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Many-Body Perturbation Theory (MBPT) and Time-Dependent Density-Functional Theory (TD-DFT): MBPT Insights About What Is Missing In, and Corrections To, the TD-DFT Adiabatic Approximation

  • Mark E. CasidaEmail author
  • Miquel Huix-Rotllant
Chapter
Part of the Topics in Current Chemistry book series (TOPCURRCHEM, volume 368)

Abstract

In their famous paper, Kohn and Sham formulated a formally exact density-functional theory (DFT) for the ground-state energy and density of a system of N interacting electrons, albeit limited at the time by certain troubling representability questions. As no practical exact form of the exchange-correlation (xc) energy functional was known, the xc-functional had to be approximated, ideally by a local or semilocal functional. Nowadays, however, the realization that Nature is not always so nearsighted has driven us up Perdew’s Jacob’s ladder to find increasingly nonlocal density/wavefunction hybrid functionals. Time-dependent (TD-) DFT is a younger development which allows DFT concepts to be used to describe the temporal evolution of the density in the presence of a perturbing field. Linear response (LR) theory then allows spectra and other information about excited states to be extracted from TD-DFT. Once again the exact TD-DFT xc-functional must be approximated in practical calculations and this has historically been done using the TD-DFT adiabatic approximation (AA) which is to TD-DFT very similar to what the local density approximation (LDA) is to conventional ground-state DFT. Although some of the recent advances in TD-DFT focus on what can be done within the AA, others explore ways around the AA. After giving an overview of DFT, TD-DFT, and LR-TD-DFT, this chapter focuses on many-body corrections to LR-TD-DFT as one way to build hybrid density-functional/wavefunction methodology for incorporating aspects of nonlocality in time not present in the AA.

Keywords

Electronic excited states Many-body perturbation theory Photochemistry Time-dependent density-functional theory 

Notes

Acknowledgements

We thank Andrei Ipatov, Mathias Ljungberg, Hemanadhan Myneni, Valerio Olevano, Giovanni Onica, Lucia Reining, Pina Romaniello, Angel Rubio, Davide Sangalli, Jochen Schirmer, and Eric Shirley for useful discussions. M. H. R. would like to acknowledge an Allocation de Recherche from the French Ministry of Education. Over the years, this work has been carried out in the context of several programs: the French Rhône-Alpes Réseau thématique de recherche avancée (RTRA): Nanosciences aux limites de la nanoélectronique, the Rhône-Alpes Associated Node of the European Theoretical Spectroscopy Facility (ETSF), and, most recently, the grant ANR-12-MONU-0014-02 from the French Agence Nationale de la Recherche for the ORGAVOLT project (ORGAnic solar cell VOLTage by numerical computation).

