Vietnam National University, Ho Chi Minh City, Vietnam Chemical Sciences
Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA

# Exploring Ligand-to-Metal Charge-transfer States in the Photo-Ferrioxalate
System using Excited-State Specific Optimization

Lan Nguyen Tran<EMAIL_ADDRESS>Department of Physics, International
University, Ho Chi Minh City, Vietnam  Eric Neuscamman
<EMAIL_ADDRESS>Department of Chemistry, University of California,
Berkeley, California, 94720, USA

###### Abstract

The photo-ferrioxalate system (PFS), [Fe(III)(C2O4)]3-, more than an exact
chemical actinometer, has been extensively applied in wastewater and
environment treatment. Despite many experimental efforts to improve clarity,
important aspects of the mechanism of ferrioxalate photolysis are still under
debate. In this paper, we employ the recently developed W$\Gamma$-CASSCF to
investigate the ligand-to-metal charge-transfer states key to the ferrioxalate
photolysis. This investigation provides a qualitative picture of these states
and key potential energy surface features related to the photolysis. Our
theoretical results are consistent with the prompt charge transfer picture
seen in recent experiments and clarify some features that are not visible in
experiments. Two ligand-to-metal charge-transfer states contribute to the
photolysis of ferrioxalate, and the avoided crossing barrier between them is
low compared to the initial photoexcitation energy. Our data also clarify that
one Fe–O bond cleaves first, followed by the C–C bond and the other Fe–O bond.

Photo-ferrioxalate system (PFS) [Fe(III)(C2O4)]3- has been used innumerable
times for the chemical actinometry in liquid systems1, 2, 3, 4, 5. Recently,
it has been extensively applied in wastewater and environment treatment6, 7,
8, 9. The photolysis of PFS generates various reactive intermediates, such as
Fe(II) and reactive oxygen species (ROS), including O∙-, ∙OH and H2O2, that
are significant for the oxidative decontamination of organic substances in the
environment and engineered systems. This photoreduction is conventionally
expressed as

$\displaystyle 2[\text{Fe(III)(C}_{2}\text{O}_{4})_{3}]^{3-}+h\nu$
$\displaystyle\rightarrow
2[\text{Fe(II)(C}_{2}\text{O}_{4})_{2}]^{2-}+2\text{CO}_{2}+(\text{C}_{2}\text{O}_{4})^{2-}.$
(1)

Understanding its prime mechanisms is crucial to optimize the reactive paths
and to control their outcomes. However, while the photoreaction intermediates
and products are known, the initial charge-transfer mechanisms that play a
pivotal role in the photochemistry of PFS are still under debate10.

Two mechanisms have been proposed for the photolysis of PFS11, 12, 13, 14, 15:
(i) a prompt reduction and (ii) a slow reduction. In the first mechanism, a
ligand-to-metal charge-transfer (LMCT) state is promptly excited, resulting in
the reduction of the metal center from Fe(III) to Fe(II)13, 15. The second
mechanism has no intramolecular electron transfer, with the metal center
instead reduced later on by an electron transfer from the environment. 11, 12
While numerous experiments support the prompt reduction mechanism, specific
interpretations of its nature differ. A more comprehensive review of these
experimental results is given in the Supporting Information (SI), and we only
summarize some important conclusions here. Pozdnyakov et al. 13 proposed that
electron transfer from oxalate to iron forms an oxalate radical that initially
remains bound to the metal and then diffuses into a solution. Ogi et al. 16
proposed that electron transfer from oxalate to iron produces a bound excited
molecule that dissociates as CO${}_{2}^{\bullet-}$ and CO2. The conclusions of
Mangiante et al. 17 and O’Neil et al. 18 mostly agree with those of Ogi et al.
Pilz and coworkers 19 showed evidence that, after the LMCT excitation, the
excited molecules dissociate into a single CO2 and the complex
[CO2Fe(II)(C2O4)2]3-, which has been further supported by work by Longetti et
al. 20.

