Large-scale numerical investigation of excited states in poly (para-phenylene)

Physical Review B - Condensed Matter and Materials Physics

Volumn 66, Issue 20, 2002, Pages 2051121-2051128

  Bursill, Robert J ^a   Barford, William ^b  

^a UNIVERSITY OF NEW SOUTH WALES   (Australia)

^b UNIVERSITY OF SHEFFIELD   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

OLIGOMER;

ABSORPTION; ARTICLE; DENSITY; ENERGY; FILM; MATHEMATICAL ANALYSIS; NONLINEAR SYSTEM;

EID: 0037113465     PISSN: 01631829     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (61)

1. R.H. Friend, R.W. Gymer, A.B. Holmes, J.H. Burroughes, R.N. Marks, C. Taliani, D.D.C. Bradley, D.A. Dos Santos, J.L. Bredas, M. Logdlund, and W.R. Salaneck, Nature (London) 397, 121 (1999).
2. P. Gomes da Costa and E.M. Conwell, Phys. Rev. B 48, 1993 (1993); R. Kersting, U. Lemmer, M. Deussen, H.J. Bakker, R.F. Mahrt, H. Kurz, V.I. Arkhipov, H. Bassler, and E.O. Gobel, Phys. Rev. Lett. 73, 1440 (1994); M. Chandross, S. Mazumdar, S. Jeglinski, X. Wei, Z.V. Vardeny, E.W. Kwock, and T.M. Miller, Phys. Rev. B 50, 14 702 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, Phys. Rev. Lett. 72, 156 (1994); E.M. Conwell, I.J. Rothberg, and H.A. Mizes, ibid. 73, 3179 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, ibid. 73, 3180 (1995); S. Barth and H. Bassler, ibid. 79, 4445 (1997); D. Moses, J. Wang, G. Yu, and A.J. Heeger, ibid. 80, 2685 (1998); S.F. Alvarado, P.F. Seidler, D.G. Lidzey, and D.D.C. Bradley, ibid. 81, 1082 (1998); V.I. Arkhipov, E.V. Emelianova and H. Bassler, ibid. 82; 1321 (1999).
3. P. Gomes da Costa and E.M. Conwell, Phys. Rev. B 48, 1993 (1993); R. Kersting, U. Lemmer, M. Deussen, H.J. Bakker, R.F. Mahrt, H. Kurz, V.I. Arkhipov, H. Bassler, and E.O. Gobel, Phys. Rev. Lett. 73, 1440 (1994); M. Chandross, S. Mazumdar, S. Jeglinski, X. Wei, Z.V. Vardeny, E.W. Kwock, and T.M. Miller, Phys. Rev. B 50, 14 702 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, Phys. Rev. Lett. 72, 156 (1994); E.M. Conwell, I.J. Rothberg, and H.A. Mizes, ibid. 73, 3179 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, ibid. 73, 3180 (1995); S. Barth and H. Bassler, ibid. 79, 4445 (1997); D. Moses, J. Wang, G. Yu, and A.J. Heeger, ibid. 80, 2685 (1998); S.F. Alvarado, P.F. Seidler, D.G. Lidzey, and D.D.C. Bradley, ibid. 81, 1082 (1998); V.I. Arkhipov, E.V. Emelianova and H. Bassler, ibid. 82; 1321 (1999).
4. P. Gomes da Costa and E.M. Conwell, Phys. Rev. B 48, 1993 (1993); R. Kersting, U. Lemmer, M. Deussen, H.J. Bakker, R.F. Mahrt, H. Kurz, V.I. Arkhipov, H. Bassler, and E.O. Gobel, Phys. Rev. Lett. 73, 1440 (1994); M. Chandross, S. Mazumdar, S. Jeglinski, X. Wei, Z.V. Vardeny, E.W. Kwock, and T.M. Miller, Phys. Rev. B 50, 14 702 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, Phys. Rev. Lett. 72, 156 (1994); E.M. Conwell, I.J. Rothberg, and H.A. Mizes, ibid. 73, 3179 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, ibid. 73, 3180 (1995); S. Barth and H. Bassler, ibid. 79, 4445 (1997); D. Moses, J. Wang, G. Yu, and A.J. Heeger, ibid. 80, 2685 (1998); S.F. Alvarado, P.F. Seidler, D.G. Lidzey, and D.D.C. Bradley, ibid. 81, 1082 (1998); V.I. Arkhipov, E.V. Emelianova and H. Bassler, ibid. 82; 1321 (1999).
