. An accurate and efficient scheme for propagating the time-dependent Schrödinger equation. J. Chem. Phys.
**1984**, 81, 3967. [Google Scholar] [CrossRef] - Feit, M.D.; Fleck, J.A., Jr.; Steiger, A. Solution of the Schrödinger equation by a spectral method. J. Comput. Phys.
**1982**, 47, 412–433. [Google Scholar] [CrossRef] - Fleck, J.A., Jr.; Morris, J.R.; Feit, M.D. Time-dependent propagation of high energy laser beams through the atmosphere. Ann. Phys.
**1976**, 10, 129–160. [Google Scholar] - Gray, S.K.; Balint-Kurti, G.G. Quantum dynamics with real wave packets, including application to three-dimensional (J = 0) D + H2 —> HD + H reactive scattering. J. Chem. Phys.
**1998**, 108, 950. [Google Scholar] [CrossRef] - Chen, R.; Guo, H. The Chebyshev propagator for quantum systems. Comput. Phys. Commun.
**1999**, 119, 19–31. [Google Scholar] [CrossRef] - Sun, Z.; Lee, S.Y.; Guo, H.; Zhang, D.H. Comparison of second-order split operator and Chebyshev propagator in wave packet based state-to-state reactive scattering calculations. J. Chem. Phys.
**2009**, 130, 174102. [Google Scholar] [CrossRef] [PubMed] - Blanes, S.; Cases, F.; Oteo, J.A.; Ros, J. The Magnus expansion and some of its applications. Phys. Rep.
**2009**, 450, 151–238. [Google Scholar] [CrossRef] - Bandrauk, A.D.; Dehghanuan, E.; Lu, H. Complex integration steps in decomposition of quantum exponential evolution operators. Chem. Phys. Lett.
**2006**, 419, 346–350. [Google Scholar] [CrossRef] - Bandrauk, A.D.; Shen, H. Exponential split operator methods for solving coupled time-dependent Schrödinger equations. J. Chem. Phys.
**1993**, 99, 1185. [Google Scholar] [CrossRef] - Bandrauk, A.D.; Shen, H. High-order split-step exponential methods for solving coupled nonlinear Schrodinger equations. J. Phys. A
**1994**, 27, 7147–7155. [Google Scholar] [CrossRef] - Gray, S.K.; Manolopoulos, D.E. Symplectic integrators tailored to the time-dependent Schrödinger equation. J. Chem. Phys.
**1996**, 104, 7099. [Google Scholar] [CrossRef] - Bader, P.; Blanes, S.; Fernando, C. Solving the Schrödinger eigenvalue problem by the imaginary time propagation technique using splitting methods with complex coefficients. J. Chem. Phys.
**2013**, 139, 124117. [Google Scholar] [CrossRef] [PubMed] - Bandrauk, A.D.; Lu, H.Z. Exponential propagators (integrators) for the time-dependent Schrödinger equation. J. Theor. Comput. Chem.
**2013**, 12, 1340001. [Google Scholar] [CrossRef] - Thalhammer, M.; Caliari, M.; Neuhauser, C. High-order time-splitting Hermite and Fourier spectral methods. J. Comput. Phys.
**2009**, 228, 822–832. [Google Scholar] [CrossRef] - Antoine, X.; Bao, W.Z.; Besse, C. Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations. Comput. Phys. Commun.
**2013**, 184, 2621–2633. [Google Scholar] [CrossRef] - Alvermann, A.; Fehske, H.; Littlewood, P.B. Numerical time propagation of quantum systems in radiation fields. New J. Phys.
**2012**, 14, 105008. [Google Scholar] [CrossRef] - Alvermann, A.; Fehske, H. High-order commutator-free exponential time-propagation of driven quantum systems. J. Comput. Phys.
**2011**, 230, 5930–5956. [Google Scholar] [CrossRef] - Staruch, F.W. Any-order propagation of the nonlinear Schrödinger equation. Phys. Rev. E
**2007**, 76, 046701. [Google Scholar] [CrossRef] - Sofroniou, M.; Spaletta, G. Derivation of symmetric composition constants for symmetric integrators. Optim. Methods Softw.
**2005**, 20, 597–613. [Google Scholar] [CrossRef] - Schlier, C.; Seiter, A. High-order symplectic integration: An assessment. Comput. Phys. Commun.
**2000**, 130, 176–189. [Google Scholar] [CrossRef] - McLachlan, R.I. Error Bounds for Dynamic Responses in Forced Vibration Problem. SIAM J. Sci. Comput.
**1995**, 15, 1–15. [Google Scholar] - Omelyan, I.P.; Mrygold, I.M.; Folk, R. Construction of high-order force-gradient algorithms for integration of motion in classical and quantum systems. Phys. Rev. E
**2002**, 66, 026701. [Google Scholar] [CrossRef] [PubMed] - Sun, Z.G.; Zhang, D.H.; Yang, W.T. Higher-order split operator schemes for solving the Schrödinger equation in the time-dependent wave packet method: Applications to triatomic reactive scattering calculations. Phys. Chem. Chem. Phys.
**2012**, 14, 1827–1845. [Google Scholar] [CrossRef] [PubMed] - Li, W.T.; Zhang, D.H.; Sun, Z.G. Efficient Fourth-Order Split Operator for Solving the Triatomic Reactive Schrödinger Equation in the Time-Dependent Wavepacket Approach. J. Phys. Chem. A
**2014**, 14, 9801–9810. [Google Scholar] [CrossRef] [PubMed] - Smith, F.T. A Symmetric Representation for Three-Body Problems. I. Motion in a Plane. J. Math. Phys.
**1962**, 3, 735. [Google Scholar] [CrossRef] - Whitten, R.C.; Smith, R.T. Symmetric Representation for Three-Body Problems. II. Motion in Space. J. Math. Phys.
**1968**, 9, 1103. [Google Scholar] [CrossRef] - Kuppermann, A. A useful mapping of triatomic potential energy surfaces. Chem. Phys. Lett.
**1975**, 32, 374–375. [Google Scholar] [CrossRef] - Johnson, B.R. On hyperspherical coordinates and mapping the internal configurations of a three body system. J. Chem. Phys.
**1980**, 73, 5051. [Google Scholar] [CrossRef] - Brink, D.M.; Satchler, G.R. Angular Momentum, 2nd ed.; Clarendon: Oxford, UK, 1968. [Google Scholar]
- Suzuki, M. Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Phys. Lett. A
**1990**, 146, 319–323. [Google Scholar] [CrossRef] - Youshida, H. Construction of higher order symplectic integrators. Phys. Lett. A
**1990**, 150, 262–268. [Google Scholar] [CrossRef] - Greutz, M.; Gocksch, A. Higher-order hybrid Monte Carlo algorithms. Phys. Rev. Lett.
**1989**, 63, 9. [Google Scholar] - McLachlan, R.I.; Quispel, G.R.W. Splitting methods. Acta Numer.
**2002**, 11, 341–434. [Google Scholar] [CrossRef] - McLachlan, R.I. Composition methods in the presence of small parameters. BIT Numer. Math.
**1995**, 35, 258–268. [Google Scholar] [CrossRef] - Monovasillis, T.; Simos, T.E. Symplectic methods for the numerical integration of the Schrödinger equation. Comput. Mater. Sci.
**2007**, 38, 526–532. [Google Scholar] [CrossRef] - Hairer, E.; Wanner, G.; Lubich, C. Geometric Numerical Integration; Springer: Berlin/Heidelberg, Germany, 2006. [Google Scholar]
- Boothroyd, A.I.; Keogh, W.J.; Martin, P.G.; Peterson, M.R. A refined H
_{3}potential energy surface. J. Chem. Phys.**1996**, 104, 7139. [Google Scholar] [CrossRef] - Babikov, D.; Kendrick, B.K.; Walker, P.B.; Pack, R.T. Metastable states of ozone calculated on an accurate potential energy surface. J. Chem. Phys.
**2003**, 118, 6298. [Google Scholar] [CrossRef] - Fu, B.; Zhang, D.H. A hierarchical construction scheme for accurate potential energy surface generation: An application to the F + H
_{2}reaction. J. Phys. Chem.**2008**, 129, 011103. [Google Scholar] [CrossRef] [PubMed] - Ren, Z.; Che, L.; Qiu, M.; Wang, X.; Dong, W.; Dai, D.; Wang, X.; Yang, X.; Sun, Z.; Fu, B.; et al. Probing the resonance potential in the F atomreaction with hydrogen deuteride withspectroscopic accuracy. Proc. Natl. Acad. Sci. USA
**2008**, 105, 12662–12666. [Google Scholar] [CrossRef]