References

  1. 1.
    Rowlinson JS (2009) The border between physics and chemistry. Bull Hist Chem 34:1Google Scholar
  2. 2.
    Casida ME, Jamorski C, Casida KC, Salahub DR (1998) Molecular excitation energies to high-lying bound states from time-dependent density-functional response theory: characterization and correction of the time-dependent local density approximation ionization threshold. J Chem Phys 108:4439CrossRefGoogle Scholar
  3. 3.
    Casida ME (2002) Jacob’s ladder for time-dependent density-functional theory: some rungs on the way to photochemical heaven. In: Hoffmann MRH, Dyall KG (eds) Accurate description of low-lying molecular states and potential energy surfaces. ACS, Washington, p 199CrossRefGoogle Scholar
  4. 4.
    Doltsinis NL, Marx D (2002) First principles molecular dynamics involving excited states and nonadiabatic transitions. J Theo Comput Chem 1:319CrossRefGoogle Scholar
  5. 5.
    Cordova F, Doriol LJ, Ipatov A, Casida ME, Filippi C, Vela A (2007) Troubleshooting time-dependent density-functional theory for photochemical applications: oxirane. J Chem Phys 127:164111CrossRefGoogle Scholar
  6. 6.
    Tapavicza E, Tavernelli I, Rothlisberger U, Filippi C, Casida ME (2008) Mixed time-dependent density-functional theory/classical trajectory surface hopping study of oxirane photochemistry. J Chem Phys 129(12):124108CrossRefGoogle Scholar
  7. 7.
    Casida ME, Natarajan B, Deutsch T (2011) Non-Born-Oppenheimer dynamics and conical intersections. In: Marques M, Maitra N, Noguiera F, Gross EKU, Rubio A (eds) Fundamentals of time-dependent density-functional theory, Lecture Notes in Physics, vol 837. Springer, Berlin, p 279Google Scholar
  8. 8.
    Casida ME, Huix-Rotllant M (2012) Progress in time-dependent density-functional theory. Annu Rev Phys Chem 63:287CrossRefGoogle Scholar
  9. 9.
    Hohenberg P, Kohn W (1964) Inhomogenous electron gas. Phys Rev 136:B864CrossRefGoogle Scholar
  10. 10.
    Kohn W, Sham LJ (1965) Self-consistent equations including exchange and correlation effects. Phys Rev 140:A1133CrossRefGoogle Scholar
  11. 11.
    Parr RG, Yang W (1989) Density-functional theory of atoms and molecules. Oxford University Press, New YorkGoogle Scholar
  12. 12.
    Dreizler DM, Gross EKU (1990) Density functional theory, an approach to the quantum many-body problem. Springer, New YorkGoogle Scholar
  13. 13.
    Koch W, Holthausen MC (2000) A chemist’s guide to density functional theory. Wiley-VCH, New YorkGoogle Scholar
  14. 14.
    Perdew JP, Schmidt K (2001) Jacob’s ladder of density functional approximations for the exchange-correlation energy. In: Doren VEV, Alseoy KV, Geerlings P (eds) Density functional theory and its applications to materials. American Institute of Physics, Melville, New York, p 1Google Scholar
  15. 15.
    Perdew JP, Ruzsinsky A, Constantin LA, Sun J, Csonka GI (2009) Some fundamental issues in ground-state density functional theory: a guide for the perplexed. J Chem Theor Comput 5:902CrossRefGoogle Scholar
  16. 16.
    Perdew JP, Constantin LA (2007) Laplacian-level density functionals for the kinetic energy density and exchange-correlation energy. Phys Rev B 75:155109CrossRefGoogle Scholar
  17. 17.
    Gill PM (2001) Obituary: density-functional theory (1927–1993). Aust J Chem 54:661CrossRefGoogle Scholar
  18. 18.
    Becke A (1993) A new mixing of HartreeFock and local density functional theories. J Chem Phys 98:1372CrossRefGoogle Scholar
  19. 19.
    Perdew JP, Ernzerhof M, Burke K (1996) Rationale for mixing exact exchange with density functional approximations. J Chem Phys 105:9982Google Scholar
  20. 20.
    Savin A (1995) Beyond the Kohn–Sham determinant. In: Chong DP (ed) Recent advances in density functional theory. World Scientific, Singapore, p 129CrossRefGoogle Scholar
  21. 21.
    Baer R, Livshits E, Salzner U (2010) Tuned range-separated hybrids in density functional theory. Annu Rev Phys Chem 61:85CrossRefGoogle Scholar
  22. 22.
    Marques MAL, Ullrich C, Nogueira F, Rubio A, Gross EKU (eds) (2006) Time-dependent density-functional theory, Lecture Notes in Physics, vol 706. Springer, BerlinGoogle Scholar
  23. 23.
    Marques M, Maitra N, Noguiera F, Gross EKU, Rubio A (2011) Fundamentals of time-dependent density-functional theory, Lecture Notes in Physics, vol 837. Springer, BerlinGoogle Scholar
