CASTEP > Theory in CASTEP > CASTEP background theory > Pseudopotentials > Norm-conserving pseudopotentials

Norm-conserving pseudopotentials

The main requirement of the pseudopotential approach is that it reproduces the valence charge density associated with chemical bonds. It has been shown (Hamann et al., 1979) that for pseudo and all-electron wavefunctions to be identical beyond the core radius, Rc, it is necessary for the integrals of squared amplitudes of the two functions be the same. This is equivalent to requiring norm-conservation from pseudo wavefunctions, i.e. that each of them should carry exactly one electron. This condition ensures that the scattering properties of the pseudopotential are reproduced correctly.

Figure 1
Figure 1. Schematic representation of the all-electron and pseudized wavefunctions and potentials

The typical method for generating pseudopotentials is as follows. All-electron calculations are carried out for an isolated atom in a chosen electronic configuration (not necessarily in the ground state). This provides valence electron eigenvalues and valence electron wavefunctions for the atom (shown as ψ in Figure 1). A parameterized form for the ionic pseudopotential (or the pseudo wavefunction) is chosen. The parameters are then adjusted, so that a pseudoatom calculation with the same exchange-correlation potential as in the all-electron atom gives pseudo wavefunctions, ψps (Figure 1), that match the valence wavefunctions outside some cutoff radius, Rc, and pseudoeigenvalues that are equal to the valence eigenvalues. This procedure involves direct inversion of the radial Kohn-Sham equation in the case when the pseudo wavefunction and not the pseudopotential itself are parameterized. If each wavefunction, pseudo and all-electron, is normalized to one, then the norm-conservation constraint is automatically satisfied as a result of matching the wavefunctions outside Rc.

The ionic pseudopotentials are constructed with Rc ranging from one to two times the value of the physical core radius. The smaller the value of Rc, the harder and more transferable the potential. The conflicting effect of Rc creates an obvious trade-off between accuracy and efficiency.

Optimized pseudopotentials

A number of recipes exist for producing pseudopotentials that are optimized with respect to the energy cutoff required in solid-state calculations.

Lin et al. (1993) suggested the following generation scheme based on the earlier work of Rappé et al. (1990):

  1. The pseudo wavefunction ψl(r) inside a cutoff radius is expressed as:

    Eq. CASTEP 5

    CASTEP Equation 5

    where jl(qir) are spherical Bessel functions with (i-1) zeroes between r=0 and r=Rc. The cutoff radius value is as large as possible, consistent with satisfactory transferability of the potential.

  2. The coefficients αi are determined from minimizing the kinetic energy beyond the cutoff vector qc:

    Eq. CASTEP 6

    CASTEP Equation 6

    where qc is chosen to be equal to q4 from Eq. CASTEP 5. Three additional constraints that are satisfied using Lagrange multipliers are the normalization of the pseudo wavefunction and the continuity of the first two derivatives of the pseudo wavefunction at Rc.

  3. The standard step of inverting the radial Kohn-Sham equation produces a smooth pseudopotential with optimal convergence properties.

Lee (1996) suggested further enhancements and it is essentially this scheme that was used to generate the bulk of norm-conserving pseudopotentials in the CASTEP database. This generation method eliminates the condition that the second derivative of the pseudo wavefunction must be continuous at a given cutoff radius because it automatically satisfies the second derivative constraint. As a result, the scheme allows one to tune qc for a given Rc to optimize the accuracy and efficiency of a pseudopotential.

USP implementation allows CASTEP calculations to be run with a lower energy cutoff than their NCP counterparts producing a clear advantage in terms of the calculation time. However, USP formalism is more complex and becomes close to intractable for such complex concepts as linear response implementation for phonons or NMR properties, or for nonlocal exchange-correlation functionals. As a result there is a number of tasks and properties that CASTEP can address only with norm-conserving potentials. The database of NCPs available in Materials Studio is rather old and in some cases might contain potentials that have not been tested sufficiently. An alternative to using this database is to generate NCPs using the open source software package Opium, The Optimized Pseudopotential Interface / Unification Module. This package allows you to generate files in .recpot format using the latest pseudopotential generation technology.

Accelrys Materials Studio 8.0 Help: Wednesday, December 17, 2014
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