Theoretical Chemistry Accounts

, 131:1083 | Cite as

Calculation of the zero-point energy from imaginary-time quantum trajectory dynamics in Cartesian coordinates

  • Sophya GarashchukEmail author
Regular Article
Part of the following topical collections:
  1. 50th Anniversary Collection


The imaginary-time quantum dynamics is implemented in Cartesian coordinates using the momentum-dependent quantum potential approach. A nodeless wavefunction, represented in terms of quantum trajectories, is evolved in imaginary time according to the quantum-mechanical Boltzmann operator in the Eulerian frame-of-reference. The quantum potential and its gradient are determined approximately, from the global low-order (quadratic) polynomial fit to the trajectory momenta, which makes the approach practical in high dimensions. Implementation in the Cartesian coordinates allows one to work with the Hamiltonian of the simplest form, to setup calculations in the molecular dynamics-compatible framework and to naturally mix quantum and classical description of particles. Localization of wavefunctions in the center-of-mass degrees of freedom and in the overall rotation, which makes the quadratic polynomial fitting in Cartesian coordinates accurate, is accomplished by the addition of a quadratic constraining potential, and its contribution to the zero-point energy is analytically subtracted. For illustration, the zero-point energies are computed for model clusters consisting of up to 11 atoms (33 dimensions).


Quantum dynamics Quantum trajectories Zero-point energy Boltzmann operator Imaginary time 



This material is based on work partially supported by the South Carolina Research foundation and by the National Science Foundation under Grant No. CHE-1056188. The author is grateful to V. A. Rassolov for many stimulating discussions.


