On the prospects of optical cycling in diatomic cations: effects of transition metals, spin–orbit couplings, and multiple bonds

Molecules with optical cycling centres (OCCs) are highly desirable in the context of fundamental studies as well as applications (e.g. quantum computing) because they can be effectively cooled to very low temperatures by repeated absorption and emission (hence, cycling). Charged species offer additional advantages for experimental control and manipulation. We present a systematic computational study of a series of diatomic radical-cations made of a d-block metal and a p-block ligand, that are isoelectronic (in their valence shell) to the successfully laser-cooled neutral molecules. Using high-level electronic structure methods, we characterise state and transition properties of low-lying electronic states and compute Franck–Condon factors. The computed branching ratios and radiative lifetimes reveal that the electronic transitions analogous to those successfully used in the laser cooling of neutral molecules are less than optimal in the cations. We propose alternative transitions suitable for optical cycling and highlight trends that could assist future designs of OCCs in charged or neutral molecules. GRAPHICAL ABSTRACT


Introduction
Optical cycling centre (OCC) is a chromophore moiety that supports electronic transitions closed for electronic and vibrational decay to other channels. It means that the excited state should decay primarily to the ground electronic state. The transition should also be closed at the vibrational level, such that the excited-state decay populates primarily the ground vibrational state. In this case, the system can be subjected to repeated absorption-emission cycles without exciting molecular CONTACT Anna I. Krylov krylov@usc.edu Department of Chemistry, University of Southern California, Los Angeles, CA 90089, USA This article has been corrected with minor changes. These changes do not impact the academic content of the article.
Supplemental data for this article can be accessed here. https://doi.org/10.1080/00268976.2022.2107582 vibrations. By tuning the photon energy to be slightly below the electronic transition energy, each cycle results in slowing the translational motion of the system. This is the essence of laser cooling. The lower-energy spectral features, such as rotational and fine structure, must also be considered, but it is the closure on the vibronic (vibrational and electronic) level that is the chief prerequisite for optical cycling. The excited electronic state must also be short lived (lifetime < 10-100 μs) to afford a rapid photon scattering rate necessary for effective laser cooling (this condition, however, is not critical for working with cations [1]). These requirements for molecules suitable for laser cooling are well known from earlier studies [1][2][3][4][5]. Potential applications of laser-coolable species include quantum computing [6,7] and sensing, as well as precision measurements of fundamental physical constants [8][9][10][11]. Charged particles are particularly attractive for experimental applications because they can be controlled by electric fields, motivating intense experimental and theoretical research in this area. Heavy molecular ions have been investigated as promising platforms for precision measurements [8][9][10][11]. In charged molecular species, an OCC is useful for system preparation and measurement [12]. Much experimental and theoretical effort has been focused on the SiO + cation [13][14][15][16]. Prospective advantages in state control motivate an active search for molecular ions with a large ground-state dipole moment and a splitting of the ground state into opposite parity states in the radio frequency range [17][18][19][20]. A closed optical cycle with a radiative lifetime of approximately 110 μ s was predicted for the AcOH + cation [21]. Computational studies have shown that because of high electronic transition energies RaF + and RaH + cations are not well suited for laser manipulations [22]. Favourable vibrational branching ratios were reported for a transition of TlF + lying in the UV region [23]. In the context of precision measurements, highly charged heavy cations offer additional benefits. In this domain, the PaF 3+ cation was identified as a promising candidate for cooling [24].
The prospects of laser-cooling anions have been discussed. Anions could afford sympathetic cooling of other negatively charged particles [25]. However, in contrast to neutral and cationic species, which support many bound excited states, anions rarely have bound excited states [26][27][28]. Among promising exceptions [25,29] are C − 2 and BN − , which are isoelectronic to SiO + . Molecular anions of alkali metals were also investigated computationally [30,31], however, the authors were not able to establish whether their electronic states were bound or metastable with respect to electron detachment.
The search for optically cyclable molecular cations is ongoing. Earlier, we investigated diatomic ions of the main group elements with a goal to find transitions suitable for cycling [1]. The results were disappointingcompared to the isoelectronic neutral species with good OCCs, the transitions in the cations generally featured less favourable properties. We attributed this difference to a more delocalised electronic structure and higher density of states in the cations, which present challenges in the design of cationic OCCs [1]. Other computational studies have shown that BO + and SiBr + are not promising for laser cooling [32,33]. Here, we extend our search for cyclable molecular cations to a series of diatomic radical-cations comprising a d-block metal and a p-block ligand that are isoelectronic (in their valence shell) to the successfully laser-cooled neutral molecules.
To assess the suitability of these species for optical cycling, we compute potential energy curves of their low-lying electronic states. We then use these curves to compute respective equilibrium geometries and Franck-Condon factors (FCFs), defined here as the squared overlap between the vibrational states: where |ν I and |ν F are the vibrational states of the initial and final electronic states, respectively. Within the Born-Oppenheimer and Condon approximations, FCFs determine vibrational branching of the electronic transition. The most desired property for optical cycling is a diagonal FCF matrix -so that there is no vibrational branching during electronic decay. Because repumping schemes can mitigate the effect of leaking population, cumulative FCFs -the sum of the first few FCFs with fixed ν I -are also valuable in evaluating quality of an OCC [34].
In diatomic molecules, FCFs are determined by the change in the position of the minimum and the curvature of the potential energy curves of the ground and excited states. The position of the minimum (i.e. equilibrium bond length) depends on the bonding pattern, determined by the shape of occupied molecular orbitals. Hence, the change in electron density upon excitation provides a visual cue on whether the bonding pattern changes and whether or not one may expect diagonal FCF -i.e. large changes in the interatomic charge distribution are expected to give rise to a significant bond length change and, consequently, non-diagonal FCFs.
The low-lying electronic transitions in doublet electronic states, such as radical-cations studied here, are transitions of the unpaired electron. Hence, we use Dyson orbitals to visualise the states of the unpaired electron in the ground and excited states [35]: where N and N−1 are the solution of the electronic Schrödinger equation for a cation (one unpaired electron) and a dication (closed-shell), respectively. The pair of states with Dyson orbitals localised outside of the bond area are likely to have diagonal FCFs. The idea of rational engineering of OCCs by seeking molecules with localised atomic-like transitions was first described in a break-through paper by Isaev and Berger [36], who also proposed several promising candidates. Later, neutral molecules featuring such an electronic structure have been successfully laser-cooled [37][38][39]. Following this strategy, more polyatomic molecules were proposed as prospective candidates for laser cooling [40][41][42][43][44].
In this study, we follow the same design paradigm as before [1] -seeking molecular cations with a single unpaired electron localised on a metal atom so that electronic transitions resemble atomic transitions on the metal. The atomic-like transitions localised on the metal do not disturb the bonding pattern in a molecule, which should lead to diagonal FCFs [36]. Figure 1 illustrates the design principle. We first consider cations of third group elements (yttrium or scandium). These metals lose one of their three valence electrons by bonding to a 17th group atom, which requires one electron to reach a closed-shell configuration. To make a cation, the second valence electron is removed leading to a molecule with a positive charge and the desired unpaired electron on the metal. We follow the same strategy to construct other pairs of a metal with n valence electrons and a ligand missing n−2 electrons to a closed-shell, as shown in Figure 1.
The structure of the paper is the following. The next section describes theoretical methods and computational details. Section 3 presents the results for the diatomic cations in which we identified atomic-like transitions on the metal atom and reports our best candidate molecules for optical cycling. Our concluding remarks are given in Section 4.

