Simulating nonadiabatic dynamics in benzophenone: Tracing internal conversion through photoelectron spectra

Abstract

Benzophenone serves as a prototype chromophore for studying the photochemistry of aromatic ketones, with applications ranging from biochemistry to organic light-emitting diodes. In particular, its intersystem crossing from the first singlet excited state to triplet states has been extensively studied, but experimental or theoretical studies on the preceding internal conversion within the singlet manifold are very rare. This relaxation mechanism is particularly important because direct population transfer of the first singlet excited state from the ground state is inefficient due to its low oscillator strength. In this work, our aim is to fill this gap by employing mixed quantum-classical and full quantum dynamics simulations and time-resolved photoelectron spectroscopy for gas-phase benzophenone and meta-methyl benzophenone. Our results show that nonadiabatic relaxation via conical intersections leads to an increase in the population of the first singlet excited state, which appears linear within the simulation time of 500 fs. This population transfer due to conical intersections can be directly detected by a bifurcation of the photoelectron signal. In addition, we discuss to clarify the role of the third singlet excited state degenerate to the second excited state—a topic that remains largely unexplored in the existing literature on benzophenone.

I. Introduction

Benzophenone (BP) is an organic chromophore frequently employed as a prototype for investigating the photochemistry of more complex aromatic ketones. This type of chromophore has a rich photochemistry under ultraviolet (UV) light, which involves several nonadiabatic steps: a conical intersection (CoIn) between the first two singlet excited states, S₂ and S₁, and subsequent efficient intersystem crossing processes between the S₁ and several triplet states. In particular, due to the latter, BP and many of its derivatives has significant applications in technological and biological contexts, including organic light-emitting diodes,^1,2 UV filters,^3,4 and photosensitizers.^5,6 The existing literature on the photochemistry of BP is extensive, but most experimental and theoretical studies^7–12 focus on the mechanisms that lead to a direct and indirect population of the lowest triplet excited state, T₁, starting from S₁.¹¹ However, the S₁ state has a negligible cross section and cannot be efficiently populated directly from the ground state. The experimentally more realistic pathway is to excite the system to higher lying singlet states of ππ* character, which have a stronger oscillator strength, and populate S₁ via internal conversion.

The S₂/S₁ CoIn has been the subject of two experimental investigations by Shah et al.¹³ and Spighi et al.⁸ In their study, Shah et al. employed time-resolved absorption spectroscopy with a full width at half maximum (FWHM) of ≈150 fs pump pulse centered at 267 nm (≈ 4.66 eV). They measured a lifetime of 0.53 ps for the S₂ state in acetonitrile following the growth in absorption of the S₁ state and a kinetic rate of 1.7–1.9 × 10¹² s⁻¹ for the S₂ → S₁ internal conversion. Spighi et al. investigated the excited-state dynamics of gas-phase BP via femtosecond and nanosecond resonance-enhanced multiphoton ionization, employing a pump pulse centered at 266 nm (≈ 4.66 eV), with a pump–probe cross correlation of ∼100 fs. The authors observed an exponential rise and decay of photoelectron bands with a time constant of 150 fs, which they attributed to a corresponding increase in the S₁ population. However, as the authors stated, the multiphoton ionization probe made it impossible to unambiguously assign each band in the photoelectron spectra to a single electronic state. The resonance-enhanced multiphoton ionization scheme involves several intermediate states that provide multiple pathways and resonance conditions, further complicating the interpretation.

Both experiments provide evidence of the ultrafast nonradiative decay from the S₂ state, indicating the presence of a conical intersection between S₂ and S₁. However, the temporal resolution of the two experiments cannot resolve the intricate ultrafast excited-state dynamics occurring within the first 100 fs. Moreover, the existence of the S₃ state, which is almost degenerate to the S₂ state in the Franck−Condon (FC) region, has not been taken into account in the interpretation of the experimental results. Figure 1 shows the natural transition orbitals (NTOs) with the highest weight, characterizing the three lowest singlet excited states in BP. The S₁ state has nπ* character (see Fig. 1) and belongs to the irreducible representation A under C₂ symmetry. The S₀ → S₁ transition predominantly involves the promotion of an electron from the oxygen lone pair to the anti-bonding π* orbital of the carbonyl group. Both S₂ and S₃ have ππ* characters and belong to the irreducible representations A and B, respectively. This pair of ππ* transitions is nearly degenerate in the FC region. As shown in Fig. 1, their NTOs describing the hole are identical except for their symmetry/phase and are delocalized over both phenyl rings. In contrast, the corresponding NTOs describing the electron are barely distinguishable and more localized in the carbonyl group. Given that the energy difference between S₂ and S₃ is predicted on the order of 10⁻² eV at the FC point, the experimental pump pulse resolution cannot differentiate them. Consequently, both states can be expected to be populated under experimental conditions.^8,13 In this work, we determine time-resolved photoelectron spectra based on the excited-state dynamics in the singlet manifold of the gas phase BP and its methylated derivative meta-methyl benzophenone (m-BP), displayed in Fig. 2. The choice of m-BP was driven by challenges in modeling the excited-state dynamics of BP, caused by the near-degenerate electronic states. Adding a methyl group in the meta position of one of the two phenyl rings breaks the symmetry and increases the energy gap between the S₂ and S₃ states, thus reducing their coupling while preserving comparable photochemical behavior. Based on our dynamics simulation, we show that (i) the increase in the population of the S₁ state appears linear within our simulation time of 500 fs, and it is accompanied by a similar linear increase in the corresponding photoelectron band. Beyond that, the internal conversion can be associated with a bifurcation of the photoelectron signal; and (ii) the inclusion of the S₃ state reveals ultrafast population transfer to S₂, due to the presence of a conical intersection in the vicinity of the FC point.