  1. 1.
    Czako G, Bowman JM (2009) J Chem Phys 131Google Scholar
  2. 2.
    Zhang W, Kawamata H, Liu K (2009) Science 325:303CrossRefGoogle Scholar
  3. 3.
    Dekker C, Ratner MA (2001) Phys World 14:29Google Scholar
  4. 4.
    Lear JD, Wasserman ZR, DeGrado WF (1988) Science 240:1177CrossRefGoogle Scholar
  5. 5.
    Cha Y, Murray CJ, Klinman JP (1989) Science 243:1325CrossRefGoogle Scholar
  6. 6.
    Knapp MJ, Klinman JP (2002) Eur J of Biochem 269:3113CrossRefGoogle Scholar
  7. 7.
    Prezhdo OV, Rossky PJ (1997) J Chem Phys 107:5863CrossRefGoogle Scholar
  8. 8.
    Brooksby C, Prezhdo O, Reid P (2003) J Chem Phys 119:9111CrossRefGoogle Scholar
  9. 9.
    Prezhdo OV, Duncan WR, Prezhdo VV (2008) Acc Chem Res 41:339CrossRefGoogle Scholar
  10. 10.
    Light JC, Carrington T Jr (2000) Adv Chem Phys 114:263CrossRefGoogle Scholar
  11. 11.
    Meyer HD, Manthe U, Cederbaum LS (1990) Chem Phys Lett 165:73CrossRefGoogle Scholar
  12. 12.
    Meyer HD, Worth GA (2003) Theor Chem Acc 109:251CrossRefGoogle Scholar
  13. 13.
    Wang HB, Thoss M (2003) J Chem Phys 119:1289CrossRefGoogle Scholar
  14. 14.
    Shalashilin DV, Child MS (2004) J Chem Phys 121:3563CrossRefGoogle Scholar
  15. 15.
    Wu YH, Batista VS (2006) J Chem Phys 124:224305CrossRefGoogle Scholar
  16. 16.
    Ben-Nun M, Quenneville J, Martinez TJ (2001) J Chem Phys 104:5161Google Scholar
  17. 17.
    Kim SY, Hammes-Schiffer S (2006) J Chem Phys 124:244102CrossRefGoogle Scholar
  18. 18.
    Prezhdo O, Kisil V (1997) Phys Rev A 56:162CrossRefGoogle Scholar
  19. 19.
    Hone TD, Izvekov S, Voth GA (2005) J Chem Phys 122:054105CrossRefGoogle Scholar
  20. 20.
    Gao J, Truhlar DG (2002) Annu Rev Phys Chem 53:467CrossRefGoogle Scholar
  21. 21.
    Náray-Szabó, G, Warshel, A (eds) (1997) Computational approaches to biochemical reactivity, vol. 19 of Understanding chemical reactivity. Kluwer Academic Publishers, DordrechtGoogle Scholar
  22. 22.
    Gindensperger E, Meier C, Beswick JA (2000) J Chem Phys 113:9369CrossRefGoogle Scholar
  23. 23.
    Meier C, Manthe U (2001) J Chem Phys 115:5477CrossRefGoogle Scholar
  24. 24.
    Karplus M, Sharma RD, Porter RN (1964) J Chem Phys 40:2033CrossRefGoogle Scholar
  25. 25.
    Rassolov VA, Garashchuk S (2008) Chem Phys Lett 464:262CrossRefGoogle Scholar
  26. 26.
    Schatz GC, Bowman JM, Kuppermann A (1975) J Chem Phys 63:685CrossRefGoogle Scholar
  27. 27.
    Miller WH (2001) J Phys Chem A 105:2942CrossRefGoogle Scholar
  28. 28.
    Wyatt RE (2005) Quantum dynamics with trajectories: introduction to quantum hydrodynamics. Springer, New YorkGoogle Scholar
  29. 29.
    Madelung E (1927) Z Phys 40:322CrossRefGoogle Scholar
  30. 30.
    de Broglie L (1930) An introduction to the study ot wave mechanics. E. P. Dutton and Company Inc., New YorkGoogle Scholar
  31. 31.
    Bohm D (1952) Phys Rev 85:166CrossRefGoogle Scholar
  32. 32.
    Garashchuk S (2010) J. Chem. Phys. 132:014112CrossRefGoogle Scholar
  33. 33.
    Garashchuk S, Rassolov V, Prezhdo O (2011) Reviews in computational chemistry, vol 27, chap. Semiclassical Bohmian dynamics. Wiley, Hoboken, pp 111–210Google Scholar
  34. 34.
    Ramond P (1990) Field theory: a modern primer. Addison-Wesley, ReadingGoogle Scholar
  35. 35.
    Feynman RP, Hibbs AR (1965) Quantum mechanics and path integrals. McGraw-Hill, New YorkGoogle Scholar
  36. 36.
    Frantsuzov PA and Mandelshtam VA (2008) J Chem Phys 128Google Scholar
  37. 37.
    Chen X, Wu YH, Batista VS (2005) J Chem Phys 122Google Scholar
  38. 38.
    Cartarius H and Pollak E (2011) J Chem Phys 134Google Scholar
  39. 39.
    Miller WH (1971) J Chem Phys 55:3146CrossRefGoogle Scholar
  40. 40.
    Blume D, Lewerenz M, Niyaz P, Whaley KB (1997) Phys Rev E 55:3664CrossRefGoogle Scholar
  41. 41.
    Ceperley DM, Mitas L (1996) Advances in chemical physics, chap. Monte Carlo methods in quantum chemistry. Wiley, LondonGoogle Scholar
  42. 42.
    Lester WA Jr, Mitas L, Hammond B (2009) Chem Phys Lett 478:1CrossRefGoogle Scholar
  43. 43.
    Viel A, Coutinho-Neto MD, Manthe U (2007) J Chem Phys 126:024308CrossRefGoogle Scholar
  44. 44.
    Hinkle CE, McCoy AB (2008) J Phys Chem A 112:2058CrossRefGoogle Scholar
  45. 45.
    Miller WH, Schwartz SD, Tromp JW (1983) J Chem Phys 79:4889CrossRefGoogle Scholar
  46. 46.
    Rassolov VA, Garashchuk S, Schatz GC (2006) J Phys Chem. A 110:5530CrossRefGoogle Scholar
  47. 47.
    Garashchuk S, Vazhappilly T (2010) J Phys Chem C 114:20595CrossRefGoogle Scholar
  48. 48.
    Garashchuk S, Mazzuca J, Vazhappilly T (2011b) J Chem Phys 135:034104CrossRefGoogle Scholar
  49. 49.
    Liu J, Makri N (2005) Mol Phys 103:1083CrossRefGoogle Scholar
  50. 50.
    Goldfarb Y, Degani I, Tannor DJ (2007) Chem Phys 338:106CrossRefGoogle Scholar
  51. 51.
    Garashchuk S (2010b) Chem Phys Lett 491:96CrossRefGoogle Scholar
  52. 52.
    Trahan CJ, Wyatt RE (2003) J Chem Phys 118:4784CrossRefGoogle Scholar
  53. 53.
    Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1992) Numerical recipes: the art of scientific computing. 2nd edn. Cambridge University Press, CambridgeGoogle Scholar
  54. 54.
    Reed SK, González-Martínez ML, Rubayo-Soneira J, and Shalashilin DV (2011) J Chem Phys 134Google Scholar
  55. 55.
    Dubbeldam D, Oxford GAE, Krishna R, Broadbelt LJ, and Snurr RQ (2010) J Chem Phys 133Google Scholar
  56. 56.
    Ochterski JW (1999) Vibrational analysis in Gaussian,
  57. 57.
    Meyer H (2002) Annu Rev Phys Chem 53:141CrossRefGoogle Scholar
  58. 58.
    M. A. Ratner, Gerber RB (1986) J Phys Chem 90:20CrossRefGoogle Scholar
  59. 59.
    Carter S, Culik SJ, Bowman JM (1997) J Chem Phys 107:10458CrossRefGoogle Scholar
  60. 60.
    Morse PM (1929) Phys Rev 34:57CrossRefGoogle Scholar
  61. 61.
    Feit MD, Fleck JA Jr, Steiger A (1982) J Comp Phys 47:412CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of Chemistry and BiochemistryUniversity of South CarolinaColumbiaUSA

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