Harnessing the Spin-Flip Radiative Lifetimes of Optically Addressable Molecular Qubits

Abstract

Optically addressable molecular qubits based on spin-flip (SF) emissive transitions are promising candidates for quantum technologies due to their sharp luminescence lines and tunable optical-spin interfaces. Yet, the microscopic mechanisms controlling the spin-flip radiative lifetime of SF emitters, a key property for efficient spin readout, remain largely unexplored. Here, we present a computational study of several Cr⁴⁺ and Mo⁴⁺ pseudotetrahedral molecular qubits, and we identify chemical and structural features that influence the transition dipole moment associated with the SF emission, which, in turn, governs the SF radiative lifetime. We find that the magnitude of the dipole moment is governed by the multireference character of the spin-flip excited-state wave function, which can be modulated by tuning the energy separation between the d orbitals of the metal and the spin-pairing energy. Both parameters are sensitive to molecular symmetry, metal–ligand bond covalency, and bond anisotropy and leave room for modulation via ligand and metal design, as well as applied strain, which is relevant for sensing applications. Our findings provide a mechanistic framework for understanding and tuning the spin-flip radiative behavior of molecular qubits and SF emitters that may guide future advances in quantum information science.

Introduction

Photoactive transition metal (TM) complexes play an important role in solar energy conversion, photocatalysis, − and sensing. − While many of their applications rely on charge-transfer (CT) excited states, , spin-flip (SF) metal-centered excitations also hold promise in several fields, including photoredox catalysis, − optical oxygen, , temperature, , and pressure , sensing, as well as biosensing and telecommunication technologies that rely on circularly polarized SF luminescence. −

SF excited states originate from an intraconfigurational excitation accompanied by the spin inversion of one of the valence electrons; hence SF excitations do not lead to substaintial reorganization of the electron density across ligand scaffolds, resulting in a markedly different photophysical behavior compared to CT excitations. CT photoluminescence often leads to broad bands with large vibronic coupling, due to large geometrical distortions in the excited state, and short lifetimes. In contrast, in SF processes, the ground and excited states exhibit nested potential energy surfaces, thereby minimizing vibrational relaxation (see Figure A) and resulting in sharper emission bands, longer excited state lifetimes and high quantum yields, if nonradiative pathways are efficiently suppressed. ,

1.

(A) Schematic representation of the potential energy surfaces of the triplet ground state (³GS) and the lowest-lying singlet (¹ES) and triplet (³ES) excited states of a tetrahedral spin-flip emitter with a d² valence electron configuration. The arrows indicate the main optical transitions between the triplet and singlet states (green arrow: resonant excitation to ¹ES; blue arrow: off-resonant excitation; purple wavy arrow: intersystem crossing; red arrow: spin-flip emission). Insets: intraconfigurational microstates relevant for the ground triplet and first excited singlet states of a d² tetrahedral spin-flip emitter. (B) Structure of the pseudotetrahedral optically addressable Cr­(o-tolyl)₄ molecular qubit. Gray and cyan spheres represent chromium and carbon atoms, respectively. Hydrogen atoms have been omitted for clarity.

A well-known example of SF emitters is that of Cr³⁺ octahedral complexes (d³ electron configuration), or “molecular rubies”, that remain the most extensively studied SF systems to date. ,− However, spin-flip luminescence has been explored across a relatively wide range of TM complexes featuring various d electron configurations, including d³ (e.g., Mo³⁺, W³⁺, Mn⁴⁺, Re⁴⁺), − d⁸ (e.g., Ni²⁺) and d² (e.g., Cr⁴⁺, V³⁺). −

Optically addressable molecular spin qubitsalso known as molecular color centers (MCCs)represent a fairly new class of leading materials for scalable quantum technologies, where tailored spin-optical interfaces enable the initialization and readout of ground state molecular electron spins. − Particularly, pseudotetrahedral Cr⁴⁺ d² spin-flip emitters are emerging as highly promising candidates: , the molecular architecture of Cr⁴⁺ MCCs (e.g., Cr­(o-tolyl)₄, Figure B) is specifically tailored to emulate the spin-optical initialization and readout mechanism of the anionic nitrogen vacancy (NV^–) in diamond, a solid state color center widely studied for applications in quantum technologies. , Particularly, Cr⁴⁺ MCCs are designed to ensure that the strong field of the aryl ligands favors the presence of an emissive spin-flip singlet excited state (¹ES) at an energy lower than that of the lowest-lying triplet state (³ES). Cr⁴⁺ MCCs also display relatively long spin–lattice and spin–spin relaxation times (T₁ and T₂, respectively), which are critical properties to ensure the storage and manipulation of quantum information before decoherence occurs.

A distinguishable feature of MCCs lies in their extensive chemical modularity, a property that color centers in bulk semiconductors lack. Specifically, the spin structure and coherence of Cr⁴⁺ molecular qubits can be tuned by chemical modification of their ligand scaffold and environment, offering the appealing perspective of chemical design of their properties for different applications. For example, cocrystallizing Cr­(o-tolyl)₄ with Sn­(o-tolyl)₄ minimizes spin interactions, , and fluorinating the host matrix significantly increases spin coherence (from 2.9 to 10.6 μs), leading to clock transitions via a large transverse zero-field splitting (E). In addition, through local symmetry manipulation of hybrid host matrices, one can vary the zero-field splitting and spin coherence of molecular qubits. Notably, Laorenza et al. synthesized Cr­(alkyl)₄ qubits, showing that stronger field alkyl ligands reduce the axial zero-field splitting (D) and red-shift the emission zero-phonon line, enabling control of optical properties for applications such as biological sensing.

