# On the short and long phosphorescence lifetimes of aromatic carbonyls

Saikat Mukherjee, Moumita Kar, Mansi Bhati, Xing Gao, Mario Barbatti

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Saikat Mukherjee, Moumita Kar, Mansi Bhati, Xing Gao, Mario Barbatti. On the short and long phosphorescence lifetimes of aromatic carbonyls. Theoretical Chemistry Accounts: Theory, Computation, and Modeling, 2023, 142 (9), pp.85. 10.1007/s00214-023-03020-w. hal-04183783

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# On the short and long phosphorescence lifetimes of aromatic carbonyls

Saikat Mukherjee,<fnref n="1" />,ᵃ* Moumita Kar,<fnref n="1" />,ᵇ Mansi Bhati,<fnref n="2" />,ᶜ Xing Gao,<fnref n="3" />,ᵈ Mario Barbatti<fnref n="1" />,<fnref n="4" />,ᵉ*

Corresponding authors: SM: saikat.mukherjee@univ-amu.fr

MB: mario.barbatti@univ-amu.fr, www.barbatti.org

**Abstract:** This work applies theoretical and computational methods to investigate the relationship between phosphorescence lifetime and the electronic character of the lowest triplet state of aromatic carbonyls. A formal analysis of the spin-perturbed wave functions shows that phosphorescence is due to a direct spin-orbit coupling mechanism modulated by permanent dipoles when the T₁ minimum is ³nπ*. If the minimum is a totally symmetric <fnref n="3" />ππ*, phosphorescence is due to an indirect spin-orbit coupling mechanism involving transition dipole moments with other excited states. The magnitude difference between permanent and transition dipoles leads to a much faster ³nπ* phosphoresce than <fnref n="3" />ππ*. These predictions were verified with phosphorescence lifetime simulations of benzaldehyde and its three derivatives in the gas phase employing a vertical approximation and the nuclear ensemble approaches. Both predict ³nπ* emission within a few tens of milliseconds. While the vertical approach indicates a <fnref n="3" />ππ* emission within a few seconds, vibronic corrections bring this value down to about 200 ms.

**Keywords:** Phosphorescence, Organic chromophores, Benzaldehyde, Nuclear ensemble approach, Spin forbidden transitions, Selection rules

**Version of record:** S. Mukherjee, M. Kar, M. Bhati, X. Gao, M. Barbatti, **On the short and long phosphorescence lifetimes of aromatic carbonyls**, Theor. Chem. Acc. (2023). DOI: 10.1007/s00214-023-03020-w

<footnote n="1">¹Aix Marseille University, CNRS, ICR, Marseille, France</footnote>

<footnote n="2">²Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA</footnote>

<footnote n="3">³School of Materials, Sun Yat-sen University, 518100 Shenzhen, Guangdong, China</footnote>

<footnote n="4">⁴Institut Universitaire de France, 75231 Paris, France</footnote>

<footnote>ᵃORCID: 0000-0002-0025-4735</footnote>

<footnote>ᵇORCID: 0000-0001-5728-4989</footnote>

<footnote>ᶜORCID: 0000-0002-8029-0516</footnote>

<footnote>ᵈORCID: 0000-0002-7726-2717</footnote>

<footnote>ᵉORCID: 0000-0001-9336-6607</footnote>
# 1 Introduction

Since the 1960s, it has been well-established that the phosphorescence lifetime of aromatic carbonyl compounds is shorter for $^3n\pi*$ than for $^3\pi\pi*$ triplet minima [1, 2]. Harrigan and Hirota experimentally estimated that the ratio between those lifetimes should be of a factor of five or bigger for aromatic carbonyls of benzaldehyde type [3]. This difference between phosphorescence lifetimes is often experimentally employed to assign the triplet state character [3], and several papers discuss when this assignment may break due to vibronic couplings [4, 5]. Moreover, it is at the basis of strategies for designing efficient pure organic room-temperature phosphorescent chromophores [6]. Nevertheless, the reason for existing such a significant difference in the first place is not usually explicitly discussed.

Naturally, we may expect the shorter phosphorescence lifetime of the $^3n\pi*$ compared to the $^3\pi\pi*$ in aromatic carbonyls to be rooted in the symmetry of the Hamiltonian terms. Some excellent reference texts discuss this topic in general terms [7, 8] and applied to other heteroaromatic molecules [9, 10], but how this knowledge transfers to aromatic carbonyl compounds is not directly evident.

The reason for the phosphorescence lifetime difference is often considered to be related to the El-Sayed rule [11]. During the research for this article, we asked social networks what the cause of this phenomenon would be. The El-Sayed rule was the most common answer, indicating this explanation's popularity among the chemistry community. Nevertheless, despite exhaustive research, we could not locate any reference that convincingly showed how the El-Sayed rule would apply in this case. The closest we found was Olmsted and El-Sayed's analysis of BA phosphorescence [4]. They attributed the benzaldehyde's $^3n\pi*$ phosphorescence to spin-orbit coupling (SOC) between the triplet $^3n\pi*$ and the singlet $^1\pi\pi*$ states, but, as we will discuss later, this seems not to be the case.

We perused 65 years of literature unsuccessfully searching for a theoretical analysis of phosphorescence lifetime in aromatic carbonyls. The only discussion we found is in the 1974 paper by Cheng and Hirota [12], in which all elements are laid out to address the problem. However, they did not explicitly discuss the difference in phosphorescence lifetimes, instead focusing on the vibronic couplings and zero-field splittings. Thus, we decided to fill this knowledge gap by surveying how molecular symmetry impacts phosphorescence from $^3n\pi*$ and $^3\pi\pi*$ in these molecules.

We show that the difference in the phosphorescence lifetimes is due to a selection rule related to (but more general than) the El-Sayed rule, controlling the coupling mechanism in each type of state. In particular, the lifetime difference occurs if the difference between the permanent dipole moments of S₀ and T₁ is significant and $^3\pi\pi*$ is totally symmetric. We also show that it should strictly occur only for molecules attaining C₂v and D₂ symmetries, although it may occasionally be observed in other point groups. Finally, we discuss how vibronic effects impact the phosphorescence lifetimes.

The conventional approach to estimating the spontaneous emission lifetime is to compute transition dipole moments at the minimum of the source state and feed them to Fermi's golden rule [13].
However, when dealing with phosphorescence, the transition is spin-forbidden, rendering null
transition dipole moments. This restriction is usually overcome by generating first-order spin-
perturbed wave functions for the source and target states, which yields non-null transition dipole
moments [7, 8, 14-17]. Our analysis was entirely based on this spin-perturbed procedure, which is
straightforward to apply if SOC elements are available.

Phosphorescence can also have significant contributions from vibronic couplings arising from
symmetry reduction during vibrational motion. Different methods are available to incorporate
vibronic couplings into the simulations, such as the nuclear ensemble approach (NEA) [18], direct
vibronic coupling [19], adiabatic Hessian [20], and vertical gradient [21]. Torres and co-authors
recently compared them, and the interested reader may know more about the pros and cons of each
one in their paper [22]. We particularly favor NEA, which we have been developing and applying for
many years to simulate different spectrum types and generate initial conditions for dynamics [18, 23,
24].

NEA simulates steady-state and time-resolved spectra by performing an incoherent sum over
independent transitions from different nuclear configurations [18]. These configurations are sampled
to represent the nuclear wave function distribution in the source state (the lowest triplet state for
phosphorescence). Standard NEA, however, does not include any information about the target state
wave function (the ground state for emission spectra) [25]. Therefore, the band shape is entirely
determined by the source-state features, and vibrational structures are neglected. Moreover, NEA has
a high intrinsic cost of a few hundred single-point calculations, which can be alleviated using
machine learning [26].

One advantage of employing NEA for phosphorescence analysis is that it allows decoupling
perturbative terms acting in the symmetric structure from those paying a role through vibronic
couplings. For example, Cheng and Hirota [12] included vibronic coupling terms directly in the
perturbative expansion, making distinguishing each effect's origin difficult.

We implemented a strategy for simulating steady-state phosphorescence spectra with NEA to
verify our formal analysis. The results are compared to the conventional vertical approximation. This
initial work is restricted to TDDFT, but generalizing it to other methods is straightforward if SOC
elements are available.

We simulated the vertical and NEA phosphorescence of four prototypical chromophores in the gas
phase (see Figure 1): benzaldehyde (BA), 4-methylbenzaldehyde (MeBA), 4-methoxybenzaldehyde
(MoBA), and 4-dimethylaminobenzaldehyde (DMABA). The small size of these molecules and the
abundance of previous experimental [27-37] and theoretical [38-40] results for BA and its
derivatives make them ideal for demonstrating the method. The motivation for choosing these
molecules is that the functional groups in the series BA, MeBA, MoBA, and DMABA have an
increasing electron donor character, stabilizing the ³ππ* state, as well known [41]. Thus, we can
probe phosphorescence under different relative energies between the ³nπ* and ³ππ* states.
<figure><img src="image_2.png" /><figcaption>Figure 1. Benzaldehyde (BA) 4-methylbenzaldehyde (MeBA), 4-methoxybenzaldehyde (MoBA), and 4-dimethylaminobenzaldehyde (DMABA).</figcaption></figure>

The lowest triplet electronic state (T₁) of BA can be directly photoexcited [27], activated via collisional processes [28], or, more usually, populated after photoabsorption into the singlet manifold [29-31]. In the latter case, the efficiency of singlet→triplet intersystem crossing is near unity [29, 32]. Thus, BA has weak fluorescence and shows phosphorescence either in a solid matrix [33] or in the gas phase [27, 30, 31, 42]. At zero pressure, the phosphorescence takes between 2 and 4.3 ms [29, 30]. The phosphorescence quantum yield is 0.60, while the T₁→S₀ intersystem crossing quantum yield is 0.40. The phosphorescence quantum yield decreases on excitation from S₀ to S₁, S₂, and S₃ states, indicating competitive relaxation pathways through nonradiative processes in each excited state. Moreover, a triplet excited state (T₂) also plays a vital role in quenching S₂ and S₃ states [31]. A monotonic decrease of the quantum yield for BA phosphorescence with increasing excitation wavenumber is also reported confirming the complex decay mechanism [30].

