# Full Potential Overview

### Table of Contents

- Overview of the full-potential method
*Smoothed Hankel functions**Local Orbitals**Augmented Plane Waves**Augmentation and Representation of the charge density**Connection to the ASA packages**Primary executables in the FP suite*- References

### Overview of the full-potential method

The full-potential program *lmf* is an augmented-wave electronic structure package. It solves the Schrodinger equation in solids by partitioning space into spheres centered at atoms, where partial waves can be efficiently evaluated numerically, and an interstitial region, where the wave functions are represented by smooth, analytic envelope functions (smooth Hankel functions). It is a descendent of an electronic structure code *nfp* written by M. Methfessel and M. van Schilfgaarde in the 1990’s. The original method was described in some detail in Ref. 1. It has been greatly expanded and the method is documented as part of the Questaal suite in Ref. 2.

*lmf* has many of the functionalities found in popular DFT packages, and it has some unique ones as well, in particular related to many-body perturbation theory. See this page for a small survey.

*Smoothed Hankel functions*

The envelope functions are smoothed generalizations of Hankel functions that are found in **LMTO** programs. Unlike the normal Hankel functions, the smoothed versions — convolutions of ordinary Hankel functions and Gaussian functions — are regular at the origin. Their smoothness is controlled by an extra degree of freedom, the gaussian width or “smoothing” radius . This page defines them and outlines some of their mathematical properties. The most complete description can be found in Ref. 3; see also Sec. 3.1 of Ref. 2.

Smooth Hankels are a significantly better choice of envelope functions than the customary **LMTO** basis set constructed of normal Hankels. However, smoothing introduces complications because the augmentation of a smoothed Hankel function is less straightforward than it is for an ordinary Hankel. The envelope functions, while an improvement over the traditional **LMTO** basis are not yet optimal. They are not screened into a tight-binding representation, as in the second-generation and later generation **LMTO** methods; thus wave functions are evaluated by Ewald summation. A new basis, “Jigsaw Puzzle Orbitals,” makes use of screening and some other tricks to construct a short ranged, minimal basis of envelope functions. They are highly accurate because they are tailored to the potential, and accomplish a level of precision in the interstitial approaching that of the augmented parts. Thus the basis set should be close to complete in the energy window where linearization is valid.

*Local Orbitals*

This package extends the linear method through the use of local orbitals. Augmented wave methods substitute radial solutions of the Schrödinger equation with combinations of partial wave of angular quantum number *l* inside the augmentation region. Linear methods used a fixed radial function (more precisely, a pair of functions), which has validity over only a certain energy window. With local orbitals, a third radial function is added to the basis, which greatly extends the energy window over which energy eigenvalues can be calculated. It is necessary, for example, to obtain the proper **LDA** band gap in GaAs, both the Ga *3d* and *4d* partial waves are important. Local Orbitals are explained in Sec. 3.7.3. of Ref. 2. To see how to include them in the basis, see this tutorial.

*Augmented Plane Waves*

In 2010 Takao Kotani added augmented plane waves (**APWs**) as additional envelope functions, which can increase the flexibility of the basis. The combination of smooth Hankel functions and APW’s is described in Ref. 4. One can view PMT’s as an extension of a conventional **LAPW** method, enabling through the use of a few MTOs with much faster convergence in APW energy cutoff. Alternatively, PMT can be viewed as an extension of the original MTO method. A principal advantage of the conventional APW basis is that it is easier to make it complete. Here addition of APW’s offer a systematic way of converging the combined MTO + APW basis in a systematic and reliable manner, to an almost arbitrarily high accuracy. This is particularly important when reliable eigenvalues far above the Fermi level are needed, and to check the accuracy of a given MTO basis. To include APW’s in the basis, see here for a tutorial.

*Augmentation and Representation of the charge density*

The charge density representation is unique to this method. It consists of three parts: a smooth density carried on a uniform mesh, defined everywhere in space ( is *not* augmented; inside the augmentation spheres it is present as a “pseudodensity”); the true density expressed in terms of spherical harmonics inside each augmentation sphere; and finally a one-center expansion of the smooth density in , inside each augmentation sphere. The total density is expressed as a sum of the three independent densities: .

This turns out to be an extremely useful way to carry out the augmentation procedure. and approximately cancel in the augmentation spheres; but it is the residual that greatly accelerates he convergence with and makes the method much more efficient than standard augmented-wave methods. The analysis is a little subtle; see Sec. 3.6 of Ref. 2.

