Skip to main content
SpringerLink
Log in

Theoretical Background

  • Chapter
  • First Online:
Book cover Polaritonic Chemistry

Part of the book series: Springer Theses ((Springer Theses))

  • 374 Accesses

Abstract

This chapter presents the essential theoretical background necessary to explain some of the most important concepts discussed throughout this thesis. The aim is to provide the reader with the basic tools to understand the many fundamental equations and approximations used in the contexts of cavity quantum electrodynamics (CQED) and quantum chemistry. We start by addressing the question of what is the quantum Hamiltonian for the light–matter interaction and illustrating what approximations play an important role in its definition. We then focus on the matter part of the light–matter Hamiltonian in order to provide the best possible description of a complex molecule. In this section we address the Born–Oppenheimer approximation, widely used in molecular and solid-state physics and in quantum chemistry. Additionally, we present the description of different characteristic phenomena of organic molecules such as chemical structure and reactions, and their response to the electromagnetic field. Then, we focus on this last part, discussing the important features of CQED and the different theoretical descriptions that study them. Finally we present the fundamentals of the two different regimes of light–matter interaction: weak and strong coupling.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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.

    This expression is valid both in the presence and absence of sources. It is worth mentioning that in the case of the free field it is possible to write each term as \(A_{\mathbf {k},\lambda }(\mathbf {r},t) = A_{\mathbf {k},\lambda }(\mathbf {r})e^{-i\omega (\mathbf {k})t}\), where the \(A_{\mathbf {k},\lambda }(\mathbf {r})\) satisfy the homogeneous Helmholtz equation \((\varvec{\nabla }^2 + k^2)A_{\mathbf {k},\lambda }(\mathbf {r})=0\). In this case the explicit time dependence of \(\mathcal {H}_\mathrm {EM}\) disappears, as the energy is conserved.

  2. 2.

    We see that in the vacuum state of the system the energy is \(E_\mathrm {0} = \frac{1}{2}\sum _{\mathbf {k},\lambda } \hbar \omega (\mathbf {k})\). The frequencies \(\omega (\mathbf {k})\) have no upper bound, so \(E_\mathrm {0}\) diverges. However, this is not a problem since expectation values only depend on energy differences and not absolute energies, so the divergence of the vacuum state does not appear in any physical observable.

  3. 3.

    The more fundamental definition for the polarization field is \(\hat{\mathbf {P}}(\mathbf {r}) = \sum _{i,j} q_i^{(j)}(\mathbf {r}_i-\mathbf {r}_j) \int _0^1 ds \delta ^3 \left[ \mathbf {r} - \mathbf {r}_j -s(\mathbf {r}_i-\mathbf {r}_j) \right] \). This is more cumbersome and less intuitive, so we instead present \(\hat{\mathbf {P}}(\mathbf {r})\) directly as a multipole expansion. The connection between each expression can be found in [2].

  4. 4.

    This approximation is completely equivalent to the long-wavelength approximation, in which the charges conforming each dipole are very close compared to the EM wavelength and thus experience the same fields, i.e. for \(\varvec{\mu } = q_i\mathbf {r}_i + q_j\mathbf {r}_j\) the fields satisfy \(\mathbf {A}(\mathbf {r}_i) \approx \mathbf {A}(\mathbf {r}_j)\). This is analogous to neglect the effects of higher electric multipoles, as they are more significant as the spatial structure of the collection of charges becomes important.

  5. 5.

    Note that, depending on the context, we use the indices i and j to represent either individual charged particles or molecules.

  6. 6.

    This effect emerges because the speed of light is finite. When the information of a particular charge configuration reaches an emitter situated far away, these charges have already rearranged, so that the emitter response is no longer in phase. This arises for distances much larger than the wavelength corresponding to the characteristic absorption frequency of the emitters [20]. In the systems that we are concerned with in this thesis, these wavelengths are of the order of hundreds of nanometers, so we can completely disregard retardation effects when dealing with dipole–dipole interactions.

  7. 7.

    Note that while in one dimension the transition state is a local maximum, in general multi-dimensional scenarios this is a saddle point.