Studies about the phosphorescence of the other derivatives are sparser [1, 3, 35, 43, 44]. A fascinating case is MoBA, which phosphoresces within 1 ms in vapor [43], but within 120 ms within a rigid glass at 77 K.

All BA derivatives investigated in this work are small molecules with highly structured spectra. As mentioned, NEA entirely neglects those vibrational structures, only delivering a smoothed band shape. Despite this limitation, NEA yields valuable information, particularly the lifetime, beyond the vertical approximation. Moreover, NEA is intended as a practical methodology aiming at medium to large molecules, where such vibrational structures become less prominent.

## 2 Theory

For phosphorescence, the NEA differential emission rate (dimensionless) from T₁ into S₀ is [18]

$$ \Gamma_{rad}(E) = \frac{2\alpha}{m_e c^2} [1 - H(E - h\nu_a)] \frac{1}{N_p} \sum_n^{N_p} \Delta E_{10}(\mathbf{R}_n)^2 |\tilde{f}_{10}(\mathbf{R}_n)|^2 w_s(E - \Delta E_{10}(\mathbf{R}_n), \delta) \quad (1) $$

where α is the fine-structure constant, m_e is the electron mass, c is the speed of light. H(E - hν_a) is the Heaviside step function, ensuring that the emission energy E is smaller than the excitation energy hν_a. N_p is the total number of geometries R_n in the ensemble. ΔE₁₀ is the transition energy between
$T_1$ and $S_0$ computed for each geometry $R_n$. $\tilde{f}_{10}$ is the oscillator strength between the spin-perturbed $\tilde{T}_1$ and $\tilde{S}_0$ states for each geometry. Its calculation is discussed later in this section. $w_s$ is a normalized sharp line shape (a Gaussian function, for instance) centered at $\Delta E_{10}$ and with width $\delta$. The nuclear geometries $R_n$ are sampled with a Wigner distribution for the quantum harmonic oscillator at the $T_1$ minimum.

The phosphorescence rate $\kappa_{rad}^{(NEA)}$ and phosphorescence lifetime $\tau_{rad}^{(NEA)}$ are

$$ \kappa_{rad}^{(NEA)} = \frac{1}{\tau_{rad}^{(NEA)}} = \frac{1}{\hbar} \int \Gamma_{rad}(E) dE \qquad (2) $$

For comparison, the vertical approximation for the phosphorescence rate and lifetime is

$$ \kappa_{rad}^{(VER)} = \frac{1}{\tau_{rad}^{(VER)}} = \frac{2\alpha}{\hbar m_e c^2} \Delta E_{10} (\mathbf{R}_{T_1 \min})^2 |\tilde{f}_{10}(\mathbf{R}_{T_1 \min})| \qquad (3) $$

In Eqs. (1) and (3), the oscillator strength between pure-spin $T_1$ and $S_0$ is null. However, due to SOC, the singlet states contain some triplet state character, and at the same time, the $T_1$ state is contaminated by singlet wave functions [8, 14-16]. In first-order perturbation theory, the spin-perturbed states are [8, 16]

$$ |\tilde{S}_0\rangle = |S_0\rangle + \sum_{K}^{\{triplets\}} \sum_{\alpha} \frac{\langle T_K^{(\alpha)} | \hat{H}^{SO} | S_0 \rangle}{E_0 - E_K} |T_K^{(\alpha)}\rangle \qquad (4) $$

and

$$ |\tilde{T}_1^{(\alpha)}\rangle = |T_1^{(\alpha)}\rangle + \sum_{L}^{\{singlets\}} \frac{\langle S_L | \hat{H}^{SO} | T_1^{(\alpha)} \rangle}{E_1 - E_L} |S_L\rangle \qquad (5) $$

In Eq. (4), the sums run over the adiabatic triplet states $K$ with energy $^3E_K$ and wave function $|T_K^{(m)}\rangle$ and over the triplet sublevels $\alpha$. In Eq. (5), the sum runs over the adiabatic singlet states $L$ with energy $^1E_L$ and wave function $|S_L\rangle$. $\hat{H}^{SO}$ denotes the SOC operator.

The transition dipole moment is

$$ \begin{aligned} \tilde{\mu}_{10}^{(\alpha)} &= \langle \tilde{T}_1^{(\alpha)} | \hat{\mu} | \tilde{S}_0 \rangle \\ &= \sum_{L}^{\{singlets\}} \frac{\langle T_1^{(\alpha)} | \hat{H}^{SO} | S_L \rangle}{E_1 - E_L} \langle S_L | \hat{\mu} | S_0 \rangle + \sum_{K}^{\{triplets\}} \frac{\langle T_K^{(\alpha)} | \hat{H}^{SO} | S_0 \rangle}{E_0 - E_K} \langle T_1 | \hat{\mu} | T_K \rangle \end{aligned} \qquad (6) $$

where $\hat{\mu}$ is the dipole moment operator and we used $\langle S_L | \hat{H}^{SO} | T_1^{(\alpha)} \rangle^* = \langle T_1^{(\alpha)} | \hat{H}^{SO} | S_L \rangle$. We can rewrite the transition dipole moment as
$$ \tilde{\mu}_{10}^{(\alpha)} = \frac{\langle T_1^{(\alpha)} | \hat{H}^{SO} | S_0 \rangle}{^3E_1 - ^1E_0} [\langle S_0 | \hat{\mu} | S_0 \rangle - \langle T_1 | \hat{\mu} | T_1 \rangle] + \sum_{\{\text{singlets}\}} \frac{\langle T_1^{(\alpha)} | \hat{H}^{SO} | S_L \rangle}{^3E_1 - ^1E_L} \langle S_L | \hat{\mu} | S_0 \rangle + \sum_{K \neq 1} \frac{\langle T_K^{(\alpha)} | \hat{H}^{SO} | S_0 \rangle}{^1E_0 - ^3E_K} \langle T_1 | \hat{\mu} | T_K \rangle \quad (7) $$

evidencing S₀ and T₁ permanent dipole moments in the first term.

Each of the three terms on the right side of Eq. (7) corresponds to a distinct coupling mechanism contributing to the transition dipole moment. The first term is a direct SOC between T₁ and S₀ modulated by these states' permanent dipoles (direct SOC mechanism):

$$ T_1 \stackrel{\text{SOC}, \mu_p}{\leftrightarrow} S_0 $$

In the second term, T₁ couples via spin-orbit to the S_L excited singlet states, then those singlet states couple to S₀ via transition dipole (indirect SOC-dipole mechanism):

$$ T_1 \stackrel{\text{SOC}}{\leftrightarrow} S_L \stackrel{\mu_r}{\leftrightarrow} S_0 $$

Finally, the third term couples T₁ to the excited T_K triplet states via transition dipole, and then these triplet states couple to S₀ via SOC (indirect dipole-SOC mechanism):

$$ T_1 \stackrel{\mu_r}{\leftrightarrow} T_K \stackrel{\text{SOC}}{\leftrightarrow} S_0 $$

To get the oscillator strength, we compute the mean transition dipole moment squared

$$ \tilde{\mu}_{10}^2 = \frac{1}{3} \sum_{\alpha} |\tilde{\mu}_{10}^{(\alpha)}|^2 \quad (8) $$

Finally, the oscillator strength is [13]

$$ \tilde{f}_{10} = \frac{2m_e}{3\hbar^2 e^2} \Delta E_{10} \tilde{\mu}_{10}^2 \quad (9) $$

### 3 Computational details

The NEA phosphorescence spectrum [Eq. (1)] was simulated with 1000 geometries sampled from a quantum harmonic oscillator Wigner distribution in the T₁ state taking δ = 0.05 eV. For each geometry, the oscillator strength between T̃₁ and S̃₀ was computed with Eq. (9). The perturbative expansion [Eqs. (4) and (5)] included 20 singlet and 20 triplet states. S₀ is determined with density functional theory (DFT) with the B3LYP functional [45] and the 6-31G(d,p) basis set [46]. All other states were computed with linear-response time-dependent DFT (TDDFT) [47], with triplet states treated as excitations of the singlet reference.
Supplementary calculations were done with TDDFT with the CAM-B3LYP [48] and ωB97X [49] functionals. The results with TD-B3LYP were systematically superior, particularly predicting the ³nπ*/³ππ* energy splitting in MeBA and MoBA in excellent agreement with the experiments [35, 44]. Moreover, despite multiple attempts, we have not found the ³nπ* minimum for BA and MeBA with TD-ωB97X, for which T₁ was always ³ππ*. With TD-CAM-B3LYP, the T₁ optimization of BA was unsuccessful, systematically overstretching the carbonyl group and yielding negative excitations. These results are in line with those reported by Sears et al.[50], who demonstrated that the large amount of Hartree-Fock exchange in range-separated functionals could degrade TDDFT predictions of triplet states in π-conjugated molecules. Although the Tamm-Dancoff approximation could alleviate these problems, we decided to limit our simulations to TDDFT with B3LYP.

