Skip to main content
SpringerLink
Log in

Confined Quantum Systems Using the Finite Element and Discrete Variable Representation Methods

  • Chapter
  • First Online:
Electronic Structure of Quantum Confined Atoms and Molecules

Abstract

Confined quantum systems, particles or systems of particles that have their movements limited to a determined region of the space, have received attention since the beginning from the Quantum Theory. The physical and chemical properties of confined objects are modified with respect to free objects such as due the spatial confinement as to other factors as, for example, the electromagnetic field, not saturated chemical bounds, etc. In recent years, the interest in this area has grown sufficiently due the great set of phenomena and physical processes which can be characterized or be understood as confined quantum systems and that have various technological applications. The confined quantum systems are important, for example, in the embedding of atoms and molecules inside cavities such as zeolite molecular sieves, fullerenes, or solvent environments; in bubbles formed around foreign objects in the liquid helium or neutral plasma; in semiconductors structures in the mesoscopic-scale, as artificial atoms and molecules, or quantum dots; in atoms under pressure that are important for the agreement of the interior of planets, among others. Thus, different theoretical and computational methodologies have been used to study confined quantum systems. In particular, methods based on the variational formalism that expand the wave function in a set of basis functions as, for example, the discrete variable representation and the finite element method have been used successfully to treat systems with few electrons. In the present work our major aim is to present a review of applications of this class of methods to study typical confined quantum systems as the hydrogen atom and the two electron quantum dot, discussing the advantages and disadvantages of each one of them.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 219.99
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

Notes

  1. 1.

    Note that for a repulsive spherical cage the natural position of an atom is in the center, but for attractive potentials, this is not a general situation. However, even if the atom is off-center, it is reasonable first solve the problem with spherical symmetry, and then develop expansions that represent the effect of the displacement of the atom to some other position. Similarly, the surface of confinement need not be spherical, however, it is reasonable to start with a sphere, then consider how the system is modified by the distortion of the confining surface.

References

  1. Jaskólski W (1996) Phys Rep 271:1

    Google Scholar 

  2. Dolmatov VK, Baltenkov AS, Connerade J-P, Manson ST (2004) Rad Phys Chem 70:417

    CAS  Google Scholar 

  3. Sabin J, Brandas E, Cruz SA (2009) Advances in quantum chemistry, vol 57. Academic, Oxford

    Google Scholar 

  4. Sabin J, Brandas E, Cruz SA (2009) Advances in quantum chemistry, vol 58. Academic, Oxford

    Google Scholar 

  5. Jacak L, Hawrylak O, Wojs A (1998) Quantum dots. Springer, Berlin

    Google Scholar 

  6. Ram-Moham LR (2002) Finite element and boundary element applications in quantum mechanics. Oxford University Press, New York