Controlling the spin-flip emission lifetime (or rate) of optically addressable molecular qubits is essential for quantum computing and sensing applications. In the qubits, the emission bandwidth is dominated by inhomogeneous line broadening and is therefore largely independent from the radiative lifetime. Hence, a large spin-flip emission rate k_SF (see Figure A), combined with a sharp emission band, is key to enabling the qubit’s spin-selective readout mechanism. Furthermore, the spin-flip emission lifetime of the qubits $({\tau }_{SF}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$k_{SF}$}\right.)$ should be shorter than the spin–lattice relaxation time (T₁), to ensure that the optical excitation and emission occur faster than thermal equilibration of the ground-state spin sublevels, thus enabling optical spin polarization. , Therefore, rational design strategies aimed not only at enhancing T₁, but also at decreasing τ_SF are highly desirable. More broadly, controlling τ_SF could benefit applications in photoredox catalysis and sensing involving bright emission of SF complexes.

In recent years, many experimental and theoretical efforts have been devoted to controlling nonradiative decay pathways to shorten τ_SF in SF emitters. Design strategies for enhancing k_SF have primarily focused on preventing the nonradiative depopulation of the spin-flip state. For example, in Cr³⁺ octahedral complexes, the emitters are designed to ensure a large energy gap between the emissive ²E and higher-lying ⁴T₂ states, thus avoiding detrimental reverse-intersystem crossing (reverse-ISC). ,,, Other strategies involve avoiding vibrational quenching (occurring, for instance, via overlap with C–H overtones) by deuteration or by inducing structural rigidity. , Vibrational quenching has been invoked, for example, to explain the lack of detectable SF emission in V³⁺ and Mo⁴⁺ (d²) pseudotetrahedral complexes, where a low-lying ¹ES state may either promote vibrational quenching or increase nonradiative decay by reducing the energy gap between the two electronic states involved in the transition. Increased metal–ligand covalency is known to lower the energy of the SF state and suppress reverse-ISC, as it reduces electron–electron repulsion in the doubly occupied excited-state closed-shell singlet configuration (see inset in Figure A). ,−

However, design strategies aimed directly at tuning the intrinsic spin-flip radiative lifetimes remain largely unexplored. Known effects are mostly limited to enhancements in spin–orbit coupling, such as the incorporation of heavy atoms. For instance, Ye et al. recently showed that introducing iodine into Cr³⁺ ligands nearly tripled the SF emission rate of the ²MC state. More recently, increasing metal–ligand covalency was shown to reduce the dissymmetry factor (g_lum) of circularly polarized luminescence of Cr³⁺ emitters, providing indirect evidence of an increased transition dipole moment for the spin-flip transition. The authors suggested stronger metal–ligand orbital mixing and therefore relaxed selection rules as plausible explanations of the observed effect, even though further investigation of this phenomenon was deemed necessary.

Here we investigate a series of M­(L)₄ (M = Cr⁴⁺, Mo⁴⁺; L = aryl, alkyl) pseudotetrahedral optically addressable molecular spin qubit candidates, with the goal of understanding how structural and bonding features influence the intrinsic radiative spin-flip behavior. By combining DFT and multireference computational approaches, we analyze the origin of the spin-flip transition dipole moment by isolating contributions arising from the chemical nature of the metal–ligand bond and from the rearrangement of the ligand scaffold via applied strain. By inspecting the nature of the spin-flip wave function, we propose a mechanistic model for understanding how the spin-flip emission transition dipole moment varies across different molecular systems, thus providing a foundation for the rational control of the spin-flip radiative lifetime of SF emitters by design.

Results and Discussion

The Impact of Metal/Ligand Nature

As mentioned in the Introduction, the optical readout mechanism of the SF-emitting qubits critically depends on the radiative lifetime of the spin-flip state. Additionally, the chemical identity of ligands has been demonstrated to be critical to control the electronic structure and quantum properties of molecular qubits. ,, To gain insights into the ligand and metal effect on the emission intensity, using a multireference SA-CASSCF-NEVPT2 computational approach (see Methods section), we computed the electronic and optical properties of six Cr⁴⁺ and Mo⁴⁺ molecular qubits (Cr1–Cr5 and Mo1, Figure A). This set of molecules spans pseudotetrahedral Cr­(aryl)₄ complexes (Cr1–Cr3), Cr⁴⁺ qubits bearing stronger-field alkyl ligands (Cr4, Cr5), and an isostructural pseudotetrahedral Mo­(aryl)₄ complex (Mo1). Notably, Cr1–Cr5 have been experimentally characterized, , and spin-flip luminescence was detected for all of them. Mo1 represents the only example of experimentally characterized Mo⁴⁺ qubit; however, unlike Cr⁴⁺ qubits, Mo1 showed no detectable spin-flip emission, likely due to nonradiative pathways quenching the emissive channel. We included this qubit in our work since it provides a valuable proxy to isolate the effect of the transition metal nature on the radiative lifetime.

2.

(A) Schematic structures of the CrL₄ (L = o-tolyl, 2,3-dimethylphenyl, 2,4-dimethylphenyl, (trimethylsilyl)­methyl, cyclohexyl) and MoL₄ (L = o-tolyl) molecular qubits investigated in this work. (B) Computed relative squared transition dipole moment ( ${|M|}_{rel.}^2$ ; see eq ) for all molecular qubits.