In TDDFT, the $|S_L\rangle$ and $|T_K^{(m)}\rangle$ wave functions can be written as single excitations of the Kohn-Sham ground-state singlet determinant as described in Ref. [51]. SOC was determined with a single-electron Breit-Pauli operator with an effective charge approximation (BP1e-eff) [51, 52].

We implemented the calculation of the oscillator strengths in a new version of PySOC [51], which calls Gaussian 16 [53] for TDDFT calculations, and MolSOC [54] to provide atomic integrals for SOC calculations. The NEA calculations were done with Newton-X [24], which we interfaced with PySOC, to deliver all results in a simple workflow. Technical details of how the transition dipole elements are computed in the frame of TDDFT are given in SI-1.

The convergence of the first-order perturbative approach was tested for the ³nπ* T₁ minimum geometry of BA (vertical emission) by varying the number of states used in Eq. (6). The results for the oscillator strength [Eq. (9)] and vertical phosphorescence lifetime [Eq. (3)] are presented in Table 1, indicating a reasonable convergence is achieved for 20 triplet and 20 singlet states.

<table><caption>Table 1. Convergence of the perturbative treatment as a function of the number of singlet and triplet states considered. The zero-order T₁ vertical excitation energy is 2.336 eV.</caption><thead><tr><td>N<sub>states</sub></td><td>Oscillator Strength (×10<sup>-7</sup>)</td><td>Lifetime (ms)</td></tr></thead><tbody><tr><td>2</td><td>2.90</td><td>15</td></tr><tr><td>5</td><td>0.65</td><td>65</td></tr><tr><td>10</td><td>1.05</td><td>40</td></tr><tr><td>15</td><td>1.26</td><td>33</td></tr><tr><td>20</td><td>1.09</td><td>39</td></tr><tr><td>25</td><td>1.09</td><td>39</td></tr><tr><td>30</td><td>1.13</td><td>38</td></tr><tr><td>35</td><td>1.12</td><td>38</td></tr><tr><td>40</td><td>1.13</td><td>38</td></tr><tr><td>45</td><td>1.11</td><td>38</td></tr><tr><td>50</td><td>1.10</td><td>38</td></tr></tbody></table>
4 Results

4.1 Electronic states of benzaldehyde

BA is perfectly planar in the ground singlet and triplet states (S₀ and T₁) and low-lying singlet excited states (S₁ and S₂). It has a global T₁ minimum with a ³nπ* character. With TD-B3LYP, we could neither find a ³ππ* minimum in T₁ nor T₂. Nevertheless, a ³ππ* stationary structure lies 0.09 eV above the ³nπ* minimum. We will refer to this structure as ³ππ* T₁ minimum, but note that it has an imaginary frequency of 171i cm⁻¹. State energies at the S₀ and S₁ minima are given in Table 2. Energies at the T₁ and T₂ minima are in Table 3. The electronic excitations and molecular orbitals to identify the state characters are given in the SI-2.

Figure 2 shows the linearly interpolated potential energy profiles between the ³nπ* and ³ππ* minima. Because the ³nπ* minimum is more stable and should couple to the ground state more strongly than ³ππ*, we will focus on this minimum.

The most notable geometrical difference between S₀ and T₁ ³nπ* minima lies in the carbonyl angle (O-C7-H), which decreases by almost 10° in ³nπ* compared to S₀. The carbonyl bond length (C7-O) increases by 0.1 Å from S₀ to ³nπ*. On the other hand, the bond between carbonyl and ring carbon (C1-C7) also decreases by about 0.1 Å from S₀ to ³nπ*. The ring C-C and C-H distances remain approximately the same in both structures. The relevant internal coordinates in S₀, S₁, and T₁ minima are compared with previously reported experimental and theoretical data in SI-3. The vertical excitation energies at S₀, S₁, and T₁ minima of benzaldehyde are compared with previous results in SI-4. Generally, the present results calculated with the TD-B3LYP agree reasonably well with the reported CASPT2 energies [55]. However, the energies of the triplet manifold reported in Ref. [40] are closer to the CASPT2 ones than in the present case. The discrepancies seem to be due to our treatment of the triplets as excitations of the singlet reference.

<figure><img src="image_3.png" /><figcaption>Figure 2. Potential energy scan of T₁, T₂ adiabatic states and ³nπ*, ³ππ* diabatic states at the linear interpolation coordinates between ³nπ* and ³ππ* minima geometries of benzaldehyde. The horizontal axis is the mass-weighted distance from the ³nπ* minimum geometry.</figcaption></figure>
## 4.2 Electronic states of the BA derivatives

The energetics of the BA derivatives are given in Table 2 and Table 3. The S<sub>1</sub> <sup>1</sup>nπ* minimum of MeBA and MoBA is above the <sup>3</sup>ππ* state by 0.35 and 0.45 eV, respectively, being a door for populating the triplet manifold via an intersystem crossing. In DMABA, this gap is 0.64 eV, making ISC less likely according to the El-Sayed rule [56].

<table><caption>Table 2. Ground and excitation energies (eV) at the S<sub>0</sub> and S<sub>1</sub> minima of BA and its derivatives calculated with TD-B3LYP/6-31G(d,p). The state character is also indicated. The excitation energies are relative to the ground-state minimum energy.</caption><thead><tr><th rowspan="2">State</th><th colspan="2">BA</th><th colspan="2">MeBA</th><th colspan="2">MoBA</th><th colspan="2">DMABA</th></tr><tr><th>S<sub>0</sub> min<br>(<sup>1</sup>nπ*)</th><th>S<sub>1</sub> min<br>(<sup>1</sup>nπ*)</th><th>S<sub>0</sub> min<br>(<sup>1</sup>nπ*)</th><th>S<sub>1</sub> min<br>(<sup>1</sup>nπ*)</th><th>S<sub>0</sub> min<br>(<sup>1</sup>nπ*)</th><th>S<sub>1</sub> min<br>(<sup>1</sup>nπ*)</th><th>S<sub>0</sub> min<br>(<sup>1</sup>nπ*)</th><th>S<sub>1</sub> min<br>(<sup>1</sup>nπ*)</th></tr></thead><tbody><tr><td>S<sub>0</sub></td><td>0.00</td><td>0.40</td><td>0.00</td><td>0.41</td><td>0.00</td><td>0.41</td><td>0.00</td><td>0.42</td></tr><tr><td>T<sub>1</sub></td><td>3.12<br>(<sup>3</sup>nπ*)</td><td>2.77<br>(<sup>3</sup>nπ*)</td><td>3.15<br>(<sup>3</sup>nπ*)</td><td>2.80<br>(<sup>3</sup>nπ*)</td><td>3.16<br>(<sup>3</sup>ππ*)</td><td>2.87<br>(<sup>3</sup>nπ*)</td><td>2.88<br>(<sup>3</sup>ππ*)</td><td>2.82<br>(<sup>3</sup>ππ*)</td></tr><tr><td>T<sub>2</sub></td><td>3.30<br>(<sup>3</sup>ππ*)</td><td>3.05<br>(<sup>3</sup>ππ*)</td><td>3.23<br>(<sup>3</sup>ππ*)</td><td>3.01<br>(<sup>3</sup>ππ*)</td><td>3.23<br>(<sup>3</sup>nπ*)</td><td>2.98<br>(<sup>3</sup>ππ*)</td><td>3.27<br>(<sup>3</sup>nπ*)</td><td>2.92<br>(<sup>3</sup>nπ*)</td></tr><tr><td>S<sub>1</sub></td><td>3.68<br>(<sup>1</sup>nπ*)</td><td>3.33<br>(<sup>1</sup>nπ*)</td><td>3.70<br>(<sup>1</sup>nπ*)</td><td>3.36<br>(<sup>1</sup>nπ*)</td><td>3.78<br>(<sup>1</sup>nπ*)</td><td>3.43<br>(<sup>1</sup>nπ*)</td><td>3.81<br>(<sup>1</sup>nπ*)</td><td>3.46<br>(<sup>1</sup>nπ*)</td></tr><tr><td>S<sub>2</sub></td><td>4.78<br>(<sup>1</sup>ππ*)</td><td>4.97<br>(<sup>1</sup>ππ*)</td><td>4.77<br>(<sup>1</sup>ππ*)</td><td>4.99<br>(<sup>1</sup>ππ*)</td><td>4.81<br>(<sup>1</sup>ππ*)</td><td>5.03<br>(<sup>1</sup>ππ*)</td><td>4.30<br>(<sup>1</sup>ππ*)</td><td>4.58<br>(<sup>1</sup>ππ*)</td></tr></tbody></table>