  24. 24.
    Ullrich CA (2012) Time-dependent density-functional theory: concepts and applications. Oxford University Press, OxfordGoogle Scholar
  25. 25.
    Runge E, Gross EKU (1984) Density functional theory for time-dependent systems. Phys Rev Lett 52:997CrossRefGoogle Scholar
  26. 26.
    van Leeuwen R (1999) Mapping from densities to potentials in time-dependent density-functional theory. Phys Rev Lett 82:3863CrossRefGoogle Scholar
  27. 27.
    Maitra NT, Todorov TN, Woodward C, Burke K (2010) Density-potential mapping in time-dependent density-functional theory. Phys Rev A 81:042525Google Scholar
  28. 28.
    Ruggenthaler M, van Leeuwen R (2011) Global fixed-point proof of time-dependent density-functional theory. Europhys Lett 95:13001CrossRefGoogle Scholar
  29. 29.
    Ruggenthaler M, Glesbertz KJH, Penz M, van Leeuwen R (2012) Density-potential mappings in quantum dynamics. Phys Rev A 85:052504CrossRefGoogle Scholar
  30. 30.
    Ruggenthaler M, Nielsen SEB, van Leeuwen R (2013) Analytic density functionals with initial-state dependence. Phys Rev A 88:022512CrossRefGoogle Scholar
  31. 31.
    Vignale G (2008) Real-time resolution of the causality paradox of time-dependent density-functional theory. Phys Rev A 77(6):1. doi: 10.1103/PhysRevA.77.062511
  32. 32.
    Messud J, Dinh PM, Reinhard P, Suraud E (2011) The generalized SIC-OEP formalism and the generalized SIC-Slater approximation (stationary and time-dependent cases). Ann Phys (Berlin) 523:270CrossRefGoogle Scholar
  33. 33.
    Rajagopal AK (1996) Time-dependent variational principle and the effective action in density-functional theory and Berrys phase. Phys Rev A 54:3916CrossRefGoogle Scholar
  34. 34.
    van Leeuwen R (1998) Causality and symmetry in time-dependent density-functional theory. Phys Rev Lett 80:1280CrossRefGoogle Scholar
  35. 35.
    van Leeuwen R (2001) Key concepts in time-dependent density-functional theory. Int J Mod Phys 15:1969CrossRefGoogle Scholar
  36. 36.
    Mukamel S (2005) Generalized time-dependent density-functional-theory response functions for spontaneous density fluctuations and nonlinear response: resolving the causality paradox. Phys Rev A 024503Google Scholar
  37. 37.
    Mosquera MA (2013) Action formalism in time-dependent density-functional theory. Phys Rev B 88:022515CrossRefGoogle Scholar
  38. 38.
    Casida ME (1995) Time-dependent density-functional response theory for molecules. In: Chong DP (ed) Recent advances in density functional methods, Part I. World Scientific, Singapore, p 155Google Scholar
  39. 39.
    Casida ME (1996) Time-dependent density functional response theory of molecular systems: theory, computational methods, and functionals. In: Seminario J (ed) Recent developments and applications of modern density functional theory. Elsevier, Amsterdam, p 391Google Scholar
  40. 40.
    Löwdin PO (1964) Studies in perturbation theory. Part VI. Contraction of secular equations. J Mol Spectr 14:112Google Scholar
  41. 41.
    Onida G, Reining L, Rubio A (2002) Electronic excitations: density-functional versus many-body Greens-function approaches. Rev Mod Phys 74:601CrossRefGoogle Scholar
  42. 42.
    Reining L, Olevano V, Rubio A, Onida G (2002) Excitonic effects in solids described by time-dependent density-functional theory. Phys Rev Lett 88:066404CrossRefGoogle Scholar
  43. 43.
    Sottile F, Olevano V, Reining L (2003) Parameter-free calculation of response functions in time-dependent density-functional theory. Phys Rev Lett 91:056402CrossRefGoogle Scholar
  44. 44.
    Marini A, Sole RD, Rubio A (2003) Bound excitons in time-dependent density-functional theory: optical and energy-loss spectra. Phys Rev Lett 91:256402CrossRefGoogle Scholar
  45. 45.
    Stubner R, Tokatly IV, Pankratov O (2004) Excitonic effects in time-dependent density-functional theory: an analytically solvable model. Phys Rev B 70:245119CrossRefGoogle Scholar
  46. 46.
    von Barth U, Dahlen NE, van Leeuwen R, Stefanucci G (2005) Conserving approximations in time-dependent density functional theory. Phys Rev B 72:235109CrossRefGoogle Scholar
  47. 47.
    Romaniello P, Sangalli D, Berger JA, Sottile F, Molinari LG, Reining L, Onida G (2009) Double excitations in finite systems. J Chem Phys 130:044108CrossRefGoogle Scholar
