Wave Function Theories and Electronic Structure Methods: Quantum Chemistry, from Atoms to Molecules

  • Fanica Cimpoesu
  • Mihai V. Putz
  • Marilena Ferbinteanu
Chapter

Abstract

The chapter continues the discussion of the atomic structure, at the level of clarifying concreteness and details, presenting also the basic methods of electronic structure theories, constituting the keystone of the developments debated in the next chapters. The Slater determinants are introduced as primitives for the construction of many-electrons wave functions, using the one-electron orbital functions as basic ingredients. The orbitals, known from the introductory part of atom theory, are generalized at the molecular level. The so-called Slater rules for handling the Hamiltonian matrix elements of the Slater determinants constructed from orthogonal orbitals are presented as the basic algorithm for the practical approach of quantum chemistry. Besides, the rules are generalizable to other operators working with one- and two-electron terms. The Slater rules are generalized, at the end of the chapter, for the case of non-orthogonal orbitals, implying non-orthogonal Slater determinants bases. With the help of Slater rules and symmetry dichotomy of the two-electron integrals in the atomic shells (the Slater–Condon parameters), several poly-electronic atoms are analyzed in quite advanced detail, writing down analytical formulas for the energies of their states and relating to the experimental data on spectral terms. This exercise is important, beyond the domain of free atoms, since in large classes of compounds and materials with technical applicability one deals with embedded ions, whose properties are approachable by an atom-alike phenomenology. An exemplification on lanthanide emission spectra, used in domestic lighting devices, occasions a quick excursus in the open challenges of current materials sciences and the rational design of properties. Going from atoms to molecules, the use of atomic basis sets as the background of quantum chemistry is debated in detail, with hands-on illustration of the various options: Slater-type orbitals, Gaussian-type bases, plane waves, and numerical bases. A critical eye is turned upon Gaussian-type orbitals, signaling certain significant failures of bases rated as rich and accurate. The underperformance is determined by the lack of appropriate polynomial factors in the definition of atomic orbitals assignable to high quantum numbers, an intrinsic design deficiency in the customary implementation and use of Gaussians. The common belief is that the limitation of Gaussians consists in the assumed use of  \( \text{exp}(-\alpha r^2)\) –type functions instead of more physical  \( \text{exp}(-\alpha r)\) ones, while the lack of proper polynomial factors can cause more severe drawbacks. In turn, numeric basis sets are observed as surprisingly good performers and possible alternative technical options. The final part of the chapter presents the fundamental electronic structure methods based on wave functions and first principles operators: the Hartree–Fock technique, introducing self-consistency, brought to a higher level by the multi-configurational approach, and the Valence Bond theory. The so-called Complete Active Space Self-Consistent Field methods are near the top of powerfulness among actually available methods, which, with broad conceptual scope and flexible technical leverages, allow the approach to a large number of problems, with the right picture of their mechanisms and manifestations. Certain other methodological varieties, such as second-order perturbation corrections to self-consistent Hartree–Fock or multi-configurational techniques, or the Coupled Cluster expansion are discarded from the actual synopsis. From our perspective, such procedures can bring only incremental changes to the physical picture, sometimes not in a well-tempered manner, while their non-variational nature is a hidden drawback, at least in conceptual respects, and a heavy burden to the computation routines. Somewhat greater attention is paid to the Valence Bond frame, acknowledging its merit as a foundational model of the chemical bond and also its potential in a modern methodological reshaping.

