Mind the GAP: quantifying the breakdown of the linear vibronic coupling Hamiltonian

Abstract

Excited state dynamics play a critical role across a broad range of scientific fields. Importantly, the highly non-equilibrium nature of the states generated by photoexcitation means that excited state simulations should usually include an accurate description of the coupled electronic–nuclear motion, which often requires solving the time-dependent Schrödinger equation (TDSE). One of the biggest challenges for these simulations is the requirement to calculate the PES over which the nuclei evolve. An effective approach for addressing this challenge is to use the approximate linear vibronic coupling (LVC) Hamiltonian, which enables a model potential to be parameterised using relatively few quantum chemistry calculations. However, this approach is only valid provided there are no large amplitude motions in the excited state dynamics. In this paper we introduce and deploy a metric, the global anharmonicity parameter (GAP), which can be used to assess the accuracy of an LVC potential. Following its derivation, we illustrate its utility by applying it to three molecules exhibiting different rigidity in their excited states.

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1 Introduction

Quantum dynamics simulations that seek to solve the time-dependent Schrödinger equation (TDSE) are crucial for describing processes occurring in electronically excited states and therefore are the key to understanding and refining the properties of functional molecules and materials across a range of applications, such as organic light emitting diodes (OLEDS)^1,2 and photovoltaics.^3–5 The equations of motion governing the dynamics occurring over excited state potential energy surfaces (PES) can be solved using either traditional grid based methods, such as the multi-configurational time-dependent Hartee (MCTDH) method^6,7 or trajectory based methods such as Tullys Trajectory Surface Hopping (TSH)^8,9 or Gaussian basis functions (GBFs) methods.^10–16 The latter being motivated by the original work of Heller,^17–19 which in contrast to TSH,^20,21 is able to retain a fully quantum description of the nuclear motion.

Independent of the method used to evolve the nuclear wavepacket, one of the major challenges in all of these simulations is obtaining an accurate description of the high-dimensional excited state PES upon which the nuclei evolve. One approach, primarily used for traditional grid based methods is to determine global analytic functions using quantum chemistry calculations.^22–24 These can be viewed as the non-Born Oppenheimer version of classical force fields and may be refined using, for example, experimental data. This is the most accurate approach, but time consuming and extremely challenging for more than a few atoms due to the size of nuclear configuration space. For trajectory based methods, most approaches exploit the spatial locality of the trajectory making it possible to calculate PES on-the-fly, and so the complicated multidimensional potential can be calculated as and when it is required²⁵ removing the significant challenge of a priori computation of a PES. However, especially for larger molecules, this can still be challenging due to the computational scaling of the quantum chemistry methods and the large number of calculations required. The final approach, which is the focus of the present work, is to develop a model Hamiltonian, which are parameterised to match the PES around the important point on the PES, such as the Franck–Condon (FC) point. This reduces the number of quantum chemistry calculations required to map the excited state potential and can be a reliable way to calculate the excited state dynamics, especially when large amplitude motions in the excited state are absent.

In this work, we focus on the linear vibronic coupling (LVC) scheme developed by Köppel et al.^26,27 which describes the excited states of a system by a series of coupled shifted harmonic oscillators. Importantly, the simplicity of this approach makes it possible parameterise an effective Hamiltonian with very few quantum chemistry calculations,²⁸ providing a mechanism to achieve a detailed understanding of excited state dynamics through simplified models, which ultimately facilitates extraction of the critical parameters affecting function. These properties have recently been exploited to perform excited state dynamics using LVC models across a range of difference materials.^1,2,29–43 While the advantage of the LVC is that it can be relatively straightforward to obtain, the disadvantage is that it becomes inaccurate for larger displacements away from the starting geometry when anharmonic effects become more important. Consequently, in the present work introduce and deploy a metric, the Global Anharmonicity Parameter (GAP), which can used to assess the accuracy of an LVC potential. Following its derivation, we illustrate and discuss its utility by applying the method to three molecular systems.