We computed the excited-state properties (i.e., vertical excitation energies and transition dipole moments) for all qubits. As shown in Figure S1, the computed excitation energies are consistent with the chemical nature of both ligands and metals and in agreement with experimental reports. ,,

The inverse lifetime of the spin-flip singlet excited state can be expressed as the sum of the inverse radiative and nonradiative lifetimes:

$${\tau }_{SF}^{-1}={\tau }_R^{-1}+{\tau }_{NR}^{-1}$$ 1

Since, as we mention in the Introduction, strategies to mitigate the nonradiative deactivation of spin-flip emissive states are widely explored, we focus exclusively on the radiative lifetime (τ _R ) of the ¹ES (first singlet excited state). The latter, under the assumption of fast thermalization among the ground triplet state sublevels, can be expressed as

$${\tau }_R=\frac{3t_0}{4{\alpha }_0^3}\frac{3}{{(\mathrm{\Delta }E)}^3{|M|}^2}$$ 2

with ΔE and M being the energy and the transition dipole moment of the spin-flip transition averaged over the triplet sublevels, $t_0={(4\pi {\epsilon }_0)}^2{\hslash }^3/m_e{e}^4$ and α₀ is the fine structure constant.

As shown by eq , the radiative SF lifetime is inversely proportional to the squared transition dipole moment of the spin-flip transition, in addition to the emission energy. Here, we use the transition dipole moment as the primary descriptor of the spin-flip emissive behavior of the systems, not only because it is an observable that directly reflects the intrinsic emissive behavior of SF emitters (and its relationship with their electronic structure), but also because radiative lifetimes would additionally depend on the refractive index of the medium they are embedded in and are therefore less suitable for direct systematic comparison in the present context.

Hence, we computed the transition dipole moments (|M|²) associated with the ³GS → ¹ES excitation for all qubit candidates. Specifically, we evaluated the relative squared transition dipole moment $({|M|}_{rel.}^2)$ for each qubit (X) using Cr1 as a reference:

$${|M|}_{rel}^2=\frac{{|M_X|}^2}{{|M_{Cr1}|}^2};X=\mathit{Cr}1-\mathit{Cr}5,\mathit{Mo}1$$ 3

The computed ${|M|}_{rel.}^2$ values (bar charts in Figure B) suggest that the nature of the ligands and the metal influences the transition dipole moment (and, therefore, the radiative lifetime) of the qubits: Cr­(aryl)₄ qubits (Cr1–Cr3) display larger ${|M|}_{rel.}^2$ values (1.00, 0.70, and 1.03 for Cr1, Cr2 and Cr3, respectively) compared to Cr­(alkyl)₄ qubits (0.24 and 0.15 for Cr4 and Cr5, respectively). Interestingly, we find that Mo1 (i.e., the only complex in the series for which no emission was detected experimentally) displays a larger ${|M|}_{rel.}^2$ value of 4.39 compared to the isostructural Cr1 (1.00). However, the computed low ¹ES energy (Figure S1) is consistent with the hypothesized inhibition of detectable emission via nonradiative decay. We also find that the relative SF oscillator strength of Mo1 (which accounts for both |M|² and (ΔE), see eq ) is strongly suppressed relative to its large transition dipole moment, as reflected by its much smaller $f_{os}/{|M|}^2$ ratio compared to Cr1–Cr5 (see Figure S2 and Table S1 in the Supporting Information). This indicates that its low transition energy strongly suppresses the radiative channel relative to its large transition dipole moment. Therefore, the absence of detectable emission from Mo1 is still consistent with its large computed transition dipole moment, since it reflects the effect of the low-energy ¹ES state on the overall oscillator strength (used as a proxy for radiative lifetimes) and the enhanced efficiency of nonradiative deactivation pathways.

It is interesting to disentangle the factors contributing to the different ${|M|}_{rel.}^2$ values along the series, as shown in Figure B. The magnitude of |M|² for the spin-flip transition is mainly affected by the extent of mixing between singlet and triplet states via spin–orbit coupling (SOC). As shown in Figure S3, the values of the spin–orbit coupling matrix elements (SOCMEs) between ³GS and ¹ES partially explain the exceptionally large ${|M|}_{rel.}^2$ for Mo1. Given that Mo⁴⁺ is heavier than Cr⁴⁺, stronger SOC is expected for Mo1 and, consistently, the latter features the largest SOCME (1.54 cm^–1), while the SOCME for Cr1–Cr5 is far below 1 cm^–1 in all cases. Interestingly, however, SOCMEs do not qualitatively explain the ${|M|}_{rel.}^2$ trends reported in Figure B: Cr4 and Cr5, despite displaying the lowest ${|M|}_{rel.}^2$ , have higher SOCMEs than Cr1 and Cr2, which display higher ${|M|}_{rel}^2$ values. Hence, this suggests that the SOC-induced wave function mixing is not the only property responsible for the trends observed in Figure B.

To further understand the observed patterns, we inspected the nature of the wave function of the triplet ground state (³GS) and the lowest-lying spin-flip excited states. ³GS is predominantly a single reference state, involving semioccupied $d_{x^2‐y^2}$ and $d_{z^2}$ orbitals. Consistent with the expected ligand-field ordering for strong-field d² tetrahedral complexes, we found three relevant low-lying spin-flip states in the excited-state manifold (Figure S4 in the SI). We analyzed the coefficients associated to the most relevant configuration state functions (CSFs) contributing to the multireference singlet wave functions (see Table S2 in the SI). In agreement with the symmetry-adapted linear combinations expected from ligand field theory, the lowest-lying (emissive) spin-flip state (¹ES) is found to mostly result from an antisymmetric combination of two closed-shell singlet configuration state functions (CSF₁ and CSF₂ in Figure A); these involve a doubly occupied orbital (either $d_{x^2‐y^2}$ or $d_{z^2}$ ) mixed with an additional open-shell configuration (CSF₃) featuring semioccupied $d_{x^2‐y^2}$ and $d_{z^2}$ orbitals. Again, in agreement with the expected ligand field ordering, CSF₃ is dominant for the second-lowest-lying singlet state, while the highest spin-flip state mostly arises from the symmetric combination of CSF₁ and CSF₂ (Table S2).

