192
M. Lundberg and M. G. Delcey
The orbital diagrams of ferricyanide and ferric chloride are shown in Fig. 3b. An
active space for valence calculations of FeCl 6 would include the five 3d orbitals
and the two e g ligand orbitals combinations forming the σ bond with the metal. For
accurate energy calculations, it is recommended to include an additional set of metal
d-type orbitals, the double shell, for a total of twelve active orbitals [69]. However,
for X-ray calculations, neglecting the double shell gives small differences in the
spectrum [71]. As the complex is relatively ionic, the ligand-dominated σ orbitals
have limited metal d character, and one could imagine limiting the active space to
the five 3d orbitals. Yet, with this choice of orbitals, the subsequent second-order
perturbation theory calculation fails to converge (see Sect. 3.5), a typical sign of an
ill-balanced active space. This example thus shows that the final choice of active
space is usually a dialogue between the user and the program.
For Fe(CN) 6 , the situation is similar except that this time, the ligands have empty
π
∗ orbitals able to form a bond with the metal t 2g orbitals through π -backdonation.
This stabilizes the t 2g orbitals and thus increases the gap with the e g , which favors the
low-spin configuration. As the π
∗ orbitals are empty and have some 3d character, they
can be populated by X-ray absorption and are needed in the active space, consisting
thus of ten orbitals.
In some cases, the X-ray process involves two different core levels. One example
is 1s2p RIXS, where 1s → 3d absorption is followed by monitoring the strongest
emission channel, 2p → 1s, see Fig. 1. Modeling of this process requires independent
control of the core-hole occupations. In a RAS calculation, this can be achieved by
placing the two sets of core orbitals in different ras spaces, typically 2p in ras1 and
1s in ras3 and thus, both 1s and 2p core-hole states can be computed with the same
active space, see Fig. 3a [32].
3.3 Generating Core-Hole States
To obtain a spectrum, a large number of excited states needs to be computed. As
mentioned in the previous section, the target is to generate specific core-excited
states, without computing all possible valence states. In some cases, such as L-edge
spectra of centrosymmetric complexes, the 2p core-hole states can be in different
symmetries (ungerade) than the valence states (gerade) and the separation is trivial.
For the general case, a simple technique is to remove from the configuration interaction all configurations with fully occupied core orbitals, the so-called core–valence
separation (CVS) [14]. For active-space methods, the CVS is closely related to the
generalized active-space method [33, 59].
Orbital optimization is typically done using state-average orbitals. This avoids the
separate optimization of every state, while ensuring a balanced description of all the
states. The main drawback is that the results depend strongly on the number of states,
which needs to be taken into account, in particular when comparing calculations with
different number of states. During orbital optimization, the algorithm may lower the
state average energy by replacing the core orbital by an occupied orbital of higher
M. Lundberg and M. G. Delcey
The orbital diagrams of ferricyanide and ferric chloride are shown in Fig. 3b. An
active space for valence calculations of FeCl 6 would include the five 3d orbitals
and the two e g ligand orbitals combinations forming the σ bond with the metal. For
accurate energy calculations, it is recommended to include an additional set of metal
d-type orbitals, the double shell, for a total of twelve active orbitals [69]. However,
for X-ray calculations, neglecting the double shell gives small differences in the
spectrum [71]. As the complex is relatively ionic, the ligand-dominated σ orbitals
have limited metal d character, and one could imagine limiting the active space to
the five 3d orbitals. Yet, with this choice of orbitals, the subsequent second-order
perturbation theory calculation fails to converge (see Sect. 3.5), a typical sign of an
ill-balanced active space. This example thus shows that the final choice of active
space is usually a dialogue between the user and the program.
For Fe(CN) 6 , the situation is similar except that this time, the ligands have empty
π
∗ orbitals able to form a bond with the metal t 2g orbitals through π -backdonation.
This stabilizes the t 2g orbitals and thus increases the gap with the e g , which favors the
low-spin configuration. As the π
∗ orbitals are empty and have some 3d character, they
can be populated by X-ray absorption and are needed in the active space, consisting
thus of ten orbitals.
In some cases, the X-ray process involves two different core levels. One example
is 1s2p RIXS, where 1s → 3d absorption is followed by monitoring the strongest
emission channel, 2p → 1s, see Fig. 1. Modeling of this process requires independent
control of the core-hole occupations. In a RAS calculation, this can be achieved by
placing the two sets of core orbitals in different ras spaces, typically 2p in ras1 and
1s in ras3 and thus, both 1s and 2p core-hole states can be computed with the same
active space, see Fig. 3a [32].
3.3 Generating Core-Hole States
To obtain a spectrum, a large number of excited states needs to be computed. As
mentioned in the previous section, the target is to generate specific core-excited
states, without computing all possible valence states. In some cases, such as L-edge
spectra of centrosymmetric complexes, the 2p core-hole states can be in different
symmetries (ungerade) than the valence states (gerade) and the separation is trivial.
For the general case, a simple technique is to remove from the configuration interaction all configurations with fully occupied core orbitals, the so-called core–valence
separation (CVS) [14]. For active-space methods, the CVS is closely related to the
generalized active-space method [33, 59].
Orbital optimization is typically done using state-average orbitals. This avoids the
separate optimization of every state, while ensuring a balanced description of all the
states. The main drawback is that the results depend strongly on the number of states,
which needs to be taken into account, in particular when comparing calculations with
different number of states. During orbital optimization, the algorithm may lower the
state average energy by replacing the core orbital by an occupied orbital of higher