Generating pseudopotentials for electronic structure calculations1

1
Generating pseudopotentials for electronic
structure calculations1

• By-
• Mohan Varanasi.
• Advisor Dr. Ronald Cosby
• Centre for Computational NanoScience,
• Department of Physics and Astronomy,
• Ball State University, Muncie, IN. 47306
• Contents
• Introduction Basics of pseudopotentials
• Check for norm-conserving condition
  satisfaction of 5 desired properties
• Generation of all-electron wavefunctions
• Generating pseudopotential
• Choosing the matching radii for wavefunctions
• Generation of fully seperable pseudopotentials
  KB-potentials.
• Ghost state analysis.
• Example output
• Conclusions and future work

2
Idea of pseudopotential

• The primary application in electronic structure
  is to replace the strong Coulomb potential of the
  nucleus and the effects of the tightly bound core
  electrons by an effective ionic potential acting
  on the valence electrons.
• A pseudopotential is constructed to replace the
  atomic all-electron potential such that core
  states are eliminated and the valence electrons
  are described by nodeless pseudo wave-functions.
• The advent of ab initio norm-conserving and
  ultrasoft pseudopotentials has led to accurate
  calculations which form basis for much of current
  research and development of new methods in
  electronic structure calculations.
• Most modern pseudopotential calculations are
  based upon ab initio norm-conserving
  pseudopotentials.

3
General pseudopotential theory

• Majority of the pseudopotentials are generated
  from all-electron atomic calculations. This is
  done by assuming a spherical screening
  approximation and self-consistently solving the
  radial Kohn-Sham equation2
• Once the pseudo-wavefunction is obtained, the
  screened potential is found by the inversion of
  the radial Schrödinger equation
• 2 P. Honenberg and W. Kohn, Phys. Rev. 136, B864
  (1964) W. Kohn and L.J. Sham, ibid . 140, A1133
  (1965).

4

• The constructed pseudopotentials must satisfy
  four general conditions
• The pseudo wavefunction contains no radial nodes.
• The normalized atomic radial pseudo-wave-function
  (PP) with angular momentum l is equal to the
  normalized radial all-electron wave-function (AE)
  beyond a chosen cutoff radius rc
• The charge enclosed within rc for the two wave
  functions must be equal
• the pseudo wavefunction and the valence
  all-electron wavefunction correspond to the same
  eigen value
• If the pseudopotential meets the conditions
  outlined above, then it is referred to as a
  norm-conserving pseudopotential
• Choosing the matching radii
• At matching radius rc the AE and PP wavefunction
  of angular momentum l match with at least
  continuous first derivative. The criteria for
  selecting the cutoff radius rc is
• The rc must be larger than the outermost node of
  the wavefunction for any given l
• A typical rc is the outermost peak, beyond if
  needed
• The larger rc , the softer the potential (less PW
  needed), but also less transferable.

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Use of norm-conserving pseudopotentials

• The norm-conserving pseudopotential concept
  allows for efficient and accurate ab initio
  electronic structure calculations of poly atomic
  systems.
• The requirement of norm-conservation is the key
  step in making accurate, transferable
  pseudopotentials, which is essential so that a
  pesudopotential constructed in one environment
  can faithfully describe the valence properties in
  different environments including atoms, ions,
  molecules.
• The key features are
• Only the valence states need to be calculated.
• The valence electrons move in a pseudopotential
  which is much smoother than the true potential
  inside small core regions around the nuclei,
  while reproducing it outside.
• The norm-conserving constraint ensures that
  outside the core the pseudo wavefunction behave
  like their all-electron counterparts over a wide
  range of different chemical situations.
  

7
Construction of Pseudopotentials

• Steps to be followed for the generation of
  pseudopotentials
• Density-functional calculation of the
  all-electron atom in a reference state and a
  chosen approximation for exchange-correlation
• Construction of pseudo valence orbitals and
  pseudopotential components, observing the
  norm-conserving constraints
• Different schemes for pseudopotential generation
  are Hamann scheme, Troullier and Martins scheme,
  Kleinman and Bylander scheme
• Assessment of pseudopotentials transferability
• Transformation to the fully separable
  Kleinman-Bylander form and the exclusion of
  unphysical ghost states in the valence spectrum

8

• Construction of pseudopotentials (Contd)
• The initial step in constructing the
  pseudopotentials is an all-electron calculation
  of the free atom usually in its neutral ground
  state configuration
• Regarding exchange-correlation, there is a choice
  of commonly used parameterizations of the local
  density approximation (LDA) and of the
  generalized gradient approximations (GGA).
• In LDA, the exchange-correlation energy is the
  parameterizations of Ceperely and Alders exact
  results for the uniform electron gas.
  
