The second-order Ehrenfest method

  • Morgane Vacher
  • David Mendive-Tapia
  • Michael J. Bearpark
  • Michael A. RobbEmail author
Regular Article
Part of the following topical collections:
  1. Shavitt Memorial Festschrift Collection


This article describes the Ehrenfest method and our second-order implementation (with approximate gradient and Hessian) within a CASSCF formalism. We demonstrate that the second-order implementation with the predictor–corrector integration method improves the accuracy of the simulation significantly in terms of energy conservation. Although the method is general and can be used to study any coupled electron–nuclear dynamics, we apply it to investigate charge migration upon ionization of small organic molecules, focusing on benzene cation. Using this approach, we can study the evolution of a non-stationary electronic wavefunction for fixed atomic nuclei, and where the nuclei are allowed to move, to investigate the interplay between them for the first time. Analysis methods for the interpretation of the electronic and nuclear dynamics are suggested: we monitor the electronic dynamics by calculating the spin density of the system as a function of time.


Ehrenfest method CASSCF Coupled electron–nuclear dynamics Charge migration Charge transfer 



This work was supported by UK-EPSRC Grant EP/I032517/1. All calculations were run using the Imperial College High Performance Computing service. The original SA CP-MCSCF programs were written by Thom Vreven. The work on the Ehrenfest programs was initiated by Patricia Hunt (see the supplementary information of reference [9]).