  8. 8.

    In Sect. 2.3 we analyze the relevance of this term when not all EM modes are explicitly considered.

  9. 9.

    In this thesis we are mainly interested in molecules, but this occurs for any set of charges such as atoms, quantum dots, nanoparticles, etc.

  10. 10.

    The Franck–Condon principle states that the intensity of a vibronic transition (i.e., a change of electronic and vibrational states) is directly proportional to the overlap between the corresponding nuclear wavefunctions. This is based on the assumption of a vertical transition, i.e., that there is some adiabatic separation between nuclear and electronic timescales, similarly as in the Born–Oppenheimer approximation.

  11. 11.

    We note that theoretical efforts in this regard have been made for small plasmonic nanoparticles using time-dependent density functional theory [38].

  12. 12.

    This also includes the emitter–emitter interactions in the multiple-emitter case.

  13. 13.

    The exact diagonalization of Eq. (2.63) can be done without invoking the RWA. However, for the sake of brevity we present the diagonalization within the single-excitation subspace.

  14. 14.

    Note that this also depends on the choice of quantum emitter, as very large Q factors are not useful if the emitter has a broader linewidth than the cavity.

  15. 15.

    This is equivalent to treating the many-particle TC Hamiltonian with collective operators, thus representing the coupling between the bright state and the cavity mode.

  16. 16.

    This is equivalent to consider a Lindblad master equation approach and neglecting the excitation refilling terms, making the ground state a population reservoir [93, 94].

References

  1. Cohen-Tannoudji C, Dupont-Roc J, Grynberg G (1997) Photons and atoms

    Google Scholar 

  2. Steck DA (2007) Quantum and atom optics, vol 47

    Google Scholar 

  3. Dirac PAM (1927) The quantum theory of the emission and absorption of radiation. Proc R Soc A Math Phys Eng Sci 114:243

    ADS  MATH  Google Scholar 

  4. Dirac PAM (1981) The principles of quantum mechanics, vol 27. Oxford University Press

    Google Scholar 

  5. Majety VP, Zielinski A, Scrinzi A (2014) Mixed gauge in strong laser-matter interaction. J Phys B At Mol Opt Phys 48:025601

    Google Scholar 

  6. Göppert-Mayer M (1931) Über Elementarakte mit zwei Quantensprüngen. Annalen der Physik 401:273

    ADS  MATH  Google Scholar 

  7. Power EA, Zienau S (1959) Coulomb gauge in non-relativistic quantum electro-dynamics and the shape of spectral lines. Philos Trans R Soc Lond Ser A Math Phys Sci 251:427

    Google Scholar 

  8. Woolley RG (1971) Molecular quantum electrodynamics. Proc R Soc Lond A 321:557

    ADS  MathSciNet  Google Scholar 

  9. Power E, Thirunamachandran T (1982) Quantum electrodynamics in a cavity. Phys Rev A 25:2473

    Google Scholar 

  10. Power EA, Thirunamachandran T (1980) The multipolar Hamiltonian in radiation theory. Proc R Soc Lond A Math Phys Sci 372:265

    Google Scholar 

  11. Born M, Oppenheimer R (1927) Zur Quantentheorie der Molekeln. Annalen der Physik 389:457

    Google Scholar 

  12. Tully JC (2000) Perspective on “Zur Quantentheorie der Molekeln”. Theor Chem Acc Theory Comput Model (Theoretica Chimica Acta) 103:173

    Google Scholar 

  13. Born M, Huang K (1954) Dynamical theory of crystal lattices. Clarendon Press

    Google Scholar 

  14. Worth GA, Cederbaum LS (2004) Beyond Born-Oppenheimer: molecular dynamics through a conical intersection. Annu Rev Phys Chem 55:127

    Google Scholar 

  15. Levine BG, Martínez TJ (2007) Isomerization through conical intersections. Annu Rev Phys Chem 58:613

    Google Scholar 

  16. Mead CA, Truhlar DG (1982) Conditions for the definition of a strictly diabatic electronic basis for molecular systems. J Chem Phys 77:6090