5. P. Gomes da Costa and E.M. Conwell, Phys. Rev. B 48, 1993 (1993); R. Kersting, U. Lemmer, M. Deussen, H.J. Bakker, R.F. Mahrt, H. Kurz, V.I. Arkhipov, H. Bassler, and E.O. Gobel, Phys. Rev. Lett. 73, 1440 (1994); M. Chandross, S. Mazumdar, S. Jeglinski, X. Wei, Z.V. Vardeny, E.W. Kwock, and T.M. Miller, Phys. Rev. B 50, 14 702 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, Phys. Rev. Lett. 72, 156 (1994); E.M. Conwell, I.J. Rothberg, and H.A. Mizes, ibid. 73, 3179 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, ibid. 73, 3180 (1995); S. Barth and H. Bassler, ibid. 79, 4445 (1997); D. Moses, J. Wang, G. Yu, and A.J. Heeger, ibid. 80, 2685 (1998); S.F. Alvarado, P.F. Seidler, D.G. Lidzey, and D.D.C. Bradley, ibid. 81, 1082 (1998); V.I. Arkhipov, E.V. Emelianova and H. Bassler, ibid. 82; 1321 (1999).
6. P. Gomes da Costa and E.M. Conwell, Phys. Rev. B 48, 1993 (1993); R. Kersting, U. Lemmer, M. Deussen, H.J. Bakker, R.F. Mahrt, H. Kurz, V.I. Arkhipov, H. Bassler, and E.O. Gobel, Phys. Rev. Lett. 73, 1440 (1994); M. Chandross, S. Mazumdar, S. Jeglinski, X. Wei, Z.V. Vardeny, E.W. Kwock, and T.M. Miller, Phys. Rev. B 50, 14 702 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, Phys. Rev. Lett. 72, 156 (1994); E.M. Conwell, I.J. Rothberg, and H.A. Mizes, ibid. 73, 3179 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, ibid. 73, 3180 (1995); S. Barth and H. Bassler, ibid. 79, 4445 (1997); D. Moses, J. Wang, G. Yu, and A.J. Heeger, ibid. 80, 2685 (1998); S.F. Alvarado, P.F. Seidler, D.G. Lidzey, and D.D.C. Bradley, ibid. 81, 1082 (1998); V.I. Arkhipov, E.V. Emelianova and H. Bassler, ibid. 82; 1321 (1999).
7. P. Gomes da Costa and E.M. Conwell, Phys. Rev. B 48, 1993 (1993); R. Kersting, U. Lemmer, M. Deussen, H.J. Bakker, R.F. Mahrt, H. Kurz, V.I. Arkhipov, H. Bassler, and E.O. Gobel, Phys. Rev. Lett. 73, 1440 (1994); M. Chandross, S. Mazumdar, S. Jeglinski, X. Wei, Z.V. Vardeny, E.W. Kwock, and T.M. Miller, Phys. Rev. B 50, 14 702 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, Phys. Rev. Lett. 72, 156 (1994); E.M. Conwell, I.J. Rothberg, and H.A. Mizes, ibid. 73, 3179 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, ibid. 73, 3180 (1995); S. Barth and H. Bassler, ibid. 79, 4445 (1997); D. Moses, J. Wang, G. Yu, and A.J. Heeger, ibid. 80, 2685 (1998); S.F. Alvarado, P.F. Seidler, D.G. Lidzey, and D.D.C. Bradley, ibid. 81, 1082 (1998); V.I. Arkhipov, E.V. Emelianova and H. Bassler, ibid. 82; 1321 (1999).
8. P. Gomes da Costa and E.M. Conwell, Phys. Rev. B 48, 1993 (1993); R. Kersting, U. Lemmer, M. Deussen, H.J. Bakker, R.F. Mahrt, H. Kurz, V.I. Arkhipov, H. Bassler, and E.O. Gobel, Phys. Rev. Lett. 73, 1440 (1994); M. Chandross, S. Mazumdar, S. Jeglinski, X. Wei, Z.V. Vardeny, E.W. Kwock, and T.M. Miller, Phys. Rev. B 50, 14 702 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, Phys. Rev. Lett. 72, 156 (1994); E.M. Conwell, I.J. Rothberg, and H.A. Mizes, ibid. 73, 3179 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, ibid. 73, 3180 (1995); S. Barth and H. Bassler, ibid. 79, 4445 (1997); D. Moses, J. Wang, G. Yu, and A.J. Heeger, ibid. 80, 2685 (1998); S.F. Alvarado, P.F. Seidler, D.G. Lidzey, and D.D.C. Bradley, ibid. 81, 1082 (1998); V.I. Arkhipov, E.V. Emelianova and H. Bassler, ibid. 82; 1321 (1999).