Fig. 1.

Diagram of the natural transition orbitals (NTOs) characterizing the three lowest singlet excited states in BP, calculated at the TDA/ωB97X-D4/6-311G* level of theory. The diagram shows only the largest contributions (w) for each transition. An isovalue of 0.05 was used.

Fig. 2. Molecular structures of benzophenone and meta-methyl benzophenone.

II. Computational Details

All calculations were performed in a reproducible compute environment using the Nix package manager together with NixOS-QChem¹⁴ (commit 9b40085ca).

A. Ab initio level of theory

All calculations were performed in the gas phase, in C₁ symmetry. Geometry optimizations and frequency analysis were carried out at the ωB97X-D2^15,16/6-311G*¹⁷ level of theory using the Gaussian 16 RevC.01¹⁸ quantum chemistry program. The excited states of BP and m-BP were determined by linear response time-dependent density functional theory (LR-TDDFT), complete active space self-consistent field (CASSCF), as well as density functional theory/multi-reference configuration interaction (DFT/MRCI) theories. The LR-TDDFT calculations were carried out within the Tamm–Dancoff approximation (TDA)¹⁹ using the ωB97X-D4²⁰/6-311G* level of theory in the ORCA 5.0.4 program package.^21,22 The state-average SA-CASSCF calculations were performed using OpenMolcas v.24.02,^23,24 with the ANO-L-VDZP²⁵ basis set, averaging over 5 states. After the CASSCF calculation, dynamic electron correlation was added using extended multistate complete active space second order perturbation theory (XMS-CASPT2).^26,27

The configuration interaction (CI) expansion for CASSCF scales factorially with the number of active orbitals and electrons. The full π-system of BP requires an active space of 16 electrons in 15 orbitals. The full CAS(16,15) active space is shown in Fig. S1 of the supplementary material. This computation is time-consuming, and complete active space second order perturbation theory (CASPT2) becomes impractical for the purpose of calculating excited-state dynamics. Although this active space can still be used to benchmark some critical points, it is not suitable for constructing global potential energy surfaces. For this reason, we employed a DFT/MRCI approach to achieve the balance between computational costs and accuracy.

Two-dimensional potential energy surfaces, diabatic couplings, and transition dipole moments (TDMs) of the ground state and three singlet excited states of m-BP were calculated at the DFT/MRCI(2)^28–30 level with the GRACI program,³¹ using the QTP17³² xc-functional, the QE8 Hamiltonian,³⁰ and the 6-311G* basis set. The quasi-diabatic states and potentials were obtained by a procedure using quasi-degenerate perturbation theory³³ as implemented in GRACI.³¹ The active space used in the DFT/MRCI calculations is shown in Figs. S4 and S5 of the supplementary material. The raw values (energies, couplings, and transition dipole moments) were then interpolated with polyharmonic splines³⁴ onto a fine grid (N₁ = N₂ = 128 points) for the use with wave packet dynamics.

B. Surface hopping dynamics

The molecular dynamics of BP and m-BP were simulated using the SHARC ab initio dynamics package version 3.0.^35,36 The necessary energies, gradients, and non-adiabatic couplings were calculated on-the-fly at the TDA/ωB97X-D4/6-311G* level of theory. In total, 1000 initial conditions for the dynamics simulations were generated based on a Wigner distribution computed from harmonic vibrational frequencies in the optimized ground state equilibrium geometry. The underlying frequency calculations were performed at the ωB97X-D2/6-311G* level of theory. Altogether, 160 initial geometries and velocities were chosen stochastically, using SHARC’s default selection criteria based on the oscillator strength. For BP 68, trajectories started in state S₂ and 92 in state S₃, while for m-BP, 62 started in state S₂ and 98 in state S₃.

The standard SHARC ab initio surface-hopping algorithm^37,38 was used for the mixed quantum-classical dynamics. The integration of the nuclear motion was done with the velocity-Verlet algorithm with a maximal time of 0.5 ps using a time step of 0.5 fs. In each time step, gradients were calculated for all states that were closer than 0.001 eV to the active state. The coefficients of the electronic wave function are propagated on interpolated intermediates with a time step of 0.02 fs, applying a local diabatization technique³⁹ in combination with the WFoverlap code⁴⁰ to compute the wave function overlaps. Decoherence correction was taken into account using the energy-based method of Granucci and Persico with the parameter α = 0.1 a.u.⁴¹ We restricted our analysis to trajectories with a maximum change in the total energy of 0.3 eV and a maximum change in the total energy per step of 0.2 eV. All trajectories are included in the analysis as long as both criteria are met. At 500 fs, 55% of the BP trajectories and 50% of the m-BP trajectories satisfied the energy conservation criteria. The uncertainty⁴² associated with the ensemble of trajectories as a function of time for both BP and m-BP is depicted in Fig. S5 of the supplementary material. At 500 fs, the maximum statistical error was 10.4% for BP and 11.2% for m-BP.

For each geometry, Dyson norms were calculated between four neutral and four cationic states using the WFoverlap code^40,43,44 at the TDA/ωB97X-D4/6-311G* level of theory. The Dyson norms provide an adequate estimate of the photoionization yield for single-photon ionization.^45,46 Excitation energies and corresponding Dyson norms were employed to generate a photoelectron spectrum, which was then broadened by a Gaussian function with a FWHM of 0.4 eV. In addition, the spectra were convoluted with a Gaussian function (4 fs FWHM) in the time domain.

