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Introduction to DFT (and QE):�.�A review of the approximations used in the plane-wave pseudopotential DFT

Abdul Muhaymin

Graduate student, MSN, Bilkent University

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Outline

  • Many body problem
  • Born-Oppenheimer approximation
  • Pseudopotential
  • HK theorem
  • Hartree and Hartree-Fock approaches
  • KS equation
  • XC functionals
  • QE

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Many-body problem

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  • Not solvable analytically – empirical impossibility
  • Not solvable numerically – technological impossibility
  • So approximations are must

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BOA (M >> me)

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  • Still neither solvable analytically nor numerically
  • The culprit is the e-e interaction term

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Frozen core

  • Core electrons+nucleus=frozen
  • Considering only valence electrons
  • Core electrons have little to no contribution in chemical bonding
  • Computationally beneficial

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  • Reduction of computation time by 1/3 in this example
  • Even better for heavier atoms
  • Hf (72)

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Pseudopotential

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  • Many wiggles of the valence electrons and sharp peak of the core electrons near nucleus are replaced by smooth curve
  • So no need for high number of Fourier components
  • Can mitigate relativistic effect
  • All electron codes do not pseudize (so much more accurate)
  • Various types:
    • Norm-conserving
    • Projector Augmented Wave
    • Ultrasoft

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HK Theorems

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  • Different external potential generate different electron density
  • Two different Hamiltonian differ only by the external potential
  • So, electron density uniquely determines external potential, Hamiltonian, wave functions, and all other ground state properties
  • Search for the ground state energy functional:
    • Variational Principle
    • The electron density which minimizes the system energy is the ground state electron density and vice-versa

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HK Theorems - Summary

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Hartree and Hartree-Fock approaches

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  • Hartree
    • Interacting system to non interacting system
    • Each electrons are individual and only interact with a mean-field (average density)
    • No antisymmetry principle, no exchange energy, no correlation energy, no exclusion principle

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  • Hartree-Fock
    • Incorporate antisymmetry using Slater’s determinant
    • But still cannot incorporate correlation energy

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KS ansatz

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  • Moving from a interacting many-body problem to a non-interacting one body problem (auxiliary system)
  • The only term to be approximated is the exchange-correlation energy term
  • Less than 10% of the total energy for most systems, sometimes even 0.3% (He)
  • KS orbitals vs real orbitals

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KS equation

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  • We will not derive this equation but if you are interested, see, for example, (Hung et al, 2021) or (Lee, 2016)

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XC functional

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  • The better the approximation is, the better the result (accuracy)
  • LDA, GGA, meta-GGA, hyper-GGA, RPA, RPA+ - Jacob’s ladder
  • Starting point is the HEG

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Summary

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Quantum ESPRESSO

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  • A collection of FORTRAN code
  • Centered on PWscf code
  • Lots of modules
  • Almost 30 years of development
  • Free, open-source
  • Responsive community
  • Not as optimized as some other commercial code but pretty fast for everyday purpose
  • Some features are not yet implemented or implemented only for a particular type of pseudopotential, especially in the context of optical or spectroscopic properties