2 Methodology

2.1 Computational details

Fig. 1 shows the molecules considered in the present work. Fig. 1a is a narrowband width blue fluorescent emitter used in OLEDS, named 2,5,8,11-tetra-tert-butylperylene (TBPe).⁴⁴ The rigidity of the molecule required to ensure the narrow emission spectrum makes it very suitable for the LVC treatment. Fig. 1b is a thermally activated delayed fluorescence^45,46 emitter exhibiting a D–A₃ structure with a triazatruxene central donor functionalized with three peripheral dibenzothiophene-S,S-dioxide acceptors abbreviated as TAT-3DBTO₂ from now on.^47,48Fig. 1c is Maleimide a small heterocyclic organic molecule previously studied using a quadratic vibronic coupling (QVC) Hamiltonian model.⁴⁹ The level of theory used for each molecule was chosen to correspond to previous works^48–50 and is described below.

Fig. 1 Structure of (a) TAT-3DBTO₂ (b) TBPe and (c) Maleimide.

All parameters required for the LVC Hamiltonian, described in Section 2.2, were extracted using the VCMaker software developed in-house and available for download at ref. 51. All quantum chemistry calculations described in the following subsections were all performed with the Q-Chem 5.0 software.^52,53

2.1.1 TBPe . The minimum energy of the ground, first excited singlet (S₁) and triplet (T₁) states for TBPe (Fig. 1a) were optimised using density functional theory (DFT) and time-dependent density functional theory (TD-DFT) within the approximation of the PBE0 exchange and correlation functional.⁵⁴ Throughout the def2-TZVP basis set was used⁵⁵ and all TD-DFT simulations exploited the Tamm-Dancoff approximation.⁵⁶

The Hessian of the electronic ground state and the Cartesian gradients of the first excited singlet (S₁) and triplet (T₁) states required for the LVC were computed at the ground state optimised geometry using the same level of theory described above. All calculations were performed in the gas phase. Cartesian coordinates of the ground and excited states optimised geometries are provided in Tables S1, S2 and S4 (ESI†).

2.1.2 TAT-3DBTO₂ . The electronic structure of TAT-3DBTO₂ (Fig. 1b) has been previously studied by Eng et al.⁴⁸ Here the ground and excited state geometry optimisations were performed with the DFT and TD-DFT approaches within the approximation of the range-separated LRC-ωPBEh functional.⁵⁷ A 6-31G* basis set^58–60 was used throughout and all of the TD-DFT simulations were performed within the Tamm–Dancoff approximation.⁵⁶ Throughout the range-separation parameter ω was fitted using the optimal tuning approach described in ref. 61–64. The geometry optimisation, Hessian and gradient calculations were all performed in the gas phase. Cartesian coordinates of the ground and excited states optimised geometries of TAT-3DBTO₂ are reported in Tables S10, S12 and S14 (ESI†).

2.1.3 Maleimide . Geometry optimisations and excited state calculations of Maleimide (1c) have followed the recent work of Lehr et al.⁴⁹ and have been performed using the Equation of Motion Coupled Cluster Singles and Doubles (EOM-CCSD) approach and a cc-pVDZ basis set.⁶⁵ Cartesian coordinates of the ground and excited states optimised geometries of Maleimide are reported in Tables S17 and S18 (ESI†).

2.2 Linear vibronic coupling

The electronic diabatic Hamiltonian elements W_n,n can be expressed by expanding W − V₀1 as a Taylor series expansion of the diabatic potential around a reference nuclear geometry (Q₀), usually the FC geometry. V₀ is a reference potential chosen accordingly to the problem. It is not restricted to a particular form but is often defined as harmonic with vibrational frequencies ω_i along dimensionless normal coordinates Q_i. In this case, Hamiltonian elements are expressed:where δ_nm is the Kronecker delta. Q_i denotes the 3N − 6 dimensionless normal coordinates related to the normal modes of vibration, N being the number of atoms. V₀, within the harmonic approximation is expressed:and the kinetic energy operator therefore takes the form:Within the LVC scheme, the Taylor expansion of the diabatic potential energy to the first order, the so-called LVC model Hamiltonian, is written:The diabatic electronic states are built such as they coincide to the adiabatic electronic states at FC (Q₀). We introduce the notation:andwhere κ_i and λ^n,m_j are the intra- and inter-state coupling, respectively. The on-diagonal elements (κ_i) are the forces acting within an electronic surface and are responsible for structural changes of excited-state potentials compared to the ground state. They are therefore responsible for the reorganization energy and often referred to as the tuning modes. The off-diagonal elements (λ^n,m_j) are the couplings responsible for transferring wavepacket population between different excited states and consequently are usually referred to as coupling modes. For m = n, eqn (4) can then be reorganised such as:This equation emphasizes the roles of the intrastate coupling constant κ⁽ⁿ⁾_i in shifting the diabatic potential in energy and in position .