    Google Scholar 

  7. Bacic Z, Light JC (1999) Ann Rev Phys Chem 40:469

    Google Scholar 

  8. Light JC, Carrington T Jr (2000) Adv Chem Phys 114:263

    Google Scholar 

  9. Guimarães MN, Prudente FV (2005) J Phys B At Mol Opt Phys 38:2811

    Google Scholar 

  10. Zicovich-Wilson C, Planelles JH, Jaskólski W (1994) Int J Quant Chem 50:429

    CAS  Google Scholar 

  11. Costa LS, Prudente FV, Acioli PH, Soares Neto JJ, Vianna JDM (1999) J Phys B At Mol Opt Phys 32:2461

    CAS  Google Scholar 

  12. Adamowski J, Sobkowicz M, Szafran B, Bednarek S (2000) Phys Rev B 62:4234

    CAS  Google Scholar 

  13. Sako T, Diercksen GHF (2003) J Phys Condens Matter 15:5487

    CAS  Google Scholar 

  14. Carvalho CR, Jalbert G, Rocha AB, Brandi HS (2003) J Appl Phys 94:2579

    CAS  Google Scholar 

  15. Connerade JP, Dolmatov VK, Lakshmi PA, Manson ST (1999) J Phys B At Mol Opt Phys 32:L239

    CAS  Google Scholar 

  16. Amusia MY, Baltenkov AS, Becker U (1998) Phys Lett A 243:99

    CAS  Google Scholar 

  17. Nascimento EM, Prudente FV, Guimarães MN, Maniero AM (2011) J Phys B At Mol Opt Phys 44:015003

    Google Scholar 

  18. Saha B, Mukherjee PK, Diercksen GHF (2002) Astron Astrophys 396:337

    CAS  Google Scholar 

  19. Bhattacharyya S, Sil AN, Fritzsche S, Mukherjee PK (2008) Eur Phys J D 46:1

    CAS  Google Scholar 

  20. Pfannkuche D, Gudmundsson V, Maksym PA (1993) Phys Rev B 47:2244

    CAS  Google Scholar 

  21. Creffield CE, Jefferson JH, Sarkar S, Tipton DLJ (2000) Phys Rev B 62:7249

    CAS  Google Scholar 

  22. Yao W, Yu Z, Liu Y, Jia B (2010) J Nanosci Nanotech 10:7612

    CAS  Google Scholar 

  23. Thompson DC, Alavi A (2005) J Chem Phys 122:124107

    Google Scholar 

  24. De Giovannini U, Cavaliere F, Cenni R, Sassetti M, Kramer B (2008) Phys Rev B 77:035325

    Google Scholar 

  25. Yakar Y, Cakir B, Ozmen A (2011) Int J Quantum Chem 111:4139

    CAS  Google Scholar 

  26. Odriazola A, Ervasti MM, Makkonen I, Delgado A, Gonzalez A, Rasanen E, Harju A (2013) J Phys Cond Matt 25:505504

    Google Scholar 

  27. Bryant GW (1987) Phys Rev Lett 59:1140

    CAS  Google Scholar 

  28. Jung J, Alvarellos JE (2002) J Chem Phys 118:10825

    Google Scholar 

  29. Jiang TF, Tong X, Chu S (2001) Phys Rev B 63:045317

    Google Scholar 

  30. Räsänen E, Harju A, Puska MJ, Nieminen RM (2004) Phys Rev B 69:165309

    Google Scholar 

  31. Jung J, García-González P, Alvarellos JE, Godby RW (2004) Phys Rev A 69:052501

    Google Scholar 

  32. Akyuz GB, Akgungor K, Sakiroglu S, Siddiki A, Sokmen I (2011) Physica E 43:1514

    Google Scholar 

  33. Avali A (2000) J Chem Phys 113:7735

    Google Scholar 

  34. Xie W (2006) Phys Rev B 74:115305

    Google Scholar 

  35. Cipriani G, Rosa-Clot M, Taddei S (2000) Phys Rev B 61:7536

    CAS  Google Scholar 

  36. Harting J, Mülken O, Borrmann P (2000) Phys Rev B 62:10207

    CAS  Google Scholar 

  37. Moreira NL, Cândido L, Rabelo JNT, Marques GE (2009) Semicond Sci Technol 24:075009

    Google Scholar 

  38. Taut M (1993) Phys Rev A 48:3561

    CAS  Google Scholar 

  39. Kestner NR, Sinanoglu O (1962) Phys Rev 128:2687

    Google Scholar 

  40. Dineykhan M, Nazmitdinov RG (1997) Phys Rev B 55:13707

