Real-time (RT) electronic structure methods provide a natural framework for describing light–matter interactions in arbitrary time-dependent electromagnetic fields (EMF). Optically induced excited state transitions are of particular interest, which require tuned EMF to drive population transfer to and from the specific state(s) of interest. Intersystem crossing, or spin-flip, may be driven through shaped EMF or laser pulses. These transitions can result in long-lived “spin-trapped” excited states, which are especially useful for materials requiring charge separation or protracted excited state lifetimes. Time-dependent configuration interaction (TDCI) is unique among RT methods in that it may be implemented in a basis of eigenstates, allowing for rapid propagation of the time-dependent Schrödinger equation. The recent spin–orbit TDCI (TD-SOCI) enables a real-time description of spin-flip dynamics in an arbitrary EMF and, therefore, provides an ideal framework for rational pulse design. The present study explores the mechanism of multiple spin-flip pathways for a model transition metal complex, FeCO, using shaped pulses designed to drive controlled intersystem crossing and charge transfer. These results show that extremely tunable excited state dynamics can be achieved by considering the dipole transition matrix elements between the states of interest.

## I. INTRODUCTION

Transition metal complexes are paramount to a number of electronic devices and light-harvesting materials. Charge transfer (CT) states are key in enabling the unique properties of these systems. Chemical potential, used to perform reactions and produce current, can be generated by optically driving a separation of charge.^{1} Of particular interest are metal–ligand and ligand–metal charge transfer states (MLCT and LMCT, respectively), which utilize electronic transitions between metal- and ligand-centered orbitals to spatially separate electron density. Many CT pathways involve a transition from a low-spin to a high-spin state (or vice versa), referred to as intersystem crossing (ISC) or spin-flip (SF).^{2–4} These spin-forbidden transitions result in excited states with protracted lifetimes, allowing chemical work to be done using the separated charge. Such optically driven molecular processes, or quantum control, provide a path to fine-tuned electron dynamics. These processes are essential to photochemistry,^{5} photocatalysis,^{6} laser-controlled magnetic switching,^{7,8} and spin dynamics,^{9–11} among others. A review of the rich history of quantum control can be found in Ref. 12.

Within the dipole approximation, predicting the appearance of CT states produced by field-induced ISC or SF is made possible by a treatment of spin–orbit coupling (SOC).^{13,14} A number of time-independent methods for describing ISC in transition metals exist, with relativistic extensions to density functional theory (DFT),^{15,16} complete active space perturbation theory (CASPT2), or self-consistent field (CASSCF) methods^{17} comprising the most common approaches. The spin-flip approach^{18} extends coupled cluster methods to treatments of ISC as well. Each method has its disadvantages—the accuracy of DFT is not systematically improvable, CAS methods depend strongly on the active space selected, and coupled cluster solutions are often prohibitively expensive for large systems. Further still, investigation of the nuanced dynamics of ultrafast SF and ISC processes requires time-dependent solutions.

Time-dependent approaches implemented in the frequency domain, such as time-dependent DFT (TD-DFT) or response theory, generally necessitate certain restrictions: Specifically, fields are limited to weak perturbations and narrow frequency domains.^{19–22} Explicitly time-dependent approaches, on the other hand, do not suffer from these limitations and also allow complete control over time-domain properties, such as the form of a perturbing electric field.^{23,24} So-called real-time extensions to the above methods include real-time TD-DFT (RT-TDDFT),^{25–29} real-time coupled cluster [both the “classic” RTCC^{30–36} and equation-of-motion (RT-EOM-CC)^{37–41} variants], and time-dependent Hartree–Fock (TDHF)^{42} and configuration interaction (TDCI)^{43} approaches. Multiconfigurational extensions have also been explored.^{44–46} Notably, RT-TDDFT has been extended to relativistic systems^{47,48} as has TDCI through the recent time-dependent spin–orbit configuration interaction method (TD-SOCI).^{49}