Although the experimental studies above have provided significant evidence for
a prompt LMCT photoexcitation followed by the dissociation of the oxalate
anion, the nature of the initial charge transfer (CT) and its immediate
effects on the complex is still under debate. 20 It remains unclear if the
species remains bound to the metal center after the photoinduced CT or whether
it immediately dissociates from the metal to diffuse into the solution. 19, 20
Certainly, a detailed ordering of the Fe–O and C–C bond dissociations and any
related transition states on the LMCT surface are also lacking.

Regarding theoretical studies, density functional theory (DFT) and time-
dependent DFT (TD-DFT) have been used to investigate the LMCT state of PFS 11,
12, 21, 22. While DFT is only able to optimize the lowest state of a given
spin multiplicity, TD-DFT faces well-known challenges when modeling charge-
transfer excitations. 23, 24 While the complete active space self-consistent
field (CASSCF) method with state-averaging optimization has been widely used
for excited states. It also has serious shortcomings for CT states whose
important orbital relaxations and dipole moment changes are resisted by the
other states in the state average. 25, 26, 27, 28 A theoretical tool more
robust to the challenges of LMCT is thus highly desirable in analyzing the
photolysis of PFS.

In the present study, we employ a recently-introduced excited-state specific
approach 27, 28 to CASSCF to investigate PFS’s LMCT states. This method
targets the state of interest during iterative CASSCF optimization according
to its physical characteristics, such as its energy and density matrices. The
approach is similar in spirit to the maximum overlap method (MOM) developed
for Hartree-Fock and DFT. 29, 30 While its current form does not make the
approach amenable to simulating the explicit dynamics of the ferrioxalate
photolysis, it can provide a qualitative picture of the key features of the
LMCT potential energy surface (PES) that explain in clearer detail how an
immediate post-excitation Fe–O bond dissociation is then followed by a low-
barrier avoided crossing into a dissociative state in which the C–C bond and
then the other Fe–O bonds rupture. These findings are consistent with the
experimentally-supported prompt LMCT picture while also revealing features
that are not visible in experiments.

Combining experiment and theoretical simulation, Ogi and coworkers 16
attributed the LMCT state after the photoexcitation to a transition from an
oxalate $\sigma-$bonding to a singly-occupied Fe 3$d$ orbital (see Figure 9 in
Ref. 16). In the present study, we start with a (7e,6o) active space including
the same ligand orbital as Ogi et al. and five Fe 3$d$ orbitals (see Figure S1
of the SI). As we will discuss, post-excitation orbital relaxations lead to a
modified character for this ligand orbital, which in turn changes the optimal
excited state geometry. We have verified that the same holds in larger active
spaces that we discuss below, but for simplicity’s sake we will begin with
this (7e,6o) minimal active space.

Let us first consider the LMCT state at the ground state (GS) equilibrium
geometry. In the top panel of Figure 1, we plot the change of LMCT energy
during SS-CASSCF(7e,6o) optimization with and without $W\Gamma$ selection.
While SS-CASSCF without targeting exhibits a strong oscillation after several
ten iterations and cannot converge, SS-CASSCF with the $W\Gamma$ selection
converges to the excitation energy of 4.80 eV, which is quite consistent with
the excitation wavelength ($\sim$266/267 nm) observed in experiment 13. To
check whether $W\Gamma$-CASSCF converges to the right state, we plot the
change in the dipole moment of the targeted state during optimization in the
middle panel. All three dipole moment components do not change significantly,
indicating that the converged state is the targeted state. As in the bottom
panel, we can see that the shape of the ligand’s singly-occupied molecular
orbital (SOMO) that loses an electron to an iron $3d$ orbital gradually
changes during the SS-CASSCF iteration. While it spreads over the oxalate for
the GS, it is dominated by the oxygen lone-pair orbital for the LMCT state.
Because of the significant difference between the orbital shape of the ground
and LMCT states, SA-CASSCF may be inadequate.

