Skip to main content
SpringerLink
Log in

Advanced Car–Parrinello Techniques: Path Integrals and Nonadiabaticity in Condensed Matter Simulations

  • Chapter
Computer Simulations in Condensed Matter Systems: From Materials to Chemical Biology Volume 2

Part of the book series: Lecture Notes in Physics ((LNP,volume 704))

Abstract

Extensions of Car–Parrinello (CP) ab initio molecular dynamics are presented for efficient treatments of nuclear quantum effects and electronically nonadiabatic processes in the realm of condensed matter simulations. Ab initio path integrals, being a combination of CP propagation of the electrons in conjunction with path integral MD sampling of the nuclei, allow to investigate quantum phenomena, such as the influence of zero-point motion and proton tunneling, in chemically complex systems. Nonadiabatic ab initio simulations rely on the coupling of the Kohn-Sham ground state, S 0, and the first excited electronic state, S 1, obtained within the restricted open-shell Kohn- Sham (ROKS) approach using Tully’s surface hopping algorithm. The efficient evaluation of the nonadiabatic couplings together with an “on-the-fly” updating scheme makes possible nonadiabatic ab initio simulations of systems of similar complexity as those typically studied by ground-state CP methods. This method is thus ideally suited to study photoinduced reactions of large molecular systems, particularly in condensed phases.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. N. Barnett, H.-P. Cheng, H. Hakkinen, and U. Landman (1995) J. Phys. Chem. 99, p. 7731

    Article  Google Scholar 

  2. M. Benoit, D. Marx, and M. Parrinello (1998) Comp. Mat. Sci. 10, p. 88

    Article  Google Scholar 

  3. M. Benoit, D. Marx, and M. Parrinello (1998) Nature 392, p. 258; see also J. Teixeira, Nature (1998) 392, p. 232

    Google Scholar 

  4. M. Benoit, D. Marx, and M. Parrinello (1999) Solid State Ionics 125, p. 23

    Article  Google Scholar 

  5. M. Benoit and D. Marx (2005) Chem. Phys. Chem. 6, p. 1738

    Google Scholar 

  6. S. Biermann, D. Hohl, and D. Marx (1998) J. Low Temp. Phys. 110, p. 97

    Article  Google Scholar 

  7. S. Biermann, D. Hohl, and D. Marx (1998) Solid State Commun. 108, p. 337

    Article  ADS  Google Scholar 

  8. S. R. Billeter and A. Curioni (2005) J. Chem. Phys. 122, p. 034105

    Article  ADS  Google Scholar 

  9. P. E. Blöchl and M. Parrinello (1992) Phys. Rev. B 45, p. 9413

    Article  ADS  Google Scholar 

  10. F. A. Bornemann and C. Schütte (1998) Numer. Math. 78, p. 359

    Article  MATH  MathSciNet  Google Scholar 

  11. D. J. E. Callaway and A. Rahman (1982) Phys. Rev. Lett. 49, p. 613

    Article  ADS  Google Scholar 

  12. J. Cao and B. J. Berne (1993) J. Chem. Phys. 99, p. 2902

    Article  ADS  Google Scholar 

  13. J. Cao and G. A. Voth (1993) J. Chem. Phys. 99, p. 10070

    Article  ADS  Google Scholar 

  14. J. Cao and G. A. Voth (1994) J. Chem. Phys. 100, p. 5106

    Article  ADS  Google Scholar 

  15. J. Cao and G. A. Voth (1994) J. Chem. Phys. 101, p. 6157

    Article  ADS  Google Scholar 

  16. J. Cao and G. A. Voth (1994) J. Chem. Phys. 101, p. 6168

    Article  ADS  Google Scholar 

  17. J. Cao and G. J. Martyna (1996) J. Chem. Phys. 104, p. 2028

    Article  ADS  Google Scholar 

  18. R. Car and M. Parrinello (1985) Unified Approach for Molecular Dynamics and Density-Functional Theory. Phys. Rev. Lett. 55, pp. 2471-2474; this article belongs to the most quoted ”Physical Review Letters, Top 10”, see http://www.aps.org/apsnews/topten.html, and led in 1996 to the definition of its own Pacs number, 71.15.Pd.