Of course, these methods inherit the challenges associated with their time-independent counterparts.^{23,24,50,51} Computational expense is especially prohibitive for wave function-based methods. To combat the increased computational cost, real-time coupled cluster (RTCC) and a specific form of time-dependent complete active space CI (TD-CASCI) have recently been implemented using graphics processing units (GPUs).^{52,53} While GPU acceleration significantly improves performance, time propagation is still the rate-limiting step in these approaches. TDCI may be implemented in a number of ways; however, the more common variant expands the wave function in terms of stationary states, with the time-dependence carried by the coefficients. This approach is in contrast to other real-time electronic structure algorithms, such as RTCC and RT-TDDFT, which propagate the ground state wave function directly. Propagating in an eigenstate basis avoids the costly re-evaluation of the CI Hamiltonian at each time step, in exchange for pre-computing the CI eigenstates, in turn allowing rapid propagation of the time-dependent wave function in the presence of a perturbing field. As such, the time-independent calculation becomes the rate-limiting step. Furthermore, transition dipole matrices can be pre-computed, allowing for rational design of excited state pathways to CT and SF states as well as symmetry breaking and restoration.^{54–57} Finally, the TDCI method is readily modified to include additional time-independent effects—such as the Breit–Pauli SOC Hamiltonian^{58} to account for relativistic effects, resulting in the aforementioned TD-SOCI.^{49} Through a combination of SOC, pulse shaping, and multi-state hopping, this work demonstrates the efficacy of controlled CT and ISC mechanisms in a model transition metal complex by successive optical excitations.

An alternative TDCI scheme exists that has been applied to optical excitations, such as that used in TD-CASCI.^{53} In this alternative formulation, only the ground state CI wave function is required. Time propagation is instead carried out by recomputing the roots of the Hamiltonian at every time step. This propagation can be done by recasting the Hamiltonian to symplectic form, then the time-dependent CI coefficients are determined in a process analogous to the iterative Davidson method, incurring additional cost at each step. Removing the necessity to solve multiple roots of the time-independent Hamiltonian reduces the initial cost of the calculation, but greatly increases the cost of propagation, making this approach well-suited for systems for which calculating many low-lying excited states is not computationally tractable. Since the goal of the current study is to examine many different pulses and simulation lengths for a small system, the more common TDCI approach—that is, pre-computing the eigenstates of the time-independent Hamiltonian—is utilized.

Herein, the theory of TDCI is presented, including the spin–orbit TDCI (TD-SOCI) method developed by Ulusoy and Wilson.^{49} Different pathways for achieving ISC, including shaped pulses and multi-state hopping, are illustrated by subjecting a model transition metal complex, FeCO, to different electric field pulses, with the goal of targeted excitations into states of varying CT and SF character. FeCO has long been of interest in the astrophysics community for its potential to be detected in circumstellar gasses,^{59} and it has also been reported to be a product of photodissociation of the “iconic”^{60} transition metal complex Fe(CO)_{5}.^{61} FeCO presents itself as a challenging case for electronic structure theory, with many studies incorrectly predicting the energetic ordering of the *X*^{3}Σ ground and ^{5}Σ first excited state.^{59} Ulusoy and Wilson^{49} previously showed that ISC between the *X*^{3}Σ ground state and a ^{5}Π excited state may be driven by an electric field pulse, suggesting an opportunity to dynamically control CT and SF behavior. The current study shows that, given careful consideration of the coupling strength between states, extremely tunable ISC processes can be realized.

## II. THEORY

*t*)⟩ is expanded using a linear combination of

*n*CI stationary states, i.e., eigenfunctions of the time-independent Hamiltonian |Ψ

_{n}⟩, given by

*a priori*using any manner of CI. Previously, many different flavors of time-independent CI have been used as the basis for TDCI. These include configuration interaction singles (TD-CIS),

^{43}singles with full (TD-CISD)

^{62}or approximate [TD-CIS(D)]

^{63}doubles, and static polarizabilities of small molecules determined with up to full triple and quadruple excitations (TD-CISDTQ).

^{64}Scalar relativistic and spin–orbit corrections have also been considered (TD-SOCI),

^{49}and the effects of ionization and dissipation can also be included, see, e.g., Ref. 65 and references therein.

*O*(Δ

*t*

^{2}) error, where Δ

*t*is the discretized integration step. Assuming a small enough Δ

*t*, the integral can now be rewritten as

^{66,67}$H\u0302$ is diagonal in the basis of time-independent eigenfunctions, and exponentiating the sparse perturbation matrix can be avoided by transforming into a basis in which $\mu \u0302$ is diagonal with eigenvalues

*ν*and eigenvectors

*U*,

*a priori*generation of time-independent states.

^{68}Spin–orbit coupling is included using the Breit–Pauli operator,

^{58}

^{,}

*i*,

*j*, … and nuclei

*A*.