C. Quantum dynamics simulations

The wave packet dynamics of the photoexcited m-BP was calculated by solving the time-dependent (non-relativistic) Schrödinger equation numerically with the Fourier method,^47–49 using the software package QDng.⁵⁰ To propagate the wave packet on the potential energy surfaces, we used a Hamiltonian on a diabatic basis and coordinates q,

$$\widehat{H}=\left(\begin{array}{cccc}\widehat{T}+{\widehat{V}}_{S_0}(q)& 0& 0& 0\\ 0& \widehat{T}+{\widehat{V}}_{S_1}(q)& {\widehat{V}}_{S_1{S}_2}(q)& 0\\ 0& {\widehat{V}}_{S_1{S}_2}(q)& \widehat{T}+{\widehat{V}}_{S_2}(q)& {\widehat{V}}_{S_2{S}_3}(q)\\ 0& 0& {\widehat{V}}_{S_2{S}_3}(q)& \widehat{T}+{\widehat{V}}_{S_3}(q)\end{array}\right),$$ (1)

where $\widehat{T}$ is the kinetic energy operator; and ${\widehat{V}}_{S_0},{\widehat{V}}_{S_1},{\widehat{V}}_{S_2}$, and ${\widehat{V}}_{S_3}$ are the diabatic potentials for the states S₀, S₁, S₂, and S₃, respectively. The diabatic couplings are given by ${\widehat{V}}_{S_1{S}_2}$ and ${\widehat{V}}_{S_2{S}_3}$. The kinetic operator is in the G-matrix form⁵¹

$\widehat{T}=-\frac{{ħ}^2}{2}{\displaystyle \sum _{r=1}^M{\displaystyle \sum _{s=1}^M}}\left[G_{rs}\frac{{\partial }^2}{\partial q_r\partial q_s}\right],$ (2)

where M is the number of coordinates taken into consideration. The expression in (2) derives from the assumption that the elements of the Wilson G-matrix are constant. The term G_rs indicates the element of the kinetic energy matrix equal to

$G_{rs}={\displaystyle \sum _{i=1}^{3N}\frac{1}{m_i}\frac{\partial q_r}{\partial x_i}\frac{\partial q_s}{\partial x_i}.}$ (3)

Here, N is the total number of atoms and m_i is the mass of atom i. In Eq. (3), diagonal entries are reciprocals of generalized reduced masses, and the off-diagonal entry is the kinetic coupling term. The calculated values are given here in atomic units: G_rr = 0.000 144 7, G_ss = 0.000 169 24, and G_rs = −5.101 169 × 10⁻⁶ a.u. The short iterative Lanczos scheme^52–54 was used to propagate the nuclear wave packet and the Hamiltonian form Eq. (1) with a time step of Δt = 1 au (24.2 as). The grid-based approach used in this work is only possible for reduced geometric subspaces and not for the full 3N-6 degrees of freedom (DOF). Consequently, the quantum dynamics simulations are performed in a low-dimensional subspace formed by carefully chosen DOF to describe the processes of interest.⁵⁵ The total wave function was expressed as a linear combination,

$\text{Ψ}={\displaystyle \sum _{i=0}^4{c}_i{\varphi }_i}$ (4)

with coefficients $c_0=c_1=0,c_2=\sqrt{0.4}$, and $c_3=\sqrt{0.6}$, at time t = 0. These values were specifically chosen to replicate the initial diabatic populations of the surface hopping dynamics.

III. Results and Discussion

A. Validation of the level of theory

Before we introduce the molecular dynamics and the simulated time-resolved photoelectron spectra for BP and m-BP, we discuss the vertical UV excitations and ionization energies of the two molecules. Table I provides a benchmark of calculated vertical excitations for BP against the experimental values taken from Ref. 56. This includes the level of theory used in surface hopping dynamics (TDA), the level of theory used to construct potential energy surfaces (PESs) (DFT/MRCI), and the multireference method (CASPT2) that employs the full π system of BP [CAS(16,15) with 5-state averaging]. An even more extensive benchmark, including CASPT2 with different sizes of the active space and RASPT2 calculations, can be found in Tables S1–S4 of the supplementary material. Additional computational details are given in the table caption and Sec. II. An extensive benchmark of CASPT2 and RASPT2 of BP at the FC point in the singlet manifold can be found in Table I of Ref. 57. Our results are in good agreement with those presented there. As expected, the CASPT2 calculations predict vertical excitations very close to the experimental values and are, therefore, well suited as a general criterion for validating the other theoretical methods applied. In contrast, DFT/MRCI and especially TDA calculations generally overestimate the excitation energies. The DFT/MRCI results are slightly more accurate but still produce vertical transitions that are systematically blueshifted by ∼ 0.4 eV. Meanwhile, TDA calculations tend to over-estimate ππ* transitions by nearly 1 eV. Nonetheless, all methods capture the near degeneracy between S₂ and S₃ at the FC point. Both DFT/MRCI and TDA predict an energy difference between S₃ and S₂ of 0.04 eV. According to CASPT2 calculations, the energy difference between S₂ and S₁ at the FC point is 0.95 eV, which is slightly above the experimental value of 0.79 eV. Likely due to error compensation, the DFT/MRCI method predicts a 0.9 eV gap, performing better than CASPT2. The TDA calculations yield the largest energy gap at 1.32 eV. In fact, TDA overestimates both the ππ* transitions (S₂ and S₃) by about 0.8 eV in the FC region, but also the nπ* transition (S₁) is slightly blue shifted by about 0.4 eV compared to the CASPT2 values. As a result, when the system starts in S₂/S₃, its initial internal energy is higher, leading to a slightly greater energy gap between S₁ and S₂/S₃ for TDA. Both factors can affect the simulated dynamics. On the one hand, the larger energy gap can slow down relaxation; and on the other hand, the molecule will have more internal energy when reaching S₁. All methods consistently predict the right trend for the oscillator strengths, identifying S₀ → S₁ as the weakest transition.