9. P. Gomes da Costa and E.M. Conwell, Phys. Rev. B 48, 1993 (1993); R. Kersting, U. Lemmer, M. Deussen, H.J. Bakker, R.F. Mahrt, H. Kurz, V.I. Arkhipov, H. Bassler, and E.O. Gobel, Phys. Rev. Lett. 73, 1440 (1994); M. Chandross, S. Mazumdar, S. Jeglinski, X. Wei, Z.V. Vardeny, E.W. Kwock, and T.M. Miller, Phys. Rev. B 50, 14 702 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, Phys. Rev. Lett. 72, 156 (1994); E.M. Conwell, I.J. Rothberg, and H.A. Mizes, ibid. 73, 3179 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, ibid. 73, 3180 (1995); S. Barth and H. Bassler, ibid. 79, 4445 (1997); D. Moses, J. Wang, G. Yu, and A.J. Heeger, ibid. 80, 2685 (1998); S.F. Alvarado, P.F. Seidler, D.G. Lidzey, and D.D.C. Bradley, ibid. 81, 1082 (1998); V.I. Arkhipov, E.V. Emelianova and H. Bassler, ibid. 82; 1321 (1999).
10. P. Gomes da Costa and E.M. Conwell, Phys. Rev. B 48, 1993 (1993); R. Kersting, U. Lemmer, M. Deussen, H.J. Bakker, R.F. Mahrt, H. Kurz, V.I. Arkhipov, H. Bassler, and E.O. Gobel, Phys. Rev. Lett. 73, 1440 (1994); M. Chandross, S. Mazumdar, S. Jeglinski, X. Wei, Z.V. Vardeny, E.W. Kwock, and T.M. Miller, Phys. Rev. B 50, 14 702 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, Phys. Rev. Lett. 72, 156 (1994); E.M. Conwell, I.J. Rothberg, and H.A. Mizes, ibid. 73, 3179 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, ibid. 73, 3180 (1995); S. Barth and H. Bassler, ibid. 79, 4445 (1997); D. Moses, J. Wang, G. Yu, and A.J. Heeger, ibid. 80, 2685 (1998); S.F. Alvarado, P.F. Seidler, D.G. Lidzey, and D.D.C. Bradley, ibid. 81, 1082 (1998); V.I. Arkhipov, E.V. Emelianova and H. Bassler, ibid. 82; 1321 (1999).
11. P. Gomes da Costa and E.M. Conwell, Phys. Rev. B 48, 1993 (1993); R. Kersting, U. Lemmer, M. Deussen, H.J. Bakker, R.F. Mahrt, H. Kurz, V.I. Arkhipov, H. Bassler, and E.O. Gobel, Phys. Rev. Lett. 73, 1440 (1994); M. Chandross, S. Mazumdar, S. Jeglinski, X. Wei, Z.V. Vardeny, E.W. Kwock, and T.M. Miller, Phys. Rev. B 50, 14 702 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, Phys. Rev. Lett. 72, 156 (1994); E.M. Conwell, I.J. Rothberg, and H.A. Mizes, ibid. 73, 3179 (1994); J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, and S. Mazumdar, ibid. 73, 3180 (1995); S. Barth and H. Bassler, ibid. 79, 4445 (1997); D. Moses, J. Wang, G. Yu, and A.J. Heeger, ibid. 80, 2685 (1998); S.F. Alvarado, P.F. Seidler, D.G. Lidzey, and D.D.C. Bradley, ibid. 81, 1082 (1998); V.I. Arkhipov, E.V. Emelianova and H. Bassler, ibid. 82, 1321 (1999).
12. M. Chandross, M. Liess, P.A. Lane, Z.V. Vardeny, M. Hamaguchi, and K. Yoshino, Phys. Rev. B 55, 1486 (1997).
13. M. Chandross and S. Mazumdar, Phys. Rev. B 55, 1497 (1997).
14. M.J. Rice and Yu.N. Gartstein, Phys. Rev. Lett. 73, 2504 (1994)
15. Yu.N. Gartstein, M.J. Rice, and E.M. Conwell, Phys. Rev. B 52, 1683 (1995).
16. N. Kirova, S. Brazovskii, and A.R. Bishop, Synth. Met. 100, 29 (1999); ibid. 101, 271 (1999).
17. N. Kirova, S. Brazovskii, and A.R. Bishop, Synth. Met. 100, 29 (1999); ibid. 101, 271 (1999).
18. J.D. Weibel and D. Yaron, J. Chem. Phys. 116, 6846 (2002).
19. S.R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys. Rev. B 48, 10 345 (1993); Density Matrix Renormalization, edited by I. Peschel, X. Wang, M. Kaulke, and K. Hallberg (Springer, Berlin 1999).