*Z*

_{A}is the atomic number of nucleus

*A*, and

*r*

_{iA}is the distance between the center of orbital

*i*and nucleus

*A*. As reported in a previous study,

^{49}this relativistic treatment is sufficient to reproduce the correct electronic energy ordering of the low- and high-spin ground states, with energetics and electric dipole moments in good agreement with literature results.

^{59}To include this effect in the propagation, $H\u0302BP$ is added to the ground state Hamiltonian, which is then re-diagonalized to obtain spin-mixed states. By propagating in the spin-mixed basis, the TD-SOCI procedure requires only this one additional step before propagation. Populations can be rotated back into the original spin-pure eigenstate basis to obtain the admixture of states present at any time step.

*α*can be computed in the spin-free or spin-mixed eigenstate basis as

*m*and

*n*is proportional to the magnitude of the transition dipole matrix element between the two states. Therefore, states with no dipole coupling will be inaccessible by optical excitation with an electric field (within the dipole approximation). In the rotating wave approximation (RWA), the optimal pulse parameters for a two-level resonant excitation may be obtained using

*f*(

*t*), pulse area

*σ*, and maximum pulse height

*ϵ*. The optimal excitation is achieved (for a two-state system) when the area under the pulse envelope,

*σ*, is equal to

*π*. This is commonly known as the

*π*-pulse condition.

^{67}Using this relationship, states that couple weakly to the ground state may be accessible in theory, but significant population transfer would require relatively intense fields (in excess of 1 × 10

^{15}W cm

^{−2}). Realistically, this may result in bond breaking or ionization, while weaker fields would necessitate timescales at which excited state decay, nuclear motion, and tunnel ionization are no longer negligible.

^{67}

An alternative to resonant excitation to a given state is multi-state hopping: successive excitation and deexcitation pathways to more strongly coupled states, which also couple strongly to the desired state.^{69,70} This allows for an increased final excited state population at a given field strength and, in some cases, reduced timescales. This is especially helpful for spin-flip processes, which often occur at protracted timescales compared to relaxation effects.^{71} Fortunately, the TDCI procedure is uniquely optimized for this task—since the excited state wave functions are available, the transition dipole matrix may be computed before any dynamics are performed. Additionally, the propagation is significantly cheaper computationally than other real-time electronic structure methods. This makes TDCI the ideal candidate for pulse optimization. The present study focuses on rational pulse design through the *π*-pulse condition coupled with frequency chirping and multi-state hopping, ultimately achieving selective CT and ISC dynamics in a model transition metal complex.

## III. COMPUTATIONAL DETAILS

Excited states of FeCO were computed using spin–orbit CISD in an active space of 18 electrons in 18 orbitals, SO-CISD(18,18), as it is implemented in the GAMESS-US package version 2018 (R1).^{72} Orbitals were computed using multiconfigurational SCF [MC-SCF(14,18)] beginning from a restricted open-shell Hartree–Fock reference wave function for the triplet state at the linear experimental equilibrium geometry.^{73} A total of 20 singlet, triplet, and quintet spin-pure states were computed, and all 180 resulting spin-coupled states are included in the TD-SOCI propagations. Dunning’s cc-pVTZ basis set^{74,75} was used for all atoms.

*ω*, maximum field strength $\u03f5\u20d7\alpha =0.01$ a.u. or 3.5 × 10

^{12}W cm

^{−2}, polarization direction

*α*, center

*t*

_{c}, and field envelope

*f*(

*t*).

*ω*is set equal to the energy difference between states of interest unless otherwise noted. The envelope is a cos

^{2}pulse given by

*σ*

_{c}being determined by the

*π*-pulse condition, Eq. (10), unless otherwise stated. The time step Δ

*t*remains fixed at 0.05 a.u. or about 1.21 × 10

^{−3}fs. Additionally, “chirped” pulses are used to facilitate certain state transitions by replacing

*ω*with

*ω*′ =

*ω*+

*β*/2(

*t*−

*t*

_{c}) in Eq. (11), where the sign of the chirp parameter

*β*determines whether the frequency increases or decreases in time. Fields are polarized in the direction that induces the strongest state coupling. While isotropic fields are more realistic for free gas- or solution-phase molecules, these details are excluded for simplicity in demonstrating the capabilities of pulse-optimized TD-SOCI on a prototype system.