**Figure 1.**The Total reaction probabilities for the O + O ${}_{2}$ (upper) and F+HD (bottom) reactions for J = 0, calculated using high-order SOs using very small time step (Convergent) and using the time steps which are capable of giving results with errors of about 1%.

**Figure 2.**${log}_{10}$(error) vs. ${log}_{10}$(effective time step) of different high-order SOs in the TVT form for the H + H ${}_{2}$ reaction with total angular momentum J = 0. In Panel (

**A**) results of 4th order SO using A-class in TVT form with efficiency less than SSO propagator; panel (

**B**) show the results obtained by using 4th order SO of S-class and only 4S5b converges in a high order way; Panel (

**C**) represent the results of other high order split operator obtained with A and S-class and only 6S7 converges in a 2nd order way but less efficient.

**Figure 3.**${log}_{10}$(error) vs. ${log}_{10}$(effective time step) of different high-order SOs in the VTV form for the H + H ${}_{2}$ reaction with total angular momentum J = 0. In Panel (

**A**) results of 4th order SO of A-class with efficiency less than SSO propagator these results are consistent with TVT form; panel (

**B**) show the results obtained by using 4th order SO of S-class and again 4S5b converges in a high order way but less efficient than SSO; Panel (

**C**) represent the results of other high order split operator with A and S-class and results are similar to those given in TVT form.