3.

(A) Configuration state functions (CSFs) describing the lowest-lying spin-flip excited singlet state in a pseudotetrahedral ML₄ qubit with S₄ symmetry. (B) Coefficients of the closed-shell (|c ₁| – |c ₂|) CSFs composing the ¹ES of Cr1–Cr5 and Mo1 (C) Schematic illustration of the competition between the splitting of the nonbonding $d_{x^2‐y^2}$ and $d_{z^2}$ orbitals (Δ) and the spin-pairing energy penalty (P) in determining the predominant CSFs (i.e., with the largest CSF coefficients) composing ¹ES. (D) Nephelauxetic parameter (β), computed with ab initio ligand field theory, and Δ, computed at the SA-CASSCF-NEVPT2 level, for Cr1–Cr5 and Mo1.

As a result, within a spin-free framework, (see SI for an extended derivation), the transition dipole moment (M) can be reasonably approximated as

$$M\approx (c_1-c_2)M_1+c_3{M}_3$$ 4

where M ₁ and M ₃ are the spin-free dipole contributions from CSF₁ and CSF₃, respectively, and c ₁, c ₂ and c ₃ are the respective CSF coefficients.

Eq shows that in order to maximize M, the contribution of CSF₁ should outweigh that of CSF₂ (i.e., |c ₁| > |c ₂|). Additionally, we note that M ₃ is expected to be smaller than M ₁ and M ₂ since the transition from an open-shell singlet (CSF₃) to a triplet formally involves no change in the angular momentum of the electron, and small c ₃ values would ensure larger values of M. As expected, due to the lower energy of the $d_{x^2‐y^2}$ orbital in S₄ symmetry, the weight of CSF₁ (in absolute value, |c ₁|) is consistently larger than |c ₂| for all qubits (Table S2). However, we compute the |c ₁| – |c ₂| difference to evaluate how much CSF₁ outweighs CSF₂ and, in agreement with eq , the trends we find for Cr1–Cr5 and Mo1 qualitatively correlate with the computed ${|M|}_{rel.}^2$ values (Figure B). Cr1–Cr3 with intermediate ${|M|}_{rel.}^2$ values (0.70–1.03), also display wave functions with intermediate (0.15–0.19) |c ₁| – |c ₂| values, consistent with the predominant contribution of CSF₁ in all three cases. Mo1, displaying the largest (4.39) ${|M|}_{rel.}^2$ value, also displays the largest |c ₁| – |c ₂| (0.21). In all cases, c ₃ = 0. Interestingly, Cr4 and Cr5, displaying the lowest ${|M|}_{rel.}^2$ (0.24 and 0.15, respectively) feature the lowest |c ₁| – |c ₂| (0.09 and 0.08, respectively) values and nonzero c ₃ values (0.04 and 0.38, respectively).

Further analysis elucidates the chemical origin of the observed trends. In principle, as shown in Figure C, the magnitude of c ₁ and c ₂ can be controlled by two parameters, namely the energy separation between the $d_{x^2‐y^2}$ and $d_{z^2}$ orbitals (Δ) and the spin-pairing energy penalty (P) arising from placing two electrons in the same orbital. Particularly, increasing Δ and decreasing P would favor the placing of two electrons in the lowest-lying $d_{x^2‐y^2}$ orbital, thus favoring CSF₁ and enhancing c ₁. Conversely, smaller Δ and larger P would increase the contributions of CSF₂ and CSF₃, therefore enhancing c ₂ and c ₃.

We find that the nature of the metal–ligand bonds in the six qubits studied here, and the symmetry of the ligand environment can rationalize the trends discussed above. The nature of the Cr-alkyl bond and Cr-aryl bonds, for instance, is intrinsically different. Aryl ligands, as previously discussed, form both π- and σ-bonding with the chromium center, while stronger field alkyl ligands can only form σ-type bonds with Cr⁴⁺, with the Cr-aryl bond being more covalent than the Cr-alkyl one. This difference in bonding is expected to lead to a higher P value for Cr4 and Cr5, explaining the larger c ₂ and c ₃ and smaller ${|M|}_{rel.}^2$ . As for Mo1, the larger size and more diffuse nature of the Mo⁴⁺ ion compared to Cr⁴⁺ is likely responsible for a decrease in P, due to the lower interelectronic repulsion and improved overlap with ligand orbitals.

To provide a more quantitative interpretation, we computed the nephelauxetic parameter β using ab initio ligand field theory (AILFT) at the SA-CASSCF-NEVPT2 level (see Methods section). The parameter β serves as a measure of the nephelauxetic effect in TM complexes, indicating the decrease of the Racah interelectronic repulsion parameter B (related to the pairing penalty P) arising from an increased metal–ligand bond covalency. The values of β decrease below one as covalency increases and interelectronic repulsion becomes weaker. As shown in Figure D (red plot), the β values decrease following the order Cr5 > Cr4 > Cr1 ∼ Cr2 ∼ Cr3 ≫ Mo1, in agreement with the expected increased bond covalency for aryl systems compared to alkyl systems and for Mo compared to Cr. In spite of being consistent with trends observed for c ₁ – c ₂ and ${|M|}_{rel.}^2$ , increasing in the order Cr5 < Cr4 < Cr2 < Cr1 ∼ Cr3 ≪ Mo1, the trends in the values of β values do not provide a complete rationalization of our results. For instance, despite having the same β, Cr2 has lower c ₁ – c ₂ and ${|M|}_{rel.}^2$ than Cr1 and Cr3 and β values for Cr4 and Cr5 (0.83 and 0.85, respectively) are only slightly higher than those of Cr1–Cr3 (0.79).