Keywords

Atomic and molecular orbitals Spin-orbitals Poly-electronic wave functions Indiscernibility Anti-symmetrization Slater determinants Slater rules Slater–Condon integrals Atomic spectra Spectral terms Optical properties Basis sets Slater-type orbitals Gaussian-type orbitals Plane waves bases Numerical basis sets Self-consistent methods Hartree–Fock theory Multi-configurational methods Complete active space Valence bond theory Koopmans theorem 

References

  1. ADF (2015) SCM, Theoretical chemistry. Vrije Universiteit, Amsterdam. http://www.scm.com
  2. Andersson K, Malmqvist PÅ, Roos BO, Sadlej AJ, Wolinski K (1990) Second-order perturbation-theory with a CASSCF reference function. J Phys Chem 94:5483–5488CrossRefGoogle Scholar
  3. Bachmann V, Jüstel T, Meijerink A, Ronda C, Schmidt PJ (2006) Luminescence properties of SrSi2O2N2 doped with divalent rare earth ions. J Lumin 121:441–449CrossRefGoogle Scholar
  4. Blochl PE (1994) Projector augmented-wave method. Phys Rev B 50:17953–17979CrossRefGoogle Scholar
  5. Collins JB, Schleyer PVR, Binkley JS, Pople JA (1976) Self-consistent molecular orbital methods. 17. Geometries and binding energies of second-row molecules: a comparison of three basis sets. J Chem Phys 64:5142–5151CrossRefGoogle Scholar
  6. Condon EU (1930) The theory of complex spectra. Phys Rev 36:1121–1133CrossRefGoogle Scholar
  7. Cooper DL (1998) Thorstein Thorsteinsson Joseph Gerratt, Modern VB representations of CASSCF wave functions and the fully-variational optimization of modern VB wave functions using the CASVB strategy. Adv Quantum Chem 32:51–67CrossRefGoogle Scholar
  8. Cooper DL (ed) (2002) Valence bond theory. Elsevier, AmsterdamGoogle Scholar
  9. Cundari TR, Stevens WJ (1993) Effective core potential methods for the lanthanides. J Chem Phys 98:5555–5565CrossRefGoogle Scholar
  10. de Castro EVR, Jorge FE (1998) Accurate universal Gaussian basis set for all atoms of the periodic table. J Chem Phys 108:5225–5229CrossRefGoogle Scholar
  11. Delley B (2000) From molecules to solids with the DMol3 approach. J Chem Phys 113:7756–7764CrossRefGoogle Scholar
  12. Di Sipio L, Tondello E, De Michelis G, Oleari L (1970) Slater–Condon parameters for atoms and ions of the second transition metal series. Inorg Chem 9:927–930CrossRefGoogle Scholar
  13. Dorenbos P (2003) Energy of the first 4f7 → 4f65d transition of Eu2+ in inorganic compounds. J Lumin 104:239–260CrossRefGoogle Scholar
  14. Dunning TH Jr (1989) Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J Chem Phys 90:1007–1023CrossRefGoogle Scholar
  15. Francl MM, Pietro WJ, Hehre WJ, Binkley JS, DeFrees DJ, Pople JA, Gordon MS (1982) Self-consistent molecular orbital methods. 23. A polarization-type basis set for 2nd-row elements. J Chem Phys 77:3654–3665CrossRefGoogle Scholar
  16. Fuchs M, Scheffler M (1999) Ab initio pseudopotentials for electronic structure calculations of poly-atomic systems using density-functional theory. Comput Phys Commun 119:67–98CrossRefGoogle Scholar
  17. Gallup GA (2002) Valence bond methods: theory and applications. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  18. Gonze X, Beuken JM, Caracas R, Detraux F, Fuchs M, Rignanese GM, Sindic L, Verstraete M, Zerah G, Jollet F, Torrent M, Roy A, Mikami M, Ghosez Ph, Raty JY, Allan DC (2002) First-principles computation of material properties: the ABINIT software project. Comput Mater Sci 25:478–492CrossRefGoogle Scholar