  48. 48.
    Oddershede J, Jørgensen P (1977) An order analysis of the particle-hole propagator. J Chem Phys 66:1541CrossRefGoogle Scholar
  49. 49.
    Nielsen ES, Jørgensen P, Oddershede J (1980) Transition moments and dynamic polarizabilities in a second order polarization propagator approach. J Chem Phys 73:6238CrossRefGoogle Scholar
  50. 50.
    Nielsen ES, Jørgensen P, Oddershede J (1980) J Chem Phys 75:499; Erratum (1980): J Chem Phys 73:6238Google Scholar
  51. 51.
    Jørgensen P, Simons J (1981) Second quantization-based methods in quantum chemistry. Academic, New YorkGoogle Scholar
  52. 52.
    Schirmer J (1982) Beyond the random phase approximation: a new approximation scheme for the polarization propagator. Phys Rev A 26:2395CrossRefGoogle Scholar
  53. 53.
    Trofimov AB, Stelter G, Schirmer J (1999) A consistent third-order propagator method for electronic excitation. J Chem Phys 111:9982CrossRefGoogle Scholar
  54. 54.
    Fetter AL, Walecka JD (1971) Quantum theory of many-particle systems. McGraw-Hill, New YorkGoogle Scholar
  55. 55.
    Kobe DH (1966) Linked cluster theorem and the Green’s function equations of motion for a many-fermion system. J Math Phys 7(10):1806CrossRefGoogle Scholar
  56. 56.
    Wilson S (1984) Electron correlation in molecules. Clarendon, OxfordGoogle Scholar
  57. 57.
    Sangalli D, Romaniello P, Colò G, Marini A, Onida G (2011) Double excitation in correlated systems: a many-body approach. J Chem Phys 134:034115CrossRefGoogle Scholar
  58. 58.
    Casida ME (2005) Propagator corrections to adiabatic time-dependent density-functional theory linear response theory. J Chem Phys 122:054111CrossRefGoogle Scholar
  59. 59.
    Hirata S, Ivanov S, Bartlett RJ, Grabowski I (2005) Exact-exchange time-dependent density-functional theory for static and dynamic polarizabilities. Phys Rev A 71:032507CrossRefGoogle Scholar
  60. 60.
    Görling A (1998) Exact exchange kernel for time-dependent density-functional theory. Int J Quant Chem 69:265CrossRefGoogle Scholar
  61. 61.
    Maitra NT, Zhang F, Cave RJ, Burke K (2004) Double excitations within time-dependent density functional theory linear response theory. J Chem Phys 120:5932CrossRefGoogle Scholar
  62. 62.
    Cave RJ, Zhang F, Maitra NT, Burke K (2004) A dressed TDDFT treatment of the 1Ag states of butadiene and hexatriene. Chem Phys Lett 389:39CrossRefGoogle Scholar
  63. 63.
    Mazur G, Włodarczyk R (2009) Application of the dressed time-dependent density functional theory for the excited states of linear polyenes. J Comput Chem 30:811CrossRefGoogle Scholar
  64. 64.
    Gritsenko OV, Baerends EJ (2009) Double excitation effect in non-adiabatic time-dependent density functional theory with an analytic construction of the exchange-correlation kernel in the common energy denominator approximation. Phys Chem Chem Phys 11:4640CrossRefGoogle Scholar
  65. 65.
    Huix-Rotllant M, Ipatov A, Rubio A, Casida ME (2011) Assessment of dressed time-dependent density-functional theory for the low-lying valence states of 28 organic chromophores. Chem Phys 391:120CrossRefGoogle Scholar
  66. 66.
    Schreiber M, Silva-Junior MR, Sauer SPA, Thiel W (2008) Benchmarks for electronically excited states: CASPT2, CC2, CCSD, and CC3. J Chem Phys 128:134110CrossRefGoogle Scholar
  67. 67.
    Hsu CP, Hirata S, Head-Gordon M (2001) Excitation energies from time-dependent density functional theory for linear polyene oligomers: butadiene to decapentaene. J Phys Chem A 105:451CrossRefGoogle Scholar
  68. 68.
    Maitra NT, Tempel DG (2006) Long-range excitations in time-dependent density functional theory. J Chem Phys 125:184111CrossRefGoogle Scholar
  69. 69.
    Huix-Rotllant M (2011) Improved correlation kernels for linear-response time-dependent density-functional theory. Ph.D. thesis, Université de GrenobleGoogle Scholar
  70. 70.
    Bokhan D, Schweigert IG, Bartlett RJ (2005) Interconnection between functional derivative and effective operator approaches in ab initio density functional theory. Mol Phys 103:2299CrossRefGoogle Scholar
  71. 71.
    Bokhan D, Bartlett RJ (2006) Adiabatic ab initio time-dependent density-functional theory employing optimized-effective-potential many-body perturbation theory potentials. Phys Rev A 73:022502CrossRefGoogle Scholar