    Google Scholar 

  19. D. M. Ceperley (1995) Rev. Mod. Phys. 67, p. 279; for errata and updates see http://archive.ncsa.uiuc.edu/Science/CMP/papers/cep95a/cep95a.html

    Google Scholar 

  20. Ch. Chakravarty (1997) Int. Rev. Phys. Chem. 16, p. 421

    Article  Google Scholar 

  21. D. Chandler (1991) In Liquids, Freezing, and Glass Transition, eds. J. P. Hansen, D. Levesque, and J. Zinn-Justin Elsevier, Amsterdam

    Google Scholar 

  22. B. Chen, I. Ivanov, M. L. Klein, and M. Parrinello (2003) Phys. Rev. Lett. 91, p. 215503

    Article  ADS  Google Scholar 

  23. H.-P. Cheng, R. N. Barnett, and U. Landman (1995) Chem. Phys. Lett. 237, p. 161

    Article  ADS  Google Scholar 

  24. R. D. Coalson (1986) J. Chem. Phys. 85, p. 926

    Article  ADS  Google Scholar 

  25. C. Daul (1994) Int. J. Quantum Chem. 52, p. 867

    Article  Google Scholar 

  26. N. L. Doltsinis, Nonadiabatic Dynamics: Mean-Field and Surface Hopping, in [53]

    Google Scholar 

  27. N. L. Doltsinis and D. Marx (2002) Phys. Rev. Lett. 88, p. 166402

    Article  ADS  Google Scholar 

  28. N. L. Doltsinis and D. Marx (2002) J. Theor. Comput. Chem. 1, p. 319; see http://www.theochem.rub.de/go/cprev.html

    Google Scholar 

  29. N. L. Doltsinis (2004) Mol. Phys. 102, p. 499

    Article  ADS  Google Scholar 

  30. N. L. Doltsinis (2004) Faraday Discuss. 127, p. 231

    Google Scholar 

  31. N. L. Doltsinis and K. Fink (2004) Unpublished calculations

    Google Scholar 

  32. N. L. Doltsinis and K. Fink (2005) J. Chem. Phys. 122, p. 087101; Comment to [114], for Reply see [41]

    Article  ADS  Google Scholar 

  33. W. Domcke and G. Stock (1997) Adv. Chem. Phys. 100, p. 1

    Article  Google Scholar 

  34. R. M. Dreizler and E. K. U. Gross (1990) Density-Functional Theory. Springer, Berlin Heidelberg

    MATH  Google Scholar 

  35. R. P. Feynman and A. R. Hibbs (1965) Quantum Mechanics and Path Integrals. McGraw-Hill, New York

    MATH  Google Scholar 

  36. R. P. Feynman (1972) Statistical Mechanics. Addison-Wesley, Redwood City

    Google Scholar 

  37. M. Filatov and S. Shaik (1998) Chem. Phys. Lett. 288, p. 689

    Article  ADS  Google Scholar 

  38. M. Filatov and S. Shaik (1999) J. Chem. Phys. 110, p. 116

    Article  ADS  Google Scholar 

  39. M. Filatov and S. Shaik (1999) Chem. Phys. Lett. 304, p. 429

    Article  ADS  Google Scholar 

  40. M. Filatov and S. Shaik (2000) Chem. Phys. Lett. 332, p. 409

    Article  ADS  Google Scholar 

  41. C. Filippi and F. Buda (2005) J. Chem. Phys. 122, p. 087102; Reply to Comment [32]

    Article  ADS  Google Scholar 

  42. J. Ford (1992) Phys. Rep. 213, p. 271

    Article  ADS  MathSciNet  Google Scholar 

  43. I. Frank, J. Hutter, D. Marx, and M. Parrinello (1998) J. Chem. Phys. 108, p. 4060

    Article  ADS  Google Scholar 

  44. G. Galli and M. Parrinello (1991) In Computer Simulations in Materials Science, p. 282, eds. M. Meyer and V. Pontikis Kluwer, Dordrecht

    Google Scholar 

  45. R. Gaudoin and K. Burke (2004) Phys. Rev. Lett. 93, p. 173001; Publisher’s Note (2005) Phys. Rev. Lett. 94, p. 029901