  1. 1.
    Breidbach J, Cederbaum LS (2005) Universal attosecond response to the removal of an electron. Phys Rev Lett 94:033901CrossRefGoogle Scholar
  2. 2.
    Kling MF, Vrakking MJJ (2008) Attosecond electron dynamics. Annu Rev Phys Chem 59:463–492CrossRefGoogle Scholar
  3. 3.
    Krausz F, Ivanov M (2009) Attosecond physics. Rev Mod Phys 81:163–234CrossRefGoogle Scholar
  4. 4.
    Tully JC (2012) Perspective: nonadiabatic dynamics theory. J Chem Phys 137(22):22A301CrossRefGoogle Scholar
  5. 5.
    Joubert-Doriol L, Lasorne B, Gatti F, Schröder M, Vendrell O, Meyer H-D (2012) Suitable coordinates for quantum dynamics: applications using the multiconfiguration time-dependent Hartree (MCTDH) algorithm. Comput Theor Chem 990:75–89 Chemical reactivity, from accurate theories to simple models, in honor of Professor Jean-Claude RayezCrossRefGoogle Scholar
  6. 6.
    Vibók Á, Csehi A, Gindensperger E, Köppel H, Halász GJ (2012) Quantum dynamics through conical intersections: combining effective modes and quadratic couplings. J Phys Chem A 116(11):2629–2635CrossRefGoogle Scholar
  7. 7.
    Worth GA, Meyer H-D, Köppel H, Cederbaum LS, Burghardt I (2008) Using the MCTDH wavepacket propagation method to describe multimode non-adiabatic dynamics. Int Rev Phys Chem 27(3):569–606CrossRefGoogle Scholar
  8. 8.
    Klein S, Bearpark MJ, Smith BR, Robb MA, Olivucci M, Bernardi F (1998) Mixed state ‘on the fly’ non-adiabatic dynamics: the role of the conical intersection topology. Chem Phys Lett 292(3):259–266CrossRefGoogle Scholar
  9. 9.
    Blancafort L, Hunt P, Robb MA (2005) Intramolecular electron transfer in bis(methylene) adamantyl radical cation: a case study of diabatic trapping. J Am Chem Soc 127(10):3391–3399 PMID: 15755157CrossRefGoogle Scholar
  10. 10.
    Mendive-Tapia D, Vacher M, Bearpark MJ, Robb MA (2013) Coupled electron–nuclear dynamics: charge migration and charge transfer initiated near a conical intersection. J Chem Phys 139(4):044110CrossRefGoogle Scholar
  11. 11.
    Roos BO, Taylor PR, Siegbahn PE (1980) A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach. Chem Phys 48(2):157–173CrossRefGoogle Scholar
  12. 12.
    Roos BO (1987) The complete active space self-consistent field method and its applications in electronic structure calculations. Adv Chem Phys 69:399–446Google Scholar
  13. 13.
    Millam JM, Bakken V, Chen W, Hase WL, Schlegel HB (1999) Ab initio classical trajectories on the Born–Oppenheimer surface: Hessian-based integrators using fifth-order polynomial and rational function fits. J Chem Phys 111(9):3800–3805CrossRefGoogle Scholar
  14. 14.
    Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, Scalmani G, Barone V, Mennucci B, Petersson GA, Nakatsuji H, Caricato M, Li X, Hratchian HP, Izmaylov AF, Bloino J, Zheng G, Sonnenberg JL, Hada M, Ehara M, Toyota K, Fukuda R, Hasegawa J, Ishida M, Nakajima T, Honda Y, Kitao O, Nakai H, Vreven T, Montgomery JA Jr, Peralta JE, Ogliaro F, Bearpark M, Heyd JJ, Brothers E, Kudin KN, Staroverov VN, Kobayashi R, Normand J, Raghavachari K, Rendell A, Burant JC, Iyengar SS, Tomasi J, Cossi M, Rega N, Millam JM, Klene M, Knox JE, Cross JB, Bakken V, Adamo C, Jaramillo J, Gomperts R, Stratmann RE, Yazyev O, Austin AJ, Cammi R, Pomelli C, Ochterski JW, Martin RL, Morokuma K, Zakrzewski VG, Voth GA, Salvador P, Dannenberg JJ, Dapprich S, Daniels AD, Farkas Ö, Foresman JB, Ortiz JV, Cioslowski J, Fox DJ (2010) Gaussian development version, revision h.10. Gaussian Inc., Wallingford, CTGoogle Scholar
  15. 15.
    Teller E (1937) The crossing of potential surfaces. J Phys Chem 41(1):109–116CrossRefGoogle Scholar
  16. 16.
    Amarouche M, Gadea F, Durup J (1989) A proposal for the theoretical treatment of multi-electronic-state molecular dynamics: hemiquantal dynamics with the whole dim basis (HWD). A test on the evolution of excited ar3+ cluster ions. Chem Phys 130:145–157CrossRefGoogle Scholar
  17. 17.
    Delos JB, Thorson WR, Knudson SK (1972) Semiclassical theory of inelastic collisions. I. Classical picture and semiclassical formulation. Phys Rev A 6:709–720CrossRefGoogle Scholar
  18. 18.