    Google Scholar 

  17. Novotny L, Hecht B (2009) Principles of nano-optics. ISBN: 9781107005464

    Google Scholar 

  18. Leonas V et al (2000) Photosynthetic excitons. World Scientific

    Google Scholar 

  19. Weiss U (2012) Quantum dissipative systems, vol 10. World Scientific

    Google Scholar 

  20. Stone A (2013) The theory of intermolecular forces, 2nd edn. OUP Oxford

    Google Scholar 

  21. Sheppard D, Terrell R, Henkelman G (2008) Optimization methods for finding minimum energy paths. J Chem Phys 128:134106

    ADS  Google Scholar 

  22. Truhlar DG, Gordon MS (1990) From force fields to dynamics: classical and quantal paths. Science 249:491

    Google Scholar 

  23. Prezhdo OV, Rossky PJ (1997) Mean-field molecular dynamics with surface hopping. J Chem Phys 107:825

    Google Scholar 

  24. Barbatti M (2011) Nonadiabatic dynamics with trajectory surface hopping method. Wiley Interdisc Rev Comput Mol Sci 1:620

    Google Scholar 

  25. Eyring H (1935) The activated complex in chemical reactions. J Chem Phys 3:107

    ADS  Google Scholar 

  26. Arrhenius S (1889) Über die dissociationswärme und den einfluss der temperatur auf den dissociationsgrad der elektrolyte. Zeitschrift für physikalische Chemie 4:96

    Google Scholar 

  27. Miller WH (1993) Beyond transition-state theory: a rigorous quantum theory of chemical reaction rates. Acc Chem Res 26:174

    Google Scholar 

  28. Yamamoto T (1960) Quantum statistical mechanical theory of the rate of exchange chemical reactions in the gas phase. J Chem Phys 33:281

    ADS  MathSciNet  Google Scholar 

  29. Miller WH (1974) Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants. J Chem Phys 61:1823

    ADS  Google Scholar 

  30. Miller WH, Schwartz SD, Tromp JW (1983) Quantum mechanical rate constants for bimolecular reactions. J Chem Phys 79:4889

    ADS  Google Scholar 

  31. Jackson JD (2007) Classical electrodynamics. Wiley

    Google Scholar 

  32. Bonin KD, Kresin VV (1997) Electric-dipole polarizabilities of atoms, molecules, and clusters. World Scientific Publishing Co. Pte. Ltd

    Google Scholar 

  33. Rescigno TN, McKoy V (1975) Rigorous method for computing photoabsorption cross sections from a basis-set expansion. Phys Rev A 12:522

    Google Scholar 

  34. Kasha M (1950) Characterization of electronic transitions in complex molecules. Discuss Faraday Soc 9:14

    Google Scholar 

  35. Nitzan A, Jortner J, Rentzepis PM (1971) Internal conversion in large molecules. Mol Phys 22:585

    ADS  Google Scholar 

  36. May V, Kühn O (2011) Charge and energy transfer dynamics in molecular systems. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany

    Google Scholar 

  37. Nitzan A (2006) Chemical dynamics in condensed phases: relaxation, transfer and reactions in condensed molecular systems. Oxford University Press

    Google Scholar 

  38. Barbry M, Koval P, Marchesin F, Esteban R, Borisov AG, Aizpurua J, Sánchez-Portal D (2015) Atomistic near-field nanoplasmonics: reaching atomic-scale resolution in nanooptics. Nano Lett 15:3410

    ADS  Google Scholar 

  39. Huttner B, Barnett SM (1992) Quantization of the electromagnetic field in dielectrics. Phys Rev A 46:4306

    Google Scholar 

  40. Scheel S, Buhmann SY (2008) Macroscopic quantum electrodynamics—concepts and applications. Acta Physica Slovaca 58:675

    Google Scholar 

  41. Buhmann SY (2007) Casimir-polder forces on atoms in the presence of magnetoelectric bodies. Thesis (PhD), Friedrich-Schiller-Universität Jena