• In a GGA, the exchange-correlation functional
  depends on the density and its gradient.
• The GGA functionals are given by Perdew and
  Wang(PW91) and by Perdew, Burke, and Ernzerhof
  (PBE)
• Pseudopotentials must be generated with the SAME
  functional that will be later used in
  calculations. The use of, for instance,
  gradient-corrected functionals with local-density
  approximation (LDA) PPs is inconsistent.

9
Transferability considerations

• Test of scattering properties
• The logarithmic derivatives of the radial
  wave-functions agree for the pseudo and the
  all-electron atom, as a function of energy at
  some diagnostic radius rdiag outside the core
  region, typically half of the inter-atomic
  distance
• Test of excitation energies
• Verify that the pseudopotential reproduces the
  all-electron results for the atomic excitation or
  ionization energies. Typically, the difference in
  the excitation energies could be around 1mRy or
  few 10mev.

10
Fully separable pseudopotentials and ghost states

• In transforming a semilocal component to the
  corresponding KB-pseudopotential, make sure that
  the KB-form does not lead to unphysical ghost
  states.
• Ghost states are the states with the wrong number
  of nodes that are absent in the all-electron
  atom, that make the PP completely useless.
• A ghost state is indicated by a marked deviation
  of the logarithmic derivatives of the
  KB-pseudopotential
• Ghost states below the valence states are
  identified by a rigorous criterion by Gonze et al.

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• To eliminate ghost states for some l we may
• Change lloc i.e., use different component of the
  semi local pseudopotential as the local potential
• Adjust the core cutoff radii rc of the offending
  component.

12
Current work on cadmium

• Input file using fhi98PP (Fritz-Haber_Institute)
  package
• 48.00 8 3 8 0.00 z nc nv iexc rnlc
• 1 0 2.00 n l f
• 2 0 2.00
• 2 1 6.00
• 3 0 2.00
• 3 1 6.00
• 3 2 10.00
• 4 0 2.00
• 4 1 6.00
• 4 2 10.00
• 5 0 2.00
• 5 1 0.00
• 2 t lmax s_pp_def

• Input file using SIESTA (SpanishInitiative for
  Electronic simulations using Thousands of Atoms)
• ae Cd Ground state all-electron
• Cd ca
• 0.0
• 8 2
• 5 0 2.00
• 4 2 10.00
• Pseudopotential generation for cadmium
• pg simple generation
• pg cadmium
• tm2 3.0 PS flavor,
  logder R
• nCd ccar Symbol, XC flavor,
  rs
• 0.0 0.0 0.0 0.0
  0.0 0.0
• 8 4 norbs_core,
  norbs_valence
• 5 0 2.00 0.00 5s2
• 5 1 0.00 0.00 5p0
• 4 2 10.00 0.00 4d10
• 4 3 0.00 0.00 4f0

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Matching of wave-functions and logarithmic
derivatives
14
MATCHING OF EXCITATION ENERGIES AND EIGEN
VALUES(error in units of mev)

• index total energy, error relative to
  all-electron calc ----
• pseudo1 0 GS Kr5s24d10 2 0.27
  Kr5s1.754d105p0.253 0.27 Kr5s1.504d105p0.5
  0 4 0 Kr5s1.254d105p0.75 5 0.27
  Kr5s1.004d105p1.00 6 7.62
  kr5s14d9.755p1.25

• - index p eigenvalue, error relative to
  all-electron calc ----
• pseudo
• 1 0
• 2 0.5
• 3 1.1
• 4 1.7
• 5 2.4
• 6 1.7

• index s eigenvalue, error relative to
  all-electron calc ----
• pseudo
• 1 0
• 2 -1.8
• 3 -4.1
• 4 -7
• 5 -10.1
• 6 -42.5

• - index d eigenvalue, error relative to
  all-electron calc ----
• pseudo
• 1 0
• 2 0.3
• 3 0.6
• 4 1
• 5 1.2
• 6 4.9

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• Conclusions
• We had obtained the pseudopotentials which
  satisfy the norm-conserving conditions
• The difference in the errors for excitation
  energies and eigen values is acceptable
• Future work
• Trying to understand the spikes in logarithmic
  derivatives
• Criterion for eliminating the ghost states
• Use the pseudopotentials obtained in
  optimization of the CdSe clusters
• References
• G.B.Bachlet, D.R.Hamann and M.Schluter, Phys.
  Rev. B 26, 4199 (1982)
• X.Gonze, R.Stumpf, and M.Scheffler, Phys. Rev. B
  44, 8503 (1991)
• N.Troullier and J.L.Martins, Phys. Rev. B 43,
  1993 (1991)
• L. Kleinman and D.M. Bylander, Phys. Rev. Lett.
  48, 1425 (1982)
• D.R.Hamann, Phys. Rev. B 40, 2890 (1989)
• M.Fuchs and M.Scheffler, Comput. Phys. Commun.
  119, 67 (1999)