<table><caption>Table 3. Ground and excitation energies (eV) at the lowest <sup>3</sup>nπ* and <sup>3</sup>ππ* triplet minima of BA and its derivatives calculated with TD-B3LYP/6-31G(d,p). The state character is also indicated. In BA, MeBA, and MoBA, both minima are T<sub>1</sub>. In DMABA, they are T<sub>1</sub> and T<sub>2</sub>. The excitation energies are relative to the ground-state minimum energy.</caption><thead><tr><th rowspan="2">State</th><th colspan="2">BA</th><th colspan="2">MeBA</th><th colspan="2">MoBA</th><th colspan="2">DMABA</th></tr><tr><th>T<sub>1</sub> min<br>(<sup>3</sup>nπ*)</th><th>T<sub>1</sub> min<br>(<sup>3</sup>ππ*)</th><th>T<sub>1</sub> min<br>(<sup>3</sup>nπ*)</th><th>T<sub>1</sub> min<br>(<sup>3</sup>ππ*)</th><th>T<sub>1</sub> min<br>(<sup>3</sup>ππ*)</th><th>T<sub>1</sub> min<br>(<sup>3</sup>nπ*)</th><th>T<sub>1</sub> min<br>(<sup>3</sup>ππ*)</th><th>T<sub>2</sub> min<br>(<sup>3</sup>nπ*)</th></tr></thead><tbody><tr><td>S<sub>0</sub></td><td>0.42</td><td>0.61</td><td>0.42</td><td>0.52</td><td>0.43</td><td>0.42</td><td>0.28</td><td>0.43</td></tr><tr><td>T<sub>1</sub></td><td>2.76<br>(<sup>3</sup>nπ*)</td><td>2.85<br>(<sup>3</sup>ππ*)</td><td>2.78<br>(<sup>3</sup>nπ*)</td><td>2.79<br>(<sup>3</sup>ππ*)</td><td>2.81<br>(<sup>3</sup>ππ*)</td><td>2.86<br>(<sup>3</sup>nπ*)</td><td>2.65<br>(<sup>3</sup>ππ*)</td><td>2.83<br>(<sup>3</sup>ππ*)</td></tr><tr><td>T<sub>2</sub></td><td>3.05<br>(<sup>3</sup>ππ*)</td><td>2.96<br>(<sup>3</sup>nπ*)</td><td>3.00<br>(<sup>3</sup>ππ*)</td><td>3.01<br>(<sup>3</sup>nπ*)</td><td>3.05<br>(<sup>3</sup>nπ*)</td><td>2.99<br>(<sup>3</sup>ππ*)</td><td>3.09<br>(<sup>3</sup>nπ*)</td><td>2.91<br>(<sup>3</sup>nπ*)</td></tr><tr><td>S<sub>1</sub></td><td>3.34<br>(<sup>1</sup>nπ*)</td><td>3.52<br>(<sup>1</sup>nπ*)</td><td>3.36<br>(<sup>1</sup>nπ*)</td><td>3.55<br>(<sup>1</sup>nπ*)</td><td>3.59<br>(<sup>1</sup>nπ*)</td><td>3.43<br>(<sup>1</sup>nπ*)</td><td>3.61<br>(<sup>1</sup>nπ*)</td><td>3.47<br>(<sup>1</sup>nπ*)</td></tr><tr><td>S<sub>2</sub></td><td>4.95<br>(<sup>1</sup>ππ*)</td><td>5.07<br>(<sup>1</sup>ππ*)</td><td>4.97<br>(<sup>1</sup>ππ*)</td><td>4.99<br>(<sup>1</sup>ππ*)</td><td>4.87<br>(<sup>1</sup>ππ*)</td><td>5.03<br>(<sup>1</sup>ππ*)</td><td>4.38<br>(<sup>1</sup>ππ*)</td><td>4.58<br>(<sup>1</sup>ππ*)</td></tr></tbody></table>
The most prominent feature in the state levels of BA and its derivatives is the relative energy gap between $^3n\pi*$ and $^3\pi\pi*$ states. In the sequence of increasing electron donation strength $-H < -CH_3 < -OCH_3 < -N(CH_3)_2$, the $^3n\pi*$ state is destabilized while the $^3\pi\pi*$ state is stabilized, as shown in Figure 3. All four species have a $^3n\pi*$ and a $^3\pi\pi*$ triplet minimum. The lowest triplet minimum in BA and MeBA is $^3n\pi*$. In MoBA and DMABA, the lowest minimum is $^3\pi\pi*$. The inversion occurs around MeBA, where the two minima are nearly degenerated, with $^3n\pi*$ only 0.01 eV more stable than $^3\pi\pi*$. This result nicely agrees with the measurements reported in Ref. [35]. In MoBA, the $^3\pi\pi*$ minimum is lower than the $^3n\pi*$ minimum by 0.05 eV, also in excellent agreement with the experimental estimates between 0.04 and 0.11 eV [44]. For BA, MeBA, and MoBA, both $^3n\pi*$ and $^3\pi\pi*$ minima are in T₁. In DMABA, the $^3n\pi*$ minimum is in T₂.

<figure><img src="image_4.png" /><figcaption>Figure 3. Energy gaps between S₀, $^3n\pi*$, and $^3\pi\pi*$ states in BA and derivatives. The lowest triplet state is set to zero, and the arrow indicates the vertical gap to S₀ computed at this triplet minimum. For BA, MeBA, and MoBA, the green and red levels correspond to the energy gap between the two T₁ minima. For DMABA, those two levels indicate the adiabatic energy gap between T₁ and T₂ minima.</figcaption></figure>

## 4.3 BA phosphorescence spectrum

It is out of the scope of this paper to discuss the photophysics of BA until T₁ is populated. The reader interested in this aspect may check Ref. [40]. Here, our focus is the phosphorescence simulation. Thus, we assumed the $^3n\pi*$ T₁ minimum is prepared at its vibrational ground state. Under this condition, the simulated BA phosphorescence spectrum in the gas phase is shown in Figure 4. It is compared with the experimental phosphorescence spectrum in vapor [57]. The theoretical spectrum shows no vibrational structure as expected for NEA and is redshifted due to the TDDFT treatment.

The area under the phosphorescence spectrum in terms of $\Gamma_{rad}$ does not represent quantum yield but a lifetime. Employing Eq. (2), we obtain $\tau_{rad} = 46 \pm 1$ ms, a little longer than the vertical value computed at the T₁ minimum, 39 ms. Both values are above the experimental phosphorescence lifetime, which lies between 2 and 4.3 ms, depending on the setup [29, 30]. The only other theoretical estimate of the phosphorescence lifetime is by Ou and Subotnik [40], 1.81 ms. Their
calculation is equivalent to our vertical lifetime [Eq. (3)] with the $^3n\pi^*\rightarrow S_0$ transition calculated at TDDFT with the ωB97X functional, but we have not been able to determine the reason for the difference between our results. All these values are summarized in Table 4.

<table><caption>Table 4. Theoretical and experimental phosphorescence lifetime of benzaldehyde.</caption><thead><tr><th></th><th>k (s<sup>-1</sup>)</th><th>τ (ms)</th><th>Reference</th></tr></thead><tbody><tr><td><strong>Theory</strong></td><td></td><td></td><td></td></tr><tr><td>k<sub>p</sub>: NEA</td><td>22</td><td>46 ± 1</td><td>Present work</td></tr><tr><td>k<sub>p</sub>: Vertical emission</td><td>26</td><td>39</td><td>Present work</td></tr><tr><td>k<sub>p</sub>: Vertical emission</td><td>553</td><td>1.81</td><td>[40]</td></tr><tr><td><strong>Experimental (at zero pressure)</strong></td><td></td><td></td><td></td></tr><tr><td>k<sub>p</sub>: S<sub>0</sub>→S<sub>1</sub>(nπ*) 0-0 (3.34 eV)</td><td>235 ± 19</td><td>4.3 ± 0.3</td><td>[29]</td></tr><tr><td>k<sub>0</sub>: S<sub>0</sub>→S<sub>1</sub>(nπ*) 0-0 (3.34 eV)</td><td>392 ± 24</td><td>2.6 ± 0.2</td><td>[29]</td></tr><tr><td>k<sub>0</sub>: S<sub>0</sub>→T<sub>1</sub>(nπ*) (3.11 – 3.15 eV)</td><td>1410</td><td>0.712</td><td>[27]</td></tr><tr><td>k<sub>p</sub>: S<sub>0</sub>→S<sub>2</sub> (4.37, 4.49 eV)</td><td>500</td><td>2.0</td><td>[30]</td></tr><tr><td>k<sub>0</sub>: S<sub>0</sub>→S<sub>2</sub> (4.37, 4.49 eV)</td><td>690</td><td>1.5</td><td>[30]</td></tr><tr><td>k<sub>0</sub>: 9-eV e<sup>-</sup>-impact, 3.12-eV emission</td><td>600</td><td>1.7</td><td>[28]</td></tr><tr><td>k<sub>0</sub>: 9-eV e<sup>-</sup>-impact, 3.32-eV emission</td><td>1300</td><td>0.8</td><td>[28]</td></tr></tbody></table>

Many experimental results are available for benzaldehyde (see Table 4), which may lead to misunderstandings regarding which values we should take for comparison with theory. Two types of reaction rates are reported—the phosphorescence rate ($k_p$) and decay rate ($k_0$). The relation between them is [29]

$$k_p = k_0 - k_{\text{ISC}} = \Phi_p k_0 \qquad (10)$$

where $k_{\text{ISC}}$ is the nonradiative intersystem crossing rate, and $\Phi_p$ is the phosphorescence quantum yield. Brühlmann et al. [29] obtained the phosphorescence lifetime by measuring the spectrally integrated $T_1$ decay following the $S_0\rightarrow S_1$ excitation. This decay contains information on radiative and nonradiative components, and they report both $k_p$ and $k_0$. Similarly, Hirata and Lim reported radiative and nonradiative decay rates [30]. In turn, Biron and Longin [27] and Inoue and Ebara [28] fitted the phosphorescence intensity signal. However, they did not explicitly consider nonradiative components. Thus, their reported rate should correspond to $k_0$. From the theory perspective, rates computed with either Eq. (2) or (3) must be exclusively compared with $k_p$.