  72. 72.
    Talman JD, Shadwick WF (1976) Optimized effective atomic central potential. Phys Rev A 14:36CrossRefGoogle Scholar
  73. 73.
    Talman JD (1989) A program to compute variationally optimized effective atomic potentials. Comp Phys Commun 54:85CrossRefGoogle Scholar
  74. 74.
    Görling A (1999) New KS method for molecules based on an exchange charge density generating the exact local KS exchange potential. Phys Rev Lett 83:5459CrossRefGoogle Scholar
  75. 75.
    Ivanov S, Hirata S, Bartlett RJ (1999) Exact exchange treatment for molecules in finite-basis-set Kohn–Sham theory. Phys Rev Lett 83:5455CrossRefGoogle Scholar
  76. 76.
    Casida ME (1995) Generalization of the optimized effective potential model to include electron correlation: a variational derivation of the Sham–Schlüter equation for the exact exchange-correlation potential. Phys Rev A 51:2505CrossRefGoogle Scholar
  77. 77.
    Casida ME (1999) Correlated optimized effective potential treatment of the derivative discontinuity and of the highest occupied Kohn–Sham eigenvalue: a Janak-type theorem for the optimized effective potential method. Phys Rev B 59:4694CrossRefGoogle Scholar
  78. 78.
    Hirata S, Ivanov S, Grabowski I, Bartlett RJ (2002) Time-dependent density functional theory employing optimized effective potentials. J Chem Phys 116:6468CrossRefGoogle Scholar
  79. 79.
    Bokhan D, Barlett RJ (2007) Exact-exchange density functional theory for hyperpolarizabilities. J Chem Phys 127:174102CrossRefGoogle Scholar
  80. 80.
    Tokatly IV, Pankratov O (2001) Many-body diagrammatic expansion in a Kohn–Sham basis: implications for time-dependent density functional theory of excited states. Phys Rev Lett 86:2078CrossRefGoogle Scholar
  81. 81.
    Tokatly IV, Stubner R, Pankratov O (2002) Many-body diagrammatic expansion of the exchange-correlation kernel in time-dependent density-functional theory. Phys Rev B 65:113107CrossRefGoogle Scholar
  82. 82.
    Gonze X, Scheffler M (1999) Exchange and correlation kernels at the resonance frequency: implications for excitation energies in density-functional theory. Phys Rev Lett 82:4416CrossRefGoogle Scholar
  83. 83.
    Harriman JE (1983) Geometry of density-matrices. 4. The relationship between density-matrices and densities. Phys Rev A 27:632CrossRefGoogle Scholar
  84. 84.
    Harriman JE (1986) Densities, operators, and basis sets. Phys Rev A 34:29CrossRefGoogle Scholar
  85. 85.
    Heßelmann A, Ipatov A, Görling A (2009) Charge-transfer excitation energies with a time-dependent density-functional method suitable for orbital-dependent exchange-correlation functionals. Phys Rev A 80:012507CrossRefGoogle Scholar
  86. 86.
    Filippi C, Umrigar CJ, Gonze X (1997) Excitation energies from density functional perturbation theory. J Chem Phys 107(23):9994CrossRefGoogle Scholar
  87. 87.
    Görling A (1996) Density-functional theory for excited states. Phys Rev A 54(5):3912CrossRefGoogle Scholar
  88. 88.
    Li SL, Marenich AV, Xu X, Truhlar DG (2014) Configuration interaction-corrected Tamm-Dancoff approximation: a time-dependent density functional method with the correct dimensionality of conical intersections. J Chem Phys Lett 5:322Google Scholar
  89. 89.
    Fromager E, Knecht S, Jensen HJA (2013) Multi-configuration time-dependent density-functional theory based upon range separation. J Chem Phys 138:084101CrossRefGoogle Scholar
  90. 90.
    Seidu I, Krykunov M, Ziegler T (2014) The formulation of a constricted variational density functional theory for double excitations. Mol Phys 112:661CrossRefGoogle Scholar
  91. 91.
    Böhm M, Tatchen J, Krügler D, Kleinermanns K, Nix MGD, LaGreve TA, Zwier TS, Schmitt M (2009) High-resolution and dispersed fluorescence examination of vibronic bands of tryptamine: spectroscopic signatures for L a/L b mixing near a conical intersection. J Phys Chem A 113:2456CrossRefGoogle Scholar
  92. 92.
    Minezawa N, Gordon MS (2009) Optimizing conical intersections by spin-flip density-functional theory: application to ethylene. J Phys Chem A 113:12749CrossRefGoogle Scholar
  93. 93.
    Huix-Rotllant M, Natarajan B, Ipatov A, Wawire CM, Deutsch T, Casida ME (2010) Assessment of noncollinear spin-flip Tamm-Dancoff approximation time-dependent density-functional theory for the photochemical ring-opening of oxirane. Phys Chem Chem Phys 12:12811CrossRefGoogle Scholar
