Abstract
The time-dependent expectation value of the electron momentum is zero if determined within the Born-Oppenheimer (BO) approximation. We show that one arrives at non-zero values, if the mean momentum is calculated from the time-derivative of the electronic coordinate. This is illustrated for the case of a coupled electron-nuclear motion in a one-dimensional model system. Regarding a situation where the BO approximation is valid, we find that the BO results are in perfect agreement with those derived from the numerically exact calculations. This holds for cases where the involved wave packets are well localized and also if a large spreading of the probability density occurs.
Similar content being viewed by others
References
M. Born, R. Oppenheimer, Ann. Phys. 389, 457 (1927)
M. Born, K. Huang, Theory of crystal lattices (Oxford University Press, London, 1954)
A. Abedi, N.T. Maitra, E.K.U. Gross, Phys. Rev. Lett. 105, 123002 (2010)
A. Abedi, N.T. Maitra, E.K.U. Gross, J. Chem. Phys. 137, 22A530 (2012)
H. Köppel, W. Domcke, L.S. Cederbaum, Adv. Chem. Phys. 57, 59 (1984)
W. Domcke, G. Stock, Adv. Chem. Phys. 100, 1 (1997)
G.A. Worth, L.S. Cederbaum, Annu. Rev. Phys. Chem. 55, 127 (2004)
W. Domcke, D.R. Yarkony, H. Köppel (Eds.), Conical intersections: theory, computation and experiment (World Scientific, Singapore, 2011)
H. Nakamura, Nonadiabatic transition, 2nd edn. (World Scientific, Singapore, 2012)
F. Gatti (Ed.), Molecular quantum dynamics (Springer, Heidelberg, 2014)
I. Barth, H.C. Hege, H. Ikeda, A. Kenfack, M. Koppitz, J. Manz, F. Marquardt, G.K. Paramonova, Chem. Phys. Lett. 481, 118 (2009)
T. Bredtmann, D.J. Diestler, S.D. Li, J. Manz, J.F. Perez-Torres, W.J. Tian, Y.B. Wu, Y. Yang, H.J. Zhai, Phys. Chem. Chem. Phys. 17, 29421 (2015)
A. Schild, F. Agostini, E.K.U. Gross, J. Phys. Chem. A 120, 3316 (2016)
C.A. Mead, A. Moscowitz, Int. J. Quantum Chem. 1, 243 (1967)
N.V. Cohan, H.F. Hameka, J. Am. Chem. Soc. 88, 2136 (1966)
P.J. Stephens, J. Phys. Chem. 89, 748 (1985)
S. Shin, H. Metiu, J. Chem. Phys. 102, 9285 (1995)
S. Shin, H. Metiu, J. Phys. Chem. 100, 7867 (1996)
M. Erdmann, P. Marquetand, V. Engel, J. Chem. Phys. 119, 672 (2003)
A. Abedi, F. Agostini, Y. Suzuki, E.K.U. Gross, Phys. Rev. Lett. 110, 263001 (2013)
F. Agostini, A. Abedi, Y. Suzuki, E.K.U. Gross, Mol. Phys. 111, 3625 (2013)
F. Agostini, A. Abedi, Y. Suzuki, S.K. Min, N.T. Maitra, E.K.U. Gross, J. Chem. Phys. 142, 084303 (2015)
J. Albert, D. Kaiser, V. Engel, J. Chem. Phys. 144, 171103 (2016)
M. Erdmann, V. Engel, J. Chem. Phys. 120, 158 (2004)
M. Falge, V. Engel, S. Gräfe, J. Chem. Phys. 134, 184307 (2011)
M. Falge, V. Engel, S. Gräfe, J. Phys. Chem. Lett. 3, 2617 (2012)
J. Albert, M. Falge, S. Gomez, I.R. Sola, H. Hildenbrand, V. Engel, J. Chem. Phys. 143, 041102 (2015)
M. Erdmann, E.K.U. Gross, V. Engel, J. Chem. Phys. 121, 9666 (2004)
T. Burnus, M.A.L. Marques, E.K.U. Gross, Phys. Rev. A 71, 010501 (2005)
S.K. Min, A. Abedi, K. Kim, E.K.U. Gross, Phys. Rev. Lett. 113, 263004 (2014)
K. Hader, J. Albert, E.K.U. Gross, V. Engel, J. Chem. Phys. 146, 074304 (2017)
J. Albert, K. Hader, V. Engel, J. Chem. Phys. 147, 064301 (2017)
M.D. Feit, J.A. Fleck, A. Steiger, J. Comput. Phys. 47, 412 (1982)
J. Manz, J.F. Perez-Torres, Y. Yang, Phys. Rev. Lett. 111, 153004 (2013)
H. Ding, D. Jia, J. Manz, Y. Yang, Mol. Phys. 115, 1813 (2017)
G. Herzberg, Spectra of diatomic molecules (van Nostrand Reinhold, New York, 1950)
J. Tellinguisen, in Photodissociation and photoionization, edited by K.P. Lawley, 3rd edn. (Wiley, New York, 1985)
E. Kolba, J. Manz, H.J. Schreier, I. Trisca, Chem. Phys. Lett. 189, 505 (1992)
M. Eichelsdörfer, V. Engel, Chem. Phys. Lett. 263, 640 (1996)
R. Schinke, J. Chem. Phys. 85, 5049 (1986)
S. Hennig, V. Engel, R. Schinke, J. Chem. Phys. 84, 5444 (1986)
R. Schinke, Photodissociation dynamics (Cambridge University Press, Cambridge, 1993)
R. Kosloff, H. Tal-Ezer, Chem. Phys. Lett. 127, 223 (1986)
J. Manz, J.F. Perez-Torres, Y. Yang, J. Phys. Chem. A 118, 8411 (2014)
J. Albert, K. Hader, V. Engel, J. Chem. Phys. 147, 241101 (2017)
Author information
Authors and Affiliations
Corresponding author
Additional information
Contribution to the Topical Issue “Special issue in honor of Hardy Gross”, edited by C.A. Ullrich, F.M.S. Nogueira, A. Rubio, and M.A.L. Marques.
Rights and permissions
About this article
Cite this article
Schaupp, T., Albert, J. & Engel, V. Time-dependent electron momenta from Born-Oppenheimer calculations. Eur. Phys. J. B 91, 97 (2018). https://doi.org/10.1140/epjb/e2018-90113-x
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2018-90113-x