First Principles Prediction of Wavelength-Dependent Product Quantum Yields

Abstract

We present a method to predict wavelength-dependent product quantum yields (PQYs) for photochemical reactions and applied it to Z/E-isomerization and several ring-closing reactions of Z-2,5-dimethyl-1,3,5-hexatriene and truncated previtamin D. Using branching ratios from surface hopping molecular dynamics, individual trajectories are correlated with the absorption spectra of their initial structures. The wavelength-dependent PQYs are computed by dividing the average spectrum of the initial structures of the product-forming trajectories by the average spectrum of all initial structures. Accurate absorption spectra are calculated using the correlated ADC(2) method with an implicit solvent. Calculations reproduce the experimentally found trend of increasing six-ring formation and decreasing Z/E-isomerization on the red side of the spectrum. Over all seven reactions studied, the mean absolute error (MAE) between experimental and calculated PQYs (MAE) amounts to 8.1%, with the largest MAE of 18.6%. For four reactions, predicted values agree quantitatively with experiments within 5.6%.

Graphical Abstract

Photochemical switches offer a variety of applications for nano devices, including molecular motors and nano transistors.^1,2 Particularly interesting for these applications are systems that undergo different photochemical reactions depending on the wavelength used to induce the switching process. This has been exploited in a variety of systems.^2–6 Furthermore, the wavelength-dependent photochemistry plays a key role in the intrinsic regulation of epidermal vitamin D photosynthesis,⁷ preventing vitamin D overproduction. The mechanism of this process is still under debate.⁸ Several mechanisms can cause a wavelength-dependent photochemistry. On one hand, electronically excited states with different orbital character can lead to different products depending on the excitation wavelength.⁹ On the other hand, the conformation adopted by a molecule can determine the product of a photochemical reaction. If the different conformers have different absorption wavelengths, a wavelength-dependent photochemistry results^10,11 (conformational control). Here, we address the latter case of conformational control. Our motivation is to develop a computational method that is able to quantitatively predict wavelength-dependent product quantum yields (PQYs) with high accuracy compared to experimental results.

The wavelength-dependent product quantum yield Φ(λ) of a photochemical reaction is defined¹² as the number of product molecules formed upon excitation at wavelength λ $\left(N_{\text{molecules}}^{\text{P}}(\lambda )\right)$ divided by the total number of photons of wavelength λ absorbed $\left(N_{\text{photons}}^{\text{tot}}(\lambda )\right)$ :

$$\mathrm{\Phi }(\lambda )=\frac{N_{\text{molecules}}^{\text{P}}(\lambda )}{N_{\text{photons}}^{\text{tot}}(\lambda )}.$$ (1)

Hence, to predict Φ(λ) from first principles it is necessary to know two contributions: a) the efficiency of light absorption (absorption cross section σ(λ)) and b) the probability that the specific product is formed once the molecule is excited, to which we will refer to as the branching ratio. For conformationally flexible molecules, the macroscopic absorption cross section often strongly depends on the equilibrium distribution of conformers present in the ground state. The branching ratio of a photochemical reaction is governed by the dynamics induced by photoexcitation, which depends on the excited and ground state potential energy surfaces (PESs) and the temperature. Conformationally flexible molecules often exhibit fundamentally different photodynamics for different conformers. Thus, in order to predict wavelength-dependent PQYs, the computational approach must take into account the dependency of the molecular conformation on the absorption properties, as well as the dependency of the molecular conformation on the photodynamical behavior. The presented approach based on trajectory surface hopping is well-suited to account for both contributions. Surface hopping has been widely used to predict PQYs, but usually the raw branching rations of the simulations are simply interpreted as the quantum yields.^5,10,13–16 This interpretation only approximates the definition of the PQY (Equation 1), because the number of photons absorbed is not taken into account and the wavelength dependency of the PQYs is neglected. By specifically taking into account the relationship between product formation and absorption properties, the presented method allows to compute the PQY consistent with Equation 1 and also yields the wavelength dependency.

