Chiral molecule candidates for trapped ion spectroscopy by ab initio calculations: From state preparation to parity violation

Following the observation of parity non-conservation (PNC) in β decay¹ and atomic spectroscopy,^2,3 the symmetry between the two mirror configurations of a chiral molecule was also predicted to be broken by the weak interaction.^4,5 In most chiral molecules, the weak interaction is supposed to make one enantiomer slightly more energetically stable. Consequently, the effect alters the tunneling dynamics between enantiomers^5–7 and creates enantiomer specific vibrational transition frequencies.⁸ This effect may also pertain to the mystery of how life originated with a specific handedness, with many examples of a single naturally occurring enantiomer.^4,9–11 Observation of PNC in molecules, which has so far not been achieved in the laboratory, will improve our understanding of such phenomena and is a fundamentally important question in chemistry.

The majority of efforts to measure PNC in molecules have focused on neutral species,^12–18 despite the advantages associated with charged molecules. However, recent advances in precision spectroscopy with trapped molecular ions that leverage long coherence times encourage the development of this platform.^19–23 Moreover, the generality of ion trapping allows the selection of a molecular ion with optimized sensitivity toward the question at hand. This capability has been exploited in searches for CP (charge and parity) violations,^19,21 searches for dark matter,²⁴ and precision rotational spectroscopy,²⁵ for example. Furthermore, methods for the production of molecular ions that are cold internally and externally have been demonstrated.^21,26 Finally, chiral molecules with particularly asymmetric electronic wavefunctions, such as radical states, are predicted to have enhanced PNC by reducing the cancellation of PNC contributions.^9,27 Such radicals have improved lifetimes as radical ions due to the suppression of reactions between the molecules.

The advantages associated with trapped ions have led to a few proposals to utilize chiral molecular ions to observe PNC.^5,9,28–30 However, the scarcity of existing theoretical modeling for chiral molecular ions makes it challenging to choose a suitable candidate for such an experiment. One challenge is that molecular ions are often prepared through the ionization of neutrals and thus have a low dissociation threshold. The susceptibility to predissociation may shorten the interrogation times of vibrationally excited states.

Here, we conduct an ab initio study in search of suitable chiral molecular ions that can be prepared through state-selective near threshold photo-ionization (STPI) and have a sufficiently high dissociation threshold to facilitate vibrational precision spectroscopy with long interrogation times. We propose STPI as a pathway to create internally cold, chiral molecular ions, as it has been applied successfully to a few diatomic species.^19,26,31,32 Using STPI on a cold neutral molecular beam only allows molecules in certain quantum states to enter the trapped ion ensemble. This step is important since the complexity of such polyatomic molecules makes them particularly difficult to cool, and the hope for a cycling transition diminishes when fewer symmetries exist.^33–36 

The following four separate stages are envisioned for the future experiment:

1. Generation and cooling of neutral molecules in a supersonic expansion or other generic beam method.

2. Ionization of the molecules via 1 + 1′—two-photon resonant ionization.

3. Ion trapping and spectroscopy experiment.

4. State-selective photo-dissociation spectroscopy with time-of-flight mass spectrometry for detection.

This paper focuses on finding suitable chiral molecular ion candidates for stages 2 and 4. A plan for step 3 is discussed in Ref. 37, but many other alternatives could be applied. Step 1 is straightforward for some of the candidates discussed in this work and more involved for others. Since the scheme addresses the same molecule in neutral and ionized forms, we refer to the neutral molecule and its companion charged molecule interchangeably. However, the precision spectroscopy stage is focused on the cation version of the molecule.

To limit the complexity of molecules, we study chiral molecules with up to five-atoms in a tetrahedral structure. Four-atom chiral molecules are not considered here as these are less likely to be chiral in both the neutral and charged configuration simultaneously, often having low transition state barriers between enantiomers.^5,38

We find several suitable candidates with various spin multiplicities of the form CHXBrI⁺ where X ∈ {D, Ca} and isotopically chiral CHX⁷⁹Br⁸¹Br⁺ where X ∈ {D, Ca, Yb}. All of these candidates are chiral in both neutral and charged forms. We discuss how our results indicate that these molecules fit stages 2 and 4. Additionally, we compute the magnitude of the expected PNC for some of these candidates here and in Ref. 39. We show, for example, that candidates where X ∈ {Li, Na} fit most but not all of the requirements and thus are not suitable candidates. Despite not explicitly investigating CHYbBrI⁺, we can infer that it is also amenable to the experiment by considering similar molecules such as $CHYbBr2+$ examined in the study.

Polyatomic molecules have many degrees of freedom, making a general method to internally cool them challenging to find. Most cold polyatomic molecule experiments resort to generic collision-based cooling methods such as supersonic expansion⁴⁰ or buffer gas cooled beams,⁴¹ but these methods lower the internal temperature of molecules to a few kelvins only. Recent advances in laser cooling schemes suggest a promising pathway toward cold polyatomic molecules.^{33–35,42,43} In fact, the Franck–Condon factors for some isotopically chiral molecules have been found to be favorable for laser cooling,³³ which places cold and controlled neutral chiral molecules within reach. For molecular ions that originate as neutrals, there is an alternative avenue by which they may be prepared, populating a low number of states through a filtering step discussed here.