Within the LVC, the minimum of any excited state PES n is given by

and in the limit that the harmonic approximation is valid, corresponds to the exact minimum of the PES Q⁽ⁿ⁾_Min. If there is significant anharmonicity, Q⁽ⁿ⁾_LVC and Q⁽ⁿ⁾_Min becomes significantly different.

Eqn (8) can be rewritten in term of κ:

This demonstrates that an alternative method for estimating κ⁽ⁿ⁾ can be obtained as the difference between the excited state harmonic minimum geometry and the FC geometry:If Q⁽ⁿ⁾_LVC is replaced by Q⁽ⁿ⁾_Min, i.e. the exact minimum of excited state n, one obtains a second set of κ parameters: :While the estimation method of κ⁽ⁿ⁾_i based on the gradient at FC probes vertically the topology of the PES, estimating based on the excited state minima provides information at potentially larger displacements from FC. Crucially, comparing κ⁽ⁿ⁾_i and provides a direct estimate of the validity of the LVC model, i.e. in the limit the harmonic approximation holds, .

In this work we seek to develop a simple general parameter which can be used to characterise how suitable the LVC approach is. The global anharmonicity parameter (GAP) is defined as:

whereandΞ takes values from 0% to 100% where Ξ = 0% corresponds to a scenario where the molecule is rigid and the theoretical LVC minimum corresponds to the real minimum and Ξ = 100% where the molecule is flexible and the harmonic oscillator approximation is no longer valid.

As these parameters are all calculated in terms of individual normal mode, Ξ can be represented in term of relative single mode anharmonicity (RSMA) ξⁿ_i:

where α is a normalisation coefficient:The GAP and its individual normal mode breakdown will be used throughout this work.

3 Results

3.1 TBPe

As described above, TBPe (Fig. 1a) is a narrowband fluorescent emitter⁴⁴ and given the rigidity required in the molecular structure to achieve narrow emission spectra⁶⁶ one would expect the LVC model to provide an accurate description of TBPe excited states. In this section we focus on the lowest excited triplet (T₁) and singlet (S₁) states which both exhibit π → π* excitation localised on the central aromatic moiety of TBPe. Fig. S1 (ESI†) shows the difference of electronic density associated to the S₁ ← GS and T₁ ← GS transitions. The minimum energy geometry of both of these states exhibit very small root-mean square deviation (RMSD: Θ) with respect to the electronic ground state optimised geometry of Θ(S^Min₁,GS) = 0.025 Å and Θ(T^Min₁,GS) = 0.029 Å, respectively.

Fig. 2a shows the GAP (Ξ) and RSMA (ξ) for the S₁ and T₁ excited states of TBPe. The GAP of Ξ^S₁ = 5.2% and Ξ^T₁ = 0.4% confirms, consistent with the rigidity discussed above, the validity of the Harmonic approximation for this molecule. The RSMA shown in Fig. 2a highlights the discrete normal modes with the largest RSMA, which are in-plane motions (high frequency). Out-of-plane distortions have smaller RSMA because such motions involve a bending of the π-system which requires more energy than a collective in-plane vibration of the C–C bonds. The largest RSMA for the S₁ and T₁ states coincide, reflecting the similar character of the excited states.

Fig. 2 RSMA (ξ) for TBPe S₁ and T₁ (a) and Maleimide S₁, S₂, S₃ and S₄ (b). GAPs are shown inset. Only fully symmetric normal modes are shown. ξ for normal modes of other symmetries are shown in Tables S6 and S7 (ESI†) for TBPe and Tables S21, S22, S24 and S25 (ESI†) for Maleimide.

The validity of the GAP (Ξ) approach is confirmed by the small RMSD between the optimised structures of the S₁ and T₁ states discussed above. In addition, the theoretical minima obtained using the displacement harmonic oscillator approach: i.e. S^LVC₁ and T^LVC₁, respectively are in very close agreement with the true minima with the RMSD of the triplet state being Θ(T^Min₁,T^LVC₁) = 0.006 Å and Θ(S^Min₁,S^LVC₁) = 0.016 Å for the singlet state. A small nuclear distortion alone is not sufficient to confirm the validity of the harmonic approximation. The relaxation energy obtained from and from for S₁ and T₁ are quantitatively the same: ΔE^(S₁) = −0.157 eV, and ΔE^(T₁) = −0.272 eV, and in good agreement to the relaxation energy between the ground state optimised geometry and S^Min₁ and T^Min₁.

