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Theoretical and Computational Methods

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Topics in Theoretical and Computational Nanoscience

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Abstract

The content that appears in this chapter has been largely adapted from the following publications and manuscript in press: Zhao J, Pinchuk AO, McMahon JM, Li S, Ausman LK, Atkinson AL, Schatz GC (2008) Methods for describing the electromagnetic properties of silver and gold nanoparticles. Acc Chem Res 41:1710–1720. doi:10.1021/ar800028jAtkinson AL, McMahon JM, Schatz GC (2009) FDTD studies of metallic nanoparticle systems. In: Self organization of molecular systems, from molecules and clusters to nanotubes and proteins. NATO science for peace and security series A: chemistry and biology. Springer, Netherlands. doi:10.1007/978-90-481-2590-6

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References

  1. McMahon JM, Gray SK, Schatz GC (2011) Surface nanophotonics theory. In: Wiederrecht G (ed) Comprehensive nanoscience and technology. Elsevier, Amsterdam

    Google Scholar 

  2. McMahon JM, Henry AI, Wustholz KL, Natan MJ, Freeman RG, Van Duyne RP, Schatz GC (2009) Gold nanoparticle dimer plasmonics: finite element method calculations of the electromagnetic enhancement to surface-enhanced Raman spectroscopy. Anal Bioanal Chem 394:1825–1825

    Article  CAS  Google Scholar 

  3. Zhao J, Pinchuk AO, McMahon JM, Li S, Ausman LK, Atkinson AL, Schatz GC (2008) Methods for describing the electromagnetic properties of silver and gold nanoparticles. Acc Chem Res 41:1720–1720

    Article  CAS  Google Scholar 

  4. Atkinson AL, McMahon JM, Schatz GC (2009) FDTD studies of metallic nanoparticle systems. In: Self organization of molecular systems, from molecules and clusters to nanotubes and proteins, NATO science for peace and security series A, chemistry and biology. Springer, Netherlands

    Google Scholar 

  5. Taflove A, Hagness S (2005) Computational electrodynamics: the finite-difference time-domain method, 3rd edn. Artech House, Boston

    Google Scholar 

  6. Jin J (2002) The finite element method in electromagnetics. Wiley, New York

    Google Scholar 

  7. Mie G (1908) Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen. Ann Phys 25:377–445

    Article  CAS  Google Scholar 

  8. Bohren CF, Huffman DR (1983) Absorption and scattering of light by small particles. Wiley, New York

    Google Scholar 

  9. Peña O, Pal U (2009) Scattering of electromagnetic radiation by a multilayered sphere. Comput Phys Commun 180:2348–2354

    Article  Google Scholar 

  10. Yeh P (1988) Optical waves in layered media, 1st edn. Wiley, New York

    Google Scholar 

  11. Yee SK (1966) Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans Antennas Propagat 14:302–307

    Article  Google Scholar 

  12. Berenger JP (1994) A perfectly matched layer for the absorption of electromagnetic waves. J Comp Phys 114:185–200

    Article  Google Scholar 

  13. Roden JA, Gedney SD (2000) Convolutional PML (CPML): an efficient FDTD implementation of the CFS-PML for arbitrary media. Microw Opt Techn Let 27:334–339

    Article  Google Scholar 

  14. McMahon JM, Gray SK, Schatz GC (2009) A discrete action principle for electrodynamics and the construction of explicit symplectic integrators for linear, non-dispersive media. J Comp Phys 228:3421–3432

    Article  CAS  Google Scholar 

  15. Umashankar KR, Taflove A (1982) A novel method to analyze electromagnetic scattering of complex objects. IEEE T Electromagn Compat 24:397–405

    Article  Google Scholar 

  16. Mur G (1981) Absorbing boundary conditions for the finite difference approximation of the time-domain electromagnetic field equations. IEEE T Electromagn C 23:377–382

    Article  Google Scholar 

  17. Merewether DE, Fisher R, Smith FW (1980) On implementing a numeric Huygen’s source scheme in a finite difference program to illuminate scattering bodies. IEEE T Nucl Sci 27:1829–1833

    Article  Google Scholar 

  18. Kashiwa T, Fukai I (1990) A treatment by FDTD method of dispersive characteristics associated with electronic polarization. Microw Opt Techn Lett 3:203–205

    Article  Google Scholar 

  19. Joseph RM , Hagness SC, Taflove A (1991) Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses. Opt Lett 16:1412–1414

    Article  CAS  Google Scholar 

  20. Gray SK, Kupka T (2003) Propagation of light in metallic nanowire arrays: finite-difference time-domain studies of silver cylinders. Phys Rev B 68:045415

    Article  Google Scholar 

  21. McMahon JM, Wang Y, Sherry LJ, Van Duyne RP, Marks LD, Gray SK, Schatz GC (2009) Correlating the structure, optical spectra, and electrodynamics of single silver nanocubes. J Phys Chem 113:2731–2735

    Article  CAS  Google Scholar 

  22. Babayan Y, McMahon JM , Li S, Gray SK, Schatz GC, Odom TW (2009) Confining standing waves in optical corrals. ACS Nano 3:615–620