  19. Gordon MS, Binkley JS, Pople JA, Pietro WJ, Hehre WJ (1982) Self-consistent molecular orbital methods. 22. Small split-valence basis sets for second-row elements. J Am Chem Soc 104:2797–2803CrossRefGoogle Scholar
  20. Hay PJ, Wadt WR (1985) Ab initio effective core potentials for molecular calculations: potentials for the transition-metal atoms Sc to Hg. J Chem Phys 82:270–283CrossRefGoogle Scholar
  21. Hehre WJ, Ditchfield R, Pople JA (1972) Self-consistent molecular orbital methods. 12. Further extensions of Gaussian-type basis sets for use in molecular-orbital studies of organic-molecules. J Chem Phys 56:2257–2266CrossRefGoogle Scholar
  22. Hehre WJ, Stewart RF, Pople JA (1969) Self-consistent molecular orbital methods. 1. Use of Gaussian expansions of Slater-type atomic orbitals. J Chem Phys 51:2657–2664CrossRefGoogle Scholar
  23. Heisenberg W (1926) Multi-body problem and resonance in quantum mechanics. Z Phys 38:411–426CrossRefGoogle Scholar
  24. Heitler W, London F (1927) Interaction between neutral atoms and homopolar binding. Z Phys 44:455–472CrossRefGoogle Scholar
  25. Hirao K, Nakano H, Nakayama K, Dupuis M (1996) A complete active space valence bond (CASVB) method. J Chem Phys 105:9227–9239CrossRefGoogle Scholar
  26. Hoffmann R, Shaik S, Hiberty PC (2003) A conversation on VB vs. MO theory: a never ending rivalry? Acc Chem Res 36:750–756CrossRefGoogle Scholar
  27. Holzwarth NAW, Tackett AR, Matthews GE (2001) A Projector Augmented Wave (PAW) code for electronic structure calculations, Part I: atompaw for generating atom-centered functions. Comput Phys Commun 135:329–347CrossRefGoogle Scholar
  28. Höppe HA (2009) Recent developments in the field of inorganic phosphors. Angew Chem Int Ed 48:3572–3582CrossRefGoogle Scholar
  29. Hückel E (1930) Zur Quantentheorie der Doppelbindung. Z Phys 60:423–456CrossRefGoogle Scholar
  30. Hund F (1928) Zur Deutung der Molekelspektren. IV. Z Phys 51:759–795CrossRefGoogle Scholar
  31. Jensen F (2007) Introduction to computational chemistry. Wiley, ChichesterGoogle Scholar
  32. Jorge FE, de Castro EVR, da Silva ABF (1997) A universal Gaussian basis set for atoms cerium through lawrencium generated with the generator coordinate Hartree–Fock method. J Comp Chem 18:1565–1569CrossRefGoogle Scholar
  33. Jüstel T, Nikol H, Ronda C (1998) New developments in the field of luminescent materials for lighting and displays. Angew Chem Int Ed 37:3084–3103CrossRefGoogle Scholar
  34. Karlström G, Lindh R, Malmqvist PÅ, Roos BO, Ryde U, Veryazov V, Widmark PO, Cossi M, Schimmelpfennig B, Neogrady P, Seijo L (2003) MOLCAS: a program package for computational chemistry. Comput Mater Sci 28:222–239CrossRefGoogle Scholar
  35. Kielkopf JF, Crosswhite HM (1970) Preliminary analysis of the spectrum of triply ionized gadolinium. J Opt Soc Am 60:347–351CrossRefGoogle Scholar
  36. Koopmans T (1934) Über die Zuordnung von Wellenfunktionen und Eigenwerten zu den einzelnen Elektronen eines atoms. Physica 1:104–113CrossRefGoogle Scholar
  37. Kotani M, Amemiya A, Ishiguro E, Kimura T (1955) Table of molecular integrals. Maruzen Co., TokyoGoogle Scholar
  38. Kresse G, Furthmuller J (1996) Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput Mat Sci 6:15–50CrossRefGoogle Scholar
  39. Li J, Duke B, McWeeny R (2010) VB2000 version 2.7. SciNet Technologies, San DiegoGoogle Scholar
  40. Li J, McWeeny R (2002) VB2000: pushing valence bond theory to new limits. Int J Quantum Chem 89:208–216CrossRefGoogle Scholar