A detailed description of the time-dependent density functional theory (TDDFT) trajectory surface hopping (SH) method, which is based on Tully’s fewest switches SH,¹⁷ can be found elsewhere.¹⁸ The general idea of the method is to represent the photodynamics of the macroscopic ensemble by a finite number of trajectories, where each trajectory stems from different initial conditions, e.g. nuclear coordinates and instantaneous nuclear velocities. The initial conditions are obtained from a Boltzmann ensemble at a given temperature. The nuclear degrees of freedom of each starting structure are then propagated according to Newton’s equations of motion using nuclear forces evaluated for the electronically excited state using TDDFT. In addition, derivative couplings between ground and excited states are evaluated on-the-fly. The couplings describe the probability that the system decays to the ground state. Each individual simulation will follow a different trajectory and distinct reaction products may result after relaxation to the ground state. An ensemble of trajectories yields the raw product distribution or branching ratio. Since the initial structures are selected without consideration of their excitation wavelength and extinction coefficient, the branching ratio does not contain any wavelength dependency. By correlating the trajectories with the absorption spectra of their initial structures the wavelength dependency is introduced.

Using the definition of the PQY (Equation 1), we can replace number of product molecules $N_{\text{molecules}}^{\text{P}}(\lambda )$ by the number of photons absorbed by reactants that successfully form product $P\left(N_{\text{photons}}^{\text{P}}(\lambda )\right):$

$$\mathrm{\Phi }(\lambda )=\frac{N_{\text{photons}}^{\text{P}}(\lambda )}{N_{\text{photons}}^{\text{totons}}(\lambda )}.$$ (2)

This is justified assuming that one photon is needed to produce one product molecule. Considering monochromatic radiation of wavelength λ, the number of photons absorbed by a material is proportional to the oscillator strength, which is related through a line function to the absorption cross section σ(λ) or, equivalently, to the extinction coefficient ϵ(λ)¹⁹ Oscillator strengths for electronic transitions are routinely obtained from excited state electronic structure methods and can be converted to absorption cross sections or extinction coefficients by broadening with a line shape function. In the framework of trajectory surface hopping, $N_{\text{photons}}^{\text{P}}(\lambda )$ is proportional to the average absorption cross section of the initial structures that successfully produce product molecules σ_P(λ):

$$N_{\text{photons}}^{\text{P}}(\lambda )=k\frac{1}{N_{\text{traj}}^{\text{P}}}\sum _i^{N_{\text{traj}}^{\text{P}}}{\sigma }_i(\lambda )=k{\sigma }_P(\lambda ).$$ (3)

$N_{\text{photons}}^{\text{tot}}(\lambda )$is proportional to the average absorption cross section of the initial structures of all trajectories σ_tot(λ):

$$N_{\text{photons}}^{\text{tot}}(\lambda )=k\frac{1}{N_{\text{traj}}^{\text{tot}}}\sum _i^{N_{\text{traj}}^{\text{tot}}}{\sigma }_i(\lambda )=k{\sigma }_{tot}(\lambda ).$$ (4)

Here, k is a constant relating the number of photons to the absorption cross section, and $N_{\text{traj}}^{\text{P}}$ and $N_{\text{traj}}^{\text{tot}}$ are the number of trajectories that successfully form products and the total number of trajectories, respectively. Hence, within trajectory surface hopping, the product quantum yield can be computed as

$$\mathrm{\Phi }(\lambda )=\frac{\frac{1}{N_{\text{traj}}^{\text{P}}}{\sum }_i^{N_{\text{traj}}^{\text{P}}}{\sigma }_i(\lambda )}{\frac{1}{N_{\text{traj}}^{\text{tot}}}{\sum }_i^{N_{\text{traj}}^{\text{P}}}{\sigma }_i(\lambda )}=\frac{{\sigma }_P(\lambda )}{{\sigma }_{tot}(\lambda )}.$$ (5)

For the practical computation of the PQY, we grouped trajectories according to their products formed. Then, for each group the average absorption spectrum σ_P(λ) was computed by averaging the broadened spectra of each initial structure. Analogously, the total spectrum σ_tot(λ) is computed by averaging the spectra of all initial structures. Using Equation 5, the PQY is computed, which avoids binning of the initial structures according to their excitation energy.