The proposed scheme begins with a cold beam of the candidate molecules in neutral form. When the neutral molecules enter the ion trap, they are ionized by a two-color resonant process. The first photon frequency is resonant with a specific resolved molecular transition. Therefore, it excites only molecules in certain quantum states. The second photon ionizes the molecules that populate this intermediate state, either directly on the continuum near the ionization threshold or through an autoionizing Rydberg state. The molecular ion can only minimally heat up internally due to the low energy at which the electron is ejected or the selectivity of the autoionization process.

We discuss two avenues to achieve state selectivity. The first is to use a ro-vibronic (rovibrational and electronic) excitation for the first photon, followed by ionization by a second photon (Fig. 1, left). This places both of these photon energies in the UV (ultraviolet) or deep UV (DUV) range. The main risk with this method is that many organic molecules are vulnerable to dissociation in electronically excited states, which may occur before the second photon completes the ionization process. A second way of achieving cold molecular ions is to use a vibrational excitation in the mid-IR along the C–H stretch where the molecule is stable as the selective filtering step, followed by a second photon in the DUV or VUV (vacuum UV) to ionize the molecule (Fig. 1, right). This avoids the pre-dissociation of the molecule in the intermediate state since the process is energetically forbidden and the state density is significantly lower.

Once the molecules are ionized and internally cold, they are still moving at hundreds of meters per second due to the neutral beam velocity. The ensemble can be decelerated by applying a uniform electric field to the molecular ion cloud.^20,21 Subsequently, the ions are trapped by the same electrodes and separated from the rest of the neutral molecules. The ionization process serves as a selective step for specific quantum states since the molecules that are not excited by the first photon remain neutral and will not be decelerated and trapped, leading to a cold sample of molecular ions. This method is utilized for the HfF⁺ experiment^20,21 following state-selective 1 + 1′ ionization and produces 40k ions in fewer than 4 quantum rotational states with a translational temperature of $∼2$ K. This same method is also used to create state-selected $N2+$ with the 2 + 1′ scheme, where the ions are subsequently trapped by sympathetic cooling with laser cooled atomic ions.²⁶ Throughout the manuscript, we refer to this state-selective near threshold photoionization step as STPI.

While the first photon transitions may be challenging to fully resolve for complex polyatomic molecules, any partial selectivity that arises in this step is likely to produce molecular ions that are very cold. Since the spectral width of nanosecond pulsed dye lasers is on the order of GHz, this tool is excellent for addressing individual rotational states, but the hyperfine structure will not be resolved. Population in hyperfine states is unlikely to harm the initial forms of our precision measurement and can be overcome by depletion or tailored pulses and polarization selections.^44,45 Another advantage of trapped molecular ions is that there is sufficient time to perform multiple depletion processes, even when the available power is low.

Our scheme requires that chiral molecular ion candidates must be stable both as neutrals and as ions. For the 5-atom molecules we consider, the removal of an electron from such a small chiral molecule often makes the molecular ion very weakly bound, with a tendency to dissociate, particularly when the molecule is vibrationally excited (Sec. V A). In fact, as we will show, many candidates that are amenable to STPI dissociate far too easily to support vibrational spectroscopy. We, therefore, search for candidates with sufficiently high dissociation thresholds (>1.2 eV). We find that adding a metal atom to halogen substituted methane substantially increases the dissociation threshold while simultaneously reducing the ionization threshold of the neutral version of the species. For example, we find that for CHCaBrI⁺, the most favorable dissociation channel is I + CHCaBr⁺ at an energy of 2.55 eV as compared to CHDBrI⁺, which dissociates to I + CHDBr⁺ at 1.29 eV (Sec. V A). Simultaneously, the ionization threshold of CHCaBrI is more than 4 eV lower than that of its non-metallic counterparts (Sec. IV).

To detect the internal state of our chiral molecular ions, we turn to photodissociation due to the relatively low dissociation threshold. In an ion trap, state-selective photodissociation (SPD) serves as a highly efficient method for probing the internal states of the molecular ions by detecting the fragments via mass spectrometry.^20,46–48 An alternative we pursue involves the detection of the internal state of the molecule using single-photon photo-dissociation. Upon dissociation, any internal rotational state of the molecule will be translated into a different kinetic energy of the photo-fragments. Detection of the photofragments may be performed by coupling a velocity map imaging setup to an ion trap, for example. Our initial design is promising, but details will be reported elsewhere. In this work, we check that the wavelengths associated with the dissociation processes would be feasible by calculating the dissociation energies for the various dissociation channels (Sec. V A).

Both STPI (Sec. IV) and SPD (Sec. V A) need to be developed experimentally, but this paper aims to rule out many candidates that would not fit these schemes and suggest some that do based on ab initio calculations.

The computed properties presented in the next sections are the vertical and adiabatic ionization energies and several excitation energies (EEs) of the neutral systems, in addition to the dissociation channel energies, isotopic vibrational modes, rotational constants, and transition state energies of the cation systems. Below are the details for all of these computations.