We achieve a full rationalization of the results reported in Figures B and B when the parameter Δ is considered (Figure D, in blue): Cr2 has a smaller Δ than Cr1 and Cr3 and both Cr4 and Cr5 have even smaller Δ values. We find that the values of Δ are related to the pseudotetrahedral symmetry of the complexes. For perfectly tetrahedral complexes, $d_{x^2‐y^2}$ and $d_{z^2}$ orbitals are degenerate (hence Δ = 0), while in systems with S₄ symmetry (the nominal symmetry of the qubits studied here), the energy of $d_{x^2‐y^2}$ is lower, hence Δ ≠ 0. By computing Continuous Shape Measures − (CShM, see Table S3 and Figure S5 in the Supporting Information), we understand these differences in terms of deviation from the ideal tetrahedral symmetry: Cr1 and Cr3 display the largest deviations, consistently with the largest Δ, while Cr2, Cr4 and Cr5 display structures closer to tetrahedral symmetry, resulting in smaller Δ values. Notably, for Mo1, the value of Δ is comparable to those of Cr2 and Cr5 and is consistent with its similar pseudotetrahedral distortion; however, its much smaller β value indicates a substantially stronger nephelauxetic effect, making the reduction of the spin-pairing penalty P the dominant factor controlling the wave function composition.

For the first time, our findings show a correlation between the unique nature of the singlet wave function and the spin-flip emission intensity. We find that enforcing the double occupation of the lowest energy d orbital ( $d_{x^2‐y^2}$ in this case) in pseudotetrahedral systems, where Δ ≠ 0, leads to an increase of the SF transition dipole moment. Our results show that the nature of the SF wave function can be modulated not only by tuning the metal–ligand bond covalency but also by controlling the symmetry of the qubits. In the following Section, we further test the consistency of the proposed model by exploring the effect of applied strain on the spin-flip radiative lifetime of Cr1.

The Impact of Applied Strain

According to the scheme proposed in the previous section, it should be possible to enhance the spin-flip transition dipole moment of the qubits and hence reduce their radiative lifetime by tuning P through the nephelauxetic effect, combined with the modulation of Δ, i.e., the energy separation between the two semioccupied $d_{x^2‐y^2}$ and $d_{z^2}$ orbitals.

To identify strategies to modulate Δ in Cr⁴⁺ molecular qubits, we performed a series of calculations on strained structural models of the experimental structure. Experimentally, Cr1–Cr5 and Mo1 have been synthesized and characterized in the form of diluted cocrystals. For instance, Cr1 (Cr­(o-tolyl)₄) has been prepared via cocrystallization with the diamagnetic analogue Sn­(o-tolyl)₄ (Sn1), resulting in a diluted cocrystal with a low Cr:Sn ratio to avoid detrimental interactions between the electron spins on Cr centers (Figure A). ,

4.

(A) A 2 × 2 × 2 diluted supercell derived from the experimental structure of the Cr1Sn1 cocrystal (tetragonal unit cell, space group: P4̅2₁c). The Sn1 units are reported in gray and the Cr1 unit is reported with a different color. The side arrows indicate the [110] direction of the applied tensile (green) and compressive (purple) biaxial strain. (B) Computed relative squared transition dipole moment ( ${|M|}_{rel.}^2$ , eq , bar plot) and relative radiative spin-flip lifetime ( ${\tau }_{SF}^{rel}$ , eq , scatter plot) of Cr1 in biaxially strained Sn1 host structures along the [110] direction (positive values indicate tensile strain and negative values indicate compressive strain). ${|M|}_{rel.}^2$ and ${\tau }_{SF}^{rel}$ are computed according to eqs and , respectively, using in both cases the nonstrained structure as reference.

It is interesting to investigate the impact of applied strain on the emissive properties of the qubits for several reasons. First, applied strain on the diluted Cr1Sn1 cocrystal represents an ideal scenario to impose structural rearrangements on the ligand scaffold without formally affecting the chemical nature of the ligands. Furthermore, the strain susceptibility of the optical and spin properties of the qubits is of great relevance for their applications as quantum nanosensors; − similarly, molecular SF emitters have shown great promise as pressure sensors. ,

In our calculations, we applied nonisotropic biaxial tensile and compressive strain along the [110] direction of the cocrystal up to 2% (Figure A). We chose this direction because it is perpendicular to the S₄ symmetry axis of Cr1, thus enabling a manipulation of the ligand manifold while preserving the (S₄) symmetry of Cr1. Additionally, biaxial tensile stressing is a common testing procedure. − We used plane-wave density functional theory (DFT) at the PBE level to relax the 2 × 2 × 2 cocrystal supercell (see Methods section) and computed the optical properties of Cr1 at the SA-CASSCF-NEVPT2 level. We recently demonstrated that this combined approach provides an accurate description of the spin and optical properties of chromium molecular qubits.