To illustrate the method, we focus on the photoproducts that arise from the photoinduced Z/E-isomerization and several electrocyclic ring-closing of Z-2,5-dimethyl-hexa-1,3,5-triene (Z-DMHT) (Scheme 1) and truncated previtamin D (Pre) (Scheme 2), since experimentally measured wavelength-dependent PQYs are available for these molecules. The hexatriene photoactive unit, present in both molecules, is prominent to many molecular switches^2–6 and has served as model system in organic photochemistry leading to the formulation of the paradigm Non-Equilibrium of Excited State Rotamers (NEER) principle.^20,21

Scheme 1:

Major photoproducts of Z-DMHT observed in experiments²² and confirmed by TDDFT-SH simulations. For CP, the mechanism of its formation found in simulations is indicated. Raw branching ratios from simulations are indicated for the products.

Scheme 2:

Major reactions and their products observed in TDDFT-SH simulations of Pre. The aliphatic chain (R) has been replaced by a methyl group in our calculations. Z/Eisomerization, forming tachysterol (left) and ring-closing reactions, forming lumisterol and provitamin D (right). Raw branching ratios from simulations are indicated for the products.

The region with wavelengths larger than 240nm, for which experimental data is available for Z-DMHT and Pre, is solely governed by the first excited singlet state (S₁). Therefore, we started excited state dynamics in S₁. The product distribution of the TDDFT-SH simulations confirms the formation of the experimentally found photoproducts of Z-DMHT, for which wavelength-dependent PQYs have been measured²² (Scheme 1). In particular, we observe Z/E-isomerization producing E-DMHT, and several ring-closing reactions, forming dimethyl-cyclohexadiene (CHD), a cyclobuten derivative (CB), and a cyclopropan derivative (CP). The first three products are formed directly in the excited state or during the transition from the excited state to the ground state, but CP is formed via a [1,6]-sigmatropic hydrogen shift at the transition from the excited state to the ground state, followed by a rearrangement in the ground state (Scheme 1). For Pre, the main photochemical products (Scheme 2) are tachysterol (Tachy), resulting from Z/E-isomerization, and the two isomers, lumisterol (Lumi) and provitamin D (Pro), resulting from six-ring formation. The found products of Pre are also consistent with the major products found in experiments.^23,24 Besides, we also find a large number of unreactive trajectories and a few other minor products (Table S1, Figure S1, Figure S2) in both molecules.

To obtain the wavelength-dependent PQYs, we computed the ten lowest excitation energies and oscillator strengths of the initial structures of the TDDFT-SH simulations using TDDFT with the PBE0 approximation²⁵ (TDPBE0) and the algebraic diagrammatic construction through second-order (ADC(2)) method²⁶ in combination with the COSMO implicit solvent model.^27,28 Figure 1 and 2 confirm that the long-wavelength region beyond 240nm is solely governed by S₁. Furthermore, the magnitudes and distribution of the S₁ oscillator strengths varies if we group them according to the photoproduct that is formed by the corresponding trajectory. Notably, the subset of structures that form the cyclohexadiene derivatives CHD, Pro, and Lumi exhibit excitation energies that reach further into the red side of the spectrum than the structures that lead to Z/E-isomerization. To obtain smooth absorption spectra, we used a Gaussian line shape to convert oscillator strengths to extinction coefficients (panels A of Figures 3 and 4). The peak positions (λ_max) of the ADC(2) total absorption spectra agree within 3nm with the experimentally measured spectra for both, Z-DMHT and Pre. The λ_max values computed by TDPBE0 are red-shifted by 29 and 30nm with respect to the experimental values, for Z-DMHT and Pre, respectively (panels A of Figures S3 and S4). Because of the more accurate spectra of ADC(2) compared to TDPBE0, we present here PQYs computed by ADC(2); TDPBE0 results are given in the Supporting Information (SI). In addition to the total absorption spectra, we computed the absorption spectra of the subset of trajectories that form a given product (Figure 3, A and Figure 4, A).