The energy differences we are interested in are the ionization potentials, adiabatic and vertical, the low-energy dissociations, the activation energy between the S and R enantiomers, and the low-level EEs of the neutral systems. All the electronic configurations considered herein have either a closed-shell structure with a singlet multiplicity or a radical configuration with a doublet multiplicity.

Radical systems are frequently studied using single reference methods. In particular, density functional theory (DFT) is often used to determine the geometrical structures and energy differences of reactions that include radicals.^49–54 Sometimes, coupled clusters with singles, doubles, and perturbative triples [CCSD(T)] are used to benchmark the proper exchange-correlation functional for geometry optimization.⁵² Configuration interaction and Møller Plesset perturbation theory from second to fourth order can also be used to optimize structures.^55,56 Notice that in Ref. 50, it was shown that DFT performs significantly worse than the unrestricted second order Møller Plesset perturbation method in both geometry optimization and interaction energy calculations for radicals. However, all the above-mentioned methods represent a single-reference approach in which the radicals’ treatment is based on an unrestricted wave-function (Hartree Fock or DFT). The unrestricted approximation for the radical wave-functions suffers, for example, from spin-contamination. Note that only in some studies is the very important ⟨S²⟩ expectation value, which gauges the spin-contamination of the wave-function, reported.⁵⁶ 

Therefore, in this study, we examine the following scheme for describing the closed-shell configurations as well as the radicals: the geometry is calculated at the second-order Møller Plesset (MP2) level, while the energies are obtained at the CCSD(T) level using the same basis sets [i.e., CCSD(T)//MP2]. However, since these open-shell radical doublet state calculations are based on the unrestricted Hartree–Fock wave-function with the well-known problematic behavior that manifests in spin-contamination, we carefully monitor this approximation. In addition, we perform equation-of-motion CCSD (EOM-CCSD) calculations to verify the validity of the CCSD(T)//MP2 scheme. Indeed, we find that for some radicals, the MP2 geometries are inconsistent with the EOM-CCSD ones (see Table S3 in the supplementary material). On the other hand, EOM-CCSD for ionization potential (EOM-IP-CCSD) or electron affinity (EOM-EA-CCSD), depending on the molecule under investigation, is a proper approach for describing radicals. It is a single-reference but multi-configuration approach that operates in Fock space, in which the reference and target states are treated in a balanced fashion, and as a result, it provides a well-defined spin state.⁵⁷ Notice that EOM-CCSD analytic gradients are available within the Q-Chem program.⁵⁸ 

The eight low-lying EEs are calculated for the neutral molecules. For the molecules with a singlet state, we use EOM-CCSD for excitation energies (EOM-EE-CCSD), and for the system with a doublet neutral state, we use EOM-EA-CCSD. As discussed earlier, the EOM-CCSD energies are calculated at the EOM-CCSD geometry using the same basis set, which is suitable for doublet states, whereas for closed-shell singlet states, we use the CCSD energies at the MP2 geometry (in a few cases, we used the CCSD geometry since MP2 did not converge, and the difference between the two is negligible, as shown before in Ref. 59).

The harmonic frequencies of the optimized structures are calculated using EOM-IP-CCSD for doublet state cations and CCSD for singlets. The isotopic effect is explicitly considered in these calculations. In systems with two hydrogens, we use the masses of hydrogen and deuterium, and in the case of Br₂, we use the different Br masses, ⁷⁹Br and ⁸¹Br. In addition, we calculate these frequencies using the DFT ωB97M-V functional. The agreement between the two methods suggests that this functional is suitable for the frequency analysis of the systems at hand. Moreover, the ωB97M-V ⟨S²⟩ expectation values are found in the 0.75–0.756 range for all the reported doublet states. Therefore, we employ ωB97M-V to calculate the transition states between the S and R enantiomers in order to evaluate if the enantiomer states are time invariant. All the non-relativistic calculations are performed with the Q-Chem electronic-structure package.⁶⁰ 

We consider two basis sets of triple-zeta (TZ) and quadruple-zeta (QZ) quality. The frozen-core scheme is used within the coupled-cluster based calculations. The Dunning correlation consistent cc-pVXZ^61,62 basis sets are used for all the light atoms (H, Li, C, O, Na, and Ca) within the XZ sets, where X = T or Q. For EEs, we use the augmented versions of these basis sets,⁶³ aug-TZ and aug-QZ. For the heavier atoms (Cl, Br, and I), we use these basis sets paired with small-core pseudopotentials, i.e., cc-pVXZ-PP.^64,65 For Yb, we use def2-XZVPP paired with a small-core pseudopotential.⁶⁶ 