Table I.

Comparison between vertical excitation energies and their corresponding oscillator strength of BP, calculated at (I) TDA/ωB97X-D4/6-311G*, (II) DFT/MRCI(2)/QTP17/QE8/6-311G*, and (III) XMS-CASPT2/CAS(16,15)/ANO-L-VDPZ levels of theory.

	Energies (eV)				Oscillator strength
	(I)	(II)	(III)	Exp	(I)	(II)	(III)
S 1	4.04	4.04	3.65	3.61a	0.0010	0.0040	0.0007
S 2	5.36	4.94	4.60	4.40a	0.015	0.0213	0.0036
S 3	5.40	4.98	4.62	4.40a	0.015	0.0114	0.0023

^aLow pressure vapor, Ref. 56

Table II provides a comparison of the adiabatic energies for several TDA optimized structures away from the FC point: the minimum energy S₂ /S₁ CoIn and the S₁ and S₂ minima. The TDA calculations estimate an energy difference of 0.03 eV between S₁ and S₂ at the CoIn, while CASPT2 predicts a slightly higher value of 0.09 eV. DFT/MRCI predicts the largest energy gap, at nearly 0.3 eV. The TDA description of the S₁ minimum is in good agreement with CASPT2, while the DFT/MRCI adiabatic energies are

Table II.

Comparison between adiabatic energies of BP at the S₂/S₁ CoIn, S₁, and S₂ minimum geometries, calculated at (I) TDA/ωB97X-D4/6-311G*, (II) DFT/MRCI(2)/QTP17/QE8/6-311G*, and (III) XMS-CASPT2/CAS(16,15)/ANO-L-VDPZ levels of theory.

	CoIn S2/S1			S1 min			S2 min
	(I)	(II)	(III)	(I)	(II)	(III)	(I)	(II)	(III)
S 1	4.83	4.99	4.53	3.12	3.66	3.15	4.00	4.33	3.63
S 2	4.86	5.28	4.64	4.88	5.10	4.63	4.89	4.80	4.28
S 3	6.12	6.42	5.88	4.91	5.12	4.64	5.49	5.41	4.83

systematically blueshifted by ≈ 0.5 eV. However, the three methods consistently predict a near-degeneracy between S₂ and S₃ at the S₁ minimum geometry. The energy associated with the relaxation of the S₂ state at its minimum, compared to the FC point, is the highest for TDA (∼0.5 eV), then for CASPT2 (around 0.3 eV), and finally for DFT/MRCI (∼0.15 eV). Considering that CASPT2 aligns with TDA in the description of the S₂/S₁ CoIn geometry and the S₁ minimum, we believe that, despite its limitations, TDA is sufficient to capture the main features of the relaxation dynamics of BP.

Table III shows the vertical excitation energies calculated for m-BP. Although we were unable to find any experimental UV spectrum data for gas-phase m-BP to directly compare with our calculations, we anticipate that it will not differ significantly from that of the parent molecule. TDA and DFT/MRCI calculations yield the same value for the S₀ → S₁ transition as observed for BP, suggesting that the methyl group in meta has a negligible influence on the CO π bond. The predicted energy gap between S₃ and S₂ is 0.14 eV with TDA and 0.12 eV with DFT/MRCI, where the splitting arises mainly from a stabilization effect on the S₂ state. The energy gap between S₂ and S₁ at the FC is calculated to be smaller for m-BP than BP. DFT/MRCI calculations place it at 0.8 eV, whereas TDA estimates it at 1.21 eV. Similarly to BP, the overestimation of the energy difference between S₂ and S₁ may slow down the population transfer during the surface hopping dynamics. The energies calculated over a Wigner distribution (see Sec. II for more details) have been convoluted with a Gaussian (FWHM = 0.4 eV) to produce the UV spectra of BP and m-BP, as shown in Figs. 3(a) and 3(b), respectively. Consistent with the findings at the FC point presented in Tables I and III, the optically brighter region of the TDA spectrum shows a blue shift of 0.4 eV compared to DFT/MRCI. There are no notable distinctions between BP and m-BP.

Table III.

Comparison between vertical excitation energies and their corresponding oscillator strength of m-BP, calculated at (I) TDA/ωB97X-D4/6-311G* and (II) DFT/MRCI(2)/QTP17/QE8/6-311G* levels of theory.

	Energies (eV)		Oscillator strength
	(I)	(II)	(I)	(II)
S 1	4.04	4.00	0.0010	0.0023
S 2	5.25	4.80	0.0220	0.0279
S 3	5.39	4.92	0.0133	0.0124

Fig. 3.

Unshifted UV spectra of benzophenone (a) and meta-methyl benzophenone (b) calculated over their respective Wigner distributions with TDA (dashed line) and DFT/MRCI (solid line).