Figure 1:

Distribution of the ADC(2) oscillator strengths of the first excited singlet state of the initial structures (blue, red, magenta, green, black) of the Z-DMHT trajectories that form the indicated product. Yellow: oscillator strengths of the second excited singlet state.

Figure 2:

Distribution of the ADC(2) oscillator strengths of the first excited singlet state of the initial structures of the Pre trajectories that form the indicated product (blue, red, magenta, black). Yellow: oscillator strengths of the second excited singlet state.

Figure 3:

A: Absorption spectra of Z-DMHT. The calculated absorption spectrum of Z-DMHT of all TDDFT-SH initial structures (solid) and experimentally measured spectrum²² (dashed) are shown in black. The spectra of the initial structures of the trajectories that lead to the indicated reactions are given in color. B: Calculated and experimentally measured²² (Exp.) wavelength-dependent product quantum yields of E-DMHT (Φ_E), CHD (Φ_CHD), CB (Φ_CB), and CP (Φ_CP).

Figure 4:

A: Absorption spectra of Pre. Calculated spectrum of all TDDFT-SH initial structures (solid) and experimentally measured spectrum²⁴ (dashed) are shown in black. The spectra of the initial structures of the trajectories that lead to the indicated reactions are given in color. B: Calculated and experimentally measured²⁴ (Exp.) wavelength dependent product quantum yields of Pre²⁴ wavelength-dependent product quantum yields of Lumi (Φ_Lumi), Pro (Φ_Pro), and Tachy (Φ_Tachy).

Dividing the spectra of the product-forming trajectories by the total absorption spectra (Equation 5), we obtain the wavelength-dependent PQYs for the reactions of Z-DMHT (Figure 3, B) and Pre (Figure 4, B). Computed values, mean errors (ME), and mean absolute errors (MAE) with respect to experimental values are given in the SI (Tables S2–S5). The PQYs of E-DMHT formation (Φ_E) and CHD formation (Φ_CHD) (Figure 3, B) agree well with the experimentally measured values. In particular, the decrease of Φ_E and the increase of Φ_CHD on the red side of the experimental spectrum between 290nm and 303nm is well reproduced by the calculations. The MAE between calculated and experimental values are 5.5% and 12.0%, for Φ_CHD and Φ_E, respectively. Also the PQYs of CB formation agree well with the experimentally measured value and are underestimated by only 3% on average. For the CP formation, calculations overestimate the PQYs by a larger amount of about 12.9% on average. The agreement of TDPBE0 PQYs with experimental values is similar to ADC(2) (Figures S3 and S4, Tables S3 and S7 -S9), but we should keep in mind that the curves of the PQYs (Figures S3 and S4) should be red-shifted by about 30nm to compensate the error in the spectra.

The PQYs for the conversion of Pre to Lumi (Φ_Lumi) and Pro (Φ_Pro) quantitatively agree with the experimentally measured values over the whole spectrum where experimental data is available. MAEs between experimentally measured and calculated values are 2.9% and 2.0% for Lumi and Pro formation, respectively. The experimentally found increase of Φ_Lumi and Φ_Pro on the red side of the spectrum is also well reproduced by the calculations. The deviation between experiment and calculation is larger for Tachy formation (Φ_Tachy); here, calculations underestimate experimental values by 18.6% on average. Qualitatively, however, the calculations confirm the experimentally found decrease of Φ_Tachy with increasing wavelength.

We implemented a method based on TDDFT-SH and ADC(2) absorption spectra to compute wavelength-dependent product quantum yields (PQYs) and applied it to the Z/E-isomerization and ring-closing reactions of Z-DMHT and Pre. Over all available data points of the seven reactions studied, the mean absolute error between experimentally measured and calculated PQYs (MAE) amounts to 8.1%. For four reactions, the predicted PQYs agree quantitatively within 5.6% with the experimentally measured values. For E-DMHT and CP formation, the agreement with experiments is better than 13% and for Tachy formation we find the largest error of 18.6%. The experimentally found trend of increasing six-ring formation and decreasing Z/E-isomerization with increasing wavelength is correctly reproduced by the calculations. The good accuracy of the method is to one part due to the high accuracy of the computed absorption spectra, but it also shows the predictive power of the TDDFT-SH method, despite its approximative nature.