Our estimate of BA phosphorescence lifetime, 46 ms (Table 4), is based on a Wigner distribution at the ground vibrational $T_1$ state. Thus, it should be compared to the lifetime measured under similar conditions, phosphorescence following excitation into $T_1$. Experimentally, BA phosphorescence
lifetime after S₂ excitation is 2 ms [30]. It increases to 4.3 ms for BA excited at S₁ [29]. These values may imply that the phosphorescence lifetime after excitation into T₁ (which is lower than S₁) may be longer than 4.3 ms, but unfortunately, such an experimental result is not available. Compared to the available experiments, our value (46 ms) seems to overestimate the phosphorescence lifetime. The reason for this overestimation seems to be connected to the computational level we used. Although TD-B3LYP yielded excellent results for the triplet states (as discussed in Section 3), the relative energy between triplet and singlet states have significant deviations, as we can see in the simulated spectrum in Figure 4. The theoretical spectrum is redshifted by about 0.7 eV. Furthermore, both SOCs and transition dipole moments were computed with auxiliary multi-electron wave functions [58, 59]. The accuracy of such approaches has not yet been fully gauged, and it may be a source of errors in estimating the phosphorescence lifetimes. Despite the overestimation, we shall see that the computed phosphorescence lifetimes deliver a satisfactory qualitative picture for the differences between $^3n\pi*$ and $^3\pi\pi*$ emissions.

## 4.4 Phosphorescence of the BA derivatives

The NEA phosphorescence spectrum of BA and derivatives is shown in Figure 4. For MeBA and MoBA, we computed the spectrum at the two T₁ minima, but only the one at the lowest minimum is given in the figure. The other offers similar results, as we shall discuss. The integral of these spectra [Eq. (2)] yields the phosphorescence lifetimes, which are collected in Table 5. The vertical phosphorescence lifetimes, computed for the T₁ minimum geometry with Eq. (3), are also reported in the table.

<figure><img src="image_5.png" /><figcaption>Figure 4. BA, MeBA, MoBA, and DMABA phosphorescence spectra in the gas phase simulated with NEA. Experimental results for BA from Ref. [57] normalized to have the same area as the simulation.</figcaption></figure>
The phosphorescence lifetime of BA is discussed in Section 4.3. For all molecules, the vertical phosphorescence lifetime is between tens of ms for $^3n\pi*$ emission and grows to several seconds for $^3\pi\pi*$ emission (Table 5). This significant difference reduces when the vibronic coupling is incorporated via NEA. The $^3n\pi*$ emission lifetime is slightly elongated to values between 46 and 75 ms. Still, the $^3\pi\pi*$ emission lifetime is reduced up to a factor of 50 in DMABA, and all molecules show $^3\pi\pi*$ phosphorescence within approximately 200 ms. The reason for these differences is discussed in Section 5.1.

<table><caption>Table 5. Phosphorescence lifetime of BA derivatives in vapor simulated with the vertical approximation and NEA.</caption><thead><tr><th rowspan="2">T<sub>1</sub> minimum</th><th rowspan="2">Molecule</th><th colspan="2">Phosphorescence lifetime (ms)</th></tr><tr><th>Vertical</th><th>NEA</th></tr></thead><tbody><tr><td rowspan="3"><sup>3</sup>nπ*</td><td>BA</td><td>39</td><td>46 ± 1</td></tr><tr><td>MeBA</td><td>38</td><td>47 ± 1</td></tr><tr><td>MoBA</td><td>61</td><td>75 ± 1</td></tr><tr><td rowspan="3"><sup>3</sup>ππ*</td><td>MeBA</td><td>8380</td><td>217 ± 3</td></tr><tr><td>MoBA</td><td>6870</td><td>214 ± 3</td></tr><tr><td>DMABA</td><td>11530</td><td>231 ± 4</td></tr></tbody></table>

Unlike BA, for which experimental data is abundant, for the BA derivatives, the only available information about phosphorescence lifetime is for MoBA. MoBA in vapor has a lifetime of ~1 ms or less, obtained by indirect inference and assigned to the $^3n\pi*$ emission in Ref. [43]. In a 77-K glass, the measured phosphorescence lifetime of MoBA is much longer, about 120 ms, and has been assigned to $^3\pi\pi*$ emission [43, 57], indicating that the glass quenches the $^3n\pi*$ emission. Our NEA values, 75 ms for $^3n\pi*$ and 214 ms for $^3\pi\pi*$ (Table 5), qualitatively capture the difference between the emission from the two states.

# 5 Discussion

## 5.1 Origins of the fast and slow phosphorescence

This section addresses why phosphorescence is faster from $^3n\pi*$ than $^3\pi\pi*$ states in aromatic carbonyls. We will take BA as an example and focus our discussion on $T_1 \to S_0$ phosphorescence. First, we must remember that state symmetry should consider both the usual symmetry representation of the spatial wave function and the symmetry representation generated by the spin wave function. The spin wave function of the singlet state always transforms as the totally symmetric representation of the group ($\Gamma_1$), while the spin wave function Cartesian components belong to the representations generated by the $R_x$, $R_y$, and $R_z$ rotations [7]. Thus,

$$ \Gamma(|T_K^{(\xi)}\rangle) = \Gamma(|T_K\rangle)\Gamma(R_{\xi}) \qquad (11) $$

where $\Gamma(\cdot)$ gives the irreducible representation.
$T_1 \to S_0$ phosphorescence will be symmetry allowed if two selection rules are satisfied. The first one concerns the SOC terms appearing in the singlet-triple transition dipole moment, Eq. (7):

$$ \Gamma(\langle S_L | \hat{H}^{SO} | T_K^{(\xi)} \rangle) = \Gamma_1 \qquad (12) $$

Because $\Gamma(\hat{H}^{SO}) = \Gamma_1$, Eq. (12) simplifies to

$$ \Gamma(|S_L\rangle) = \Gamma(|T_K^{(\xi)}\rangle) \qquad (13) $$

Note that, for convenience, we wrote the states in terms of the Cartesian components $\xi$ instead of the magnetic quantum number $m$, which is actually employed in our program. The transformation between the two representations is [8]

$$ \begin{aligned} |T^{(x)}\rangle &= \frac{1}{\sqrt{2}}(|T^{(-1)}\rangle - |T^{(+1)}\rangle) \\ |T^{(y)}\rangle &= \frac{i}{\sqrt{2}}(|T^{(-1)}\rangle + |T^{(+1)}\rangle) \\ |T^{(z)}\rangle &= |T^{(0)}\rangle \end{aligned} \qquad (14) $$

Alternatively, this first selection rule can be restated in terms of the eigenvectors in the perturbative expansion in Eqs. (4) and (5). In the first-order term, only wave functions belonging to the same representation as the zero-order wave function contribute to the spin-perturbed wave function. This approach is adopted by Marian in Ref. [8].

The second selection rule concerns the dipole moments in Eq. (7). The transition can only be allowed if at least one of the elements belongs to the totally symmetric representation:

$$ \Gamma(\langle S_0 | \hat{\mu} | S_L \rangle) \text{ or } \Gamma(\langle T_K | \hat{\mu} | T_1 \rangle) \supset \Gamma_1 \qquad (15) $$

In this expression, a component $\hat{\xi}$ of a singlet-singlet transition dipole contributes to the phosphorescence if

$$ \Gamma(|S_0\rangle)\Gamma(|S_L\rangle) = \Gamma(\hat{\xi}) \qquad (16) $$

Alternatively, the component $\hat{\xi}$ of a triplet-triplet transition dipole contributes if

$$ \Gamma(|T_K\rangle)\Gamma(|T_1\rangle) = \Gamma(\hat{\xi}) \qquad (17) $$

These two selection rules are weak because the perturbative expansion sums over many $L$ and $K$ states. Thus, it is always likely that some states will satisfy them even for a highly symmetric structure. Physically, however, it is relevant whether the low-order terms satisfy the conditions or not since they contribute the most to the singlet-triplet transition dipole. Thus, for our analysis, we restrict the perturbative contributions in Eq. (6) to contain only $S_0$, $S_1$, $T_1$, and $T_2$ states ($L = 0, 1$ and $K = 1, 2$), reducing it to
$$ \tilde{\mu}_{10}^{(\xi)} \approx \frac{\langle T_1^{(\xi)} | \hat{H}^{SO} | S_0 \rangle}{^3E_1 - ^1E_0} \left[ \langle S_0 | \hat{\mu} | S_0 \rangle - \langle T_1 | \hat{\mu} | T_1 \rangle \right] + \frac{\langle T_1^{(\xi)} | \hat{H}^{SO} | S_1 \rangle}{^3E_1 - ^1E_1} \langle S_1 | \hat{\mu} | S_0 \rangle + \frac{\langle T_2^{(\xi)} | \hat{H}^{SO} | S_0 \rangle}{^1E_0 - ^3E_2} \langle T_1 | \hat{\mu} | T_2 \rangle \quad (18) $$

These three terms, corresponding to the three coupling mechanisms introduced in Section 2, are schematically illustrated in Figure 5. Because they have the smallest denominators in the series, they tend to dominate the qualitative features of the total transition dipole, even though the high-order terms are essential for the quantitative description.