    CAS  Google Scholar 

  41. Barakat T, Al-Rawaf AS (2011) Phys Scr 83:055001

    Google Scholar 

  42. Cantele G, Ninno D, Iadonisi G (2001) Phys Rev B 64:125325

    Google Scholar 

  43. Shi L, Yan Z (2011) J Appl Phys 110:024306

    Google Scholar 

  44. Klama S, Mishchenko EG (1998) J Phys Condens Matter 10:3411

    CAS  Google Scholar 

  45. Serra L, Nazmitdinov RG, Puente A (2003) Phys Rev B 035341

    Google Scholar 

  46. Sako T, Diercksen GHF (2003) J Phys B At Mol Opt Phys 36:1681

    CAS  Google Scholar 

  47. Harris DO, Engerholm GG, Gwinn WD (1965) J Chem Phys 43:1515

    Google Scholar 

  48. Dickinson AS, Certain PR (1968) J Chem Phys 49:4209

    CAS  Google Scholar 

  49. Light JC, Hamilton IP, Lill JV (1985) J Chem Phys 82:1400

    CAS  Google Scholar 

  50. Prudente FV, Costa LS, Vianna JDM (2005) J Chem Phys 123:224701

    Google Scholar 

  51. Kubota Y, Nobusada K (2007) Phys Lett A 369:128

    CAS  Google Scholar 

  52. Xu M, Sebastianelli F, Gibbons BR, Bacic Z, Lawler R, Turro NJ (2009) J Chem Phys 130:224306

    Google Scholar 

  53. Lin CY, Ho YK (2013) Few Body Syst 54:425

    CAS  Google Scholar 

  54. Parrish RM, Hohenstein EG, Martinez TJ, Sherrill CD (2013) J Chem Phys 138:194107

    Google Scholar 

  55. Lombardi M, Barletta P, Kievsky A (2004) Phys Rev A 70:032503

    Google Scholar 

  56. Szalay V (1993) J Chem Phys 99:1978

    CAS  Google Scholar 

  57. Echave J, Clary DC (1992) Chem Phys Lett 190:225

    CAS  Google Scholar 

  58. Bitencourt ACP, Prudente FV, Vianna JDM (2007) J Phys B At Mol Opt Phys 40:2075

    CAS  Google Scholar 

  59. Colbert DT, Miller WH (1992) J Chem Phys 96:1982

    CAS  Google Scholar 

  60. Jin J (1993) The finite element method in electromagnetics. Wiley, New York

    Google Scholar 

  61. Zienkiewicz OC (1971) The finite element method in engineering science. McGraw-Hill, New York

    Google Scholar 

  62. Soares Neto JJ, Prudente FV (1994) Theor Chim Acta 89:415

    CAS  Google Scholar 

  63. Prudente FV, SoaresNeto JJ (1999) Chem Phys Lett 302:43

    CAS  Google Scholar 

  64. Pask JE, Klein BM, Sterne PA, Fong CY (2001) Comput Phys Commun 135:1

    CAS  Google Scholar 

  65. Pask JE, Sterne PA (2005) Model Simul Mater Sci Eng 13:R71

    CAS  Google Scholar 

  66. Salci M, Levin S, Elander N, Yarevsky E (2008) J Chem Phys 129:134304

    Google Scholar 

  67. Alizadegan R, Hsia K, Martinez T (2010) J Chem Phys 132:034101

    CAS  Google Scholar 

  68. Guimarães MN, Ragni M, Bitencourt ACP, Prudente FV (2013) Eur Phys J D 67:253

    Google Scholar 

  69. Qu F, Alcalde AM, Almeida CG, Dantas NO (2003) J Appl Phys 94:3462

    CAS  Google Scholar 

  70. Ramírez HY, Santana A (2012) Comp Phys Commun 183:1654

    Google Scholar 

  71. Jurczak G, Young TD (2012) Appl Surf Sci 260:59

    CAS  Google Scholar 

  72. Linderberg J, Padkjær SB, Öhrn Y, Vessal B (1989) J Chem Phys 90:6254

    CAS  Google Scholar 

  73. Jaquet R, Schnupf U (1992) Chem Phys 165:287

    CAS  Google Scholar 

  74. Prudente FV, Soares Neto JJ (1999) Chem Phys Lett 309:471

    CAS  Google Scholar 

  75. Chuluunbaatar O, Gusev AA, Kaschiev MS, Kaschieva VA, Amaya-Tapia A, Larsen SY, Vinitsky SI (2006) J Phys B At Mol Opt Phys 39:243