The ionization spectrum of gas-phase BP has been the subject of several experimental and theoretical studies.^58–61 The experimental spectrum measured by Centineo et al.⁵⁸ shows an overlapping band in the 8−10 eV range with two distinguishable peaks at ∼ 9.0 and 9.45 eV, along with two shoulders at higher energies. The calculated static photoelectron spectrum of BP is shown in Fig. 4. The CASPT2 calculations, including the complete π system of BP, are in good agreement with the experiment. The lack of a shoulder at higher binding energies is due to the absence of the corresponding electronic states in the SA-CASSCF calculations, which were averaged over four states of the cation. The photoelectron spectrum calculated with TDA accurately models the D₀ state, but it overestimates the ionization energies for D₁, D₂, and D₃, which are blueshifted by ∼ 0.5 eV. One explanation for the observed blueshift comes from the quantum chemistry calculations that were used to derive these energies. The ionization energy for D₀ is calculated from the singlet and doublet ground state energies that were obtained by DFT calculations. In contrast, the ionization energies for the excited states (S₀ to D₁, D₂, etc.) are computed using an unrestricted TDDFT approach, which yields energies that are less accurate compared to D₀.

Fig. 4.

Static photoelectron spectrum of ground-state BP calculated with TDA/ωB97X-D4 (red color scheme) and XMS-CASPT2 (blue color scheme). To prevent overlapping features, the CASPT2 spectrum is shown up−side−down.

Overall, although there are some quantitative differences, we are convinced that TDA and the chosen functionals are adequate to describe the mixed quantum-classical dynamics of BP and m-BP.

B. Mixed quantum-classical dynamics and time-resolved photoelectron spectra

Having validated the ab initio method, we will now discuss the excited-state dynamics of BP and m-BP, starting with the parent molecule. Note that the number of trajectories that meet the total energy conservation criteria decreases with time. This time-dependent number of trajectories was used to construct the photoelectron spectra and the populations of the singlet manifold. The on-the-fly selection criteria are discussed in Sec. II. Upon excitation to the S₂ or S₃ state, for all analyzed trajectories, nonadiabatic processes are observed.

The time evolution of the average adiabatic populations along the BP trajectories is shown in the top panel of Fig. 5. The second y-axis represents the fraction of trajectories (gray line) that satisfy the total energy conservation criteria at each time step. The trajectories reveal a rapid exponential decay of the S₃ population to S₂ within the initial 10 fs of the dynamics, and the nonadiabatic relaxation is completed within 50 fs. The strong depletion in the S₃ population suggests that S₃ and S₂ are strongly coupled and that there is a favorable gradient toward S₂. After 40 fs, an almost linear increase in the S₁ population can be observed, along with a corresponding decrease in the S₂ population. The normal mode analysis of the SHARC trajectories within the first 150 fs reveals that the dihedral movement between the two aryl rings is predominant. Nevertheless, a direct correlation between the dihedral oscillations and the photoelectron observable could not be established. The normal mode analysis is available in Fig. S6 of the supplementary material. As more population enters S₁ of nπ* anti-bonding character, an elongation of the CO bond can be observed, as shown in Fig. S7 of the supplementary material.

Fig. 5.

Time-resolved photoelectron spectrum of BP computed at the TDA/ωB97X-D4/6-311G* level of theory from the combined trajectories of S₂ and S₃. Upper panel: time evolution of the average adiabatic populations along the BP trajectories. The second y-axis shows the fraction of trajectories (gray line) meeting the total energy conservation criteria at each time step, used for calculating the spectrum and time-dependent populations of the singlet manifold. A unitary fraction corresponds to 160 trajectories, with a minimum fraction of ≈0.55 equivalent to 88 trajectories. Lower left panel: temporal slice of the time-resolved photoelectron spectrum selected at 0 fs. Dashed lines indicate the binding energies at the FC point, using the color scheme of the population plot. Lower right panel: full time-resolved photoelectron spectrum.

The lower left panel of Fig. 5 displays a snapshot of the time-resolved photoelectron spectrum taken at time zero. The dashed lines indicate the binding energies at the FC point, with the same color scheme as the population plot in Fig. 5. The snapshot spectrum consists of two bands, approximately centered at 3.6 and 4.7 eV. With the temporal evolution of the populations in mind, we are now able to discuss the time-resolved photoelectron spectrum of BP, shown in the lower right panel of Fig. 5. Note that due to the different characteristics of the probe (multi-photon vs single-photon ionization), we cannot directly compare our results to the experiment of Spighi et al.⁸ In fact, Dyson norms can only be used to describe single-photon ionization steps. During the initial 15 fs of the dynamics, a distinct blueshift of the two bands is observed in the time-resolved photoelectron spectrum, along with a notable inversion in intensity. The band originating at 3.6 eV experiences a blueshift of ∼ 0.4 eV and an increase in intensity, while the band at 4.7 eV experiences a blueshift of ∼0.3 eV and a decrease in intensity. The internal conversion from S₃ to S₂ is responsible for this feature of the time-resolved photoelectron spectrum. The observed intensity exchange matches the exponential decay of the S₃ population. Moreover, the blueshift suggests a favorable gradient toward S₂. After 50 fs, the bifurcation of the band centered at 3.6 eV becomes increasingly more evident. The signal covers a wide spectral range, from 4 to 5.5 eV. Ultimately, two distinct bands can be identified after 190 fs. As will be shown later, this spectral feature is even more pronounced in the case of m-BP. In addition to the splitting of the signal, a distinctive change in intensity is also observed. In particular, the intensity of the new band emerging at 5.3 eV appears to increase linearly within 500 fs, while the other decreases steadily. The nearly linear increase in the intensity profile of the new band is consistent with the time-dependent evolution of the S₁ population and its timescale. Therefore, we attribute the bifurcation in the photoelectron signal to the population transfer from S₂ to S₁.