The frozen-nuclei approximation is assumed throughout the study. While the vibrational motion of FeCO has been experimentally shown to occur within the time scales of the following simulations,^{76} nuclear motion is not expected to play a significant role in the low-lying electronic excited states, which exhibit minimal changes in geometry.^{59} Furthermore, modern iron-based chromophores are designed to target ultrafast ISC by careful consideration of ligand field splitting, where the main concern is short ($<$100 fs) excited state lifetimes^{71} rather than nuclear reorganization. Finally, while the choice of laser intensity has been fixed in the present study for simplicity, the ISC time scales presented here could be reduced in practice by increasing the maximum field strength by up to an order of magnitude before reaching the strong-field regime.^{77} As the focus of the present study is to demonstrate the general efficacy of the TD-SOCI approach for tuning excited state dynamics in a simple model compound, these topics are beyond the scope of the current manuscript.

All propagations were carried out using an in-house python module, rtci. This module has been developed internally for TDCI propagations agnostic to the time-independent states used or the electronic structure package used to obtain them. Transition dipole moments and electronic energies (in the CI eigenstate basis) are the only input from external codes. Multiple laser pulses, including Gaussian and cos^{2} envelope functions for sin and cos carrier pulses, linear and quadratic pulse chirping, as well as delta pulses for broadband excitations (used for, e.g., absorption spectra) are available. rtci depends on the NumPy^{78} and SciPy^{79} python libraries. The package is available upon request.

## IV. RESULTS

Here, we present the results for FeCO, a prototype transition metal complex with multiple bound states with spin-flip character. The energies of all electronically coupled excited states (i.e., nonzero transition electric dipole moment from the triplet ground state) are plotted in Fig. 1 against their *static* z-direction dipole moments. The magnitude of the ground state dipole coupling, $|\u3008\Psi 0|\mu \u0302\alpha |\Psi n\u3009|$, is represented by the relative size of the circles. The strongest ground state coupling is to states above the Koopmans theorem ionization potential, 0.078 E_{h}, denoted by a dashed line. A total of 66 electronic states lie below this threshold; however, only 21 have nonzero electric dipole coupling to the ground state. Of these, 16 states are predominantly quintet in character. This suggests that many spin-flips from the triplet ground state are possible. Three quintets are labeled in Fig. 1: two ^{5}Π states and a ^{5}Σ state. Laser-driven excitations into these three electronic states from the triplet ground state, *X*^{3}Σ_{0}, are considered.

The dominant electron configurations for the three target excited states are shown in Fig. 2, with paths A, B, and C each leading to a different target state. These configurations constitute three possible triplet–quintet spin-flips—two single-excitations (A and C), and one double-excitation (B). The relative energy ordering, state labels (for the determinant with the largest contribution to the spin-mixed state), excitation energies (in *E*_{h}), and static z-direction electric dipole moments (in a.u.) are given for these and other relevant states in Table I. State labels containing *a*, *b*, and *c* are for convenience only and do not denote symmetry labeling.

State . | Label . | ΔE/E_{h}
. | μ_{z}/a.u.
. |
---|---|---|---|

0 | X^{3}Σ_{0} | 0.000 | −0.85 |

4 | a^{5}Σ_{1} | 0.002 | −0.56 |

5 | b^{5}Σ_{1} | 0.002 | −0.56 |

10 | a^{5}Π_{2} | 0.026 | 0.74 |

11 | b^{5}Π_{2} | 0.026 | 0.74 |

14 | a^{5}Π_{0} | 0.027 | 0.74 |

46 | b^{5}Π_{0} | 0.066 | −0.15 |

State . | Label . | ΔE/E_{h}
. | μ_{z}/a.u.
. |
---|---|---|---|

0 | X^{3}Σ_{0} | 0.000 | −0.85 |

4 | a^{5}Σ_{1} | 0.002 | −0.56 |

5 | b^{5}Σ_{1} | 0.002 | −0.56 |

10 | a^{5}Π_{2} | 0.026 | 0.74 |

11 | b^{5}Π_{2} | 0.026 | 0.74 |

14 | a^{5}Π_{0} | 0.027 | 0.74 |

46 | b^{5}Π_{0} | 0.066 | −0.15 |

### A. Path A: Spin-flip to state *a*^{5}Π_{0}

As reported previously,^{49} the strongest dipole coupling (0.081 in the z-direction) is for Path A, the (predominantly) quintet state *a*^{5}Π_{0}, which also has a static electric dipole moment of the opposite sign as the ground sate, signifying strong metal–ligand charge transfer character. This transition corresponds to one electron being promoted from a 4*π* MO to the 12*σ* MO, which is dipole-allowed. The results of applying a *π*-pulse with a cos^{2} envelope in the z-direction are shown in Fig. 3. The population of each bound state (computed as the square of the state coefficients, |*C*_{n}(*t*)|^{2}) is plotted against simulation time. The electric field strength vs time is also plotted on the right y-axis. Only states whose populations exceed 1% at any point during the simulation are shown. The sum of the populations of all states above the ionization threshold are also shown.