    Google Scholar 

  46. M. J. Gillan (1990) In Computer Modelling of Fluids, Polymers, and Solids, eds. C. R. A. Catlow, S. C. Parker, and M. P. Allen Kluwer, Dordrecht

    Google Scholar 

  47. S. Goedecker and C. J. Umrigar (1997) Phys. Rev. A 55, p. 1765

    Article  ADS  Google Scholar 

  48. A. Görling (2000) Phys. Rev. Lett. 85, p. 4229

    Article  ADS  Google Scholar 

  49. J. Gräfenstein, E. Kraka, and D. Cremer (1998) Chem. Phys. Lett. 288, p. 593

    Article  ADS  Google Scholar 

  50. J. Gräfenstein and D. Cremer (2000) Phys. Chem. Chem. Phys. 2, p. 2091

    Article  Google Scholar 

  51. S. Grimm, C. Nonnenberg, and I. Frank (2003) J. Chem. Phys. 119, p. 11574

    Article  ADS  Google Scholar 

  52. E. K. U. Gross and W. Kohn (1990) Adv. Quant. Chem. 21, p. 255

    Article  Google Scholar 

  53. J. Grotendorst, D. Marx, and A. Muramatsu, eds., Quantum Simulations of Complex Many-Body Systems: From Theory to Algorithms. (John von Neumann Institute for Computing, Forschungszentrum Jülich 2002); for Electronic Version and Audio-Visual Lecture Notes see: http://www.theochem.rub.de/go/cprev.html

    Google Scholar 

  54. R. W. Hall and B. J. Berne (1984) J. Chem. Phys. 81, p. 3641

    Article  ADS  Google Scholar 

  55. A. Hellmann, B. Razaznejad, and B. I. Lundqvist (2004) J. Chem. Phys. 120, p. 4593

    Article  ADS  Google Scholar 

  56. J. Hutter et al., CPMD: Car-Parrinello Molecular Dynamics: An Ab Initio Electronic Structure and Molecular Dynamics Program; IBM Zurich Research Laboratory (1990-2005) and Max-Planck-Institut für Festkörperforschung (1997-2001); for downloads see: http://www.cpmd.org/

    Google Scholar 

  57. F. Illas, I. de P. R. Moreira, C. de Graaf, and V. Barone (2000) Theor. Chem. Acc. 104, p. 265

    Google Scholar 

  58. S. Ismail Beigi and S. G. Louie (2003) Phys. Rev. Lett. 90, p. 076401

    Article  ADS  Google Scholar 

  59. H. Kitamura, S. Tsuneyuki, T. Ogitsu, and T. Miyake (2000) Nature 404, p. 259

    Article  ADS  Google Scholar 

  60. H. Kleinert (2004) Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets. World Scientific, Singapore

    MATH  Google Scholar 

  61. M. Klessinger and J. Michl (1995) Excited States and Photochemistry of Organic Molecules. VCH, New York

    Google Scholar 

  62. L. Knoll, Z. Vager, and D. Marx (2003) Phys. Rev. A 67, p. 022506

    Article  ADS  Google Scholar 

  63. W. Kołos (1970) Adv. Quant. Chem. 5, p. 99

    Article  Google Scholar 

  64. H. Köppel, W. Domcke, and L. S. Cederbaum (1984) Adv. Chem. Phys. 57, p. 59

    Article  Google Scholar 

  65. K. Laasonen, A. Pasquarello, R. Car, C. Lee, and D. Vanderbilt (1993) Phys. Rev. B 47, pp. 10-142

    Article  Google Scholar 

  66. H. Langer and N. L. Doltsinis (2004) Phys. Chem. Chem. Phys. 6, p. 2742

    Article  Google Scholar 

  67. H. Langer, N. L. Doltsinis, and D. Marx (2005) Chem. Phys. Chem. 6, p. 1734

    Google Scholar 

  68. M. A. L. Marques and E. K. U. Gross (2004) Annu. Rev. Phys. Chem. 55, p. 427

    Article  ADS  Google Scholar 

  69. G. J. Martyna, M. L. Klein, and M. Tuckerman (1992) J. Chem. Phys. 97, p. 2635

    Article  ADS  Google Scholar 

  70. G. J. Martyna (1996) J. Chem. Phys. 104, p. 2018

    Article  ADS  Google Scholar 

  71. G. J. Martyna, A. Hughes, and M. E. Tuckerman (1999) J. Chem. Phys. 110, p. 3275

    Article  ADS  Google Scholar 

  72. D. Marx and M. Parrinello (1994) Z. Phys. B (Rapid Note) 95, p. 143; a misprinted sign in the Lagrangian is correctly given in (21) of the present review

