Analysis of the quantum-classical Liouville equation in the mapping basis

Journal of Chemical Physics

Volumn 133, Issue 13, 2010, Pages

  Nassimi, Ali ^a   Bonella, Sara ^b   Kapral, Raymond ^a  

^a UNIVERSITY OF TORONTO   (Canada)

^b UNIVERSITÀ DI ROMA LA SAPIENZA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

ALTERNATE ROUTES; BACK REACTION; COMPLEX TERM; COUPLING TERMS; CURVE-CROSSING MODELS; EVOLUTION EQUATIONS; EXPECTATION VALUES; LIOUVILLE EQUATION; NEWTONIANS; NON-ADIABATIC; POISSON BRACKETS; QUANTUM COHERENCE; QUANTUM STATE; QUANTUM SUBSYSTEMS; QUANTUM-CLASSICAL; QUANTUM-CLASSICAL LIOUVILLE DYNAMICS;

MAPPING; POISSON EQUATION;

QUANTUM THEORY;

EID: 79953767090     PISSN: 00219606     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.3480018     Document Type: Article

References (49)

1. G. D. Billing, The Quantum Classical Theory (Oxford University Press, Toronto, 2003).
2. J. C. Tully, in Modern Methods for Multidimensional Dynamics, Computations in Chemistry, edited by, D. Thompson, (World Scientific, London, 1998), p. 34.
3. M. F. Herman, Annu. Rev. Phys. Chem. 0066-426X 45, 83 (1994). 10.1146/annurev.pc.45.100194.000503
4. A. W. Jasper, C. Zhu, S. Nangia, and D. G. Truhlar, Faraday Discuss. 0301-7249 127, 1 (2004). 10.1039/b405601a (Pubitemid 39369435)
5. R. Kapral, Annu. Rev. Phys. Chem. 0066-426X 57, 129 (2006). 10.1146/annurev.physchem.57.032905.104702
6. K. mre, E. Ozizmir, M. Rosenbaum, and P. Zweifel, J. Math. Phys. 0022-2488 8, 1097 (1967). 10.1063/1.1705323
7. M. Hillery, R. O'Connell, M. Scully, and E. Wigner, Phys. Rep. 0370-1573 106, 121 (1984). 10.1016/0370-1573(84)90160-1
8. S. Nielsen, R. Kapral, and G. Ciccotti, J. Chem. Phys. 0021-9606 115, 5805 (2001). 10.1063/1.1400129 (Pubitemid 33006178)
9. A. Donoso and C. Martens, J. Phys. Chem. A 1089-5639 102, 4291 (1998). 10.1021/jp980219o
10. A. Donoso, Y. Zheng, and C. Martens, J. Chem. Phys. 0021-9606 119, 5010 (2003). 10.1063/1.1597496
11. C. Martens and J. Fang, J. Chem. Phys. 0021-9606 106, 4918 (1997). 10.1063/1.473541
12. J. M. Riga, E. Fredj, and C. C. Martens, J. Chem. Phys. 0021-9606 124, 064506 (2006). 10.1063/1.2162878 (Pubitemid 43248032)
13. M. Santer, U. Manthe, and G. Stock, J. Chem. Phys. 0021-9606 114, 2001 (2001). 10.1063/1.1336576 (Pubitemid 32873180)
14. I. Horenko, C. Salzmann, B. Schmidt, and C. Schutte, J. Chem. Phys. 0021-9606 117, 11075 (2002). 10.1063/1.1522712
15. Y. Tanimura and S. Mukamel, J. Chem. Phys. 0021-9606 101, 3049 (1994). 10.1063/1.467618
16. M. Toutounji, J. Chem. Phys. 0021-9606 125, 194520 (2006). 10.1063/1.2382916 (Pubitemid 44867366)
17. G. Hanna and R. Kapral, J. Chem. Phys. 0021-9606 122, 244505 (2005). 10.1063/1.1940051 (Pubitemid 41012654)
18. G. Hanna and R. Kapral, J. Chem. Phys. 0021-9606 128, 164520 (2008). 10.1063/1.2907847 (Pubitemid 351620547)
19. G. Hanna and R. Kapral, Acc. Chem. Res. 0001-4842 39, 21 (2006). 10.1021/ar030281q
20. H. Kim, G. Hanna, and R. Kapral, J. Chem. Phys. 0021-9606 125, 084509 (2006). 10.1063/1.2336778 (Pubitemid 44320069)
21. G. Hanna and E. Geva, J. Phys. Chem. B 1089-5647 112, 4048 (2008). 10.1021/jp076155b
22. R. Grunwald, A. Kelly, and R. Kapral, in Energy Transfer Dynamics in Biomaterial Systems, edited by, I. Burghardt, V. May, D. Micha, and, E. Bittner, (Springer, Berlin, 2009).