    CAS  Google Scholar 

  76. Guimarães MN, Prudente FV (2011) Eur Phys J D 64:287

    Google Scholar 

  77. Babuška I, Guo BQ (1992) Adv Eng Softw 15:159

    Google Scholar 

  78. Soares Neto JJ, Costa LS (1998) Braz J Phys 28:1

    CAS  Google Scholar 

  79. Curry HB, Schoenberg IJ (1947) Bull Am Math Soc 53:1114

    Google Scholar 

  80. de Boor C (1978) A practical guide to splines. Springer, New York

    Google Scholar 

  81. Ndengué SA, Motapon O (2008) J Phys B At Mol Opt Phys 41:045001

    Google Scholar 

  82. Kang S, Li J, Shi T (2006) J Phys B At Mol Opt Phys 39:3491

    CAS  Google Scholar 

  83. Bhatti ML, Coleman KD, Perger WE (2003) Phys Rev A 68:044503

    Google Scholar 

  84. Xi J, Wu L, He X, Li B (1992) Phys Rev A 46:5806

    CAS  Google Scholar 

  85. Johnson WR, Sapirstein J (1986) Phys Rev Lett 57:1126

    CAS  Google Scholar 

  86. Shore BW (1973) J Chem Phys 58:3855

    CAS  Google Scholar 

  87. Bachau H, Cormier E, Decleva P, Hansen JE, Martín F (2001) Rep Prog Phys 64:1815

    CAS  Google Scholar 

  88. Martín F (1999) J Phys B At Mol Opt Phys 32:R197

    Google Scholar 

  89. Prudente FV, Acioli PH (1999) Chem Phys Lett 302:249

    CAS  Google Scholar 

  90. Prudente FV, Costa LS, Acioli PH (2000) J Phys B At Mol Opt Phys 33:R285

    CAS  Google Scholar 

  91. Costa LS, Prudente FV, Acioli PH (2000) Phys Rev A 61:012506

    Google Scholar 

  92. Grinberg M, Jaskólski W, Kopke C, Plannelles J, Jnowicz M (1994) Phys Rev B 50:6504

    CAS  Google Scholar 

  93. Goodfriend PL (1990) J Phys B At Mol Opt Phys 23:1373

    CAS  Google Scholar 

  94. Adams JE, Miller WH (1977) J Chem Phys 67:5775

    CAS  Google Scholar 

  95. Consortini A, Frieden BR (1976) Nuovo Cimento B 35:153

    Google Scholar 

  96. Almeida MM, Guimarães MN, Prudente FV (2005) Rev Bras Ens Fís 27:395

    CAS  Google Scholar 

  97. Michels A, de Boer J, Bijl A (1937) Physica 4:981

    CAS  Google Scholar 

  98. Sommerfeld A, Welker H (1938) Ann Phys (Leipzig) 32:56

    CAS  Google Scholar 

  99. Guillot T (1999) Space Sci 47:1183

    CAS  Google Scholar 

  100. Guillot T (2005) Annu Rev Earth Planet Sci 33:493

    CAS  Google Scholar 

  101. Tabbert B, Günther H, Zu Putlitz G (1997) J Low Temp Phys 109:653

    Google Scholar 

  102. Aquino N (1995) Int J Quantum Chem 54:107

    Google Scholar 

  103. Goldman S, Joslin C (1992) J Phys Chem 96:6021

    CAS  Google Scholar 

  104. Varshni YP (1997) J Phys B At Mol Opt Phys 30:L589

    CAS  Google Scholar 

  105. Zicovich-Wilson C, Jaskólski W, Planelles JH (1995) Int J Quant Chem 54:61

    CAS  Google Scholar 

  106. Dutt R, Mujherjee A, Varshni YP (2001) Phys Lett A 280:318

    CAS  Google Scholar 

  107. Banerjee A, Sen KD, Garza J, Vargas R (2002) J Chem Phys 116:4054

    CAS  Google Scholar 

  108. Laughlin C (2004) J Phys B At Mol Opt Phys 37:4085

    CAS  Google Scholar 

  109. Dillon AC, Jones KM, Bekkedahl TA, Kiang CH, Bethune DS, Heben MJ (1997) Nature 386:377

    CAS  Google Scholar 