The quality of the structure and energy of the state of interest is important, however the overall electronic structure at the LVC minimum is also required to be in good agreement to the optimised minimum. Table 1 shows the comparison between the electronic structure of TBPe at the two optimised geometries S^Min₁ and T^Min₁ and their respective LVC geometries: S^LVC₁ and T^LVC₁. At both S^Min₁ and T^Min₁ the energy of S₁ is very well reproduced by the LVC generated geometry with a difference of energy <0.005 eV. The energy of T₁ also shows good agreement between the optimised and the LVC minima with only a small difference in energy of 0.01 eV between S^Min₁ and S^LVC₁. The maximal difference of energy is observed for the ground state energy S^Min₁ and S^LVC₁ with a difference of 0.02 eV. This is due mainly to the difference in the shape of the PES where S₁ and T₁ might feature shallower PES than the ground state. While this difference is negligible, it shows that for less rigid system the energetics of neighbouring states might not be is such a good agreement to the optimised minimum.

Table 1 The excited state energies and oscillator strengths of TBPe at the true and LVC minimum energy geometries of the S₁ and T₁ states

State	S^Min1		S^LVC1
	f	ΔE/eV	f	ΔE/eV
S0	—	0.13	—	0.15
T1	—	1.65	—	1.64
S1	0.587	2.98	0.594	2.98
T2	—	3.08	—	3.09
T3	—	3.31	—	3.32

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State	T^Min1		T^LVC1
	f	ΔE/eV	f	ΔE/eV
S0	—	0.26	—	0.27
T1	—	1.62	—	1.62
S1	0.594	3.01	0.617	3.01
T2	—	3.13	—	3.13
T3	—	3.38	—	3.39

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3.2 TAT-3DBTO₂

Triazatruxene tris(dibenzothiophene-S,S-dioxide) TAT-3DBTO₂ is a promising candidate for application in TADF OLEDs.⁴⁷ It consists of a central donor moiety linked to three peripherals acceptor units through carbon–nitrogen single bonds (see Fig. 1). This type of donor–acceptor molecule features low-lying charge transfer (CT) states that typically exhibit small singlet–triplet energy gap, a requirement for the up-conversion of non-emissive triplet states in TADF.⁴⁵ Indeed, after excitation the acceptor units can rotate with respect to the central donor moiety around the C–N bond (φ_i, see Fig. S2, ESI†) to accommodate the excited state electronic density and minimise the orbital overlap. This relaxation effectively reduces the energy gap between singlet and triplet CT states that is desirable for efficient TADF emitters.

Large amplitude motions, and in particular rotations, are typically situations where harmonic potentials breakdown. At the geometry of the ground state minimum, TAT-3DBTO₂ features a C₃ symmetry where all three acceptors are tilted by φ = 57.29 deg with respect to the central moiety. At this geometry, the lowest triplet state T₁ at E(T₁) = 3.09 eV is a mix of local excitation focused on the donor and charge transfer (CT) towards all the acceptors units. In contrast, the S₁ state (E(S₁) = 3.62 eV) is a pure CT from the donor to all acceptors. The electronic structure of TAT-3DBTO₂ at the ground state optimised geometry is reported in Table 2 and the difference of electronic density of S₁ and T₁ are shown in Fig. S3 (ESI†). Relaxation after excitation into S₁ leads to a localisation⁶⁷ of the CT towards a single acceptor and a stabilisation of the state to E(S₁) = 3.32 eV (see Table 2 top right). The acceptor involved in the CT of S₁ rotates around the donor–acceptor bond to a more perpendicular orientation with φ₁= 77.47 deg. The other two acceptors are not involved in the excitation, remaining tilted with φ₂ = 61.04 deg and φ₃ = 59.84 deg. The nature of T₁ changes during the relaxation in S₁ to become a mixed charge transfer/locally excited state based upon the acceptor, ³ CT/³ LE(A) state at E(T₁) = 3.14 eV. We report the reader to the work of Eng et al.⁴⁸ for a more detailed analysis of the electronic structure.