    Article  CAS  Google Scholar 

  23. Ringe E, McMahon JM, Sohn K, Cobley C, Xia Y, Huang J, Schatz GC, Marks LD, Van Duyne RP (2010) Unraveling the effects of size, composition, and substrate on the localized surface plasmon resonance frequencies of gold and silver nanocubes: a systematic single particle approach. J Phys Chem C 114:12511–12516

    Article  CAS  Google Scholar 

  24. McMahon JM, Henzie J, Odom TW, Schatz GC, Gray SK (2007) Tailoring the sensing capabilities of nanohole arrays in gold films with Rayleigh anomaly-surface plasmon polaritons. Opt Express 15:18119–18129

    Article  Google Scholar 

  25. Schatz GC, McMahon JM, Gray SK (2007) Tailoring the parameters of nanohole arrays in gold films for sensing applications. In: Mark I Stockman (ed) Plasmonics: metallic nanostructures and their optical properties V, pp 664103(1–8)

    Google Scholar 

  26. Gao H, McMahon JM, Lee MH, Henzie J, Gray SK, Schatz GC, Odom TW (2009) Rayleigh anomaly-surface plasmon polariton resonances in palladium and gold subwavelength hole arrays. Opt. Express 17:2334–2340

    Article  CAS  Google Scholar 

  27. Odom TW, Gao H, McMahon JM, Henzie J, Schatz GC (2009) Plasmonic superlattices: hierarchical subwavelength hole arrays. Chem Phys Lett 483:187–192

    Article  CAS  Google Scholar 

  28. Stewart ME, Mack NH, Malyarchuck V, Soares JANT, Lee TW, Gray SK, Nuzzo RG, Rogers JA (2006) Quantitative multispectral biosensing and 1D imaging using quasi-3D plasmonic crystals. Proc Natl Acad Sci USA, 103:17143–17148

    Article  CAS  Google Scholar 

  29. Chang SH, Gray SK, Schatz GC (2005) Surface plasmon generation and light transmission by isolated nanoholes and arrays of nanoholes in thin metal films. Opt Express 13:3150–3165

    Article  Google Scholar 

  30. Schwartz M (1987) Principles of electrodynamics, 1st edn. Dover Publications, New York

    Google Scholar 

  31. Jackson JD (1998) Classical electrodynamics, 3rd edn. Wiley, New York

    Google Scholar 

  32. Manry CW, Broschat SL, Schneider JB (1995) Higher-order FDTD methods for large problems. J Appl Comput Electromagnet Soc 10:17–29

    Google Scholar 

  33. Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1988) Numerical recipes in C: the art of scientific computing, 1st edn. Cambridge University Press, Cambridge

    Google Scholar 

  34. Verlet L (1967) Computer “experiments” on classical fluids. I. Thermodynamical properties of Lennard–Jones molecules. Phys Rev 159:98–103

    Article  CAS  Google Scholar 

  35. Goldstein H (1980) Classical mechanics, 2nd edn. Addison-Wesley, MA

    Google Scholar 

  36. Ruth RD (1983) A canonical integration technique. IEEE Trans Nucl Sci NS-30:2669–2671

    Google Scholar 

  37. Yoshida H (1990) Construction of higher order symplectic integrators. 150:262–268

    Google Scholar 

  38. Suris YB (1990) Hamiltonian methods of Runge–Kutta type and their variational interpretation. Math Mod 2:78–87

    Google Scholar 

  39. Candy J, Rozmus W (1991) A symplectic integration algorithm for separable Hamiltonian functions. J Comp Phys 92:230–256

    Article  Google Scholar 

  40. McLachlan RI, Atela P (1992) The accuracy of symplectic integrators. Nonlinearity 5:541–562

    Article  Google Scholar 

  41. Okunbor DI, Skeel RD (1992) Explicit canonical methods for Hamiltonian systems. Math Comput 59:439–455

    Article  Google Scholar 

  42. Sanz-Serna JM (1992) The numerical integration of Hamiltonian systems. In: Cash JR, Gladwell I (eds) Computational ordinary differential equations. Clarendon Press, Oxford

    Google Scholar 

  43. Tuckerman M, Berne BJ, Martyna GJ (1992) Reversible multiple time scale molecular dynamics. J Chem Phys 97:1990–2001

    Article  CAS  Google Scholar 

  44. Calvo MP, Sanz-Serna JM (1993) The development of variable-step symplectic integrators with applications to the two-body problem. Siam J Sci Comput 14:936–952

    Article  Google Scholar 

  45. Sanz-Serna JM, Calvo MP (1993) Numerical Hamiltonian problems, 1st edn. Chapman and Hall, London

    Google Scholar 

  46. Gray SK, Noid DW, Sumpter BG (1994) Symplectic integrators for large scale molecular dynamics simulations: a comparison of several explicit methods. J Chem Phys 101:4062–4072

    Article  CAS  Google Scholar 

  47. Gray SK, Manolopoulos DE (1996) Symplectic integrators tailored to the time-dependent Schrödinger equation. J Chem Phys 104:7099–7112