  41. Li YQ, de With G, Hintzen HT (2008) The effect of replacement of Sr by Ca on the structural and luminescence properties of the red-emitting Sr2Si5N8:Eu2+ LED conversion phosphor. J Solid State Chem 181:515–524CrossRefGoogle Scholar
  42. Li YQ, van Steen JEJ, van Krevel JWH, Botty G, Delsing ACA, DiSalvo FJ, de With G, Hintzen HT (2006) Luminescence properties of red-emitting M2Si5N8:Eu2+ (M = Ca, Sr, Ba) LED conversion phosphors. J Alloys Compd 417:273–279CrossRefGoogle Scholar
  43. Martin WC, Zalubas R, Hagan L (1978) Atomic energy levels: the rare-earth elements. In: National standard reference data series-NSRDS-NBS, vol 60. National Bureau of Standards, USAGoogle Scholar
  44. Morton DC, Wu Q, Drake GWF (2006) Energy levels for the stable isotopes of atomic helium (4He I and 3He I). Can J Phys 84:83–105CrossRefGoogle Scholar
  45. Mukai T, Yamada M, Nakamura S (1998) Current and temperature dependences of electroluminescence of InGaN-based UV/blue/green light-emitting diodes. Jpn J Appl Phys 37:L1358–L1361CrossRefGoogle Scholar
  46. Mullliken RS (1928a) The assignment of quantum numbers for electrons in molecules. I. Phys Rev 32(2):186–222CrossRefGoogle Scholar
  47. Mullliken RS (1928b) The assignment of quantum numbers for electrons in molecules. II. The correlation of molecular and atomic states. Phys Rev 32:761–772CrossRefGoogle Scholar
  48. Nakamura S, Fasol G (1997) The blue laser diode. Springer, BerlinCrossRefGoogle Scholar
  49. Nakamura S, Senoh M, Iwasa N, Nagahama S (1995) High-brightness InGaN blue, green and yellow light-emitting diodes with quantum well structures. Jpn J Appl Phys 34:L797–L799CrossRefGoogle Scholar
  50. Nakano H (1993) Quasidegenerate perturbation theory with multiconfigurational self-consistent-field reference functions. J Chem Phys 99:7983–7992CrossRefGoogle Scholar
  51. Nakano H, Nakayama K, Hirao K, Dupuis M (1997) Transition state barrier height for the reaction H2CO–H2 + CO studied by multireference Moller-Plesset perturbation theory. J Chem Phys 106:4912–4917CrossRefGoogle Scholar
  52. NIST: National Institute of Standards and Technology (2015) Atomic spectra database levels data. http://physics.nist.gov/cgi-bin/ASD/energy1.pl
  53. Nugent LJ, Baybarz RD, Burnett JL, Ryan JL (1973) Electron-transfer and f-d absorption bands of some lanthanide and actinide complexes and the standard (II–III) oxidation potential for each member of the lanthanide and actinide series. J Phys Chem 77:1528–1539CrossRefGoogle Scholar
  54. Pantazis AD, Neese F (2009) All-electron scalar relativistic basis sets for the lanthanides. J Chem Theory Comput 5:2229–2238CrossRefGoogle Scholar
  55. Paschen F (1919) Das spektrum des neon. Ann Phys (Leipzig) 365:405–453CrossRefGoogle Scholar
  56. Pauling L (1931) The nature of the chemical bond: application of results obtained from the quantum mechanics and from a theory of paramagnetic susceptibility to the structure of molecules. J Am Chem Soc 53:1367–1400CrossRefGoogle Scholar
  57. Pauling L (1939) The nature of the chemical bond, 3rd ed., 1960. Cornell University Press, Ithaca, NYGoogle Scholar
  58. Pou-Amérigo R, Merchán M, Nebot-Gil I, Widmark PO, Roos BO (1995) Density matrix averaged atomic natural orbital (ANO) basis sets for correlated molecular wave functions. Theor Chim Acta 92:149–181CrossRefGoogle Scholar
  59. Ramanantoanina H, Cimpoesu F, Gottel C, Sahnoun M, Herden B, Suta M, Wickleder C, Urland W, Daul C (2015) Prospecting lighting applications with ligand field tools and density functional theory: a first-principles account of the 4f(7)-4f(6)5d(1) luminescence of CsMgBr3:Eu2+. Inorg Chem 54:8319–8326CrossRefGoogle Scholar