As shown in Figure B, applied biaxial tensile strain has a remarkable effect on the relative transition dipole moment associated with the spin-flip transition. We find an almost linear relationship between the applied strain and ${|M|}_{rel.}^2$ . In fact, when we progressively apply compressive strain (up to 2%) to Cr1Sn1, we observe a decrease of the transition dipole moment of Cr1 compared to the nonstrained structure. The values of ${|M|}_{rel.}^2$ decreases from 1.00 for the nonstrained structure to 0.91 for the 2% compressively strained structure. Conversely, applying tensile strain (up to 2%) to Cr1Sn1 leads to an almost symmetrical increase of ${|M|}_{rel.}^2$ of Cr1, to 1.11 for the maximally strained structure. Additionally, we observe an effect of the applied strain on the energies of the lowest-lying singlet and triplet states (see Figure S6); however, the strain susceptibility of vertical excitation energies is found to be smaller compared to ${|M|}_{rel.}^2$ .

While the variations found here are smaller than those discussed in the previous sections for Cr⁴⁺ aryl and alkyl complexes ( ${|M|}_{rel.}^2$ for alkyl qubits is below 0.30, see Figure B), our results are significant, considering that no changes to the qubit and host chemical composition are involved. Furthermore, a ±2% applied strain corresponds to a roughly ±10% change in ${|M|}_{rel.}^2$ . The strain susceptibility of the radiative emissive properties of Cr1 is significant because it suggests a mechanism for signal amplification that could be relevant to quantum sensing; in addition, it highlights the intrinsic high sensitivity of ${|M|}_{rel.}^2$ with respect to minor geometrical rearrangements of the ligand scaffold.

Additionally, since the composition of the qubit’s host is unchanged and assuming that applied strain negligibly affects the refractive index of the material, we can also compute the relative radiative spin-flip lifetimes for each strained structure (X) using the nonstrained structure (ref) as a reference as follows:

$${\tau }_{SF}^{rel}\approx \frac{{\tau }_{SF}^X}{{\tau }_{SF}^{ref}}=\frac{{(\mathrm{\Delta }{E}_{ref})}^3{|M_{ref}|}^2}{{(\mathrm{\Delta }{E}_X)}^3{|M_X|}^2}$$ 5

As shown in Figure B (scatter plot) and consistently with the minor effect of applied strain on the energy of the spin-flip singlet state (Figure S6), the qualitative trend for spin-flip lifetimes mirrors that of the relative transition dipole moments: upon application of compressive (tensile) strain, the relative spin-flip radiative lifetime almost linearly becomes longer (shorter) by roughly 10%, essentially indicating the transition dipole moment as the core control parameter for the relative radiative lifetimes upon applied strain.

To understand the origin of the high strain susceptibility of ${|M|}_{rel.}^2$ (and ${\tau }_{SF}^{rel}$ ) we analyzed the nature of the spin-flip wave function in all strained Cr1 structures. As expected from the results of the previous section, applying compressive/tensile strain along the [110] direction leads to a change in the relative weight of the closed-shell configurations CSF₁ and CSF₂ (Figure ). As shown in Figure S7, c ₁ – c ₂ increases almost linearly with tensile strain and similarly decreases with biaxial compressive strain. This result further corroborates the connection between the composition of the spin-flip wave function, and the transition dipole moments discussed earlier.

Since we do not expect major changes in terms of overall nephelauxetic effect (at least formally, vide infra), we turn to analyze the energy separation between the two semioccupied ground state $d_{x^2‐y^2}$ and $d_{z^2}$ orbitals (Δ), which, as shown in Figure , plays a role in enhancing c ₁ and, in turn, increasing ${|M|}_{rel.}^2$ . Indeed, as shown in Figure A, the energy separation between the two orbitals linearly correlates with ${|M|}_{rel.}^2$ . Particularly, upon tensile strain, Δ and ${|M|}_{rel.}^2$ (and, consistently c ₁ – c ₂) increase, while Δ and ${|M|}_{rel.}^2$ linearly decrease upon compressive strain (as does c ₁ – c ₂).

5.

(A) The energy separation between $d_{x^2‐y^2}$ and $d_{z^2}$ orbitals (Δ) as a function of the transition dipole moment $({|M|}_{rel.}^2)$ of Cr1 in biaxially strained Sn1 host structures along the [110] direction. Pearson’s correlation coefficient (R²) is 0.98. (B) Iso-surfaces of the weakly antibonding $d_{x^2‐y^2}$ and $d_{z^2}$ active SA-CASSCF orbitals of nonstrained Cr1 (iso-value: 15 me/a₀ ³). (C) Schematic illustration of the effect of the chromium-ligand bond length (r_CrL , top) and of the C–Cr–C–C characteristic torsional angle (ω, bottom) on the value of Δ and P. (D) The average chromium-ligand bond length (r_CrL ) and the torsional angle ω as a function of applied biaxial tensile strain along the [110] direction for Cr1.

The complex relationship between ${|M|}_{rel.}^2$ , the composition of the spin-flip wave function and the electronic structure of Cr⁴⁺ molecular color centers is further highlighted by the centers’ behavior under applied strain. To gain a more thorough understanding, we analyze the nature of the $d_{x^2‐y^2}$ and $d_{z^2}$ orbitals. As shown in Figure B, both orbitals present a weak antibonding character as a result of the interactions with the π system of the o-tolyl ligands, in agreement with molecular orbital theory for tetrahedral complexes. These interactions are crucial in determining the relative energies of the two orbitals, as minor geometrical rearrangements significantly affect the electronic structure of the qubits. We find that there are mainly two geometrical descriptors that can be used to tune this interaction. First, the average metal–ligand bond length (r _CrL , Figure C), which can affect the extent of the antibonding character: shorter (longer) r _CrL leads to an increase (decrease) of the strength of the antibonding character of the two orbitals by increasing (decreasing) the overlap between the metal and ligand orbitals. Furthermore, decreasing r _CrL leads to an increase in the interelectronic repulsion in the complex, and thus to a higher energy penalty P. As shown in Figure D, and as expected, r _CrL varies almost linearly with strain: longer bond lengths are observed upon the application of tensile strain, and consistently are accompanied by lower P, and higher c ₁ – c ₂ and ${|M|}_{rel.}^2$ .

