On Hagedorn wavepackets associated with different Gaussians (2405.07880v4)
Abstract: Hagedorn functions are carefully constructed generalizations of Hermite functions to the setting of many-dimensional squeezed and coupled harmonic systems. Wavepackets formed by superpositions of Hagedorn functions have been successfully used to solve the time-dependent Schr\"{o}dinger equation exactly in harmonic systems and variationally in anharmonic systems. For evaluating typical observables, such as position or kinetic energy, it is sufficient to consider orthonormal Hagedorn functions with a single Gaussian center. Here, we instead derive various relations between Hagedorn bases associated with different Gaussians, including their overlaps, which are necessary for evaluating quantities nonlocal in time, such as time correlation functions needed for computing spectra. First, we use the Bogoliubov transformation to obtain commutation relations between the ladder operators associated with different Gaussians. Then, instead of using numerical quadrature, we employ these commutation relations to derive exact recurrence relations for the overlap integrals between Hagedorn functions with different Gaussian centers. Finally, we present numerical experiments that demonstrate the accuracy and efficiency of our algebraic method as well as its suitability to treat problems in spectroscopy and chemical dynamics.
- Heller E J 1975 J. Chem. Phys. 62 1544–1555
- Heller E J 1981 Acc. Chem. Res. 14 368–375
- Hagedorn G A 1981 Ann. Phys. (NY) 135 58–70
- Herman M F and Kluk E 1984 Chem. Phys. 91 27–34
- Ben-Nun M, Quenneville J and Martínez T J 2000 J. Phys. Chem. A 104 5161–5175
- Worth G A, Robb M A and Burghardt I 2004 Faraday Discuss. 127(0) 307–323
- Miller W H 2001 J. Phys. Chem. A 105 2942
- Werther M and Großmann F 2020 Phys. Rev. B 101 174315
- Lasser C and Lubich C 2020 Acta Numerica 29 229–401
- Vaníček J J L 2023 J. Chem. Phys. 159 014114
- Heller E J 2018 The semiclassical way to dynamics and spectroscopy (Princeton, NJ: Princeton University Press)
- Heller E J 1976 J. Chem. Phys. 64 63–73
- Coalson R D and Karplus M 1990 J. Chem. Phys. 93 3919–3930
- Lubich C 2008 From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis 12th ed (Zürich: European Mathematical Society) ISBN 978-3037190678
- Grossmann F 2006 J. Chem. Phys. 125 014111
- Wehrle M, Šulc M and Vaníček J 2014 J. Chem. Phys. 140 244114
- Begušić T, Cordova M and Vaníček J 2019 J. Chem. Phys. 150 154117
- Golubev N V, Begušić T and Vaníček J 2020 Phys. Rev. Lett. 125 083001
- Begušić T and Vaníček J 2020 J. Chem. Phys. 153 024105
- Begušić T, Tapavicza E and Vaníček J 2022 J. Chem. Theory Comput. 18 3065–3074
- Scheidegger A, Vaníček J and Golubev N V 2022 J. Chem. Phys. 156 034104
- Moghaddasi Fereidani R and Vaníček J J L 2023 J. Chem. Phys. 159 094114
- Klētnieks E, Alonso Y C and Vaníček J J L 2023 J. Phys. Chem. A 127 8117–8125
- Moghaddasi Fereidani R and Vaníček J J L 2024 J. Chem. Phys. 160 044113
- Poulsen J A and Nyman G 2024 Entropy 26 412
- Dirac P 1947 The Principles of Quantum Mechanics International series of monographs on physics (Clarendon Press)
- Tannor D J 2007 Introduction to Quantum Mechanics: A Time-Dependent Perspective (Sausalito: University Science Books) ISBN 978-1891389238
- Hagedorn G A 1985 Ann. Henri Poincaré 42 363–374
- Hagedorn G A 1998 Ann. Phys. (NY) 269 77–104
- Faou E, Gradinaru V and Lubich C 2009 SIAM J. Sci. Comp. 31 3027–3041
- Gradinaru V, Hagedorn G A and Joye A 2010 J. Chem. Phys. 132 184108 (pages 8)
- Zhou Z 2014 J. Comput. Phys. 272 386–407
- Gradinaru V and Rietmann O 2021 J. Comput. Phys. 445 110581
- Ohsawa T and Leok M 2013 J. Phys. A 46 405201
- Gradinaru V and Hagedorn G A 2014 Numer. Math. 126 53–73
- Lasser C and Troppmann S 2014 J. Fourier Anal. Appl. 20 679–714
- Hagedorn G A 2015 Ann. Phys-new. York. 362 603–608
- Ohsawa T 2015 J. Math. Phys. 56 032103
- Punoševac P and Robinson S L 2016 J. Math. Phys. 57 092102
- Dietert H, Keller J and Troppmann S 2017 J. Math. Anal. Appl. 450 1317–1332
- Bourquin R 2017 Numerical Algorithms for Semiclassical Wavepackets Ph.D. thesis ETH Zürich
- Hagedorn G A and Lasser C 2017 SIAM J. Matrix Anal. Appl. 38 1560–1579
- Lasser C, Schubert R and Troppmann S 2018 J. Math. Phys. 59 082102
- Ohsawa T 2018 Nonlinearity 31 1807–1832
- Punoševac P and Robinson S L 2019 J. Math. Phys. 60 052106
- Ohsawa T 2019 J. Fourier Anal. Appl. 25 1513–1552
- Blanes S and Gradinaru V 2020 J. Comput. Phys. 405 109157
- Arnaiz V 2022 J. Spectr. Theory 12 745–812
- Miao B, Russo G and Zhou Z 2023 IMA J. Numer. Anal. 43 1221–1261
- Kargol A 1999 Annales de l’I.H.P. Physique théorique 71 339–357
- Hagedorn G and Joye A 2000 Ann. Henri Poincaré 1 837–883
- Gradinaru V, Hagedorn G A and Joye A 2010 J. Phys. Math. Theor. 43 474026
- Gradinaru V, Hagedorn G A and Joye A 2010 J. Chem. Phys. 132 184108
- Kieri E, Holmgren S and Karlsson H O 2012 J. Chem. Phys. 137 044111
- Bourquin R, Gradinaru V and Hagedorn G A 2012 J. Math. Chem. 50 602–619
- Gradinaru V and Rietmann O 2024 J. Comput. Phys. 509 113029
- Begušić T and Vaníček J 2020 J. Chem. Phys. 153 184110
- Wehrle M, Oberli S and Vaníček J 2015 J. Phys. Chem. A 119 5685
- Begušić T and Vaníček J 2021 Chimia 75 261
- Ma X and Rhodes W 1990 Phys. Rev. A 41(9) 4625–4631
- Feit M D, Fleck Jr J A and Steiger A 1982 J. Comp. Phys. 47 412
- Kosloff D and Kosloff R 1983 J. Comp. Phys. 52 35–53
- Tapavicza E 2019 J. Phys. Chem. Lett. 10 6003–6009
- Patoz A, Begušić T and Vaníček J 2018 J. Phys. Chem. Lett. 9 2367–2372
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.