  60. Ramanantoanina H, Urland W, Cimpoesu F, Daul C (2013a) Ligand field density functional theory calculation of the 4f2-4f15d1 transitions in the quantum cutter Cs2KYF6:Pr3+. Phys Chem Chem Phys 15:13902–13910CrossRefGoogle Scholar
  61. Ramanantoanina H, Urland W, García-Fuente A, Cimpoesu F, Daul C (2013b) Calculation of the 4f1 → 4f0d1 transitions in Ce3+ -doped systems by ligand field density functional theory. Chem Phys Lett 588:260–266CrossRefGoogle Scholar
  62. Ramanantoanina H, Urland W, García-Fuente A, Cimpoesu F, Daul C (2014) Ligand field density functional theory for the prediction of future domestic lighting. Phys Chem Chem Phys 16:14625–14634CrossRefGoogle Scholar
  63. Riblet P, Hirayama H, Kinoshita A, Hirata A, Sugano T, Aoyagi Y (1999) Determination of photoluminescence mechanism in InGaN quantum wells. Appl Phys Lett 75:2241–2243CrossRefGoogle Scholar
  64. Roos BO, Andersson K, Fulscher MK, Malmqvist PA, Serrano-Andres L, Pierloot K, Merchan M (1996) Multiconfigurational perturbation theory: applications in electronic spectroscopy. Adv Chem Phys 93:219–331Google Scholar
  65. Roos BO, Lindh R, Malmqvist PÅ, Veryazov V, Widmark PO (2005a) Main group atoms and dimers studied with a new relativistic ANO basis set. J Phys Chem A 108:2851–2858CrossRefGoogle Scholar
  66. Roos BO, Lindh R, Malmqvist PÅ, Veryazov V, Widmark PO (2005b) New relativistic ANO basis sets for actinide atoms. Chem Phys Lett 409:295–299CrossRefGoogle Scholar
  67. Roos BO, Lindh R, Malmqvist PÅ, Veryazov V, Widmark PO (2008) New relativistic atomic natural orbital basis sets for lanthanide atoms with applications to the Ce diatomic abd LuF3. Phys Chem A 112:11431–11435CrossRefGoogle Scholar
  68. Roos BO, Veryazov V, Widmark PO (2003) Relativistic atomic natural orbital type basis sets for the alkaline and alkaline-earth atoms applied to the ground-state potentials for the corresponding dimers. Theor Chem Acc 111:345–351CrossRefGoogle Scholar
  69. Roos O, Lindh R, Malmqvist PÅ, Veryazov V, Widmark PO (2005c) New relativistic ANO basis sets for transition metal atoms. J Phys Chem A 109:6575–6579CrossRefGoogle Scholar
  70. Rumer G (1932) Zum Theorie der Spinvalenz. Nachrichten der Akademie der Wissenschaften in Göttingen. Mathematisch-Physikalische Klasse, Göttingen, pp 337–341Google Scholar
  71. Saloman EB, Sansonetti CJ (2004) Wavelengths, energy level classifications, and energy levels for the spectrum of neutral neon. J Phys Chem Ref Data 33:1113–1158CrossRefGoogle Scholar
  72. Schäfer A, Horn H, Ahlrichs R (1992) Fully optimized contracted Gaussian basis sets for atoms Li to Kr. J Chem Phys 97:2571–2577CrossRefGoogle Scholar
  73. Schäfer A, Huber C, Ahlrichs R (1994) Fully optimized contracted Gaussian basis sets of triple zeta valence quality for atoms Li to Kr. J Chem Phys 100:5829–5835CrossRefGoogle Scholar
  74. Schmidt MW, Baldridge KK, Boatz JA, Elbert ST, Gordon MS, Jensen JH, Koseki S, Matsunaga N, Nguyen KA, Su S, Windus TL, Dupuis M, Montgomery JA Jr (1993) General atomic and molecular electronic structure system. J Comput Chem 14:1347–1363CrossRefGoogle Scholar
  75. Schwarz K, Blaha P (2003) Solid state calculations using WIEN2k. Comput Mat Sci 28:259–273CrossRefGoogle Scholar
  76. Serber R (1934) Extension of the Dirac vector model to include several configurations. Phys Rev 45:461–467CrossRefGoogle Scholar
  77. Shaik S, Hiberty PC (2008) A chemist’s guide to valence bond theory. Wiley, Hoboken, NJGoogle Scholar