Table 2 Electronic structure of TAT-3DBTO₂ at the ground state optimised geometry (top left), the geometry of S₁ minimum (top right), the C₃ symmetry minimum of S₁ (bottom left) and the S₁ LVC minimum (bottom right). All energies are given in eV and relative to the energy of the electronic ground state at the ground states optimised geometry. f stands for oscillator strength

GS			S^Min1
State	f	ΔE/eV	State	f	ΔE/eV
S0	—	0.00	S0	—	0.42
T1	—	3.09	T1	—	3.14
T2/T3	—	3.19	S1	0.004	3.32
T4	—	3.23	T2	—	3.36
T5/T6	—	3.42	T3	—	3.45
T7	—	3.51	T4	—	3.61
S1	0.005	3.62	T5	—	3.62
T8/T9	—	3.67	T6	—	3.65
T10	—	3.67	S2	0.013	3.68
S2/S3	0.184	3.67	T7	—	3.78

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S^Sym1			S^LVC1
State	f	ΔE/eV	State	f	ΔE/eV
S0	—	0.18	S0	—	0.46
T1	—	3.19	T1	—	3.39
T2/T3	—	3.23	T2	—	3.44
T4	—	3.30	T3	—	3.47
T5/T6	—	3.48	T4	—	3.54
S1	0.000	3.50	T5	—	3.71
S2/S3	0.046	3.52	T6	—	3.73
T7	—	3.53	S1	0.036	3.74
T8/T9	—	3.60	T7	—	3.78
			S2	0.048	3.80
			S3	0.092	3.83

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The GAP for S₁ is Ξ_S1 = 86% and therefore clearly reflects a significant breakdown of the LVC model. Due to symmetry, only fully symmetric normal modes can have κ ≠ 0, i.e. a non-zero gradient at the FC geometry and therefore the κ values calculated using the LVC will not be able to capture the symmetry breaking. In addition, because the κ* values are estimated at a structure with no symmetry (S^Min₁) it does not have the same constraint and may exhibit non-zero gradient along any normal modes. This is demonstrated in Fig. S4 (ESI†) as normal modes that are not a₁ show non negligible RSMA. Consequently, we could expect the LVC minimum to be similar to the optimised geometry of S₁ within the C₃ point group symmetry (S^Sym₁), discussed in ref. 67. The electronic structure at S^Sym₁ is shown in Table 2 bottom left. The electronic structure at this geometry is similar as the electronic structure at GS but with the lowest excited states closer in energy. The interest of this critical structure for TADF is discussed in more details in the work of Eng et al.⁶⁷ The degeneracy of states due to the C₃ point group symmetry is maintained and S₁ is stabilised to E(S₁) = 3.50 eV and T₁ is found at E(T₁) = 3.19 eV. The GAP of S₁ considering the set κ* taken from the S^Sym₁ geometry is slightly lower than Ξ_S1 but not sufficiently to validate the harmonic approximation.

This analysis demonstrates that symmetry breaking is not the only reason for the breakdown of the harmonic approximation in TAT-3DBTO₂. The rotational motion involved in the relaxation is not well described by normal modes of vibrations and the normal modes at the S₁ minimum are too different from the set of normal modes at the FC geometry. The LVC minimum of S₁ (S^LVC₁) shows torsion angles (φ₁ = 66.59 deg, φ₂ = 62.88 deg and φ₃ = 60.98 deg) closer to the ones of the structure of S^Min₁. While S^LVC₁ shows an electronic structure that retains, considering numerical errors, the degeneracy pattern of the C₃ symmetry with the good ordering of the electronic states, see Table 2 bottom right, the energies of the electronic states, and especially the energy of S₁, are however aberrant and reflect the non-physical nature of the structure.