These pseudopotentials were optimized to provide an accurate description of the Pauli repulsion of the cores, their Coulomb and exchange effects on the valence space, and scalar-relativistic corrections.^64,65 Therefore, we also examined the spin-orbit effect by taking the difference between the fully relativistic calculations using the Dirac-Coulomb Hamiltonian including the Gaunt interaction, and the spin-free version,⁶⁷ which provides results without spin-orbit coupling for the four-component Hamiltonian. The spin-orbit contributions to the adiabatic ionization potential (AIP) and the vertical ionization potential (VIP) of the ten molecules were calculated using the dyall.cv2z basis set within EOM-IP-CCSD. The eight low-lying EEs of the neutral molecules (²CHCaBr₂ and ²CHCaBrI) with doublet multiplicity were calculated via EOM-EA-CCSD and the dyall.av2z basis set, except for Ca, for which we used dyall.v2z. For neutral molecules with singlet multiplicity, we used EOM-EE-CCSD/dyall.v2z; these include ¹CH₂Br₂ (four excited states), ¹CHLiBr₂ (four excited states), ¹CHLiBrI (four excited states), ¹CHNaBr₂ (five excited states), and ¹CHNaBrI (two excited states). All in all, we calculated the spin-orbit effects for ten VIPs and AIPs, twenty-four EEs using EOM-EA-CCSD, and nineteen EEs using EOM-EE-CCSD. The maximal spin-orbit contribution among the molecules reduces the AIPs by 0.02 eV, the VIPs by 0.08 eV, the EEs using EOM-EA-CCSD by 0.003 eV, and the EOM-EE-CCSD EEs by 0.05 eV. The average shifts are at least fourfold smaller and have the same sign for each molecule. We conclude that the spin-orbit effect is negligible and cancels out since we investigated the energy difference between very similar chemical systems. Therefore, we report the values obtained by Q-Chem.⁶⁰ All the relativistic calculations are performed using the relativistic electronic structure package DIRAC22.⁶⁸ 

Finally, we evaluate the stability of the selected cations by calculating their low lying dissociation channel energies. For fragments with singlet multiplicity, we use CCSD, and for doublet fragments, we use EOM-IP-CCSD or EOM-EA-CCSD using the QZ basis set and the Stuttgart/Cologne pseudopotentials. The spin-orbit corrections in this case are not negligible and are added to the non-relativistic dissociation energies. The spin-orbit effect was calculated only for the lowest dissociation channel of each system using the dyall.cv2z basis set with the same methodology. The average spin-orbit effect for the 11 dissociation energies is −0.15 eV, with the maximum value −0.27 eV, constituting less than 10% of the calculated values that neglect relativistic effects.

Benchmark calculations are presented in the supplementary material. In summary, the presented values and geometries were obtained using the following schemes: EOM-CCSD/QZ is used for the ionization energies (Sec. IV), and EOM-CCSD/aug-QZ for the excitation energies (Sec. IV). The cation’s vibrational frequencies using EOM-CCSD/TZ and ωB97M-V/TZ (Sec. V B). Transition states and their vibrational analysis using ωB97M-V/TZ (Sec. V B). Dissociation energies are calculated using EOM-CCSD/QZ; in addition, spin-orbit corrections are added via EOM-CCSD/dyall.cv2z (Sec. V A).

State-selective ionization combined with control of the excess energy in the process can be achieved using a resonant 1 + 1′ process, where 1 and 1′ denote the number of photons involved in each of the two stages in the process, and the “prime” tag indicates that the two stages use different photon energies. This notation is common for such REMPI (resonance enhanced multi-photon ionization) methods. The advantage of a 1 + 1′ process is that the power in each stage can be carefully tuned to avoid the competing 1 + 1 process. This is in contrast to the 2 + 1′ and 3 + 1′ processes, where the simultaneous interaction of multiple photons in the first stage usually necessitates focusing the laser beam, which precludes fine control of the first photon beam power since the ionization volume and power are coupled. To approximate the amenability of our candidates to 1 + 1′ resonant photo-ionization, we calculate the ionization threshold of the molecule as well as the electronically excited states for the resonant transition of the first photon.

The ionization threshold has an adiabatic and vertical component, and we calculate both for each of the initial sets of candidates: CHXBrI, where X ∈ {D, Li, Na, Ca}, and CHX⁷⁹Br⁸¹Br, where X ∈ {D, Li, Na, Ca, Yb} (Table I). Between the AIP and the VIP, we expect to find resonant autoionizing states that are often used to control the emitted electron energy and quantum states while maintaining a large coupling to the continuum. Controlled ionization is also possible by choosing a near threshold energy for the 1′ photon.

To estimate the resonant, filtering, and transition energies, we calculate the energies of the first eight vertical electronic excitations. For each candidate, these are listed in Table II. The first photon’s (1) frequency would need to be resonant with one of these states. Naturally, there are many specific rovibrational transitions associated with each state to choose from, but these energies provide a rough estimate of where these transitions with maximal Frank–Condon factors are expected.

Once excited to the intermediate state, we can use the AIP and VIP to estimate the energy of the 1′ photon that is needed to ionize the molecules by subtracting the respective electronic state energy. Figure 2 shows the predicted photon energies needed in the 1 + 1′ process for each candidate. The energies of the electronically excited states are depicted by circles [first photon energy (1)], and the ionizing photon energy (1′) is depicted by an error bar that stretches between the AIP and the VIP. The color-coding of the markers of the 1 and 1′ photons corresponds to different electronic excitations (Table II). Excitations one through eight are plotted in ascending order and colored black, red, light green, blue, orange, purple, magenta, and dark green, respectively. The excitation energy and first photon energy are equal, which serves as a key to help distinguish between the different intermediate states.

