Nonadiabatic dynamics in open quantum-classical systems: Forward-backward trajectory solution

Journal of Chemical Physics

Volumn 137, Issue 22, 2012, Pages

  Hsieh, Chang Yu ^a   Kapral, Raymond ^a  

^a UNIVERSITY OF TORONTO   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATE SOLUTION; BACKWARD TRAJECTORY; COHERENT STATE; DIFFERENTIAL FORMS; DIMENSIONAL PHASE SPACES; FORMAL SOLUTION; NON-ADIABATIC DYNAMICS; QUANTUM-CLASSICAL; QUANTUM-CLASSICAL SYSTEMS; TIME EVOLUTIONS;

DYNAMICS; LIOUVILLE EQUATION; PHASE SPACE METHODS;

TRAJECTORIES;

ARTICLE; QUANTUM THEORY; TIME;

QUANTUM THEORY; TIME FACTORS;

EID: 84871222513     PISSN: 00219606     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.4736841     Document Type: Article

References (66)

1. J. C. Tully, J. Chem. Phys. 93, 1061 (1990). 10.1063/1.459170
2. J. C. Tully, in Modern Methods for Multidimensional Dynamics Computations in Chemistry, edited by, D. L. Thompson, (World Scientific, New York, 1998), p. 34.
3. F. Webster, E. T. Wang, P. J. Rossky, and R. A. Friesner, J. Chem. Phys. 100, 4835 (1994). 10.1063/1.467204
4. O. V. Prezhdo, J. Chem. Phys. 111, 8366 (1999). 10.1063/1.480178
5. A. W. Jasper, C. Zhu, S. Nangia, and D. G. Truhlar, Faraday Discuss. 127, 1 (2004). 10.1039/b405601a
6. M. J. Bedard-Hearn, R. E. Larsen, and B. J. Schwartz, J. Chem. Phys. 123, 234106 (2005). 10.1063/1.2131056
7. S. A. Fischer, C. T. Chapman, and X. Li, J. Chem. Phys. 135, 144102 (2011). 10.1063/1.3646920
8. W. H. Miller and C. W. McCurdy, J. Chem. Phys. 69, 5163 (1978). 10.1063/1.436463
9. H. D. Meyer and W. H. Miller, J. Chem. Phys. 70, 3214 (1979). 10.1063/1.437910
10. X. Sun, H. B. Wang, and W. H. Miller, J. Chem. Phys. 109, 7064 (1998). 10.1063/1.477389
11. G. Stock and M. Thoss, Phys. Rev. Lett. 78, 578 (1997). 10.1103/PhysRevLett.78.578
12. U. Müller and G. Stock, J. Chem. Phys. 108, 7516 (1998). 10.1063/1.476184
13. M. Thoss and G. Stock, Phys. Rev. A 59, 64 (1999). 10.1103/PhysRevA.59.64
14. K. Thompson and N. Makri, J. Chem. Phys. 110, 1343 (1999). 10.1063/1.478011
15. W. H. Miller, J. Phys. Chem. A 105, 2942 (2001). 10.1021/jp003712k
16. M. Thoss and H. B. Wang, Annu. Rev. Phys. Chem. 55, 299 (2004). 10.1146/annurev.physchem.55.091602.094429
17. G. Stock and M. Thoss, Adv. Chem. Phys. 131, 243 (2005). 10.1002/0471739464
18. S. Bonella and D. F. Coker, J. Chem. Phys. 122, 194102 (2005). 10.1063/1.1896948
19. E. Dunkel, S. Bonella, and D. F. Coker, J. Chem. Phys. 129, 114106 (2008). 10.1063/1.2976441
20. P. Huo and D. F. Coker, J. Chem. Phys. 135, 201101 (2011). 10.1063/1.3664763
21. For a review with references see, [R. Kapral, Ann. Rev. Phys. Chem. 57, 129 (2006)]. 10.1146/annurev.physchem.57.032905.104702
22. S. Hammes-Schiffer and J. C. Tully, J. Chem. Phys. 101, 4657 (1994). 10.1063/1.467455
23. E. R. Bittner and P. J. Rossky, J. Chem. Phys. 103, 8130 (1995). 10.1063/1.470177
24. J. E. Subotnik and N. Shenvi, J. Chem. Phys. 134, 024105 (2011). 10.1063/1.3506779
25. N. Shenvi, J. E. Subotnik, and W. Yang, J. Chem. Phys. 134, 144102 (2011). 10.1063/1.3575588
26. Quantum Dissipative Systems, edited by, U. Weiss, (World Scientific, Singapore, 1999).
27. G. Lindblad, Commun. Math. Phys. 48, 119 (1976). 10.1007/BF01608499
28. A. Redfield, in Advances in Magnetic Resonance, edited by, J. Waugh, (Academic, New York, 1965), Vol. 1.
29. Density Matrix Theory and Applications, edited by, K. Blum, (Plenum, New York, 1981).
30. H.-D. Meyer, U. Manthe, and L. Cederbaum, Chem. Phys. Lett. 165, 73 (1990). 10.1016/0009-2614(90)87014-I
31. M. Beck, A. Jckle, G. Worth, and H.-D. Meyer, Phys. Rep. 324, 1 (2000). 10.1016/S0370-1573(99)00047-2
32. M. Marques and E. Gross, Annu. Rev. Phys. Chem. 55, 427 (2004). 10.1146/annurev.physchem.55.091602.094449
33. O. E. Alon, A. I. Streltsov, and L. S. Cederbaum, J. Chem. Phys. 127, 154103 (2007). 10.1063/1.2771159
34. H. Wang and M. Thoss, J. Chem. Phys. 131, 024114 (2009). 10.1063/1.3173823
35. M. Casida and M. Huix-Rotllant, Annu. Rev. Phys. Chem. 63, 287 (2012). 10.1146/annurev-physchem-032511-143803