The top panel of Fig. 6 shows the time evolution of the average adiabatic populations along the m-BP trajectories. Although the observed features in the population dynamics are very similar to the parent molecule, some small differences also emerge. The depletion of the S₃ population remains exponential. However, a comparison between the two molecules reveals that, during the first 60 fs, the population transferred from S₃ to S₂ is about 10% greater in BP than in m-BP. After this initial phase, the populations in S₃ for both molecules become approximately the same. These features can be better appreciated in Fig. S11 of the supplementary material. These differences can be attributed to a larger energy gap between S₃ and S₂, resulting in weaker coupling of the states in m-BP. After a simulation time of 500 fs, 60% of the total population resides in S₁ for both molecules. We conclude that the presence of a methyl group in the meta position does not significantly affect internal conversion from S₂ to S₁. Similarly to the parent molecule, there is a direct correlation between the increase in the population of S₁ and the elongation of the CO bond (Fig. S7 of the supplementary material).

Fig. 6.

Time-resolved photoelectron spectrum of m-BP computed at the TDA/ωB97X-D4/6-311G* level of theory from the combined trajectories of S₂ and S₃. Upper panel: time evolution of the average adiabatic populations along the m-BP trajectories. The secondary y-axis shows the fraction of trajectories (gray line) meeting the total energy conservation criteria at each time step. A unitary fraction corresponds to 160 trajectories, with a minimum fraction of ≈0.5 equivalent to 76 trajectories. Lower left panel: temporal slice of the time-resolved photoelectron spectrum selected at 0 fs. Dashed lines indicate the binding energies at the FC point, maintaining the color scheme of the population plot. Lower right panel: full time-resolved photoelectron spectrum.

The spectral snapshot of the time-resolved photoelectron spectrum at 0.0 fs, shown in the left panel of Fig. 6, displays an intensity profile different from that of BP. In particular, the bands centered at 3.6 and 4.7 eV now show an equal intensity. The time-resolved photoelectron spectrum of m-BP is displayed in Fig. 6. As in the BP situation, the m-BP spectrum exhibits a blueshift of ≈ 0.4 eV of these two bands during the initial 15 fs of dynamics, paired with an exponential change in their intensity. Following this, the signal at 3.9 eV splits, resulting in a linear rise in the intensity of the S₁ signal, while the intensity of the S₂ signal decreases. Similarly to the parent molecule, the blueshift, accompanied by the exponential decay/rise in the intensity profile, mirrors the nonadiabatic relaxation from S₃ to S₂. The signal bifurcation and the linear increase in intensity of the newly emerged band correspond to the nonadiabatic relaxation from S₂ to S₁, and they appear more pronounced here than BP. There is no clear connection between the dihedral oscillations of the aryl rings and the photoelectron spectrum.

To provide a more complete picture of the relaxation mechanism from S₃/S₂ to S₁, we have investigated the mechanistic details that drive nonadiabatic processes for both BP and m-BP. Through the characterization of the conical intersection and hopping geometries for the S₂ → S₁ decay, the puckering of one of the two phenyl rings was identified as the primary driving force of the relaxation mechanism. The puckering of the phenyl ring was calculated according to the Cremer–Pople representation,⁶² which defines three puckering coordinates for a six-membered ring. For our purposes, we limit our discussion to the ring puckering amplitude $\overline{Q}$, which is the average of the amplitudes of the two phenyl rings. Panels (a) and (c) of Fig. 7 show the correlation between the puckering amplitude averaged between the two phenyl rings, indicated with $\overline{Q}$, and the phenyl rings dihedral angle, Φ, defined by the carbon atoms 1−2−3−4 in Fig. 2. Panels (b) and (d) of Fig. 7 show the correlation between $\overline{Q}$ and the carbonyl CO bond length. We found that, for both molecules, the optimized CoIn and the hopping geometries are characterized by larger values of the puckering amplitude $(\overline{Q}>0.1)$ compared to other critical points, such as the FC point or the S₁ and S₂ minima. Moreover, this puckering motion is accompanied by a change in the dihedral angle between the two rings. Figure 7 suggests that there is no direct link between the S₂ → S₁ hopping geometries and the CO bond length, indicating that the elongation of the carbonyl bond primarily results from relaxation on the S₁ PES.

Fig. 7.

Critical geometrical parameters of the S₂ → S₁ hopping geometries and relevant minimum structures for BP (left column) and m-BP (right column). Panels (a) and (c) show the correlation between the phenyl rings dihedral angle, Φ, and the average puckering amplitude, $\overline{Q}$. Panels (b) and (d) show the correlation between the carbonyl CO bond length and the average puckering amplitude.

C. Construction of potential energy surfaces

The construction of diabatic PESs for BP was unattainable with DFT/MRCI due to numerical problems in the diabatization process, attributed to the degeneracy of S₂ and S₃, and the resulting coupling already at the FC point. To address this issue, we opted for methylated benzophenone as a candidate to construct diabatic PESs and carry out subsequent quantum dynamics (QD) simulations. Indeed, despite the presence of a methyl group, the photochemistry of m-BP does not change significantly, making the two systems quite comparable, as demonstrated by the UV spectrum based on a Wigner sampling in Fig. 3 and its surface-hopping dynamics discussed in Sec. III B. Positioning the methyl group in meta yields a sufficiently large splitting between the S₃ and S₂ states enough to reduce their coupling at the FC point, without introducing a strong steric effect on the relaxation dynamics. This, in turn, facilitates the construction of diabatic PESs. Moreover, the asymmetric substitution of the aryl rings breaks the C₂ symmetry of BP.