Figure 1: Upper panel: the excitation energy (relative the GS energy,
$-64956.96$ eV) of the first LMCT state during SS-CASSCF optimization with and
without state targeting using $W\Gamma$ criterion. Midle panel: the change of
dipole moment, $\vec{\mu}=\\{\mu_{x},\mu_{y},\mu_{z}\\}$, of the LMCT state
during targeted SS-CASSCF. Lower panel: Two ligand’s orbitals of LMCT state at
different stages of SS-CASSCF optimization.

The question is whether the oxalate anion completely dissociates from the Fe
center or remains bound to it after the photoexcitation. To answer this
question, we fully relax the geometry of the LMCT state using $W\Gamma$-CASSCF
and TD-DFT. Geometries and important parameters are presented in Figure 2 and
Tabel 1, and Cartesian coordinates are given in the SI. Both methods predict
that the LMCT state has a bound minimum, and so we refer to this excited state
as b-LMCT. Interestingly, however, the two methods have qualitatively
different minimum geometries, with TD-DFT showing the C1–C2, Fe–O1, and Fe–O3
bonds all elongated but intact, and $W\Gamma$-CASSCF showing cleavage of the
Fe–O3 bond, slight elongation of Fe–O1, and essentially no change to C1–C2.
TD-DFT’s C1–C2 elongation agrees with the picture presented by Ogi et al, 16
but the reasons behind these geometric differences, to which we now turn,
casts doubt on the TD-DFT prediction.

To understand why the minimum geometries differ so greatly, it is crucial to
look at both the donor hole orbital (b-LMCT’s ligand SOMO) and the relaxation,
or artificial lack thereof, of the other ligand orbitals. Figure 2 shows that
$W\Gamma$-CASSCF’s hole orbital is localized on an oxygen lone pair, while TD-
DFT’s is spread across the ligand in an orbital with substantial $\sigma$
bonding character (hence the weakened C–C bond). While localizing the hole may
at first glance seem less stable than spreading it out, the lower panel of
Figure 1 reveals that localization is stabilized by strong relaxations in the
shape of a key doubly occupied orbital. Indeed, even at the GS geometry, these
relaxations allow the hole to localize without creating undue concentrations
of charge on either end of the ligand. This localization affects the geometry
by creating different environments for O1 and O3, evidently making it
preferable for O3 to elongate its bond to Fe to the point of cleavage. In TD-
DFT, in contrast, the adiabatic approximation introduces an artifact in which
doubly occupied orbitals do not relax after the excitation. 31, 32 It thus
cannot stabilize a localized hole and instead places it in the delocalized,
partially-$\sigma$-bonding orbital. As seen in Figure 1, $W\Gamma$-CASSCF
initially does the same, but, after state-specific orbital optimization,
adopts a relaxation that stabilizes the localized hole. Given that TD-DFT’s
lack of localization seems driven by an artifact of its approximations, we
find the picture painted by $W\Gamma$-CASSCF to be more persuasive.

We report the vertical and adiabatic excitation energies from
$W\Gamma-$CASSCF(7e,6o) and TD-DFT/B3LYP in Table 2. We also consider the
effect of solvent on these energy levels using the COSMO solvation model with
the water dielectric constant ($\epsilon=80$). We used geometries optimized
without solvent for adiabatic excitation energies with solvent. We can see
that TD-DFT excitation energies are lower than $W\Gamma-$CASSCF. Although the
solvent generally reduces the excitation energies, its effect on the adiabatic
excitation energies of $W\Gamma-$CASSCF is negligible.