36. E. Bukhman and N. Makri, J. Phys. Chem. A 113, 7183 (2009). 10.1021/jp809741x
37. J. C. Tully, in Classical and Quantum Dynamics in Condensed Phase Simulations, edited by, B. J. Berne, G. Ciccotti, and, D. F. Coker, (World Scientific, Singapore, 1998).
38. M. F. Herman, Annu. Rev. Phys. Chem. 45, 83 (1994). 10.1146/annurev.pc. 45.100194.000503
39. R. Kapral and G. Ciccotti, J. Chem. Phys. 110, 8919 (1999). 10.1063/1.478811
40. Q. Shi and E. Geva, J. Chem. Phys. 121, 3393 (2004). 10.1063/1.1771641
41. S. Bonella, G. Ciccotti, and R. Kapral, Chem. Phys. Lett. 484, 399 (2010). 10.1016/j.cplett.2009.11.056
42. C. C. Martens and J. Y. Fang, J. Chem. Phys. 106, 4918 (1996). 10.1063/1.473541
43. A. Donoso and C. C. Martens, J. Phys. Chem. A 102, 4291 (1998). 10.1021/jp980219o
44. D. MacKernan, R. Kapral, and G. Ciccotti, J. Phys.: Condens. Matter 14, 9069 (2002). 10.1088/0953-8984/14/40/301
45. D. MacKernan, G. Ciccotti, and R. Kapral, J. Phys. Chem. B 112, 424 (2008). 10.1021/jp0761416
46. I. Horenko, C. Salzmann, B. Schmidt, and C. Schutte, J. Chem. Phys. 117, 11075 (2002). 10.1063/1.1522712
47. C. Wan and J. Schofield, J. Chem. Phys. 113, 7047 (2000). 10.1063/1.1313525
48. C. Wan and J. Schofield, J. Chem. Phys. 116, 494 (2002). 10.1063/1.1425835
49. J. Schwinger, in Quantum Theory of Angular Momentum, edited by, L. C. Biedenharn, and, H. V. Dam, (Academic, New York, 1965), p. 229.
50. H. Kim, A. Nassimi, and R. Kapral, J. Chem. Phys. 129, 084102 (2008). 10.1063/1.2971041
51. A. Nassimi, S. Bonella, and R. Kapral, J. Chem. Phys. 133, 134115 (2010). 10.1063/1.3480018
52. A. Kelly, R. van Zon, J. M. Schofield, and R. Kapral, J. Chem. Phys. 136, 084101 (2012). 10.1063/1.3685420
53. S. Nielsen, R. Kapral, and G. Ciccotti, J. Chem. Phys. 115, 5805 (2001). 10.1063/1.1400129
54. R. Glauber, Phys. Rev. 131, 2766 (1963). 10.1103/PhysRev.131.2766
55. In the present formulation, since we are interested in constructing an algorithm for the evolution of the QCLE in the subsystem basis and its mapping onto the harmonic oscillator single-excitation subspace, the coherent states are introduced in the mapping-transformed space. An interesting alternative approach is to introduce the coherent states directly in the subsystem space and bypass the mapping transformation. This approach, however, requires one to work with a microscopic model in which the Hamiltonian is conveniently represented in the second quantized form with the help of fermionic or bosonic creation and annihilation operators. We shall not pursue this alternative further here.
56. I. V. Aleksandrov, Z. Naturforsch. 36, 902 (1981).
57. V. I. Gerasimenko, Theor. Math. Phys. 50, 77 (1982). 10.1007/BF01027604
58. W. Boucher and J. Traschen, Phys. Rev. D 37, 3522 (1988). 10.1103/PhysRevD.37.3522
59. W. Y. Zhang and R. Balescu, J. Plasma Phys. 40, 199 (1988). 10.1017/S0022377800013222
60. Y. Tanimura and S. Mukamel, J. Chem. Phys. 101, 3049 (1994). 10.1063/1.467618
61. Although Eqs. are not in Hamiltonian form, through a simple non-canonical transformation where the coherent state variables are scaled by a constant factor, they may be cast into Hamiltonian form.
62. The use of the traceless form of the Hamiltonian from the outset simplifies the calculation. In particular, when the anti-normal form of the annihilation and creation operator product is introduced to evaluate the propagator, the trace term does not appear and the structure of the equations is simpler. We have chosen to carry out the derivation with the usual form of the Hamiltonian to stress that the final results are independent of how one chooses to write the Hamiltonian.
63. N. Ananth, C. Venkataraman, and W. H. Miller, J. Chem. Phys. 127, 084114 (2007). 10.1063/1.2759932
64. X. Sun and W. H. Miller, J. Chem. Phys. 110, 6635 (1999). 10.1063/1.478571
65. M. Thoss, H. B. Wang, and W. H. Miller, J. Chem. Phys. 114, 9220 (2001). 10.1063/1.1359242
66. X. Sun and W. H. Miller, J. Chem. Phys. 106, 916 (1997). 10.1063/1.473171