Theoretical methods and computational details
We solve the electronic structure problem using highlevel ab initio methods following the protocols from our previous studies on laser-coolable species [1,40,41,43]. We use the equation-of-motion coupled-cluster method for electron attachment (EOM-EA-CC) to describe the ground and excited electronic states of doublet radicalcations [50,51]. In these calculations, the target states (with +1 charge) are obtained by attaching an electron to a closed-shell +2 reference state described by the coupled-clusters singles doubles (CCSD) method. This approach leads to a balanced description of the ground and excited electronic states and is naturally spin-pure [51]. Table 1 lists the basis sets used in our calculations [52]. Because the ordering of the electronic states is different in different cations (and sometimes changes at different levels of theory), we do not use spectroscopic notations (i.e. in which the ground state is labelled by 'X'), but instead, simply number each state from 1 in each symmetry irrep (i.e. angular momentum projection group). This convention allows us to use a consistent notation for all molecules.
We include the effect of spin-orbit couplings (SOCs) using the state interaction scheme (sometimes referred to as a perturbative approach): we first compute a set of low-lying non-relativistic EOM-EA-CCSD states and use them to evaluate matrix elements of the Breit-Pauli Hamiltonian (we use mean-field approach to include two-electron contributions) [53,54]. We then construct and diagonalise the matrix of the SOC-perturbed Hamiltonian: where H EOM 0 is a diagonal matrix composed of EOM-EA energies and H BP is the matrix of the Breit-Pauli Hamiltonian in the basis of the EOM-EA states. The resulting energies E SOC are energies of the SOC-perturbed EOM states and matrix U contains the respective eigenstates. This matrix can be used to obtain properties of the SOperturbed states, such as permanent and transition dipole moments: where μ 0 is the matrix of dipole moments calculated in the basis of the non-relativistic EOM-EA-CCSD states. The state-interaction treatment of SOCs becomes exact in the limit where all electronic states are included in the calculation. In the present study, we considered the eight lowest electronic states. We use the computed spin-orbit corrected potential energy curves to determine equilibrium geometries and to compute FCFs. We do not use harmonic approximation [55] but solve the vibrational problem exactly. Numerical integration of the overlaps between eigenfunctions gives the FCFs.
Einstein's A IF coefficient gives the probability of a spontaneous decay from an initial state I to a final electronic state F: where ω IF is the transition frequency and μ IF is the transition dipole moment. The radiative lifetime of a state I is obtained by summing over all decay channels where the index F goes over all states below state I.
All electronic structure calculations (including energies, state and transition properties) were carried out using Q-Chem [56,57]. We computed FCFs with the LEVEL16 program [58], which solves the quantum vibrational problem numerically using potential energy curves provided by the user. In these calculations, the interatomic potential is represented by analytic functions: exponential functions (fit to the provided potential energy curves) represent asymptotes and the regions in between the sampled points are interpolated with cubic splines. The complete potential energy curves for both non-relativistic and spin-orbit corrected states are given in the Supplementary Information (SI).

