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  • This story begins mostly with one person, who has ties to MIT.
  • Mulliken was born in Newburyport, MA. His father was a professor of organic chemistry at MIT. Mulliken is perhaps most famous for his population analysis scheme, which is perhaps the most widely used method of deriving charges from quantum chemical calculations today.
  • There is is a long and fascinating history to the problem of defining atomic charges, some of which predates quantum mechanics. The possibility that atoms in molecules, in contrast with isolated atoms/ions, may have fractional charges can be traced back to Lewis. The first quantitative use of partial charges that I am aware of is in Bernal and Fowler’s 1933 paper describing their water model.
  • It is precisely in bonding configurations that the spatial distribution of electrons, and hence the values of atomic charges, are the most interesting; these are also precisely the cases where the definition of atomic charges becomes the most ambiguous and problematic.
  • In a sense, the easiest way to define atomic charges is by their effect on the properties of atoms and molecules. As a quantitative measure of charge polarization (imbalance), the atomic charges are vitally related to many chemical concepts, most notably that of electronegativity. Shown here, for fun, is a picture from Abegg’s 1904 paper showing the periodic trends in electronegativity.
  • It seems impossible to discuss electronegativity in any detail without paying homage to Linus Pauling. Pauling’s scale was initially proposed as an empirical quantity designed to explain why the observation of bond additivity broke down, i.e. why E(H-H) + E(F-F) =/= 2 E(H-F). Although Pauling was the first chemist to popularize the use of valence bond theory, and hence quantum mechanics, it is more instructive to study the contributions of Robert Mulliken to the theory of electronegativity and charge analysis.
  • In a follow-up paper (JCP 3 (1935), 573), Mulliken discusses in detail the errors introduced by the LCAO approximation in assuming that the orbitals of atoms in molecules are the same as the orbitals, and relates the difference in orbitals to charge screening. Mulliken also develops the theory for calculating electronegativities from spectroscopic data and quantum mechanical grounds, with particular emphasis that the states (electronic configurations) that ought to go into the scale are not isolated atom states, but those of atoms in molecules.
  • Mulliken’s early work on defining an electronegativity scale played a pivotal role in his later development of his famous population analysis scheme. In this 1935 paper, the fundamental idea behind Mulliken population analysis had already been described.
  • The general case appears to have been independently rediscovered by Chirgwin and Coulson in 1950, and McWeeny in 1951.
  • In practice, this problem does not crop up often, which could be because most population analyses are performed on the ground state. Sometimes the population of individual orbitals can become negative, but this is usually explained away as an artifact of the analysis scheme that does not affect the interpretation of the gross atomic charge.
  • The new density will not in general transform this simply. Nevertheless we can find the nearest such density that will behave like this. We expect that in the complete basis set limit, the two densities will coincide almost everywhere (except possibly at points of degeneracies, e.g. conical intersections).
  • Here the matrices in blackboard bold are the block supermatrices and dagger is transposition.
  • To understand this formula, first note that 2 S Z S is what would be obtained from the derivative d/dZ (up to a factor of 2). The application of the projection matrices extract the upper right and lower left quadrants, those being the only quadrants which contain A. The overall minus sign comes from the chain rule.

Transcript

  • 1. Quantum mechanical definitions of atomic charge Do we really need another one? Do we really need another one? Do we really need another one?
    • Jiahao Chen
    • Group meeting
    • 2011.03.28
  • 2. Thanks to:
  • 3. BS, MIT, 1917 PhD, Chicago, 1923 Nobel Laureate, Chemistry, 1966
  • 4. BS, MIT, 1917 PhD, Chicago, 1923 Nobel Laureate, Chemistry, 1966 Robert S. Mulliken
  • 5. What is an atomic charge?
    • Easy to define for isolated atoms or ions.
    • For atoms in molecules, difficult to quantify rigorously.
      • The charges of atoms in molecules may be fractional, which reflects how electrons are redistributed when a bond forms.