The values of the bond length alone, however, do not rationalize the trends we find as a function of Δ. We find that, in pseudotetrahedral TM complexes, the extent of the separation (and order) of the two $d_{x^2‐y^2}$ and $d_{z^2}$ orbitals is governed by the characteristic C–Cr–C–C dihedral angle formed between the ligands comprising the S₄ symmetry axis (ω, Figure C). As reported by numerous experimental and computational works, this angle is related to the anisotropy of the π metal–ligand interactions: − specifically, the energy separation between the $d_{x^2‐y^2}$ and $d_{z^2}$ orbitals (with $d_{x^2‐y^2}$ being the lowest-energy orbital in systems with S₄ symmetry, e.g., Cr1) has been reported to increase as a function of ω, becoming maximal at ω = 90°. , As shown in Figure D, we confirm that this is the case: applied tensile (compressive) strain leads to an increase (decrease) in ω, and to larger (smaller) anisotropies in π bonding, resulting in increasingly (decreasingly) larger Δ values and, in turn, ${|M|}_{rel.}^2$ .

In summary, applied strain leads to subtle changes both in the Cr-ligand bond length and the characteristic C–Cr–C–C dihedral angle of Cr1. These minor geometrical rearrangements of the ligand coordination sphere affect the antibonding character of the orbitals and their interelectronic repulsion, leading to variations in Δ and P. As successfully demonstrated by our model, these minor variations lead to alterations of the spin-flip wave function composition that, in turn, result in a relatively high tunability of the spin-flip radiative lifetime and transition dipole moment. These findings provide a mechanistic framework for understanding how structural perturbations can modulate the spin-flip radiative lifetime of SF emitters and highlight the potential of these systems as highly tunable and versatile platforms for applications in quantum information science.

Conclusions

In this study, we identified the key factors governing the radiative spin-flip (SF) behavior of optically addressable Cr⁴⁺ and Mo⁴⁺ molecular qubits, thus offering a new perspective on how their emissive properties can be tuned by chemical and structural design. Through a combined multireference and density functional theory approach, we elucidated how the SF transition dipole moment is related to the multireference character of the excited singlet state.

Our findings revealed that due to the symmetry of the wave function, the SF transition dipole moment increases when a closed-shell configuration featuring a doubly occupied, lower-energy $d_{x^2‐y^2}$ orbital, is a dominant component of the wave function. Further analysis showed that the wave function composition can be engineered by tuning two key quantities: the energy separation between the $d_{x^2‐y^2}$ and $d_{z^2}$ orbitals (Δ), and the spin-pairing energy (P). By comparing the electronic structure of Cr­(alkyl)₄ and Cr­(aryl)₄ qubits, as well as of a Mo­(aryl)₄ complex, we find that P can be modulated by the metal/ligand bond covalency (nephelauxetic effect), while Δ is efficiently modulated by the extent of the distortion from an ideal tetrahedral molecular symmetry. Furthermore, we found that Δ can be modulated by perturbing the ligand scaffold via applied strain. Applied tensile strain increases the bond length, as well as the characteristic ligand–ligand torsional angle comprising the S₄ symmetry axis. The resulting perturbation of the metal–ligand bond anisotropy and the decrease of electron–electron repulsion lead to an increase in Δ and a decrease in P, resulting in a roughly 10% increase of the transition dipole moment under 2% applied tensile strain. Hence, our results show that strain can reshape the SF wave function and radiative properties, without altering the chemical composition of the system.

While a full assessment of the strain susceptibility of the optical and spin properties of the qubits is outside the scope of the present work, our analysis provides a qualitative assessment of the sensitivity of the electronic and optical properties of Cr⁴⁺ molecular qubits to applied strain. Our results are expected to be relevant for strain-sensitive applications such as quantum sensing, where external stimuli are translated into optical signals.

More broadly, our study presents a novel and physically grounded framework for the mechanistic understanding of the SF radiative behavior of transition metal-based optically addressable molecular qubits and, more generally, SF emitters. Our results highlight the sensitivity of the spin-flip emission to metal–ligand bonding, ligand field symmetry, and structural distortions, identifying chemically actionable parameters that enable the design of controlled spin-flip radiative lifetimes. The spin-pairing energy P and the orbital energy separation Δ, which control the character of the SF wave function and hence the spin-flip transition dipole moment, act as independent parameters to influence the radiative spin-flip behavior. The energy P can be easily decreased by increasing metal–ligand covalency, for instance by choosing ligands enabling σ- and π-bonding (e.g., aryl ligands) or by employing 4d metal centers, which induce stronger SOC and a stronger nephelauxetic effect. The energy separation Δ, instead, can be readily manipulated via symmetry control or structural distortions. For instance, heteroleptic or lower-symmetry systems with tunable spin-flip properties are worth exploring, since minor deviations from perfect S₄ pseudotetrahedral symmetry lead to an increase of Δ. Additionally, our results on applied strain highlight the possibility of modulating SF radiative emission by host matrix-induced perturbations: for instance, by designing nonisostructural hosts where the symmetry of the qubit is lowered in a controlled way without altering the qubit’s ligand scaffold. We note that the specific design rules identified here are tailored to spin-flip emission in transition-metal complexes and should not be directly transferred to other ODMR-active systems (e.g, organic radicals, acenes, rare-earth coordination complexes), where different readout mechanisms are employed. Nevertheless, the broader electronic-structure perspective adopted here may be useful for identifying the system-specific parameters that control optical spin readout in other molecular platforms.

