3.15 Delta-SCF Occupations

Example:
deltascf_basis 3 1 atomic 2 1 1 0.3333 6

In this example, we have chosen the basis function that resides on the third atom (atom=3), as listed in the geometry.in in the first spin channel. Since we are interested in constraining an atomic-like orbital, we chosen one that is part if the minimal basis (basis_type=atomic). A constraint of 0.3333 is applied onto the 2$p$ level, $m$=1 orbital given by the quantum numbers 2 1 1. Only Kohn-Sham orbitals up to the 6th state will be allowed to be chosen for the constraint to occupy.

In general, it is a good idea to constrain the orbital in question out of the occupied space, i.e., choose a value for max_KS_state that indicates a state above the Fermi level.

This assumption will work well for localized basis functions such as the 1$s$ levels of most elements. As a rule, the constraint is expected to be less and less unique if applied to more delocalized basis functions – there can even be multiple different self-consistent constrained solutions for the same formal constraint. For instance, Si 2$p$ basis functions can exhibit this problem if the basis sets on the surrounding atoms – which overlap with the Si atom – become too large. Here, a smaller basis set (tier 1) can indeed be the way to keep a qualitatively meaningful Mulliken-type constraint.

For an short how to calculate core-hole spectroscopy see deltascf_projector.

Example:
deltascf_projector 1 1 0.0 1 1
deltascf_projector 8 2 0.5 6 10

The first example enforces an occupation of 0.0 onto state 1 in spin channel 1. The constraint is not allowed to move between SCF steps as both KS_start and KS_stop are the same as the chosen KS_state. As an example this would be used for constraining the oxygen 1s state in a water molecule as there are no similar KS states the core-hole could go.

For the second example, an electron occupation of 0.5 is constrained onto KS state 9 in spin channel 2. Here a range of KS states, between 6 and 10 are checked for overlap between SCF steps. This would be chosen for a hydrocarbon molecule where you are constraining one particular carbon 1s state but the range of states are the other carbon 1s states in the molecule.

To simulate XPS energies with $n$ inequivalent atoms of the same species a total of $n+1$ single runs is required: One ground state calculation and one deltascf_basis/deltascf_projector calculation for each of the eindividual atoms.

An initial ground state calculation is needed to calculate the ground state energy, and to create a restart file which is needed for a deltascf_projector calculation (one isn’t needed for deltascf_basis). This can be done by including the restarr_write_only or restart flags. Using this it is therefore important to ensure that the geometry.in and all other parameters (except the charge) need to be kept the same. I can be very benefical to use output to view either the cube eigenstate or cube spin_density to confirm the core-hole has localized properly.

For the simulation of the XPS spectra typically a full core hole is introduced (for example in ref. [75, 168]). Relying on initial state effects alone (i.e., defining the ionization energies using ground state eigenvalues) neglects the screening of the core hole by valence electrons [203] and is therefore not a good approximation for XPS. For example, to force the occupation of eigenstate 3 to 1.0, where states 1-4 are (near-)degenerate or at least very similar in energy and type:

deltascf_projector 3 1 1.0 1 4

Results in the following occupation (the introduction of the core hole leads to a re-ordering):

Note that charge was set to 1 to take into account the reduced electron number and a restart from the ground state run was made using restart.

XPS energies can then be calculated as the difference of the total energy obtained in the ground state calculation and the total energy of the core-hole excited simulation (corresponding to the definition of the ionization energy). That means that for each excitation center an ionization energy is calculated. For the core level shifts only relative energy differences are relevant, which are already directly reflected in the differences of total energies of the core-hole excited states. If, however, absolute energies are of interest, note that experiments are referenced to either the vacuum level or the Fermi level, and that simulations including an extended surface might differ by the workfunction from those for isolated molecules. The ionization energies can then by broadened with Gaussian functions of same amplitude (assuming no preferential direction, especially valid for $1s$ spectra) and summed up to obtain the total XPS spectrum.

Simulating NEXAFS spectra can be less straightforward as there are different approximations to account for the core hole and the excited electron. One possibility is to use the transition potential approach [301], where instead of a full core only half a core hole is used, i.e., $n=0.5$ in one spin channel. Independently from the approximation used for the core hole: To obtain the dipole matrix elements that give information about the transition probability the flag compute_dipolematrix needs to be used. Note that to use this option the FHI-aims binary has to be compiled enabling hdf5, as the output is a hdf5 container containing eigenvalues and matrix elements. It is recommended to include additional empty_states, depending on the amount of unoccupied states you want to probe. In this case the ground state calculation has already to include the same number of empty states, otherwise a restart is not possible.

Example:
delta_scf_occ_non_aufbau k_space_aware 1
delta_scf_occ_non_aufbau k_space_aware 0.1

In the first example a single electron is excited from the valence band to the conduction band. Similarly, the second example follows the same with a fractional 0.1 electrons.

The static exciton calculation is calculated with the $\mathrm{\Delta }$-SCF method similar to the previous sections. The core subroutines are implemented in ELSI and are linked to FHIaims through the ELSI interface.

At each SCF iteration, single-particle Kohn-Sham eigenvalues, ${ϵ}_i$ are determined for a given density where the subscript $i$ corresponds to the index of the single-particle orbital. The occupation number matrix, $f_i^n$ for each orbital and $n$ electrons in the ground state are calculated. This is followed by two additional calculations for the fixed set of single-particle energies ${ϵ}_i$, one for $n-a$ and another with $n+a$ number of electrons where $a$ is the number of excited electrons. This provides the chemical potentials ${\mu }_{n-a}$ and ${\mu }_{n+a}$ for the same single-particle orbitals filled with $n-a$ and $n+a$ electrons, respectively, along with their occupation number matrices, $f_i^{n-a}$ and $f_i^{n+a}$. The final occupation number matrix for the excited state single-particle configuration is achieved by setting $f_i$ = $f_i^{n-a}$ + $f_i^{n+a}$ - $f_i^a$. This mechanism allows the excitation of $a$ electrons from the VBM to the CBM. Once the system achieves its new self-consistent field (SCF) under these conditions, band structures, total energies, electron densities and forces of the excited system can be obtained.