20. S.R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys. Rev. B 48, 10 345 (1993); Density Matrix Renormalization, edited by I. Peschel, X. Wang, M. Kaulke, and K. Hallberg (Springer, Berlin 1999).
21. S.R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys. Rev. B 48, 10 345 (1993); Density Matrix Renormalization, edited by I. Peschel, X. Wang, M. Kaulke, and K. Hallberg (Springer, Berlin 1999).
22. R. Pariser and R.G. Parr, J. Chem. Phys. 21, 446 (1953); J.A. Pople, Trans. Faraday Soc. 42, 1375 (1953).
23. R. Pariser and R.G. Parr, J. Chem. Phys. 21, 446 (1953); J.A. Pople, Trans. Faraday Soc. 42, 1375 (1953).
24. R.J. Bursill, C. Castleton, W. Barford, Chem. Phys. Lett. 294, 305 (1998). The P-P-P model is parametrized by fitting the dominant absoption peaks of benzene to determine the Coulomb parameter U and the phenyl bond transfer integral. The two strong, longitudinally polarized absorption peaks of the two unit oligomer (biphenyl) are then used to parametrize the single-bond transfer integral.
25. S. Ramasesha, D.S. Galvao, and Z.G. Soos, J. Phys. Chem. 97, 2823 (1993); A. Chakrabarti and S. Ramasesha, Int. J. Quantum Chem. 60, 381 (1996); Z.G. Soos, S. Ramasesha, D.S. Galvão, and S. Etemad, Phys. Rev. B 47, 1742 (1993).
26. S. Ramasesha, D.S. Galvao, and Z.G. Soos, J. Phys. Chem. 97, 2823 (1993); A. Chakrabarti and S. Ramasesha, Int. J. Quantum Chem. 60, 381 (1996); Z.G. Soos, S. Ramasesha, D.S. Galvão, and S. Etemad, Phys. Rev. B 47, 1742 (1993).
27. S. Ramasesha, D.S. Galvao, and Z.G. Soos, J. Phys. Chem. 97, 2823 (1993); A. Chakrabarti and S. Ramasesha, Int. J. Quantum Chem. 60, 381 (1996); Z.G. Soos, S. Ramasesha, D.S. Galvão, and S. Etemad, Phys. Rev. B 47, 1742 (1993).
28. D. Yaron, E.E. Moore, Z. Shuai, and J.L. Bredas, J. Chem. Phys. 108, 7451 (1998); G. Fano, F. Ortolani, and L. Ziosi, ibid. 108, 9246 (1998); A. Race, W. Barford, and R.J. Bursill, Phys. Rev. B 64, 035208 (2001).
29. D. Yaron, E.E. Moore, Z. Shuai, and J.L. Bredas, J. Chem. Phys. 108, 7451 (1998); G. Fano, F. Ortolani, and L. Ziosi, ibid. 108, 9246 (1998); A. Race, W. Barford, and R.J. Bursill, Phys. Rev. B 64, 035208 (2001).
30. D. Yaron, E.E. Moore, Z. Shuai, and J.L. Bredas, J. Chem. Phys. 108, 7451 (1998); G. Fano, F. Ortolani, and L. Ziosi, ibid. 108, 9246 (1998); A. Race, W. Barford, and R.J. Bursill, Phys. Rev. B 64, 035208 (2001).
31. R.J. Bursill and W. Barford, Phys. Rev. Lett. 82, 1514 (1999); W. Barford, R.J. Bursill, and M.Yu. Lavrentiev, Phys. Rev. B 65, 075107 (2002).
32. R.J. Bursill and W. Barford, Phys. Rev. Lett. 82, 1514 (1999); W. Barford, R.J. Bursill, and M.Yu. Lavrentiev, Phys. Rev. B 65, 075107 (2002).
33. W. Barford, R.J. Bursill, and M.Yu. Lavrentiev, cond-mat/0003189 (unpublished).
34. C. Zhang, E. Jeckelmann, and S.R. White, Phys. Rev. Lett. 80, 2661 (1998).
35. A different optimized local Hilbert space is developed each time a different calculation (with different target states) is performed. For instance, different calculations will be performed when determining dipole allowed (one-photon absorption) states, as opposed to determining triplet states. In general, the local Hilbert spaces derived in each case will differ.
