# cPES & ligand field

asked by Francesco Borgatti (2023/09/05 15:21)

Dear Quanty developers,

I kindly ask for indications on setting up core-level photoemission calculations. In the example file about 2p core-level photoemission from NiO including ligand field effects, the energy balance for the final states is reported as :

-- The 2p^5 L^10 d^n+1 configuration has an energy 0
-- The 2p^5 L^9  d^n+2 configuration has an energy Delta + Udd - Upd
-- The 2p^5 L^8  d^n+3 configuration has an energy 2*Delta + 3*Udd - 2*Upd

These configurations correspond to the expressions
-- 5 ep + 10 eL + (n+1) ed + (n+1)(n)   Udd/2 + 5 (n+1) Upd == 0
-- 5 ep +  9 eL + (n+2) ed + (n+2)(n+1) Udd/2 + 5 (n+2) Upd == Delta+Udd-Upd
-- 5 ep +  8 eL + (n+3) ed + (n+3)(n+2) Udd/2 + 5 (n+3) Upd == 2*Delta+3*Udd-2*Upd

My question: is this treatment correct for cPES or, since the $2p$ electron is photoemitted rather than promoted in the $d$ states as for XAS, I have to reduce the final state occupation of the $d$ states by one electron (e.g. $2p^5 L^{10} d^n$, $2p^5 L^9 d^{n+1}$, etc…)?

, 2023/09/05 17:25

Dear Francesco,

That is a nice question.

First of all: If I would write a script for cPES only then indeed it would make more sense to define the zero of energy for the final state to be such that the $p^5 d^n L^{10}$ configuration has zero energy. Now let me explain why the current definition is also correct and why, if one calculates both XAS and cPES (which our tutorials do) using one “zero” of energy for the core excited state is useful.

For the following one should realise that when we make models we can freely define the zero of energy. We define the number of electrons to be fixed so we do not even need to worry about the chemical potential. We also use that we can define the zero of energy for the configuration without a core hole and the configuration with a core hole separately. i.e. we do not try to calculate the absolute energy of an edge, only the difference.

There are different ways one can define the zero of energy for a particular configuration. We in our tutorials follow the nomenclature of Zaanen, Sawatzky and Allen. This sets for the initial state the energy of the configuration corresponding to the formal valences of the elements to zero, i.g. the energy of the $2p^6 3d^n L^{10}$ configuration for transition metal compounds has an average of zero energy. We furthermore define the energy of the configuration $2p^6 3d^{n+1} L^{9}$ to $\Delta$. It then follows that the configuration $2p^6 3d^{n+2} L^{8}$ has an energy of $2*\Delta + U$. The first two configurations fix the onsite energy of the $d$ ($e_d$) and $L$ ($e_L$) shell. The difference between them $e_d - e_L$ is important for the physics. The sum defines the zero of energy.

In spectroscopy we set the energy of the excited state configuration one excites into to zero. You can either take the energy of the $2p^5 3d^{n+1} L^{10}$ to have zero energy (XAS) or the configuration $2p^5 3d^{n} L^{10}$ to have zero energy (cPES). Both will not change the energy difference between the $d$ and $L$ shell. The onsite energy of the core shell ($e_p$) will change and as such the position of the spectrum. We however do not calculate the absolute position of the spectrum.

If you only calculate cPES or only XAS both definitions are fine and will yield the same spectrum. If you calculate both cPES and XAS it is nice to use the same definition for both as one then can compare the relative positions of the spectra.

Best wishes, Maurits

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