Y and Sc series
Yttrium and scandium cations feature the sought-after atomic-like spectrum, as clearly illustrated by the Dyson orbitals in Figure 2. In these representative examples of the YCl + and NbN + cations, Dyson orbitals strongly resemble atomic D states. The 1 2 and 2 2 states correspond to the cation with a singly occupied sor ptype orbital, respectively, the 1 2 , 1 2 , and 2 2 correspond to a singly occupied d-type orbital. Although atomic states originating from a singly occupied d-type orbital have the same electronic angular momentum, the diatomic states vary with respect to its projection on the molecular axis, giving rise to three types of molecular states. Table 2 and Figure 3 show non-relativistic EOM-EA-CCSD excitation energies computed at the groundstate equilibrium geometries for ScF + , ScCl + , ScBr + , YF + , YCl + , and YBr + .
All states consistently appear in the same order except for the two lowest-lying states. The 1 2 is the ground state for scandium cations whereas it lies above the 1 2 state for yttrium cations. By inspecting Table 2 and Figure 3, we observe a general trend -that the energy gaps increase with the increasing electron-withdrawing strength of the halogen ligand. This ability to tune up OCC properties by tuning the ligand's electronwithdrawing strength was discussed in previous computational studies [40,44].
We first assess whether the 2 2 → 1 2 transition (analogue of the atomic 2 P → 2 S transition) is suitable for optical cycling. Just as in the case of neutral molecules, this transition is characterised by a highly diagonal FCFs (see Table S1 in the SI). The large excitation energy (≈ 4 eV) and a strong transition dipole moment give rise to a short radiative lifetime of the excited state. Unfortunately, these advantageous properties come together with undesired ones. First, there are intermediate states, which can  [59]). In the hydrogen-like orbital labels, the principal quantum number n equals 4 for the Sc, Ti, and V metal cations and 5 for the Y, Zr, and Nb cations.
drain the populations out of the OCC. Second, the excitation energy of this transition is in an undesirable UV range. This transition is even less attractive in the case of the scandium cations, where the 1 2 state is an excited state. Our calculations indicate that the YF + cation, the best candidate in this group shown in Figure 4, is unlikely to form a good OCC.
The simplest way to avoid electronic branching decays is to use the transition between the ground and the first excited electronic states. Unfortunately, the transition between 1 2 and 1 2 states is dipole forbidden. However, the SOCs can mix non-relativistic states, giving rise to a non-vanishing transition dipole moment, which might be sufficiently large to allow for optical cycling.
Tables S3 and S4 show the SOC-corrected states with their energies, transition dipole moments, and radiative lifetimes. SOCs mix the non-relativistic states and the extent of mixing is determined by the energy gaps between the states (which varies between the cations) and the magnitude of the Breit-Pauli Hamiltonian matrix elements (which are larger for cations containing heavier atoms). The energy shifts due to the SOCs do not exceed 0.1 eV and cause splittings of the and states. Crucially, the SOC-perturbed states show non-vanishing transition dipole moments between the and states in the range of 0.01 a.u.