<figure><img src="image_6.png" /><figcaption>Figure 5. Each diagram illustrates one of the three terms in Eq. (18). The first is a direct SOC mechanism, the second is an indirect SOC-dipole mechanism, and the third is an indirect dipole-SOC mechanism. The double arrow indicates the SOC divided by the singlet-triplet energy difference; the dashed circle indicates a permanent dipole moment, and the dashed line is a transition dipole moment.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 6. Spin density of the T₁ state of benzaldehyde (left). Two Cartesian axes are indicated, corresponding to the C–C and C=O representations.</figcaption></figure>
Let us now apply these conditions to BA. Although the T₁ minimum has Cₛ symmetry, we should at least work with C₂ᵥ point group to be able to distinguish between allowed and forbidden matrix elements. The T₁ spin density shown in Figure 6 indicates that BA can be approximately treated as a C₂ᵥ system, with the principal axis either along the C–C direction or the C=O direction (see both representations in the figure).

We first assume that the principal axis is oriented along C–C. As discussed in the previous section, BA has two T₁ minima, with ³nπ* (~B₁) and ³ππ* (~A₁) characters (see excitations and molecular orbitals in SI-2). In both minima, S₀ is ~A₁ and S₁ is ~B₁ (¹nπ*). At the ³nπ* minimum, T₂ is a ³ππ* (~A₁) while at ³ππ* minimum, T₂ is an ³nπ* (~B₁). In C₂ᵥ, x, y, and z transform as B₁, B₂, and A₁, respectively. In turn, Rₓ, Rᵧ, and R_z transform as B₂, B₁, and A₂, respectively.

For the ³nπ* minimum, the first selection rule [Eqs. (13)] imposes that the only non-null SOC elements in Eq. (18) are ⟨T₁(y)|ŷSO|S₀⟩, appearing in the first term. (Applying Eq. (11), the symmetry of T₁(y) is Γ(³nπ*) × Γ(Rᵧ) = B₁ × B₁ = A₁.) Thus, the short expansion of the singlet-triplet transition dipole [Eq. (18)] simplifies to

$$ \tilde{\mathbf{\mu}}_{n\pi^* \to S_0}^{(y)[C-C]} \approx \frac{\langle T_1^{(y)} | \hat{H}^{SO} | S_0 \rangle}{E_1 - E_0} [\langle S_0 | \hat{\mu} | S_0 \rangle - \langle T_1 | \hat{\mu} | T_1 \rangle] \quad (19) $$

corresponding to the direct SOC mechanism.

The second selection rule [Eq. (15)] limits the contribution to the z component of the permanent dipoles. Therefore, for the ³nπ* minimum, emission occurs exclusively oriented along z,

$$ \tilde{\mathbf{\mu}}_{n\pi^* \to S_0}^{(y)[C-C]} \approx \frac{\langle T_1^{(y)} | \hat{H}^{SO} | S_0 \rangle}{E_1 - E_0} [\langle S_0 | \hat{z} | S_0 \rangle - \langle T_1 | \hat{z} | T_1 \rangle] \quad (20) $$

The singlet-triplet transition dipole moments considering the complete first-order perturbative expansion in C₂ᵥ are given in SI-5.

Comparatively, if we apply these selection rules to the ³ππ* minimum, once more, only sublevel y has a non-null transition dipole in the short expansion, given by

$$ \tilde{\mathbf{\mu}}_{n\pi^* \to S_0}^{(y)[C-C]} \approx \frac{\langle T_1^{(y)} | \hat{H}^{SO} | S_1 \rangle}{E_1 - E_0} \langle S_1 | \hat{x} | S_0 \rangle + \frac{\langle T_2^{(y)} | \hat{H}^{SO} | S_0 \rangle}{E_0 - E_2} \langle T_1 | \hat{x} | T_2 \rangle \quad (21) $$

and oriented in the x direction. In this case, only the indirect SOC-dipole and dipole-SOC mechanisms contribute.

These results in Eqs. (20) and (21) explain why ³nπ*→S₀ phosphorescence is so much faster than ³ππ*→S₀ one for aromatic carbonyls. The spin SOC elements have similar magnitudes in both expressions (about 20 cm⁻¹). Nevertheless, the permanent dipoles in Eq. (20) are about 40 times bigger than the transition dipoles in Eq. (21) (SI-6).
Olmsted and El-Sayed proposed that BA phosphorescence could be analyzed using a C=O diatomic model [4]. Suppose we follow their suggestion and rotate the z-axis to coincide with the carbonyl bond direction, as in Refs. [3, 12]. The axes convention for this C=O representation are indicated in Figure 6. In that case, the approximated C2v assignment of the states change (ππ* and nπ* become ~A1 and ~A2, respectively; see symmetry analysis in SI-2), but the conclusions above remain valid.

To see that, note that for the ³nπ* (~A2) T₁ minimum, T₂ ³ππ* is ~A1 and S₁ ¹nπ* is ~A2. The second T₁ minimum is ³ππ* (~A1), where T₂ ³nπ* is ~A2 and S₁ ¹nπ* is ~A2. S₀ is ~A1 in both minima. For the ³nπ* minimum, the only non-null SOC term in the short expansion is in sublevel z with emission along z through the permanent dipole terms, that is

$$ \tilde{\mu}_{10}^{(z)}[\text{C=O}] \approx \frac{\langle T_1^{(z)} | \hat{H}^{\text{SO}} | S_0 \rangle}{E_1 - E_0} [\langle S_0 | \hat{z} | S_0 \rangle - \langle T_1 | \hat{z} | T_1 \rangle] \quad (22) $$

For the ³ππ* minimum, the SOC term in the z sublevel is allowed in the indirect coupling terms. However, all transition dipole terms are forbidden in the short expansion, rendering a null singlet-triplet transition dipole moment from this state. Thus, once again, we obtained that emission from ³nπ* should be faster than from ³ππ*.

In both symmetry orientations, the crucial distinction between ³nπ* and ³ππ is their spatial symmetry, Γ(³ππ*) = Γ(S₀) = Γ₁ while Γ(³nπ*) ≠ Γ(S₀). For the C₂v and D₂ groups, for which Γ(Rξ) ≠ Γ₁, the first relation implies that ³ππ* cannot satisfy the first selection rule [Eq. (13)] and ⟨³ππ*(ξ)| ˆH<sup>SO</sup> |S₀⟩ = 0. Because these groups are Abelian with order 4, one of the Cartesian components ξ yields Γ(³nπ*) Γ(Rξ) = Γ₁. Therefore, they satisfy the first selection rule and give ⟨³nπ*(ξ)| ˆH<sup>SO</sup> |S₀⟩ ≠ 0.

For Dₙ (n > 2), Cₙᵥ (n > 2), Dₙh, Dₙd, and cubic groups, ⟨³ππ*(ξ)| ˆH<sup>SO</sup> |S₀⟩ = 0 for the same reason, but we cannot guarantee that ⟨³nπ*(ξ)| ˆH<sup>SO</sup> |S₀⟩ ≠ 0. For Cₙ (n > 2), Cₙh, and Sₙ, at least one Γ(Rξ) is Γ₁, and ⟨³ππ*(ξ)| ˆH<sup>SO</sup> |S₀⟩ ≠ 0 for these components. In principle, this is also the case for C₁, C₂, and C₃, but molecules belonging to these groups may have ⟨³ππ*(ξ)| ˆH<sup>SO</sup> |S₀⟩ ≈ 0, if they are close to C₂v (like BA is, for example). A survey of these conditions is given in Table 6.

The analysis in this section allows concluding that ³nπ* phosphorescence will be much faster than ³ππ* one if:

1. ³ππ* is totally symmetric, and the molecule (at least approximately) attains C₂v or D₂ symmetry. It may also be valid for other point groups, as indicated in Table 6, but it depends on the specific irreducible representation of the states.

