Full Potential Overview

Table of Contents

• Overview of the full-potential method
• Questaal’s Basis Functions
• Augmented Wave Methods
  • Questaal’s Augmentation
  • Linear Methods in Band Theory
• Smoothed Hankel functions
• Local Orbitals
• Augmented Plane Waves
• Augmentation and Representation of the charge density
• The Atomic Spheres Approximation
  • 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.

Questaal’s Basis Functions

The primary code in the density-functional package is lmf, though there special-purpose forms as explained below and in the Questaal overview. lmf uses atom-centered functions for envelope functions. They are “smoothed Hankel functions”, constructed from a convolution of a Hankel and Gaussian function centred at the nucleus. Thus, in contrast to ordinary Hankel functions which are singular at the origin (the envelope functions of the LMTO method), they resemble Gaussian functions for small r and are smooth everywhere. For large r they behave like ordinary Hankel functions and are better approximations to the wave function than Gaussian orbitals. The mathematical properties of these functions are described in some detail in this J. Math. Phys. paper. The envelope functions are augmented by partial waves inside augmentation spheres, as is customary for all-electron methods. The basis set is described in more detail here.

Such a basis has significant advantages: basis sets are much smaller for a given level of precision. On the other hand they are also more complex.

It is also possible to combine smoothed Hankels and plane waves : the “Planar Muffin Tin” (PMT) basis is another unique feature of this package.

Partially completed is a new basis of “Jigsaw Puzzle Orbitals”. JPO’s are based on smooth Hankel functions, but advantageous in two respects: The kinetic energy is by construction made continuous everywhere, significantly increasing the quality of the basis and also first they are combined to make short-ranged functions. Thus they result in a minimal, yet highly accurate basis approaching LAPW precision over an energy window of about 1 Ry around the Fermi level. At present the “screening” part of the JPOs is operational, but the continuous matching of the kinetic energy has yet to be completed.

Augmented Wave Methods

Augmented Wave Methods, originally developed by Slater, partitions space into spheres enclosing around each nucleus, and an “interstitial” region. Basis functions used to solve Schrödinger’s equation consist of a family of smooth envelope functions which carry the solution in the interstitial, and are “augmented” with solutions of the Schrödinger equation (aka partial waves) inside each sphere. The reason for augmentation is to enable basis functions to vary rapidly near nuclei where they must be orthogonalized to core states.

Augmented-wave methods consist of an “atomic” part and a “band” part. The former takes as input a density and finds the partial waves $\phi(\varepsilon,r)$ on a numerical radial mesh inside each augmentation sphere and makes the relevant matrix elements needed, e.g. for the hamiltonian or some other property (e.g. optics). The “band’’ part constructs the hamiltonian and diagonalizes the secular matrix made by joining the partial waves to the envelopes.

Solutions of the Schrödinger equations are then piecewise: the envelope functions must be joined differentiably onto the partial waves. Matching conditions determine a secular matrix, so solution of the Schrödinger equation in the crystal for a given potential reduces to an eigenvalue problem.

The choice of envelope function defines the method (Linear Muffin Tin Orbitals, Linear Augmented Plane Waves, Jigsaw Puzzle Orbitals); while partial waves are obtained by integrating the Schrödinger equation numerical on a radial mesh inside the augmentation sphere.

Questaal’s Augmentation

lmf carries out augmentation in a manner different than standard augmented wave methods. It somewhat resembles the PAW method, though in the limit of large angular momentum cutoff it has exactly the same behaviour that standard augmented-wave methods do. Thus this scheme is a true augmented wave method, with the advantage that it converges more rapidly with angular momentum cutoff than the traditional approach. For details see Sec. 3.6 of Ref. 1.

Linear Methods in Band Theory

Nearly all modern electronic structure methods make use of the linear method pioneered by O.K. Andersen. Partial waves $\phi_l(\varepsilon,r)$, are solutions to the radial Schrodinger equation inside a spherically symmetric potential, subject to some boundary condition to suit a particular purpose. One boundary condition can be a “linearization energy” $\varepsilon_\nu$ chosen to be near the states of interest (typically the middle of the occupied part of the energy bands). The linear approximation abounts to expanding $\phi_l(\varepsilon,r)$ in a Taylor series to first order about $\varepsilon_\nu$. This is explained in detail in Richard Martin’s book, Electronic Structure.

The linear approximation rests on the fact that a partial wave $\phi_l(\varepsilon,r)$ for an atom centered at the origin varies slowly with $\varepsilon$. $\phi_l(\varepsilon,r)$ is expanded in a Taylor series about a linearization energy $\varepsilon_\nu$ $$\phi_l(\varepsilon,r) \approx \phi_l(\varepsilon_\nu,r) + (\varepsilon-\varepsilon_\nu)\dot\phi_l(\varepsilon_\nu,r) \quad\quad\quad\quad (1)$$

Note: in practice the vast majority of methods construct the partial waves from the spherical part of the potential, so that l is a good quantum number. Then each $\phi_l$ can be integrated independently of the others. Matrix elements of the partial waves are calculated in the full, nonspherical potential.