    Google Scholar 

  73. D. Marx and M. Parrinello (1995) Nature 375, p. 216

    Article  ADS  Google Scholar 

  74. D. Marx and M. Parrinello (1996) J. Chem. Phys. 104, p. 4077

    Article  ADS  Google Scholar 

  75. D. Marx and M. Parrinello (1996) Science 271, p. 179

    Article  ADS  Google Scholar 

  76. D. Marx and A. Savin (1997) Angew. Chem. Int. Ed. Engl. 36, p. 2077

    Article  Google Scholar 

  77. D. Marx (1998) In Classical and Quantum Dynamics in Condensed Phase Simulations. Chap. 15, eds. B. J. Berne, G. Ciccotti, and D. F. CokerWorld Scientific, Singapore

    Google Scholar 

  78. D. Marx and M. H. Müser (1999) J. Phys. Condens. Matter 11, p. R117

    Article  ADS  Google Scholar 

  79. D. Marx, M. E. Tuckerman, J. Hutter, and M. Parrinello (1999) Nature 397, p. 601; see also J. T. Hynes (1999) Nature 397, p. 565

    Google Scholar 

  80. D. Marx, M. E. Tuckerman, and G. J. Martyna (1999) Comput. Phys. Commun.118, p. 166; the misprinted definition of the fictitious normal mode masses given in (2.51) is corrected in (40) of the present review. However, the correct definition was implemented in the CPMD package [56] so that all data reported in the paper are unaffected

    Google Scholar 

  81. D. Marx and M. Parrinello (1999) Science 286, p. 1051

    Article  Google Scholar 

  82. D. Marx, M. E. Tuckerman, and M. Parrinello (2000) J. Phys.: Condens. Matter 12, p. A153

    Article  ADS  Google Scholar 

  83. D. Marx and J. Hutter, “Ab Initio Molecular Dynamics: Theory and Implementation,” In Modern Methods and Algorithms of Quantum Chemistry, pp. 301-449 (.rst edition, paperback, ISBN 3-00-005618-1) or pp. 329-477 (second edition, hardcover, ISBN 3-00-005834-6), ed. J. Grotendorst, (John von Neumann Institute for Computing, Forschungszentrum Jülich 2000); see: http://www.theochem.rub.de/go/cprev.html

