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 today, a new, non-empirical, density functional
theory (DFT)-based ligand field model [1] [2] 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
(microstates) 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 wavenumbers 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) [3], Zeeman interaction [4], hyperfine
splitting (HFS) [5], magnetic exchange coupling [5] [6], and shielding
constants [7].

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. [8] using an effective Hamiltonian is more advantageous and
yields more insight for experimentalists. The LFDFT model was extended
to two open-shell systems [9]. 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. [9] [10] [11].


Input description
=================


Average of Configuration run
----------------------------

From a defined atomic structure (for example, obtained from geometry
optimization or a crystal .cif structure file), run a DFT calculation
representing the electron configuration of the system to 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 the
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 scalar relativistic ZORA Hamiltonian
should be used.

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

   $AMSBIN/ams << eor

   System
     Atoms
       Co 0.0 0.0 0.0
       ...
     End
   End

   ...

   Engine ADF
     ...
     IrrepOccupations
       A 56 1.4 1.4 1.4 1.4 1.4
     End
     Relativity Level=Scalar
     Symmetry NOSYM
   EndEngine
   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: open AMSinput and
go to the LFDFT section (**Properties → Ligand Field DFT**). Then it
will automatically be installed via AMSpackages if needed. You can
also install it from the command line using "$AMSBIN/amspackages
install lfdft", or with the AMSpackages GUI "$AMSBIN/amspackages gui".

Alternatively, you can download the LFDFT atomic database LDFDT from
https://downloads.scm.com/Downloads/lfdft/LFDFT.zip, and save the
unzipped file in a different 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

* d^9 f^n, n=1,..,14


LFDFT run
---------

The adf.rkf (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 adf.rkf. 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. A
finite magnetic field can be included.

   $AMSBIN/lfdft << eor
   ADFFILE adffile
   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
   BField bx by bz
   DegeneracyThreshold threshold
   eor

"ADFFILE adffile"
   Path to adf.rkf (TAPE21) file from which lfdft reads data and to
   which lfdft writes data. Default TAPE21.

"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 numbers 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 numbers 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, the 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.

"BField bx by bz"
   Include a finite magnetic field (Tesla). For MCD calculations
   include a magnetic field in the z-direction. For example, "BField
   0.0 0.0 1.0" is a magnetic field of 1.0 Tesla in the z-direction.
   By default, no magnetic field is included. The DegeneracyThreshold
   should be small to see the splitting of levels due to the magnetic
   field.

"DegeneracyThreshold threshold"
   Energy difference threshold (eV) to determine degenerate levels.

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

   $AMSBIN/lfdft <<eor
   ADFFile ams.results/adf.rkf
   NSHELL 1
   NLVAL1 3 2
   MOIND1 29 30 31 32 33
   SOC    1
   SOCTYPE
      shell1 zora
   End
   eor


EPR (ESR) g-tensor
------------------

LFDFT can be used to calculate the EPR (ESR) g-tensor, see Ref. [4].
In the LFDFT implementation in ADF the EPR (ESR) g-tensor will be
calculated only for Kramers doublet states. Results of the
calculations should normally only be used for an odd number of
electrons. Spin-orbit coupling should be included, but a finite
magnetic field should not be included, and no other keywords are
needed. One should be careful when interpreting the g-tensor if two or
more Kramers doublets are close in energy. In that case, in the
effective Hamiltonian used to interpret EPR (ESR) experiments an
effective spin higher than S=1/2 might be used, which is not taken
into account in the calculations.


LFDFT intensities
-----------------

One can calculate excitation energies and oscillator strengths between
two atomic multiplet states which come from different electron
configurations of the same molecule 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.

   $AMSBIN/lfdft_tdm << eor
   STATE1 file1
   STATE2 file2
   eor

"STATE1 file1"
   Filename file1 should be a result adf.rkf of a lfdft calculation,
   that contains the ground state electron configuration.