  78. Slater J (1929) The theory of complex spectra. Phys Rev 34:1293–1322CrossRefGoogle Scholar
  79. Slater J (1931) Directed valence in polyatomic molecules. Phys Rev 37:481–489CrossRefGoogle Scholar
  80. Slater J (1932) Analytic atomic wave functions. Phys Rev 42:33–43CrossRefGoogle Scholar
  81. Slater J, Verma HC (1929) The theory of complex spectra. Phys Rev 34:1293–1295CrossRefGoogle Scholar
  82. Song L, Chen Z, Ying F, Song J, Chen X, Su P, Mo Y, Zhang Q, Wu W (2012) XMVB 2.0: an ab initio non-orthogonal valence bond program. Xiamen University, XiamenGoogle Scholar
  83. Song L, Mo Y, Zhang Q, Wu W (2005) XMVB: a program for ab initio nonorthogonal valence bond computations. J Comput Chem 26:514–521CrossRefGoogle Scholar
  84. Stevens WJ, Basch H, Krauss M (1984) Compact effective potentials and efficient shared-exponent basis sets for the first- and second-row atoms. J Chem Phys 81:6026–6033CrossRefGoogle Scholar
  85. Stevens WJ, Krauss M, Basch H, Jasien PG (1992) Relativistic compact effective potentials and efficient, shared-exponent basis-sets for the 3rd-row, 4th-row, and 5th-row atoms. Can J Chem 70:612–630CrossRefGoogle Scholar
  86. te Velde G, Bickelhaupt FM, van Gisbergen SJA, Fonseca Guerra C, Baerends EJ, Snijders JG, Ziegler TJ (2001) Chemistry with ADF. Comput Chem 22:931–967CrossRefGoogle Scholar
  87. Tondello E, De Michelis G, Oleari L, Di Sipio L (1967) Slater–Condon parameters for atoms and ions of the first transition period. Coord Chem Rev 2:53–63CrossRefGoogle Scholar
  88. Vanderbilt D (1990) Soft self-consistent pseudopotentials in a generalized eigenvalue formalism. Phys Rev B 41:7892–7895CrossRefGoogle Scholar
  89. Weber V, Daul C, Baltensperger R (2004) Radial numerical integrations based on the sinc function. Comput Phys Commun 163:133–142CrossRefGoogle Scholar
  90. Weigend F, Ahlrichs R (2005) Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: design and assessment of accuracy. Phys Chem Chem Phys 7:3297–3305CrossRefGoogle Scholar
  91. Weigend F, Furche F, Ahlrichs R (2003) Gaussian basis sets of quadruple zeta valence quality for atoms H–Kr. J Chem Phys 119:12753–12762CrossRefGoogle Scholar
  92. Widmark PO, Malmqvist PÅ, Roos BO (1990) Density-matrix averaged atomic natural orbital (ANO) basis-sets for correlated molecular wave-functions 1: 1st row atoms. Theor Chim Acta 77:291–306CrossRefGoogle Scholar
  93. Wolfram Research Inc. (2014) Mathematica. Champaign, IllinoisGoogle Scholar
  94. Wolfram S (2003) The mathematica book, 5th edn. Wolfram-Media, Champaign, IllinoisGoogle Scholar
  95. Woon DE, Dunning TH Jr (1993) Gaussian-basis sets for use in correlated molecular calculations. 3. The atoms aluminum through argon. J Chem Phys 98:1358–1371CrossRefGoogle Scholar
  96. Žukauskas A, Vaicekauskas R, Ivanauskas F, Vaitkevicius H, Shur MS (2008) Spectral optimization of phosphor-conversion light-emitting diodes for ultimate color rendering. Appl Phys Lett 93:051115CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Fanica Cimpoesu
    • 1
  • Mihai V. Putz
    • 2
  • Marilena Ferbinteanu
    • 3
  1. 1.Institute of Physical Chemistry “Ilie Murgulescu”BucharestRomania
  2. 2.West University of Timisoara & National Institute of Research and Development for Electrochemistry and Condensed Matter Timisoara (INCEMC)TimisoaraRomania
  3. 3.Department of Inorganic ChemistryUniversity of BucharestBucharestRomania

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