    Article  CAS  Google Scholar 

  48. Saitoh I, Suzuki Y, Takahashi N (2001) The symplectic finite difference time domain method. IEEE T Magn 37:3251–3254

    Article  Google Scholar 

  49. Huang Z, Wu X (2005) Symplectic partitioned Runge–Kutta scheme for Maxwell’s equations. Int J Quant Chem 106:839–842

    Article  Google Scholar 

  50. Sha W, Huang Z, Wu X , Chen M (2007) Application of the symplectic finite-difference time-domain scheme to electromagnetic simulation. J Comp Phys 225:33–50

    Article  Google Scholar 

  51. Anderson N, Arthurs AM (1978) A variational principle for Maxwell’s equations. Int J Electron 45:333–334

    Article  Google Scholar 

  52. Masiello D, Deumens E, Öhrn Y (2005) Dynamics of an atomic electron and its electromagnetic field in a cavity. Phys Rev A 71:032108

    Article  Google Scholar 

  53. Marsden JE, West M (2001) Discrete mechanics and variational integrators. Acta Numer 10:357–514

    Article  Google Scholar 

  54. Qin H, Guan X (2008) Variational symplectic integrator for long-time simulations of the guiding-center motion of charged particles in general magnetic fields. Phys Rev Lett 100:035006

    Article  Google Scholar 

  55. McLachlan RI, Gray SK (1997) Optimal stability polynomials for splitting methods, with application to the time-dependent Schrödinger equation. Appl Numer Math 25:275–286

    Google Scholar 

  56. Hairer E, Lubich C, Wanner G (2002) Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, 1st edn. Springer, Berlin

    Google Scholar 

  57. Liu Y (1996) Fourier analysis of numerical algorithms for the Maxwell equations. J Comp Phys 124:396–416

    Article  Google Scholar 

  58. Hoffman J (1992) Numerical methods for engineers and scientists, 1st edn. McGraw-Hill, New York

    Google Scholar 

  59. Berenger JP (1996) Perfectly matched layer for the FDTD solution of wave-structure interaction problems. IEEE Trans Antennas Propagat 51:110–117

    Article  Google Scholar 

  60. Willets KA, Van Duyne RP (2007) Localized surface plasmon resonance spectroscopy and sensing. Annu Rev Phys Chem 58:267–297

    Article  CAS  Google Scholar 

  61. Elghanian R, Storhoff JJ, Mucic RC, Letsinger RL, Mirkin CA (1997) Selective colorimetric detection of polynucleotides based on the distance-dependent optical properties of gold nanoparticles. Science 277:1078–1081

    Article  CAS  Google Scholar 

  62. Borisov AG, Shabanov SV (2005) Lanczos pseudospectral method for initial-value problems in electrodynamics and its applications to ionic crystal gratings. J Comp Phys 209:643–664

    Article  Google Scholar 

  63. Car R, Parrinello M (1985) Unified approach for molecular dynamics and density-functional theory. Phys Rev Lett 55:2471–2474

    Article  CAS  Google Scholar 

  64. Runge E, Gross EKU (1984) Density-functional theory for time-dependent systems. Phys Rev Lett 52:997

    Article  CAS  Google Scholar 

  65. Nedelec JC (1980) Mixed finite elements in \({\mathbf{R}}^3\). Numer Meth 35:315–341

    Article  Google Scholar 

  66. Graglia RD, Wilton DR, Peterson AF (1997) Higher order interpolatory vector bases for computational electromagnetics. IEEE Trans Antennas Propagat AP-45:329–342

    Article  Google Scholar 

  67. Dunvant DA (1985) High degree efficient symmetrical Gaussian quadrature rules for the triangle. Int J Numer Meth Eng 29:1129–1148

    Article  Google Scholar 

  68. Keast P (1986) Moderate degree tetrahedral quadrature formulas. Comput Meth Appl Mech Eng 55:339–348

    Article  Google Scholar 

  69. Saxon DS (1955) Lecture on the scattering of light. UCLA Department meteorological science report 9

    Google Scholar 

  70. Moharam MG, Grann EB, Pommet DA, Gayrold TK (1995) Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings. J Opt Soc Am A 12:1068–1076

    Article  Google Scholar 

  71. Lalanne P, Morris GM (1996) Highly improved convergence of the coupled-wave method for TM polarization. J Opt Soc Am A 13:779–784

    Article  Google Scholar 

  72. Benabbas A, Halte V, Bigot JY (2005) Analytical model of the optical response of periodically structured metallic films. Opt Express 13:8730–8745

    Article  CAS  Google Scholar 

  73. McMahon JM, Gray SK, Schatz GC (2008) Dephasing of electromagnetic fields in scattering from an isolated slit in a gold film. In: Kawata S (ed) Plasmonics: nanoimaging, nanofabrication, and their applications IV, pp 703311/1–6

    Google Scholar 

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  1. Northwestern University, Evanston, IL, 60208, USA

    Jeffrey Michael McMahon

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  1. Jeffrey Michael McMahon
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McMahon, J.M. (2011). Theoretical and Computational Methods. In: Topics in Theoretical and Computational Nanoscience. Springer Theses. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8249-0_3

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