To understand if it is feasible to realize the wavelengths in the lab, we compare the energies of the photons to the tuning range of a dye laser pumped by a 532 nm source. The choice of dye lasers originates in their broad tunability over the range and relatively narrow linewidth that commonly allows the resolution of individual rotational lines. The maximal lasing frequencies for tunable dye lasers are depicted by vertical dashed lines including their frequency conversion add-ons such as doubling and tripling.

In the second approach discussed to achieve the filtering step, the molecule would be excited along a specific rovibrational transition (Fig. 1 right). The vibrational excitation needs to be sufficiently high to ensure that the intermediate state is not thermally populated. The relevant energies for the second transition are the AIPs and VIPs (Table I) minus the vibrational excitation. This means that the second photon would need to be in the VUV range for most of the molecules except for the Ca and Yb substituted candidates, where the transition is reachable with a tripled dye laser.

A clear trend emerges from Table II and Fig. 2. Substituting the deuterium with an alkali atom reduces the energies of the excited electronic states by more than 1 eV. Replacing deuterium with Ca or Yb further reduces this energy since these candidates are now radicals in the neutral state. Furthermore, these substitutions cause the ionization energy to dramatically drop (see Table I). This makes this subset of candidates particularly appealing from a state preparation perspective.

For the scheme to work, one of the important steps is for the intermediate transition to be resolved. This naturally becomes challenging with the growth of molecular complexity, and certainly, the molecules discussed here are far from simple. However, any sort of partially resolved transition should already assist in reducing the target molecular ion’s internal temperature in the process. Moreover, rovibrational transitions tend to be resolved for similar molecules.⁶⁹ 

In this section, we consider the candidates in charged form to test their amenability for precision spectroscopy in search of PNC. We discuss the binding energy of the candidates (Sec. V A) as well as the structure of the molecules (Sec. V B).

Since the candidate cations are produced by shedding an electron, as we describe in Sec. IV, we suspect that their binding energy might be relatively low. For the chiral molecular ion to be a viable candidate, we must verify that it will not fragment too easily, and certainly not when undergoing vibrational excitation. This is because one of the main avenues to search for PNC is through vibrational spectroscopy and comparison between enantiomers.⁸ If the molecular ion unintentionally dissociates during the spectroscopy stage, the precision of the measurement will be substantially reduced, which lowers its appeal as a candidate.

Here, we calculate the dissociation threshold for the various dissociation channels for the candidates. The resulting thresholds for the lowest energy dissociative channel are given in Table III. The dissociation channels were calculated by taking the difference between the cation energy and the sum of the fragment energies. EOM-CCSD/QZ//EOM-CCSD/QZ (IP, ionization potential, or EA, electron attachment) is used for systems with doublet multiplicity and CCSD/QZ//MP2/QZ for singlet state molecules. The relativistic spin-orbit coupling correction is added on top of the non-relativistic results. This correction is obtained by taking the difference between results obtained using the full relativistic Hamiltonian and its spin-free version; the same approach but a smaller basis set was used for this calculation (see Sec. III for details). For $CHLiBr2+$ and $CHNaBr2+$⁠, we find two channels with similar dissociation energies, and both are listed. For $CHYbBr2+1$⁠, the EOM-CCSD calculation of ²CHYbBr⁺ did not converge, so we present the CCSD(T) results instead. The spin multiplicity of the system, which is either singlet or doublet, is noted in a superscript before the molecular formula.

Additional, higher dissociation channel energies are listed in Table VI.

To estimate the minimal acceptable dissociation threshold, we can consider, for example, the worst-case scenario of a vibrational excitation along the C–H stretch mode where the photon energy is roughly 0.4 eV (explicit calculations in Sec. V B). Therefore, the energy of the first vibrational excitation for the C–H stretch is $∼0.4$ eV above the minimum electronic energy of CHDBrI⁺. In addition, the presence of the excitation laser may dissociate the molecule in the first excited vibrational state if it couples between that state and the continuum. This would cause a loss of population during the spectroscopy. However, this loss is manageable since, in a Ramsey spectroscopy experiment, this dissociation would be limited to the $π2$ pulses³⁷ but could also occur from black body radiation. We would nonetheless prefer that the dissociation threshold be higher than 0.8 eV, or about double the highest vibrational mode energy. Alternatively, using a lower energy vibrational mode is also possible. For a dissociation channel such as CH₂BrI⁺ → CH₂Br⁺ + I, the zero point energies (ZPEs) of CH₂BrI⁺ and the CH₂Br⁺ fragment are similar since the iodine is associated with the lowest energy vibrational modes. Therefore, we neglect the contribution of the ZPEs to the reduction of the dissociation threshold of the molecules since their difference is small relative to the threshold. The energies for the vibrational modes for the cation-candidates are shown in Table IV. For the metal substituted candidates, we can use the same cutoff of 0.8 eV as the molecular properties and dissociation channels are similar to the other systems. The vibrational energies were calculated at the coupled-cluster level as well as using density functional theory with the ωB97M-V functional. The agreement between the two approaches suggests that ωB97M-V is also suitable for calculating the energies and frequencies of transition states between the two enantiomers.