Other groups: Ti or Zr with chalcogens, and V or Nb with N or P
The spectra of the multiply bonded cations are similar to those of the scandium cations discussed above. All species feature the 1 2 ground state followed by the 1 2 and 1 2 states. Again, we observe an increase in the energy gaps with an increasing electron withdrawing strength of the ligand. Additionally, a new trend becomes apparent -the excitation energies increase with the number of bonds in the cation, as shown in Figure 5. In the case of the transition from yttrium to zirconium and niobium cations, where the order of the ground and the first excited states changes, taking the energy of 1 2 state as the reference point allows to observe a consistent increase. This new trend is particularly apparent for the series F, O, and N as the ligands or Cl, S, and P, shown in Figures S1 and S2 in the SI. In this set of cations, we again inspect the transition between the ground and the first excited states. The increased energy gaps in Ti and V relative to Sc improve the radiative lifetimes. The heavy atoms Zr and Nb also have stronger SOCs, which increases the transition dipole moment, but an improvement in the radiative lifetime is achieved only for Nb as in the Zr cations the 1 2 and 1 2 states become nearly degenerate. The small improvement in the radiative lifetime of the 1 2 → 1 2 3/2 transition in multiply bonded cations is accompanied by a large improvement in the diagonality of the corresponding FCFs, Sec. S3 in the SI, making this transition a very promising OCC. The best candidate in this group is the NbN + cation, with FCFs summarised in Table 3 and the radiative lifetime τ = 650 μs.
In search for a shorter radiative lifetime, we focus on the dipole-allowed transition. As the last candidate for an OCC we consider a three-level cycle with the 1 2 excited state decaying to the ground 1 2 and an intermediate 1 2 states. The FCFs on the 1 2 1/2 → 1 2 3/2 transition single out zirconium and scandium cations where the sum of the first four FCFs reaches 0.9999; the titanium cations also reach such saturation by adding the fifth FCF (see Section S4 in the SI). In all these cations, the excited state decays significantly (> 10%) to the intermediate 1 2 state. To take advantage of the rapid cycling offered by the short lifetime of the excited state, this intermediate state must be repumped. Diagonal FCFs, necessary for an efficient repumping, render the ZrO + cation to be the most appealing candidate for the three-level OCC.
We do not report details about other screened cations because our exploratory calculations suggest that they do not feature the localised transitions. Cations of Cr or Mo with a C or Si have a very different electronic structure with significant multi-configurational character and their  dications do not converge to a closed-shell singlet state.
We detected similar problems for Ti or Zr with Se cations as well as V and Nb with As cations. This is not surprising, as transition metals with more than three d-electrons are known to result in complicated multi-configurational structures and dense manifolds of electronic states.

FCFs, electronic charge displacement, and Dyson orbitals
The transition between the 2 2 and 1 2 states is known for its diagonal FCFs, which are attributed to the localised character of the transition. The localised transition preserves the charge distribution in the interatomic area of both states leading to similar bond lengths. We interpret the bond length changes of other electronic transitions using similar analysis applied to the changes in shapes of the Dyson orbitals. The 1 2 → 1 2 transition is most promising as it involves the ground and the first excited electronic states avoiding branching to intermediate levels. The electronic density changes from a spherical distribution of the sorbital to a clover-shaped shape of the d orbital, see Figure 2. For most of the yttrium and scandium cations, this transition features FCFs indicative of the expected significant vibrational branching (see Tables S5 and S6 in the SI). For all multiply bonded cations except for ZrS + this is one of the most diagonal transitions that we found (Tables S7-S10 in the SI).
In the dipole-allowed 1 2 → 1 2 transition, the dorbital on a metal atom changes its alignment from the one that is along the bond to the perpendicular one ( Figure 2). This charge displacement leads to the expected poor FCFs, with a few exceptions of scandium or zirconium cations where the sum of the first four FCFs differs from unity by less than 10 −4 (see Section S4 in the SI).
This analysis suggests that a disruption in the electronic density of the bond area does not necessarily render the transition to be unusable for optical cycling. One possible reason is that the presence of a multiple bond is a structurally stabilising factor.

