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# Ligand field

In ligand field theory the covalent interaction between a correlated (d) shell and a ligand shell is explicitly taken into account by adding hopping between the correlated (d) shell and a ligand shell. This hopping can be seen as an effective potential coupling two different shells. The potential can similar as in the case of crystal field theory be expanded on spherical harmonics and parameterized by $A_{k,m}$.

In Quanty the ligand interaction is given as:

Example.Quanty
-- ligand field operator in Oh
-- symmetry acting on a d shell
NF = 20
NB = 0
IndexDn_3d = { 0, 2, 4, 6, 8}
IndexUp_3d = { 1, 3, 5, 7, 9}
IndexDn_Ld = {10,12,14,16,18}
IndexUp_Ld = {11,13,15,17,19}
Veg     =  2.06
Vt2g    =  1.21
Akm = {{0, 0,0.4*Veg+0.6*Vt2g},
{4, 0,(21/10)*(Veg-Vt2g)},
{4,-4,(3/2)*math.sqrt(7/10)*(Veg-Vt2g)},
{4, 4,(3/2)*math.sqrt(7/10)*(Veg-Vt2g)}}
OppLF = NewOperator("CF", NF, IndexUp_3d,IndexDn_3d, IndexUp_Ld,IndexDn_Ld,Akm)
+ NewOperator("CF", NF, IndexUp_Ld,IndexDn_Ld, IndexUp_3d,IndexDn_3d,Akm) 

Note that one needs to add both the hopping from d to ligand states as well as from the ligand to the d states.

## Onsite energy, $\Delta$ and $U$

In Quanty the onsite energy of the Ligand and d orbital need to be specified. In the literature you will often find the energy $\Delta$ which refers to this difference. In a many body calculation $\Delta$ is not (IS NOT!) defined as $\epsilon_d - \epsilon_p$. The effect of $U$ enters. (note that $U \neq F[0]$, but $U = F[0] - (2/63)(F[2]+F[4])$)

We follow the energy definitions as introduced in the group of G.A. Sawatzky (Groningen) (See paper of Zaanen Sawatzky and Allen). For parameters of specific materials see A.E. Bockquet et al. After some initial discussion the energies $U$ and $\Delta$ refer to the center of a configuration, including the d orbitals as well as the a linear combination of the Ligand orbitals that interact directly with these d orbitals:

• The $L^{10} \, d^{n}$ configuration has an energy $0$.
• The $L^{9} \, d^{n+1}$ configuration has an energy $\Delta$.
• The $L^{8} \, d^{n+2}$ configuration has an energy $2\times\Delta+U$.

If we relate this to the onsite energy of the Ligand ($\epsilon_L$) and d ($\epsilon_d$) orbitals we find: \begin{align} 10 \epsilon_L &+& n \epsilon_d &+& n(n-1) U/2 &=& 0\\ \nonumber 9 \epsilon_L &+& (n+1) \epsilon_d &+& (n+1)n U/2 &=& \Delta\\ \nonumber 8 \epsilon_L &+& (n+2) \epsilon_d &+& (n+1)(n+2) U/2 &=& 2\times\Delta+U \end{align} 3 equations with 2 unknowns, but with interdependence yield: \begin{align} \epsilon_d &=& (10\times\Delta-n\times(19+n)*U/2)/(10+n)\\ \nonumber \epsilon_L &=& n\times((1+n)\times U/2-\Delta)/(10+nd) \end{align}

Note that $\epsilon_d-\epsilon_L = \Delta - n \times U$ and not $\Delta$. Note furthermore that $\epsilon_L$ and $\epsilon_d$ here are defined for the onsite energy if the system had locally $n$ electrons in the d-shell. In DFT or Hartree Fock the d occupation is in the end not $n$ but larger due to covalence. The onsite energy of the Kohn-Sham orbitals is not equal to $\epsilon_L$ and $\epsilon_d$ in model calculations.