All the molecules CHXBrI⁺ where X ∈ {D, Li, Ca, Na} and $CHXBr2+$ where X ∈ {D, Li, Ca, Na, Yb} have dissociation thresholds above 0.8 eV. The higher the threshold, the less susceptible the molecule will be to spontaneous dissociation by absorbing a photon from the black body. This is in contrast to the other chiral molecular ions we have computed, such as CHCl₂F⁺, CHBrClF⁺, CHBr₂F⁺, and CHIBrCl⁺ (Table V), which range from instability with a negative energy for dissociation to weakly bound molecular ions with thresholds below the 0.8 eV cutoff. These molecules would likely not support spectroscopy of the C–H stretch transition and other lower energy modes without compromising the molecule’s lifetime. Notably, we find that all the metal substituted molecules are very stable with a high dissociation threshold. This high stability, combined with the predicted convenient wavelengths for STPI (Sec. IV), makes them particularly appealing for precision spectroscopy.

The dissociation energies for the candidates (Table V) are computed at the CCSD(T) at the MP2 geometries. Since these values are well below our 0.8 eV cutoff and these candidates are unlikely to be suitable, we did not perform the EOM-CCSD calculations as performed for the candidates in Table III.

With molecule stability established, we turn to discuss dissociation for state-selective detection. Using dissociation for state-selective detection is common in trapped molecular ion experiments due to the low dissociation thresholds.^20,46–48 Since the fragments have a different mass than the parent molecular ion, they are straightforward to distinguish with high quantum efficiency. The rise of ion traps that are coupled to mass spectrometers^72,73 facilitates a high quantum efficiency avenue to detect these fragments and thus measure the internal state distribution of the parent molecular ion.

It is challenging to predict the dominant dissociation channel for these complicated five-atom molecules that have a large density of states originating in multiple channels above the dissociation threshold. However, if we consider the different dissociation channels, we see that the wavelengths for a two photon resonant dissociation process should be feasible. For example, for CHDBrI⁺, the dissociation energy is 1.29 eV, which is comparable to commercially available diode lasers in the near infrared, which range from 0.8 to 1.8 eV. These lasers can be combined with a selective excitation step realized with microwaves or a second diode laser to create the state-selective 1 + 1′ dissociation process, similarly to the STPI described in Sec. IV. The 1′ photon can also be tuned to the most favorable dissociation process for detection, which is not necessarily immediately above the dissociation threshold. In Table VI, we show additional dissociation channel thresholds for the promising candidates to help guide the experiment. In this case, the spin-orbit contribution is not included; however, we expect a similar shift of the dissociation energies to the shifts noted in Table III. Therefore, these values represent an estimate for experimental studies; a more refined computation will be necessary in the future.

A second avenue we consider for detection is dissociation by a single photon process. In this approach, the fingerprints of the internal state of the molecular ion will appear in the kinetic energies and angular distributions of the photo-fragments. The parent molecule’s internal state will affect the kinetic energy of the photofragments due to energy conservation since a single photon is absorbed. We aim to probe two rotational states in this manner.³⁷ For distinguishing between vibrational states, this method may not be as straightforward since the energy difference may remain within one of the photofragments. While measurement of the kinetic energy of photo-fragments has been achieved when dissociating trapped molecular ions,²⁰ the energy resolution needed to resolve individual rotational states of the parent molecule would need to be below 10 m/s for the candidates considered. An ion trap that is optimized for the detection of photofragment energies by coupling to a velocity map imaging detector should be able to reach the required resolutions. Currently, we are building such an apparatus. The ion trajectory simulations, which will be reported elsewhere, indicate that this is feasible.

As indicated in Table III for the single photon dissociation process, realizing the required wavelengths is also possible with a single photon for a wide range of diode and pulsed dye lasers.

The geometry of the molecular cation is one of the most important properties that must be verified to support a measurement of PNC through a comparison of vibrational transition frequencies. For many 4-atom candidates, such as hydrogen peroxide, the chiral geometry does not survive the removal of an electron because the cation has a planar structure.³⁸ However, even the neutral form of hydrogen peroxide does not support time-invariant chiral states since it has a relatively low transition state between enantiomers relative to twist mode energy splitting. The eigenstates of such a system are the symmetric and anti-symmetric superposition states of the S and R molecule configurations.⁵ 

For our candidates, we search for stable chiral geometry, which can be determined if the transition state between the two enantiomers is sufficiently high. First, we calculate the minimal-energy geometry of the molecular ion candidates to verify that they are indeed chiral. These geometries can be seen in Fig. 3. While all of these molecules have a chiral structure, the geometries of both the lithium and sodium substituted molecules are very close to planar. This near planar geometry hints at a low barrier between the two enantiomers. Indeed, a search for a planar transition state (TS) reveals that its energy is only 0.01 and 0.03 eV for CHLiBrI⁺ and CHNaBrI⁺, respectively. The transition states are planar and have a single imaginary frequency out of the nine normal modes. Therefore, we can conclude through comparison with the zero point energies of the molecules that the chiral ground state of these molecules will not be time invariant.