    Google Scholar 

  84. H. S. W. Massey (1949) Rep. Prog. Phys. 12, p. 248

    Article  ADS  Google Scholar 

  85. S. Miura, M. E. Tuckerman, and M. L. Klein (1998) J. Chem. Phys. 109, p. 5290

    Article  ADS  Google Scholar 

  86. T. Miyake, T. Ogitsu, and S. Tsuneyuki (1998) Phys. Rev. Lett. 81, p. 1873

    Article  ADS  Google Scholar 

  87. T. Miyake, T. Ogitsu, and S. Tsuneyuki (1999) Phys. Rev. B 60, pp. 14–197

    Article  Google Scholar 

  88. Á. Nagy (1998) Phys. Rev. A 57, p. 1672

    Article  ADS  MathSciNet  Google Scholar 

  89. H. Nakamura (2002) Nonadiabatic Transition: Concepts, Basic Theories and Applications. World Scientific, Singapore

    Book  Google Scholar 

  90. C. Nonnenberg, S. Grimm, and I. Frank (2003) J. Chem. Phys. 119, p. 11585

    Article  ADS  Google Scholar 

  91. L. Noodleman (1981) J. Chem. Phys. 74, p. 5737

    Article  ADS  Google Scholar 

  92. M. Odelius, D. Laikov, and J. Hutter (2003) J. Mol. Struc. (Theochem.) 630, p. 163

    Article  Google Scholar 

  93. R. G. Parr and W. Yang (1989) Density-Functional Theory of Atoms and Molecules. Oxford University Press, Oxford

    Google Scholar 

  94. M. Parrinello and A. Rahman (1984) J. Chem. Phys. 80, p. 860

    Article  ADS  Google Scholar 

  95. M. Parrinello (1997) Solid State Commun. 102, p. 107

    Article  ADS  Google Scholar 

  96. G. Pastore, E. Smargiassi, and F. Buda (1991) Phys. Rev. A 44, p. 6334

    Article  ADS  Google Scholar 

  97. G. Pastore (1996) In Monte Carlo and Molecular Dynamics of Condensed Matter Systems, Chap. 24, p. 635, eds. K. Binder and G. Ciccotti Italien Physical Society SIF, Bologna

    Google Scholar 

  98. M. Pavese, D. R. Berard, and G. A. Voth (1999) Chem. Phys. Lett. 300, p. 93

    Article  ADS  Google Scholar 

  99. M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias, and J. D. Joannopoulos (1992) Rev. Mod. Phys. 64, p. 1045

    Article  ADS  Google Scholar 

  100. Á. J. Pérez Jiménez, J. M. Pérez Jordá, and F. Illas (2004) J. Chem. Phys.120, p. 18

    Article  ADS  Google Scholar 

  101. J. Polonyi and H. W. Wild (1983) Phys. Rev. Lett. 51, p. 2257

    Article  ADS  Google Scholar 

  102. B. De Raedt, M. Sprik, and M. L. Klein (1984) J. Chem. Phys. 80, p. 5719

    Article  ADS  Google Scholar 

  103. R. Ramírez, T. López Ciudad, and J. C. Noya (1998) Phys. Rev. Lett. 81, p. 3303

    Article  ADS  Google Scholar 

  104. R. Ramírez and T. López Ciudad (1999) J. Chem. Phys. 111, p. 3339

    Article  ADS  Google Scholar 

  105. R. Ramírez and T. López Ciudad (1999) Phys. Rev. Lett. 83, p. 4456

    Article  MATH  ADS  MathSciNet  Google Scholar 

  106. R. Ramírez and T. López Ciudad, Dynamic Properties via Fixed Centroid Path Integrals, in [53]

    Google Scholar 

  107. D. K. Remler and P. A. Madden (1990) Molec. Phys. 70, p. 921

    Article  ADS  Google Scholar 

  108. M. A. Robb, M. Garavelli, M. Olivucci, and F. Bernardi (2000) Rev. Comp. Ch. 15, p. 87

    Article  Google Scholar 

  109. C. C. J. Roothaan (1960) Rev. Mod. Phys. 32, p. 179

    Article  MATH  ADS  MathSciNet  Google Scholar 

  110. R. Rousseau and D. Marx (1998) Phys. Rev. Lett. 80, p. 2574

    Article  ADS  Google Scholar 

  111. R. Rousseau and D. Marx (1999) J. Chem. Phys. 111, p. 5091

    Article  ADS  Google Scholar 

  112. E. Runge and E. K. U. Gross (1984) Phys. Rev. Lett. 52, p. 997

    Article  ADS  Google Scholar 

  113. F. Della Sala, R. Rousseau, A. Görling, and D. Marx (2004) Phys. Rev. Lett.92, p. 183401

    Article  ADS  Google Scholar 

  114. F. Schautz, F. Buda, and C. Filippi (2004) J. Chem. Phys. 121, p. 5836; see also a Comment [32] and the Reply [41]

    Article  ADS  Google Scholar 

  115. K. E. Schmidt and D. M. Ceperley (1992) In The Monte Carlo Method in Condensed Matter Physics, p. 205, ed. K. Binder Springer-Verlag, Berlin Heidelberg

    Google Scholar 

  116. I. Štich, D. Marx, M. Parrinello, and K. Terakura (1997) Phys. Rev. Lett. 78, p. 3669

    Article  ADS  Google Scholar 

  117. I. Štich, D. Marx, M. Parrinello, and K. Terakura (1997) J. Chem. Phys. 107, p. 9482

    Article  ADS  Google Scholar 

  118. G. Stock and M. Thoss (2005) Adv. Chem. Phys. 131, p. 243

    Article  Google Scholar 

  119. A. C. Stückl, C. A. Daul, and H. U. Güdel (1997) Int. J. Quantum Chem. 61, p. 579

    Article  Google Scholar 

  120. A. Szabo and N. S. Ostlund (1989) Modern Quantum Chemistry - Introduction to Advanced Electronic Structure Theory (McGraw-Hill Publishing Company, New York