Compared to the S₁ state, the GAP of the T₁ state is smaller: Ξ_T1 = 53%. This is due to the predominantly LE nature of T₁ at the ground state optimised geometry. π → π excitation typically lead to bond length reorganisation that involves small nuclear distortions resulting in small GAP as observed in TBPe. The mixed nature of the state means the excitation is delocalised over both the acceptors and the donor units. Modulation of the orbital overlap to minimise the state energy occurs via donor–acceptor rotation. This effect alone is not sufficient to explain the large Ξ_T1, in particular as torsion angle is similar for the ground state optimised geometry and the minimum of T₁, i.e. φ_1,2,3 = 57.29 deg and φ₁ = 52.51 deg, φ₂ = 54.47 deg, φ₃ = 50.88 deg, respectively. The value of the torsion coordinates φ is reported for all geometries in Table S9 (ESI†). We attribute the large anharmonicity to the change of nature of the state from a LE/CT mixed nature at FC geometry to a LE(A) at the T₁ minimum. The κ parameters extracted from the electronic gradient at FC describe the mixed nature of T₁, while the κ* extracted from the difference of geometry, describe the diabatic ³LE(A) excited state.

The analysis of the RSMA (Fig. S4, ESI†) confirms the difference nature of S₁ and T₁. Indeed the normal modes with the largest ξ in S₁ are normal modes within the 1000 cm⁻¹ to 1200 cm⁻¹ (from mode Q₁₆₃ to Q₂₀₁) range that are typically out of plane motions such as rotations and bending. Such motions are representative of the large localisation of the excitation occurring in S₁. Conversely, the largest ξ in the T₁ state are found to be normal modes with frequencies between 1600 cm⁻¹ and 1700 cm⁻¹ (from mode Q₁₆₅ to Q₁₉₄) and are C–C and C=C bond stretching vibrations that are involved in the π system reorganisation that occurs during the relaxation in a π → π* electronic state.

3.3 Maleimide

Small heterocyclic organic compounds are ideal test systems for probing theoretical concepts in photochemistry and photophysics. In this context maleimides have gained increasing interest over the past years due to a wide variety of potential applications.^68–71 Recently Worth et al.⁴⁹ have studied the excited state dynamics of Maleimide (see Fig. 1c) using a QVC Hamiltonian. By including four low-lying singlet excited states within the manifold of 24 vibrational modes they were able to provide a detail insight into the excited state relaxation dynamics and the vibrational modes involved.

The electronic structure of Maleimide is described in detailed in ref. 49. Briefly, at the ground state optimised geometry the two lowest excited singlet states S₁ and S₂ are at E(S₁) = 4.07 eV and E(S₂) = 4.84 eV, respectively and are excitations from the oxygen lone pairs to a π* anti-bonding orbital. S₃ and S₄ are at E(S₃) = 5.23 eV and E(S₂) = 6.70 eV and correspond to excitations from a π orbital to the same π* anti-bonding orbital. The electronic structure at the ground state minimum is reported in Table S19 (ESI†), while the orbitals involved are shown in Fig. S4 (ESI†).

To estimate the GAP from the reported excited states, special attention needs to be taken while optimising each excited states. Indeed, κ and κ* must be computed at the Franck–Condon and minimum geometries of a same diabatic state, i.e. the electronic nature of the state must be the same. As reported by Worth et al. S₂ and S₃ electronic states cross and therefore generate two minima of different nature in the S₂ PES. In order to avoid any confusion, we will refer to the adiabatic computed states as S₁, S₂, S₃ and S₄ and to the diabatic states by their electronic nature. To alleviate the notations and because the electronic transition associated to each diabatic states is population the same π orbital, we will name the diabatic states by the orbital the electron is excited from, i.e. S_n2, S_n1, S_π1 or S_π2.

Scans along interpolated coordinates between the ground state and optimised S^Min_k geometries are shown in Fig. S5 (ESI†) and emphasize the crossing between S_n1 and S_π2. Other crossings between states can be observed, but only at higher energies, therefore not affecting the minima of the potentials. The electronic structure of Maleimide at each of the excited states' minima is shown in Table S20 (ESI†).

The anharmonicity analysis yields small GAPs for each of the lowest singlet states, namely, , , and . These values are slightly larger than the values for TBPe reflecting an increase in anharmonicity despite the apparent rigidity of Maleimide. Fig. S6 (ESI†) shows the potential of the ground and four lowest excited states along the a₁ normal modes, i.e. with κ ≠ 0. The curvature of the ground electronic states is systematically overestimated, especially for low-frequency normal modes. While a deviation of the curvature of the electronic excited states from the ground states frequency, can be due to vibronic coupling between excited states, a deviation of the curvature of the ground state from the LVC model is a direct signature of anharmonicity. The RSMA analysis of Maleimide is shown in Fig. 2c and d. The normal mode exhibiting the largest ξ is mode ν₂₁ that corresponds to a symmetric stretching of C=O. The small ξ for non fully symmetric normal modes reflects the quasi-C_2v symmetry of the four lowest excited state minima.