Potential sources of errors in our calculations possibly arise from the neglect of the solvent, in both the generation of the Boltzmann ensembles as well as in the excited state SH simulations. In the spectra calculations, the solvent has been approximated by an electrostatic continuum. Approximations introduced by the use of TDDFT and the approximate hybrid exchange-correlation functional possibly affect the accuracy of the branching ratio due to erroneous description of conical intersections or the neglect of double excitations^29,30 during excited state dynamics.

However, an appealing factor of our method is that it is easily extendible by step-wise improvements to achieve higher accuracy. For instance, explicit solvent molecules could be included in ground and excited state simulations using a hybrid quantum mechanical/molecular mechanics^31,32 or even a full quantum³³ approach. Furthermore, the quantum nature of the nuclei could be considered in the initial sampling by either using a Wigner distribution³⁴ or path-integral molecular dynamics.³⁵ More accurate branching ratios could possibly be computed using correlated wavefunction based electronic structure methods,^32,36–38 if the increased computational cost is affordable. Lastly, the accuracy of absorption spectra could further be improved by taking into account vibronic effects^39,40 and effects of an explicit solvent.⁴⁰ To include reactions of higher excited states, the method can be extended by starting dynamics in higher excited states.

Overall the present method constitutes an efficient tool to predict wavelength-dependent PQYs in conformationally controlled photochemical reactions. The method can be applied to assist the design and optimization of photochemical switches.

Computational Details

We use ab-initio molecular dynamics (AIMD)⁴¹ to sample the ground-state ensemble and to model photodynamics of the molecules. All AIMD simulations were performed with the density functional theory (DFT)⁴² and TDDFT⁴³ modules of the TURBOMOLE V6.3.⁴⁴ Simulations employ the def2-SVP⁴⁵ basis set. Ab initio Replica exchange molecular dynamics (REMD)^11,46 is used to enhance the sampling of ground state molecular conformations and employ the PBE functional,⁴⁷ accelerated using the resolution of identity approximation.⁴⁸ REMD was performed in the canonical ensemble at four temperatures (300K, 600K, 900K, 1200K) controlled by a Nosé-Hoover thermostat.^49,50 Switches between the trajectories at different temperatures were evaluated every 200 molecular dynamics steps. A time step of 50 au was used to propagate the nuclear degrees of freedom. Non-adiabatic molecular dynamics simulations were carried out with the TDDFT-SH method⁵¹ as implemented in TURBOMOLE.^10,18 From the ground-state Boltzmann ensemble at 300K generated by REMD, we sequentially choose several hundreds of snapshot structures (Z-DMHT: 991, Pre: 684) and their instantaneous velocities every 80 time steps (96.7fs) as starting geometry for an excited state dynamics simulation. Initial structures were propagated in the photochemically active S₁ state. The PBE0 approximation to the exchange-correlation functional²⁵ was used in TDDFT-SH simulations, since it produces accurate PES for hexatriene systems.¹⁰ To minimize problems with instabilities at conical intersections,^9,30 we used the Tamm-Dancoff approximation.⁵² Each TDDFT-SH simulation was propagated for 900fs allowing the full excited-state decay and subsequent hot ground-state reactions of each molecule. The total energy was conserved in TDDFT-SH simulations (NVE ensemble). The ten lowest singlet excitation energies and oscillator strengths were calculated for each initial structure of the trajectories by ADC(2)²⁶ and TDPBE0. ADC(2) was applied within the resolution of identity approximation,^53,54 employing the def2-TZVP basis set,⁴⁵ as previously described.⁵⁵ TDPBE0 calculations make use of the full linear response and the def2-SVP basis.⁴⁵ Oscillator strengths were converted into molar decadic extinction coefficients by broadening with a Gaussian function with a full width at half maximum of 0.15eV and averaged to obtain the macroscopic absorption spectrum.^39,56 The conductor-like screening (COSMO) model was used in ADC(2) calculations to account for solvent effects;^27,28 a dielectricity constant of 1.844 and a refractive index of 1.3575, corresponding to n-pentane at 20°C was used.⁵⁷