# Full Potential LMTO Capabilities

### Introduction

This page should serve as a demonstration of a number of the capabilities of the lmf code and can be used as a reference for the corresponding procedures.

##### Energy bands

You can run lmf in band mode to generate energy bands along lines or planes, for generating, e.g. Fermi surfaces. Particularly useful are the color weights.

##### Partial DOS and Mulliken analysis

lmf can generate partial DOS within augmentation spheres, and construct a Mulliken analysis. DOS can be resolved by site, by site and l, or by site and lm. These options are invoked through command-line switches. For illustrations, invoke

fp/test/test.fp co 2

fp/test/test.fp fe 2


##### Charge density

lmf can generate the charge density (smooth or not, with or without cores), and the contribution to the density from a selected window of states. See –wden and –window in command-line switches

##### Core-level spectroscopy

lmf can generate EELS spectra, which involve matrix elements between core and valence electrons. The EELS option is invoked with a command-line switches. For illustrations, invoke

fp/test/test.fp fe 2

fp/test/test.fp crn 2


##### LDA+U

The LDA + U functional was built into lmf in v6.15 and later, by Walter Lambrecht. LDA + U needs in addition to the LDA, parameters U and J for selected orbitals, which are empirical. The LDA + U constructs an additional potential for a particular l subblock (m = -l..l) from the U supplied by the user, and the density-matrix, which is generated by lmf.

Two modifications must be added to the input, which are described here:

In a strictly LDA calculation, complete information is carried by the density, contained in the restart file, rst.ext. In the LDA + U case, complete information is carried by density and on-site density matrices, which are contained in file dmats.ext.

An example that illustrates LDA + U method is ErAs, which you can run by

fp/test/test.fp eras


ErAs is an interesting case because LDA puts all 4 minority f electrons in a single extremely narrow band at the Fermi level. In LDA + U the minority f is split into a 4 - and 3-manifold; see PRB 67, 035104 (2003).

##### GW

lmf is designed to work in coordination with a GW package by T. Kotani (the GW package comes separately). lmf acts both as a driver for the GW package and can also be used in a self-consistent GW cycle. An extra driver lmfgw is compiled as part of this extension. Use of this driver is described in the GW driver documentation. You need the extension package GW.version.tar.gz. Also, you will need the GW package itself. For illustrations of the driver invoke

gw/test/test.gw si

gw/test/test.gw gas


##### Spin-Orbit coupling

lmf solves the scalar Dirac hamiltonian. The dominant difference between the full Dirac hamiltonian and the scalar one is the spin orbit coupling, which can be added as a term $\lambda L \cdot S$ to the scalar Dirac Hamiltonian.

Starting with v6.15, lmf can add λL·S to the scalar Dirac hamiltonian (courtesy of A. Chantis). It is possible to include the full $L \cdot S$ or just the $L_z\cdot S_z$ part. Beginning with v7.9, $L_z\cdot S_z + (L \cdot S - L_z \cdot S_z)$ can be added where the last term is treated in an approximate manner. The approximation turns out to be rather good. See here for some description and analysis of all three approximations. $L \cdot S$ in all three forms can be combined with the self-energy read by a QSGW calculation. In the $L_z\cdot S_z$ and approximate $L \cdot S$ forms the effect of $S_O$ coupling can be passed through to the GW code, to include its effect on the self-energy.

Use HAM_SO to turn on SO coupling.

For illustrations of all three kinds of approaches, try

fp/test/test.fp felz

fp/test/test.fp gasls


##### Local orbitals

In v6.12 and later, local orbitals may be added to the basis set. These orbitals are important when energy bands over a very wide energy window are required, when high accuracy is needed for shallow (semi-)core states, or for energies far above the Fermi energy. Examples of the former occur in oxides: bond lengths are small and cations with shallow p orbitals extend somewhat beyond the augmentation radius.

Local orbitals are presented in one of two flavors. The first, conventional type of local orbital is constructed by solving the radial Schrödinger at a different linearization energy than the usual valence states, and then subtracting off a particular linear of the valence wave function $\phi$ and energy derivative $\dot{\phi}$ such that the local orbital’s value and slope vanish at the augmentation radius. Thus

• A local orbital is strictly confined to the augmentation sphere, and has no envelope function at all.
• When taken in linear combination with the valence augmentation functions, it can solve the Schrödinger equation exactly for linearization the energy chosen (that is, for the spherical potential that defines the wave functions).
• It turns the linear method into a quadratic one. Of course, it would be possible, and more accurate, to construct a LMTO orbital of a different principal quantum number complete with envelope function; however, there is a corresponding loss in efficiency because the additional matrix elements of the envelope function must be evaluated.
• for states with energies in the vicinity of the local orbital energy, the envelope functions of the regular valence states combine with the local orbital to make the interstitial part of the wave function.

Examples that illustrate local orbitals of this type are

  fp/test/test.fp gas

fp/test/test.fp cu


The Ga 3d semi-core the high-lying As 5s state are included as local orbitals. In the Cu case, the high-lying Cu 4d is included, which is important in GW calculations.

The second, extended, kind of local orbital can only be used for semi-core states. Instead of artificially subtracting off some linear of the phi and phi-dot to make the orbital vanish at the augmentation radius, a smooth Hankel tail is attached to the orbital. The smoothing of tail is constructed to match as well as possible the kinetic energy of the semi-core state. This type of orbital has the advantage that the valence envelope function need not “carry” the tail of the semi-core state. Its drawback is that more things can “go wrong,” namely it may fail to do a good job of fitting the kinetic energy.

An example that illustrates the second kind of local orbital is

fp/test/test.fp srtio3


The Sr 4p and Ti 3p semi-core states are included as local orbitals. In the first part of the test, they are included as local orbitals of the first type; then the last step is recalculated using local orbitals of the second type.

##### Floating Orbitals

In v6.15 and later, floating orbitals may be added to the basis set. These orbitals can be important when very accurate calculations are needed in open systems, e.g. when reliable energy bands are needed for a wide energy window. These orbitals differ from the usual smooth Hankels in that they are not centered at an atom. They are augmented just as the other orbitals, but there is no augmentation radius and thus no “head” sphere.

The following illustrate the inclusion of floating orbitals in the basis:

fp/test/test.fp te

fp/test/test.fp gaslc


##### Augmented Plane Waves

Starting with v6.17, Augmented Plane Waves can also be included in the basis. They play the same role as floating orbitals do, but APWs are superior because they are simpler to use (there are no parameters and they do not need to be located at any particular site), and the control over convergence is more systematic.

The following illustrate the inclusion of APWs in the basis:

fp/test/test.fp te

fp/test/test.fp srtio3

fp/test/test.fp felz 4


##### Parallel Implementation

Two separate parallel versions of lmf have been made (courtesy of A. T. Paxton). One parallelizes over k-points, and is the most efficient for scaling; the other parallelizes many points at a lower level.