**Figure 4.**${log}_{10}$(error) vs. ${log}_{10}$(effective time step) of different high-order SOs in the VTV form for the O + O ${}_{2}$ reaction with total angular momentum J = 0. In panel (

**A**) results obtained using 4th order SO of A-class are given with most of these are less efficient than SSO; Panel (

**B**) show the results of 4th order SO of S-class and they all converges in their higher order way and all are more efficient than SSO; Panel (

**C**) show the results using 6th order SO of A and S-class, only 6S7 converges in higer order way and 2× times efficient than SSO; Panel (

**D**) represent the results of 8th order SO of A and S-class, all S-class operators converges in their higher order way and more efficient than any other SO used in calculation.

**Figure 5.**${log}_{10}$(error) vs. ${log}_{10}$(effective time step) of different high-order SOs in the TVT form for the O + O ${}_{2}$ reaction with total angular momentum J = 0. In panel (

**A**) results of 4th order SO of A-class are given, similar to VTV form most are these operators less efficient than SSO; Panel (

**B**) show the results of 4th order SO of S-class and again they all converges in their higher order way and all are almost 3× time more efficient than SSO except 4S5a; Panel (

**C**) show the results using 6th order SO of A and S-class, again similiar to VTV form only 6S7 converges in higer order way; Panel (

**D**) show the results of 8th order SO of A and S-class, except A-class operator, all S-class operators converges in their higher order way and more efficient than SSO.

**Figure 6.**${log}_{10}$(error) vs. ${log}_{10}$(effective time step) of different high-order SOs in the TVT form for the F + HD reaction with total angular momentum J = 0. Panel (

**A**) represents results for the A class operators, while panel (

**B**) shows results for the S class operators.

**Figure 7.**${log}_{10}$(error) vs. ${log}_{10}$(effective time step) of different high-order SOs in the VTV form for the F + HD reaction with total angular momentum J = 0. Panel (

**A**) represents results for the A class operators, while panel (

**B**) shows results for the S class operators.

**Table 1.**Reactions, Number of Collision Energies M and Range of energy. Collision energies are evenly distributed in the given range.

Reactions | M | Range of Energies |
---|---|---|

H + H ${}_{2}$ | 700 | [0.3, 1.0] |

O + O ${}_{2}$ | 701 | [0.02, 0.16] |

F + HD | 341 | [0.01, 0.035] |

**Table 2.**Comparison of the most efficient propagators in the TVT form in the Jacobi and the APH coordinates.

Jacobi | APH | Jacobi | APH | |
---|---|---|---|---|

Reactions | A Class/Time Step (a.u) | A Class/Time Step (a.u) | S Class/Time Step (a.u) | S Class/Time Step (a.u) |

H + H ${}_{2}$ | 4A4b/5.0, 4A6b/3.3 | 4A4b/11.9, 4A6b/13.1, 6A6/14.91 | 4S5b/4.5 | 4S5a/12.1, 4S5b/8.7 |

O + O ${}_{2}$ | - | 4A4b/5.25 | - | 4S9/16.41 |

F + HD | 4A6a/9.0 | 4A6a/4.08 | 4S7, 4S9/5.1 | 4S5b/5.99 |

**Table 3.**Comparison of the most efficient propagators in the VTV form in the Jacobi and the APH coordinates.

Jacobi | APH | Jacobi | APH | |
---|---|---|---|---|

Reactions | A Class/Time Step (a.u) | A Class/Time Step (a.u) | S Class/Time Step (a.u) | S Class/Time Step (a.u) |

O + O ${}_{2}$ | 4A6a/20.0 | 4A4b/5.05 | - | 4S11/16.96 |

F + HD | 6A8/10.0 | 4A6a/3.41 | 4S7/4.5 | 4S5b/5.94 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Umer, U.; Zhao, H.; Usman, S.K.; Sun, Z.
High Order Split Operators for the Time-Dependent Wavepacket method of Triatomic Reactive Scattering in Hyperspherical Coordinates. *Entropy* **2019**, *21*, 979.
https://doi.org/10.3390/e21100979

**AMA Style**

Umer U, Zhao H, Usman SK, Sun Z.
High Order Split Operators for the Time-Dependent Wavepacket method of Triatomic Reactive Scattering in Hyperspherical Coordinates. *Entropy*. 2019; 21(10):979.
https://doi.org/10.3390/e21100979

**Chicago/Turabian Style**

Umer, Umair, Hailin Zhao, Syed Kazim Usman, and Zhigang Sun.
2019. "High Order Split Operators for the Time-Dependent Wavepacket method of Triatomic Reactive Scattering in Hyperspherical Coordinates" *Entropy* 21, no. 10: 979.
https://doi.org/10.3390/e21100979