The number of vibrational DOF for m-BP is 3N − 6 = 75, where N is the number of atoms in the molecule. In grid-based methods, the size of the nuclear wave function increases exponentially with the number of nuclear DOF, which makes the task computationally prohibitive to solve unless we restrict ourselves to the reactive subspace of the multidimensional problem. In such cases, a choice of reaction coordinate space is required. To describe the initial dynamics associated with the transfer of population from S₂ to S₁, the difference vector between the FC and CoIn geometries was selected as the primary coordinate,

$v_1={\mathbf{R}}^{\text{FC}}-{\mathbf{R}}^{\text{CoIn}}.$ (5)

This ensures the presence of the S₂/S₁ CoIn in the coordinate space by construction. The primary motion of this coordinate qualitatively describes a ring-puckering vibration. The analysis of the BP and m-BP SHARC trajectories revealed that, within the first 150 fs, the dominant motion involved the dihedral angle between the two aryl rings. The normal mode analysis is available in Fig. S6 of the supplementary material. For this reason, the vector describing this motion was chosen as the second coordinate (v₂). The two vectors used as reaction coordinates are depicted in Fig. S9 of the supplementary material.

After removing the purely translational and rotational components of the two vectors by satisfying the Eckart conditions,^63,64 we orthonormalized the two coordinates using Löwdin’s orthogonalization scheme. We then calculated a 2D rectangular grid by scanning the two vectors using the FC point as reference geometry,

${\mathbf{R}}_{xyz}^{new}=R_{ref}+q_1{\mathbf{v}}_1+q_2{\mathbf{v}}_2$ (6)

with coefficients q₁/q₂ in the range (− 3, 3). We conducted an unrelaxed scan of the reaction coordinates. This choice is supported by the timescale of the process we intend to describe, specifically the early sub-150-fs dynamics, during which we assume that additional relaxation does not play a dominant role. The diabatic PESs for m-BP are displayed in Fig. 8.

Fig. 8.

2D diabatic potential energy surfaces of m-BP. The white dashed line indicates the diabatic intersection seam between S₂ and S₁, including the optimized CoIn. A blue dashed line indicates the diabatic intersection seam between S₃ and S₂.

D. Quantum dynamics

The mixed quantum−classical excited-state dynamics based on independent trajectories provide a clear picture of the nonadiabatic relaxation events in BP and m-BP. The surface hopping methodology has the ability to tackle the full dimensionality of the nuclear problem; however, it cannot correctly describe coherent superpositions of electronic states. In fact, all dynamics start from a single adiabatic electronic state and not from a superposition of states, as created by a laser pulse. Moreover, it does not adequately account for the decoherence in the nuclear subspace, caused by the nuclear motion. In contrast, an explicit full quantum dynamics simulation correctly describes coherent superpositions and the wave packet branching near a conical intersection, producing a short-lived electronic coherence. Information about the energy gap between the coupled electronic states is encoded in the oscillation period of these transient electronic coherences, and their detection can be used as a direct signature of CoIns. However, the grid-based approach we use is only possible for reduced geometric subspaces and not for the full 3N-6 degrees of freedom. For this reason, we have constructed global PESs in a reduced subspace of the 3N-6 nuclear DOF of m-BP and used them for full QD simulations. Note that, for numerical reasons, the diabatic picture is the more convenient choice for grid-based quantum dynamics simulations. However, the surface hopping dynamics of both BP and m-BP was performed and discussed in the adiabatic picture. In order to compare the two simulations, we also computed the diabatic populations of the surface-hopping dynamics as implemented in SHARC. Based on these diabatic populations, we constructed the coherent superposition defined in Eq. (4) as the initial wave packet for the quantum dynamics simulation.

Figure 9 displays the temporal evolution of diabatic populations and electronic coherences, ρ, obtained from the QD simulations. The chosen reduced reaction coordinate space can capture the general trend for the S₃ to S₂ internal conversion, as shown in Fig. S12 of the supplementary material. However, the S₃/S₂ dynamics are likely washed out by laser pump-pulses longer than 15 fs. Moreover, the electronic coherence generated by the S₃/S₂ CoIn would be largely concealed by the one induced by the pump pulse. Therefore, the populations of S₃ and S₂ are summed in Fig. 9. The QD simulations can qualitatively reproduce the population transfer from S₂ + S₃ to S₁ predicted by the surface hopping dynamics. In particular, the nonadiabatic relaxation into S₁ begins at 40 fs and displays a similar linear growth. Despite this, there is a quantitative discrepancy in the total population transfer, as QD predicts a 0.05 S₁ population at 100 fs, while the surface hopping dynamics predicts 0.1. A number of factors may account for this: the underlying electronic structure method used in the two approaches (TDA-LR-DFT vs DFT/MRCI) is different, as discussed in Sec. III A. The lack of electronic decoherence in semiclassical trajectory-based dynamics may influence the population transfer directly. The reaction coordinate space used for QD is two-dimensional and does not include all DOF that the system may explore during the time interval of interest. Beyond 120 fs, these differences become more pronounced as the S₁ state is unable to relax within the chosen 2D reaction coordinate space.