*Connection to the ASA packages*

The full-potential program builds on the ASA suite which contains an implementation of a tight-binding **LMTO** program in the Atomic Spheres Approximation (**ASA**), and shares most things in common with it, including a number of auxiliary programs useful to both **ASA** and **FP**. For example, both methods are linear augmented-wave methods, and the wave functions inside the augmentation spheres are equivalent in the two cases. You may find that the ASA overview is helpful even if you will not be using the ASA package. Most input is common to both methods, but there are some differences, e.g. the selection of sphere radii. The **FP** code requires some additional information, but most of it can be generated automatically, as explained in the introductory tutorial or in more detail in this tutorial for PbTe. It is interesting to compare that tutorial with an ASA tutorial on the same material.

A description of the input system and tags needed for each method are found in the input file guide.

One important difference between the **ASA** and **FP** methods is that the **FP** method has no simple parametrization of total density in terms of the **ASA** energy moments , , , or the representation of the potential by a few potential parameters, as in the **ASA** (see **ASA** overview). However, the basis within the augmentation spheres is defined from the spherical average of the potential, just as in the **ASA**, the linearization proceeds in the same way. Both use the continuously variable principal quantum numbers *P* to establish a mapping between the linearization energy and logarithmic derivative at the augmentation sphere boundary, and to float the linearization energy to the center-of-gravity of the occupied states.

A second important difference is that the basis set is more complicated, and in its current form, the user must choose parameters defining the basis. This complication is the most onerous part of the present method (a new “Jigsaw Puzzle Orbital basis” will automatically tailor the basis set shape to the given potential) but at present the basis set is determined by hand, or only semi-automatically.

*Primary executables in the FP suite*

*lmfa*makes each species self-consistent for the free atom and writes the atomic density, plus a fit to the tail beyond the the augmentation sphere radius (fit as a linear combination of smooth Hankel functions) to an atom file. This is used to overlap atomic densities to make a trial density for the crystal.*lmfa*serves other important purposes, e.g. as an automatic generator of parameters for the basis set, as explained in this tutorial.*lmf*is the program used for self-consistent full-potential calculations. It requires a starting density, which it obtains either from a restart file (typically generated by a prior invocation of*lmf*) or by a superposition of free-atom densities generated by*lmfa*.*lmgw*is a script that performs the main computational steps in acalculation, linking the*GW**GW*and one-body parts*lmgwsc*is a higher level script that performs self-consistent QS*GW*calculations*lmfgwd*supplies an interface to Questaal’s*GW*code, supplying one-body input to it. It is not usually run on its own, but is invoked inside*lmgw*or*lmgwsc*.*lmfdmft*supplies an interface to**DMFT**solvers.*lmfgws*is a post-processing utility that generates useful information from a dynamical self-energy generated either by*GW*or by DMFT.

### References

[1] M. Methfessel, Mark van Schilfgaarde, and R. A. Casali, “A full-potential **LMTO** method based on smooth Hankel functions,” in Electronic Structure and Physical Properties of Solids: The Uses of the **LMTO** Method, *Lecture Notes in Physics 535. H. Dreysse, ed. (Springer-Verlag, Berlin)* 2000.

[2] Dimitar Pashov, Swagata Acharya, Walter R. L. Lambrecht, Jerome Jackson, Kirill D. Belashchenko, Athanasios Chantis, Francois Jamet, Mark van Schilfgaarde, *Questaal: a package of electronic structure methods based on the linear muffin-tin orbital technique*, Comp. Phys. Comm. **249**, 107065 (2020).

[3] E. Bott, M. Methfessel, W. Krabs, and P. C. Schmidt, *Nonsingular Hankel functions as a new basis for electronic structure calculations*, Journal of Mathematical Physics 39, 3393 (1998).

[4] T. Kotani and M. van Schilfgaarde, *A fusion of the LAPW and the LMTO methods: the augmented plane wave plus muffin-tin orbital (PMT) method*, Phys. Rev. B81, 125117 (2010).

See also T. Kotani, H. Kino, H. Akai, *Formulation of the Augmented Plane-Wave and Muffin-Tin Orbital Method*, J. Phys. Soc. Jpn. 84, 034702 (2015), and also Sec. 3.10 of Ref. [2].