While the GAP remains small, anharmonicity is clearly playing a role in these potentials. Consequently, to assess the effect of the anharmonicity especially on the excited state dynamics, we build two model diabatic Hamiltonians using the LVC approach and compare these to the Hamiltonians previously published by Worth et al.⁴⁹ and constructed using a fitting procedure of the QVC approximation. The Hamiltonians include the lowest five singlet states, i.e. the ground and the four lowest excited states and either 6 or 12 normal modes, full details of the two Hamiltonian are given in Tables S25–S30 (ESI†).

Fig. 3 shows the population kinetics along the dynamics using the Hamiltonians containing 6 (dashed lines) and 12 (full lines) normal modes. In both cases, the simulations are initiated with the wavepacket vertically excited from the ground state geometry vertically into the S₃ state. In the 6-modes model, ultrafast population from S₃ to S₂ is observed within 10 fs. The population then oscillates between the S₃ and S₂ states as there are no normal modes present which couple these states to the lowest S₁ state. S₄ is only marginally populated through the dynamics as it can only receive population from the energetically distant S₂ and S₁ as the S₃ state is of same symmetry as S₄ (A₁) and therefore coupling is symmetry forbidden. In the 12-modes dynamics, the initial population transfer from S₃ to S₂ is still observed, but due to the inclusion of b₂ mode Q₂₀, population within S₂ is quickly transferred into the S₁ state. As before, S₄ receives only a very small amount of population. An equilibrium between the three (S₁, S₂ and S₃) states is reached within 50 fs with the S₂ and S₃ state both exhibiting ≃15% of the total population each, reflecting their near degeneracy.

Fig. 3 Excited state population dynamics of maleimide after excitation in S₃ (¹B₂) state. The two models include 6 (dashed lines) and 12 (full lines) nuclear degrees of freedom.

Both the 6- and 12-modes dynamics offer good agreement with dynamics reported by Worth et al.⁴⁹ using the QVC approach. For the 6 model model, small differences are observed in the frequency of the oscillations observed between the S₃ and S₂ states which are slightly slower and larger in amplitude than those observed within the QVC approach.⁴⁹ In the 12 mode model, similar differences are observed, especially in the oscillations in the population of the S₁ state. While this observation could justify the usage of the LVC model for the construction of the model Hamiltonian, the topology of the excited states must be investigated.

Fig. 4 shows the comparison between the computed adiabatic and the diabatic potential energy curves built using the LVC approach of S₁ to S₄ along normal mode Q₁₈. This normal mode is especially important for the dynamics as it creates a conical intersection (CI) between the PES of S₂ and S₃. The topology of the CI (Fig. 4a) obtained from the calculated adiabatic excited state energies is peaked^72,73 and consequently, a wavepacket passing through the CI originating from the FC region would end up in the minimum of diabatic state with a similar nature according to the Landau–Zener principle.^74,75 In contrast the LVC potential (Fig. 4b) along this mode generates a CI topology which is sloped.^72,73 Consequently, a wavepacket would end up in a different minimum exhibiting different character and dynamics. This is highlighted in Fig. 4 where the schematic arrows represent the main direction of the wavepacket upon relaxation. This emphasises that while the population kinetics of the LVC model may appear unchanged compared to a higher order expansion, suggesting that a GAP of ∼10% corresponds to a valid LVC, care must be already be taken during the interpretation of these dynamics that the similarity observed in the populations kinetics occurs for the same reason. Indeed, while quantities such as population kinetics are useful, a more rigorous assessment of the accuracy of a potential energy surface is often achieved by calculating observables such as absorption spectra. Fig. 5 shows the second absorption band of Maleimide obtained by the Fourier transform of the autocorrelation function of a wavepacket initially placed in S₄. It is compared to the experimental spectrum⁴⁹ and the spectrum obtained with the 12 modes Hamiltonian in ref. 49. While the spectrum obtained within the LVC approximation shows qualitative agreement with the spectrum obtained from Worth et al. it fails to reproduce the relative intensity between the peaks. The absorption peaks are also sightly shifted and the Gaussian envelope of the absorption band is not well reproduced. This can be attributed to the absence of anharmonicity in the LVC model. In this case, the calculated spectra incorporates peaks >0.5 eV above the absorption band onset, where anharmonicity will play an increasing important role, this is highlighted by the disagreement between the experimental and both calculated spectra with appears to increase with energy.