Fig. 9. Temporal evolution of diabatic populations and coherences in BP and m-BP during the initial 120 fs.

(a) Diabatic populations as a function of time, derived from surface hopping (solid and dashed lines) and QD (dashed-dotted line) simulations. The populations of S₂ and S₃ are summed together. (b) Magnitude of the electronic coherences between S₂ and S₃ as well as S₁ and S₂. Subscripts indicate the electronic states involved.

Figure 10 shows the time-dependent evolution of the nuclear wave packet on different PESs. Snapshots of the nuclear wave packet density (∣Ψ∣)² are captured at t = 0, 30, 60, and 80 fs. Since any experimental pump pulse would populate both states, the QD simulations start in a coherent superposition of S₂ and S₃, thus generating a strong electronic coherence ρ₃₂ at 0 fs. This coherence rapidly decays over 20 fs, after which it remains relatively constant around 0.1, with minor oscillations, as illustrated in Fig. 9(b). More interesting is the case of ρ₂₁ between S₂ and S₁. In fact, ρ₂₁ is zero for the first 35 fs of the dynamics and begins to increase as the wave packet approaches the CoIn, peaking around 70 fs. The magnitude of ρ₂₁ is remarkably large and long-lived (>50 fs), especially considering that electronic coherences generated by CoIns are typically on the order of 10⁻³ or less⁶⁵ and decay very fast. Therefore, the calculated 2D PESs show promise for the following studies where ρ₂₁ could be probed using time-resolved x-ray techniques,⁶⁶ such as TRUECARS^67,68 or time-resolved x-ray absorption spectroscopy. Detection of ρ₂₁ could serve as a direct spectroscopic signature of the conical intersection.

Fig. 10.

Snapshots of wave packets (gold contour lines) at 0, 30, 60, and 80 fs. First row: wave packet evolution on S₁, second row: wave packet evolution on S₂, and third row: wave packet evolution on S₃. The white dashed line indicates the diabatic intersection seam between S₂ and S₁, including the optimized CoIn. A blue dashed line indicates the diabatic intersection seam between S₃ and S₂.

IV. Conclusion and Outlook

The excited-state dynamics of gas-phase benzophenone and its alkyl derivative, meta-methyl benzophenone, have been investigated using mixed quantum-classical and full quantum dynamics simulations, focusing on nonadiabatic processes in the singlet manifold. The mixed quantum-classical simulations were carried out at the Tamm−Dancoff approximation linear response time-dependent density functional theory using a surface hopping approach. The electronic structure method was benchmarked against experimental and theoretical data, assuring a qualitatively correct description of all states involved. We were able to analyze the excited-state dynamics for benzophenone and meta-methyl benzophenone for 500 fs, observing both the non-adiabatic relaxation process from S₃ to S₂ and then from S₂ to S₁. In addition to discussing the geometrical relaxation and the population dynamics, we also simulated an experimentally accessible observable, namely, time-resolved photoelectron spectra for both molecules.

In both cases, we observe an exponential decay of the population in the S₃. This behavior is observed in the time-resolved photoelectron spectra as a blueshift of 0.4 eV within the first 15 fs, immediately followed by an interchange in the intensity profile of the photoelectron bands. The population transfer to S₁ starts around 40 fs, and the increase/decrease in the population of S₁/S₂ is nearly linear. As the system passes through the S₂/S₁ conical intersection, a bifurcation in the photoelectron signal appears in the spectra, followed by an almost linear increase in the S₁ band and a decrease in the intensity of the S₂ signal within the simulation time of 500 fs.

We investigated the mechanistic details that lead to the S₂ → S₁ nonradiative decay to provide a complete picture of the relaxation event. We found that for both molecules, the puckering of one of the two phenyl rings is the primary driving force of the relaxation mechanism. This puckering motion is accompanied by a change in the dihedral angle between the two rings.

To gain more insight and to correctly describe electronic coherences and superpositions of electronic states, we computed two-dimensional potential energy surfaces for meta-methyl benzophenone to perform full quantum dynamics simulations. The presence of the methyl group in the meta position increases the energy gap between the S₃ and S₂ states, which, however, only marginally affects the excited-state dynamics compared to BP. This separation is in turn beneficial because it leads to a slight decoupling of the two states at the Franck–Condon point, allowing for reasonable diabatization. The quantum dynamics simulations, carried out in a subspace of the full dimensional problem and in a diabatic basis, qualitatively reproduce the surface hopping dynamics over the first 120 fs of propagation. While the relaxation timescale and population increase are accurately depicted, surface hopping and quantum dynamics simulations differ quantitatively. The former predicts a population of 0.1 in S₁ at 100 fs, whereas the latter predicts 0.05. The discrepancy between the two approaches may be attributed to a number of factors, including the differing electronic structure theories employed and the reduced reaction coordinate space utilized in quantum dynamics. The electronic coherence generated by the bifurcation of the nuclear wave packet at the S₂/S₁ conical intersection is remarkably large and survives for more than 50 fs. Consequently, the selected coordinate subspace appears promising for future investigations in which the electronic coherence might be probed with more advanced time-resolved x-ray techniques, which rely on electronic coherences.

Supplementary Material

See the supplementary material for additional benchmarking of vertical excitations of benzophenone using CASSCF and RASSCF, natural orbital plots of the active spaces used in the calculations, normal mode analysis of SHARC trajectories for the first 150 fs, and average temporal evolution of selected geometric parameters of benzophenone and meta-methyl benzophenone.

Data Availability

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