On the other hand, the rest of the candidates whose chiral structure is more acute survived this test. Their geometries and transition state energies are shown in Fig. 3. For CHDBrI⁺ and $CHDBr2+$⁠, the transition state energies at 1.30 and 1.39 eV are significantly higher than all the vibrational mode energies in the system, including the first excited state of the C–H stretch mode. For the other metal substituted candidates, the transition states are all above 0.7 eV, which is higher than the zero point energy for all the molecules. In particular, all the bending modes that overlap with the enantiomer mutation coordinate have at least 5 states below the transition state barrier, and this barrier scale is approximately equal to the energy of the first excited state of the C–H stretch mode.

Another effect that might limit the lifetime of excited vibrational states is known as intra-molecular vibrational redistribution (IVR).⁷⁴ To a certain extent, the rate of IVR is determined by the vibrational state density at the excited energy. Taking all the different vibrational combinations, we find that the state density at the energy of the first excited state of the C–H stretch mode (v₉ = 1, where v₉ is the vibrational quantum number of mode No. 9) is 1.3 and 0.9 states per cm⁻¹ for CHDBrI⁺ and $CHDBr2+$⁠, respectively. At the energy of the excited state of the C–D stretch mode (v₈ = 1), the densities are 3 times lower. For CHCaBrI⁺ and $CHCaBr2+$⁠, the density at v₉ = 1 is significantly higher, ranging from 35 to 25 states per cm⁻¹, but for the C–H bend modes (v₈ = 1), the density drops below 1 state per cm⁻¹ for both calcium substituted molecules. These densities are much higher relative to the effective density that should be considered as redistribution is less likely to proceed to combinations of more than two modes. An accidental overlap is unlikely for these modes when compared to the natural linewidth of the vibrational excited state, even when considering the densities resulting from combinations of all modes.

These are good indications for long-lived excited vibrational states. We do not explicitly estimate the state lifetimes in the current manuscript. However, spectroscopy using lower energy vibrational modes would enhance the lifetimes with respect to dissociation.

Additionally, in order to support 3-wave mixing schemes such as the one presented in Ref. 37, we also need the rotational constants of these molecular ions. These are shown in Table VII. The two deuterated molecules have approximately prolate symmetric tops, while the Yb substituted molecule may be approximated as having an oblate top.

Finally, we discuss the sensitivity of our candidates to Zeeman shifts. The chiral molecular ions have two spin multiplicities for the candidates. Closed-shell molecules such as CHCaBrI⁺ will have a small magnetic moment, which makes them favorably immune to magnetic field drifts.

Although we have skipped the computations for CHYbBrI⁺ in this work, we can compare the results between BrI-containing molecules and Br₂-containing molecules. In particular, we observe the similarities between $CHCaBr2+$ and CHCaBrI⁺ as well as between $CHCaBr2+$ and $CHYbBr2+$ with respect to the ionization wavelengths, dissociation threshold, and transition state energy. These similarities in molecules that are explicitly examined lead us to infer that CHYbBrI⁺ is also a promising candidate from an experimental perspective.

Sections IV and V discussed pathways to create cold molecules, which affect the contrast and the stability of the molecular ions, which relate to the accessible coherence time in Eq. (1). However, to estimate the number of molecules that need to be measured in a precision measurement assuming that the quantum projection limit is achieved,³⁷ we must compare the magnitude of the expected shift due to PNC in the vibrational frequency to the expected precision δf.

Here, we present the PNC calculations for the different vibrational modes of CHCaBrI⁺. Table VIII shows the PNC shifts expected for the various modes, with 100 and 29 mHz shifts for the most relevant modes for precision spectroscopy, the C–H bend and stretch modes (No. 8 and 9 in Table IV, respectively). We also computed the PNC shifts for the doubly isotopically chiral CHD⁷⁹Br⁸¹Br⁺, which are significantly lower, probably due to the relatively low mass of its constituents. However, its symmetric structure may simplify the molecule’s spectrum, promoting other aspects of the experiment. In contrast, the PNC frequency shift in CHDBrI⁺ is very large, on the order of 1 Hz for most of its higher energy modes, and is fully reported in Ref. 39.

For the PNC calculations, the molecular geometry was optimized on the ωB97M-V/Def2-TZVPP level of theory using Q-Chem 5.2.2.⁷⁵ 

In order to obtain the PV contributions to the total energies, we carried out single point relativistic DFT calculations using the DIRAC23 program.⁷⁶ In order to conserve computational effort, we replaced the 4-component Dirac Hamiltonian by the exact 2-component (X2C) Hamiltonian, where the large and small components are exactly decoupled and the positive energy spectrum of the 4-component Hamiltonian is reproduced within numerical precision. In this scheme, the spin-same-orbit interactions are introduced in a mean-field fashion by using the AMFI procedure.⁷⁷ We used the CAM-B3LYP* functional, whose parameters were adjusted by Thierfelder et al., to reproduce the PV energy shifts obtained using coupled cluster calculations.⁷⁸ Dyall’s v4z basis sets were used for all the elements.⁷⁹ Alongside the relativistic absolute energy at each geometry, these calculations also yield the PV energy contribution, E^PV.