Fig. 4 PES of the ground and four lowest excited singlet states along normal mode Q₁₈. (a) Calculated adiabatic PES. The change of the electronic nature of S₂ and S₃ is emphasized by the change of the diffuse color. (b) Diabatic electronic states built from the LVC approach. The dashed lines qualitatively represent the diabatic states built from following the nature of the electronic states along the adiabatic states shown in (a).

Fig. 5 Experimental and theoretical absorption spectra for the second excitation band of Maleimide. The spectra obtained in this work is shown in blue. The spectrum extracted from the 12 modes dynamics of ref. 49 is shown in turquoise. All spectra have been shifted and scaled to align the first absorption peak in position and intensity to the experimental value.

4 Conclusion

Quantum dynamics simulations are a key theoretical tool for describing processes occurring in electronically excited states, but one of the key challenges for these simulations is obtaining an accurate description of the high-dimensional excited state PES upon which the nuclei evolve. The LVC model is a very compact and efficient model which can be parameterised from relatively few quantum chemistry calculations. However, despite these advantages, the model is based upon coupled-shifted Harmonic oscillators and therefore its validity will be severely tested when larger amplitude motions are present in the excited state dynamics.

For clarity shows some limiting cases of the LVC are shown in Fig. 6. Initially, Fig. 6a illustrates a situation where the LVC approach will remain valid; i.e where small anharmonicity of the potential that leads to a small Ξ. In contrast Fig. 6b–e shows four cases that will lead to a large Ξ and the breakdown of the LVC approach, namely (b) a large anharmonicity of the potential, (c) a symmetry breaking in the excited states, (d) a breakdown of the normal mode representation, i.e. the normal modes change nature between the FC and the excited state minimum geometries, and (e) the crossing of electronic states.

Fig. 6 Possible anharmonicity scenarios. The full black line represents the adiabatic energy of the state of interest (V_i) and a full gray line represents any other state involved in the scenario (V_x). Dashed blue and red lines represent the diabatic states built from either κ (W) or the minimum of V_i (W*). Green arrows represent the electronic gradients at FC and correspond to κ and κ*. Q⁰, Q^LVC and Q^MIN correspond to the FC, to the minimum of W and the minimum of V_i, respectively. ε^vert is the vertical excitation energy of V_i and ε^LVC is the energy at FC of W*.

In this paper we have introduced and tested a metric, the Global Anharmonicity Parameter (GAP), which can be used to assess the accuracy of an LVC potential. This is based upon assessing the differences between the intrastate coupling constant, κ which can be calculated in two ways. If both yield the same results, the harmonic approximation upon which the LVC is based is valid. We have assessed the performance of this metric using three molecules, namely TBPe, TAT-3DBTO₂ and Maleimide. The first two illustrate two clear examples when the LVC is valid and breakdowns, respectively. The potential for Maleimide illustrates an interesting case. Indeed, it displays a relatively small GAP, ∼10% and the excited state dynamics calculated using the LVC potential is in good agreement with those previously obtained using the QVC approach:⁴⁹ However, the approximations made in the LVC change the nature of the interaction between the coupled states meaning that although the population dynamics remains very similar to higher level approaches, the mechanism leading to the dynamics is different.

In conclusion, we have demonstrated that the GAP is a useful metric for assessing the validity of the LVC approximation. From the present work, it appears that a GAP of ≥10% means any simulations using the LVC should be treated with caution. The validity of the LVC and impact of the corresponding GAP value can be assessed by calculating experimental observables, such as absorption spectra. Although further work on a broader range of systems would be required to provide insights into the effect of the magnitude of this metric, this provides an important basis for future works using the LVC approximation.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

JE would like to thank Dr Thomas J. Pope, Dr Conor D. Rankine and Clelia Middleton for helping with the beta test of VCMaker. The authors would like to acknowledge the EPSRC for funding via project EP/T022442/1 and EP/V010573/1.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d2cp05576g

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