# Electric fields gradients

### This page gives hints on how to calculate electric fields gradients with the ABINIT package.

##### Copyright (C) 2016-2017 ABINIT group (JZ)

Mentioned in help_features#2.5.

### Table of content:

- 1. Introduction.
- 2. Related lesson(s) of the tutorial.
- 3. Related input variables.
- 4. Selected input files.
- 5. References.

### 1. Introduction.

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.
Go to the top

### 2. Related lesson(s) of the tutorial.

- The lesson on the properties of the nuclei shows how to compute the electric field gradient. Prerequisite : PAW1.
Go to the top

### 3. Related input variables.

Basic input variables:

... prtefg [PRint Electric Field Gradient]
... prtfc [PRinT Fermi Contact term]
... ptcharge [PoinT CHARGEs]
... quadmom [QUADrupole MOMents]
Go to the top

### 4. Selected input files.

WARNING : as of ABINITv8.6.x, the list of input files provided in the specific section of the topics Web pages is still to be reviewed/tuned. In some cases, it will be adequate, and in other cases, it might be incomplete, or perhaps even useless.
The user can find some related example input files in the ABINIT package in the directory /tests, or on the Web:

tests/v5/Input: t31.in t32.in t33.in t34.in t35.in t36.in

tests/v6/Input: t44.in t49.in
Go to the top

### 5. References.

[Petrilli1998] Helena M. Petrilli, Peter E. Blöchl, Peter Blaha and Karlheinz Schwarz, "Electric-field-gradient calculations using the projector augmented wave method", Phys. Rev. B 57, 14690–14697 (1998).
DOI: 10.1103/PhysRevB.57.14690.
[Pickard2001] Chris J. Pickard and Francesco Mauri, "All-electron magnetic response with pseudopotentials: NMR chemical shifts", Phys. Rev. B 63, 245101 (2001).
DOI: 10.1103/PhysRevB.63.245101.
[Profeta2003] M. Profeta, F. Mauri and C. J. Pickard, "Accurate First Principles Prediction of 17 O NMR Parameters in SiO2: Assignment of the Zeolite Ferrierite Spectrum", J. Am. Chem. Soc. 125, 541 (2003).
DOI: 10.1021/ja027124r.
[Zwanziger2008] J. W. Zwanziger and M. Torrent, "First-Principles Calculation of Electric Field Gradients in Metals, Semiconductors, and Insulators", Applied Magnetic Resonance 33, 447–456 (2008).
DOI: 10.1007/s00723-008-0080-1.
[Zwanziger2009] J. W. Zwanziger, "Computation of Mössbauer isomer shifts from first principles", Journal of Physics: Condensed Matter 21, 195501 (2009).
DOI: 10.1088/0953-8984/21/19/195501.
[Zwanziger2009a] J. W. Zwanziger, "First-principles study of the nuclear quadrupole resonance parameters and orbital ordering in LaTiO3", Phys. Rev. B 79, 033112 (2009).
DOI: 10.1103/PhysRevB.79.033112.

Go to the top