### Local orbital Setup

In Fleur a local orbital (LO) is given by an energy parameter, an angular momentum quantum number, and a definition
of the kind of radial function used to construct the LO. Within the `inp.xml`

file LOs are defined for certain species
within the `atomSpecies`

section. Some examples for such definitions are:

```
<lo type="SCLO" l="1" n="3" eDeriv="0"/>
```

An LO definition like this is typically used to define a local orbital used to represent semicore states within the
valence electron framework in an FLAPW calculation. Sometimes it is even better to add another LO to describe such a
state as the energy parameter might not be perfectly adjusted. In such a case one might add an LO with the same parameters
except the degree of the energy derivative (`eDeriv`

) which would be 1.

```
<lo type="HELO" l="2" n="4" eDeriv="0"/>
```

An LO definition typically used to define local orbitals with radial functions at energy parameters in the range of the unoccupied states. Such LOs are typically used whenever the performed calculation explicitly considers the unoccupied states, e.g., in calculations employing the GW approximation to many-body perturbation theory. Another use for such LOs is the elimination of the linearization error within the FLAPW method.

```
<lo type="SCLO" l="0-3" n="4,4,3,4" eDeriv="2"/>
```

A definition of a set of local orbitals for the angular momentum quantum numbers 0 to 3 and corresponding main quantum numbers 4,4,3, and 4. Each of the LOs uses the second energy derivative of the solution to the radial scalar-relativistic approximation (SRA) to the Dirac equation as third radial function. Such sets of LOs can be used to overcome the linearization error in all relevant l channels. However, one has to be careful not to obtain a numerically singular overlap matrix for the radial functions in one of the l channels. If energy parameters for unoccupied states are used, this way of defining sets of LOs is very practical to cover a large range of energy and l quantum numbers in only a few lines in the input file.

In detail, the energy parameter for the LO is given by the LO type and the main quantum number. The main quantum number `n`

defines the number of nodes (n-l-1) of the additional radial function constructed for the LO. The energy parameter is then
obtained by solving the radial problem under certain boundary conditions defined by the type attribute:

LO-Type | Description |
---|---|

SCLO | A semicore local orbital. The spherical part of the potential in the MT sphere is extended by an artificial confining potential outside the MT sphere. The energy parameter then is the eigenenergy belonging to the solution to this problem with the given l and n quantum numbers. |

HELO | A higher energy local orbital. Here the SRA to the radial Dirac equation is solved for different test energies as a differential equation outwards starting at the atomic nucleus. The energy parameter then is that energy whose solution yields the correct number of nodes and a logarithmic derivative of -(l+1) at the MT boundary. It is found by a bisection search algorithm. |

The angular momentum quantum number l and the main quantum number n are defined by the associated attributes of the lo XML element. The entries for these values can either be single integer values or sequences of values. For the l quantum number these sequences can be defined by two numbers and a "-" in between or by comma separated values. For the n quantum number only comma separated values are allowed. Note that l and n quantum numbers are used in pairs: The i-th l quantum number together with the i-th n quantum number are used to define the i-th local orbital.

Note that if an enpara file is present the energy parameters defined in that file override the definitions in the inp.xml file. If the energy parameters are to be obtained by the energy center of mass (ECM) method, this additional file has to be used.

The kind of the additional radial function is given by the `eDeriv`

attribute. If this is set to 0 the solution of the SRA to
the radial Dirac equation at the given energy parameter is used. If it is set to finite positive integers the energy derivative
of this solution of degree eDeriv is used to construct an HDLO (higher derivative local orbital).

**Further reading**

- Local orbitals for the representation of semicore states have originally been introduced by Singh et al.
- In the context of GW calculations local orbitals employing higher energy derivatives have been introduced by Friedrich et al.
- For the representation of unoccupied states local orbitals on the basis of the HELO -(l+1) criterion have been used by Betzinger et al.
- An analysis about the usefulness of different types of local orbitals to eliminate the linearization error for the representation of valence electrons has been performed by Michalicek et al.