    Google Scholar 

  42. Kavokin A, Baumberg J, Malpuech G, Laussy F (2008) Microcavities

    Google Scholar 

  43. Tame MS, McEnery K, Özdemir Ş, Lee J, Maier S, Kim M (2013) Quantum plasmonics. Nat Phys 9:329

    Google Scholar 

  44. Hopfield J (1958) Theory of the contribution of excitons to the complex dielectric constant of crystals. Phys Rev 112:1555

    ADS  MATH  Google Scholar 

  45. Dung HT, Knöll L, Welsch D-G (1998) Three-dimensional quantization of the electromagnetic field in dispersive and absorbing inhomogeneous dielectrics. Phys Rev A 57:3931

    ADS  Google Scholar 

  46. Van Vlack C, Kristensen PT, Hughes S (2012) Spontaneous emission spectra and quantum light-matter interactions from a strongly coupled quantum dot metal-nanoparticle system. Phys Rev B 85:075303

    ADS  Google Scholar 

  47. Alpeggiani F, Andreani LC (2014) Quantum theory of surface plasmon polaritons: planar and spherical geometries. Plasmonics 9:965

    Google Scholar 

  48. Li R-Q, Hernángomez-Pérez D, García-Vidal FJ, Fernández-Domínguez AI (2016) Transformation optics approach to plasmon-exciton strong coupling in nanocavities. Phys Rev Lett 117:107401

    ADS  Google Scholar 

  49. Savage KJ, Hawkeye MM, Esteban R, Borisov AG, Aizpurua J, Baumberg JJ (2012) Revealing the quantum regime in tunnelling plasmonics. Nature 491:574

    ADS  Google Scholar 

  50. Zhang P, Feist J, Rubio A, García-González P, García-Vidal FJ (2014) Ab initio nanoplasmonics: the impact of atomic structure. Phys Rev B 90:161407(R)

    ADS  Google Scholar 

  51. Varas A, García-González P, Feist J, García-Vidal FJ, Rubio A (2016) Quantum plasmonics: from jellium models to ab initio calculations. Nanophotonics 5:409

    Google Scholar 

  52. Vukics A, Grießer T, Domokos P (2014) Elimination of the \({{A}}\)-square problem from cavity QED. Phys Rev Lett 112:073601

    ADS  Google Scholar 

  53. Keeling J (2007) Coulomb interactions, gauge invariance, and phase transitions of the Dicke model. J Phys Condens Matter 19:295213

    Google Scholar 

  54. Kirton P, Roses MM, Keeling J, Dalla Torre EG (2018) Introduction to the Dicke model: from equilibrium to nonequilibrium, and vice versa. Adv Quantum Technol 1800043

    Google Scholar 

  55. De Bernardis D, Jaako T, Rabl P (2018) Cavity quantum electrodynamics in the nonperturbative regime. Phys Rev A 97:043820

    ADS  MathSciNet  Google Scholar 

  56. Rokaj V, Welakuh DM, Ruggenthaler M, Rubio A (2018) Light-matter interaction in the long-wavelength limit: no ground-state without dipole self-energy. J Phys B 51:034005

    ADS  Google Scholar 

  57. Todorov Y, Sirtori C (2014) Few-electron ultrastrong light-matter coupling in a quantum LC circuit. Phys Rev X 4:041031

    Google Scholar 

  58. De Bernardis D, Pilar P, Jaako T, De Liberato S, Rabl P (2018) Breakdown of gauge invariance in ultrastrong-coupling cavity QED. Phys Rev A 98:053819

    ADS  Google Scholar 

  59. Andrews DL, Jones GA, Salam A, Woolley RG (2018) Perspective: quantum Hamiltonians for optical interactions. J Chem Phys 148:040901

    ADS  Google Scholar 

  60. Stokes A, Nazir A (2019) Gauge ambiguities imply Jaynes-Cummings physics remains valid in ultrastrong coupling QED. Nat Commun 10:499

    Google Scholar 

  61. Kónya G, Vukics A, Domokos P. The equivalence of the Power-Zineau-Woolley picture and the Poincaré gauge from the very first principles