        • The pair of electrons which constitutes the bond may lie between two atomic centers in such a position that there is no electric polarization, or it may be shifted toward one or the other atom in order to give to that atom a negative, and consequently to the other atom a positive charge. But we can no longer speak of any atom as having an integral number of units of charge, except in the case where one atom takes exclusive possession of the bonding pair, and forms an ion.
            • - Lewis, 1923
    Lewis, G. N. J. Am. Chem. Soc. 54 (1932), p. 83; quoted in Jensen, W. B. J. Chem. Educ. 86 (2009), 545.
  • 6. What is an atomic charge?
    • Easy to define for isolated atoms or ions.
    • For atoms in molecules, difficult to quantify rigorously.
      • The charges of atoms in molecules may be fractional, which reflects how electrons are redistributed when a bond forms.
      • Electrons are indistinguishable; this poses a problem in apportioning the charge density (or density matrix).
      • What observable operator (if any) is an atomic charge?
    A A B B ?
  • 7. Charge polarization and chemistry
    • Charge polarization arises from electronegativity differences Avogadro, 1809; Berzelius, 1819; Abegg, 1899; Lewis, 1916; Pauling, 1932;...
    • Related to:
      • chemical reactivity (Davy, 1806)
      • oxidation state (Avogadro, 1809)
      • heats of reaction (Berzelius, 1819)
      • ionization potentials (Stark, 1913)
      • electrophilicity (Lewis, 1916)
      • metal workfunctions (Gordy and Orville Thomas, 1955)
      • electrode potentials (Kaputinskii, 1960)
    “ Gradation of electroaffinity in the Periodic Table” Abegg, R. Z. Anorg. Chem. 39 (1904), 330. Historical review: Jensen, W. B. J. Chem. Ed. 37 (1996), 10; 80 (2003), 279.
  • 8. 1932: Pauling’s electronegativity Pauling, L. J. Am. Chem. Soc. 54 (1932), 3570. Pauling developed his concept of electronegativity as an empirical additive correction to reaction enthalpies.
  • 9. 1934: Mulliken’s electronegativity Mulliken. R. S. J. Chem. Phys. 2 (1934), 782. The first serious attempt to justify an electronegativity scale using quantum mechanical arguments. *in energies + * +
  • 10. 1935: Mulliken’s charges for diatomics Mulliken. R. S. J. Chem. Phys. 3 (1935), 573.
  • 11. 1935: Mulliken’s charges for diatomics i.e. in modern terms , (McWeeny, 1951) density matrix overlap matrix Mulliken, R. S. J. Chim. Phys. 46 (1949) 675. canonical reference: Mulliken, R. S. J. Chem. Phys. 23 (1955) 1833; 1841; 2338; 2343. The general (polyatomic) case was worked out by Mulliken in 1949. McWeeny, R. J. Chem. Phys. 19 (1951) 1614.
  • 12. A problem with Mulliken charges
    • Consider the wavefunction of minimal basis HeH 2+
    • Then
    localized more bonding more antibonding Attributed to K. Ruedenberg in Mulliken, R.S; Ermler, W. C. Diatomic molecules: results of ab initio calculations . Academic Press, NY, 1977, pp. 33-38.
  • 13. A problem with Mulliken charges
    • Consider the wavefunction of minimal basis HeH 2+
    • Then
    localized more bonding more antibonding Attributed to K. Ruedenberg in Mulliken, R.S; Ermler, W. C. Diatomic molecules: results of ab initio calculations . Academic Press, NY, 1977, pp. 33-38. physically unreasonable region
  • 14. Other charge definitions
    • Other population analysis schemes: Coulson, Löwdin, natural bond order (NBO/NPA),...
    • Density fitting: distributed multipole analysis (DMA),...
    • Density partitioning: Hirshfeld, Bader, Voronoi-deformed...
    • Electrostatic potential (ESP) fitting: CHELP, RESP,...
    • Experimentally derived charges: Szegeti, ESCA, Born,...
    • Empirical charge models: Gasteiger-Marsili, QEq, fluc -q, ...