Figure 2: Geometries of b-LMCT equilibrium optimized using $W\Gamma$-CASSCF(7e,6o) (left) and TD-DFT (right). See Table 1 for important parameters. The shapes of $W\Gamma-$CASSCF oxalate’s SOMO and TD-DFT oxalate’s natural orbital are also presented. Table 1:  Some important parameters of the GS geometry and b-LMCT equilibrium geometries optimized using $W\Gamma-$CASSCF and TD-DFT. Parameters | GS | b-LMCT
---|---|---
CAS(7e,6o) | CAS(9e,13o) | TD-DFT
Fe–O1 (Å) | 2.04 | 2.21 | 2.19 | 2.28
Fe–O3 (Å) | 2.04 | 4.19 | 4.10 | 2.28
C1–C2 (Å) | 1.57 | 1.56 | 1.58 | 1.76
$\alpha$(O1,C1,C2,O3) (∘) | 179.83 | 46.14 | 45.62 | 164.50
$\alpha$(O1,C1,C2,O4) (∘) | 0.14 | 141.16 | 142.54 | 15.53
Table 2:  Energies in eV for $W\Gamma-$CASSCF(7e,6o) and TD-DFT/B3LYP with and without solvent (water). See text for a description of the TS barrier calculation. Method | GS energy | Excitation energy | TS barrier
---|---|---|---
Vertical | Adiabatic
$W\Gamma-$CASSCF(7e,6o) |  |  |  |
Without solvent |  $-$64956.96 | 4.80 | 2.58 | 0.31
With solvent |  $-$64971.02 | 3.69 | 2.56 | 0.64
TD-DFT/B3LYP |  |  |  |
Without solvent |  $-$65132.99 | 2.55 | 0.64 |
With solvent |  $-$65147.99 | 2.16 | 0.30 |
Figure 3: Converged natural orbitals and occupations from the
$W\Gamma$-CASSCF(9e,13o) b-LMCT geometry optimization.

To check whether our conclusions are biased by a too-small active space, we
have also employed the expanded (9e,13o) active space shown in Figure 3. This
expansion does two things. First, by including the ligand’s $\sigma$ bonding
and antibonding orbitals, it ensures that CASSCF has the flexibility to break
the C1-C2 bond if it wants to. Second, the inclusion of the Fe 4d orbitals
guards against the double d shell effect, in which 4d orbitals can play a
larger-than-expected role in transition metal correlation effects (indeed,
Figure 3 shows one of the 4d having an occupation similar to the valence
antibonding orbital). We felt this effect was important to be cautious of in
this LMCT excitation, as the two states involved have differing numbers of d
electrons and so will have differing amounts of d electron correlation. As
seen in Figure 3 and Table 1, neither of these extensions made meaningful
changes to the b-LMCT state geometry or the shape of the donor orbital, giving
us additional confidence in our conclusions above.

We note that our finding of the Fe–O bond cleavage due to intra-molecular CT
contrasts with some recent works. Combining experimental spectroscopy and DFT
calculations, Chen et al.11, 12 suggested that the primary reaction is Fe–O
cleavage, but without intra-molecular CT. Although Ogi et al.16 provided
experimental evidence that the intra-molecular LMCT initiates the photolysis
of ferrioxalate, those authors proposed a different primary reaction where the
C1–C2 bond weakens and subsequently dissociates, which is consistent with our
present TD-DFT results depicted in Table 1 and Figure 2. Recently, O’Neil and
coworkers 18 reported an ultrafast time-resolved X-ray absorption spectroscopy
measurement that showed evidence of the elongation of the Fe–O bond. However,
the change in the Fe–O bond predicted by those authors is only $0.14\pm
0.06$Å, which is comparable to our prediction for Fe–O1 but much smaller than
our prediction for Fe–O3.

Figure 4: Potential energy curves showing the $W\Gamma-$CASSCF(7e,6o) energy
of the d-LMCT state after relaxing the geometry with C1–C2 (left panel) and
Fe–O1 (right panel) bond distances held fixed. The Fe–O1 bond is constrained
at 2.2Å for the C1–C2 curve, whereas the C1–C2 bond is constrained at 1.56Å
for the Fe–O1 curve. The leftmost points of both curves correspond to b-LMCT
equilibrium. Ligand’s SOMO is dominated by C1–C2 $\sigma-$bonding. Energies
are relative to the b-LMCT minimum energy ($-64954.38$ eV), indicated by the
horizontal line.