The linear approximation is usually quite accurate over an energy window where the valence partial wave is “active” (1 or a few Ry for typical s and p states, a few eV for d states of the transition metals). An estimate for this window is given by $1/p_l$ where potential parameter $p_l = \int_{\mathrm{sphere}} \dot\phi_l^2 d^3r$ is called the “small parameter.” Linearization greatly simplifies the secular matrix : it reduces to a linear algebraic eigenvalue problem, which greatly simplifies practical solutions of the Schrödinger equation.

Some elements possess partial waves of very different energies that are both relevant to the total energy or states near the Fermi level. The classic examples of this are Ga and In: both 3d (4d) and 4d (5d) states are relevant. To obtain accurate calculations a third partial wave must be added to the pair in Eq. (1) constituting the linear method. In practice additional partial waves are incorporated by turning them into local orbitals which are confined to the augmentation sphere. This is accomplished by adding a judicious amount of $\phi_l(\varepsilon_\nu,r)$ and $\dot\phi_l(\varepsilon_\nu,r)$ to the third partial wave, so that its value and slope vanish at the augmentation radius and not spill out into the interstitial. These modified waves are called $\phi_z$ in the Questaal suite.

Extension of the linear approximation through local orbitals ensures that the eigenvalue problem remains a linear one, albeit at the expense of an increase in the rank of the hamiltonian.

The importance of the linear method to electronic structure cannot be overstated. Slater’s X−α method to approximate the difficult Fock exchange with a simpler functional of the density, which was subsequently formalized into rigorous density-functional theory by Hohenberg and Kohn, taken in combination of the linear method, form the backbone for most of the practical modern electronic structure methods in condensed matter.

The linearization energy of a partial wave $\phi_l(\varepsilon,r)$ of angular momentum l is usually parameterized by codes in the Questaal package through the “continuous principal quantum number” P_l, as described on this page.

Smoothed Hankel functions

The envelope functions are “smoothed Hankel functions,” 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 $r_{s}$. 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 $n_0$ carried on a uniform mesh, defined everywhere in space ($n_0$ is not augmented; inside the augmentation spheres it is present as a “pseudodensity”); the true density $n_1$ expressed in terms of spherical harmonics $Y_{lm}$ inside each augmentation sphere; and finally a one-center expansion $n_2$ of the smooth density in $Y_{lm}$, inside each augmentation sphere. The total density is expressed as a sum of the three independent densities: $n = n_0 + n_1 - n_2$.

This turns out to be an extremely useful way to carry out the augmentation procedure. $n_0$ and $n_2$ approximately cancel in the augmentation spheres; but it is the residual $n_0{-}n_2$ that greatly accelerates he convergence with $\ell$ 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.

The Atomic Spheres Approximation

The classical LMTO method of Andersen was coupled to the Atomic Spheres Approximation or ASA: in addition to the LMTO basis sets it makes a shape approximations to the potential. That is, it takes only the spherical part of the potential inside each augmentation sphere, and neglecting the interstitial. Because of the latter, and because the system must conserve charge, the sum-of-sphere volumes must equal the unit cell volume. Some of Questaal’s density functional codes (lm, lmgf, lmpg) make this approximation. The problem with the ASA is that the need to fill spheres causes geometry violations, which can become severe unless sites are reasonably close packed. If this happens you will need to alleviate the problem by including additional fictitious atoms (“empty spheres”) with atomic number zero.

Connection to the ASA packages

The full-potential builds on the ASA suite. The latter 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 has a better basis set with a proper interstitial and does not make the ASA. This 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 $Q_0$, $Q_1$, $Q_2$, 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 between ASA and FP is that the latter basis set is more complicated in that the envelopes’ form must be specified. Envelopes are the smooth Hankel functions noted above — with the Gaussian smoothing radius and Hankel energy to be specified per basis function. Typically the atomic density code lmfa automatically generates a reasonable set of values, but you can create them by hand, or tinker with the autogenerated ones. When it is operational, the “Jigsaw Puzzle Orbital” basis noted above will make the kinetic energy everywhere continuous and significantly improve the quality of envelope functions.

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 a GW calculation, linking the GW and one-body parts

• lmgwsc is a higher level script that performs self-consistent QSGW 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].

[5] Brian Cunningham, Myrta Gruening, Dimitar Pashov, Mark van Schilfgaarde, QSGW: Quasiparticle Self consistent GW with ladder diagrams in W, Phys. Rev. B 108, 165104 (2023)