    Some useful reviews: Bachrach, S. M. Rev. Comp. Chem. 5 (1994), 171 Meister, J.; Schwarz, W. H. E. J. Phys. Chem. 98 (1994), 8245 Francl, M. M.; Chirlian, L. E. Rev. Comp. Chem. 14 (2000), 1 - ESP charges Rick. S.W.; Stuart, S. J. Rev. Comp. Chem., 18 (2002), 89 - empirical models
  • 15. Can we do better?
    • Want a definition that has a clear physical basis
    • Other desiderata:
      • Should “give stable and rational results”
      • Does not require additional fragment calculation
      • Works well with constrained DFT
      • Based purely on the density, so that the charges are true density functionals
  • 16. A A B B Rigid response to perturbation A A B B A’ A’
  • 17. A density matrix in general, has the derivative Introducing the projection matrix for the basis functions on atom (fragment) A, can rewrite the derivative in block matrix form We want to find the nearest density matrix that has the form
  • 18. We can find such a density that minimizes where
  • 19. The density matrix A that minimizes satisfies where are projection matrices Recover the population from the fragment density and finally calculate the charge as which is clearly minimized by This yields Mulliken populations If we neglect S x and S xx , then this reduces to Mulliken charges!
  • 20. STO-3G [HeH] + , R=0.7882, B3LYP essentially converged essentially converged convergence unclear 6-311G ++ ** ++** 6-31G ** ++** ++ 3-21G cc-pVDZ aug- d-aug- aug- d-aug- aug- d-aug- cc-pVTZ cc-pVQZ
  • 21. HeH + (Pople bases) Basis set size Sticky Mulliken Löwdin STO-3G 1 0.41494 0.03273 0.28708 0.04627 0.39419 3-21G 2 0.83336 0.05509 0.29233 0.08790 0.42185 6-31G 2 0.83729 0.05481 0.30041 0.08929 0.40847 6-31++G 3 0.71980 0.05360 0.31986 0.08970 0.39491 6-31G** 5 0.88949 0.04083 0.32363 0.09502 0.49733 6-31++G** 6 0.78046 0.03941 0.33764 0.09504 0.48215 6-311G 3 0.76358 0.05113 0.31566 0.08699 0.41799 6-311++G 4 0.68874 0.05034 0.31954 0.08706 0.40355 6-311++G** 6 0.77650 0.03425 0.32544 0.08370 0.54197 6-311++G** 7 0.68375 0.03333 0.32976 0.08283 0.52623
  • 22. HeH + (Dunning bases) Basis set size Sticky Mulliken Löwdin cc-pVDZ 5 1.11660 0.04885 0.39045 0.11513 0.56447 aug-cc-pVDZ 9 0.95481 0.04759 0.44084 0.13322 0.58819 d-aug-cc-pVDZ 13 0.84586 0.04346 0.41126 0.12381 0.58564 cc-pVTZ 14 0.97298 0.03321 0.37769 0.09676 0.65809 aug-cc-pVTZ 23 0.88933 0.02533 0.34723 0.08810 0.69507 d-aug-cc-pVTZ 32 0.89358 0.02569 0.32296 0.09286 0.69504 cc-pVQZ 30 0.85875 0.02890 0.37490 0.09393 0.72658 aug-cc-pVQZ 46 0.92001 0.02128 0.36010 0.08541 0.75868 d-aug-cc-pVQZ 62 0.92469 0.01984 0.35451 0.08967 0.75877
  • 23.  
  • 24.  
  • 25.  
  • 26. Reviewing our design criteria
    • Want a definition that has a clear physical basis
    • Other desiderata:
      • Should “give stable and rational results”
      • Does not require additional fragment calculation
      • Works well with constrained DFT
      • Based purely on the density, so that the charges are true density functionals
    • Disadvantages
      • More costly: need density and overlap derivatives
      • Don’t know analytic formula, must solve numerically
    ■ ☺ ? Possible to write down such a variant; haven’t done it ■ ☺