2. The difference between permanent dipole moments in the triplet and singlet states is much bigger than the transition dipole moments between singlets and between triplet states (which is a consequence of Eqs. (20) and (21)).
<table><caption>Table 6. Status of the first selection rule concerning the SOC matrix elements. $\phi$ and $\varphi$ are the spatial configurations of the singlet and triplet states, respectively. $\Gamma(R_{\xi})$ is the irreducible representation of the Cartesian rotation in $\xi$. If the spatial configurations of the singlet and triplet states in relation 1 are identical, this table corresponds to the El-Sayed rule.</caption><thead><tr><th></th><th>Relation 1<br>⟨<sup>1</sup>φ|<span style="text-decoration: overline">H</span><sup>SO</sup>|<sup>3</sup>φ<sup>(ξ)</sup>⟩ = 0<br>with Γ(φ) = Γ(φ)</th><th>Relation 2<br>⟨<sup>1</sup>φ|<span style="text-decoration: overline">H</span><sup>SO</sup>|<sup>3</sup>φ<sup>(ξ)</sup>⟩ ≠ 0<br>with Γ(φ) ≠ Γ(φ)</th></tr></thead><tbody><tr><td>The relation is valid if:</td><td>Γ(R<sub>ξ</sub>) ≠ Γ<sub>1</sub> (all ξ)</td><td>Γ(φ)Γ(R<sub>ξ</sub>) = Γ(φ) (at least one ξ)</td></tr><tr><td>Point group</td><td></td><td></td></tr><tr><td>C<sub>2v</sub>, D<sub>2</sub></td><td>True</td><td>True</td></tr><tr><td>D<sub>n</sub> (n > 2), C<sub>nv</sub> (n > 2), D<sub>nh</sub>, D<sub>nd</sub>, cubic groups</td><td>True</td><td>Not necessarily true</td></tr><tr><td>C<sub>1</sub>, C<sub>2</sub>, C<sub>s</sub></td><td>False</td><td>True</td></tr><tr><td>C<sub>n</sub> (n > 2), C<sub>nh</sub>, S<sub>n</sub></td><td>False</td><td>Not necessarily true</td></tr></tbody></table>

Note that the first point implies that if, for a C<sub>2v</sub> molecule, the <sup>3</sup>ππ* state is B<sub>2</sub>, its phosphorescence lifetime is not necessarily longer than that of a <sup>3</sup>nπ*.

The phosphorescence polarization is (with the phosphorescence lifetime) another indicator of the emitting state character [60]. Experimentally, one expects a <sup>3</sup>nπ* emission in the molecular plane and a <sup>3</sup>ππ* emission perpendicular to the molecular plane [2, 61]. Eqs. (20) and (21) for C–C representation corroborate these emission directions, with <sup>3</sup>nπ* and <sup>3</sup>ππ* emissions polarized in the z- and x-directions, respectively. However, these polarization directions are only significantly preferential in <sup>3</sup>nπ*, for which the permanent dipole terms dominate. $\tilde{T}_1^{(z)}$ emits polarized in y, but it is proportional to small transition dipole terms involving states beyond S<sub>1</sub> and T<sub>2</sub> (SI-5). In the <sup>3</sup>ππ* emission, perturbative terms in $\tilde{T}_1^{(x)}$ (also involving states beyond S<sub>1</sub> and T<sub>2</sub>) introduce significant emission polarized along y. Similar conclusions are reached with the C=O representation. Although the triplet sublevel changes from y to z when changing from C–C to C=O representation, the emission is still primarily in plane, as shown in Eqs. (20) and (22).

## 5.2 Phosphorescence lifetime connection to the El-Sayed rule

The El-Sayed rule is often invoked to explain the difference between <sup>3</sup>nπ* and <sup>3</sup>ππ* phosphorescence lifetimes in aromatic carbonyls [11]. A possible source for this prevailing association between the El-Sayed rule and phosphorescence lifetime may be the 1971 paper by Olmsted and El-Sayed on benzaldehyde [4], where they claimed that “Most of the phosphorescence emission arises because of the direct spin-orbit interaction: <sup>3</sup>A"(nπ*)<sup>SOC</sup>↔<sup>1</sup>A'(ππ*),” evoking the El-Sayed rule [56]. However, this specific SOC term (appearing in the ⟨T<sub>1</sub><sup>(m)</sup>|<span style="text-decoration: overline">H</span><sup>SO</sup>|S<sub>2</sub>⟩⟨S<sub>2</sub>|<span style="text-decoration: overline">μ</span>|S<sub>0</sub>⟩(<sup>3</sup>E<sub>1</sub> − <sup>1</sup>E<sub>2</sub>)<sup>−1</sup> contribution of the singlet-triplet transition dipole moment [Eq. (7)]) should have a minor contribution.
In any case, is there a connection between the El-Sayed rule and the phosphorescence lifetimes? We anticipate that not unless we adopt a loose definition of the rule. Let us see the El-Sayed rule in the context and notation used in our work.

The El-Sayed rule states that the magnitude of the SOC matrix elements between triplet and singlet states with the same spatial configurations is much smaller than for states with different configurations [56]. It is rooted in the first selection rule discussed above [Eq. (13)], and we can express the El-Sayed rule as

$$ \frac{\left| \sum_{\xi=x,y,z} \langle 1\phi | \hat{H}^{SO} | 3\phi(\xi) \rangle \right|}{\left| \sum_{\xi=x,y,z} \langle 1\phi | \hat{H}^{SO} | 3\phi(\xi) \rangle \right|} \ll 1 \qquad (23) $$

where $\phi$ and $\varphi$ represent the spatial configuration of the states. The validity of the El-Sayed rule for different point groups follows the conditions in Table 6. Thus, it is strictly valid for $C_{2v}$ and $D_2$ molecules only. It may occasionally be valid for $D_n$ ($n > 2$), $C_{nv}$ ($n > 2$), $D_{nh}$, $D_{nd}$, and cubic groups, depending on the particular state's irreducible representations. Depending on how close the molecule is to a higher symmetry group, it may be approximately invoked for $C_1$, $C_2$, and $C_s$ groups.

The El-Sayed rule is distinct from the rule we discussed in Section 5.1 because it imposes restrictions on the spatial configurations $\phi$ and $\varphi$, which are not required when discussing the phosphorescence lifetime. However, the El-Sayed rule is a particular case of the discussion in Section 5.1.

We started this subsection by asking whether there was a connection between the El-Sayed rule and the phosphorescence lifetimes. The difference between phosphorescence lifetimes depends on a SOC selection rule that is slightly more general than the El-Sayed rule, as it does impose any restriction on the spatial configurations. Moreover, the phosphorescence lifetimes also depend on the singlet-triplet dipole moment difference. Therefore, it seems inadequate to attribute the phosphorescence lifetime difference to a consequence of the El-Sayed rule.

## 5.3 Contributions for phosphorescence

The previous sections focused on the phosphorescence from the $T_1$ minimum geometry. However, the vibrational freedom allows vibronic couplings that may impact the emission. We have seen in Section 4.4 that how strongly they impact depends on the molecule and state character. For all molecules, vibronic couplings increased the $^3n\pi*$ phosphorescence lifetime by about 20% (Table 5). On the other hand, vibronic coupling reduced the $^3\pi\pi*$ phosphorescence lifetime by a factor of 32 in MoBA, 39 in MeBA, and 50 in DMABA, causing all three molecules to have similar vibronically corrected lifetimes of about 200 ms.
We can understand the vibronic contributions by looking at the three mechanisms composing the singlet-triplet transition dipole moment in Eq. (7). Figure 7 shows the contributions of each of these mechanisms to the singlet-triplet oscillator strength for emission from the lowest T₁ minimum of BA (³nπ*), MeBA (³nπ*), MoBA (³ππ*), and DMABA (³ππ*). The first row of graphs contains the final oscillator strength computed with Eq. (9). The next three rows have the oscillator strength

$$ \tilde{f}_x = \frac{2m_e}{3\hbar^2 e^2} \Delta E_{10} \tilde{M}_x^2 \qquad (24) $$

where $\tilde{M}_x^2 = \frac{1}{3} \sum_{\alpha} |\tilde{M}_x^{(\alpha)}|^2$ is the partial triplet-singlet transition dipole moment for the direct SOC (DS), indirect SOC-dipole (ISD), and indirect dipole-SOC (IDS) mechanisms, computed respectively as

$$ \begin{aligned} \tilde{M}_{DS}^{(\alpha)} &= \frac{\langle T_1^{(\alpha)} | \hat{H}^{SO} | S_0 \rangle}{E_1 - E_0} [\langle S_0 | \hat{\mu} | S_0 \rangle - \langle T_1 | \hat{\mu} | T_1 \rangle] \\ \tilde{M}_{ISD}^{(\alpha)} &= \sum_{\substack{\{\text{singlets}\} \\ L \neq 0}} \frac{\langle T_1^{(\alpha)} | \hat{H}^{SO} | S_L \rangle}{E_1 - E_L} \langle S_L | \hat{\mu} | S_0 \rangle \\ \tilde{M}_{IDS}^{(\alpha)} &= \sum_{\substack{\{\text{singlets}\} \\ L \neq 0}} \frac{\langle T_1^{(\alpha)} | \hat{H}^{SO} | S_L \rangle}{E_1 - E_L} \langle S_L | \hat{\mu} | S_0 \rangle \end{aligned} \qquad (25) $$

The histograms give the oscillator strength distribution of all points in the nuclear ensemble. The vertical and horizontal solid lines in each graph indicate the mean value and standard deviation for the distributions, respectively. The vertical dashed line marks the vertical oscillator strength in each case. The mean and vertical values are also collected in Table 7. The analysis, in terms of partial contributions, neglects crossing terms between mechanisms. Thus, the sum over DS, ISD, and IDS should not recover the total oscillator strength.

The total oscillator strength distributions (first row in Figure 7) show that BA and MeBA, with a ³nπ* T₁ minimum, have NEA distributions with a broad peak with a non-null maximum. As expected, the vertical and NEA mean value oscillator strengths are similar, implying that the nuclear ensemble distributes around the ³nπ* minimum. The distribution shows a second narrow peak at zero for both molecules, meaning that the ensemble also partially covers the ³ππ* region.

