# Ligand Field and Density Functional Theory (LFDFT)¶

## Introduction¶

Hans Bethe and John Hasbrouck van Vleck have introduced the Ligand Field (LF) model simultaneously in the 1930s. This theory often plays a central role for molecules containing d- and/or f-elements with open shells. It is a semi-empirical model with adjustable parameters. Twenty years later Christian Klixbüll Jorgensen and Klaus Erik Schäffer proposed the Angular Overlap Model (AOM) that is a transcription of LF theory using molecular orbitals, still using adjustable parameters. Starting from the very beginning of the twenty first century until nowadays, a new, non-empirical, Density Functional Theory (DFT) based Ligand Field model [465, 466] was developed, called LFDFT. LFDFT emerged through the collaboration of many European scientists during these years.

The key feature of the LFDFT approach is the explicit treatment of near degeneracy correlation using ad hoc Configuration Interaction (CI) within the active space of Kohn-Sham (KS) orbitals with dominant p-, d- or f-character. The calculation of the CI-matrices is based on a symmetry decomposition in the full rotation group and/or on a ligand field analysis of the energies of all single determinants (micro-states) calculated according to Density Functional Theory (DFT) for frozen KS-orbitals corresponding to the averaged configuration, possibly with fractional occupations, of the p-, d- or f-orbitals. In the literature one can find cases where this procedure yields multiplet energies with an accuracy of a few hundred wave numbers and fine structure splitting accurate to less than a tenth of this amount. This procedure was used to calculate different molecular properties, e.g. Zero Field Splitting (ZFS) [467], Zeeman interaction [468], Hyper-Fine Splitting (HFS) [469], magnetic exchange coupling [469, 470], and shielding constants [471].

Strictly speaking, LF theory is nothing but the consideration of active p-, d- and/or f-electrons moving in the potential of a passive chemical environment. A rigorous formulation of a passive chemical environment could be given using a frozen density embedding (FDE) scheme. However, in practice, it turns out that the method described in Ref. [472] using an effective Hamiltonian is more advantageous and yields more insights to the experimentalists. The LFDFT model was extended to two-open-shell systems [473]. This has relevance for inter-shell transitions in lanthanides, important for understanding the optical and magnetic properties of rare-earth materials. It could also be used, for example, to calculate multiplet effects in X-ray absorption spectroscopy. Theory and applications for f → d transitions can be found in Refs. [473-475].

## Input description¶

**Average of Configuration run**

From a defined atomic structure (for example, obtained from geometry optimization or crystal .cif structure file), run a DFT calculation representing the electron configuration system from which the metal ion belongs.
This DFT run is achieved by the Average of Configuration (AOC) calculation with fractional occupations of s, p, d or f orbitals, i.e. for
3d^{7} electron configuration of Co^{2+}, 7 electrons are evenly distributed in the molecular orbitals having dominant cobalt character.
Choosing which orbitals have to be fractionally occupied may not always be trivial, since there may be several close-lying levels with different character.
This may lead to problematic SCF convergence.
One should check that the final fractionally occupied molecular orbitals indeed have the expected metal character, since otherwise the subsequent LFDFT calculation will be meaningless.
It is required that the metal ion is placed as first atom in the coordinate system.
Besides, the calculation implies a single-point spin restricted SCF without symmetry constraint (C1 point group = SYMMETRY NOSYM).
A non-relativistic or scalar relativistic Hamiltonian should be used.

For a Co^{2+} d^{7} electron configuration the input for ADF could be something like:

```
$ADFBIN/adf << eor
...
Atoms
Co 0.0 0.0 0.0
...
End
IrrepOccupations
A 56 1.4 1.4 1.4 1.4 1.4
End
relativistic scalar ZORA
SYMMETRY NOSYM
eor
```

**LFDFT atomic database**

The LFDFT atomic database is *not* included in the ADF distribution.

The easiest way to install it is using the ADF-GUI and open ADFinput and go to the LFDFT section. Then it will be installed in $ADFHOME/atomicdata or in “$HOME/Library/Application Support/SCM” (on a mac). If you do not have an ADF-GUI download the LFDFT atomic database LDFDT from http://downloads.scm.com/Downloads/lfdft/LFDFT.zip, and save the unzipped file in the aforementioned location. If you save this LFDFT atomic database in a different location, make sure to point the environment variable SCM_LFDFT to this directory.