  110. Dolmatov VK (2009) In: Sabin JR, Brändas E, Cruz SA (eds) Advances in quantum chemistry: theory of confined quantum systems, vol 58. Academic Press, Oxford, p 13

    Google Scholar 

  111. Xu YB, Tan MQ, Becker U (1996) Phys Rev Lett 76:3538

    CAS  Google Scholar 

  112. Yuan L, Yang J, Deng K, Zhu Q (2000) J Phys Chem A 104:6666

    CAS  Google Scholar 

  113. Piskoti C, Yarger J, Zettl A (1998) Nature 393:771

    CAS  Google Scholar 

  114. Ichimaru S (1982) Rev Mod Phys 54:1017

    CAS  Google Scholar 

  115. Sil AN, Canuto S, Mukherjee PK (2009) In: Sabin JR, Brändas E, Cruz SA (eds) Advances in quantum chemistry: theory of confined quantum systems, vol 58. Academic Press, Oxford, p 115

    Google Scholar 

  116. Ichimaru S (1992) Statistical plasma physics, volume I: basic principles. Addison-Wesley Publishing Company, New York

    Google Scholar 

  117. Guimarães MN, Prudente FV (2012) In: Apduhan B, Ragni M, Misra S, Gervasi O, Taniar D, Murgante B (eds) Proceedings of the 12th international conference on computational science and its applications, IEEE Computer Society’s Conference Publishing Services, Los Alamitos, p 167

    Google Scholar 

  118. Varshni YP (1998) Superlattices Microstruct 23:145

    CAS  Google Scholar 

  119. Lee WC, Lee TK (2002) J Phys Condens Matter 14:1045

    CAS  Google Scholar 

  120. Pino R, Villalba VM (2001) J Phys Condens Matter 13:11651

    CAS  Google Scholar 

  121. Imbo T, Pagnamenta A, Sukhatme U (1984) Phys Rev D 29:1669

    Google Scholar 

  122. Schwartz C (1985) J Math Phys 26:411

    Google Scholar 

  123. Friedberg R, Lee TD, Zhao WQ (1999) Nuovo Cimento A 112:1195

    Google Scholar 

  124. Prudente FV, Riganelli A, Varandas AJC (2001) Rev Mex Fis 47:568

    CAS  Google Scholar 

Download references

Acknowledgments

This work has been supported by the following Brazilian National Research Councils: Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Fundação de Amparo à Pesquisa do Estado da Bahia (FAPESB) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).

Author information

Authors and Affiliations

  1. Instituto de Física, Universidade Federal Da Bahia, Salvador, BA, 40170-115, Brazil

    Frederico V. Prudente

  2. Centro de Formação de Professores, Universidade Federal Do Recôncavo Da Bahia, Amargosa, BA, 45300-000, Brazil

    Marcilio N. Guimarães

Authors
  1. Frederico V. Prudente
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcilio N. Guimarães
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Frederico V. Prudente .

Editor information

Editors and Affiliations

  1. School of Chemistry, University of Hyderabad, Hyderabad, India

    K.D. Sen

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Prudente, F.V., Guimarães, M.N. (2014). Confined Quantum Systems Using the Finite Element and Discrete Variable Representation Methods. In: Sen, K. (eds) Electronic Structure of Quantum Confined Atoms and Molecules. Springer, Cham. https://doi.org/10.1007/978-3-319-09982-8_5

Download citation

Publish with us

Policies and ethics

Search

Navigation