Because the PAW formalism provides a robust way to reconstruct the all-electron wavefunctions in the valence space, it is suitable for computing expectation values of observables localized even very close to the nuclei. Obtaining equivalent accuracy within the norm-conserving pseudopotential framework would require very small atomic radii for the pseudization procedure, and concomitantly high planewave cutoff energies and lengthy calculations. There remains the question of whether even all-electron accuracy in the valence space is sufficient for accurate representation of observables close to the nuclei, where conventional wisdom would suggest that deep core polarizations might be quite significant for properties such as the electric field gradient or Fermi contact interaction. Such concerns turn out to be unwarranted, however, as our experience and others have shown that the PAW formalism together with a typical chemical valence/core separation are sufficient for accurate description of nuclear point properties such as the electric field gradient [Petrilli1998],[Profeta2003],[Zwanziger2008], Fermi contact interaction [Zwanziger2009] and magnetic chemical shielding [Pickard2001].
Both the electric field gradient and Fermi contact interaction are ground-state observables, and their computation adds negligible time to a normal self-consistent ground state calculation. The total charge density in the PAW formalism contains the pseudovalence density, the nuclear ionic charges, and the all-electron and pseudo charge densities within the PAW spheres. As done in earlier work, the electric field gradient due to the pseudovalence density is computed in reciprocal space, and the gradient due to the (fixed) ionic charges is computed with an Ewald sum approach. The PAW sphere charge densities contribute matrix elements of (3xα xβ -r2δαβ)/r5, weighted by the charge densities in each channel determined by the self-consistent minimization procedure. This treatment [Zwanziger2008] is more flexible than for example assuming all bands are doubly occupied, and permits the current approach to be used with more complex electronic and magnetic states than just insulators.
Within ABINIT, the electric field gradient computation is invoked with the key word prtefg (for Print EFG), together with the key word quadmom. The prtefg key word takes the values 1--3. For value 1, the electric field gradient coupling in MHz is reported, where the conversion is made by atom by combining the gradient with the nuclear quadrupole moments supplied by quadmom. When prtefg is input as 2, the additional breakdown of the field gradient in terms of valence and on-site PAW terms is reported, along with the eigenvectors of the EFG matrix, so that principal directions may be determined with respect to the crystal axes. Finally, prtefg=3 allows additional computation of a point-charge model of the gradient, for comparison purposes. The point charges by atom are supplied through the additional variable ptcharge. Detailed examples of the use of ABINIT to compute EFG's can be found in [Zwanziger2008],[Zwanziger2009a].
The Fermi contact interaction, which is just the electron density evaluated precisely at the nuclear location, is an important component of the isomer shift measured in Mössbauer spectroscopy. The isomer shift is directly proportional to nabs(R)-nsrc(R), the difference in electron density at the absorber (sample) and the source. Evaluating the density at a nuclear position can be accomplished by treating δ(r-R) as the observable, that is, the three-dimensional Dirac delta function centered on the nuclear position R. Because of the short-range nature of the delta function, in the PAW-transformed version of the observable only matrix elements of the on-site all-electron valence functions are required [Zwanziger2009], and these are evaluated from a linear fit to the first few points of the PAW radial functions.
Within ABINIT the Fermi contact interaction is invoked with the key word prtfc (for Print Fermi Contact), which only takes the value 1. When called, the electron density at each nuclear position is reported, in atomic units (electrons per cubic Bohr). The isomer shift as measured in Mössbauer spectroscopy is typically reported in velocity units and is obtained from the formula
δ = (2πcZe2/3Eγ) [nabs(R)-nsrc(R)] Δ⟨r2⟩
where c is the speed of light, Eγ the γ-ray energy, Z the atomic number,
e the electron charge, and Δ⟨r2⟩ the change in the mean square nuclear
radius for the transition. The electronic densities nabs and nsrc refer
to the absorber and source respectively. Because of the linearity of this formula in the density
at the absorber (sample) nucleus, the only unknown (Δ⟨r2⟩) can be obtained
by comparing the calculated values in several standards to experiment and then the computations
can be used to interpret the measurements of new materials. In [Zwanziger2009]
it is showed how to perform such studies on a variety of compounds.
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Basic input variables:
... prtefg [PRint Electric Field Gradient]
... prtfc [PRinT Fermi Contact term]
... ptcharge [PoinT CHARGEs]
... quadmom [QUADrupole MOMents]
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tests/v5/Input: t31.in t32.in t33.in t34.in t35.in t36.in
tests/v6/Input: t44.in
t49.in
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