36. R.J. Bursill, Phys. Rev. B 60, 1643 (1999); R.J. Bursill, R.H. McKenzie, and C.J. Hamer, Phys. Rev. Lett. 80, 5607 (1998); ibid. 83, 408 (1999).
37. R.J. Bursill, Phys. Rev. B 60, 1643 (1999); R.J. Bursill, R.H. McKenzie, and C.J. Hamer, Phys. Rev. Lett. 80, 5607 (1998); ibid. 83, 408 (1999).
38. R.J. Bursill, Phys. Rev. B 60, 1643 (1999); R.J. Bursill, R.H. McKenzie, and C.J. Hamer, Phys. Rev. Lett. 80, 5607 (1998); ibid. 83, 408 (1999).
39. Y. Anusooya, S.K. Pati, and S. Ramasesha, J. Chem. Phys. 106, 10230 (1997).
40. Although extensive convergence tests have not been carried out for the even N cases, in the noninteracting limit (U = 0) agreement with exact results is as good as that obtained for the case of odd N (see Table I). Further, in the interacting case, energies are found to be smooth functions of N, so we are reasonably confident that our results for even N are comparable in accuracy to those for odd N.
41. A. Niko, E. Zojer, F. Meghdadi, C. Ambrosch-Draxl, and G. Leising, Synth. Met. 101, 662 (1999).
42. E. Zojer, N. Koch, P. Puschnig, F. Meghdadi, A. Niko, R. Resel, C. Ambrosch-Draxl, M. Knupfer, J. Fink, J.L. Bredas, and G. Leising, Phys. Rev. B 61, 16538 (2000).
43. A.J. Cadby, P.A. Lane, H. Mellor, S.J. Martin, M. Grell, C. Giebeler, D.D.C. Bradley, M. Wohlgenannt, C. An, and Z.V. Vardeny, Phys. Rev. B 62, 15 604 (2000); H. Mellor, Ph.D. thesis, University of Sheffield, 2000.
44. A.J. Cadby, P.A. Lane, H. Mellor, S.J. Martin, M. Grell, C. Giebeler, D.D.C. Bradley, M. Wohlgenannt, C. An, and Z.V. Vardeny, Phys. Rev. B 62, 15 604 (2000); H. Mellor, Ph.D. thesis, University of Sheffield, 2000.
45. P.A. Lane, M. Liess, Z.V. Vardeny, M. Hamaguchi, M. Ozaki, and K. Yoshino, Synth. Met. 84, 641 (1997).
46. - result of 4.88 eV for N = 6 (see Fig. 5).
47. E.K. Miller, G.S. Maskel, C.Y. Yang, and A.J. Heeger, Phys. Rev. B 60, 8028 (1999).
48. W. Barford, R. J. Bursill, and R. W. Smith, Phys. Rev. B 66, 115205 (2002).
49. A. Abe, J. Yu, and W.P. Su, Phys. Rev. B 45, 8264 (1992).
50. Mapping the low-energy physics of poly(para-phenylene) onto a two-band dimerized P-P-P model suggest a U of around 3-4 eV and δ = 0.2, which lies in the weak-coupling limit (Ref. 29).
51. W. Barford, R.J. Bursill, and M.Yu. Lavrentiev, J. Phys.: Condens. Matter 10, 6429 (1998).
52. + state visible in electroabsortion.
53. E.E. Moore and D. Yaron, J. Chem. Phys. 109, 6147 (1998).
54. B.E. Kohler, J. Chem. Phys. 88, 2788 (1988).
55. W. Barford, Phys. Rev. B 65, 205118 (2002).
56. R. J. Bursill, E. Jeckelmann, and W. Barford (unpublished).
57. 2 gives estimates of the bulk energies, at least for the lowest-lying state in a given symmetry sector, and for the intraphenyl exciton, to within confidence limits dictated by the DMRG truncation error (around 0.02 eV).
58. -, Whence the ratio of B values is reasonably close to the expected ratio of 1:9. However, a detailed investigation of Eq. (2) for the whole series requires substantially larger values of N and probably substantially higher DMRG accuracy and is beyond the scope of this paper.
59. - exciton.
60. - energies very much (Ref. 39).
61. C. Castleton and W. Barford, J. Chem. Phys. 117, 3570 (2002).