Best candidates for optical cycling
The variations of the spectra introduced by states derived from a singly occupied d-type orbital led us to consider four types of OCCs. The molecular analogue of the atomic 2 P → 2 S transition was deemed not promising (see Section 3.1 in the SI). Among the low-lying states, Figure 6. Three types of optical cycling schemes exemplified by the best candidate cations. In the group with the 1 2 ground state, YCl + shows the most diagonal transition. In the group with the 1 2 ground state and highly diagonal transitions, NbN + stands out with the shortest radiative lifetime. In the third group, the FCFs for the formally dipole-allowed transition between the ground and the second excited state favour scandium and zirconium cations. Large decay to the intermediate state requires also the side branch 1 2 1/2 → 1 2 transition to show quickly saturating FCFs. This second requirement renders the ZrO + cation as the best candidate in this group.
we considered the transitions between the first excited and the ground states, separately for ground 1 2 and for ground 1 2 cations, as well as a three-level scheme involving the second excited 1 2 state. We present the best candidates in each of these groups in Figure 6; Table 3 summarises the corresponding FCFs.
The observed trends can help in designing transitions with desired excitation energy. In multiply bonded cations, we observe persistent diagonal FCFs that are promising for optical cycling, on the 1 2 1/2 → 1 2 3/2 transition. Finally, our analysis of FCFs of selected transitions together with an inspection of the Dyson orbitals of involved states suggests that although the charge displacement often makes the transition unlikely for optical cycling, it is not the crucial design factor. The most promising candidates in the three identified optical cycling schemes (discussed in Section 3.4) are: 1 2 3/2 → 1 2 in YCl + , 1 2 1/2 → 1 2 with a side branch 1 2 1/2 → 1 2 in ZrO + and 1 2 → 1 2 in NbN + .
The most promising optical cycling scheme is that of the NbN + cation. The calculated FCFs show that with lasers addressing the ν = 0 → ν = 0 and ν = 1 → ν = 0 branches this molecule should be able to scatter on average more than 10 5 photons before population leaks to ν > 1 states. This number of scattering events should be sufficient for laser cooling, however, direct laser cooling on a transition with a radiative lifetime of the order of hundreds microseconds is not attractive.
On the other hand, diatomic cations held in an ion trap can be sympathetically cooled by co-trapped, lasercooled atomic ions [76]. Therefore, it is not always necessary that a fast cycling transition be used for cooling the motion of the molecular ion, but is only needed to enable quantum state preparation and measurement [13,77]. This significantly relaxes the constraints on the lifetime of the excited electronic state of the diatomic cation. In this use case, the optical cycling lifetime needs only be significantly faster than the population redistribution rate from the environment, typically black-body radiation.
While the black-body redistribution rate is dependent on molecular structure, state lifetimes < 1 ms would appear to allow for quantum state preparation and measurement in most scenarios.
Further, longer lifetime species with cycling transitions may also be useful for experiments using quantum logic spectroscopy [78,79]. Here, the cycling transition can be used to optically pump the molecular internal states to greatly improve the rate of the heralded state preparation and measurement used in quantum logic spectroscopy. The cations presented in this manuscript show properties that make them promising candidates for use in these applications. The range of described properties allows one to select the cations that match the best desired experimental requirements for a particular application.
Our results can assist the engineering of OCCs in other neutral or charged molecules. In particular, a shortening of the radiative lifetime in the most promising 1 2 → 1 2 scheme appears possible. The radiative lifetime of an excited state becomes shorter with an increase in either the excitation energy or the transition dipole moment. A use of polyatomic ligands of an electronwithdrawing strength exceeding that of atoms appears as the most practical way of raising transition energy. The excitation energy can be further increased with the growing number of the metal-ligand bonds. Similarly, a formally dipole forbidden transition can become sufficiently bright in molecules with heavy nuclei due to increased SOCs. We expect that the heavy metal analogues of the systems studies in this work (Lu, Hf, Ta cations) are likely to feature fast OCCs on a transition between its low-lying doublet states and that the design principle outlined above should help in its fine-tuning.

Disclosure statement
The authors declare the following competing financial interest(s): A.I.K. is the president and a part-owner of Q-Chem, Inc.

Funding
This study was funded through the 'Molecules Functionalized with Optical Cycling Centers' collaboration, supported by the U.S. Department of Energy (Award DE-SC0019245), and through 'CCI Phase I: NSF Center for Advanced Molecular Architectures for Quantum Information Science' (Award CHE-2221453), supported by the National Science Foundation.