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Density Functional Calculations

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Computational Chemistry

Abstract

Density functional theory is based on the two Hohenberg-Kohn theorems, which state that the ground-state properties of an atom or molecule are determined by its electron density function, and that a trial electron density must give an energy greater than or equal to the true energy (the latter theorem is true only if the exact functional could be used). In the Kohn-Sham approach the energy of a system in formulated as a deviation from the energy of an idealized system with noninteracting electrons. From the energy equation, by minimizing the energy with respect to the Kohn-Sham orbitals the Kohn-Sham equations can be derived, analogously to the Hartree-Fock equations. Finding good functionals is the main problem in DFT. Various levels of DFT and kinds of functionals are discussed. The mutually related concepts of electronic chemical potential, electronegativity, hardness, softness, and the Fukui function are discussed.

My other hope is that…a basically new ab initio treatment capable of giving chemically accurate results a priori, is achieved soon.

M.J.S. Dewar , A Semiempirical Life, 1992.

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Notes

  1. 1.

    Walter Kohn, born in Vienna 1923. B.A., B. Sc., University of Toronto, 1945, 1946. Ph.D. Harvard, 1948. Instructor in physics, Harvard, 1948–1950. Assistant, Associate, full Professor, Carnegie Mellon University, 1950–1960. Professor of physics, University of California at Santa Diego, 1960–1979; University of California at Santa Barbara 1979-present. Nobel Prize in chemistry 1998. Died Santa Barbara, CA, 2016.

  2. 2.

    M06: :”M zero six", or colloquially “M oh six”. A descendant of M05, Minnesota ‘05 (2005): Y. Zhao, N. E. Schultz, D. E. Truhlar, J. Chem. Phys., 2005, 123, 161103.

  3. 3.

    Personal communication from Professor J. P. Perdew, 2009 November 7. As of 2015 February, TPSS evidently still held essentially this position, although a nonempirical meta-GGA close to TPSS for molecules, but more accurate for solids, revTPSS (revised TPSS) had been developed (personal communication from Professor J. P. Perdew, 2015 February 25).

  4. 4.

    Kenichi Fukui, born Nara, Japan, 1918. Ph.D. Kyoto Imperial University 1948, Professor Kyoto Imperial University 1951. Nobel Prize 1981. Died 1998.

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Authors and Affiliations

  1. Trent University, Peterborough, ON, Canada

    Errol G. Lewars

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Appendices

Easier Questions

  1. 1.

    State the arguments for and against regarding DFT as being more a semiempirical than an ab initio-like theory.

  2. 2.

    What is the essential difference between wavefunction theory and DFT? What is it that, in principle anyway, makes DFT simpler than wavefunction theory?

  3. 3.

    Why can’t current DFT calculations be improved in a stepwise, systematic way, as can ab initio calculations?

  4. 4.

    Which of these prescriptions for dealing with a function are functionals: (1) square root of \( f(x).\;(2) \sin f(x).(3){\displaystyle \sum_{x=1}^3f(x)\cdot (4)}{\displaystyle \int f(x)dx}\cdot (5)\; \exp \left(f(x)\right). \)

  5. 5.

    For which class(es) of functions is the nth derivative of f(x) a functional?

  6. 6.

    Explain why a kind of molecular orbital is found in current DFT, although DFT is touted as an alternative to wavefunction theory.

  7. 7.

    What is fundamentally wrong with functionals that are not gradient-corrected?

  8. 8.

    The ionization energy of a molecule can be regarded as the energy required to remove an electron from its HOMO. How then would a pure density functional theory, with no orbitals, be able to calculate ionization energy?

  9. 9.

    Label these statements true or false: (1) For each molecular wavefunction there is an electron density function. (2) Since the electron density function has only x, y, z as its variables, DFT necessarily ignores spin. (3) DFT is good for transition metal compounds because it has been specifically parameterized to handle them. (4) In the limit of a sufficiently big basis set, a DFT calculation represents an exact solution of the Schrödinger equation. (5) The use of very big basis sets is essential with DFT. (6) A major problem in density functional theory is the prescription for going from the molecular electron density function to the energy.

  10. 10.

    Explain in words the meaning of the terms electronegativity, hardness, and the Fukui function.

Harder Questions

  1. 1.

    It is sometimes said that electron density is physically more real than a wavefunction. Do you agree? Is something that is more easily grasped intuitively necessarily more real?

  2. 2.

    A functional is a function of a function. Explore the concept of a function of a functional.

  3. 3.

    Why is it that the Hartree-Fock Slater determinant is an inexact representation of the wavefunction, but the DFT determinant for a system of noninteracting electrons is exact for this particular wavefunction?

  4. 4.

    Why do we expect the “unknown” term in the energy equation (E xc [ρ 0], in Eq. (7.21)) to be small?

  5. 5.

    Merrill et al. have said that “while solutions to the [HF equations] may be viewed as exact solutions to an approximate description, the [KS equations] are approximations to an exact description!” Explain.

  6. 6.

    Electronegativity is the ability of an atom or molecule to attract electrons Why then is it then (from one definition) the average of the ionization energy and the electron affinity (Eq. (7.32)), rather than simply the electron affinity?

  7. 7.

    Given the wavefunction of a molecule, it is possible to calculate the electron density function. Is it possible in principle to go in the other direction? Why or why not?

  8. 8.

    The multielectron wavefunction Ψ is a function of the spatial and spin coordinates of all the electrons. Physicists say that Ψ for any system tells us all that can be known about the system. Do you think the electron density function ρ tells us everything that can be known about a system? Why or why not?

  9. 9.

    If the electron density function concept is mathematically and conceptually simpler than the wavefunction concept, why did DFT come later than wavefunction theory?

  10. 10.

    For a spring or a covalent bond, the concepts of force and force constant can be expressed in terms of first and second derivatives of energy with respect to extension. If we let a “charge space” N replace the real space of extension of the spring or bond, what are the analogous concepts to force and force constant? Using the SI, derive the units of electronegativity and of hardness.

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Lewars, E.G. (2016). Density Functional Calculations. In: Computational Chemistry. Springer, Cham. https://doi.org/10.1007/978-3-319-30916-3_7

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