"STATE2 file2"
   Filename file2 should be a result adf.rkf 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:

   $AMSBIN/lfdft_tdm <<eor
   STATE1 f2.results/adf.rkf
   STATE2 f1d1.results/adf.rkf
   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.


MCD
---

LFDFT can be used to calculate magnetic circular dichroism (MCD)
effect in X-ray absorption spectroscopy, see Ref. [12]. By using the
same methodology as for an X-ray absorption calculation, one needs two
LFDFT calculations for two atomic multiplet states which come from
different electron configurations of the same molecule. Additionally,
in both calculations the same finite magnetic field in the z-direction
(for example, Bfield 0 0 1) should be included. Next the module
lfdft_tdm can be used to calculate besides the total intensity, also
the polarized part that is perpendicular to the magnetic axis, the
circular-left and circular-right part.

[1] M. Atanasov, C.A. Daul, C. Rauzy, *New insights into the effects
    of covalency on the ligand field parameters: a DFT study*,
    Chemical Physics Letters 367, 737 (2003)

[2] M. Atanasov, C.A. Daul, C. Rauzy, *A DFT Based Ligand Field
    Theory*, Structure & Bonding 106, 97 (2004)

[3] A. Borel, C.A. Daul, L. Helm, *Hybrid ligand-field theory/quantum
    chemical calculation of the fine structure and ZFS in
    lanthanide(III) complexes*, Chemical Physics Letters 383, 584
    (2004)

[4] M. Atanasov, E.J. Baerends, P. Beattig, R. Bruyndockx, C. Daul, C.
    Rauzy, *The calculation of ESR parameters by density functional
    theory: the g- and A-tensors of Co(acacen)*, Chemical Physics
    Letters 399, 433 (2004)

[5] M. Atanasov, C.A. Daul, *A DFT based ligand field model for
    magnetic exchange coupling in transition metal dimmer complexes:
    (i) principles*, Chemical Physics Letters 379, 209 (2003)

[6] M. Atanasov, C.A. Daul, *A DFT based ligand field model for
    magnetic exchange coupling in transition metal dimer complexes::
    (ii) application to magnetic systems with more than one unpaired
    electron per site*, Chemical Physics Letters 381, 584 (2003)

[7] F. Senn, C.A. Daul, *Calculation of*   ^59Co *shielding tensor*
    \sigma  *using LF-DFT*, Journal of Molecular Structure: THEOCHEM
    954, 105 (2010)

[8] C. Daul, *Non-empirical Prediction of the Photophysical and
    Magnetic Properties of Systems with Open d- and f-Shells Based on
    Combined Ligand Field and Density Functional Theory (LFDFT)*,
    Chimia 68 (2014)

[9] H. Ramanantoanina, W. Urland, F. Cimpoesu, and C. Daul, *Ligand
    field density functional theory calculation of the* 4f^2 → 4f^1
    5d^1 *transitions in the quantum cutter* Cs_2KYF_6:Pr^3+, Physical
    Chemistry Chemical Physics 15, 13902 (2013).

[10] H. Ramanantoanina, W. Urland, A. Garcia-Fuente, F. Cimpoesu, and
     C. Daul, *Ligand field density functional theory for the
     prediction of future domestic lighting*, Physical Chemistry
     Chemical Physics 16, 14625 (2014).

[11] H. Ramanantoanina, M. Sahnoun, A. Barbiero, M. Ferbinteanu, F.
     Cimpoesu, *Development and applications of the LFDFT: the non-
     empirical account of ligand field and the simulation of the f–d
     transitions by density functional theory*, Physical Chemistry
     Chemical Physics 17, 18547 (2015).

[12] H. Ramanantoanina, M. Studniarek, N. Daffé, J. Dreiser, *Non-
     empirical calculation of X-ray magnetic circular dichroism in
     lanthanide compounds*, Chemical Communications 55, 2988 (2019).