23. R. Kapral and G. Ciccotti, J. Chem. Phys. 0021-9606 110, 8919 (1999). 10.1063/1.478811
24. A. Sergi, D. MacKernan, G. Ciccotti, and R. Kapral, Theor. Chem. Acc. 1432-881X 110, 49 (2003). 10.1007/s00214-003-0456-0 (Pubitemid 37229639)
25. D. MacKernan, G. Ciccotti, and R. Kapral, J. Phys. Chem. B 1089-5647 112, 424 (2008). 10.1021/jp0761416 (Pubitemid 351184616)
26. C. Wan and J. Schofield, J. Chem. Phys. 0021-9606 113, 7047 (2000). 10.1063/1.1313525 (Pubitemid 32021201)
27. C. Wan and J. Schofield, J. Chem. Phys. 0021-9606 116, 494 (2002). 10.1063/1.1425835 (Pubitemid 34064584)
28. H. Kim, A. Nassimi, and R. Kapral, J. Chem. Phys. 0021-9606 129, 084102 (2008). 10.1063/1.2971041
29. A. Nassimi and R. Kapral, Can. J. Chem. 0008-4042 87, 880 (2009). 10.1139/V09-041
30. J. Schwinger, in Quantum Theory of Angular Momentum, edited by, L. Biedenham, (Academic, New York, 1965), p. 229.
31. T. Holstein and H. Primakoff, Phys. Rev. 0031-899X 58, 1098 (1940). 10.1103/PhysRev.58.1098
32. W. Miller and C. McCurdy, J. Chem. Phys. 0021-9606 69, 5163 (1978). 10.1063/1.436463
33. X. Sun, H. Wang, and W. Miller, J. Chem. Phys. 0021-9606 109, 7064 (1998). 10.1063/1.477389 (Pubitemid 128674371)
34. W. Miller, J. Phys. Chem. A 1089-5639 105, 2942 (2001). 10.1021/jp003712k (Pubitemid 35373060)
35. G. Stock and M. Thoss, in Advances in Chemical Physics, edited by, S. Rice, (Wiley-IEEE, New York, 2005), Vol. 131, p. 560. 10.1002/0471739464.ch5
36. M. Thoss and G. Stock, Phys. Rev. A 1050-2947 59, 64 (1999). 10.1103/PhysRevA.59.64
37. U. Müller and G. Stock, J. Chem. Phys. 0021-9606 108, 7516 (1998). 10.1063/1.476184
38. G. Stock and M. Thoss, Phys. Rev. Lett. 0031-9007 78, 578 (1997). 10.1103/PhysRevLett.78.578
39. S. Bonella and D. Coker, J. Chem. Phys. 0021-9606 118, 4370 (2003). 10.1063/1.1542883
40. S. Bonella and D. Coker, J. Chem. Phys. 0021-9606 122, 194102 (2005). 10.1063/1.1896948 (Pubitemid 40914299)
41. E. Dunkel, S. Bonella, and D. F. Coker, J. Chem. Phys. 0021-9606 129, 114106 (2008). 10.1063/1.2976441
42. Since the mapping oscillators are fictitious, we set mω=1 so that the mapping coordinates and momenta are defined to have dimensions of.
43. This set of evolution equations appears in Ref. where a mapping description of the dynamics was investigated in the context of a semiclassical formulation. In this reference these equations of motion were obtained by replacing the Heisenberg equations of motion with Hamilton's equations for classical variables. Here, they result from an approximation to the QCLE in the mapping basis.
44. A. Kelly, R. van Zon, J. Schofield, and R. Kapral (2010) (unpublished).
45. Efficient algorithms can be constructed to simulate these equations (Ref.) that account for the time scale disparity between the classical and mapping quantum degrees of freedom.
46. J. Tully, J. Chem. Phys. 0021-9606 93, 1061 (1990). 10.1063/1.459170
47. D. Kosloff and R. Kosloff, J. Comput. Phys. 0021-9991 52, 35 (1983). 10.1016/0021-9991(83)90015-3
48. Depending on the nature of the potential and the choice of initial conditions, there are situations where the dynamics described by this set of equations will yield unphysical results unless modifications to the dynamics are introduced (Refs.).
49. S. Bonella and D. Coker, J. Chem. Phys. 0021-9606 114, 7778 (2001). 10.1063/1.1366331 (Pubitemid 32486691)