Here, the incomplete population inversion suggests there is a breakdown of the *π*-pulse condition due to deviations from the two-state RWA model. Despite this, since no significant population is lost to other states and ionization is not yet a concern at this time scale (as evidenced by the low population of unbound states), the excited state yield may be improved simply by adjusting the resonant frequency. Scaling the original *π*-pulse frequency by a factor of 0.98 produces near-complete excitation into the *a*^{5}Σ_{0} state within 200 fs (Fig. 4). This constitutes both spin-flip and dipole switching dynamics, with total inversion of the electric dipole moment. The relative ease of this transition can be attributed to the moderate dipole coupling between these states combined with the low density of (accessible) states within 0.05 *E*_{h} of the ground state.

### B. Path B: Spin-flip to state *b*^{5}Π_{0}

The low-lying *a*^{5}Π_{0} state is not the only accessible quintet state. There is a high-lying quintet state, *b*^{5}Π_{0}, with a z-direction ground state transition dipole strength magnitude of 0.023. Additionally, this state is several milliHartree removed from the next state with dipole coupling greater than 1 × 10^{−3}. This corresponds to Path B in Fig. 2. Unfortunately, the coupling for Path B suggests a *π*-pulse width of over 600 fs. This can be rationalized by observing that direct excitation from the ground state would require a two-electron excitation: either 4*π* to 12*σ* and 11*σ* to 1*δ*, or 4*π* to 1*δ* and 11*σ* to 12*σ*, the latter of which is dipole-forbidden. While a pulse can be shaped to realize this transition (see the supplementary material), there are several obstacles. Other than the neglected nuclear motion, at this timescale population transfer would compete with multi-photon ionization effects. Furthermore, if the excited state does not have a protracted lifetime, electronic decay would be expected to occur before significant population was reached. If population of the state is to be useful for charge transfer, for example, it must be populated within a shorter timeframe.

Alternatively, these excitations may be driven separately, beginning with 4*π* to 12*σ*, which corresponds to Path A. Through examination of the coupling between the *b*^{5}Π_{0} state to all other states, an alternative path becomes clear: The *b*^{5}Π_{0} state couples nearly five times stronger to the *a*^{5}Π_{0} state than to the *X*^{3}Σ_{0} ground state, with a transition dipole magnitude of 0.437. This means that, given a total transition from *X*^{3}Σ_{0} to *a*^{5}Π_{0}, subsequent excitation to *b*^{5}Π_{0} should take less than 50 fs, assuming the *π*-pulse condition. Figure 5 shows that, after applying two appropriate pulses, nearly 98% population transfer is achieved in under 250 fs. The sign of the electric dipole switches twice (Fig. 6), suggesting rapid charge oscillations within the molecule. The transition may be further accelerated by overlapping the second pulse with the first, such that transfer from *a*^{5}Π_{0} to *b*^{5}Π_{0} begins at 150 fs, before fully depleting the ground state, as shown in Fig. 7.

### C. Path C: Spin-flip to state *b*^{5}Σ_{1}

While these augmentations improve the likelihood of significant population of the *a*^{5}Σ_{0} state, there is still the potential issue of ionization when exciting to such a high-energy state. To avoid this, we may wish to populate a lower-energy quintet state. The *b*^{5}Σ_{1} state lies just 1.5 mE_{h} above the ground state—however, direct excitation from *X*^{3}Σ_{0} to *b*^{5}Σ_{1} (Path C in Fig. 2) is spin-forbidden, and hence, very slow and difficult to achieve (see the supplementary material). Fortunately, as before, this transition may be driven by coupling through the intermediate state *a*^{5}Π_{0}, with a transition dipole magnitude of 0.075 in the x-direction. Utilizing the same scaled z-polarized *π*-pulse for exciting into the *a*^{5}Π_{0} state, followed by an x-polarized pulse tuned to state *b*^{5}Σ_{1}, partial population of this state is achieved as shown in Fig. 8(a). In this case, we face a different challenge—coupling to another nearly degenerate *a*^{5}Σ_{1} state, as well as the population of two ^{5}Π_{2} states. The latter are likely due to two-photon effects, evidenced by their delayed population relative to the near-degenerate ^{5}Σ_{1} states. Additionally, the intermediate *a*^{5}Π_{0} state is not completely depleted over the duration of the second pulse.