    Google Scholar 

  62. George J, Chervy T, Shalabney A, Devaux E, Hiura H, Genet C, Ebbesen TW (2016) Multiple Rabi splittings under ultrastrong vibrational coupling. Phys Rev Lett 117:153601

    ADS  Google Scholar 

  63. Chikkaraddy R, de Nijs B, Benz F, Barrow SJ, Scherman OA, Rosta E, Demetriadou A, Fox P, Hess O, Baumberg JJ (2016) Single-molecule strong coupling at room temperature in plasmonic nanocavities. Nature 535:127

    ADS  Google Scholar 

  64. Chikkaraddy R, Turek VA, Kongsuwan N, Benz F, Carnegie C, van de Goor T, de Nijs B, Demetriadou A, Hess O, Keyser UF, Baumberg JJ (2018) Mapping nanoscale hotspots with single-molecule emitters assembled into plasmonic nanocavities using DNA origami. Nano Lett 18:405

    ADS  Google Scholar 

  65. Autore M, Li P, Dolado I, Alfaro-Mozaz FJ, Esteban R, Atxabal A, Casanova F, Hueso LE, Alonso-González P, Aizpurua J, Nikitin AY, Vélez S, Hillenbrand R (2018) Boron nitride nanoresonators for phonon-enhanced molecular vibrational spectroscopy at the strong coupling limit. Light Sci Appl 7:17172

    Google Scholar 

  66. Zengin G, Wersäll M, Nilsson S, Antosiewicz TJ, Käll M, Shegai T (2015) Realizing strong light-matter interactions between single-nanoparticle plasmons and molecular excitons at ambient conditions. Phys Rev Lett 114:157401

    ADS  Google Scholar 

  67. Gubbin CR, Maier SA, De Liberato S (2017) Theoretical investigation of phonon polaritons in SiC micropillar resonators. Phys Rev B 95:035313

    ADS  Google Scholar 

  68. Jaynes ET, Cummings FW (1963) Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proc IEEE 51:89

    Google Scholar 

  69. Tavis M, Cummings FW (1968) Exact solution for an N-molecule-radiation-field Hamiltonian. Phys Rev 170:379

    Google Scholar 

  70. Garraway BM (2011) The Dicke model in quantum optics: Dicke model revisited. Philos Trans R Soc A Math Phys Eng Sci 369:1137

    Google Scholar 

  71. Michetti P, Mazza L, La Rocca GC (2015) Strongly coupled organic microcavities. In: Zhao YS (ed) Organic nanophotonics, nano-optics and nanophotonics, vol 39. Springer, Berlin, Heidelberg

    Google Scholar 

  72. Agranovich V, Gartstein YN, Litinskaya M (2011) Hybrid resonant organic-inorganic nanostructures for optoelectronic applications. Chem Rev 111:5179

    Google Scholar 

  73. Spano FC (2015) Optical microcavities enhance the exciton coherence length and eliminate vibronic coupling in J-aggregates. J Chem Phys 142

    Google Scholar 

  74. Herrera F, Spano FC (2016) Cavity-controlled chemistry in molecular ensembles. Phys Rev Lett 116:238301

    Google Scholar 

  75. Ćwik JA, Kirton P, De Liberato S, Keeling J (2016) Excitonic spectral features in strongly coupled organic polaritons. Phys Rev A 93:33840

    ADS  Google Scholar 

  76. Zeb MA, Kirton PG, Keeling J (2018) Exact states and spectra of vibrationally dressed polaritons. ACS Photonics 5:249

    Google Scholar 

  77. Wu N, Feist J, Garcia-Vidal FJ (2016) When polarons meet polaritons: exciton-vibration interactions in organic molecules strongly coupled to confined light fields. Phys Rev B 94:195409

    ADS  Google Scholar 

  78. Herrera F, Spano FC (2017) Dark vibronic polaritons and the spectroscopy of organic microcavities. Phys Rev Lett 118

    Google Scholar 

  79. Herrera F, Spano FC (2018) Theory of nanoscale organic cavities: the essential role of vibration-photon dressed states. ACS Photonics 5:65