In conclusion, our work bridges the gap between chemical understanding and functional design in spin-active molecular systems, providing a useful theoretical foundation for the future design and interpretation of emissive systems for and beyond applications in quantum technologies.

Methods

We compute the optical properties of all qubits using a combination of Density Functional Theory (DFT) and multireference methods. We recently showed that this approach can accurately reproduce experimental properties of MCCs and capture trends of optical and spin properties that are sensitive to the qubit structure.

The structure of isolated Cr1–Cr5 and Mo1 has been optimized in their triplet ground state using unrestricted DFT as implemented in the quantum chemistry code ORCA (5.0.3 version), , using the PBE exchange-correlation functional, the triple-zeta quality def2-TZVPP basis set, and the D3 atom-pairwise dispersion correction with the Becke-Johnson damping scheme (D3BJ). , The impact of the exchange-correlation functional in determining the structural properties of the qubits was evaluated for a subset of the systems studied here and, as shown in Table S4 in the SI, we found negligible differences between the results of several different functionals.

The structure of the Cr1 qubit embedded in the isostructural tin matrix Sn1 (Cr1Sn1) was obtained by building a 2 × 2 × 2 supercell of the experimental cocrystal structure. The latter was relaxed (both atomic positions and cell parameters) using plane-wave DFT as implemented in Quantum Espresso, , using the PBE functional, a kinetic energy cutoff of 100 Ry and SG15 pseudopotentials for all elements. The supercell Brillouin zone was sampled with the Γ point. The total energy and force convergence thresholds were set to 10^–5 and 10^–4 Ry, respectively. The strained Cr1Sn1 structures have been obtained following the same computational protocol. In the case of tensile strain along the [110] direction, the a and b cell parameters were increased by 0.5%, 1.0%, 1.5% and 2.0%. Analogously, compressed structures were obtained by reducing the a and b cell parameters by 0.5%, 1.0%, 1.5% and 2.0% and relaxing the atomic positions. In all cases, only the Cr1 unit was extracted from the relaxed (strained and nonstrained) cocrystal structures to evaluate the optical properties of the qubit.

The optical properties were computed using multireference approaches as implemented in ORCA. State-averaged complete active space self-consistent field (SA-CASSCF) calculations were performed including scalar relativistic effects using a second-order Douglas-Kroll-Hess (DKH) Hamiltonian − together with the cc-pVTZ-DK basis set. , The active space consists of 8 electrons and 13 orbitals, and includes the nd and (n+1)­d orbitals of Cr (n = 3) and Mo (n = 4), as well as the t₂ metal–ligand bonding orbitals (see Figure S8 for a graphical representation of the active space). The chosen active space is consistent with previous works, ,, and it was also confirmed as the optimal active space for all complexes by the atomic valence active space (AVAS) automated method. Additionally, the balance of the active space was confirmed by evaluating convergence of properties for Cr1 (see Figure S9), as well as from the qualitative agreement of the computed vertical excitations with experimental zero-phonon lines for Cr1–Cr5 (see Figure S1 in the SI). State-averaged CASSCF calculations were performed using equal weights over seven triplet and nine singlet roots, chosen, in agreement with choices adopted in previous works, , to provide a balanced description of the low-energy metal-centered d² ligand-field manifold predicted by the related Tanabe–Sugano diagram. Higher-energy ligand-field states are well separated from this manifold and were therefore not included in the state averaging. All energies were further refined by including dynamic correlation with N-electron valence-state second-order perturbation theory (NEVPT2) calculations. − Spin–orbit coupling (SOC) was explicitly included in all calculations using quasi-degenerate perturbation theory (QDPT), − which accounts for SOC effects to infinite order. The “PrintWF csf” keyword was used to print each wave function in the basis of the configuration state functions in the output (see a sample of the ORCA input file used in the SI). Using this keyword, the code prints the signed CSF coefficients used in this work (c₁, c₂, c₃) for each electronic state. A snapshot of the related section in the output file is shown in Figure S10. The transition dipole moment for each qubit and for each strained structure was computed at the SA-CASSCF-NEVPT2-QDPT level (see a snapshot of the related section of the output file in Figure S11).

The orbital energy separation Δ was computed using SA-CASSCF orbital energies. We emphasize that these orbital energies are not observables, but are used here as a relative ligand-field descriptor across the series. This use is justified by the fixed active space, fixed orbital convention, and identical state-averaging protocol used for all complexes, which provide a balanced representation of the low-energy d² manifold and allow for a consistent tracking of changes in the relative splitting of the frontier metal-centered active orbitals. Similar uses of SA-CASSCF-derived d-orbital splittings as qualitative descriptors for interpreting ligand-field, optical, and magnetic trends in transition-metal complexes have been reported previously. − Ab-initio Ligand Field Theory (AILFT) calculations , were carried out to compute the Racah parameter B of free Cr⁴⁺ and Mo⁴⁺ ions and of Cr1–Cr5 and Mo1 at the SA-CASSCF-NEVPT2 level, using the same computational settings described above. A (2,5) active space has been used and ten triplet and fifteen singlet states have been included in the state-averaging procedure.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/jacsau.6c00464.

• Additional figures and tables describing orbital analyses; strain effects; structure–property correlations for Cr⁴⁺ and Mo⁴⁺ molecular qubits (PDF)

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

The authors declare no competing financial interest.