The presence of a bound minimum on this b-LMCT state’s PES implies that it has
at least a small barrier for the photolysis process, leading to the question
of which bond elongation, Fe–O1 or C1–C2 (see Figure 2 for numbering), is more
likely to occur first during the photolysis process. If, upon Fe–O1 or C1–C2
stretching, a higher state with dissociative character mixed with the b-LMCT
in an avoided crossing, a low barrier to dissociation would be possible. To
check whether such a dissociative state is present, we located higher-lying
LMCT states at the b-LMCT minimum geometry and performed unconstrained
$W\Gamma-$CASSCF geometry optimizations on them. These also employed (7e,6o)
active spaces as for b-LMCT, but each had a different ligand donor orbital.
One such optimization was indeed dissociative. The dissociative state (d-LMCT)
appeared to initially favor C1–C2 stretching, presumably due to the $\sigma$
character of its donor orbital (see inset images in Figure 4). We note that
initial C1–C2 elongation is consistent with the recent suggestion of Mangiante
et al.17, who combined pump/probe mid-infrared transient absorption
spectroscopy and theoretical simulations of a single C2O${}_{4}^{\bullet-}$
dissociation (see Figure 5 in Ref. 17). To clarify that d-LMCT indeed prefers
initial C1–C2 stretching instead of Fe-O1 stretching, we plot in Figure 4
potential energy curves during geometry optimizations of this state in which
either the C1–C2 or Fe–O1 bond distance is held fixed. Specifically, we
constrained one optimization to have $d_{\text{Fe--O1}}=2.21$Å and the other
to have $d_{\text{C1--C2}}=1.56$Å. The d-LMCT state showed dissociative
character under both of these constraints, but the rate at which bond
elongation lowered the energy during the initial bond stretching was much
greater when stretching C1–C2 than when stretching Fe–O1, suggesting that a
molecule placed on the d-LMCT surface at the b-LMCT minimum would see its
C1–C2 bond elongate first. This picture is consistent with recent experiments
20, 22, 21, 19 showing that a CO2 molecule is released first, leaving behind
[Fe(ox)2(CO2)]3-. However, the photochemical process initiated by excitation
to b-LMCT does not start in the dissociative d-LMCT state, so we must now turn
to a more careful analysis of how easy it is for the state character to
switch.

Figure 5: Top left panel: SA(3)-CASSCF(11e,8o) state energies during a geometry interpolation between the b-LMCT minimum ($x=0.0$) and the minimum geometry for the d-LMCT state when the C1–C2 distance is constrained to $R_{\mbox{\small}{CC}}=2.2$ Å ($x=1.0$). Three states include b-LMCT, d-LMCT states, and one LMCT state between the two. Energies are relative to the lowest-state energy of SA(3)-CASSCF(11e,8o) at the b-LMCT equilibrium ($-64954.22$ eV). The donor ligand’s natural orbitals for different states are also depicted. Top right panel: transition-state (TS) geometry optimized using SA(3)-CASSCF(11e,8o). Bottom panel: ligand’s SOMO of the lowest state in SA(3)-CASSCF (the blue solid curve) at different interpolation points $x$. Table 3:  Important geometrical parameters of transition-state (TS) geometry in comparison to the b-LMCT minimum. Parameters | b-LMCT | TS
---|---|---
Fe–O1 (Å) | 2.21 | 2.32
C1–C2 (Å) | 1.56 | 1.68
$\alpha$(O1,C1,C2,O3) (∘) | 46.14 | 99.9
$\alpha$(O1,C1,C2,O4) (∘) | 141.16 | 82.43