  94. 94.
    Rinkevicius Z, Vahtras O, Ågren H (2010) Spin-flip time dependent density functional theory applied to excited states with single, double, or mixed electron excitation character. J Chem Phys 133:114104CrossRefGoogle Scholar
  95. 95.
    Minezawa N, Gordon MS (2011) Photoisomerization of stilbene: a spin-flip density functional theory approach. J Phys Chem A 115:7901CrossRefGoogle Scholar
  96. 96.
    Casanova D (2012) Avoided crossings, conical intersections, and low-lying excited states with a single reference method: the restricted active space spin-flip configuration interaction approach. J Chem Phys 137:084105CrossRefGoogle Scholar
  97. 97.
    Huix-Rotllant M, Filatov F, Gozem S, Schapiro I, Olivucci M, Ferré N (2013) Assessment of density functional theory for describing the correlation effects on the ground and excited state potential energy surfaces of a retinal chromophore model. J Chem Theory Comput 9:3917CrossRefGoogle Scholar
  98. 98.
    Minezawa N (2014) Optimizing minimum free-energy crossing points in solution: linear-response free energy/spin-flip density functional theory approach. J Chem Phys 141:164118CrossRefGoogle Scholar
  99. 99.
    Harabuchi Y, Keipert K, Zahariev F, Taketsugu T, Gordon MS (2014) Dynamics simulations with spin-flip time-dependent density functional theory: photoisomerization and photocyclization mechanisms of cis-stilbene in (π, π*) states. J Phys Chem A 118:11987CrossRefGoogle Scholar
  100. 100.
    Nikiforov A, Gamez JA, Thiel W, Huix-Rotllant M, Filatov M (2014) Assessment of approximate computational methods for conical intersections and branching plane vectors in organic molecules. J Chem Phys 141:124122CrossRefGoogle Scholar
  101. 101.
    Gozem S, Melaccio F, Valentini A, Filatov M, Huix-Rotllant M, Ferré N, Frutos LM, Angeli C, Krylov AI, Granovsky AA, Lindh R, Olivucci M (2014) Shape of multireference, equation-of-motion coupled-cluster, and density functional theory potential energy surfaces at a conical intersection. J Chem Theory Comput 10:3074Google Scholar
  102. 102.
    Zhang X, Herbert JM (2014) Analytic derivative couplings for spin-flip configuration interaction singles and spin-flip time-dependent density functional theory. J Chem Phys 141:064104CrossRefGoogle Scholar
  103. 103.
    Frank I, Damianos K (2007) Restricted open-shell Kohn–Sham theory: simulation. J Chem Phys 126:125105CrossRefGoogle Scholar
  104. 104.
    Friedrichs J, Darnianos K, Frank I (2008) Solving restricted open-shell equations in excited state molecular dynamics simulations. J Chem Phys 347:17Google Scholar
  105. 105.
    Filatov M (2015) Spin-restricted ensemble-referenced Kohn–Sham method: basic principles and application to strongly correlated ground and excited states of molecules. Comput Mol Sci 5:146CrossRefGoogle Scholar
  106. 106.
    Shibuya T, Rose J, McKoy V (1973) Equations-of-motion method including renormalization and double-excitation mixing. J Chem Phys 58:500Google Scholar
  107. 107.
    Jørgensen P, Oddershede J, Ratner MA (1975) Two-particle, two-hole corrections to a self-consistent time-dependent Hartree-Fock scheme. Chem Phys Lett 32:111CrossRefGoogle Scholar
  108. 108.
    Oddershede J, Sabin JR (1983) The use of modified virtual orbitals in perturbative polarization propagator calculations. J Chem Phys 79:2295Google Scholar
  109. 109.
    Oddershede J, Jørgensen P, Yeager DL (1984) Polarization propagator methods in atomic and molecular calculations. Comp Phys Rep 2:33CrossRefGoogle Scholar
  110. 110.
    Oddershede J, Jørgensen P, Beebe NHF (1978) Analysis of excitation energies and transition moments. J Phys B Atom Mol Phys 11:1CrossRefGoogle Scholar
  111. 111.
    Trofimov AB, Schirmer J (1995) An efficient polarization propagator approach to valence electron excitation spectra. J Phys B At Mol Opt Phys 28:2299CrossRefGoogle Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Département de Chimie Moléculaire, Institut de Chimie Moléculaire de GrenobleUniversité Joseph Fourier (Grenoble I)Grenoble Cedex 9France
  2. 2.Institut für Physikalische und Theoretische ChimieUniversität Frankfurt am MainFrankfurtGermany

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