Multiconfigurational Approach to X-ray Spectroscopy …
197
on the π backbonding peak actually indicates a missing orbital in the active space,
specifically the ligand π which strongly correlates with the π
∗ . In ferrocyanide
([Fe(CN) 6 ]
4− ), the error for the corresponding π
∗ peak drops from 2.0 to 0.6 eV
with PT2, while the error in the relative energy of different states in the e g peak
decrease from 0.6 to 0.1 eV [32].
A well-known problem with CASPT2/RASPT2 is the presence of possible
intruder states. To reduce this problem, an imaginary shift of 0.3–0.5 hartree can
be applied [24]. It is still important to check that the reference weights, i.e., the
weight of the RASSCF state in the total correlated wavefunction, are consistent for
all core-hole states. Despite being a perturbative approach, the RASPT2 equations
are solved iteratively and low reference weights often translate into convergence difficulties and/or inaccurate results. Low weights despite a small imaginary shift most
commonly stem from improper active-space selection.
The formulation of CASPT2/RASPT2 also includes one empirical parameter
called the ionization potential electron affinity (IPEA) shift, which was introduced to
fix a systematic error when dealing with open-shell configurations [26]. There is no
consensus on the optimal value of this shift and recent studies suggest that it strongly
depends on system, active space and basis set [97]. The IPEA should not be used as
an empirical parameter to improve the match with the experimental spectrum, and
a large effect of changing the IPEA value indicates that the active space may be too
limited, as can be seen for the previously discussed π
∗ peak in ferricyanide [73].
NEVPT2 does not include any such shift in its formulation and is also less sensitive
to intruder states. However, the correlation contribution to the spin-state energetics
of some transition metal complexes showed larger deviations for NEVPT2 compared
to CASPT2 [70].
Second-order perturbation theory is a correlated method, and thus in theory displays a slow basis set convergence. However, for X-ray calculations no major changes
in the spectrum have been observed going beyond a triple-zeta basis set, and in many
cases good results are obtained already at the double-zeta level [73]. On the other
hand, standard contracted basis sets do not provide much flexibility for core electrons
to contract upon excitation or for core correlation. This leads to errors in absolute
edge positions of around 3–4 eV for L-edges and up to 18–20 eV for K edges when
using a triple-zeta basis set, slightly depending on the active space and the number
of states [33, 49, 73]. The errors in the absolute L-edge position can be improved to
0.75 eV with the use of an uncontracted basis set [16], but this is very expensive and
only applicable to small systems. In the frozen-core approximation, the quality of
the core basis set is less important. However, relative energies between complexes
with similar ligand environments and active-space selections can still be reproduced
within 0.3 eV [33, 49].
When it comes to spectral shape, RASPT2 calculations typically predict all major
peaks in the L 3 edge with at most 30% error in intensity. The largest error in relative
energy, around 1 eV, is seen for cases with incomplete active spaces, as in the ferricyanide L-edge XAS π
∗ peak. That energetic error might seem large, but should
197
on the π backbonding peak actually indicates a missing orbital in the active space,
specifically the ligand π which strongly correlates with the π
∗ . In ferrocyanide
([Fe(CN) 6 ]
4− ), the error for the corresponding π
∗ peak drops from 2.0 to 0.6 eV
with PT2, while the error in the relative energy of different states in the e g peak
decrease from 0.6 to 0.1 eV [32].
A well-known problem with CASPT2/RASPT2 is the presence of possible
intruder states. To reduce this problem, an imaginary shift of 0.3–0.5 hartree can
be applied [24]. It is still important to check that the reference weights, i.e., the
weight of the RASSCF state in the total correlated wavefunction, are consistent for
all core-hole states. Despite being a perturbative approach, the RASPT2 equations
are solved iteratively and low reference weights often translate into convergence difficulties and/or inaccurate results. Low weights despite a small imaginary shift most
commonly stem from improper active-space selection.
The formulation of CASPT2/RASPT2 also includes one empirical parameter
called the ionization potential electron affinity (IPEA) shift, which was introduced to
fix a systematic error when dealing with open-shell configurations [26]. There is no
consensus on the optimal value of this shift and recent studies suggest that it strongly
depends on system, active space and basis set [97]. The IPEA should not be used as
an empirical parameter to improve the match with the experimental spectrum, and
a large effect of changing the IPEA value indicates that the active space may be too
limited, as can be seen for the previously discussed π
∗ peak in ferricyanide [73].
NEVPT2 does not include any such shift in its formulation and is also less sensitive
to intruder states. However, the correlation contribution to the spin-state energetics
of some transition metal complexes showed larger deviations for NEVPT2 compared
to CASPT2 [70].
Second-order perturbation theory is a correlated method, and thus in theory displays a slow basis set convergence. However, for X-ray calculations no major changes
in the spectrum have been observed going beyond a triple-zeta basis set, and in many
cases good results are obtained already at the double-zeta level [73]. On the other
hand, standard contracted basis sets do not provide much flexibility for core electrons
to contract upon excitation or for core correlation. This leads to errors in absolute
edge positions of around 3–4 eV for L-edges and up to 18–20 eV for K edges when
using a triple-zeta basis set, slightly depending on the active space and the number
of states [33, 49, 73]. The errors in the absolute L-edge position can be improved to
0.75 eV with the use of an uncontracted basis set [16], but this is very expensive and
only applicable to small systems. In the frozen-core approximation, the quality of
the core basis set is less important. However, relative energies between complexes
with similar ligand environments and active-space selections can still be reproduced
within 0.3 eV [33, 49].
When it comes to spectral shape, RASPT2 calculations typically predict all major
peaks in the L 3 edge with at most 30% error in intensity. The largest error in relative
energy, around 1 eV, is seen for cases with incomplete active spaces, as in the ferricyanide L-edge XAS π
∗ peak. That energetic error might seem large, but should