If b-LMCT were to smoothly convert into d-LMCT through an avoided crossing,
the energy barrier associated with this crossing would determine how
accessible this this pathway was. To determine if such an avoided crossing
exists at all and to get a good guess for a corresponding transition state
(TS) geometry, we have performed SA-CASSCF calculations along a linear
interpolation between the Cartesian geometries of the b-LMCT minimum and the
2.2Å point from the left panel of Figure 4 (see SI for explicit coordinates
for these two geometries). Since we found that there is in fact another LMCT
state in between b-LMCT and d-LMCT, we have included all three of these LMCT
states in the SA in equal weights. As each state corresponds to excitation
from a different ligand donor orbital (one lone pair, one $\sigma$, and one
$\pi$ as shown in the insets to Figure 5), we employ an (11e,8o) active space
containing the Fe 3d orbitals and these three ligand orbitals (see Figure S2
in the SI). As seen in Figure 5, the lower adiabat in this interpolation does
indeed show avoided crossing behavior between b-LMCT and d-LMCT: its donor
orbital has clear lone pair character at the b-LMCT geometry (x = 0), which
changes smoothly into clear $\sigma$ bonding character on the other side of
the barrier (x = 1).

Figure 6: The change of energy, geometry, and ligand’s SOMO during the full
$W\Gamma-$CASSCF(7e,6o) geometry optimization without constraints starting
from the TS geometry (cycle = 1). The energy is relative to the energy at the
cycle 100 ($-64958.60$ eV).

Using the geometry at the avoided crossing in the interpolation as an initial
guess, we searched for TS geometry using SA(3)-CASSCF(8e,11o). The TS geometry
is depicted in the right panel of Figure 5, and some critical geometrical
parameters are summarized in Table 3. While both C1–C2 and Fe–O1 bonds
elongated slightly compared to the b-LMCT minimum, the oxalate anion is
twisted, preparing for the dissociation. As it is not balanced to compare SA-
CASSCF and SS-CASSCF energies, we then evaluated the $W\Gamma$-CASSCF(7e,6o)
energy at the TS and compared it to the corresponding energy at the b-LMCT
minimum. The results in Table 2 show that this TS barrier is small relative to
the energy of the initial photoexcitation and that the same is true under
implicit solvent. Such a small barrier relative to the photoexcitation energy
suggests that the CO2 detachment can proceed rapidly following
photoexcitation, as has been measured in recent experiments. 22, 19, 20

In Figure 6, we plot the change of energy, geometry, and the ligand SOMO
during a full relaxation of the geometry on $W\Gamma$-CASSCF’s representation
of the the d-LMCT surface starting from the TS geometry. Although both the
C1–C2 and Fe–O1 bonds eventually rupture, one’s elongation precedes the other.
For example, in geometry optimization cycle 4, C1–C2 has elongated by 14%,
whereas Fe–O1 has stretched by just 2%, again implying that C1–C2 cleavage
somewhat precedes Fe–O1 cleavage, as was suggested by our constrained
optimizations in Figure 4. That said, Figure 6 also shows that, once the
avoided crossing barrier has been crossed, the dissociation of these two bonds
can occur in a single, concerted process.

We have applied the recently developed $W\Gamma$-CASSCF method to investigate
the LMCT states governing ferrioxalate’s photolysis while fully accounting for
post-excitation orbital relaxations, overcoming a key shortcoming of many
existing theoretical approaches. Many recent experiments have provided
evidence that LMCT states play the pivotal role in the photolysis of
ferrioxalate, which makes the lack of proper excited-state-specific orbital
relaxation in TD-DFT and SA-CASSCF a severe concern. The present study shows
that CASSCF geometry optimization on the LMCT state is possible with
$W\Gamma$-CASSCF despite SA-CASSCF’s difficulties in this regard. Although we
have not simulated the explicit dynamics of ferrioxalate photolysis, our
results do clarify key qualitative features on the LMCT potential energy
surface.