    Google Scholar 

  121. M. E. Tuckerman, B. J. Berne, G. J. Martyna, and M. L. Klein (1993) J. Chem. Phys. 99, p. 2796

    Article  ADS  Google Scholar 

  122. M. E. Tuckerman, D. Marx, M. L. Klein, and M. Parrinello (1996) J. Chem. Phys. 104, p. 5579

    Article  ADS  Google Scholar 

  123. M. E. Tuckerman, D. Marx, M. L. Klein, and M. Parrinello (1997) Science 275, p. 817

    Article  Google Scholar 

  124. M. E. Tuckerman and A. Hughes (1998) In Classical and Quantum Dynamics in Condensed Phase Simulations, Chap. 14, p. 311, eds. B. J. Berne, G. Ciccotti, and D. F. Coker World Scientific, Singapore

    Chapter  Google Scholar 

  125. M. E. Tuckerman and D. Marx (2001) Phys. Rev. Lett. 86, p. 4946; see animation of quantum fluctuations and tunneling proton http://www.theochem.rub.de/go/malon.html

    Google Scholar 

  126. M. E. Tuckerman, D. Marx, and M. Parrinello (2002) Nature 417, p. 925; see also R. Ludwig (2003) Angew. Chem. Int. Ed. 42, p. 258

    Google Scholar 

  127. J. C. Tully and R. K. Preston (1971) J. Chem. Phys. 55, p. 562

    Article  ADS  Google Scholar 

  128. J. C. Tully (1990) J. Chem. Phys. 93, p. 1061

    Article  ADS  Google Scholar 

  129. J. C. Tully (1998) In Modern Methods for Multidimensional Dynamics Computations in Chemistry, ed. D. L. Thompson World Scientific, Singapore

    Google Scholar 

  130. J. C. Tully (1998) In Classical and Quantum Dynamics in Condensed Phase Simulations. Chap. 21, p. 489, eds. B. J. Berne, G. Ciccotti, and D. F. Coker World Scientific, Singapore

    Chapter  Google Scholar 

  131. P. V. Parandekar and J. C. Tully (2005) J. Chem. Phys. 122, p. 094102

    Article  ADS  Google Scholar 

  132. G. A. Voth (1996) Adv. Chem. Phys. 93, p. 135

    Article  Google Scholar 

  133. D. R. Yarkony (1996) Rev. Mod. Phys. 68, p. 985

    Article  ADS  Google Scholar 

  134. A. H. Zewail (1994) Femtochemistry: Ultrafast Dynamics of the Chemical Bond. World Scientific, Singapore

    Book  Google Scholar 

  135. T. Ziegler, A. Rauk, and E. J. Baerends (1977) Theor. Chim. Acta 43, p. 261

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lehrstuhl für Theoretische Chemie, Ruhr–Universität Bochum, 44780, Bochum, Germany

    D. Marx

Authors
  1. D. Marx
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dipartimento di Fisica, Università di Modena e Reggio Emilia, Via Campi, 213/A, 41100, Modena, Italy

    Mauro Ferrario Professor

  2. Dipartimento di Fisica, INFN, Università di Roma La Sapienza, Piazzale Aldo Moro 2, 00185, Roma, Italy

    Giovanni Ciccotti Professor

  3. Institut für Physik, Universität Mainz, Staudinger Weg 7, 55128, Mainz, Germany

    Kurt Binder Professor

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer

About this chapter

Cite this chapter

Marx, D. (2006). Advanced Car–Parrinello Techniques: Path Integrals and Nonadiabaticity in Condensed Matter Simulations. In: Ferrario, M., Ciccotti, G., Binder, K. (eds) Computer Simulations in Condensed Matter Systems: From Materials to Chemical Biology Volume 2. Lecture Notes in Physics, vol 704. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-35284-8_19

Download citation

Publish with us

Policies and ethics

Search

Navigation