MoBA and DMABA, with a ³ππ* T₁ minimum, have total oscillator strength distributions peaked at zero. Again, this is the expected result, as the ensemble distributes around a minimum with near-zero vertical oscillator strength. The NEA mean value is significatively displaced toward larger oscillator strengths, reflecting the vibronic coupling introduced by the procedure.

As we have seen in Eq. (20), a molecule with a ³nπ* T₁ minimum is expected to have dominant vertical contributions from the direct SOC mechanism and minor contributions from the indirect SOC-dipole and dipole-SOC mechanisms in the minimum geometry. This is precisely what the second, third, and fourth rows in Figure 7 reveal for BA and MeBA. Direct SOC dominates. The
indirect SOC-dipole mechanism has significant contributions, while the indirect dipole-SOC
contribution is almost negligible.

<figure><img src="image_8.png" /></figure>

Figure 7. Histogram of the oscillator strengths and the individual contributions to the oscillator strengths of the three mechanisms in Eq. (7) for BA and its derivatives in the NEA simulations estimated with Eqs. (24) and (25). The first row shows the total oscillator strength. The second row corresponds to the direct SOC mechanism, while the third and fourth rows show the indirect SOC-dipole and dipole-SOC mechanisms, respectively. The black dashed vertical lines indicate the vertical value. The mean value and the standard deviation of each distribution are depicted by vertical and horizontal solid black lines, respectively. The values in the horizontal axes are dimensionless quantities, whereas the vertical axes show the counts. Crossing terms contributing to the transition dipole are neglected. All results include 20 singlet and 20 triplet states in the perturbative expansion.

The contributions to the singlet-triplet oscillator strength distributions are completely distinct when considering MoBA and DMABA. Both molecules have ³ππ* T₁ minima; direct SOC should not contribute, as we can see in Eq. (21). The vertical contributions are nearly null for all three terms in Figure 7, explaining the 6.87 s and 11.5 s long phosphorescence lifetimes of MoBA and DMBA, respectively (Table 5). Note, however, that when vibronic couplings are introduced via NEA, the contributions from the three mechanisms increase significantly. As a result, the NEA phosphorescence lifetime drops to 213 ms in MoBA and 231 ms in DMABA, revealing a remarkable vibronic effect. As shown in Table 7, direct and indirect couplings contribute equally to the oscillator strength in MoBA and DMABA.

Table 7. Vertical and NEA mean contributions of each coupling mechanism to the singlet-triplet oscillator strength. These quantities, computed with Eqs. (24) and (25), give only an approximate estimate of the
importance of each mechanism, as they neglect crossing terms contributing to the total oscillator strength (also shown).

<table><thead><tr><th rowspan="2">Molecule</th><th rowspan="2"></th><th colspan="3">Oscillator strength (×10<sup>-7</sup>)</th><th rowspan="2">Total</th></tr><tr><th>direct SOC</th><th>indirect SOC-dipole</th><th>indirect dipole-SOC</th></tr></thead><tbody><tr><td rowspan="2">BA (<sup>3</sup>nπ*)</td><td>vertical</td><td>2.91</td><td>1.22</td><td>0.21</td><td>1.09</td></tr><tr><td>NEA mean</td><td>2.40</td><td>0.89</td><td>0.21</td><td>1.06</td></tr><tr><td rowspan="2">MeBA (<sup>3</sup>nπ*)</td><td>vertical</td><td>3.21</td><td>1.54</td><td>0.25</td><td>1.08</td></tr><tr><td>NEA mean</td><td>2.43</td><td>1.02</td><td>0.19</td><td>0.98</td></tr><tr><td rowspan="2">MoBA (<sup>3</sup>ππ*)</td><td>vertical</td><td>0.0000</td><td>0.0024</td><td>0.0034</td><td>0.0059</td></tr><tr><td>NEA mean</td><td>0.0846</td><td>0.1027</td><td>0.0822</td><td>0.1964</td></tr><tr><td rowspan="2">DMABA (<sup>3</sup>ππ*)</td><td>vertical</td><td>0.0000</td><td>0.0008</td><td>0.0015</td><td>0.0036</td></tr><tr><td>NEA mean</td><td>0.0081</td><td>0.0226</td><td>0.0700</td><td>0.1925</td></tr></tbody></table>

## 6 Conclusions

This work analyzed the phosphorescence lifetime of aromatic carbonyl compounds. We aimed to explain the difference between $^3n\pi*$ and $^3\pi\pi*$ emissions and estimate the importance of vibronic contributions. We addressed these questions by combining formal analysis of the selection rules controlling the singlet-triplet transition dipole moments and simulating vertical and vibronically-corrected phosphorescence lifetimes for benzaldehyde (BA) and three derivatives (MeBA, MoBA, and DMABA). These systems range from $^3n\pi*$ to $^3\pi\pi*$ $T_1$ minimum, enabling a broad assessment of the different coupling mechanisms contributing to light emission.

It is well established that phosphorescence in aromatic carbonyls occurs within a few ms when $T_1$ has a $^3n\pi*$ character, but it may take much longer when this state is a $^3\pi\pi*$. We explained this effect based on analyzing the first-order perturbative expansion of triplet-singlet transition dipole moments without considering vibronic couplings. We showed that the $^3n\pi*-S_0$ transition dipole moment depends on the permanent dipoles of the unperturbed $S_0$ and $T_1$, making it much bigger than the $^3\pi\pi*-S_0$ transition dipole moment, which depends on weaker transition dipole terms between unperturbed states.

This cause, however, is symmetry dependent. It is strictly valid only for molecules with $C_{2v}$ and $D_2$ symmetries, although it can be approximately extended to other point groups. Moreover, it requires the $^3\pi\pi*$ to be totally symmetric. Therefore, while our analysis clearly explains the phosphorescence lifetime difference between a $^3n\pi*$ and a $^3\pi\pi*(A_1)$ emission, we have no reason to expect that the phosphorescence lifetimes of a $^3n\pi*$ and a $^3\pi\pi*(B_2)$ emission would significantly differ. We additionally show that the difference in the phosphorescence lifetimes is not explained by the El-Sayed rule, as sometimes stated, although it is connected to it.
We estimated vertical and vibronically corrected phosphorescence lifetimes for BA, MeBA, MoBA, and DMABA using TDDFT. The vertical values predict $^3n\pi*$ emission within 38 to 61 ms and $^3\pi\pi*$ emission within 8 to 11 s, depending on the molecule (Table 5). Vibronic coupling introduced through the NEA approach increases the vertical $^3n\pi*$ emission to 46 to 75 ms and drastically reduces the $^3\pi\pi*$ emission to 214 to 231 ms. Although our results for the $^3n\pi*$ seem to overestimate the lifetime, they deliver an excellent qualitative picture, where the $^3\pi\pi*$ emission takes three to five times longer than the $^3n\pi*$ emission. Based on experimental analysis of aromatic carbonyls, Harrigan and Hirota [3] proposed that this ratio should be about five times or bigger.

The TDDFT estimates of the phosphorescence lifetimes also corroborate our formal analysis of the reason for the lifetime difference. Decomposing the computed triplet-singlet oscillator strength in the three basic coupling mechanisms confirmed that within the vertical approximation, the direct SOC mechanism (proportional to the permanent dipole of unperturbed states) dominates $^3n\pi*$ emissions and is absent in $^3\pi\pi*$ emissions, as predicted by the formal analysis. This fact is especially significant considering that the aromatic carbonyls studied here have only a roughly approximated $C_{2v}$ symmetry. Including vibronic couplings does not change this picture for $^3n\pi*$ emissions but impacts $^3\pi\pi*$ emissions by allowing direct and indirect coupling mechanisms to contribute equally.

The theoretical and experimental records about the phosphorescence lifetime of aromatic carbonyls are profoundly incomplete, which has been a problem for our analysis. At this point, we can only draw recommendations for future work in the field. On the theory side, applying other vibronic-coupling approaches with more accurate electronic structure methods to estimate aromatic carbonyls' phosphorescence lifetime would be helpful. Such results would allow for gauging the quality of the NEA predictions and explain the lifetime overestimation. Moreover, new theoretical studies could also expand the ensemble of molecules to check which kind of chemical functionalization could extend or reduce the $^3\pi\pi*$ phosphorescence lifetimes.

On the experimental side, a systematic study of aromatic carbonyl phosphorescence using modern spectroscopic techniques and controlled conditions could help elucidate the true extent of the $^3n\pi*$ and $^3\pi\pi*$ lifetime differences.

## Acknowledgments

SM, MK, and MB thank the support of the funding provided by European Research Council (ERC) Advanced grant SubNano (Grant agreement 832237). XG thanks the National Natural Science Foundation of China, grant No. 22273122.
## Data availability

The datasets generated during the current study are available in figshare repository, Ref. [62]. They include the Cartesian coordinate of stationary structures and the raw data of the NEA simulations. The NEA phosphorescence method was implemented in Newton-X and is publicly available at www.newtonx.org. PySOC is freely available at gitlab.com/light-and-molecules/pysoc.

## Author Contribution

Conceptualization: MB; Formal Analysis: SM, MB; Funding acquisition: MB; Investigation: SM, MK, MB; Methodology: XG, MB; Project administration: MB; Software: SM, MBh; Supervision: MB; Visualization: SM, MB; Writing – original draft: SM, MB; Writing – review & editing: SM, MBh, XG, MB.

## Competing Interests

The authors have no relevant financial or non-financial interests to disclose.

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