    Google Scholar 

  80. del Pino J (2018) Vibrational and electronic strong light–matter coupling with molecular excitations. Thesis (PhD), Universidad Autónoma de Madrid

    Google Scholar 

  81. Flick J, Ruggenthaler M, Appel H, Rubio A (2015) Kohn-Sham approach to quantum electrodynamical density-functional theory: exact time-dependent effective potentials in real space. Proc Natl Acad Sci 112:15285

    ADS  Google Scholar 

  82. Flick J, Ruggenthaler M, Appel H, Rubio A (2017) Atoms and molecules in cavities, from weak to strong coupling in quantum-electrodynamics (QED) chemistry. Proc Natl Acad Sci 114:3026

    Google Scholar 

  83. Tokatly IV (2013) Time-dependent density functional theory for many-electron systems interacting with cavity photons. Phys Rev Lett 110:233001

    ADS  Google Scholar 

  84. Ruggenthaler M, Flick J, Pellegrini C, Appel H, Tokatly IV, Rubio A (2014) Quantum-electrodynamical density-functional theory: bridging quantum optics and electronic-structure theory. Phys Rev A 90:012508

    ADS  Google Scholar 

  85. Feist J, Galego J, Garcia-Vidal FJ (2018) Polaritonic chemistry with organic molecules. ACS Photonics 5:205

    Google Scholar 

  86. Flick J, Appel H, Ruggenthaler M, Rubio A (2017) Cavity Born-Oppenheimer approximation for correlated electron-nuclear-photon systems. J Chem Theory Comput 13:1616

    Google Scholar 

  87. Somaschi N, Giesz V, De Santis L, Loredo J, Almeida MP, Hornecker G, Portalupi SL, Grange T, Antón C, Demory J et al (2016) Near-optimal single-photon sources in the solid state. Nat Photonics 10:340

    ADS  Google Scholar 

  88. Purcell EM (1946) Spontaneous emission probabilities at radio frquencies. Phys Rev 69:674

    Google Scholar 

  89. Louisell WH, Louisell WH (1973) Quantum statistical properties of radiation, vol 7. Wiley, New York

    Google Scholar 

  90. Koenderink AF (2010) On the use of purcell factors for plasmon antennas. Opt Lett 35:4208

    ADS  Google Scholar 

  91. Kristensen PT, Van Vlack C, Hughes S (2012) Generalized effective mode volume for leaky optical cavities. Opt Lett 37:1649

    ADS  Google Scholar 

  92. Sauvan C, Hugonin JP, Maksymov IS, Lalanne P (2013) Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators. Phys Rev Lett 110:237401

    ADS  Google Scholar 

  93. Brecha R, Rice P, Xiao M (1999) N two-level atoms in a driven optical cavity: quantum dynamics of forward photon scattering for weak incident fields. Phys Rev A 59:2392

    ADS  Google Scholar 

  94. Sáez-Blázquez R, Feist J, Fernández-Domínguez A, García-Vidal F (2017) Enhancing photon correlations through plasmonic strong coupling. Optica 4:1363

    ADS  Google Scholar 

  95. Bozhevolnyi SI, Khurgin JB (2016) Fundamental limitations in spontaneous emission rate of single-photon sources. Optica 3:1418

    Google Scholar 

  96. Törmä P, Barnes WL (2015) Strong coupling between surface plasmon polaritons and emitters: a review. Rep Prog Phys 78:13901

    Google Scholar 

  97. Henkelman G, Uberuaga BP, Jónsson H (2000) A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J Chem Phys 113:9901

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Física teórica de la materia condensada, Universidad Autónoma de Madrid, Madrid, Spain

    Dr. Javier Galego Pascual

Authors
  1. Dr. Javier Galego Pascual
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Javier Galego Pascual .

Rights and permissions

Reprints and permissions

Copyright information

© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Galego Pascual, J. (2020). Theoretical Background. In: Polaritonic Chemistry. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-48698-3_2

Download citation

Publish with us

Policies and ethics

Search

Navigation