Figure 7: Primary processes in the ferrioxalate photolysis predicted using
$W\Gamma$-CASSCF method. The singly-occupied oxalate orbital is shown in each
image, which helps clarify the difference between the b-LMCT and d-LMCT
characters on either side of the avoided crossing.

In general, our theoretical results for ferrioxalate are consistent with the
prompt dissociation the picture suggested by multiple experimental studies.
Based on our findings, a likely photolysis pathway is summarized in Figure 7.
We see that two LMCT states contribute to the photolysis of ferrioxalate: the
bound b-LMCT and the dissociative d-LMCT states, with the low-energy pathway
from the bound excited state minimum traversing a low-barrier avoided crossing
in which the adiabatic LMCT state changes character between b-LMCT and d-LMCT.
Our results also indicate that although one Fe–O bond is likely to rupture
before the C–C bond, the other Fe–O bond then follows the C–C bond in the
latter stage of dissociation. After the photoexcitation of $\sim$4.8 eV, the
b-LMCT state is responsible for the first Fe–O bond dissociation. It has a
bound geometry minimum at $d_{\text{FeO1}}\simeq 2.2$Å and
$d_{\text{FeO3}}\simeq 4.1$Å. This finding helps clarify conflicting reports
about whether a Fe–O bond or the C–C bond cleaves first during
photoexcitation. The detachment of CO2 can then proceed via an avoided
crossing with a barrier much lower than the photoexcitation energy, through
which the b-LMCT state converts in character to the d-LMCT state, upon which
the C–C $\sigma-$bond and the second Fe–O bond then dissociate in an
essentially concerted fashion with the C–C bond leading the way.

## Method

In our state-specific approach (SS-CASSCF), we retain the two-step
optimization approach in which the orbitals and CI variables are optimized
separately. The CI variables are chosen at each macro iteration as the CASCI
root most similar to the desired state by measuring the criterion 27

$Q_{W\Gamma}=\frac{\braket{\Psi}{(\omega-\hat{H})^{2}}{\Psi}}{\braket{\Psi}{\Psi}}+\frac{\hskip
2.84526pt||\hskip 1.42262pt\Gamma_{t}-\Gamma\hskip 1.42262pt||\hskip
2.84526pt}{n_{{}_{\mathrm{CAS}}}}.$ (2)

Here $\omega$ is a target energy that is dynamically updated at each
iteration, $\Gamma$ is our state’s one-body density matrix, $\Gamma_{t}$ is a
target density matrix (taken either from an initial CASCI or the previous
iteration), and $n_{CAS}$ is the number of active orbitals. The idea is that,
after finding the spectrum of CASCI roots, we select the root that minimizes
$Q_{W\Gamma}$ and then perform an orbital optimization step to make that
root’s energy stationary with respect to orbital changes. Note that the final
energy depends only on the wave function at the stationary energy point,
independent of $\omega$ and $\Gamma_{t}$.

Once the energy is stationary, there is no need for response calculations, as
full energy stationarity eliminates the wavefunction response terms from the
gradient expression. With our analytic gradients in hand, we perform our
geometry relaxations using the geomopt module within pySCF, 33 which
interfaces with geomeTRIC 34 and PyBerny 35 for constrained and unconstrained
geometry optimizations, respectively. SA-CASSCF calculations were performed
using PySCF and Molpro 36, and TD-DFT calculations were performed using
Orca37. The def2-SVP basis set 38 and the spin state sextet ($S=5/2$) were
used for all calculations.

## Acknowledgements

This work was supported by the Early Career Research Program of the Office of
Science, Office of Basic Energy Sciences, the U.S. Department of Energy, Grant
No. DE-SC0017869. Calculations were performed using the Berkeley Research
Computing Savio cluster.

## Supporting information

An overview of recent experimental works. Orbitals selected in CAS(7e,6o) and
CAS(11e,8o). Cartesian coordinates of ground state, b-LMCT minimum optimized
by SS-CASSCF and TD-DFT, stretched CC (2.2 Å) equilibrium and TS geometries.

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