    Delos JB, Thorson WR (1972) Semiclassical theory of inelastic collisions. II. Momentum-space formulation. Phys Rev A 6:720–727CrossRefGoogle Scholar
  19. 19.
    Herman MF (1994) Dynamics by semiclassical methods. Annu Rev Phys Chem 45(1):83–111CrossRefGoogle Scholar
  20. 20.
    Jasper AW, Nangia S, Zhu C, Truhlar DG (2006) Non-Born–Oppenheimer molecular dynamics. Acc Chem Res 39(2):101–108 PMID: 16489729CrossRefGoogle Scholar
  21. 21.
    Li X, Tully JC, Schlegel HB, Frisch MJ (2005) Ab initio ehrenfest dynamics. J Chem Phys 123(8):084106CrossRefGoogle Scholar
  22. 22.
    Saita K, Shalashilin DV (2012) On-the-fly ab initio molecular dynamics with multiconfigurational ehrenfest method. J Chem Phys 137(22):22A506CrossRefGoogle Scholar
  23. 23.
    Shalashilin DV (2010) Nonadiabatic dynamics with the help of multiconfigurational ehrenfest method: improved theory and fully quantum 24d simulation of pyrazine. J Chem Phys 132(24):244111CrossRefGoogle Scholar
  24. 24.
    Takatsuka K, Yonehara T (2011) Exploring dynamical electron theory beyond the Born–Oppenheimer framework: from chemical reactivity to non-adiabatically coupled electronic and nuclear wavepackets on-the-fly under laser field. Phys Chem Chem Phys 13(11):4987–5016CrossRefGoogle Scholar
  25. 25.
    Yonehara T, Hanasaki K, Takatsuka K (2012) Fundamental approaches to nonadiabaticity: toward a chemical theory beyond the Born–Oppenheimer paradigm. Chem Rev 112(1):499–542CrossRefGoogle Scholar
  26. 26.
    Zhu C, Nangia S, Jasper AW, Truhlar DG (2004) Coherent switching with decay of mixing: an improved treatment of electronic coherence for non-Born–Oppenheimer trajectories. J Chem Phys 121(16):7658–7670CrossRefGoogle Scholar
  27. 27.
    Tully JC (1998) Mixed quantum–classical dynamics. Faraday Discuss 110:407–419CrossRefGoogle Scholar
  28. 28.
    Hack MD, Wensmann AM, Truhlar DG, Ben-Nun M, Martinez TJ (2001) Comparison of full multiple spawning, trajectory surface hopping, and converged quantum mechanics for electronically nonadiabatic dynamics. J Chem Phys 115(3):1172–1186CrossRefGoogle Scholar
  29. 29.
    Tully JC (1990) Molecular dynamics with electronic transitions. J Chem Phys 93(2):1061–1071CrossRefGoogle Scholar
  30. 30.
    Almlöf J, Taylor PR (1985) Molecular properties from perturbation theory: a unified treatment of energy derivatives. Int J Quantum Chem 27(6):743–768CrossRefGoogle Scholar
  31. 31.
    Cederbaum L, Zobeley J (1999) Ultrafast charge migration by electron correlation. Chem Phys Lett 307(3–4):205–210CrossRefGoogle Scholar
  32. 32.
    Kuleff AI, Cederbaum LS (2007) Charge migration in different conformers of glycine: the role of nuclear geometry. Chem Phys 338:320–328CrossRefGoogle Scholar
  33. 33.
    Kuleff AI, Lünnemann S, Cederbaum LS (2010) Ultrafast charge migration following valence ionization of 4-methylphenol: jumping over the aromatic ring. J Phys Chem A 114(33):8676–8679CrossRefGoogle Scholar
  34. 34.
    Kuleff AI, Lünnemann S, Cederbaum LS (2012) Ultrafast reorganization of the hole charge created upon outer-valence ionization of porphyrins. Chem Phys 399:245–251CrossRefGoogle Scholar
  35. 35.
    Lünnemann S, Kuleff AI, Cederbaum LS (2008) Ultrafast charge migration in 2-phenylethyl-n, n-dimethylamine. Chem Phys Lett 450:232–235CrossRefGoogle Scholar
  36. 36.
    Periyasamy G, Levine R, Remacle F (2009) Electronic wave packet motion in water dimer cation: a many electron description. Chem Phys 366:129–138CrossRefGoogle Scholar
  37. 37.
    Aberg T (1967) Theory of X-ray satellites. Phys Rev 156:35–41CrossRefGoogle Scholar
  38. 38.
    Aberg T (1970) Asymptotic double-photoexcitation cross sections of the helium atom. Phys Rev A 2:1726–1729CrossRefGoogle Scholar
  39. 39.
    Manne R, Åberg T (1970) Koopmans’ theorem for inner-shell ionization. Chem Phys Lett 7(2):282–284CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Morgane Vacher
    • 1
  • David Mendive-Tapia
    • 2
  • Michael J. Bearpark
    • 1
  • Michael A. Robb
    • 1
    Email author
  1. 1.Department of ChemistryImperial College LondonLondonUK
  2. 2.Laboratoire CEISAM - UMR CNR 6230Université de NantesNantes Cedex 3France

Personalised recommendations