The *a*^{5}Π_{0} to *b*^{5}Σ_{1} energy gap is roughly 0.0255 E_{h}. Meanwhile, the gap between the pairs of ^{5}Σ_{1} and ^{5}Π_{2} states is roughly 0.0246 E_{h}. Thus, a pulse resonant with ^{5}Π_{0} to ^{5}Σ_{1} will produce significant off-resonance excitation to the ^{5}Π_{2} states if the pulse is nonzero long enough to allow for two-photon excitations. Previously, it has been observed by Levine and co-workers^{53} that chirped pulses may be used to manipulate off-resonant excitations in TDCI simulations. Specifically, “up-chirped” pulses (which begin at a lower frequency and gradually increase) favor higher-energy state transfers, while “down-chirped” pulses produce the opposite result. Using this to our advantage, we may apply an up-chirped pulse with a chirping coefficient *β* = 1 × 10^{−6}, favoring the slightly higher *a*^{5}Π_{0} to *b*^{5}Σ_{1} gap. Figure 8(b) shows that, while this does succeed in suppressing the populations of the ^{5}Π_{2} states, the *b*^{5}Σ_{1} population suffers due to overshooting the resonant frequency. However, if the chirp were modified such that only the leading portion of the pulse is effected—ensuring that there is no discontinuity in the electric field—a balance can be struck between suppressing and populating the ^{5}Π_{2} and ^{5}Σ_{1} states, respectively. Figure 8(c) shows the results of a split-chirped pulse, where *β* switches from 1 × 10^{−6} to zero at the center. While some population is lost to the ^{5}Π_{2} states, significantly stronger transfer into the *b*^{5}Σ_{1} state is observed. This may be further augmented by widening the latter side of the pulse, as shown in Fig. 8(d); however, caution should be exercised in practice, as a longer pulse increases the chances of population loss due to multi-photon and ionization effects.

## V. CONCLUSIONS

Several possible mechanisms for spin-flip in a prototypical transition metal complex have been explored. Shaped pulses, as well as multi-state hopping, have been shown to enable highly tunable electron dynamics in systems with spin–orbit coupling. This is made possible by leveraging the TD-SOCI approach, which gives access to the electric dipole transition matrix with spin–orbit coupling *a priori*. Through the *π*-pulse condition, approximate timescales are readily computed, offering valuable insight into optimal excitation pathways and pulse parameters before performing dynamics simulations. Corrections for effects beyond the RWA can be accounted for using simple frequency modulation, and in several cases coupling through an intermediate state significantly improves the rate of excitation. More complex strategies are required for excitations in multi-state systems exhibiting strong off-resonant character; however, chirped and semi-chirped pulses prove to offset these effects. Once a viable route has been obtained, further pulse optimization through automated schemes, such as optimal control theory^{80} or stochastic pulse optimization,^{81} may further improve final state populations. These results show that, through a combination of pulse shaping and careful consideration of multi-state dipole coupling, optical excitations can be driven to deliver selective spin-flip and charge transfer dynamics. These insights enable faster design of materials with favorable excited state properties, such as MLCT and spin-trapped states, for charge separation and light-harvesting.

## SUPPLEMENTARY MATERIAL

x-, y-, and z-direction electric dipole transition matrix elements for relevant states and population dynamics for direct excitation from the ground state to the b5Π0 and b5Σ1 states.

## ACKNOWLEDGMENTS

This work was supported by the U.S. Department of Energy (DOE) Office of Science (Office of Basic Energy Science) under Grant No. DE-SC0017889. The authors gratefully acknowledge use of computational resources provided by the iCER computational facility at Michigan State University.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Benjamin G. Peyton**: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Writing – original draft (lead); Writing – review & editing (lead). **Zachary J. Stewart**: Investigation (supporting); Methodology (supporting); Writing – review & editing (equal). **Jared D. Weidman**: Investigation (supporting); Methodology (supporting); Writing – review & editing (equal). **Angela K. Wilson**: Conceptualization (equal); Funding acquisition (equal); Project administration (lead); Resources (lead); Supervision (lead); Writing – review & editing (equal).

## DATA AVAILABILITY

The code as well as time-independent and -dependent data that support these findings are available from the corresponding author upon reasonable request.

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