To calculate the vibrational parity violating frequency shifts, the parity violating shifts of the vibrational ground and first excited states are needed, which we obtained as follows. We performed relativistic single-point calculations at 11 equally spaced points between −0.5 and 0.5 Å along the selected normal mode. This yields the potential energy and the parity violating energy as a function of the normal coordinate q; we fitted polynomials to these points in order to create smooth potential and parity violating energy curves V(q) and V_PV(q).

Any candidate discussed here requires a pathway for its creation if it is to be used in a precision measurement. For some of the candidates, such as CHDBrI and CHDBr₂, the non-chiral counterparts, CH₂BrI and CH₂Br₂, are commercially available. The natural abundance between the two bromine isotopes is 1:1, leaving 50% of the molecules in the pro-chiral mixed isotope form. The molecules with different bromine isotopes can be chosen through mass selection. For the deuterated molecules, selection is also an option, but the very low natural abundance means that a synthesis method is preferable.

The Ca substituted molecules would need to be generated in the vacuum chamber. For example, it may be possible to generate CHCaBrI by creating a Ca plasma by laser ablation near a supersonic expansion seeded with CH₂BrI or CHBr₂I. A similar scheme⁸³ has been used to generate CH₃Ca by ablation of Ca near CH₃Cl. Another similar scheme has been proposed to create Yb substituted methanes,⁸⁴ which may be a pathway to generate CHYbBrI.

The search for PNC in molecules can benefit from the long interrogation times accessible in trapped chiral molecular ions as well as the enhanced PNC shifts they are predicted to exhibit. However, for a successful precision spectroscopy experiment with chiral molecular ions, a favorable candidate must also fulfill other criteria, including efficient state preparation, high quantum efficiency in detection, and resistance to predissociation when vibrationally excited. In this work, we investigate these properties for several five-atom, carbon-center, tetrahedral chiral cations via ab initio calculations and estimate the magnitude of the PNC shift for some of the candidates.

To this end, we calculate several electronic properties mainly using coupled-cluster based methods. We validate the chirality of the optimized cation geometries and calculate vertical and adiabatic ionization energies, dissociation channel energies, isotopic vibrational modes, rotational constants, transition state energies, several excitation energies of the neutral systems, and PNC frequency shifts where relevant.

Our in-depth study in search of candidates for trapped chiral-molecular-ion precision spectroscopy has revealed that CHXBrI⁺, where X ∈ {D, Ca}, and isotopically chiral CHX⁷⁹Br⁸¹Br⁺, where X ∈ {D, Ca, Yb}, are favorable candidates. These candidates have promising avenues toward their preparation with internally cold temperatures and are stable in the charged form. Moreover, the magnitude of vibrational frequency shifts due to PNC is shown to be significant for selected candidates.

The supplementary material Tables S1–S6 contain benchmark calculations for IPs, optimized geometries, and dissociation energies, which demonstrate the need to use EOM-CC for calculating doublet states. Table S7 demonstrates the importance of augmenting the basis sets for excitation energy calculations. Table S8 compares vibrational modes for the neutral, achiral CH2BrI and CH2Br2 with experimental measurements and introduces vibrational scaling factors. These factors can be used to retrieve realistic cation vibrational frequencies, for which experimental values do not exist to the best of our knowledge.

We acknowledge I. Gilary for useful discussions. E.E. acknowledges Peter Schwerdtfeger for useful discussions. This research was supported by the Israel Science Foundation under Grant No. 1661/19 and the Israel Science Foundation under Grant No. 1142/21. This research project was partially supported by the Helen Diller Quantum Center at the Technion. E.E. wishes to acknowledge the Indonesia Endowment Fund for Education/Lembaga Pengelola Dana Pendidikan (LPDP) for research funding. A.B., E.E., and L.F.P. acknowledge the Center for Information Technology of the University of Groningen for their support and for providing access to the Peregrine high-performance computing cluster.

The authors have no conflicts to disclose.

Arie Landau: Conceptualization (equal); Investigation (lead); Software (lead); Writing – original draft (lead); Writing – review & editing (lead). E. Eduardus: Investigation (lead); Methodology (equal); Software (equal); Writing – review & editing (equal). Doron Behar: Formal analysis (supporting); Investigation (supporting); Writing – review & editing (supporting). Eliana Ruth Wallach: Formal analysis (supporting); Investigation (supporting); Writing – review & editing (supporting). Lukáš F. Pašteka: Investigation (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). Shirin Faraji: Methodology (equal); Supervision (equal). Anastasia Borschevsky: Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Writing – review & editing (equal). Yuval Shagam: Conceptualization (lead); Funding acquisition (lead); Investigation (lead); Methodology (equal); Software (equal); Supervision (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.