Note that at present available electron configurations for the LFDFT atomic database are:

- s
^{n}, n=0,..,2 - p
^{n}, n=0,..,6 - d
^{n}, n=0,..,10 - f
^{n}, n=0,..,14 - p
^{5}d^{n}, n=0,..,10 - f
^{n}d^{1}, n=1,..,13

**LFDFT run**

The TAPE21 file of the average of configuration (AOC) run should be an input file for the program lfdft. Results of the LFDFT calculation will be put on TAPE21. The input for lfdft is keyword oriented and is read from the standard input. Spin-orbit coupling is calculated using the ZORA equation by default. In cases where this is an underestimation, the old approximate method (a simple core potential) and/or an extra multiplication factor can be included.

```
$ADFBIN/lfdft << eor
NSHELL nshell
NLVAL1 nval1 lval1
{NLVAL2 nval2 lval2}
MOIND1 MO#1 MO#2 ... MO#(2*lval1+1)
{MOIND2 MO#1 MO#2 ... MO#(2*lval2+1)}
SOC soc1 {soc2}
SOCTYPE
shell1 type1
{shell2 type2}
End
eor
```

`NSHELL nshell`

- nshell is the number of shells in the electron configuration system under consideration: for Co
^{2+}in 3d^{7}electron configuration, nshell = 1; for Fe^{2+}in 2p^{5}3d^{7}, nshell = 2. nshell should be equal to 1 or 2. `NLVAL1 nval1 lval1`

- nval1, and lval1, are the main quantum number n and l for shell 1, in case of all-electron calculations on the metal atoms. For 5d this is ‘5 2’. For frozen core calculations the number n should be reduced with the number of core levels with angular momentum l that are in the core. For example for 5d, with 3d and 4d in the core, one should use ‘3 2’.
`NLVAL2 nval2 lval2`

- If nshell=2, nval2, and lval2, are the main quantum number n and l for shell 2.
`MOIND1 MO#1 MO#2 ... MO#(2*lval1+1)`

- MO#1, ..., are the indices of the molecular orbitals which have the most dominant nval1 lval1 character, for example the most dominant 3d character. They should match the fractionally occupied orbitals that are used in the AOC run for shell 1.
`MOIND2 MO#1 MO#2 ... MO#(2*lval1+1)`

- If nshell=2, MO#1, ..., are the indices of the molecular orbitals which have the most dominant nval2 lval2 character. They should match the fractionally occupied orbitals that are used in the AOC run for shell 2.
`SOC soc1 {soc2}`

- soc1 indicates if spin-orbit coupling is considered (1) or not (0) for shell 1. If nshell=2, same applies for soc2 for shell 2. soc1 and soc2 can also be real numbers, in case one wants to scale the calculated approximate spin-orbit coupling(s) by LFDFT. By default, the spin-orbit coupling is included (soc1 = 1, soc2 = 1).

```
SOCTYPE
shell1 type1
{shell2 type2}
End
```

type1 and type2 can be either ‘zora’ or ‘core’ and indicate whether the spin-orbit coupling for the given shell is calculated using the corresponding term in the zora equation (type = zora) or using the old approximation with the core potential only (type = core). The default is set to zora.

For example for a 3d^{7} electron configuration the input for lfdft could be

```
$ADFBIN/lfdft <<eor
NSHELL 1
NLVAL1 3 2
MOIND1 29 30 31 32 33
SOC 1
SOCTYPE
shell1 zora
End
eor
```

**LFDFT intensities**

One can calculate excitation energies and oscillator strengths between two atomic multiplet states which come from different electron configurations with the module lfdft_tdm. Note that electronic transitions between two multiplet states which come from the same electron configuration are not dipole allowed. lfdft_tdm can calculate excitation energies and oscillator strengths (in the dipole approximation) from the calculated ground state multiplet of one electron configuration to all multiplet states that can be made for the other electron configuration. The calculated transition dipole moments and oscillator strengths are in arbitrary units. They are averaged over the degeneracy of the ground state as well as over the degeneracy of the excited state multiplet. The oscillator strength has to be multiplied with the degeneracy of the excited state multiplet.

The input for lfdft_tdm is keyword oriented and is read from the standard input.

```
$ADFBIN/lfdft_tdm << eor
STATE1 file1
STATE2 file2
eor
```

`STATE1 file1`

- Filename file1 should be a result TAPE21 of a lfdft calculation, that contains the ground state electron configuration.
`STATE1 file2`

- Filename file2 should be a result TAPE21 of a lfdft calculation, that contains the excited state electron configuration. Results of the lfdft_tdm calculation will be put on file2.

For example for Pr 4f^{2} → Pr 4f^{1} 5d^{1} the input could be something like:

```
$ADFBIN/lfdft_tdm <<eor
STATE1 Pr_f2.t21
STATE2 Pr_f1d1.t21
eor
```

In this case lfdft_tdm will calculate the excitations from the ground state of Pr 4f^{2} to all multiplet states that can be made with the Pr 4f^{1} 5d^{1} electron configuration.